Richard Buswell

Keywords: Additive manufacturing, concrete, 3D Printing, robotics, form-finding, funicularity, construction sequence, cantilevers

Those three years of PhD have been an amazing time for me, a time of exploration, discoveries and hard work which would not have been possible without the help of all the people that were around me every day.

First of all, I would like to thank my supervisors, Olivier Baverel, Nicolas Roussel and Romain Mesnil for giving me the opportunity to go on this PhD adventure with them, and for trusting me from the start and helping and guiding me during those years of research.

I am thankful to the Navier laboratory who welcomed me, especially Jean-François Caron and Marie-Françoise Kaspi whose precious help allowed me to solely focus on research. Thank you to all the technicians, researchers and other PhD students of the laboratory who I had the pleasure to work with or just share fruitful discussions.

I would also like to thank the members of the jury of this thesis, Antonella Mastrorilli and Ar-

Résumé

Au cours des dix dernières années, la fabrication additive en matériaux cimentaires ou impression 3D béton a connu un développement exponentiel à la fois dans les milieux académiques et industriels. Cette technologie consiste à empiler des couches de mortier frais déposées à l'aide d'un robot. Ce processus automatisé ne nécessite aucun coffrage pour soutenir la structure pendant la fabrication et offre donc beaucoup plus de liberté architecturale que les méthodes traditionnelles de mise en place du béton. Il peut donc être utilisé pour minimiser la quantité de matière utilisée tout en augmentant la productivité. Cependant, maîtriser parfaitement le processus d'impression 3D est encore un défi de nos jours. Cela nécessite une bonne compréhension des propriétés du matériau à l'état frais, du comportement mécanique de la structure pendant la fabrication et des contraintes liées à la robotisation du procédé de mise en oeuvre. Cette complexité peut expliquer pourquoi la majorité des objets imprimés aujourd'hui consiste en de simples extrusions verticales d'une géométrie dans le plan, ce qui limite grandement l'espace des formes réalisables avec ce procédé.

Cette thèse vise à explorer cet espace des géométries imprimables en étudiant la possibilité de générer des porte-à-faux, malgré les mauvaises propriétés du matériau à l'état frais. En examinant des structures existantes en maçonnerie telles que les voûtes ou les dômes, construits sans coffrage temporaire, un premier aperçu de l'interaction entre le processus de fabrication additive et les géométries résultantes est proposé. Cela conduit au développement d'un nouveau workflow pour la conception et la fabrication de structures imprimées en béton. Ce workflow différencie l'analyse à l'échelle de la structure de l'analyse à l'échelle du matériau extrudé. Dans le premier cas, une approche computationnelle pour la recherche de forme par morceaux de structures imprimables est proposée. Cette approche s'inspire du processus de conception des voûtes funiculaires en maçonnerie mais en tenant compte des propriétés du matériau frais et de leur évolution dans le temps. L'analyse à l'échelle du matériau extrudé consiste en la caractérisation de la stratégie d'écrasement des couches qui permet un excellent contrôle de la géométrie des couche et, par extension, de plus de liberté géométrique à l'échelle de la structure finale.

Mots-clefs: Fabrication additive, béton, robotique, recherche de forme, funiculaire, imprimabilité, porte-à-faux

3D Printing in a Construction Industry in transition

Concrete 3D Printing has seen its evolution take off in the last five years. Large-scale 3D Printing facilities have flourished around the world both in academic laboratories of architectural or engineering school and in industrial fields following the market interest. The success of this disruptive technology in a conservative construction industry has multiple sources, one of which lies in its capacity to address simultaneously major issues faced by the AEC Industry (Architecture, Engineering and Construction) today. Environmental impact, productivity standby and housing crisis are three recurrent topics in the Construction sector that are concerned by large-scale additive manufacturing.

A Necessary Transition

On the Environmental Impact

The AEC industry is known to have a high environmental impact that architects, engineers and construction companies work on addressing at different levels. The impact goes from the depletion of material, waste production (from the construction sites and demolition), site works pollution, building energy consumption, etc. The subject of cementitious material draws particular attention because of the gigantic quantities produced every year [Ezema, 2019]. Firstly, the production of cement is highly demanding in terms of energy. The clinkerisation which transforms the limestone into clinker process occurs at 1450°C, and result in the emission of CO 2 at the ratio of 750 kg for a ton of cement produced [Claisse, 2016]. Its carbon footprint is amplified by the fact that it is the most used construction material in the world with increasing production quantities every year.

Another ingredient in the production of concrete is sand. Sand and gravels are the most mined materials on earth [START_REF] Sutherland | A 2017 Horizon Scan of Emerging Issues for Global Conservation and Biological Diversity[END_REF]. It is used in the manufacturing of concrete but also in the production of glass or electronics for instance. Recent research aims at developing the use of dune sand for concrete production [START_REF] El-Hassan | Fresh and Hardened Properties of 3D-Printed Concrete Made with Dune Sand[END_REF], [Cisse, 2012] which can be found in great quantities and easily extracted. Today, however, sand used for construction is mainly mined underwater, in bed river, beaches and lakes. This tends to act on biological diversity by degrading or suppressing natural habitats. Besides, the increasing demand leads to the apparition of unregulated and illegal international markets for this material.

The environmental impact of the construction industry and cementitious material, in particular, is also coming from the production of waste [Osmani, 2011]. Construction is a world leader in waste production. The main factor being the demolition of old existing buildings and infrastructures, since reuse or recycling is only becoming a trend in recent years. BOCK states that 30 to 40% of world waste is due to construction [START_REF] Bock | Robot-Oriented Design: Design and Management Tools for the Deployment of Automation and Robotics in Construction[END_REF]. For instance, on a worksite, construction waste represents on average 30% of the weight of the building [Llatas, 2011]. For complex, large-scale concrete structures, the use of non-standard formworks for the material casting is also creating waste since those formworks are thrown away once the concrete has hardened. These two key environmental aspects can be addressed by the integration of automated processes such as 3D Printing in the construction phase.

Productivity in Standby

In parallel to its environmental impact, the construction sector experiences a continuous lack of productivity. A report by McKinsey [START_REF] Barbosa | Reinventing Construction: A route to higher productivity[END_REF] states that productivity (calculated as the value added by construction workers per hour of work) in construction has increased by only 1% in the last twenty years while the manufacturing industry has seen a growth of 3.6% in the same period. The main difference between the construction sector and other manufacturing industries is the possibility of mass-producing elements. In construction, each project is a prototype with a unique combination of geometry, material, systems and environment. In [START_REF] Bock | Robot-Oriented Design: Design and Management Tools for the Deployment of Automation and Robotics in Construction[END_REF], BOCK explains that the lack of production in the Construction sector also lies in the low R&D investments, poor management and operational structure due to the numbers of actors involved in a single project (this issue being addressed by the introduction of BIM) and the little automation in the sector. Regarding the latter, BOCK states that automation can impact productivity by different means, ranging from better control of the material resources (less waste and defect), a higher safety factor (reducing accidents and injuries), or a better efficiency of the process.

On the Growth of the Construction Market

Finally, with a rapid increase in the world population, the construction sector has to adapt to be able to provide shelter to everyone. Projections for 2050 say that two-third of the world population (∼ 7 billion people) will live in urban areas [START_REF] Ritchie | Urbanization. Our World in Data[END_REF]. This implicates building new infrastructures while controlling their environmental impact, as well as maintaining and renovating existing ones. The challenge is considerable for all actors of the AEC industry. In this regard, it is not surprising to witness the success of large-scale 3D Printing companies like ICON1 , specialising in providing affordable and comfortable housing.

Towards Automation in Construction Industry

From Conception to Fabrication

Within this intricate context for the construction industry, many actors are looking at automation as an opportunity to provide answers that can simultaneously address these issues. Automation in construction is not new, but it is far from prevailing on current projects in the world. In his books on robot-oriented design [START_REF] Bock | Robot-Oriented Design: Design and Management Tools for the Deployment of Automation and Robotics in Construction[END_REF], BOCK mentions the early start taken by the Japanese construction industry in terms of automation in the 1970s. Companies like Toyota with Toyota Home or Kajima Corporation [START_REF] Sekiguchi | The Development and Application of an Automatic Building Construction System Using Push-Up Machines[END_REF] have developed building factories, making mass production pre-fabrication more flexible to adapt to the Construction market. Nowadays, disruptive companies like Katerra2 are rising by embracing technology in every fabrication processes. However, only using robots is not efficient to increase productivity without a digitalisation of the entire phase of a construction project, from conception to site operations and exploitation. The practice of design for buildings and infrastructure has rapidly evolved in the last 20 years thanks mainly to the significant improvements in terms of computational power offered by computers. In [Carpo, 2013], CARPO collected texts for the last two decades illustrating the Digital Turn that is taking place in Architecture.

It is not about designing a building using digital tools, but rather to design a building that could not have been either designed or built without them.

-Mario Carpo Indeed, computational power is not only used nowadays to accelerate the traditional design process but to allow for new and different approaches. Parametric design, topological optimisation or the recent use of machine learning in architecture are examples of those methods that are inherent to computational design and can help through data to address the issues faced by the construction sector. Those recent approaches can generate geometries and structures that are more complex, aesthetically interesting and structurally efficient, taking into account environmental data, users experience and other kinds of information to optimise a design. However, fabrication and construction methods have to level up to be able to give substance to those designs without causing escalating costs and delays. The design must be aware of the fabrication constraints [Mesnil, 2017]. That is where automation can provide solutions to help building non-standard structures while remaining competitive in comparison to traditional methods.

Digitalisation at All Levels

The impact and influence of digitalisation have been identified for the conception and fabrication phases of a construction project but is not limited to those. Nowadays, almost every aspect and actor intervening in the life of a building or an infrastructure can benefit from numeric workflows. At the management level, the development of BIM (Building Information Modeling) for small to large-scale projects, helps coordinate different actors of a project, smooths the phasing of the construction and simplifies planning. During the life of the building, its exploitation can be controlled and monitored through sensors measuring the temperature, humidity, energy efficiency, structural integrity, allowing to reduce power consumption and facilitating the maintenance activities for instance. All the data collected can help to better understand the needs of an infrastructure and provide information for the design of new ones. Data collection also helps to increase the accuracy of Life Cycle Analysis which is becoming more common in the construction sector [START_REF] Kuzmenko | Environmental Impacts of 6-Axes Robotic Arm for 3D Concrete Printing[END_REF]. With more data on material, fabrication process and exploitation, the evaluation of the environmental impact of buildings and infrastructures provided by LCA can show where the efforts must be put.

Finally, digital tools are being developed to manage infrastructures beyond their end of life, through recycling and reuse. Structures can specifically be designed to allow for dismantlement when the time comes [START_REF] Bertin | A BIM-Based Framework and Databank for Reusing Load-Bearing Structural Elements[END_REF]. And algorithms are being developed to ease the design of new structures with a constraint of recycling old structural elements coming from dismantled buildings with a form follows availability approach [START_REF] Brütting | Form Follows Availability -Designing Structures Through Reuse[END_REF].

An Opportunity for 3D Printing

In this general context, driven towards automation in construction with high environmental consideration, Concrete 3D Printing has naturally evolved exponentially both in the academic and industrial fields. 3D Printing is indeed very promising in terms of process control that can lead to more geometric freedom using a minimum of material with lower cost and delays than traditional building methods.

The Evolution of a New Fabrication Process in Construction

The first glimpse at a 3D Printing process with cementitious material for construction goes back to the middle of the 20th century with URSCHELL in the United States. His invention named Machine for Building Walls was patented in 1944 [Urschel, 1944]. It consisted of a mechanical arm that could rotate at 360°, and move vertically to stack layers (Fig. 1.1, left) in a cylindrical fashion. The end-effector mounted on the arm is illustrated in Fig. 1.1, right. The material was manually poured on top of the end-effector, before being extruded in a rectangular layer as an infinite brick. This system was also capable of feeding a cable inside the extruded layer working as longitudinal reinforcement. URSCHEL interestingly described his invention in the patent as a molding machine, which was able to shape layers in the formation of a wall. A variant of his machine presented an additional degree of freedom at the connection between the vertical centre post and the rotating arm. It thus allowed a rotation of the arm not only in the horizontal plane but in the vertical plane as well. This system was used to fabricate true domes or spherical domes 3 .

Figure 1.1 -URSCHEL's Machine For Building Walls, patented in 1944, from [Urschel, 1944]. Left: the mechanical rotative arm, right: the extrusion head or end-effector with the cable feeder for longitudinal reinforcement.

Although this process seems very similar to the 3D Printing systems developed about 50 years later, several differences should be noted. First, there is no digital aspect at any level. The fabrication tool is not connected to a computer with a digital model of the final object and no monitoring or parameter setting of the process can be done during the fabrication. This leads to low freedom in term of geometry and customisation. The entire design space consists of a cylinder and a dome that can be scaled up or down by changing the arm length. Finally, the need for manual feeding of material during the entire process is conflicting with today's target of a fully automatised process with as minimum human intervention as possible.

In the second part of the 20th century, other patents were filled with similar methods of formwork free construction processes [START_REF][END_REF], [START_REF] Goldsworthy | On-site wall structure formation[END_REF], [Maddock, 1988]. But it is only in the 1990s with PEGNA's seminal article on freeform construction through additive manufacturing [Pegna, 1997] that Concrete 3D Printing really becomes a digital fabrication process. PEGNA proposes to link computer-generated design to a particle bed printer to build free-form structures for construction. It sets a path for mass customisation in the construction industry and provides a solution for a fabrication process that can align with the computational design tools being developed. PEGNA refers to 3D Printing as a new approach to masonry given the similarities both in terms of material properties and the layers stacking process.

Fig. 1.2 shows the functional diagram of Concrete 3D Printing. The process mentioned in PEGNA's article is selective deposition on a particle bed but the diagram is identical for 3D Printing by extrusion, which is the subject of this dissertation. The diagram includes not only the fabrication process but the design part as well, which results in a digital model. Finally, it shows the interactions between the digital model, the process controller and the fabrication equipment.

For previous examples like URSCHEL's machine for building walls, a similar functional diagram would simply include the last box referring to the fabrication. The process controller would be replaced by a human work-force while the CAD System and Analytical model would simply not exist.

Figure 1.2 -Functional Diagram of layered construction process, reproduced from [Pegna, 1997] Subsequently to PEGNA, KHOSHNEVIS from the University of South California started developing its Contour Crafting technology [Khoshnevis, 2004]. It is known as the first largescale system for additive manufacturing by extrusion of a cement-based material. The specificity lies in the use of side trowels just outside the extrusion nozzle that shape the extruded material in an infinite brick and smooth the visible surfaces of the printed walls (see Fig. 1.3). Contour Crafting is still being developed today with the objective to 3D Print entire buildings or houses on-site, using extruders mounted on a gantry bridge mechanism, scaled to the size of the final structure (as illustrated in Fig. 1.3,right).

Figure 1.3 -Contour Crafting process from [Khoshnevis, 2004]. Left: the extrusion nozzle and the side trowels shaping the layers, right: illustration of 3D Printing of an entire building using a large gantry.

Around the same period, DINI developed D-shape [START_REF] Dini | Method and device for building automatically conglomerate structures[END_REF], a large-scale 3D Printer for construction, using binder-jetting technology. The layers here are not extruded but are formed by injecting a binder on a bed of sand, hardening the material only where required. This technology is similar to that of PEGNA in [Pegna, 1997]. The process offers a wider range of printable geometries compared to an extrusion-based process since the structure is entirely supported at all times by the powder bed. However, it becomes impractical for very large structures due to the necessary dimension of the printing system and the quantity of material required.

It is in the early 2010s that additive manufacturing by extrusion started to expand worldwide. Universities like Loughborough and Eindhoven have initiated the European research in this field, while several companies were created around the globe (WinSun, XTreeE, ApisCor, etc.) achieving more and more projects (see Fig. 1.4). Today, more than 30 research labs are equipped to study 3D Printing for construction [START_REF] Buswell | 3D printing using concrete extrusion: A roadmap for research[END_REF], resulting in an exponential increase of scientific publications on the subject and showing the global interest given by the construction community to this rather young and disruptive technology.

The Promise of Architectural Freedom Serving the Environment

As mentioned earlier, the attractivity of concrete printing comes from the capacity of the technology to address multiple major problems simultaneously. The main characteristic of additive manufacturing is that it does not require any formwork to shape the structure. The layers are extruded on top of the others and the stability of the object relies only on the printed material. In traditional concrete casting, the formwork is holding the fresh material in place during the hardening phase. This specificity provides a serious advantage in terms of geometric freedom (see examples in Fig. 1.5).

Achievable geometries with concrete printing are therefore not constrained to flat elements. In addition, adding geometric complexity in the design has no impact on the cost of the process thanks to the robot absolute precision in space. This architectural freedom is also beneficial to the environment. Since one challenge of the construction sector is to reduce the consumption of material resources and especially concrete, being able to deposit material only when it is Figure 1.4 -Evolution of actors and projects related to Concrete 3D Printing from [START_REF] Buswell | 3D printing using concrete extrusion: A roadmap for research[END_REF] needed can lead to a huge economy at a large scale [START_REF] Alabbasi | Developing a design framework for the 3D printing production of concrete building components: A case study on column optimisation for efficient housing solutions for Saudi Arabia[END_REF]. The combination of topological optimisation and 3D Printing offers for instance promising results where the material is used only in the stressed parts of the structure [START_REF] Vantyghem | 3D printing of a post-tensioned concrete girder designed by topology optimization[END_REF].

A Global Process Born for Productivity

Additive Manufacturing can help optimise material resources by permitting the fabrication of the most complex designs, but it can also boost the productivity of the construction sector altogether. 3D Printing embraces the digital turn by involving in a linear process, parametric design and automation. It can answer the needs of the AEC industry for mass customisation. Several unique geometries can come out of a parametric design tool, and each one can be fabricated individually without rising the total costs. In an early study of Concrete 3D Printing [START_REF] Lim | Developments in construction-scale additive manufacturing processes[END_REF], the unit cost of a component with regards to the quantity being fabricated is used to illustrate where this technology can be impactful and viable economically (see Fig. 1.6). The casting method gets cheaper with each new element since the cost of the formwork, when reused, is depreciated over time. By removing the formwork and using a digital tool (the robot), the cost of the component becomes independent of the number of items printed.

In addition to the economical analysis, the use of automation can increase safety in the construction sector. The main tasks being performed by the robots, the labour force is more unlikely to face injuries on site. Besides, with the automation of all the supply chain, from fabrication to site delivery, the working schedule can also be optimised, increasing planning operations accuracy and hence productivity.

On Technological Challenges

Additive Manufacturing is a very promising technology for the construction industry, though it is still facing challenges that slow its implementation on real construction projects. From material to process, researchers and industrials are reporting issues at different levels, and solutions do not always exist yet.

Technological Complexity

The Absence of Formwork

If the architectural freedom brought by the absence of formwork for Concrete 3D Printing is an advantage of the process, it comes at the cost of removing an important element in the hardening phase of the material. Indeed, the formwork in a traditional casting method has several roles. It first shapes the material in the desired geometry (usually straight elements) taking full benefit of the phase change from liquid to solid and earning its name of liquid stone [START_REF] Cohen | Liquid Stone: New Architecture in Concrete[END_REF]. From the moment it is cast until the moment the material has set, the formwork helps to withstand the hydrostatic pressure. The fresh material has not enough internal strength to hold itself without the help of an external apparatus. Finally, the formwork is also protecting the concrete surface during the cure. It keeps the material in a protected environment, avoiding water loss that can affect the hydration process and lead to shrinking and cracking. This last Figure 1.6 -Unit cost of a component with regards to the quantity being fabricated for additive manufacturing and traditional casting method, from [START_REF] Lim | Developments in construction-scale additive manufacturing processes[END_REF]. consideration is one of the main obstacles to on-site 3D Printing. Indeed, a fresh material exposed to wind, temperature and humidity changes is more likely to behave in an uncontrolled manner and result in weak hardened properties [START_REF] Federowicz | Effect of Curing Methods on Shrinkage Development in 3D-Printed Concrete[END_REF].

On the Need for Reinforcements

Reinforcement is also one of the principal technological difficulties that come with the additive manufacturing process. Indeed in any standardised construction process using concrete as a material, the incorporation of reinforcement is compulsory to withstand the tensile loads and provide ductility. Concrete has a poor resistance in traction, yet it is often subjected to tensile stresses in construction project either in the construction phase, during lifting and installation for instance, or in its final configuration.

Adding reinforcement in a printed layer is a real technological challenge with an increasing research interest [START_REF] Asprone | Rethinking reinforcement for digital fabrication with concrete[END_REF], [START_REF] Classen | Additive Manufacturing of Reinforced Concrete-Development of a 3D Printing Technology for Cementitious Composites with Metallic Reinforcement[END_REF]. The principal solutions include short fibres added to the premix [START_REF] Li | Experimental and numerical study on tensile strength and failure pattern of high performance steel fiber reinforced concrete under dynamic splitting tension[END_REF] (similar to ultra-high-performance fiber-reinforced concrete) or long fibres added continuously at the nozzle level [START_REF] Ducoulombier | Additive Manufacturing of Anisotropic Concrete: a Flow-Based Pultrusion of Continuous Fibers in a Cementitious Matrix[END_REF], [START_REF] Bos | 3D Printing Concrete with Reinforcement[END_REF]. The latter results in an anisotropic material with high tensile resistance in the longitudinal direction of the layer but with unchanged characteristics in the transverse direction. The choice of the toolpath with regards to the final structure stress configuration becomes very relevant.

Cold joints

For multi-layer casting, ROUSSEL defined in [START_REF] Roussel | Distinct-layer casting of SCC: The mechanical consequences of thixotropy[END_REF] a critical time t c over which two successive layers of concrete do not bond correctly. It suggests that the weak bond comes from the structuration gradient between the two layers and therefore expresses the critical time in function of the structuration rate A thix , the layer height H and the casting velocity V :

t c = (ρgH) 2 12 + 2µpV H 2 A thix (1.1)
ρ is the material density and µ p is dynamic viscosity.

The same problem impacts Concrete 3D Printing where the layering process is at the heart of the technology. Weak interfaces between layers can drastically change the macroscopic structural behaviour [START_REF] Nerella | Effects of layerinterface properties on mechanical performance of concrete elements produced by extrusionbased 3D-printing[END_REF]. Eq. 1.1 can be used for the additive manufacturing process by replacing H by the layer height and V by the printing velocity. However recent research [START_REF] Keita | Weak bond strength between successive layers in extrusion-based additive manufacturing: measurement and physical origin[END_REF], [START_REF] Sanjayan | Effect of surface moisture on inter-layer strength of 3D printed concrete[END_REF] suggest that strength loss at the layers interface gets higher when the interfaces are exposed to fast surface drying and that thixotropy is not responsible alone for poor cohesion between layers. The chemistry of most printable concretes is such that they can reach temperatures, above which drying, even for a couple of minutes, cannot be neglected.

To completely remove the risk of weak bonds or cold joints, some studies look at reinforcing the interface between successive layers. In [START_REF] Marchment | Method of enhancing interlayer bond strength in construction scale 3D printing with mortar by effective bond area amplification[END_REF] the addition of an inter-layer glue removes the weakness at the interface and even enforces it. Fig. 1.7 illustrates different nailing configurations of successive layers studied by PERROT et al. [START_REF] Perrot | Nailing of Layers: A Promising Way to Reinforce Concrete 3D Printing Structures[END_REF]. The nails are intended to act as a mechanical bond between layers, creating a transverse reinforcement that can withstand tensile and shear stresses.

Figure 1.7 -Nailing of layers to enhance cohesion, published in [START_REF] Perrot | Nailing of Layers: A Promising Way to Reinforce Concrete 3D Printing Structures[END_REF].

On the Evolution of Material Properties

In classic applications of concrete (concrete spraying excluded), the material properties are only important at a very early age and in their hardened state. The properties of the fresh material must fit the requirements of the application methods, by facilitating the pumping process, extending the open-time and ensuring a good compacting of the concrete. And in the final state, the material properties are directly related to the structural behaviour. However, the interest in the material properties evolution is limited since it has no load-bearing responsibility before the formwork is removed.

In the case of additive manufacturing, the material properties evolution is guiding the entire process, from the pumping to the extrusion, to the elevation of the layers until completion of the structure. Good knowledge of concrete properties over time is crucial to ensure a good fabrication process and optimise the printing time. For mortar with slow structuration (where A thix ≈ 1P a/s), classic measurement methods can be implemented to understand how the material properties evolve [Wolfs et al., 2018a], [START_REF] Mettler | Evolution of strength and failure of SCC during early hydration[END_REF]. However, when the material presents a high structuration rate, the use of existing measurement equipment is inadequate.

New methods must then be developed to assess both the initial properties and their evolution to feed the structural analysis tools. Inline measurement methods of fresh material properties are described later in the State of the Art, but to the authors' knowledge, no reliable method exists yet that measures the evolution of fast setting mortars.

Interconnected Complexity

Due to the global aspect of the 3D Printing process, solving individually each difficulty is not a guarantee of a controlled process and a successful print. 3D Printing can not be seen as a linear optimisation problem but rather as a multi-objectives one as illustrated in Fig. 1.8. Indeed, the many parameters are intertwined, and changing them can have opposing effects on the expected results. This complexity is amplified by the diversity of technologies used by the different actors (robotic mechanisms, pumping devices, material formulation, etc.). In [START_REF] Figueiredo | Quality Assessment of Printable Strain Hardening Cementitious Composites Manufactured in Two Different Printing Facilities[END_REF], mechanical tests are performed on samples printed with the same material but in two different facilities with different equipment and printing parameters. The results show significant differences in the hardened material strength.

The understanding of the influence of each printing parameter is crucial in ensuring a controlled process and predicting correct structural behaviour. For instance, reducing the printing velocity V r (in the kinematic parameters) gives more time for the lower layers to build an internal structure that can withstand the weight of the subsequent layers. At first glance, one would say that reducing the printing speed is beneficial. However, it consequently increases the inter-layer time which can thus affect the cohesion between layers, and therefore the structural integrity.

Setting the correct parameters for Concrete 3D Printing is thus a matter of finding a good balance between all the objectives of Fig. 1.8 with a hard constraint being that the structure is eventually printable and standing at the end of the process.

A Normative Void

The increasing involvement of academic and industrial sectors in the field of Concrete 3D Printing tends to progressively bring solutions to the technical issues specific to this new technology. However, even with a robust and efficient production system, the path into actual construction projects is still complex. Indeed, the engineering behind building fabrication is very codified and normalised, from the material used, the amount of reinforcement in structural elements and the construction methodology. Following those standards and codes, specific to each region of the world, ensures the integrity of the structure depending on its design, location and purpose. Innovative technology and in this case 3D Printing, is not covered by existing construction standards.

The community gravitating around additive manufacturing for construction is working towards the creation of norms and standards [START_REF] Buswell | A process classification framework for defining and describing Digital Fabrication with Concrete[END_REF] to simplify the use of the technology in real projects. In the meantime, 3D Printing companies have managed to realise projects by working around this normative barrier in different ways. In Eindhoven, a pedestrian bridge was 3D printed and installed after validation by testing on a scaled model [START_REF] Salet | Design of a 3D printed concrete bridge by testing[END_REF]. The extensive series of test on the material and the final structure provided sufficient proof of safety and compliance with the traditional codes. However, this strategy of qualification demands much more material than necessary for the final structure and is thus not compliant with the goal of reducing the consumption of material of construction projects.

In 2016, XtreeE 3D printed a column with an organic shape for a public building [START_REF] Gaudillière | Industrialising Concrete 3D Printing:: Three Case Studies[END_REF]. The advantage of the technology for this project is showcased by its geometry that could not have been fabricated by traditional methods. To comply with construction standards, only the outer skin of the structure was 3d printed, serving as a mould for casting concrete within. And only the internal cast material was taken into account in the structural analysis. The final structure is thus oversized since the printed material is considered not to play any mechanical role, giving room again to further material optimisation.

Finding applications

Another challenge of large-scale additive manufacturing that goes beyond technological issues and normative constraints lies in finding the best applications for the technology. The ideal application makes use of the architectural freedom, reduces the material quantities that other manufacturing methods would have needed and is competitive in terms of costs and delays. The difficulty is that trying to simply replace traditional methods to fabricate already existing structures and design might not be sufficient. It is the designs and structures that need to be renewed and adapted so that 3D Printing technology can truly be considered in the construction sector.

To help with this exploration of what can be achieved with cement-based additive manufacturing, DUBALLET proposed a classification of the process based on multiple criteria related to the scale of the layer and the object, the environmental condition or the temporary supports required during the fabrication [START_REF] Duballet | Classification of building systems for concrete 3D printing[END_REF]. This classification can help researchers to explore new domains within the 3D Printing sector by simply selecting new combinations of parameters.

Multiple applications of Concrete 3D Printing for construction are worth mentioning for their design specifically thought for additive manufacturing. The Krypton column by XtreeE, delivered in 2016 is a truss-shape pillar that could not have been easily fabricated with another technology (see Fig. 1.9, top left). Additive manufacturing is used in this project to print the lost formwork which is then filled with high-performance concrete [START_REF] Gaudillière | Large-Scale Additive Manufacturing of Ultra-High-Performance Concrete of Integrated Formwork for Truss-Shaped Pillars[END_REF]. The column structure is mechanically optimised to reduce material consumption, resulting in this very organic shape.

Another promising application comes from the combination of additive manufacturing and topological optimisation. Fig. 1.9, top right shows a topologically optimised bridge, fabricated at Ghent University. Again, computational tools are used to minimise the amount of material while maintaining an efficient structural behaviour [START_REF] Vantyghem | 3D printing of a post-tensioned concrete girder designed by topology optimization[END_REF]. The bridge is divided Figure 1.9 -Top left: Krypton column by XtreeE (credits: Lisa Riciotti), top right: topologically optimised post-tensioned bridge by Ghent University (picture from [START_REF] Vantyghem | 3D printing of a post-tensioned concrete girder designed by topology optimization[END_REF]), bottom: mechanically optimised reinforced concrete beam (picture from [START_REF] Maitenaz | Sustainable Reinforced Concrete Beams: Mechanical Optimisation and 3D-Printed Formwork[END_REF]).

into several voussoirs that are printed individually. They are later assembled together and the entire structure is post-tensioned.

A third example of Concrete 3D Printing for construction is the development of a sustainable reinforced concrete beam [START_REF] Maitenaz | Sustainable Reinforced Concrete Beams: Mechanical Optimisation and 3D-Printed Formwork[END_REF]. This beam illustrated in Fig. 1.9, bottom, is mechanically optimised according to Eurocode 2 recommendations. 3D Printing is used to build the formwork of the beam. The reinforcement is then added and the formwork is filled with concrete. This project stands out by its objective to apply computational design and digital fabrication to standard structures. It could this way hope for a faster introduction in the construction world.

Those three examples illustrate how digital manufacturing can bring architectural freedom at the service of material reduction and structural efficiency. Additive manufacturing has also attracted designers out of the construction sector. The interest in the process is here related mainly to its geometric potential. Furniture like chairs, tables or stools has been produced for domestic use, embracing the new aesthetics of the layering process. Other applications have been found in urban design, for benches or flower pots for instance (see Fig. 1.10,left). Besides the aesthetics advantages, those developments allows also 3D Printing company to push the boundaries of their process by working on always more complex geometries. This exploration is therefore indirectly beneficial to the construction industry.

Beyond the aesthetics consideration, 3D Printing is also relevant for its geometric freedom in the fabrication of artificial reefs. Those sub-sea structures can be specifically designed and fabricated to provide shelter to marine life. This kind of structures are too complex to be built with traditional methods, or would be too costly and are therefore a perfect fit for 3D Printing.

Finally, the advantages of additive manufacturing can be found beyond architectural freedom. The process itself can be the source of new applications in construction. The rapidity of the process, from design to fabrication can be a leveraging factor when the urgency of the project is at stake. After the outbreak of the Coronavirus, for instance, several 3D Printed houses designed for quarantined patients were fabricated and installed near Wuhan, China. Each house is said to Figure 1.10 -Left: Pavilion in Innsbruck 3D Printed by Incremental3d (photo: Rupert Asanger, from [START_REF] Grasser | Complex Architecture in Printed Concrete: The Case of the Innsbruck University 350th Anniversary Pavilion COHESION[END_REF]), right: artificial reef 3D Printed by XtreeE (from [START_REF] Gaudillière | Industrialising Concrete 3D Printing:: Three Case Studies[END_REF]). be manufactured in two hours 4 . Beside the rapidity of the process, the fact that it requires very little human intervention can also be profitable for construction in extreme environments. 3D Printing can indeed be envisioned as a potential solution for repairing or rebuilding after natural disasters. It is already studied for construction of extra-terrestrial habitat 5 , where autonomous robots would build a base on the Moon or Mars in a foreseeable future.

Problem Statement

Structural Morphology and 3D Printing

An objective of this dissertation is to pursue the exploration of printable geometries to potentially widen the range of applications. When looking at the 3D Printed structures today, one can see that they still mainly consist of simple vertical extrusion. Their geometries are closely related to cylindrical shapes where free-form is only effective on the horizontal plane. Expanding the geometric freedom in the vertical plane involves automatically the generation of cantilevers or overhangs. Those are however difficult to achieve with additive manufacturing since they usually come at the cost of significant tensile forces in the structures which are not well managed by fresh mortars.

The exploration of printable geometries requires a deep understanding of what describes the additive manufacturing process. This analysis of a process going from the design to the fabrication of a structure is called structural morphology [Motro, 2009]. It highlights the interactions between the form of the object, the flow of forces, the material used and the way it is fabricated. The centrepiece of the diagram in Fig. 1.11, reproduced from [Bagnéris, 2009], illustrates structural morphology through those interaction. The case of Concrete 3D Printing is interesting since, as mentioned before, it is a global process that requires perfect control of each of those aspects.

Some themes of research specific to Concrete 3D Printing have been added on the outskirt of the diagram of Fig. 1.11. The problems of cold joints or reinforcements for instance are shown in the vicinity of the category of structural morphology they belong to. The technology refers here to the robotic system and the printing equipment. The goal of this dissertation is to provide a design methodology allowing the geometric exploration of printable structures, that is organised to address all the notions of structural morphology while maintaining the interactions between them.

Scope of Work and Methodology

The scope of work of this dissertation is made of large-scale structures (from a meter to larger dimensions), showing ambitious cantilever, that can be 3D Printed without temporary support in unreinforced cement-based material. The design space exploration of printable structures is thus performed on this basis. However, the use of punctual or linear external support is considered is the second part of the dissertation. And the possibility to take into account reinforcement aligned with the layers in the form-finding process is covered in the third part.

From the scope of work, one can see clear similarities between Concrete 3D Printing and masonry. Both processes involve stacking layers with a material that has poor tensile strength. More specifically, many strategies have been developed for centuries to build large scale ma-sonry structures presenting cantilever (domes, arches, vaults) without involving any formwork. Examples of those strategies are detailed in the following State of the Art. Consequently, the first part of this dissertation focuses on a deeper comparison between the two additive manufacturing processes, namely masonry and 3D Printing. This comparison leads to a classification of strategies that can be used to create cantilevers with Concrete 3D Printing while respecting the technological constraints of the fabrication. This classification is in turn the basis for the proposal of a new methodology specifically thought for Concrete 3D Printing design process.

Following this, the next two parts look at the different steps of this methodology. Respectively, the form-finding process involving global structural stability and taking into account the material properties evolution. And the characterisation of the extrusion process itself, focusing on the local stability of the layer which leads to a better understanding of the printing parameters.

Chapter 2

Litterature Review and State of the Art

Additive Manufacturing of Cantilevers

In this first section of the State of the Art, different strategies to create a cantilever with an additive manufacturing process are reviewed. Indeed, the exploration of printable geometries requires generating overhangs to go further than simply vertical extrusion. In Concrete 3D Printing, the research on the design space is still quite scarce. In [START_REF] Brun | Experimental Study on 3D Printing of Concrete with Overhangs[END_REF], BRUN et al. look at the maximum inclination angle of a flat printed wall based on the mortar formulation. The overhang is achieved by horizontally offsetting each layer, creating a corbel effect. The maximum inclination angle reached in this study is 17°from the vertical plane.

The analogy between Concrete 3D Printing and masonry, in terms of process, is at the heart of the work of BHOOSHAN [START_REF] Bhooshan | Equilibrium-aware shape design for concrete printing[END_REF]. In this work, a form-finding algorithm for printable structures is proposed based on scalar-field interpolation. The idea is that a layer is stable if it has sufficient overlap with the previous one. Complex structures with cantilevers can thus be generated with the limitation that all layers are planar [START_REF] Bhooshan | Morph & Slerp: Shape description for 3D printing of concrete[END_REF]. Besides, no consideration is given to the material properties and their evolution in time.

A recent team at Ghent University has uploaded videos of concrete printing with different orientation. In the first video, the printing is done upside-down1 . The nozzle is facing up and a few layers can be extruded before the collapse of the structure. This example clearly shows the potential of additive manufacturing from a material point of view. It is indeed possible to formulate mortars with sufficient cohesion and tensile strength to withstand their weight while suspended on an upper surface.

The following review looks at ways to create overhang with additive manufacturing, with regards to the material and technology, the structural behaviour and the slicing strategy used.

On the Evolution of the Structural Behaviour

In the absence of formwork, the structure being printed is entirely responsible for withstanding its own weight during the entire fabrication process. The load case is simple and is only composed of the weight of the material. However, the structural system can become hard to determine when the geometry moves away from a pure vertical extrusion. The internal forces are evolving at each addition of a layer and their distribution is dependent on the geometry at a considered time of the fabrication. Therefore the slicing strategy which guides this geometry evolution has a direct impact on the buildability of the structure. This phenomenon has been understood and depicted by Auguste CHOISY in [Choisy, 1883] for masonry structures built without temporary supports. Fig. 2.1 taken from this book, shows different strategies that can be used for laying brick in the construction of a niche or half-dome. The horizontal brick beds are simpler to set and ensure constant brick thickness. It however creates out of plane forces at each end of the brick beds. This force increases with the higher beds are in the half-dome configuration. The fan configuration avoids the apparition of the forces, ensuring a stable structure throughout the fabrication. but the brick beds present a thickness gradient which requires more work from the mason.

Figure 2.1 -Illustration of the brickworks used in masonry to build a half-dome, by Auguste Choisy in [Choisy, 1883]. left part: radial orientation of the bricks courses, right part: combination of horizontal and radial bricks courses.

In [START_REF] Motamedi | Supportless 3D Printing of Shells: Adaptation of Ancient Vaulting Techniques to Digital Fabrication[END_REF], MOTAMEDI looks at applying masonry strategies from Persian architecture to 3D Printing of self-supporting domes, taking into account the relation between force and form, as well as the robotic fabrication constraints.

Temporary Supports in 3D Printing

Robotic additive manufacturing or 3d printing is already used successfully in industrial sectors such as aeronautics [START_REF] Joshi | 3D printing in aerospace and its long-term sustainability[END_REF] or automotive industry [START_REF] Chua | 3D Printing and Additive Manufacturing: Principles and Applications (with Companion Media Pack)Fourth Edition of Rapid Prototyping Fourth Edition[END_REF], with different materials. Technologies to print metal or plastic have existed for quite a while now. Plastic 3d printing (PLA, ABS, etc.) has emerged at the beginning of the 21st century, opening the technology of additive manufacturing to the general public. The question of finding the design space of printable geometries has already been worked around with those different materials and solutions have been developed that address the problem of cantilevers, using mainly temporary supports.

• in Fused Deposition Modelling (FDM), using plastic material (such as PLA or ABS), the most common solution is the real-time fabrication of temporary supports, printed at the same time and with the same material as the final object, and removed afterwards. Algorithms of generation of supports minimising the amount of material used [START_REF] Cacace | A level set based method for fixing overhangs in 3D printing[END_REF], [START_REF] Barnett | Weak support material techniques for alternative additive manufacturing materials[END_REF], [START_REF] Dumas | Bridging the gap: automated steady scaffoldings for 3D printing[END_REF], play with parameters such as the maximum inclination of the admissible cantilever, the structure of the supports and its density. Another approach, closer to what is proposed in this dissertation and presented by ALLAIRE and al. in [START_REF] Allaire | Structural optimization under overhang constraints imposed by additive manufacturing technologies[END_REF], proposes to optimise a printable structure with a level-set method. The topology optimisation algorithm prevents cantilevers in the object to exceed a given value. This strategy is still difficult to apply as such with Concrete 3D Printing process due to the discontinuity of the tool path, which is a practical requirement for concrete printing when using a bi-component material. The flow is indeed hard to stop for self-compacting mortar or concrete since it is significantly impacted by gravity.

• other additive manufacturing technologies such as Selective Layer Sintering (SLS), Stereolithography (STL) or powder-bed printing for concrete (D-shape2 , ensure the fabrication of any object geometry without consideration of the cantilevers. Indeed the printing is made by selective transformation of matter in a bed or pool of material. If a cantilever were to be printed it would be supported by the material in the bed or pool that has not been transformed. These technologies have been developed for metallic material, glass, plastic or concrete. Soliquid3 , a French start-up uses this strategy to extrude concrete directly into a pool of gel. The concrete in suspension in the gel has time to set before the ambient liquid is removed, revealing the entire structure.

• 3D Printing of metal is usually performed by SLS. However, for large-scale manufacturing, MX3D4 developed another technology using 6-axis robotic arms equipped with welding systems to build large-scale stainless steel structures without temporary supports.

• Research on clay printing of cantilever is also ongoing and faces similar constraints as the printing of cement-based material. At the Institute for Advanced Architecture of Catalonia (IAAC), they are looking into the maximum printing angle of wall elements5 (see Fig. 2.2,left), as well as using linear supports and curvature to achieve impressive overhangs (see Fig. 2.2,right). Most of the design space exploration in this dissertation can be applied to this 3D Printing with clay.

Self-Supporting Masonry Structures

Masonry structures and Concrete 3D Printing share many similarities. The present section looks at existing strategies to build masonry structures with overhangs using a minimum or no support at all. The following chapter of this dissertation looks at classifying those strategies to apply them in Concrete 3D Printing.

Corbels

The simplest way to build overhangs in masonry is to use the technique of corbelling. In 2 or 3 dimensions, it consists of creating a local cantilever of each horizontal layer onto the previous one, generating thus the global cantilever. This strategy has been used in stone construction in Ancient Greece (Treasury of Atreus in Mycenes showed in Fig. 2.3,right), as well as in rural areas of France (bories illustrated in Fig. 2.3,left) and Italy (trulli [Smith, 1955]) to quickly build protective shacks the easiest way possible with available material. It is described by Cowan in [Cowan, 1977] and Choisy in [Choisy, 1883], stating that such structures have been discovered in Egypt from about 2900 B. C. This typology of structure is usually characterised by a low ratio span over height. The stability of the temporary structure is ensured if there is no opening of the bed joints during construction. This can be achieved by adding a counterweight on top of the location where joints are more likely to open. Another solution is to stack the brick in a way that the centre of mass of the overall section, when projected vertically onto the ground, is contained within the footprint of the first layer. The structure is hence always in equilibrium. This approach has serious limitations when trying to minimise material quantities and maximise its efficiency:

• in two dimensions the relation between the span S and the global height H of the structure given by Hall in [Hall, 2005], shows that:

H h ∝ exp 2S d (2.1)
where d and h are respectively the brick width and height. This equation shows that the height increases exponentially with the span, and thus limits the strategy to small structures. This strategy is used in the design and fabrication of a small bridge prototype (see Fig. 2.4). In this example, the curvature of the layers allows reaching a higher local cantilever while maintaining the elements in equilibrium.

• since the width of the bricks is determined by construction constraints, the final thickness of the vault is bigger than necessary, implying an inefficient use of material.

Figure 2.4 -Bridge prototype constructed by corbelling horizontal curve layers.

Nubian Vault

Barrel vaults around 3300 years old have been found in Egypt in the region of Gourna. Those structures built by the Nubians are composed of inclined arches laying onto each other. The construction starts by disposing the bricks on a gable wall until the first arch is set. Then a horizontal translation of this arch creates the vault. Fig. 2.5 shows the basic typology of the structure, with the gable wall and the inclined arches. From a structural point of view, the construction process can be divided into three phases.

• First the deposition of the brick is made on an inclined bed joint, meaning the stability is ensured by the mortar holding the brick in place temporarily.

• Once all the brick of an arch have been installed, a thrust line appears in the arch putting the bricks in compression and allowing them to withstand the loads from the future arches.

• Finally, when an arch is far enough from the front one (the one being built), it is not impacted anymore by the newly added layers. The principal stress directions in this area are independent of the inclination of the arches. And the process can thus continue indefinitely, meaning that there is no structural limitation for the barrel vault length. This way of building vaults is reappearing nowadays thanks to its efficiency and the absence of additional material for support. The association La Voûte Nubienne6 is for instance promoting this mode of construction that only require local material with a low environmental impact.

Dome with inclined layers

To build a dome without local cantilever, which Cowan named a true dome in [Cowan, 1977], the layers have to be inclined so that their upper and lower faces are normal to the thrust in the structure. This prevents any sliding along the interface, maximise the contact area between layers and maximise the utilisation of the material. Gaspard Monge theorised stereotomy for such structures following geometric principles as reported by Sakarovitch in [Sakarovitch, 2009] under the name constructive geometry.

The stability of the temporary structure comes from the formation of compression rings locking the bricks in position (see Fig. 2.6) without creating bending moments in the material or opening of the joint.

This strategy presents also some limitations:

• if the dome is hemispherical, the final structure presents tensile rings in its lower area which is not well supported by unreinforced material.

• when the radius of the dome is too high, the curvature of the layer is low, and so is the geometric stiffness. The compressive stress holding the ring together is then more likely to • in a true dome, the inclination of the layers increases continuously to reach a fully horizontal position at the apex. This is hardly possible when building with no temporary support (whether in masonry or 3d printing). Given that α max is the maximum inclination angle of a layer, namely the angle after which friction is not sufficient to withstand the new layer, a dome could be built in a classic way until α max is reached. Then the inclination of the layers must remain constant, giving the dome a conical shape in its upper part (see Fig. 2.7). CHOISY even suggests that this constraint may be one of the reasons behind Persian and Byzantine noticeable architecture of domes [Choisy, 1883]. This strategy recalls the levelset method of Cacace et al. in [START_REF] Cacace | A level set based method for fixing overhangs in 3D printing[END_REF] for optimisation of supports in FDM process, but with the constraints applied to the final structure instead of the temporary supports.

α max α max Figure 2.7 -Modification of a dome shape based on the maximum admissible angle α max defined by the angle after which friction cannot support the new layer.

Catalan vault

Catalan vaulting techniques as for the previous true domes, use double curvature to cover large areas without needing any support. In this system, tiles are positioned on their thin edges using fast setting plaster. Their lightness allows the structure to withstand the bending force created by the addition of a tile. As soon as a layer is built, compression is predominant. The first layer of tiles is also used to support a second and usually third one, providing larger thickness to the structure shell thus increasing the rigidity and reducing the risk of failure by buckling. The technique of the Catalan vault has been perfected by GUASTAVINO and described in [START_REF] Ochsendorf | Guastavino Vaulting: The Art of Structural Tile[END_REF]. This technique allows to build funicular vaults and domes and has recently made a come back with the apparition of new computational tools for the formfinding of masonry structures [START_REF] Rippmann | Interactive Vault Design[END_REF]. Fig. 2.8 is an illustration of a spiral staircase built by the Guastavino company, using the Catalan vaulting technique. The structure is built with tiles and does not require temporary supports.

Persian vault

The Persians were able to build domes and vaults without support by inclining layers and creating compression rings. However, the actual buildings did not always present a circular boundary to start the dome. Most vaults were supported either on four walls or four columns. Hence the development of pendentives and squinches around the years 250 AD [START_REF] Ashkan | Persian Domes: History, Morphology and Typologies[END_REF]. As illustrated in Fig. 2.9, those peculiar brickworks are self-supporting, and they allow to go from a given boundary condition to a circular layer from which it is easy to build a simple dome. They act by locally modifying the curvature of the surface. This is of course not an exhaustive description of Persian architecture but merely a mention of their contribution to the supportless fabrication of domes with squinches.

These brickworks can also be used when the base of the dome is circular but the diameter is too high to ensure the stability of the compression rings as mentioned earlier. In this case, the squinches artificially bring more curvature and thus more stability. squinch pendentive Figure 2.9 -Squinches and pendentives with discontinuous layering

Gothic vault

For the construction of Gothic cross-vaults, or rib vaults, FITCHEN described in [Fitchen, 1981] a strategy to build the vault with a simple stone-weight rope device (see Fig. 2.10). The diagonals are built first using linear supports. Then for the construction of the actual vault, each additional block is temporarily stabilised by the rope device which applies a punctual force tangent to the surface of the vault, keeping the structure in compression. This combination of linear and punctual supports prevented the need for a full formwork for the structure which would have been difficult to install at the height where these vaults usually stand.

Interlocking systems

Interlocking systems in masonry implies that each block is geometrically prevented from moving by its adjacent blocks. The most notorious example of such system is the Abeille Vault developed in the 17th century [START_REF] Brocato | A new type of stone dome based on Abeille's bond[END_REF], [START_REF] Brocato | Geometric methods and computational mechanics for the design of stone domes based on Abeille's bond[END_REF]. Since a block is stabilised as soon as the next one is set, this kind of structure can be built using temporary supports only for the elements on the edges. However, those structures require precise and timeconsuming stone cutting beforehand since the geometry of each voussoir is rather complex. In this regards, interlocking systems has made a come-back in recent years with the potential use of digital fabrication. Indeed, new technology can allow now the rapid fabrication of complex geometry with high precision. The vault of Fig. 2.11 is made of interlocked voussoirs that have been 3D Printed before assembly [START_REF] Loing | Free-form structures from topologically interlocking masonries[END_REF]. The power of digital fabrication is all the more interesting in this example since all elements are different from a layer to the other. Special discontinuous brickworks can also be used to ensure the stability of the structure at the scale of the layer. For instance, BRUNELLESCHI built the dome of Florence cathedral in the 15th century, in masonry, without support, using a herringbone brickwork [START_REF] Paris | Statics of self-balancing masonry domes constructed with a cross-herringbone spiraling pattern[END_REF], [START_REF] Bartoli | Numerical Modeling of the Structural Behavior of Brunelleschi's Dome of Santa Maria del Fiore[END_REF] that alternates vertical and horizontal elements avoiding thus the creation of sliding planes. This category of brickworks creates discontinuous layers. This feature is yet hardly applicable to Concrete 3D Printing process as mentioned earlier. It will thus not be studied further in this dissertation.

Form-Finding

The previous section gave an overview of existing strategies to build masonry vaults and domes without temporary supports. It showed examples of structures that manage to withstand loads through their geometry, in their final configuration as well as during the fabrication phase. Now, to explore the design space of self-supporting structures and printable structures, in particular, specific tools are necessary. These tools allow to performing a form-finding process, where geometries can be generated to address specific constraints. These constraints can be simply structural (all elements in compression) or related to the fabrication (all faces of a quad mesh must be flat to install glass panels).

Looking at the structural morphology diagram in Fig. 1.11, form-finding makes the interaction between Form and Force sections. The emergence of computation in architecture and engineering has permitted the development of many form-finding tools with specific applications [START_REF] Adriaenssens | Shell Structures for Architecture : Form Finding and Optimization[END_REF], [START_REF] Veenendaal | A framework for comparing form finding methods[END_REF].

Figure 2.11 -Vault built with interlocking bricks from [START_REF] Loing | Free-form structures from topologically interlocking masonries[END_REF] To explore the design space of masonry structures, namely compression only structures, the main form-finding methods are the Force Density Method (FDM) [Schek, 1974], the Surface Stress Density Method (SSDM) [START_REF] Maurin | The surface stress density method as a form-finding tool for tensile membranes[END_REF]] and the Thrust Network Analysis (TNA) [Block, 2009].

The Force Density Method was initially developed to find the equilibrium state of cable nets. In this configuration, all elements are in tension, and the system of equations of equilibrium is non-linear. By introducing the concept of force density q as the ratio between the normal force in the element F and their length L, the equilibrium system becomes linear and can be solved easily. The new constraint is that the force densities of each element are now input in the problem and must thus be chosen by the user.

This form-finding method for tensile structures can be directly applied to compression-only structure by simply changing the sign of the force density coefficients. The result is a thrust network that can be used to design masonry structures following HEYMAN's lower bound theorem [Heyman, 1967] which says that the equilibrium is ensured if the thrust network lies within the depth of the shell structure. The equations of the Force Density Method are detailed in the third part of this dissertation regarding the form-finding of printable geometries.

From the Force Density Method, MAURIN proposed an extension that considers the surface stresses instead of a network of linear forces. The force density coefficient is replaced by a surface stress density which can be used to design both tensile and compression-only structures [START_REF] Maurin | Concrete Shells Form-Finding with Surface Stress Density Method[END_REF] with a strong condition that all stresses must be isotropic.

The most recent form-finding tool for masonry structures is the Thrust Network Analysis developed by BLOCK. this method makes use of the force densities to calculate the static equilibrium of compression-only shells and combine it with features coming from graphic statics [Wolfe, 1921]. When considering that all external loads are unidirectional (vertical force of gravity), the global equilibrium can be divided into a horizontal and a vertical equilibrium. The horizontal equilibrium is governed by the duality between a form diagram (representing the geometry of the system) and the force diagram (representing the interactions in the system). Both form and force diagrams can be manipulated by the user to generate new shell geometries that are fully in compression by design. Thrust Network Analysis has led to the development of a computational framework called RhinoVault [START_REF] Rippmann | Interactive Vault Design[END_REF], allowing users to directly interact with the form and force diagrams and observe in real-time the resulting geometry.

Structural Analysis

The form-finding processes that have been mentioned above propose interesting methodologies to ensure that a structure is fully in compression. For concrete 3D Printed structures that are not reinforced, it can be seen as a first approach in the exploration of the design space of the final shapes. However, those methods do not take into account the fabrication process or the evolution of the material properties. Indeed, the final equilibrium of a shell structure does not guarantee its equilibrium at intermediate steps of the fabrication. Moreover, during the fabrication process, the properties of the fresh material are far from the properties of the hardened concrete or bricks. This invalidates HEYMAN's hypothesis of the lower bound theorem that considers the material strength to be infinite in compression. Those limitations may explain the absence of form-finding methods dedicated to Concrete 3D Printing.

The developments in terms of structural behaviour analysis of the 3D Printing process have been carried out independently from any form-finding methods. A classic finite element method have been proposed by WOLFS [Wolfs et al., 2018a], where a numeric model of the structure is calculated in Abaqus after activation of each new layer (see Fig. 2.12, right). The model takes into account the material properties with regards to the age of the layer that is deduced from the structure geometry and the printing velocity. The method has shown a good prediction of the plastic and buckling failures of the printed structure. It seems however that for largescale structures with thousands of layers, such a model may lead to extremely time-consuming calculations. Besides, the result of the analysis only provides information on structural behaviour. The user will know if the object is printable or not but no information is proposed to either improve the geometry or choose more appropriate printing parameters.

VANTYGHEM et al. have developed a plug-in for Grasshopper called VoxelPrint that automatically generates a finite element model from a 3D geometry (see Fig. 2.12, left), that can be exported in Abaqus to run the finite element analysis [START_REF] Vantyghem | 3D printing of a post-tensioned concrete girder designed by topology optimization[END_REF]. This kind of algorithm facilitates the transfer from a CAD software to a finite elements analysis software and hence allows iterating between the two models and optimise the design. Figure 2.12 -Left: from 3D geometry to a finite elements model with VoxelPrint (from [ [START_REF] Vantyghem | 3D printing of a post-tensioned concrete girder designed by topology optimization[END_REF]), right: result of a finite elements analysis of a 3D Printing process (from [Wolfs et al., 2018a])

Extrusion-based Additive Manufacturing -from Material to Technology

Continuing on the structural morphology diagram of Fig. 1.11, this section describes the process of additive manufacturing by extrusion of cement-based material, with a focus on the material and the technology. The state of the art looks here at the advances in terms of material properties understanding for 3D Printing, as well as its relation to the extrusion process from the pumping to the building of the structure.

Yield Stress and Thixotropy

Concrete 3D Printing is possible thanks to the natural properties of the material used. Fresh mortar can reversibly change from a fluid to a solid phase and its material properties increase with time when at rest. It can therefore be pumped and extruded in its fluid form and then remain solid in the shape of a layer until the final setting occurs. Fresh concrete can be seen as a visco-plastic Bingham fluid [Roussel, 2018], which behaves as an elasto-plastic solid when at rest but starts flowing if submitted to a stress higher than the threshold value called the yield stress τ c . It is also a thixotropic material meaning that its properties when at rest evolve with time [START_REF] Roussel | The origins of thixotropy of fresh cement pastes[END_REF]. Once a layer is deposited, interactions between particles help to build up an internal structure causing an evolution of the apparent yield stress of the material. Therefore, it allows the layer to withstand more and more load coming from the weight of the following layers. Concrete is thus naturally suited for additive manufacturing process.

However, thixotropic behaviour is also reversible. The yield stress of the material can decrease when submitted to a shear rate, even after a long period at rest. If the load applied to a layer that has been at rest for a certain time, causes the internal stress to reach and exceed the yield stress value, the material starts flowing again, and its yield stress will go back to its initial value. Fig. 2.13 illustrates this phenomenon with the example of a 3D Printed tower that collapses entirely after completion, due to a collision between the nozzle of the extrusion head and the last printed layers when the robot goes back to its initial position. The collision with the layer causes the phase change of the material from solid to liquid of all the layers, even the lower ones which have been at rest for the printing process (around 30 minutes in this case). The experience shows that if material starts flowing in a printed structure, the chances to save the structure are very low and the flow will most likely propagate to other layers and lead to full collapse. The only option after that is to start again from the first layer with different printing parameters.

Understanding the evolution of the material properties is a real challenge for 3D Printing since the structuration of the layers guides the speed at which the structure can be built. A first model proposed by Roussel [Roussel, 2006] considers a linear evolution of the yield stress with time:

τ c (t) = τ 0 + t • A thix (2.2)
A thix is the structuration rate of the material. This model has the advantage to be simple and accurate. However, it may show some limitations when looking at a long period or when the material used is highly accelerated, since this model was first established for concrete casting technique and not 3D Printing.

PERROT et al. proposed an extension of the previous model to take into account the nonlinear evolution of the yield stress between one hour after the casting and the setting time [ [START_REF] Perrot | Prediction of lateral form pressure exerted by concrete at low casting rates[END_REF], [START_REF] Perrot | Structural built-up of cement-based materials used for 3D-printing extrusion techniques[END_REF]]. This model offers a transition between a linear and an exponential growth of the yield stress:

τ c (t) = τ 0 + t c A thix (e t tc -1) (2.3)
where t c is a characteristic time to be determined experimentally.

Recent models have been developed to fit more closely to the requirements of 3D Printing process. JEONG et al. have considered the time between the injection of accelerant and the extrusion of material in the equation [START_REF] Jeong | Rheological Property Criteria for Buildable 3D Printing Concrete[END_REF]. KRUGER et al. proposed a bi-linear model where re-flocculation is separated from structuration period [START_REF] Kruger | A rheology-based quasistatic shape retention model for digitally fabricated concrete[END_REF]. The goal of these models is to gain accuracy at the very early age of the material, which is critical for the stability of the extruded layer. The material properties and their evolution with time depend mostly on the formulation of the mix used in the process and the use of additives. Extensive research on the formulation of mortar for 3D Printing has been conducted, specially in the early years of the process [START_REF] Ma | Role of shear stress at rest on the viscoelastic response of fresh cement pastes[END_REF], [START_REF] Buswell | A process classification framework for defining and describing Digital Fabrication with Concrete[END_REF], [START_REF] Paul | A review of 3D concrete printing systems and materials properties: current status and future research prospects[END_REF], [START_REF] Zhang | Fresh properties of a novel 3D printing concrete ink[END_REF]. The objective is to find the good ratio between water, cement and filler that are best suited for the extrusion and the building of a structure. In addition to the base ingredients, additives can be added to the mix, in the initial batch or later in the process. Those additives can increase the open-time of the material and its fluidity (retardant, superplasticizers, etc.), or increase the initial yield stress and the structuration rate (accelerant, thixotropic agent, etc.) [START_REF] Marchon | Hydration and rheology control of concrete for digital fabrication: Potential admixtures and cement chemistry[END_REF]. The mix formulation with the choice of additive is the base of any 3D Printing process. A good understanding of the influence of the mix on material behaviour allows to correctly fit the material properties with the requirements of the printing process developing a set on demand mortar [START_REF] Reiter | The role of early age structural build-up in digital fabrication with concrete[END_REF] as illustrated in Fig. 2.14.

The prediction of the structural behaviour during the fabrication process and the precision of the numerical model depends mainly on a good knowledge of the fresh material evolution properties. When the structuration of the material is rather low, classic rheological methods of measurement of the yield stress can be applied [Wolfs et al., 2018a]. The preparation time for the test samples is low in comparison to the structuration rate of the material. However, when the material presents a high structuration rate, coming from the addition to the mix of accelerant, for instance, traditional testing methods become obsolete. The evolution of the material properties Figure 2.14 -Evolution of the material yield stress with regards to an extrusion-based process by REITER [START_REF] Reiter | The role of early age structural build-up in digital fabrication with concrete[END_REF].

during the preparation of the samples are not negligible anymore compared to the actual yield stress. In extreme condition of very fast setting mortar, the measurement equipment may even be damaged. The necessity for inline methods for the measurement of material properties becomes obvious. To this end, DUCOULOMBIER proposes a slug test [Ducoulombier et al., 2020a], which measures the material initial yield stress right after leaving the extrusion nozzle. The material flowing out of the nozzle naturally falls in drops whose weight can be directly related to the yield stress following the formula:

τ 0 = 4gm s √ 3πD 2 (2.4)
where g is the constant of gravity, m s the mass or average mass of the material drops and D is the nozzle diameter. A similar approach is proposed by JACQUET for material presenting much higher initial strength [START_REF] Jacquet | Gravity Driven Tests to Assess Mechanical Properties of Printable Cement-Based Materials at Fresh State[END_REF]. The falling filament presents a more fragile behaviour with a neat fracture while the slug test works well for ductile material (with lower initial yield stress).

Beyond the properties of the fresh material and its impact on the extrusion process, particular attention is given to the properties of the hardened material [Le et al., 2012a]. Indeed, the formulation, the process and the curing conditions differ from classic applications of concrete. The absence of formwork may provide much geometric freedom but it removes the protective shield of the material during is hardening period. Surface drying, cracking and shrinking must thus be looked at carefully [START_REF] Paul | A review of 3D concrete printing systems and materials properties: current status and future research prospects[END_REF], especially when the environmental conditions cannot be perfectly controlled (on-site printing). Besides, the layering process itself can also have an impact on the properties of the final material. The layers interface and the potential apparition of cold joints usually give to the material anisotropic properties weakening its loadbearing capacity in certain directions [START_REF] Salet | Design of a 3D printed concrete bridge by testing[END_REF], [Wolfs, 2019]. The orientation of the layers can thus be determinant in the final behaviour of the structure. In this regards, LIM et al. propose a method to generate a toolpath that optimises the alignment between the layers and the principal stress direction, to maximise the load-bearing capacity [START_REF] Lim | Stresscognizant 3D printing of free-form concrete structures[END_REF].

Pumpability -Printability -Buildability

Early articles on Concrete 3D Printing [START_REF]Mix design and fresh properties for high-performance printing concrete[END_REF], [START_REF] Lim | Developments in construction-scale additive manufacturing processes[END_REF] explored in detail the requirements of the material properties at different stages of the fabrication process. They identified four key characteristics of the material that can be used to define the entire additive manufacturing process:

• Pumpability

• Printability

• Buildability

• Open time

The pumpability refers to the ease of moving the fresh material through a hose from the pump, usually fixed on the ground of the printing facility, to the extrusion head mounted on the robot. The regime flow of fresh concrete in the hose can be considered fully laminar [START_REF] Schutter | Pumping of Fresh Concrete: Insights and Challenges[END_REF]. And the relation between flow rate, pressure loss and material properties for a Bingham fluid is given by:

Q = π 3∆p 4 tot R 4 + 16τ 4 0 L 4 -8τ 0 LR 3 ∆p 3 tot 24∆p 3 tot µL (2.5)
where Q is the flow rate, ∆p tot is the pressure loss along the hose, L and R are the hose length and radius respectively, and τ 0 and µ are the material yield stress and viscosity.

It is clear from eq. 2.5 that the choice of pumping equipment depends highly on both the printing facility configuration which influences the hose length and the fresh material properties, especially the initial yield stress.

The printability applies to the material going through the extrusion nozzle and being transformed into an actual concrete layer. Therefore it refers to the control and quality of the layer deposition without consideration of previous sublayers. A good printability is characterised by a perfect control of the extruded layer cross-section geometry and good surface quality of the extruded material. The printability is thus governed by the material on one hand, and by the process itself on the other hand. It will be shown in this dissertation what are the parameters impacting the printability and they influence the complexity of the process but also the overall design space of concrete printing.

The buildability is the capacity of the material to withstand the loads coming from the extrusion of the following layers until the structure is complete. It translates the relationship between the evolution of the material properties with time on one side and the evolution of the external loads that it is subjected to on the other side. As noted in [START_REF]Mix design and fresh properties for high-performance printing concrete[END_REF], buildability is specific to 3D Printed in a sense that in traditional methods of construction with concrete, the material is cast in a formwork to which all the loads are transmitted until the concrete has set sufficiently to withstand its on weight. The printing velocity V r and the material structuration rate A thix are the principal parameters that influence the buildability.

Printability and Buildability can be seen as the characterisation of a 3D Printing process at different scales: the scale of a single layer and the scale of a structure made of multiple layers respectively.

Finally, the Open time defines how long the fresh material upstream of the pump remains suitable not to alter the pumpability or printability of the process. After this time, the material is too hard to either be pumped or be correctly deposited. In practice, the value of the open time limits the volume of the batch of material waiting to be pumped and imposes a minimum flow rate of extrusion to the process. The open time can be increased by adding superplasticizer in the material [Previte, 1977], [START_REF] Ravina | Slump loss and compressive strength of concrete made with WRR and HRWR admixtures and subjected to prolonged mixing[END_REF], [START_REF] Kim | Slump improvement mechanism of alkalies in PNS superplasticized cement pastes[END_REF] which can also ease the pumpability by decreasing the initial yield stress. However, external parameters like the temperature can also influence the open time, which makes 3D Printing process difficult to transpose on-site application.

To summarise, the properties of the material used in a 3D Printing Process are evolving through time and these properties must fit the requirements at the distinct stages of the process. Before being pumped, the properties must remain constant for as long as possible. For the pumping, the initial yield stress and viscosity should be low enough to ease the transportation of the material through the hose to the extrusion head. At the extrusion nozzle, the material has to be deformable to be shaped into a concrete layer but stiff enough to withstand its weight after the deposition. Finally, the material properties should evolve fast enough to withstand the increasing loads as the structure builds up.

Two Asymptotic Extrusion Strategies

From the key characteristics of Concrete 3D Printing described in the previous part, two strategies seen as asymptotic cases have emerged either from academic or industrial sectors. The distinction between the two strategies is first mentioned in [Roussel, 2018]. It is important to understand the differences between those printing strategies since they directly impact the layer control after the extrusion and the global structural stability. Besides, it will be shown that the design space of printable geometries is also dependent on the extrusion strategy chosen.

Figure 2.15 -Left: In the Infinite Brick Extrusion, the deposited layer geometry is given by the shape of the nozzle, right: in the Layer Pressing Strategy, the nozzle is usually circular and the extruded material is pressed onto the substrate to shape the layer.

Infinite Brick Extrusion or Extruded Lace Shaping [Duballet, 2019] defines a printing process where the extruded material is unsheared when it exits the nozzle. The initial yield stress of the material is high (around 2kP a) and the layer takes the shape of the nozzle which is usually rectangular. The material used is commonly a "mono-component" mortar, meaning that no accelerant is added to the mix and the thixotropy is rather low (1P a/s < A thix < 5P a/s). The slow structuration has a direct impact on the design space of the Infinite Brick Extrusion, with structures that usually are low in height but which can have a long linear length in plane view. Besides, a consequence of the extrusion process is the incapacity to print with an inclined nozzle. This limits the fabrication of overhangs to the corbel strategy [START_REF] Bos | Additive manufacturing of concrete in construction: potentials and challenges of 3D concrete printing[END_REF], by creating local cantilevers at the scale of the layer with the risk of local deformation as illustrated in Fig. 2.16, left. Another constraint coming from the shaping of the layer by the nozzle is the necessity for the extrusion to always be oriented in the direction of the layer to avoid any twist in the corner (see Fig. 2.16,right).

The advantages of the Infinite Brick Extrusion resides in the simplicity of the automation process. The robot used for fabrication is usually a 3-axis Cartesian mechanism taking the form of a gantry bridge that can be controlled by a G-code. This system has great potential in terms of precision and velocity. Besides, since no accelerant is added to the mix, the material consistency is more likely to remain the same during the entire process. The other asymptotic extrusion strategy is the Layer Pressing or Oriented Lace Pressing [Duballet, 2019]. In the literature, it was first mentioned in [START_REF] Gosselin | Large-scale 3D printing of ultra-high performance concrete -a new processing route for architects and builders[END_REF] which refers to the process used by large-scale 3D Printing company XtreeE. In this case, the material is "bicomponent", meaning that an accelerator is added to the mix just before the extrusion. When exiting the nozzle, the material is sheared and has low initial yield stress (200 < τ 0 < 1000P a). The layer does not take the shape of the nozzle which is usually circular. It is indeed shaped while being pressed between the nozzle and the substrate. The layer geometry can thus vary by simply modifying the printing parameters, and can adapt itself when the printing head is inclined. The design space of printable geometries is then wider than with the Infinite Brick Extrusion and the addition of accelerant in the mix allows to print tall structures such as the column of Fig. 2.13.

However, this geometric freedom comes at the cost of complexity in the setting of printing parameters, control of the material consistency and generation of the toolpath. Those issues are the subject of the last part of this dissertation which focuses on the characterisation of the extrusion process when using the Layer Pressing Strategy.

The graph in Fig. 2.17 offers a direct correlation between the two extrusion strategies and the four key characteristics of the 3D Printing process described earlier. In the case of the Infinite Brick Extrusion, the material formulation remains constant throughout the entire process so the material properties have a linear evolution following to eq. 6.1. The high initial yield stress requires a pump with sufficient power to set the material in motion within the hose. Regarding buildability, the low structuration rate can be a limitation to the fabrication of high structures.

In the Layer Pressing Strategy, the material is modified to fit the requirements of the different stages of the process. The initial mix contains retardant and plasticisers to increase the open Figure 2.17 -Material properties evolution for the two asymptotic cases of extrusion strategies time and reduce the yield stress and viscosity, therefore facilitating the pumping operation. Then an accelerator is added to the mortar right before it exits the nozzle. Therefore the material has just sufficient strength to carry its own weight after the deposition and it gives the possibility to adapt the structuration rate to the actual structure by changing the amount of accelerant added. This last characteristic is illustrated in Fig. 2.17 by the multiple dot lines representing different yield stress evolution after the addition of accelerant.

Part II

From Masonry to 3D Printing Chapter 3

Similarities of two Additive Manufacturing Processes

There is a gap of several thousand years between the birth of Masonry and Concrete Printing in Construction. Yet many similarities between the two additive manufacturing processes are clearly apparent. PEGNA stated in [Pegna, 1997] that Additive Manufacturing of Concrete is a new approach to Masonry, it can indeed be seen as an extended development of Masonry through computational design and digital fabrication. This first chapter aims at identifying in details what are the key points of the comparison between Masonry and Concrete 3D Printing in order to better understand and develop the latter. For the comparison, a virtual structure is analysed at different scales: the scale of the component or layer (micro), the scale of the temporary unfinished structure (during fabrication) and the scale of the complete structure in its final configuration. Fig. 3.1 illustrates this scale separation for both masonry and 3D Printing. From there a classification of strategies used to create cantilevers is proposed based on common criteria to masonry and 3D Printing.

Bricks and Extrusion

Both masonry and 3D printed structures are constituted of stacked layers. The smallest scale of comparison is thus the layer scale. A masonry structure is made of discrete elements -bricks or voussoirs. These elements are usually cuboids defined by three dimensions: height H, length L and width B (see Fig. 3.1 (a)). They are stacked together to form the layers and the final structure. The cohesion between elements is ensured by the addition of mortar or plaster at the interfaces. The material properties of such mortar and its setting time influence the maximal inclination that can take an individual brick, in addition to limiting the growth velocity of the structure. Dry Masonry structures can also be built without mortar between the bricks (see the Italian Trulli or french Bori in the state of the art), but the design space of geometries built with this strategy is limited.

The equivalent of the brick (and mortar) for 3D Printing is a section of a layer that is a continuous element. The layer is thus defined by two dimensions only: its height H and width B. The layer section is usually rectangular but thanks to the fresh material properties and depending on the printing strategy employed, the shaping of the layer during the extrusion can lead to a trapezoidal section (see Fig. 3.2,right). This flexibility in the layer geometry exists Figure 3.1 -Masonry wall vs. 3d printed wall at the three different scales in Masonry as well. Stereotomy or the art of cutting stone into precise shapes can provide curvature to a masonry structure by making trapezoidal voussoirs at the cost of a complex and time-consuming process [Fitchen, 1981]. Another way of generating curvature in masonry structure with bricks consists of using the mortar between two brick beds to allow for different orientations (see Fig. 3.1,left).

The cohesion between printed layers is provided by the extruded material itself. A poor interface between consecutive layers can be compared to dry masonry and will lead to low mechanical properties of the final structure. Concrete 3D Printing is similar to a Masonry process where only the mortar is used.

As for masonry where the geometric freedom and process efficiency is guided by the material properties of the mortar, the fresh material properties of the extruded concrete and its evolution are key parameters in Concrete 3D Printing. The identification and understanding of the failure modes that can occur at the scale of the component are the first steps to a successful printing process. A first idea of the potential failures happening at this scale in the material has been proposed by REITER [START_REF] Reiter | The role of early age structural build-up in digital fabrication with concrete[END_REF] with the so-called "Flow out" where the material cannot withstand its own weight right after the extrusion. WOLFS [START_REF] Roussel | Numerical simulations of concrete processing: From standard formative casting to additive manufacturing[END_REF] and CARNEAU [Carneau et al., 2020a] identified additional local printing regimes such as tearing or over-pressing for the Infinite Brick extrusion and the Layer Pressing Strategy respectively. The detailed study of the latter is presented in part IV of this dissertation.

For Concrete 3D Printing, the challenge at the scale of the component is twofold: ensuring good control over the layer section geometry, and avoiding any local failure of the material depending on its fresh properties.

Layers and Strips

The second scale to consider in the comparison between Masonry and Concrete 3D Printing is a macro-scale defined by a series of stacked layers (Fig. 3.1.b). Throughout this dissertation, a series of layers representing this macro-scale is referred to as a strip. A strip is thus a section of a discrete model of a 3D Printed structure, composed of several layers. The mechanical behaviour of the structure during the fabrication process, i. e. while the final configuration is not complete, depends on one hand on the material properties and on the other hand on the chosen slicing strategy. Ensuring the stability of the structure during the fabrication process happens at this scale. The complexity of this step is enforced by both the difficulty to apprehend the shell behaviour of the temporary structure and the evolution of material properties which creates a property gradient in the object. Indeed the material properties evolve slowly compared to the printing time. By comparison, in other additive manufacturing processes like Fused Deposition Modelling with polymers or metal, there is an evolution of several orders of magnitude of the material strength right after the deposition.

Furthermore, both strip behaviour and material property gradient are dependent on the orientation of the strips that is governed by the slicing strategy. Fig. 3.3 illustrates this aspect within the upper part, a classic horizontal slicing strategy leading to corbels in the structure. The layers are simply stacked vertically limiting the development of a shell behaviour before the completion of the structure. In the lower part, the slicing is done vertically, the layers forming planar arcs which are all supported at both ends. The structural behaviour of the strips or temporary structure is better defined with the second strategy where form and force are well combined and only compressive stresses appear in the material.

In [Choisy, 1883], CHOISY illustrates the same mechanical considerations for Masonry structures (see Fig. 3.4). He explains that ancient master builders understood how the orientation of the layers impacted the structural behaviour of the shell during construction and what choices were made to ensure stability at all times while building without any temporary supports. The stability of the structure at the strip scale is developed in detail in part III of this dissertation.

Masonry and 3D Printing share mechanical and structural aspects during the fabrication. However, they differ at this scale by the fabrication process itself. Indeed, masonry structures are assembled manually, by stacking hundreds or thousands of elements with a risk of error propagation. Robotically assembled masonry structures [START_REF] Andres | First Results of the Development of the Masonry Robot System ROCCO: a Fault Tolerant Assembly Tool[END_REF], [START_REF] Loing | Free-form structures from topologically interlocking masonries[END_REF], or augmented reality for construction [START_REF] Jahn | Holographic Construction[END_REF], are being developed but are still not the norm. 3D Printing is also characterised by a high number of simple steps, namely the extrusion of a layer but is exclusively performed by a robotised system. The automation of the process and the precision of gantry or industrial robot reduces the error propagation and increases the geometrical freedom. The type of robot used for the fabrication can directly impact the slicing and printing strategy. Most 3D Printing facilities today are equipped with one of the following:

• 3 or 4-axis Cartesian robots whose movements are translations in X, Y and Z directions and in some cases a full rotation of the nozzle (4 th axis). In practice, the printing head is connected to a gantry bridge, a very stiff structure ensuring high precision of the nozzle position even at high speeds. If curve printing is possible with this kind of mechanism [START_REF] Lim | Modelling curved-layered printing paths for fabricating large-scale construction components[END_REF], it is mostly used for so-called 2.5D printing, each layer being printed in a horizontal plane (see Fig. 3.5, left of TU Eindhoven Concrete 3D Printing facility1 ).

• 6-axis robot articulated arms with 6 rotating joints giving the nozzle 3 degrees of freedom of translation and 3 degrees of freedom of rotation. The extrusion can be done at any point in space, in any direction. An additional degree of freedom can be added to the robot to further increase its working area. 6-axis mechanisms offer more freedom of movement and orientation during the printing than 3 or 4-axis mechanisms. But this is done at the expense of computational complexity of the toolpath generation (defined as the trajectory along with information on speed, acceleration, etc.) and calibration, due to the large number of axis and the non-linearities of the rotations.

Final Structure

The third and final scale for comparison of Masonry and Concrete 3D Printing, is the scale of the entire structure, in its operational or exploitation phase. Meaning, that the material properties have reached their final value, the mortar or concrete have entirely set. Hence, this scale applies to both space and time.

Masonry and Concrete structures are characterised by their poor resistance to tensile stress. For hardened concrete, the tensile strength is around 10% of the compressive strength according to Eurocode 2. In traditional applications of this material, reinforcement is incorporated to add resistance to traction and ductility behaviour. It is still possible to use unreinforced concrete but in non-seismic regions only which is very restrictive. The technological difficulty to add reinforcements during Concrete 3D Printing as discussed in the introduction guides the exploration of application in the realm of unreinforced structures.

A first approach is to use 3D Printing in the context of confined masonry [Duballet, 2019], where the 3D Printing or masonry part has more a role of filling in a wall, and where tensile or bending actions are withstood by a reinforced concrete frame.

As shown in the state of the art, unreinforced masonry can take many forms (arch, vault or dome, etc.), with a common characteristic being the search for compression only. Research in shell and spatial structures has led to efficient form-finding tools for thin shells in concrete or masonry [Motro, 2011] allowing for final structures to work only in compression providing efficient use of the material. Fig. 3.6 is ISLER's shell typology exploration showing the wide variety of the design space. Many of those structures require a full temporary formwork for fabrication which goes against the scope of work of this dissertation. The design space of printable geometry is thus a subspace of Isler's shells, where the structural stability is ensured in its final configuration, but also at all stages of the fabrication at each of the previous scales (extruded component and strip scales).

In the rest of this dissertation, the structural analysis of the final object is not addressed, as it is an arguably well-established field of structural design, even if the structure is not reinforced. It is considered that all structural elements have been designed according to their live loads, using material properties of the hardened concrete, validating both their geometry and thickness.

Classification for Additive Manufacturing of Cantilevers

Classification of Concrete 3D Printing process has been proposed by DUBALLET et al. [START_REF] Duballet | Classification of building systems for concrete 3D printing[END_REF] to help actors in the field exploring this new design space efficiently. It is based on the work environment (on-site or off-site), the scale of the printed object (for a meter to a fullscale building), the assembly specifications, etc. Following the same methodology, this section is an attempt at classifying masonry structures that present cantilevers, with criteria describing the brick stress in its initial state, the fabrication process and the supports in use. Note that the definition of local and global cantilever refer respectively to overhang that can be observed at the layer scale (offset between two successive layers) and at the strip or final structure scale.

Initial Stress State

This criterion classifies the assembly process based on the initial stress state of the brick, just after it is set in the structure. The criterion is specific to the component scale (the smaller of the three scales described in the previous chapter), meaning that the brick is looked at independently from Figure 3.6 -Shell structure typologies by ISLER [Chilton, 2000] its neighbours of the same layer. We look here at the brick element positioned with or without mortar, before completion of a full layer that would create a compression ring, for example, changing the stress inside the element. The specification on the time dependence of the criteria comes from the assumption that most masonry vaults are designed to be funicular. In mortar 3D Printing, the initial state involves the material with its weakest properties (concrete strength only increase at rest), this step is thus decisive for the success of the fabrication process.

The criteria can be divided into three categories:

• f 0 shear stress. The element is set onto the previous layer at a certain angle using friction and shear strength to stay in place. The inclination of the layer at the location of the element follows exactly the inclination of the tangent plane to the object surface at the same position.

• f 1 bending and shear stresses. The element is set with an angle and an offset from the previous layer creating a local cantilever. The inclination of the layer is different from that of the tangent plane at the considered position.

• f 2 bending moment. The element is set on the previous layer horizontally with an offset, creating a local cantilever. By using Mohr-Coulomb theory (3.1) as it is usually the case in masonry [START_REF] Godio | Effects of the dilatancy of joints and of the size of the building blocks on the mechanical behavior of masonry structures[END_REF], a simple comparison between the initial stress in the layer in the different configurations can be made.

τ < c + σ tan(φ) (3.1)
Where the cohesion c can be assimilated to the initial yield stress of the material τ 0 . Let's consider the top layer on Fig. 3.7, inclined of an angle α, subject to its self weight only F = ρgHdL and we note A = dL the linear contact with the previous layer. We can define the normal and shear stresses of the material at the interface between previous layer:

     τ = F A sin(α) σ = F A cos(α) (3.2)
By reporting in equation ( 3.1), it yields :

F A sin(α) < c + F A cos(α) tan(φ) (3.3)
With F = ρghdl and A = dl and by introducing β = c ρgh this relation becomes:

sin(α) -cos(α) tan(φ) < β (3.4)
We see that since sin(α) < 1 and cos(α) tan(φ) > 0, the relation is always true when β > 1 and in this case α max = Π 2 .

If 0 < β < 1, we write:

sin(α) cos(φ) -sin(φ) cos(α) < β cos(φ) (3.5) sin(α -φ) < β cos(φ) (3.6) α < φ + arcsin(β cos(φ)) (3.7)
Finally the values of the critical angle α max are:

α max = φ + arcsin(β cos(φ)) if 0 < β ≤ 1 π 2 if β > 1 (3.8)
In the assumption of no cohesion (β = 0), one finds that the critical angle is simply α max = φ and does not depend on the scale of the blocks. This criterion is often used in preliminary design stages for masonry structures.

This approach used to calculate the maximum inclination of a layer can help to compare the efficiency between the pure shear (f 0 ) and pure corbels (f 2 ) strategies. Let's consider a material with a cohesion c, a density ρ and a friction angle equal to zero φ = 0. Layers are extruded with a constant height H. H and c are such that 0 < β ≤ 1.

In the pure shear case (f 0 ), the maximum angle α f 0 is directly given by (3.8):

sin(α f 0 ) = β (3.9)
In the pure corbels case, the local cantilever δ is related to the inclination of the structure α by the following: (3.8) can then be applied to the offset part of the corbelled layer only(see Fig. 3.8) by considering it as a virtual layer of height δ printed horizontally, so with an inclination of π 2 . The stability condition for this cantilever element is:

tan(α) = δ H (3.10)
β δ = c ρgδ > 1 (3.11)
By combining (3.10) and (3.11), the maximum wall inclination α f 2 becomes:

tan(α f 2 ) = c ρgH (3.12)
Note that the value of α 2 is conservative since the stresses and potential deformations in the rest of the corbelled layer are not taken into consideration. The graph on Fig. 3.9 compares maximum wall inclination in the case of a pure shear and corbels strategy. The ratio α f 0 α f 2 is always larger than 1, showing an advantage to the inclined layer strategy over the corbels when it comes to creating global cantilevers.

0 0.5 1 1.5 2 0 0.5 1 1.5 2 β α f 0 ,max α f 2 ,max
The criterion f on brick stress and the geometry at the component scale are closely related. Fig. 3.7.b and 3.7.d, show layers inclined at a constant angle. The section of all layers are rectangles rotated of an angle α. For many other geometries, this angle does not remain constant during the fabrication, as shown for example in Fig. 3.2 for a curved object. In that case, the object can not be divided into rectangles, instead, the laces have a trapezoïdal section. In masonry, this is achieved by either using the mortar to correctly orient each brick or by cutting the brick in trapezoïdal shapes beforehand. For 3D Printing, it implies that the process can shape the layer when it is extruded. The combination of a f 0 or f 1 criteria with curved geometry (α not constant) leads automatically to a variation of the shape of the section and thus to necessary additional features on the printing technology used.

Application to Horizontal Printing

Some optimisation strategies to determine the feasibility of plastic or steel printing use a simple angular criterion, corresponding to the friction between two layers [START_REF] Allaire | Structural optimization under overhang constraints imposed by additive manufacturing technologies[END_REF]. Using a simple analogy with Mohr-Coulomb material, (3.8) states that a layer can be printed horizontally with cementitious material if:

β = τ 0 ρgH > 1 (3.13)
This equation is commonly fulfilled for current technologies. For example, the infinite brick extrusion strategy employs an initial yield stress τ c 1000P a and an average layer height H 0.02m. Clay-based materials also typically fall within that range. The pressing layer strategy uses τ c 100P a and h 0.002m. The two strategies have a similar value for β = 2.2, which shows that they both can be used to print layer horizontally.

The following experiment shown in Fig. 3.10 illustrates the feasibility to stack layers horizontally, α = π/2 (each layer is printed in a vertical plane) with the layer pressing strategy. A cylinder of 25 cm radius is printed on a vertical surface in the lab of École des Ponts. The first layer is printed over staples that are protruding from the surface so it is mechanically anchored. The height of the layers is h = 4mm and the target width is B = 20mm similar to the diameter of the nozzle. Eight layers (32mm cantilever) were printed before the apparition of the first failure in traction of the lower part of the cylinder. The upper part forming an arch remained stable for another seven layers before the tensile stress at the anchoring of the first layer causes the material to come off the surface and collapse.

The criterion to evaluate the maximum layer inclination is only valid for one layer at a local scale. The previous experiment ensures that printing horizontally is possible but it does not provide information on the realisation of a cantilever in terms of a printed structure, at the scale of a strip. The importance of this maximum value for the layer inclination in additive manufacturing can be traced back to the fabrication of domes in Persian architecture. In Fig. 3.11 taken from [Choisy, 1883], the maximum layer inclination is reached, leading to a conical shape of the upper part of the dome. To improve the aesthetics of such domes, it is believed that Persian builders played with the curvature in the lower and upper part of the dome so that the maximum inclination angle is only reached at the apex. It emphasises how a local criterion can impact the final geometry of a structure.

Layer Continuity

Masonry structures are built with discrete elements. They can be assembled using complex brickworks for aesthetic or mechanical purposes. The present criterion serves at describing if the structure can be built following a linear bed of bricks, or not. The continuity is influenced by • c 0 continuous layer. This most common strategy consists of assembling a layer by setting one brick after the other continuously and stacking layers onto layers until completion of the structure.

• c 1 piecewise continuous. It refers to structures built by an assembly of separate subparts. Each sub-part taken independently is continuous c 0 . This happened in Persian vaults built on squinches for example. Each squinch is elevated separately from the others until they are all connected to form the final structure (see Fig. 3.11).

• c 2 discrete assembly. Refers to complex masonry bonds such as the herringbone bond used for the construction of the dome of Florence cathedral by BRUNELLESCHI and which impact on the structural behaviour of the dome has been recently demonstrated in [START_REF] Paris | Statics of self-balancing masonry domes constructed with a cross-herringbone spiraling pattern[END_REF]. The layers do not form a continuous alignment of bricks, preventing in this case the creation of a slipping plane that would lead to an early breakdown of the structure during construction. As stated in the introduction, the scope of this article is focused on continuously printed layers. Hence, in this case, the analogy between brickworks and concrete printing does not hold. The criterion is still mentioned as those limitations may disappear in a foreseeable future with better control of discontinuous material flow rate.

The three criteria are illustrated in Fig. 3.12 where the same wall is built with different brickworks, impacting the assembly process of the masonry structure. Unlike the first criteria present at the lower scale of the element, this continuity criterion is involved at the scale of the strips or the layers.

Support Conditions

The third and final criterion concerns the support condition during the additive manufacturing process. A support criteria s has already been introduced by Duballet in its classification [START_REF] Duballet | Classification of building systems for concrete 3D printing[END_REF] for Concrete 3D Printing. It is divided into 5 categories:

• no support

• printed supports left in place or removed afterwards • external supports left in place or removed afterwards

The present classification concerns masonry structures and thus refers only to categories s 0 (no supports) and s 4 (external supports removed afterwards). The asterisk in the following criteria shows subcategories of the existing criteria s 4 . They are illustrated in Fig. 3.13:

• s * 1 punctual supports. This refers to structures built using temporary columns or cables to maintain the bricks in place until completion of the construction. Examples can be found in arches or vault assembly using cables to keep the temporary structure funicular [START_REF] Deuss | Assembling self-supporting structures[END_REF] and illustrated in Fig. 2.10 of the State of the Art.

• s * 2 boundary linear support, such as lintel for an opening in a structure or to the gable wall supporting the starting edge in the construction of Nubian vaults.

• s * 3 internal linear supports. Used in Gothic architecture to build internal ribs, supporting later the construction of the actual vault.

• s * 4 fully supported structure. Refers to structures built onto a scaffold. The structure gains its final mechanical behaviour after the complete removal of the scaffold [START_REF] Block | THE ARMADILLO VAULT:: BALANCING COMPUTATION AND TRADITIONAL CRAFT[END_REF], [START_REF] Block | Structural design, fabrication and construction of the Armadillo vault[END_REF] show the design process of the Armadillo Vault by BLOCK et al., a very thin funicular vault in unreinforced masonry but which requires a full scaffold for the construction phase). This criterion is out of the scope of the present dissertation as defined in the problem statement. 3.5 Impact on Concrete 3D Printing

s 0 s* 1 s* 2 s* 3 s* 4

Structure Typology and Classification

The following table 3.1 regroups all vaults types mentioned in the State of the Art for masonry structures in section 2.1.2, with their respective criteria for support, brick initial stress state and layer continuity. In this list, the dome with inclined layers, or true dome stands as a reference for a structure built continuously, without any support, and preventing any local cantilever.

Masonry structure typology Support

Brick initial stress state Layer continuity

Corbels s 0 f 2 c 0 Nubian vault s * 2 f 1 c 0 Dome with inclined layers s 0 f 0 c 0 Interlocking systems s * 1 f 1 c 2 Catalan vault s 0 f 2 c 0 Persian vault s 0 f 0 c 1 Gothic vault s * 3 f 0 c 1 Table 3.

-Summary of masonry vaults typologies and their parameters

In the case of the barrel vault in masonry (see Fig. 3.14), the rigidity of the bricks and the inclination of the layers prevent a perfect contact between two successive brick beds. A small but existing local cantilever exists which is the reason for the f 1 criteria.

In the Catalan vaulting technique, the flat bricks are set on their thinner edge and hold thanks to the fast setting plaster. The stress in the brick is therefore pure bending, thus the f 2 criteria. with the initial stress state, and the global scale of the layer with the existence of a structural sub-system. The primitive sub-structural systems in question can be:

• a compression ring usually present in circular domes;

• an arch in compression, seen in Gothic vaults built on ribs or barrel vaults. Note that when successive layers are connected, the structure starts behaving as a shell and not as a series of consecutive arches, improving its mechanical performance. The pure bending strategy (f 2 ) in the right column, is hardly applicable to Concrete 3D Printing due to the poor resistance of the fresh material. For instance, a layer that is deposited horizontally but with a local cantilever is very likely to deform and resembles the case presented in the second column (f 1 ). The 3D Printed dome illustrated in the table is a result of this behaviour. The layers are printed horizontally but keep deforming toward the centre of the dome. The deformation increases as the layers go up since the local cantilever is all the more accentuated for the small radii.

The case of the barrel vault brings up the advantage that 3D Printing can have on Masonry. Indeed, as previously mentioned, a local offset from a brick bed to the other is inevitable. With 3D Printing and the Layer Pressing Strategy especially, the layers section can differ from a rectangular shape (see. Fig. 3.2 from the previous section). Therefore, if the extrusion nozzle is always aligned with the surface, the contact area between a layer and the next is optimum and no local cantilever appears. Fig. 3.16 shows the path followed by the robot for the fabrication of the 3D Printed barrel vault. Staying aligned with the surface imposes torsion in the layer which is difficult to obtain in masonry while setting the brick manually. The robot shows here advantages in precision and speed over the human hand. Another important detail is the capacity of 3D Printing to create variations in the height of the layers. This specificity is used in the case of the barrel vault to start the structure directly on the floor instead of a vertical gable as it is usually the case in Masonry (see Fig. 2.5 from the State of the Art). The result is a vault divided into a transition phase where layers gradually rise from the horizontal plane to the final inclination with a thickness gradient and a cylindrical part where all layers are identical (see Fig. 3.17).

General Results

It is now possible to relate the 3D Printing process of cement-based material and the newly presented classification for masonry structures built with a minimum of temporary supports. The correlation between the brick initial stress state criteria f and the capacity to control the extruded layer geometry has already been discussed in part 3.4.1.

TIME transition part cylindrical part

These two aspects can also be aligned with material properties. A layer printed with a material with a high initial yield stress, (case of the Infinite Brick Extrusion), takes the shape of the nozzle. It is the best choice when the layer has to withstand high initial stresses, such as bending moment caused by a local cantilever (f 1 : bending and shear, or f 2 : bending moment). Extrusion with very low initial yield stress on the other hand (with the Layer Pressing Strategy, for instance), results in a layer able to withstand only a low shear stress (f 0 : shear stress) when coming out of the nozzle. But it offers the possibility to modify its section geometry. This relation between material properties, initial stress state and layer geometry control is illustrated in Fig. 3.18.

From a technology point of view, the robotic complexity can also be added to the graph of Fig. 3.18. This robotic complexity is considered here to be proportional to the number of axis of the mechanism used. The more axis, the more freedom in the geometry, but the more difficult it is to calibrate the robot (or working area) and generate a toolpath. A Cartesian robot is not able to print an object where the orientation of the layers is changing as in Fig. 3.2. In that case, cantilevers can only be created using corbels (f 2 criteria) with a material presenting high Layer geometry control can be provided by 6 axis robot whose orientation can be set for each target. The support parameter s also has an impact on the choice of the robot. Any support brought in the working environment of the printing cell needs to be calibrated before the works start. Adding supports increases thus the preparation time and reduce the precision of the robot by amplifying the chances of error accumulation.

Finally, the same reasoning can be made to relate robotic complexity with layers continuity c. Discontinuous toolpaths of a continuous surface are more likely to involve changes in orientation of the layers (see Fig. 3.12) and additional supports (see example on Gothic vaults). Thus discontinuous layers naturally lead to higher robotic complexity.

On Tensile Stress in 3D Printed Structures

During the design phase, unreinforced concrete is commonly considered (as is masonry structure) to have zero tensile strength. However, the experience shows that, for the fresh cementitious material extruded during 3D Printing, the tensile resistance, though very low, is of the same order of magnitude as the compressive strength.

With this in mind, the arch opening of Fig. 3.19 serves as an example to show how the tensile strength of the fresh material can be exploited during the fabrication process to balance the cantilever. Once the arch is complete, the structure behaves in pure compression as it should be for unreinforced concrete. For this kind of geometry in 2D, the sub-structural systems illustrated in Fig. 3.15 cannot be developed. The solution chosen here as the structural system of each half of the wall during the fabrication (before completion of the arch) is inspired by a construction technique used to build Gothic vaults without formwork (see Fig. Another feature in this example is the inclination of the layers which do not lay in a horizontal plane as seen in Fig. 3.20. This inclination combined with a nozzle that follows the orientation of the layers two significant advantages: the height of the layer is always constant in the "arch part" of the wall (see the part in red in Fig. 3.20.b), and no local cantilever appear at the edge of the arch, thus maximising the contact area between layers and minimising the bending in the material.

Finally, the continuity of the toolpath (c 0 ) can be maintained at this scale by closing the layers path with the back wall visible in the top view of Fig. 3.20.

Summary and Contributions

This first chapter analyses the different typologies of masonry structures that can be built without temporary supports. At a local scale or the scale of the component, creating cantilevers is best performed by inclining the layers, avoiding then any local offset between layers. This strategy can be applied to Concrete 3D Printing and is demonstrated by the extrusion of successive layers on a vertical plane. By inclining the printing head, the contact between layers is optimum and bending moments in the fresh material are avoided.

At the scale of the layer, the stability of the structure fabricated without support can be ensured by the creation of compression-only arches (for vaults) or rings (for domes). These different solutions that work for masonry structures are reproduced with a Concrete 3D Printing process. The successful fabrication of a 3D Printed dome and a Nubian vault validates the approach consisting of exploring the design space of printable geometries by following directions relevant to masonry structures.

Chapter 4

Design Methodology for 3D Printed Structures

In this chapter, the question of the workflow for designing, calculating and fabricating a structure by additive manufacturing is addressed. First, the main independent steps of the workflow are identified. Then two different workflows are presented, an iterative approach commonly used today, inspired by Fused Deposition Modelling process, and a linear approach, intended for large-scale cement-based additive manufacturing where structural analysis and form-finding are both integrated within the process.

Towards Toolpath Generation

Concrete 3D Printing is a fully automated process, meaning that all parameters defining the structure geometry and thickness, the fabrication speed and flow rates must be defined beforehand. The printing process is defined by a succession of targets delimiting the trajectory of the robot. Those targets are a discretisation of the final geometry containing information on the printing parameters. Some of these parameters can vary from one target to another during the fabrication of an object, while others are set for the entire printing process:

1. Targets positions and orientations. The position is the location of the end-effector in space while the orientation is related to its inclination. For instance, the toolpath for 2.5D printing used in Fused Deposition Modelling is translated in G-code and does not contain information on the orientation. The nozzle is always contained on the XY plane (horizontal plane). By extension, the position parameter contains the information in the layer thickness H.

2. Robot velocities V r . These represent the speeds in m/s at which the end-effector passes through a target.

3. Material flow rate Q. The flow rate is imposed by a pump which can be mounted on the robot or not. The signal for the pump can be integrated into the robot control unit and set within the toolpath generation software. This way the flow rate can vary from a target to another.

4. The nozzle geometry defined by its diameter D for circular aperture or by its width b and length and h for rectangular section.

5. The material rheological properties, namely the initial yield stress τ 0 and the structuration rate A thix .

In the present dissertation, different printed examples show the possibility to vary the printing velocity V r and material flow rate Q from a layer to another, and even in a single layer. Regarding the nozzle section, some studies show how it can be modified during the printing process via mechanical actuators, for instance, to optimise the surface render [START_REF] Xu | Volumeforming 3D concrete printing using a variable-size square nozzle[END_REF]. However, all the experiments realised for this thesis are made with a circular nozzle whose diameter D remains constant during the process.

3D Printing Workflow

Before detailing the sequencing of the existing and new workflows for additive manufacturing, the main key steps are introduced.

Final Geometry and Boundary Conditions

Examples of masonry structures built without or with little temporary support showed how wide the range of geometries that can be achieved by additive manufacturing is. Fig. 4.1 presents compression-only structures generation using primitive shapes that can be translated or rotated. The surface directly comes from the boundary conditions and the choice of a profile curve. The profiles presented in the table come from masonry construction and have been developed throughout the ages along with technological advances (in stereotomy for instance) and shell theory breakthroughs. For fabrication consideration, they are regular geometries composed mainly of circular arcs. This way, the profile and by extension, the vault or dome can be entirely described using simple tools such as a mason's threads. The need for a special apparatus for the positioning of masonry elements becomes obsolete as soon as robots are involved. Indeed, robots have absolute precision in space with an extremely small error, especially for the construction industry (where achieved precision barely reaches the centimetre). Therefore, designers should target geometries that make the best of the material without consideration of fabrication constraints at this stage. In the present dissertation, the targeted geometry to be 3D Printed must be as close as possible to the funicular of the self-weight of the structure and potential external loads specific to its function. So that the final object is fully in compression. In addition to the shape of the object, its thickness must also be defined at this step following structural considerations and the hardened material properties.

Slicing

Slicing operation is specific to additive manufacturing processes. It divides the final object into layers and provides the robot with a trajectory. The slicing procedure used in most cases of concrete 3D Printing project is directly inspired by pre-existing additive manufacturing processes such as Fused Deposition Modelling for plastic or metal materials. The simple upscaling of this operation for an application on large-scale additive manufacturing is questionable. Indeed the robot used for large-scale 3D Printing often offers more than 3 degrees of freedom (which is usually the limit for small scale printers). Besides, at a large scale, the structural behaviour of the temporary object is substantial as it is directly linked to this slicing operation. A strong focus on the slicing operation for large-scale manufacturing is hence absolutely legitimate.

By considering complete freedom in position and orientation of the targets provided by the robot, the slicing problem can be decomposed into three steps:

• the determination of the extrusion direction, in a monolithic sense;

• the choice of the layers height;

• the adjustment of the orientation of the targets.

Extrusion Direction

The extrusion direction defines the sequence and the orientation of the extrusion of the structure in a global way. The final layers are always perpendicular to the extrusion direction. The choice of the extrusion direction is mainly guided by the boundary condition giving the position of the first layer. It then dictates the structural configuration of the printed object at each step of the fabrication process. It also impacts the continuity of the process according to the layer continuity criteria c of section 3.4.2.

In most cases of additive manufacturing, the extrusion direction is purely vertical, leading to layers laying in horizontal planes. Although, some geometries prevent this pure horizontal slicing. Indeed the example of the barrel vault in the previous section showed that the stability of the structure during the fabrication is ensured by the formation of arches created by the inclined layers. This is an example of a horizontal extrusion direction. [START_REF] Adiels | Brick patterns on shells using geodesic coordinates[END_REF] In practice, generating a non-horizontal slicing can be a complex task. However, the literature contains algorithms that can be adapted to 3D Printing in that regards. In [START_REF] Adiels | Brick patterns on shells using geodesic coordinates[END_REF], ADIELS et. al propose a bricklaying algorithm for masonry structures, allowing to generate constantly spaced curves on any surface (see Fig. 4.3). This geometric approach uses orthogonal curves to the geodesics of the surface, ensuring a constant geodesic distance between a layer and the other. The algorithm starts from a given curve on the surface, guiding the extrusion direction. It also allows creating patches or blocks by selecting the surface boundary within the general surface. This procedure developed for masonry can also be applied to 3D Printing. It can however become slow for large structures due to the workflow involving multiple geometric operations. It can also lead to the apparition of sharp cusps at points with large curvature (see Fig. 4.4) which can lead to large robot reorientation during the process that better be avoided. Another method to perform the slicing operation for additive manufacturing comes from [START_REF] Crane | The heat method for distance computation[END_REF], where heat equation is used to calculate efficiently geodesic distances on any given mesh. Indeed, the heat flow on a surface spread at a constant speed in the geodesic directions. Thus for a given heat source, the iso-temperature lines contain points at the same distance to the source. The numerical efficiency of the method makes it suitable to explore the results of different heat source for a given geometry. The results come as a series of curves on the geometry with a constant geodesic distance, making it perfectly applicable to additive manufacturing slicing operation.

In Fig. 4.5, this method is used on a dome, with different sources. The sources can be linear (edges of the dome), or punctual (corners of the dome). The result is a striped mesh that can be used to evaluate the structural behaviour during the fabrication process. With this method, the extrusion direction is equivalent to the direction of the virtual heat flow on the geometry. In their article, CRANE et al. also mention how the time step in the heat equation can be tweaked to smooth the eventual sharp cusps in the curves. Finally, the heat method can be used within a workflow to generate layering patterns similar to the ones found in traditional masonry construction. Indeed, the layering of squinches and pendentives typical of Persian architecture (see section 2.1.2.5) can be reproduced by changing the source points in the heat method as illustrated in Fig. 4.6. The process starts by generating strips on the entire geometry dome from the four corners. Then the first closed curve is identified and used as the new boundary condition for the rest of the geometry. The source points are moved to the new corners of the remaining dome and the layering operation is repeated. This iterative process can go on to generate multiple-layers patches similarly to what is achieved in [START_REF] Motamedi | Supportless 3D Printing of Shells: Adaptation of Ancient Vaulting Techniques to Digital Fabrication[END_REF]. The two aforementioned algorithms result in the creation of lines with constant geodesic distances, meaning that the height of the layers remains constant at all times. This is interesting for 3D Printing but is not always possible. In the case of the barrel vault of Fig. 3.16, the layers are best inclined with a certain angle, which automatically leads to variation in the height of the layers between the apex of the layer and its supports. The slicing in this case is thus generating by the intersection of the geometry and a series of parallel inclined planes.

Layers Height

When the extrusion direction is set, the next step in the slicing operation is to determine the actual height of the final layers. Literature shows a wide range of height H used by the different labs and industrials (from a few millimetres to several centimetres). The choice is so far coming from user experience with their respective equipment and printing strategies.

Part IV of this dissertation explores the characterisation of the Layer Pressing Strategy and provides insight on the influence of the height of the layers at a local scale. The choice of the final height can be made within a given range dependent on the printing parameters and the material properties. At this stage, it is important to note that the height of the layers does not have to be constant. Variation can occur from one layer to another, as well as within the same layer from a target to another.

Target Orientations

The last step of the slicing operation is to provide the target orientation which pilots the inclination of the extrusion nozzle. This orientation is closely related to the height of the layers. Fig. 4.7 shows the results of three main slicing strategies and the different target orientation. When the slicing follows horizontal planes (Fig. 4.7, left), the extrusion head stays vertical and the targets lay in the horizontal plane. The layers height is constant but not the geodesic distance between the target. This strategy leads to corbels and local cantilevers. The opposite strategy aims at creating only shear stress in the material (f 0 ), it is described by GOSSELIN et al. in [START_REF] Gosselin | Large-scale 3D printing of ultra-high performance concrete -a new processing route for architects and builders[END_REF] as Tangential Continuity Method. The layers height is equal to the geodesic distance between layers. For a circular geometry, this comes down to radial slicing as illustrated in Fig. 4.7 middle. It can be achieved using the heat method algorithm detailed in the previous section. The targets orientations are obtained from a Darboux frame at each position, made from the outgoing normal to the surface of the object, the layer longitudinal direction and the cross-product of the two (Fig. 4.8). The extrusion is made in the direction of the latter vector. In this case, a 6-axis robot is necessary to reach all positions of the toolpath. From a material point of view, the initial brick state in the layer is optimised by avoiding any local cantilever and maximising the contact area between layers.

In practice, it is not always easy to stay tangent to the surface due to possible obstruction of the robot or to the material properties preventing printing above a maximum angle for example. Those limitations can be translated into robot mechanism constraints, preventing the inclination of the extrusion head in certain directions. n s t l n s x t l n s x t l n s The following example illustrates how those constraints can be applied to the toolpath of a dome with a rectangular base of 40cm by 80cm (see Fig. 4.9). The dome geometry is generated with the Force Density Method [Schek, 1974] to ensure compression only in the structure. The layers orientation is directly given by the mesh strips geometry after the form-finding. The optimal toolpath implies that the extrusion head follows the Darboux frame at each target. This toolpath is illustrated in the upper row of Fig. 4.10. The configuration in step b. causes a clash between the extrusion head and the robot arm. The rotation around Y-axis is thus blocked, and the rotation between X-axis is limited to reduce the reorientation at the corners of the dome. In the resulting toolpath (see bottom row of Fig. 4.10), the extrusion head is purely vertical in steps b. and d., while a limited rotation around X-axis is permitted at steps a. and c..

Setting of Printing Parameters

The third key step in a 3D Printing workflow is the setting of the printing parameters. They are related by a relation of mass conservation between the material in the extrusion nozzle and the material that shapes the layer after the extrusion [Carneau et al., 2020a]:

V e S = V r S (4.1)
where V e is the extrusion velocity or the speed at which the material exits the nozzle, S is the nozzle section, V r is the robot velocity and S is the layer section. The material volumic flow rate Q can be written as a function of the extrusion velocity and the nozzle section, assuming incompressibility of the fresh mortar:

Q = V e S (4.2)
In the case of the Infinite Brick Extrusion [Roussel, 2018], the layers take the shape of the rectangular nozzle so the extrusion velocity and the robot velocity must be equal at all times. This is not always verified for other printing strategies like the Layer Pressing. In this case, the nozzle is circular so its section S can be written as a function of the nozzle diameter D following S = πD 2 4 . And the layer section S can be assimilated at first to a rectangle of height H and width B leading to the following relation of mass conservation:

V e πD 2 4 = V r BH (4.3)
In this relation, B is usually an input since it is given by the thickness of the final object. The velocities V r and V e and the height of the layers H must be set to fulfil this relation.

Iterative Workflow

The first design workflow is widely used today in the additive manufacturing field since it is directly inspired by pre-existing 3D Printing technology like Fused Deposition Modelling for plastic or metal printing. With these technologies, the scale of the printed objects is small, usually around a dozen centimetres. Besides, the material goes from a liquid phase to solid extremely fast, the issue of the structural or mechanical behaviour of the object during the fabrication is thus not addressed. Consideration is given at the local scale to determine the maximum inclination possible from one layer to the other. [START_REF] Cacace | A level set based method for fixing overhangs in 3D printing[END_REF], [START_REF] Allaire | Structural optimization under overhang constraints imposed by additive manufacturing technologies[END_REF] propose methods to optimise the printing toolpath and update the geometry or the support location based on criteria depending on the maximum angle of the geometry but not the global structural behaviour.

Yet this approach is used for Concrete 3D Printing where the stability of the structure being fabricated is of the utmost importance. Indeed, in this case, both the scale of the structure (typically around a metre) and the material behaviour of fresh concrete compels engineers to understand and predict the stability of printed object at all times to avoid early collapse.

The integration of structural analysis of the 3D Printed model becomes essential. With this first approach, the necessary structural analysis is thus added, at the end of the process. Fig. 4.11 represents a diagram of the entire workflow.

The geometry of the object and its corresponding thickness is first given as an input. The first step is to slice the object by choosing the extrusion direction, the height of the layers H and the target orientation consecutively. The layer height H is shown as an output of the slicing the operation in Fig. 4.11, but it is in practice a choice coming from user experience more than a result of process and material considerations. The following step is to set the printing parameters V e (or Q) and V r following the relation of mass conservation (4.1). A good strategy is to first set the flow rate of the material depending on the capacity of the pump and deduce the robot velocity accordingly.

At this stage, the toolpath is operational and can be fed to the robot but without any guarantee of the success of the printing process. A structural check is necessary to ensure the stability of the object during the fabrication at all times. Following the identification of the printing potential failure modes in [START_REF] Reiter | The role of early age structural build-up in digital fabrication with concrete[END_REF], SUIKER provides the analytical equation to check a vertical 3D Printed wall for plastic collapse or elastic buckling [START_REF] Suiker | Elastic buckling and plastic collapse during 3D concrete printing[END_REF]. WOLFS finally proposes a workflow and a numerical model [Wolfs et al., 2018a] that can be used for more complex geometries. First, experimental tests are performed to retrieve the material properties and their evolution in time. Then a finite element model is built with Abaqus where the layers are activated consecutively and the material properties are updated accordingly to simulate the real printing process. The model allows predicting quite precisely the occurrence of a plastic collapse of elastic buckling during the fabrication.

In case of failure, the user can modify the printing parameters, by reducing the robot velocity for instance and give more structuration time to the material. Once the structural check is successful, the real fabrication can start with good confidence in the result.

A Guide to Parameters Setting

The main characteristic of this direct approach is that the height of the layers H is set before the printing parameters V e and V r . The length L of each layer can thus be used to set the parameters.

In practice, the volume flow rate is limited by the capacity of the pump, or by the maximal speed of the robot, the second case being unlikely for large 6 axes robots or gantry mechanisms. Due to the mass conservation (4.1), this means that the maximal and minimal speeds of the robot are bounded. This bound does not depend on material properties, but rather on the printing set-up and can precisely be determined.

V - r < V r < V + r (4.4)
We shall look now at the compatibility between a toolpath and the rheological properties of the mortar. Assuming for simplification that the printing path is a collection of curves of length L i , for which the robot speed is constant V i r , the total printing time is simply defined by equation (4.5).

T = i L i V i r = i ∆t i (4.5)
The inter-layer time ∆t i between the layer i and the layer i + 1 is L i /V i r . It is constrained by the time window. To maintain a seemingly constant time window, the speed must increase proportionally to the layer length. One expects thus for the speed to be maximal for the longest layer and minimal for the shortest one. This gives a geometrical limit that relates the ratio of curve length to the time window depending on the inter-layer time.

max(L i ) min(L i ) < V + r ∆t + V - r ∆t - (4.6)
The lower boundary of the time window depends on the structuration rate of the material and its capacity to withstand the weight of successive layers. Equation (4.7) gives an expression of this time window lower bound for a straight wall printed at a constant speed and with a material having a constant structuration rate A thix [Roussel, 2018].

∆t -= ρgH √ 3 -τ 0 1 A thix (4.7)
The upper bound ∆t + depends on both the material thixotropy [Roussel, 2018], [START_REF] Zareiyan | Effects of interlocking on interlayer adhesion and strength of structures in 3D printing of concrete[END_REF] and the printing environment. High thixotropy and dry environment are factors that can lead to the creation of so-called cold joints, which characterised weak cohesion at successive layers interface. This phenomenon is increased with time and its apparition is what defines ∆t + . Roussel gives its analytical expression in [Roussel, 2018]:

∆t + = (ρgH) 2 12 + 2µpVr H 2 A thix (4.8)
where µ p is the plastic viscosity. Although recent research [START_REF] Keita | Weak bond strength between successive layers in extrusion-based additive manufacturing: measurement and physical origin[END_REF] suggests that strength loss at the layers interface gets higher when the interfaces are exposed to fast surface drying and that thixotropy is not responsible alone for poor cohesion between layers. The chemistry of most printable concretes is indeed often accelerated through the use of aluminatebased compounds. They can therefore reach temperatures, above which, drying, even for a couple of minutes, can not be neglected.

The boundaries of the time window might be difficult to assess precisely experimentally, while a satisfying time within the time window can be found. In practice, a user might want to aim for a constant inter-layer time, which simplifies then the previous estimate.

max(L i ) min(L i ) < V + r V - r (4.9)
By introducing parameter Ω =

V + r V - r min(L i ) max(L i )
, it comes directly that the object is printable by the set-up if Ω > 1.

This kind of constraint can easily be implemented in a parametric CAD environment, like Dynamo or Grasshopper. Fig. 4.12 is an illustration of implementation of this criteria on a dome. The range of the flow rate is taken between 20g.s -1 and 50g.s -1 which corresponds to the maximum and minimum capacity of the pump. The section of the layer is chosen equal to 120mm 2 -20mm width and 6mm height. It comes directly from equation (4.1): V + r = 185mm.s -1 and V - r = 74mm.s -1 . Fig. 4.12.left shows the layers length L i gradually decreasing. Fig. 4.12.middle shows the printable part in blue and non-printable part in red obtained by calculating Ω for each layer by considering min(L i ) to be the length of the actual layer and max(L i ) being always the length of the first layer.

To print the entire object without changing the overall geometry, one can either modify the section of the layers during the print or change the inter-layer time. In the first case, one can In Fig. 4.13, the domes from Fig. 4.12 are actually printed. The one on the left has a constant layer thickness H. The inter-layer time ∆t is constant until the flow rate Q reaches the minimal value achievable by the pump. The layers after that are printed at a constant flow rate and robot velocity V r , thus decreasing the inter-layer time at each new layer. In this case, the material properties are such that no failure occurred even with low inter-layer time in the upper part.

The dome on the right illustrates the capacity to modify the layer cross-section on a single object. The linear increase in thickness H of the layer allows keeping a constant inter-layer time ∆t while remaining within the working range of the equipment. It also helped to decrease the global printing time of the object from 15 minutes (for the one with constant thickness) to 10 minutes, for the same amount of material and same geometry. However, for the thicker layers (on the top part), a settlement can be observed coming for a local flow after the extrusion, leading to an error in the total height of the object.

The other strategy to ensure a good setting of the printing parameters is to reintroduce the inter-layer time ∆t in the calculation of Ω, following equation (4.6). This gives a new parameter Ω = V + ∆t + min(li)

V -∆tmax(li) . ∆t -can be chosen so that Ω > 1 for all layers. In this case, the material properties, namely the initial yield stress and structuration rate must be properly set to accommodate the smaller inter-layer time.

Limitations

This iterative approach requires by definition a certain number of loops between the setting of the parameters and the structural check. The validation of the structural integrity of the object during fabrication is done after the setting of the printing parameters which can lead, for large structures, to a long design process before reaching a valid toolpath. In the next section, a new workflow for 3D Printing is proposed which removes the need for iteration by reordering the key steps of the methodology.

Linear Workflow

This new Linear Workflow is proposed to better fit the needs of large-scale additive manufacturing. The goal is to remove the iterations in the process of setting the printing parameters and generating the robotic toolpath. This is done by integrating the mechanical and structural verification at the core of the design process instead of having it as an add-on. This way the printing parameters are not an input to the structural analysis anymore but a result of it.

Overview

This approach follows the division into levels of scale studied in section 3 that is characteristic of additive manufacturing processes. As for the Iterative Workflow, the first step is to determine and analyse the geometry and structural behaviour of the structure in its final configuration. The second step occurs at the level of the strips, during the fabrication. The object being printed is analysed at different time steps, taking into account the evolution of the material properties to determine the adequate growth velocity of the structure. The final step consists of looking at the material at the level of the extrusion to determine the final printing parameters, namely the optimal layer height, the robot velocity and the extrusion nozzle dimension. the parameters are set to avoid any local failure modes.

This linear workflow is summarised in the diagram of Fig. 4.11. The different steps are separated according to the scale level they belong to and analysed below in chronological order.

The Final Structure

The first step of the workflow starts with the final geometry as an input. This is also the case for the Iterative Workflow. The geometry includes the shape and thickness as well as the boundary Figure 4.14 -Linear Workflow conditions. This dissertation does not cover the form-finding process and structural analysis that can lead to this geometry. Although, since the scope of work targets unreinforced funicular structures in compression only, Thrust Network Analysis ( [Block, 2009]) seems a good candidate for these operations.

The structure is supposed to be designed for its live load, and the rest of the workflow aims at validating the geometry and thickness of the shell on one hand, and develop the entire fabrication process on the other hand.

Still at this level, the extrusion direction is determined for the entire geometry based on the boundary conditions. This is the first step of the slicing operation (as developed in section 8) which generates strips in the model. This can be done using the heat method for calculating geodesic distances [START_REF] Crane | The heat method for distance computation[END_REF]) for instance. The strips give some insight into the orientation of the final layer but they have no physical meaning at this stage (no information on the layer height H is provided at this point).

The Structure under Fabrication

At the intermediate scale level, the goal is to ensure the stability of the structure at each step of the fabrication process. To do so, the model is discretised using the strips created in the previous steps. Then the strips are activated consecutively while the material properties (initial yield stress τ 0 and structuration rate A thix ) are given as input and updated at each new step to take into account the time dependence of the fabrication process. The temporary structural behaviour at each time step is analysed and the flow rate is fine-tuned to prevent early collapse.

Flow Rate and Growth Velocity

In [Suiker, 2018], the concept of Growth Velocity l of a printed structure is introduced:

l = Q V r HT l (4.10)
where Q is the volume flow rate of material, V r is the robot velocity, H the layer height and T l the printing duration of a single layer. This parameter is used to evaluate the evolution of the structure and the ageing of the material. The existence of the layer height and printing speed in the formula indicates that the analysis detailed by SUIKER is part of the iterative workflow. In the case of the linear process, the layers are not formed yet, so the printing speed, layer height or layer printing time are unknown. The growth velocity can thus simply be replaced by the Flow Rate Q only or deduced for each strip using the average thickness B and length L of the strip:

l = Q L B (4.11)
However, B and L are parameters inherent to the geometry of the structure. Q is thus the only value that needs to be set. Q is thus the only parameter directly linked to the duration of the printing process.

The flow rate Q can have different values for each strip of the model. The structural analysis in question at this step can be done using Finite Elements, similarly to the methodology given in [Wolfs et al., 2018a], where the strips replace the layers in the model discretisation.

In this dissertation, this step is detailed in part III where an equilibrium analysis is preferred to Finite Elements. It allows combining the material resistance verification and a form-finding process simultaneously. This kind of analysis is commonly used in masonry structures and was thus chosen here by consideration of the similarities between the structural behaviour of the two additive manufacturing processes.

Local Scale of the Extrusion

In this final step, each strip is processed individually to perform the final layering, where the heights H and robot velocities V r of the layers are determined. From the strip geometry and the extrusion flow rate Q, the layering problem can be summarised into finding the correct number of layers N per strip. This number depends simply on the local stability of the extruded material, which is the subject of part IV of this dissertation.

Case Study

In this section, the Linear Workflow for 3D Printing is applied to the simple case of fabrication of a vertical wall by 3D Printing. The entire workflow is presented in Fig. 4.15. The wall has a width B, a length L and a total height h. The material properties (τ 0 and A thix ) are given as an input and the material evolution follows the linear relation τ c (t) = τ 0 + A thix × t. The extrusion direction is purely vertical. All layers will be horizontal in the end. For the sake of clarity, only plastic collapse is considered in this example. Therefore the critical stress in the structure is at the base of the wall and is equal to (see [Roussel, 2018]):

τ max (t) = ρgh * (t) √ 3 (4.12)
where h * (t) is the total height of the object at a given time t during the fabrication. Since the material properties and the internal stresses both evolve linearly with time, there is no need to discretise the model, and the print is successful if the material at the base of the object can withstand the weight of the entire wall at the exact end of the fabrication process. This yields the following relation:

τ c (t f ) = τ 0 + A thix × t f > ρgh * (t f ) √ 3 (4.13)
The final printing time t f can be expressed in function of the material flow rate Q and the wall geometry parameters:

t f = BhL Q (4.14)
By introducing t f expression in (4.13), the maximum value of the flow rate Q max appears:

Q max = BhLA thix ρgh √ 3 -τ 0 (4.15)
This concludes the second step of the Linear Workflow, with the setting of the flow rate and the structural analysis of the structure.

The final step is the actual layering of the wall, with the setting of the robot velocity V r and the layer height H. This problem can be solved by simply finding the optimal number of layer N and following the relation of mass conservation (4.1). It comes:

H = h N V r = QN Bh (4.16)
The choice for N depends on the printing process and the material properties. A first condition on the stability of a single layer gives a lower bound for N :

ρgH √ 3 < τ 0 ⇒ N > ρgh τ 0 √ 3 (4.17)
Some insight on finding an upper bound for N is provided in part IV of this dissertation. Fig. 4.15 shows on the right the layering result for two different values of N . The relation of mass conservation implies that an increase of N leads to smaller layer height H and higher velocity V r . It is interesting to note that in this linear workflow, choosing a higher robot velocity V r only impact the layer height H, but has no effect on the total printing duration. Besides, the final layer height H does not influence the global stability criterion given by (4.13).

Several results come from the analysis of this small example:

• the total printing time depends only on the material flow rate Q and not on the robot velocity V r . This statement may not be true for very large-scale structures. Since the velocity V r is set according to the flow rate Q, the required velocity could get higher than what the robot can reach and hence becomes the limiting parameter;

• the structural behaviour of the temporary structure under its own weight is not influenced by the height of the layers H;

• the global (structure scale) and local (extrusion scale) failure modes are independent.

Summary and Contributions

This chapter focuses on the design process leading to the 3D Printing of a structure. Specifically, the slicing strategy, the printing parameters and the robotic path. So far, this workflow has been copied from other additive manufacturing process like Fused Deposition Modelling. Therefore, a new workflow is proposed that takes into consideration the specificity of cement-based additive manufacturing. The slicing strategy is decomposed into steps that take advantage of the layer geometric freedom and the robotic arm accessibility. The setting of the printing parameters follows an analysis of the structure at different scales, namely the global structural scale and the local scale of the extrusion. These two steps are studied independently in the two following parts III and IV of this dissertation.

Eventually, the goal of the new workflow is to provide users with a methodology to generate toolpaths and the corresponding printing parameters for the most complex geometries. And so, without the need to go through trial and error before each printing session.

Part III

Global Piecewise Analysis

Chapter 5

Form-Fitting and Yield Design In this part, the focus is solely made on the global structural stability of the structure being printed (see Fig. 5.1). The characterisation of the material behaviour at the scale of the layer is the next step in the workflow and is treated in the next part of this dissertation. For large scale additive manufacturing with cement-based material, the structural integrity of the built object during the fabrication is critical. The identification of the potential failure modes at this stage have been reported in several articles [Suiker, 2018], [Wolfs et al., 2018b], [START_REF] Bhooshan | Equilibrium-aware shape design for concrete printing[END_REF], [START_REF] Reiter | The role of early age structural build-up in digital fabrication with concrete[END_REF]. Plastic collapse and elastic buckling are the two main causes of collapse during fabrication. When printing challenging geometries with cantilevers, the collapse can also come from a loss of global equilibrium as a rigid body movement (see Fig. At this stage of the process, only the slicing direction is selected with the creation of the ordered strips. However, the layers are not defined yet. Therefore, the layer cross-section geometry and especially their height H is not involved in the following analysis and neither is the robot printing velocity V r .

Problem Statement and Methodology

The scope of work of this dissertation is defined by geometries presenting cantilever built by 3D Printing without temporary supports or formwork. In the previous part, it was seen how cantilever can be achieved by the creation of shell-like structures in a similar manner it is done with masonry. The shell typology is characterised by a double curvature that can withstand the load coming from the material weight by membrane stresses only (no shear or bending). For material like clay or concrete with poor tensile strength, the membrane stresses can furthermore be constrained to compressive stresses given an appropriate geometry. Therefore using all the material thickness efficiently and justifying the absence of reinforcement in the structure.

In the case of masonry, the material properties of the brick are not evolving. The brick can withstand high stresses as soon as it is set. That way, with a smart building process, vault and domes can be built without formwork in a shape defined in consideration to the final configuration of the structure only [START_REF] Ochsendorf | Guastavino Vaulting: The Art of Structural Tile[END_REF], [Wendland, 2004]. For additive manufacturing with cement-based material, the evolution of the material properties is crucial in the analysis of the structural behaviour during the fabrication. Indeed, the extruded material has a very low initial strength (characterised by yield stress τ 0 between 200 and 2000 Pa) and a structuration rate which will lead to a material close to its final strength in hours at best, but most usually days. This high time dependence has a direct impact on the geometry that can be printed, especially when cantilevers are present.

The goal of this part is thus to provide a methodology able to represent the design space of 3D Printed geometries with cement-base material. The algorithm presented in the following sections starts with a given geometry or target shape, and proceed to a step by step analysis where the initial geometry is modified to fit as best as possible the target shape while respecting the material properties.

Heyman's Safe Theorem

The similarities between masonry and Concrete 3D Printing to build funicular structures naturally leads to the use of a similar structural analysis approach. KOOHARIAN [Kooharian, 1952] and HEYMAN [Heyman, 1995], developed theorems of limit analysis (or yield design) for masonry construction problems. For masonry arches, vaults and domes, HEYMAN proposed a Safe Theorem (or Lower Bound Theorem) that ensures the equilibrium of the structure by looking for a statically admissible stress field [Heyman, 1969], [Heyman, 1967].

The equilibrium condition will be satisfied by constructing a line of thrust in equilibrium with the loads acting on the arch (including self-weight). The yield condition will be satisfied if such a line of thrust lies everywhere within the masonry. This theorem is illustrated by Fig. 5.3. The thrust line (dashed line) in equilibrium with the loads acting on the structure (only self-weight here) lies within the envelope of the arch ensuring that all voussoirs are in compression. [Heyman, 1969] As stated by HEYMAN, the power of this theorem lies in the fact the existence of any admissible thrust line is a satisfying condition to ensure the equilibrium of the structure. The thrust line resulting from the analysis does not need to be the real thrust line. The thrust line for an arch can be extended to a thrust network for a shell with the same condition that the entire network lies within the thickness of the shell structure.

This condition is eventually a simple geometric criterion. For this reason, graphic statics is a good candidate for the analysis of the structure and the generation of the thrust lines [Wolfe, 1921]. The duality between the form and the force diagrams allows to iteratively modify one of the diagram to act on the thrust line geometry until it lies within the envelope of the structure. This 2D analysis can be extended to 3D through the Thrust Network Analysis (TNA) [Block, 2009], used for the form-finding of funicular vaults in compression or the assessment of the behaviour of existing masonry structures [START_REF] Block | Three-dimensional funicular analysis of masonry vaults[END_REF].

Extension to Cement-Based Material

HEYMAN's Safe Theorem can be applied to unreinforced concrete structures fully in compression. However, some hypotheses in the theorem are too strong when considering a 3D Printed 90 structure with cement-based material, and are not valid during the fabrication process. Indeed the Safe Theorem is governed by three hypotheses related to the material and formulated as follows:

1. The material has no tensile strength; 2. The material has infinite compressive strength; 3. Sliding failure between voussoirs cannot occur.

The first hypothesis reflects in particular the weakness of the bonds in tension rather than the actual brick material. This assumption is true for dry masonry but is conservative when using mortar or plaster between the bricks. In the case of concrete 3D Printing, it seems reasonable to keep HEYMAN's statement due to the poor strength in tension of concrete (fresh or hardened) compared to its compression strength, especially for non reinforced concrete. It must be noted however that current research on reinforcement for additive manufacturing can lead in the near future to the consideration of the tensile strength of the fresh material ( [START_REF] Ducoulombier | Additive Manufacturing of Anisotropic Concrete: a Flow-Based Pultrusion of Continuous Fibers in a Cementitious Matrix[END_REF]). A note on the extension of the design space of 3D printing taking into consideration the tensile properties of the material is done at the end of chapter 6.

Figure 5.4 -Apparition of hinges at the joints where the thrust line crosses the intrados or extrados of the arch, from [Heyman, 1969] The second assumption states that the general compressive stresses in a shell can never be high enough to break the material. Meaning that the failure of masonry structures comes from the apparition of hinges (see Fig. 5.4) creating mechanisms that can lead to collapse. The limit analysis of masonry developed by HEYMAN [Heyman, 1967] does not look at the stresses in the material but simply at the static equilibrium of the structure.

For fresh concrete, this assumption of infinite compressive strength cannot be made. The extruded material strength is driven by the evolution of its yield stress τ c over time. This is the main novelty in the presented algorithm, where the material property is fully integrated into the analysis with a chosen yield criterion.

Finally, the third assumption on the sliding of the blocks can be related to the absence of cold joints between 3D Printed layers. Indeed weak interface between successive layers can lead to an early collapse of the new layers when printing with an inclination. In addition, cold joints lead to poor global material strength of the final structure [START_REF] Nerella | Effects of layerinterface properties on mechanical performance of concrete elements produced by extrusionbased 3D-printing[END_REF], [START_REF] Kloft | Influence of process parameters on the interlayer bond strength of concrete elements additive manufactured by Shotcrete 3D Printing (SC3DP)[END_REF].

In summary, the safe theorem hypotheses for 3D printing can be reformulated as such:

1. The material has no tensile strength;

The material must verify the yield criterion at all times;

3. There is a perfect bond between layers.

On Elastic Buckling

The buckling of the 3D Printed structure is not taken into account at first in the present method. Indeed, the scope of work covers geometries whose curvature increases the geometric stiffness hence reduces the risk of buckling. However, for large-scale structures, it can still happen if the structuration of the material is not fast enough. In that case, buckling analysis of the resulting structure can be done in the last step. If the buckling load factor is not satisfactory, the layer's geometry can be modified to increase the geometric stiffness of the printed object and increase the buckling load factor.

Problem Formulation

The problem of finding a printable geometry from a target shape can be formulated as follows: at each step of the fabrication, find the best-fitting thrust network to the target geometry, in equilibrium with the self-weight of the structure and respecting the yield criterion based on the material properties evolution and the age of the structure. The algorithm look for a piecewise funicular geometry, stable at every step of the fabrication. The workflow for each step is a constrained form-fitting problem that is presented in the diagram in Fig. 5.5. This algorithm is repeated at each step of the fabrication according to the discretisation of the model and the sequencing strategy that is introduced in the following chapter. The main contribution in this section is the combination of existing form-finding and stress evaluation methods in a single algorithm applied to sequential construction. The problem of structural stability during fabrication is critical for Concrete 3D Printing due to the absence of formwork. There is however not a lot of reference on the topic. For masonry structures, DEUSS et al. look at using punctual support on the edge voussoirs to ensure the global stability of the self-supported shell, depending on the construction sequence [START_REF] Deuss | Assembling self-supporting structures[END_REF]. This strategy recalls the one presented in the State of the Art for the construction of Gothic cathedrals with the stone-weighted rope device. It is however not applicable to Concrete Printing due to the poor fresh material properties.

The other advantage of the presented method is that it is not specific to compression-only structures. Indeed it has been demonstrated that fresh mortars used for 3D Printing have nonnegligible tensile strength (see [Ducoulombier et al., 2020a]). The yield criterion used in the algorithm can thus take into account this feature.

The development of the constrained form-fitting algorithm of Fig. 5.5 is introduced step by step in the following sections. No consideration is given at this stage to the additive manufacturing process, meaning that the evolution of the material properties are not taken into account and only the geometry in its final configuration is studied. The application of the algorithm to sequential construction and specifically Concrete 3D Printing is presented in the next chapter.

Force Density Method

The core of the algorithm relies on the Force Density Method [Schek, 1974] presented in the State of the Art of this dissertation. This method linearises a system of equilibrium equations for each node of a model by introducing the concept of force density, representing the ratio between the distance between nodes and the force that interacts between them q = F L . The system has reached equilibrium when the external forces applied to each node are in equilibrium with the internal forces oriented along the edges of the model (see Fig. 5.7).

The connections between the edges at the nodes are free of all rotations. Nodes can not transmit bending moment or shear force. Therefore a system at equilibrium is by design subjected to membrane stresses only. Such a system is a proper model of a cable net structure whose elements can only transmit forces in the direction of their longitudinal direction. The method was later extended to the design of shell structures such as tensile membranes [?], [START_REF] Maurin | The surface stress density method as a form-finding tool for tensile membranes[END_REF]] and compression-only structures [START_REF] Maurin | Concrete Shells Form-Finding with Surface Stress Density Method[END_REF], [Bruggi, 2020].

Construction of the Linear System of Equations

The principle of the Force Density Method is briefly recalled hereafter. Let's consider a system of N nodes X and M elements e connecting the nodes together (see Fig. 5.7). x f refers to the vector of dimension N f of nodes free of movement while x s is the vector of dimension N s of the support nodes. The vector x is sorted so that elements 0 to N f are free and elements N f + 1 to N are supports. One can construct the matrix C (of dimension M × N ) of connectivity between the nodes and the edges. For any edge i connected to the two nodes k and l, C[i, k] = 1 and C[i, l] = -1 while all other values on the row are 0. Q is the diagonal matrix (M × M ) of force densities q and P the vector of external forces applied to the free nodes. In this entire dissertation, only the self-weight of the structure is considered. This is justified by the fact that the object is studied during its fabrication phase only, at which stage no other external load can act on the structure. Therefore the external forces p i is calculated by measuring the influence area of each node of the mesh.

The Force Density Method helps to simply write the equilibrium equations of the system under matrix form as follows:

C T QCX = P (5.1)
Eq. ( 5.1) can be decomposed into the three coordinates of space and by separating free and fixed nodes: [Schek, 1974].

C T QC f x f + C T QC s x s = x x (5.2) C T QC f y f + C T QC s y s = x y (5.3) C T QC f z f + C T QC s z s = x z (5.4)
After introducing D = C T QC and D s = C T QC s , the system can be solved by finding the coordinates of each free nodes:

x f = D -1 (p x -D s x s )
(5.5)

y f = D -1 (p y -D s y s ) (5.6) z f = D -1 (p z -D s z s ) (5.7)
And the forces in each edge can be retrieved by solving f i = q i l i where l i is the length of the edge given by the coordinates of the nodes.

In practice, when studying funicular structure under self-weight only, the projections on the horizontal plane of the external loads p x and p y are set to 0. Finally, the equations (5.5) compose a linear system that can be solved using sparse matrix and making a Cholesky decomposition of matrix C T QC f [START_REF] Nocedal | Numerical optimization[END_REF] which is positive definite. Therefore the system can be numerically solved efficiently and optimisation procedures can be performed by updating the values of the force densities to fit the constraints of the general problem.

Fitting Force Densities

The Form-Fitting algorithm involves several optimisation steps whose success and accuracy strongly depends on the input variables. As discussed before, the force density method transforms a non-linear problem into a linear system of equations at the cost of introducing a new variable, namely the force density which needs to be correctly initialised. The solution presented in this section aims at finding the force densities q i that minimise the deviation from the equilibrium of the system without moving the nodes of the model. The resulting values of q i are then used in the Form-Fitting algorithm as the initial state of the system. This procedure is similar to the first step of the form-fitting problem presented by Vouga et al. in [START_REF] Vouga | Design of Self-supporting Surfaces[END_REF]. The equilibrium at each node is illustrated in Fig. 5.7 where P i is the external force applied to the node x i and f are the internal forces in each edge applied to the same node. Using the definition of the force density, it comes q j = fj |x i -x j | . The residual force or the deviation to equilibrium at a given node, is given by the following equation, using the same notations previously introduced:

R i = j∼i q ij (x i -x j ) + P i
(5.8)

Figure 5.7 -Equilibrium of a node of the model. The external force P i act in the vertical direction, while the internal force f j to f m are aligned with the edges of the model.

The function R to minimise is the sum of the residual forces at each free node in the least square sense:

R(q) = N f i=0 |R i (q)| 2
(5.9)

In [START_REF] Vouga | Design of Self-supporting Surfaces[END_REF], this problem is solved by adding box constraints on the force densities to force all elements to be in compression following the formulation given in Eq. (5.10). The geometry is then relaxed from the initial shape by moving the free nodes to minimise the deviation to the equilibrium while keeping the constraint on the force densities. min qi R s.t. q min < q i < 0 (5.10)

In the present algorithm, the fitting of the force density is simply a step of initialisation to improve the efficiency of the following procedures. The optimisation is therefore left unconstrained to quickly reach an initial set of force densities. It can be solved using a Limited-memory BFGS (or L-BFGS ) optimisation algorithm [START_REF] Nocedal | Numerical optimization[END_REF]. In practice, the numerical optimisation procedures are computed with ALGLIB library [Bochkanov and Bystritsky, ], which allows to program constrained and unconstrained optimisation problems with a series of available algorithm (L-BFGS, Conjugate gradient, Levenberg-Marquardt, etc.).

Gradient of the FD-Fit Function

The function R for the fitting of the force densities is convex and its gradient can be calculated analytically to improve the efficiency of the optimisation algorithm. The gradient is the sum of the derivative with respect to the force densities. Let's consider the derivative of R with respect to a single force density q kl . The derivative of the residual forces with respect to q kl is zero except at the node x k and x l :

∂R ∂q kl = ∂ ∂q kl |R k | 2 + |R l | 2 (5.11)
Each term in the previous sum can be calculated separately:

∂||R k || 2 ∂q kl = ∂ ∂q kl j∈k q kj (x k -x j ) + P k 2 = ∂ ∂q kl q 2 kl (x k -x l ) 2 + 2 j =l q kl q kj (x k -x l ) • (x k -x j ) + 2q kl (x k -x l ) • P k = 2(x k -x l ) • R k
Following the same methodology to calculate the derivative of R l with respect to q kl , it comes:

∂R q kl = 2(x k -x l ) • (R k -R l ) (5.12)
∇R can finally be calculated by iterating Eq. ( 5.12) on all elements of the model. If the result of the optimisation is zero, then the model is already in equilibrium. However, the value of the force in the elements can not give at this stage information on the actual membrane stresses in the structure.

Case of the Hemispherical Dome

The example of the hemispherical dome is useful to give a first insight on the well-behaving of the algorithm, since the analytical solution for the membrane forces on the shell are well known [START_REF] Ventsel | Thin Plates and Shells: Theory: Analysis, and Applications[END_REF]. Indeed, for a given radius R and surface load p, the meridional force N 1 and circumferential force N 2 (in the direction of the parallel) only depends on the angle φ between the vertical and the position on the meridian. The forces are:

N 1 = - pR 1 + cosφ (5.13) N 2 = -pR cosφ - 1 1 + cosφ (5.14)
The meridional force is always negative, the material is in compression in this direction. Its amplitude decreases from the bottom of the dome to the top. The parallel force changes sign at some point along the meridian. In the lower part of the dome, the force is positive. The parallels are tensile rings. In the upper part of the dome, the parallel force is negative and the rings are in compression. The exact angle where N 2 changes sign is φ * = 51°49 . The part of the dome above the parallel given by φ * is thus fully in compression and can be built as such using unreinforced concrete or masonry. The amplitude of the forces is in adequacy with what is expected. The meridional forces are all negative and maximum at the bottom of the dome which is supporting all the structure. The parallel forces are in traction in the lower part and change sign to create compression rings in the upper part of the dome.

Free-Form Geometries

The two following examples show the results of the fitting of the force densities on non-standard geometries. The surface of Fig. 5.9 is a free-form shell with a complex change of curvature. In this example, it is supported along all its edges (shown by the black lines on the figure), both at the base and the upstanding arches at the ends. Nevertheless, the force-fitting algorithm manages to converge close to zero. This means that this geometry and its boundary conditions can withstand its self-weight through membrane stresses only. However, those stresses are both compressive and tensile, and the next section will show how this geometry can be transformed to include compressive stresses only.

The geometry on Fig. 5.10 is chosen explicitly for its incapacity to be in equilibrium with membrane stresses only. This S-shape surface is supported on its rectilinear edges only (shown in black on the figure). In this case, the optimisation trying to fit the force densities converges but the final result of the objective function defined in Eq. (5.9) is different from zero. The residual forces of the free nodes given by Eq. ( 5.8) are shown on Fig. 5.10 with the green spheres. The diameter of the sphere corresponds to the amplitude of the residual force at the node or the deviation from equilibrium.

Geometry Close-Fit

Once the model is initialised, the form-fitting procedure starts. This operation consists of finding the closest geometry to the target shape in equilibrium with the external load and that respects the constraints on the force densities. The fitting is evaluated by measuring the distance node to node between the current geometry and the initial one. The following form-fitting process is inspired by the one presented by Block and Lachauer in [START_REF] Block | Closest-fit, compression-only solutions for free form shells[END_REF], and detailed in the book "Shell Structures for Architecture" [START_REF] Adriaenssens | Shell Structures for Architecture : Form Finding and Optimization[END_REF], with one major difference. In the cited version, the fitting is performed by moving the nodes in the vertical direction only. The algorithm is based on the Thrust Network Analysis which provides a form-finding tool by playing with the duality between the planar form and force diagrams. The form diagram is chosen at the beginning of the process to suit the target geometry's features. After that, all the optimisation is done iteratively on the force densities and the vertical position of the nodes. This strategy is particularly adapted to the study of existing masonry structures made of large stone voussoirs. Indeed, in this case, the form diagram is directly given by the existing position of the different voussoirs and the fitting process allows to analyse the stress distribution.

However, with additive manufacturing, the form diagram and the position of the nodes cannot be directly linked to a specific part of the structure. Besides, in the piecewise analysis described in the next chapter, the 3-directional freedom given to the nodes during the form-fitting is an advantage in the exploration of the 3D Printing design space.

Close-Fit Formulation

The close-fit algorithm consists of finding the force densities q i that minimise the distance from the geometry to its initial configuration. Only the force densities are variables of the problem since for each set of force densities q, the Force Density Method provides a new position of the vertices x. The close-fit function f representing the distance between the current geometry and the initial shape can be written as:

f (q) = N f i=0 |x i (q) -x * i | 2 (5.15)
Where x i is the position in the current geometry of the i th node and x * i is the position of the same node in the target shape. The optimisation problem consists in minimising f while keeping the model in equilibrium with respect to Eq. ( 5.1) and keeping the force densities within the constrained bounds:

min f (q) s.t.
C T QCX = P q min < q < q max (5.16) This optimisation problem can be solved using the box-constrained Conjugate Gradient method. The constraint on the equilibrium of the system is integrated directly in the objective function f . For each new set of force densities q, the new system in equilibrium is calculated with Eq. (5.5) before measuring the distance from the newly found nodes to the target ones.

The gradient of f can be calculated analytically as well in order to speed the optimisation algorithm. The gradient calculation follows the methodology given in [START_REF] Adriaenssens | Shell Structures for Architecture : Form Finding and Optimization[END_REF] for the geometry close-fit using Thrust Network Analysis, but is calculated for the three space coordinates:

∇f (q) = 2(X -X * )D -1 N C T N CX
(5.17)

Influence of the Constraints

Without looking yet at the actual stresses in the structure, the constraints on the force densities can guide the structure geometry. For instance, when working with unreinforced masonry or concrete, the geometry is preferably driven towards a funicular system where the material is entirely in compression. This can simply be applied by imposing all force densities of the model to remain negative. With this boundary constraint, the optimisation problem of Eq. ( 5.16) becomes:

min f (q) s.t. C T QCX = P q < 0 (5.18)
As a result, the geometry is moving away from the target shape but the system remains in equilibrium and no tension can appear in the structure under the considered loading case. This application of the form-fitting to the hemispherical dome is given in Fig. 5.11. The final positions of the nodes prevent the apparition of the tensile rings in the bottom part of the dome as observed in Fig. 5.8.

The same optimisation method can be applied to more complex geometries like the free form surface presented in Fig. 5.9. When imposing all force densities to be negative (elements in compression), the nodes move to ensure that structure remains funicular. The convex bump in the centre of the initial shape disappears and the overall surface curvature is smoothed.

The last example in Fig. 5.13 looks at the best-fit funicular geometry starting from the Sshape surface from Fig. 5.10. The result of the optimisation is an almost flat shell, with a slight positive curvature in both longitudinal and transverse directions. The curvature ensures a good equilibrium between the vertical external forces and the internal forces given by the edges of the model.

Minimal Thickness of a Hemispherical Dome

The minimal thickness of an arch or a dome is given by the minimal envelope that can contain a thrust line for the given loads. For an arch under its self-weight, the thrust line is a catenary and the minimal thickness is the distance between two arcs of a circle that come in contact with the catenary, one from inside and one from outside. The illustration of an arch of minimal thickness following this definition is illustrated in Fig. 5.4.left. In [Fantin, 2017], Fantin makes a detailed historical analysis of the different studies regarding the minimal thickness of a circular arch and a hemispherical dome.

Heyman defines a "geometric factor of safety" by comparing the minimal theoretical thickness d and the actual thickness D of the structure [Heyman, 1969]. Fig. 5.14 taken from [O 'Dwyer, 1999] illustrate this geometric factor of safety which simply reads D d . With this definition, the determination of the minimal safety factor is an important challenge towards material optimisation and as FANTIN recalls in his thesis [Fantin, 2017], different methods of resolution have been developed to solve this problem. To evaluate the efficiency of the form-fitting algorithm, it is tested on the minimal thickness problem of a hemispherical dome. The model is initialised using the force densities fitting algorithm (cf. Eq. (5.10)). Then the form-fitting algorithm is used by constraining the force densities such that q < 0. The minimal thickness of the dome is measured by drawing arcs of a circle centred on the centre of the dome and passing through the closest and furthest node on the dome to the centre point. The difference in radius of these two arcs directly gives the minimal thickness value or the minimal dome envelope. The result of this process is illustrated in Fig. 5.15. The figure shows the resulting dome geometry (black line), as well as the initial hemispherical dome (dashed line) and the minimal envelope. The nodes of the final structure where the thrust line is in contact with the envelope are marked on the right of the figure and their angle with the vertical is measured.

The result of this process gives a minimal ratio thickness over radius d R = 0.0433. This result should be compared with the ones presented by FANTIN:

• d/R = 0.0420 [Heyman, 1967] • d/R = 0.041 [Lau, 2006] • d/R = 0.043 [Zessin, 2012] • d/R = 0.0452 [START_REF] Coccia | Minimum thrust and minimum thickness of hemispherical masonry domes[END_REF] Figure 5.14 -Minimal theoretical thickness d and actual thickness D defining the "geometric safety factor" from [O 'Dwyer, 1999] • d/R = 0.0428 [Fantin, 2017] Finally, although this approach is not directly looking at the minimal thickness of a hemispherical dome in masonry, it yields results similar to the ones observed in the literature.

Stress Evaluation

The final step in the constrained form-fitting algorithm is to evaluate the actual membrane forces in the structure to ensure stability from a material point of view. Indeed, as stated before, the analysis in masonry stops after the generation of a thrust line in equilibrium with the external loads, considering that the material has infinite compressive strength. In the present case, the evaluation of the membrane stress field is used to compare the material stress with the yield criterion.

The stress field can be evaluated directly from the network of force densities in equilibrium with the external loads. This starts with the discovery by AIRY that plane stresses can be represented by a stress function represented by a surface. The curvature of the surface gives the amplitude and direction of the stresses [Airy, 1863]. Later, PUCHER proposed a method to calculate the membrane stresses in a shell subjected to vertical loads only, using the membrane stress function [Pucher, Adolf, 1937] [Pucher, Adolf, 1937]. The membrane stresses are projected on the horizontal plane. The stress surface can be calculated from the equilibrium equations resulting in the planar stresses or Pucher stress of the initial shell. Finally, the general membrane stresses are calculated by projecting back the planar stress onto the initial shell surface.

A similar method has been developed for a discrete representation of a planar surface (trusslike surface) [START_REF] Fraternali | A lumped stress method for plane elastic problems and the discrete-continuum approximation[END_REF], [START_REF] Fraternali | On the Correspondence between 2D Force Networks and Polyhedral Stress Functions[END_REF] where the Airy stress surface is generalised to a polyhedral stress surface. The planar stress can be calculated from the discrete forces in the edges connecting the nodes. FRATERNALI uses this method for the analysis of unreinforced masonry vault [Fraternali, 2010]. PELLIS et al. generate the stress direction from the discrete Pucher stress to align them with the structure principal curvature directions [START_REF] Pellis | Aligning principal stress and curvature directions[END_REF].

The Surface Stress Density Method proposed by MAURIN and MOTRO in 1998 also allows running a form-finding process while evaluating the membrane shell stresses directly from the force densities [START_REF] Maurin | The surface stress density method as a form-finding tool for tensile membranes[END_REF]]. However, they impose that all stresses must be isotropic which is too restrictive for free form structures in compression, especially when considering the temporary structural behaviour during the fabrication.

Membrane Stress Field from Planar Stress

The planar stress field can be expressed in function of the force densities q, the node position in the horizontal plane and the influence area Ā of each node in the horizontal plane (see Fig. 5.16. In all the following, the upper bar symbol refers to the projection of the concerned object in the horizontal plane.

N (x i ) = 1 2 Āi j∼i q ij (x j -xi ) T (x j -xi ) (5.19)
This relation gives the stress tensor of the structure projected onto the horizontal plane. The following procedure explains how to project the tensor back onto the initial 3D shape to retrieve the actual membrane stresses.

By noting k ij = xi-xj lij , where lij is the length in the projection plane between nodes xi and xj , Eq. ( 5.19) becomes:

N (x i ) = 1 2 Āi j∼i q ij l 2 ij (k ij ⊗ k ij ) (5.20)
Let's introduce u an v, two orthonormal vectors laying in the tangent plane to the initial geometry at node x i , and ū and v their respective projection in the horizontal plane after normalisation. The amplitude of the in-plane stress in the direction ū is:

Nūū (x i ) = 1 2 Āi j∼i q ij l 2 ij k ū ij k ū ij (5.21)
Figure 5.16 -Detail of a node on the initial geometry and its projection on the horizontal plane

Where k ū ij is the scalar product between ū and k ij . Finally, following Pucher theory [Pucher, Adolf, 1934], the amplitude of the membrane stress on the initial geometry in the direction u is given by:

N uu (x i ) = Nūū (x i ) cos φ cos ψ (5.22)
Where ψ is the angle between u and its projection ū on the horizontal plane and φ is the angle between the vector ū⊥ orthogonal to ū in the horizontal plane and its unitised projection u ⊥ on the tangent plane. Note that u and u ⊥ are not necessarily orthogonal.

Eq. ( 5.4.1) can be expressed as:

N uu (x i ) = Nūū (x i ) ||ū ⊥ • u ⊥ || ||ū • u|| (5.23)
The same methodology is repeated to retrieve the amplitude of the membrane stress in the v direction. The shear component N uv is computed by calculating the stress component N ww in the direction w = u+v 2 and using the stress transformation formula:

σ x = σ x + σ y 2 + σ x -σ y 2 cos(2θ) + τ xy sin(2θ) (5.24)
Replacing, σ x by N ww , σ x by N uu and σ y by N vv , and setting θ = π 4 in Eq. ( 5.24), it comes:

N ww = N uu + N vv 2 + N uv ⇒ N uv = N ww - N uu + N vv 2 (5.25)
Leading finally to the membrane stress tensor on the initial geometry:

N (x i ) = N uu (x i ) N uv (x i ) N uv (x i ) N vv (x i ) (5.26)
5.4.2 Validation of the Methodology with an Analytical Formula: Principal Stresses on a Hemispherical Dome

The special case of the hemispherical dome is used again to evaluate the method introduced above to determine the membrane stress field. The analytical values of the stress in the meridional and circumferential direction on a true dome (see Eq. ( 5.13)) can be extended to a dome with an oculus (Fig. 5.17) at its top subjected to a linear loading q (see [START_REF] Ventsel | Thin Plates and Shells: Theory: Analysis, and Applications[END_REF]):

N 1 = 1 sin 2 φ
(pR(cos φcos φ 0 )q sin φ 0 ) (5.27)

N 2 = 1 sin 2 φ
(-pR(cos φ sin 2 φcos φ 0 + cos φ) + q sin φ 0 ) (5.28) Figure 5.17 -Geometry of the truncated dome from [START_REF] Ventsel | Thin Plates and Shells: Theory: Analysis, and Applications[END_REF] In this case study, the linear loading at the oculus is not considered q = 0. The opening angle at the top is set at φ 0 = 0.16rad, the radius of the dome is R = 0.5m and its thickness is t = 5cm leading to a surface load of p = 1128N/m 2 for a material density of ρ = 2300kg/m 3 . The results of the analysis are shown on 5.18 which gives the evolution of the meridional (N φ ) and circumferential (N θ ) stresses in function of the angle φ) from the analytical formula and the numerical model using the form-fitting approach.

The dashed lines represent the analytical values of the meridional and parallel stresses in the dome. The parallel force N θ changes sign for φ * = 51°49 which is shown on the graph. The numerical calculation from the force densities fits well the analytical values for both directions. A small deviation of the meridian force is visible for high values of φ corresponding to the base of the dome. In this area the tangent plane to the nodes is close to a vertical, causing a poorly structured projection of the force network on the horizontal plane. As explained by FRATERNALI et al. [Fraternali, 2007], the method to retrieve the Pucher stress from the planar force densities is accurate for structured mesh only and deviation occur otherwise. A new method to work around this issue and achieving a good result with unstructured mesh has been proposed by AMENDOLA [Amendola, 2019], it was however not implemented in this thesis.

Strength-Based Analysis

At this step in the constrained form-fitting process, a funicular geometry is found and the membrane stress field is calculated on the entire shell. If the material properties are known, a simple yield analysis can be run through the nodes of the model to avoid strength-base failure. The yield criterion must be chosen wisely.

In the literature, Von Mises yield criterion is often used to characterise the behaviour of concrete in its fresh state [Coussot, 2005], [Roussel, 2018]. However, when the initial yield stress is high (∼ 2kP a), JACQUET et al. showed that the material is not symmetrical anymore (compressive and tensile strength values are not equal) and Von Mises criterion can not be used anymore [START_REF] Jacquet | Assessment of asymmetrical rheological behavior of cementitious material for 3D printing application[END_REF] [START_REF] Kruger | 3D concrete printing: A lower bound analytical model for buildability performance quantification[END_REF], [START_REF] Jayathilakage | Yield stress criteria to assess the buildability of 3D concrete printing[END_REF]) with modified Mohr-Coulomb and Tresca criteria respectively.

In this dissertation, a simple Tresca criterion is chosen to run the yield analysis during the fabrication process. For the characterisation of the extrusion presented in the next part, the Von Mises criterion is kept as the reference. Tresca criterion characterises the maximum shear stress admissible and therefore requires the knowledge of a single material parameter, namely the shear yield stress, while Mohr-Coulomb criterion needs the cohesion and the friction angle of the material and Drucker-Prager criterion needs both compressive and tensile yield strengths. Tresca criterion can be written as a function of the two principal stresses of the membrane stress tensor:

τ c > 1 2 max (|σ 1 -σ 2 |, |σ 1 |, |σ 2 |) (5.29)
Where τ c is the shear yield stress of the material whose evolution is given by τ c (t) = τ 0 + t × A thix . This limit analysis can be incorporated into the workflow after the form-fitting optimisation. If the yield criterion is verified for all nodes, then the geometry of the structure and its associated stress field are admissible. If the yield criterion is not verified, the form-fitting step is run again after modifying the constraints on the force density values (see Eq. (5.16)). This iterative process continues until all nodes verify the yield criterion as shown in the diagram on Fig. 5.5.

Figure 5.19 -Influence of the maximum maximum shear strength of the material on the shell geometry in the constrained form-fitting approach. Fig. 5.19 shows the influence of the yield strength on the final dome geometry. In this example, the material properties are the same for the entire structure. The shear yield stress value τ c given below each dome is chosen arbitrarily. The force density bounds are then set manually by the user until the yield condition of Eq. ( 5.29) is respected everywhere in the structure. The dashed line shows the perfect hemispherical dome geometry. For the dome on the right, an additional constraint of pure compression is added by imposing all force densities to remain negative (q < 0).

Towards Stress Constrained Optimisation

The algorithm that has been presented requires the user intervention in the setting of the force densities. This process can be time-consuming and imprecise when the structure gets large and the number of nodes in the mesh gets high. A possible development of the algorithm of Fig. 5.5 would be to run the form-fitting algorithm with constraints on the yield criterion directly, instead of the force densities. The problem can be formulated as follows:

min f (σ) s.t. C T QCX = P σ ∈ D (5.30)
Where D is the material yield domain. This way the result of the optimisation is directly an admissible geometry and no iterative process is required. However, the constraints in the optimisation algorithm become non-linear while Eq. ( 5.18) introduces box-constraints only. It can be solved using an Augmented Lagrangian algorithm with non-linear constraints. It was implemented and used to solve the problem of the hemispherical dome. However, the algorithm only converged for a very small amount of nodes and edges in the model (8 nodes), after what convergence was not achieved or took too much time due to the need for a precise setting of the additional parameters involved in the Augmented Lagrangian algorithm ( [Bochkanov and Bystritsky, ]). Therefore the rest of this dissertation only consider the form-fitting with constraints on the force densities, manually set by the user.

Chapter 6

A Computational Strategy for Sequential Form-Finding

Now that the constrained form-fitting algorithm is introduced, it can be applied to the formfinding of structures to be fabricated by additive manufacturing of cement-base material. The following algorithm is an attempt at understanding the ideal geometry for 3D printing. The process consists of a step by step analysis of the structure, considering its geometric evolution in time with respect to the material properties.

It is reminded here that at this stage no consideration is given to the final layers and their thickness. Indeed the scale of the layers is small compare to the scale of the structure and has no impact on the structural stability during the fabrication process.

Principle and Main Hypothesis

The principle of the algorithm is to start from a target geometry divided into strips that can be activated one after the other, in a discrete simulation of the printing process. The strips and the final physical layers only share the same orientation, meaning that the layers will be aligned with the strips but at this point, no information on the layer geometry can be deduced from the strip dimensions. The structural stability of the temporary structure is assessed at each step after every new strip is added by applying the constrained form-fitting algorithm on the model. The process looks therefore for a funicular configuration of the geometry at each step. Before starting, some hypotheses are made on the material, the process and the structural behaviour.

Material Evolution

The evolution of the material driven by the yield stress τ c is considered to be linear in time following the model by Roussel [Roussel, 2006] used in the rest of this dissertation:

τ c (t) = τ 0 + A thix × t (6.1)
This model of the evolution of the material is conservative. Perrot proposed an extension of this relation to an exponential growth after a characteristic time t c [START_REF] Perrot | Prediction of lateral form pressure exerted by concrete at low casting rates[END_REF], however, it was not developed for accelerated material used in Concrete 3D Printing. Therefore, the simpler linear evolution of the material is preferred here. Note that if more precise models of the material evolution for accelerated mortar were to arise, they could be implemented easily without modifying the core of the algorithm.

Minimal Flow Rate

The main goal of the form-finding process presented here is to provide a printable geometry for the designer. An implied objective is to print the geometry continuously, in a minimum amount of time. This is equivalent to maximise the growth velocity introduced in the first part of this dissertation (see section 4.4.3). The growth velocity is directly related to the flow rate Q of the material which is the main global printing parameter of the printing process.

The proposition here is to set a minimal flow rate Q min for the entire process. This value is used in the algorithm to calculate the age of the material at each successive step. Depending on the results of the yield analysis at each step, the flow rate for the last strip can be adjusted upwards, thus reducing printing time.

The minimal flow rate Q min is an input parameter in the algorithm that must be set by the user. Its choice can be guided by productivity needs or by process-related constraints. Indeed in the context of industrial fabrication, the profitability of the process may impose a maximum time available for printing a given amount of material, thus imposing the minimal flow rate. Q min can also be directly given by the specification of the pumping equipment used in the process. Finally, for printing set-up without a continuous mix feeding system for the pump, the minimal flow rate can be related to the open time T open of the material and the volume of each batch Ω batch awaiting upstream. Simply put, all the material in the batch must be pumped before the open time is reached, which leads to the following expression of Q min :

Q min = Ω batch T open (6.2)

Structural Behaviour

The algorithm developed here exploits shell behaviour with membrane stresses only to ensure that the structure is fully in compression. However, when the material has just been extruded, the membrane stresses involved may be too high to consider that a thrust line or thrust network is developed in the strip. Although, the material can still hold by simple friction on the previous layers. In the barrel vault of Fig. 6.1, the strips act as arches that support their own weight by compression forces. But these forces can only exist if the material has had time to hardened. The hypothesis here is to consider that the last printed layer (red strip in Fig. 6.1) is only supported by friction on the previous strip. The weight of that strip is carried by the shell structure formed by the previous strips altogether. This reasoning is similar in masonry construction where the last brick layer usually plays its structural role only after all bricks have been set and the "keystone" allows the forces to go through. While the layer is incomplete, the bricks are held by friction on the previous layers.

This assumption is directly applied in the algorithm in the activation of the strips. The last strip activated and on which the form-fitting optimisation acts is the penultimate printed strip. The last printed strip is not activated in the model, however, its weight is applied to the last activated strip as an external load. This way, the age of the material of the last activated strip is equal to or higher than the printing time of the next strip.

Model Definition

This section details the initialisation of the model used in the algorithm. Similarly to the previous chapter, the model is composed of nodes whose positions define the overall geometry and edges transmitting the internal forces. The weight of the material is the only load acting on the structure and is applied to each node as a function of their influence area on the mesh.

Initial Quad Mesh

The first input of the model is the target geometry of the structure. It must come as a quad mesh that can be divided into strips perpendicular to the extrusion direction defined in section 4.2.2. At the time of writing this dissertation, the algorithm does not accept singularities in the mesh. For instance, the structure of a dome should come with an opening at the top to avoid the singularity that appears when closing it. The strip itself can be closed (like in a dome) or open (like in a barrel vault) (see Fig. 6.4).

The implementation of the mesh in the algorithm uses the half-edge data structure first developed by MULLER in 1978 [Muller and[START_REF] Muller | [END_REF]. It uses more memory than a classic face-vertex data structure but has the advantage to provide extensive information on the connectivity between the edges and vertices, allowing for fast and easy navigation and manipulation of the mesh.

On Mesh Density

The initialisation of the model requires the choice of the mesh density. It has been shown previously that the calculation of the membrane stresses is only valid for structured meshes. Therefore all edges lengths must be of the same magnitude. By looking at the quad mesh, it appears that the edge length determines the width of a strip in the model. The problem of finding a good strip width w is crucial for the good behaviour of the algorithm.

A first guess would be to choose w as small as possible, to create a very dense mesh as it is done for the Finite Elements method. This way the number of strips is high and the precision of the algorithm is increased. However, it was shown in the previous section on the structural behaviour of the printed structure that if the material is too young, it is not able to withstand membrane forces and must hold by friction on the previous strip. A minimal width w min of a strip is difficult to assess since it requires the determination of the actual membrane stresses which are unknown at this stage. Choosing a strip width w too small may not provide the previous strips enough time to build up an internal structure that can withstand their weight. The idea is thus to determine a maximal strip width w max that can be held by friction only and can help setting a correct mesh density. To do so, the method used in section 3.4.1.1 for the calculation of the maximum layer inclination can be used, but for a strip instead of a single layer. The strip is defined by its width w to be determined, its length L and thickness b, and its inclination α (see Fig. 6.2). 3.1) is still valid in this case but since the ratio w/b is not small, the normal stress σ must be calculated by taking into account the bending moment in the material.

     N = F cos(α) T = F sin(α) M = F w 2 sin(α) (6.3)
Where F = ρgwb is the dead load of the strip per unit length, applied to its centre of gravity. The shear and normal stresses are calculated in the upper part of the strip section, where the bending may create tension that is the most critical for the material:

τ = ρgw sin(α) σ = ρgw cos(α) -3ρgw 2 b sin(α) (6.4)
Combining equations ( 6.4) and (3.1) yields the condition:

ρgw sin(α) < c + ρgw cos(α) - 3ρgw 2 b sin(α) tan(φ) (6.5)
The cohesion c being the yield strength for pure shear failure, it can be replaced by the material yield stress τ c :

τ c > ρgw sin(α) -cos(α) - 3w b sin(α) tan(φ) (6.6)
The value of τ cmin is plotted in Fig. 6.3 for a friction angle φ = π/4, in function of the strip aspect ratio w/b for different inclinations α. Now, the yield stress of the material can be evaluated at the base of the strip by calculating its age after completion of the strip. A lower bound of t can be evaluated by introducing a maximal flow rate Q max that must not be exceeded during the printing:

t min = Lbw Q max (6.7)
Combining Eq. ( 6.6), ( 6.1) and ( 6.7) gives a yield condition with respect to the strip geometry and the material properties:

τ 0 + A thix Lbw Q max > ρgw sin(α) -cos(α) - 3w b sin(α) tan(φ) (6.8)
This last expression can be used as such to validate the choice of mesh density when creating the model. Note that after triangulation and smoothing of the mesh, the strip length L and width w might vary from a strip to the other. By implementing Eq. (6.8) in a parametric model can help validate in real-time the initial mesh geometry.

Eventually, a simplified expression of the maximum width w max can be obtained by taking conservative values for the parameters in Eq. 6.8. By neglecting the initial yield stress τ 0 = 0, taking φ = π/4 and α = π/2 (horizontal extrusion), it simply yields:

w max = b 3
LbA thix ρgQ max -1 (6.9)

The barrel vault of Fig. 6.1 with a span of 1.5m and a thickness b = 5cm is taken as an example for the determination of the maximum strip width w max . The strip length is L = 3m, the structuration rate is taken as A thix = 30P a/s and the maximum flow rate as Q max = 45g/s. The result of Eq. 6.9 is w max = 15cm so three times the shell thickness. In practice a strip width twice larger than the shell thickness is a good initial guess.

Boundary Conditions

The boundary conditions define the supports of the structure. A distinction between two different sets of supporting nodes must be done:

• the first set of nodes is composed of the supports of the first strip. They define the path for the printing of the first layer and will help to order the strips of the model following the extrusion direction as described in the next section.

• the second set of nodes contains all additional supports, that appear progressively during the fabrication process. They are not present in the first step of the algorithm. On a dome, only the first set of nodes exist (see Fig. 6.4.left). The structure is entirely supported by its first layer. In the case of a barrel vault, each layer is supported at its end, creating new supports during the fabrication process (see Fig. 6.4.right).

By default the supports act as perfect articulations, they are fixed in all directions and entirely free of rotation.

Strips Sequencing

The strips of the model are used to simulate the fabrication process by activating them successively. In a mesh, a strip usually consists of a series of faces that are aligned. In the model presented here, the strips are defined by their components, namely the nodes or vertices of the mesh and the edges that are separated into transverse and longitudinal edges.

Except for the nodes on the free boundary of the model, all nodes belong to two strips. The edges however can only be part of a single strip. Fig. 6.5 shows the division of the model into strips. Edges of the same colour belong to the same strip. The first strip called strip-0 is a single polyline made of the supports of the first layer and the edges in-between. The other strips are made of the transverse edges leaving from the previous strip and the longitudinal edges parallel to the previous strip (see Fig. 6.5).

The creation of the strips is performed by an orderly navigation the mesh and allocation of the visited nodes and edges following the following procedure:

1. Create first strip-0 from supports nodes; 2. Select any node of the first strip and find the outgoing half-edge. Add it to strip-1 as transverse element;

3. Add all parallel edges to the first half-edge to strip-1 by travelling the mesh using the half-edge sequence: next-next-pair ;

4. From the first half-edge find the next half-edge and add it as a longitudinal element of strip-1 / next;

5. Add all longitudinal edges of strip-1 by travelling the mesh using the half-edge sequence: next-pair-next;

6. Add the start vertex of each half-edge of strip-1 as node of the strip;

7. Repeat steps 2 to 6 until all strips are created. The operation on the half-edges is illustrated in Fig. 6.5. The arrows represent the halfedges. The red arrow is the first transverse half-edge of strip-1 from step 2. The numbering of the half-edges on the figure corresponds to their creation step in the sequencing procedure above.

While travelling the mesh, a verification is done that the current half-edge is not on the boundary or that it has not already been visited. If it is on the boundary, the loop stops and the mesh is travelled again from the first half-edge but in the opposite direction. If the half-edge has already been visited, that means the strip is closed and the loop stops to go to the next step.

Triangulation and Smoothing

The calculation of the membrane stresses following the procedure described in the previous chapter is valid for structured triangular meshes only [START_REF] Fraternali | A lumped stress method for plane elastic problems and the discrete-continuum approximation[END_REF]. This means that all angles between consecutive edges must be as close as possible to 60°and outgoing edges of the same vertex must be of similar length. To achieve this, the quad mesh is triangulated and then smoothed according to the following methods.

The triangulation simply consists of dividing each quad face of the mesh into two triangles by adding a diagonal edge. The triangulation process is done strip by strip so that each newly added edge can be allocated to the correct corresponding strip as a transverse element. The choice of the diagonal to create in each strip has an impact on the final triangulated mesh after smoothing. Fig. 6.7 shows the influence of the triangulation on the final mesh. When the quad faces are triangulated randomly, the smoothing leads to a very unstructured mesh and nodes with different valence from 4 to 8. On the contrary, when all faces are triangulated in the same direction, each internal node has a valence of 6 but the final mesh is not perfectly structured either. With all diagonals in the same direction, the final mesh is twisted which would lead to a poor calculation of the membrane stresses principal directions.

Finally, the solution for a correct node valence and structured mesh is to orient all diagonals of the same strip in the same direction and alternate that direction every other strip (see Fig. From the triangulated mesh, a Laplacian smoothing is performed. This is an iterative process where each vertex of the mesh is replaced by the average of all the vertices it is connected to. The new position of each vertex is thus:

x new = 1 N i∈N x i (6.10)
In the present algorithm, all boundary nodes are fixed in the smoothing procedure. This includes all support nodes and all nodes on free boundaries so that the initial geometry is maintained intact. At each iteration of the smoothing, all internal nodes x move by a vector v = x newx. However, doing so modifies significantly the initial geometry as illustrated on Fig. 6.6.right). This phenomenon is known as the shrinkage of the mesh [Taubin, 1995]. An additional constraint must be introduced to keep the nodes on the initial geometry. A straightforward approach is to project the new nodes back on the initial mesh at each step of the iteration process. This solution solves the problem but is very costly numerically. In [Taubin, 1995], Taubin proposes to multiply the vector v alternatively by a factor λ and µ with 0 < λ < -µ so that the mesh oscillates around its converging state. The setting of parameters λ and µ can depend on the mesh, its scale and its density, complexifying the overall problem. The choice made in the current algorithm is to simply prevent any displacement of the vertices along the normal direction to the mesh. The nodes are finally moved by a vector

v * = v -v • n.
The last step of the model initialisation is the calculation of the mesh vertices influence area used to determine the external loads applied to each node. The area is calculated by allocating to

Piecewise-Funicular Algorithm

Once the model is initialised, the sequencing procedure starts. The constrained form-fitting algorithm presented in the previous chapter can now be applied strip by strip. The age of the material is calculated at each step of the procedure to update the material properties correspondingly.

Asymptotic Approaches

The principle of the piecewise funicular algorithm is to run the form-fitting procedure step by step, after activation of each new strip. At this stage, a choice must be done by the user on the kinematic of the strip activation, depending on the material structuration rate, the printing equipment and the scale of the structure.

The State of the Art detailed two asymptotic printing strategies for extrusion-base additive manufacturing, namely, the Infinite Brick Extrusion (or Extrusion Lace Shaping) and the Layer Pressing Strategy (or Oriented Lace Pressing) [Duballet, 2019], [Roussel, 2018]. It was shown that the first one is adapted to slow material structuration while the latter is better suited for materials with fast thixotropy, where accelerant has been added to the mix. Fig. 6.9 illustrates the material yield stress evolution in time for both cases. Based on this, two different approaches are developed for the piecewise form-fitting algorithm. In the first case, the process considered uses a material with a fast structuration rate. The initial yield stress is low but it increases rapidly so that only the last activated strip of the model is considered. All previous strips are deactivated and it is assumed that they are stiff enough to withstand the loads coming from the new layers independently of their geometry. In the second approach, the material evolution is slow, so that the form-fitting is applied to the entire set of strips from the first one to the last activated. Figure 6.9 -The two asymptotic approaches are illustrated in terms of material properties evolution. Case A refers to a material with a high structuration rate but starting at a low initial yield stress τ a . Case B shows a more homogeneous yield stress in the entire structure.

Strip by Strip Analysis

In this first approach, the form-fitting procedure is applied to one strip at a time. The main assumption being that the material of the previous strips is stiff enough to withstand the load they are carrying. After a strip is calculated, it remains deactivated in the model until the entire procedure is complete.

Another constraint imposed to the model is that the free nodes of the new strips cannot be lower in height than the previous strip. In practice, this prevents printing with the extrusion nozzle facing upwards (see Fig. 6.10). In the algorithm, each node x is compared to its closest node x close on the previous strip, and its height is updated accordingly:

z x = max(z x , z x close ) (6.11)
This step is especially important when targeting compression-only structures. Indeed the fitting of the force densities on a strip going "down" will automatically result in elements in tension. Limiting the strip to an inclination of π/2 provides a better geometric initialisation of the model.

Based on the assumption presented in the previous section, the activated strip n is carrying the weight of the next strip n + 1 (or ghost strip) as illustrated on Fig. 6.11. The age of the free nodes of the strip n is thus T n+1 max . It corresponds to the maximum printing time of strip n + 1 and can be expressed as the ratio between its volume of the strip and the minimal flow rate Q min :

T n+1 max = Ω n+1
Q min (6.12) Figure 6.10 -Illustration of the initialisation of the strip geometry. Each free node of the current strip is projected onto the free edge of the previous strip (left). The nodes are moved vertically if the projected nodes are higher than the current ones, giving the new strip geometry (the strip free edge is in red). The dashed lines can be seen as the nozzle direction while printing. The age of the nodes is used to calculate the material properties in order to compare the yield stress at the node to the yield criterion coming from the membrane stresses.

The steps of the procedure with this first approach are:

1. Activation of the strip: the elements composing the strip are activated and so are the nodes.

The nodes in contact with the previous layer are considered fixed.

2. Application of the loads: in addition to the self-weight of the strip, the free nodes of the strip are carrying the entire weight of the next strip as assumed in the hypothesis described in the previous section.

3. Running of the constrained form-fitting algorithm on the strip: the form-fitting procedure detailed in the previous chapter is applied to the strip, starting by the force densities optimisation, the form-fitting and the calculation of the membrane stresses at the free nodes. Then the verification of the yield criterion is performed at each node. The material properties at the nodes of the strip are calculated based on their age which depends on the strip volume and the minimal flow rate Q min . If all nodes are compliant with the yield criterion, the procedure continues to the next step, otherwise, it iterates after manually modifying the force densities bounds q min and q max . 4. Calculation the optimal printing time T opt for the strip and lock all nodes: this way the supports of the next strip are automatically set.

Figure 6.12 -Strip by strip analysis: a. stresses in the initial configuration. The high values of the force density bounds allow for an unconstrained form-fitting. The red curve on the strip free edge indicates that the maximum stress in the strip exceeds the material yield strength; b. in the middle, the force densities are highly constrained, the strip has deformed and shrunk but is not yet admissible stress-wise; c. after manual fitting of the force density bounds, an admissible geometry for the strip is found, slightly away from the initial geometry but keeping the continuity with the previous strips.

Step 3. of the algorithm is illustrated in Fig. 6.12 which shows the influence of the force density bounds on the strip geometry and the membrane stresses in the structure. For better visual understanding during the process, the free edge of each strip is coloured in red if the maximum yield stress is higher than the material yield strength and green otherwise. Fig. 6.13 shows the result of the form-fitting algorithm on the entire structure. In this example, the first strips are the shortest in length (and volume), therefore they give less time to the material to increase its strength. In consequence, they are the strips that move the furthest away from the initial geometry.

The optimal printing time T n+1 opt is calculated to optimise the flow rate of the process for the strip n + 1. Indeed, at the beginning of the process, the maximum printing time for each strip is calculated from Eq. ( 6.12). If this printing time is kept, the entire structure is printed at the minimum flow rate Q min . However, if a strip is yield-compliant with enough margin at each node, the age of the material or the printing time of the next strip can be decreased so that the material yield stress equals the maximum yield criterion τ ymax . The yield criterion is calculated at each node as a function of the membrane stresses, the maximum yield criterion is the highest value among all the node. 6.13) From this optimal printing time, a new flow rate can be calculated for this strip which is higher than the minimal flow rate Q min :

T opt = τ ymax -τ 0 A thix (
Q n+1 opt = Ω n+1 T n+1
opt (6.14) This way, each strip can have a different flow rate. Fig. 6.13.right shows the final flow rates obtained for the vault structure used in the example above. Note that if the printing equipment does not allow for varying the flow rate during the process, the minimum flow rate Q min or the minimum of all optimum flow rates can be kept for all strips. The strip form-fitting procedure as such is not directly applicable for the first and last strips of the model. Indeed, strip-0 is not subjected to the form-fitting since all the nodes are fixed, so the printing time of strip-1 can not be correctly set. The proposal is to simply use Eq. ( 6.8) and keep the minimum flow rate Q min for strip-1 which by construction of the model is admissible. A similar problem exists with the last strip since there is no n + 1 strip to calculate the age of the free nodes. The first option is again to consider Eq. (6.8) using Mohr-Coulomb yield criterion without looking at the membrane stresses and verifying that the criterion is admissible for Q opt instead of Q max . The second option is to add a virtual strip in the last position of the model and run the standard procedure.

The strip-by-strip analysis on a hemispherical dome is provided as an example in Fig. 6.14. This dome has a diameter of one meter, a thickness of 5mm and the material properties are τ 0 = 250P a and A thix = 30P a/s while the minimal flow rate is kept at Q min = 20g/s. The resulting geometry is very close to the original dome in the bottom part and starts deviating upward as it goes up. Indeed the upper strips are smaller and therefore provide less time for the material to strengthen. This example shows how the curvature of the dome is modified to reduce the stresses. The cantilever is reduced by lifting the upper strips, decreasing the circumferential stresses which are predominant in this part of the dome. Fig. 6.15 shows the strip-by-strip analysis on the same hemispherical dome after variation of the initial mesh density. The analysis is performed for models made of 6, 9 and 10 strips. After the form-fitting process, it results that all three geometries have similar shapes. This interesting result shows that the algorithm is insensitive to the mesh density, at least when staying within the recommended range proposed in section 6.2.1.1.

Future works may look at optimising further the process by allowing changes in the structuration rate of the material on the go. Indeed, one may want that the material gets strength faster in the upper strip than in the lower ones. Fig. 6.16 shows an example of application of the strip by strip analysis on a barrel vault. The vault is supported at its base and on its first arch on the left part. The strips are composed of successive arches that are supported are their ends and by the previous strip. The same vault is analysed using different material properties. In the first case (a), the material strength is considered to be infinite (both in traction and compression). This is obtained by setting very large values for the force densities bounds (-∞ < q < ∞). Therefore the strip by strip analysis fits the target geometry perfectly.

In the second case (b), the compressive strength of the material is infinite but it has no tensile strength (-∞ < q < 0). These conditions are in line with HEYMAN's Safe Theorem hypotheses for Masonry structures. This leads to a small inclination of the strips so that each one is laying on the previous one. Note that the overall geometry is not impacted, only the strips geometry are slightly modified. Indeed, besides guiding the geometry of the structure, the algorithm also provides insights on its slicing and the orientation of the layers which are aligned with the strips.

In the last case (c), the properties of a mortar commonly used for 3D Printing are implemented with an initial yield stress of 300P a and a structuration rate of 30P a/s. At each step, the stresses are calculated and the force densities bounds are set so that the yield criterion is respected. The result is a vault where the strips have a much higher inclination. Once again, the overall geometry is not significantly modified, only the strips geometry is impacted and thus the printing path.

Limitations

At the end of the process, the result is a new geometry as close as possible to the initial shape, with a flow rate value for each strip. For large-scale structures, the form-fitting algorithm can significantly modify the geometry of the strips and thus the volume of extruded material. This leads to two errors in the procedure on the external load applied to each strip and the maximum printing time which are both calculated on the initial geometry. It must be noted, that the form-finding usually results in smaller strips than the initial geometry. Meaning that the initial calculation of the external loads is conservative while the calculation of the maximum printing time is not. Although this issue is not theoretically addressed, the experience has shown that running the algorithm a second time on the resulting geometry always leads to a converging state. In the second run, the fitting of the force densities leads to a statically admissible stress field all the time, preventing the need to modify the geometry of the strips. This second run helps thus to set the correct printing flow rates, however, it seems to indicate that the final geometry is not exactly the best fit.

Another limitation of the strip by strip process is the possible creation of kinks between strips. For instance, since the geometry of a strip is found independently from the geometry of its neighbours, trying to fit as best as possible each strip can lead to a non-smooth surface. The influence of the user can help to address this issue while setting the force densities bounds q min and q max , by visually looking at keeping the curvature continuity.

Finally, when using this approach, the user must be careful when choosing the initial geometry. The individual funicularity of each strip does not ensure at all the global funicularity of the structure. Moreover, full compression of the elements of the strips in the piecewise form-finding does not ensure full compression of all elements at the end, when considering the entire structure.

Accumulated Piecewise Analysis

The second approach is intended for processes using non-accelerated material. The material properties evolution is illustrated in Fig. 6.9 by the blue curve which starts high but evolves slowly. With a slow structuration rate, the entire structure must be analysed at all times. At the limit, this strategy can be used for non-thixotropic material like clay, for instance, presenting a high initial stiffness but almost no evolution in time. Since the material properties never reach a high value, this approach works best for smaller-scale structures (less than a meter) which do not develop high membrane stresses.

It must be noted that there is no restriction to use this approach with material presenting fast structuration properties, and besides it will be shown later how it can provide new information not available with the previous approach. Ultimately those two approaches can be combined all together.

The search for a funicular geometry is performed this time on all activated strips at each step. The core of the algorithm is very similar to the strip by strip analysis with some minor differences:

• in the last step, the nodes of the last activated strip are not locked. That way, the algorithm always runs on all strips that have been activated. each step adding a new active strip in the model.

• in the first step, at the activation of the strip, all active nodes are moved back to the initial geometry. At each step, the form-fitting is performed on a portion of the initial geometry and not the previous form-fitted shape.

• At the end of each form-fitting iteration, the geometry of the current state of the model is saved. Therefore, the result of this process is not a single geometry like in the previous algorithm, but a set of geometries that is used to define an envelope (see Fig. 6.17).

Each one of the resulting geometry is thus entirely funicular and can be, if wanted, fully in compression. This solves the limitation faced with the previous approach where while each Figure 6.17 -Result of the global piecewise form-fitting giving a new geometry at each step of the analysis. The initial dome geometry is shown in the dashed line.

strip is in compression, the final structure could have developed tensile forces (example of the hemispherical dome). The results of this analysis on a hemispherical dome are illustrated in Fig. 6.17 for a model with 6 strips. All the geometries given at each step are represented. They correspond to structures entirely in compression, thus the shift towards the inside in the bottom parts preventing the apparition of the tensile rings. The combination of these geometries can help define an envelope that is more detailed in the following section. Note that the initial geometry (illustrated by the dashed line) does not necessarily remain within the envelope.

The evaluation of the age of the material must be performed on all active nodes at each time. This is done at each activation of a new strip by adding to the age of all active nodes the maximum printing time of the next strip. Therefore the age of the free nodes of strip n is:

T n strip = N i=n T i+1 max (6.15)
where N is the number of active strips in the model.

This approach takes much more computational time than the previous one since the strip are not deactivated after each step. Another limitation so far is that the form-fitting is done with the same force density bounds for all elements. Therefore it does not take into account the ageing of the material. In practice, this means that changing the bounds to make the stresses admissible in one area of the model may affect the force densities in another area where the yield stress was already below the limit. This issue would be solved by solving the optimisation problem under stress constraints instead of force densities constraints according to Eq. ( 5.30) that has not been implemented yet.

Structure Envelope

The result of this approach is a series of geometries where the i th geometry is comprised of strip-0 to strip-i. Each node of the initial model has several corresponding nodes in the other geometries that can be used to define an envelope for the model. Let's consider a specific node x i on the free edge of strip-n, and that the model is made of N strips in total (counting strip-0 ). Node x i has thus Nn corresponding nodes (x n i to x N -1 i ) creating a point cloud around the position on the initial geometry (see Fig. 6.18). Figure 6.18 -Section of a model after piecewise form-fitting with the initial geometry (dashed line) and all temporary geometries corresponding to the activation of each strip. The envelope thickness d can be defined from the point cloud around the node on the initial geometry. The middle of the envelope x * i can then be deduced.

The envelope thickness d can be determined by calculating the distance between the two extreme nodes and project it on the normal direction n on the initial geometry:

d = max j∈[n;N -1] ((x j i -x i ) • n) -min j∈[n;N -1] ((x j i -x i ) • n) (6.16)
And the middle point of the envelope can be obtained by moving the initial node of a vector v so that:

v = max j∈[n;N -1] ((x j i -x i ) • n) + min j∈[n;N -1] ((x j i -x i ) • n) 2 (6.17)
With this definition, the envelope thickness and the middle point can be defined for each node of the model. For the nodes of the last strip of the model whose point clouds only contain one element, the envelope has a thickness of 0. A new geometry can be defined by connecting all the middle points. In Fig. 6.18, it is represented by the red line. If the envelope thickness at each node is smaller than the initially defined thickness B of the structure, then according to HEYMAN's Safe Theorem [Heyman, 1967], the fabrication process is safe. Meaning that at each step of the fabrication a thrust network can be found that remains within the thickness of the shell.

Geometric Safety Factor

In [Heyman, 1969], HEYMAN introduces the notion of Geometrical Factor of Safety for the stability of masonry arches as the ratio between the thickness of the arch D and the minimum thickness d that contains the thrust line. The idea of the minimum thickness has already been introduced in chapter 5. 3.3 In chapter 5.3.3, the notion of minimal thickness of a masonry structure was introduced as well as HEYMAN's Geometric Safety Factor which is the ratio between the actual thickness of the structure D and the minimal thickness d [Heyman, 1969]:

GSF = D d (6.18)
For circular arches or hemispherical dome, the Geometric Safety Factor is explicit since the thrust line is not parallel to the geometry of the structure. For D d < 1, the structure is unsafe, the thrust line goes out of the geometric envelope of the arch or the dome. For 1 < D d , the structure is theoretically safe, however, a factor of 3 is usually recommended as a safety factor. This way the thrust line lies within the middle third of the structure envelope [Block, 2009].

However if the structure is designed to be funicular under its self-weight, meaning that the thrust line or thrust network is one with the middle curve or surface of the structure, then the minimal thickness is 0 and the Geometric Safety Factor is infinite, whatever the actual thickness D. For masonry structures, a solution is to consider different load case combinations that generate different thrust lines, and take the envelope of those lines as the minimal thickness [Heyman, 1969], [O 'Dwyer, 1999].

The same principle can be applied in the case of additive manufacturing by considering that each step in the piecewise analysis is a different load case. That way the envelope of geometries determined in the previous section can be seen as the minimum thickness for the structure to be printed. The extended Geometric Safety Factor GSF * can therefore be defined as the maximum ratio between the envelope depth and the structure thickness calculated at each node of the model:

GSF * = max i D i d i (6.19)
With the determination of the Geometric Safety Factor, this accumulated strips analysis provides additional information this time on the thickness of the structure. Indeed in the first approach, the only influence of the thickness was on the age of the material, a higher thickness means more material to deposit, so more time in between strips for a constant flow rate.

To go further, this approach can be used to optimise the thickness of the structure by trying to impose a given Geometric Safety Factor on each node of the model. This way, the material quantity is fully optimised. To do so, an iterative process can be performed by updating the initial model thickness based on the result of the envelope geometry. However, doing so modifies the loads on the shell structure. The form-fitting routine has to be run again in order to ensure that the behaviour of the structure during the fabrication is not impacted. This process can thus be iterated until a satisfying convergence is reached.

Combining Both Approaches

While both approaches were first developed for different printing strategies in terms of material properties, they are not restrictive in any sense. The strip-by-strip analysis provides fast results on the potential geometry of the structure to be printed, while as it was seen in the previous section, the accumulated strip analysis can be used to validate and even optimise the thickness of the structure. Using both approaches consecutively, starting with the former can be an interesting strategy in the design of printable structures, resulting in the definition of an admissible geometry, its thickness and the optimal flow rate for each strip.

Summary and Discussions

In this chapter, the constrained form-fitting algorithm is applied to the sequential analysis of a structure being 3D Printed. At this stage of the analysis, the final layers are not defined. The discretisation of the model is based on strips that guide the construction sequence.

Two approaches of the algorithm are proposed:

• a first approach where the strips are activated one after the other and analysed individually. In this case, the previous strips are not taken into account. The material properties evolution is considered fast enough so that the previous strips would not reach the yield limit. As a result, a piecewise funicular geometry is provided with information on the optimal flow rate for each strip;

• a second approach where the entire structure at each step is analysed. This results in the generation of multiple geometries which can be used to define an envelope for the structure shell that ensures its stability during the entire fabrication process.

The advantage of this procedure is that it only uses global printing parameters (material properties and flow rate). The robot velocity and the geometry of the layers are not involved at this stage and will be set after the analyse at the local scale of the extrusion, detailed in the following part. This algorithm can help generate geometries specifically adapted to additive manufacturing when they present ambitious and challenging cantilevers. However, due to the calculation of the stresses that requires the projection of the model on the horizontal plane, it does not work for vertical extrusion only. In that case, a more classic Finite Elements Model can be used with the same discretisation in strips instead of layers.

Part IV

Local Characterisation of the Extrusion

Chapter 7

Printing Regimes

Local Stability at Extrusion Level

This chapter focuses on the characterisation of the behaviour of the material right after it exits the nozzle. Looking back at the workflow diagram (see Fig. 7.1), the present study concerns the generation of the layers. It comes after the piecewise structural analysis detailed in the previous part, and just before the generation of the toolpath that incorporates the robotic constraints. At this step, the object geometry and thickness are fixed and the macro-parameters have been set, namely, the material properties (initial yield stress τ 0 and structuration rate A thix ) and the material flow rate Q.

Before generating a complete toolpath for the robot, the height of the layers H and the printing velocity V r need to be set. These micro-parameters -as opposed to the macro-parameter Q -have no impact on the global structural stability of the printed structure. However, the following study shows how they can impact the geometry, quality and stability of the printed layers at the scale of the extrusion.

In this part, the focus is made at a much closer scale. The aim is to characterise the behaviour of the extruded material from the moment it exits the nozzle until a full layer is printed. The following study is focusing mainly on the Layer Pressing Strategy since it is the most suitable for the present scope of work. However, many concepts and relations presented in this part are also applicable to other printing strategies such as the Infinite Brick Extrusion, as well as other materials like clay.

Specifications of the Layer Pressing Strategy

Advantages

The Layer Pressing Strategy presents three main advantages that help define the design space of geometries that are printable with this process.

It provides perfect control of the position of the upper surface of the extruded layer. Indeed, when the material is extruded, it is squeezed between the nozzle and the previous layer. The upper surface of the layer is then, by fabrication, at the exact level of the nozzle. In other words, the distance between the nozzle and the substrate δ equals the layer height H, when the process is well controlled. That prevents any accumulated error on the vertical position of the printed structure that could come from deformation or creep of the sublayers during the fabrication Besides, pressing the layers provides freedom in the inclination of the extrusion head, and thus the orientation of the layers (see Fig. 7.3). Again, since the material is shaped at the extrusion thanks to its low initial yield stress, its cross-section is not restricted to a rectangle. The two planes defined by the nozzle aperture and the surface of the previous layer do not have to be parallel so the layer's cross-section can take a trapezoidal shape. Therefore, the printing head inclination can vary during the fabrication, creating room for a much larger design space [START_REF] Carneau | An exploration of 3d printing design space inspired by masonry[END_REF].

Finally, since the layer does not take the nozzle geometry but is shaped by the space between the nozzle and the previous layer, this strategy is best performed with a circular nozzle. This way a degree of freedom is added to the movement of the robot which does not have to keep the same alignment with the longitudinal direction of the layer. Issues like twisting of the layer [START_REF] Bos | Additive manufacturing of concrete in construction: potentials and challenges of 3D concrete printing[END_REF], are automatically avoided. Sharp angles can even be achieved and have been subject to multiple studies [START_REF] Comminal | Motion planning and numerical simulation of material deposition at corners in extrusion additive manufacturing[END_REF], [START_REF] Liu | Rotation nozzle and numerical simulation of mass distribution at corners in 3D cementitious material printing[END_REF].

On the Inter-Layer Bond

Pressing the extruded material on the substrate can also be beneficial for the cohesion between layers. the creation of cold joints or weak interface between subsequent layers is a risk of the additive manufacturing process. As explained in the introduction, many studies are conducted on the apparition of these weaknesses, trying to assess which parameters are involved (temperature, humidity, thixotropy, etc.). In any case, the inter-layer time is an important factor in the Figure 7.2 -3D Printing of a dome with a gradient of layer heights. The discontinuity in the middle layer is due to a local problem with the pumping equipment. [START_REF] Carneau | Additive manufacturing of cantilever -From masonry to concrete 3D printing[END_REF] apparition or not of the cold joints. In [START_REF] Roussel | Distinct-layer casting of SCC: The mechanical consequences of thixotropy[END_REF], ROUSSEL proposes an estimation of the upper bound for the inter-layer time dependant on the structuration rate of the material. The idea is that the cohesion between distinct layers is ensured if the newly deposited material can apply shear stress on the "old" material higher than its yield stress. The material of the previous layer is thus sheared which breaks the bonds between particles allowing to create new interaction with the particles of the newly deposited material.

When pressing layers, shear stress is applied to the surface of the previous material. The higher the pressing force is, the more sheared the material is, hence a better cohesion between the successive layers. This is still an assumption at this stage and is not in the scope of this dissertation. A specific experimental study should be conducted to validate this hypothesis.

Drawbacks

The Layer Pressing Strategy offers a wide design space but it comes with some drawbacks that are inherent to the shaping of the material. First, the choice of printing parameters is more difficult to make since it directly impacts the shape of the layer. Indeed in the Infinite Brick Extrusion, the layer's section is given by the nozzle geometry only and the robot velocity equals the velocity at which the material exits the nozzle.

In practice, the choice of nozzle diameter D or cross-section, layer height H and printing velocity V r is usually determined empirically and remains the same once a good combination of these parameters yields a correct layer quality and control.

Pressing the layers can also lead to new types of deformation and failure that do not occur with the other strategies. When the material is squeezed between the nozzle and the previous layer, additional stress is acting on the sub-layers right below the nozzle. This action is short in time but if the force gets too high, it can deform the sublayers, leading to discrepancies in the width of the structure and eventually to its collapse. Besides, the layer deformation induces a shearing of the material which due to thixotropy can slow down the structuration of the material and weaken the structure being printed.

Experimental Setup

To characterise the extrusion at the local scale with the Layer Pressing Strategy, different experiences are done. The first one described below aims at understanding the potential failures that can occur during a print, depending on the printing parameters values.

Printing Parameters

Table 7.1 recalls the different parameters involved in the 3D Printing process. A few remarks can be made in relation to the Layer Pressing Strategy:

• Since the extrusion nozzle is circular to retain an additional degree of freedom as described in the previous section, we introduce its diameter D as a parameter in the process, which is independent of the height of the layer H and its width B.

• The printing velocity V r , extrusion velocity V e and layer height H are directly imposed by the user as instructions to the robot or the extrusion pump, and they can vary from one layer to the other in the same print. However the layer width B is only a result of the previous parameters set by the user.

• The material properties are defined by the initial yield stress τ 0 and the structuration rate A thix and we assume as in the previous part that yield stress evolution follows the relation defined in [Roussel, 2006]:

τ c (t) = τ 0 + t × A thix
• The distance δ between the nozzle and the sublayer is equal to the layer height H. Therefore only parameter H is used in the following. For sake of simplification, the aforementioned parameters can be expressed in a dimensionless form. The new geometric and kinematic dimensionless parameters are marked by an asterisk ( * ):

Category

• H * = H D is the dimensionless height; • B * = B D is the dimensionless width; • V * = Vr
Ve is the velocity ratio. Other dimensionless parameters related to the stresses involved in the extrusion process are introduced later on.

Protocol Description

This experiment aims at identifying the printing parameters range ensuring good control of the layer's cross-section and local stability of the structure while using the Layer Pressing Strategy.

The tests consist of printing multiple series of square columns (see Fig. 7.4) made of 10 layers with a target width B of 20 mm equal to the nozzle diameter D. The material properties remain the same for all the prints. The parameters that vary from column to column are the printing velocity V r , the extrusion velocity V e and the layer height H following the values given in Table 7.2.

Since the subject of the study is the behaviour of the material at a local scale, right after it is extruded from the nozzle, the parameters are set so that the eventual failure can not come from a plastic collapse of the first layer. In practice, this is done by choosing the side length L of each column that keeps the loading rate of the first layer constant. In this sense, the loading rate is equivalent to the growth velocity defined by SUIKER [Suiker, 2018].

At the end of the printing, the width of each column is measured manually with a calliper to compare with the target value. And the different phenomena that occurred are identified and described in the following section. The following relation of mass conservation has already been introduced in section 4 (see (4.1)):

S V r = SV e (7.1)
S is the section of the nozzle. It can thus be expressed as S = πD 2 4 with D being the nozzle diameter. S is the cross-section of the layer. In the first part, the section of the layer was considered rectangular (S = BH). However, while pressing the material, it appears that the layer's cross-section takes more of an oval shape which can still be defined by its height H and width B. However, a corrective parameter β must be introduced to correctly express the cross-section area:

S = βHB (7.2)
The values and influence of β on the layer geometry is discussed in the next section.

From this (7.1) becomes:

βBHV r = πD 2 4 V e (7.3)
By using the dimensionless parameters, this relation can be written under the simpler form:

V * H * B * = π 4β (7.4)
It is clear from that equation, that the same layer width B can be achieved from different height and velocity parameters value. For example, if one decreases the dimensionless velocity ratio V * , then the dimensionless height H * has to be decreased to keep B * constant. However, not all choices of parameter values yield satisfactory results in terms of layer stability and geometry control. Indeed, the following section illustrates the different failure modes or printing regimes that can be observed. Later, the study continues with finding the bounds of each parameter admissible range.

Controlled Layer Pressing

The first regime corresponds to a layer pressing (Fig. 7.5) where the control over the geometry is perfect on both the width B and the height H. All layers have the same height and the extruded material does not present any defect. The aim is thus to find the range of parameters that restrains the printing process within this regime at all times. The pressing of the layer ensures good control over the layer cross-section geometry as well as the overall height of the structure being printed.

Layer Self-Collapse

This second regime illustrates a flow of the extruded material that cannot withstand its weight (see Fig. 7.6). This phenomenon called Flow-out by REITER in [START_REF] Reiter | The role of early age structural build-up in digital fabrication with concrete[END_REF], happens when the target height H of the layer is too high and the initial yield τ 0 stress is too low. The boundary for this regime can be easily expressed in function of these two parameters using Von Mises failure criterion as proposed by ROUSSEL [Roussel, 2018]:

τ 0 < ρgH √ 3 (7.5)
Where ρ is the volumic mass and g is the gravity constant. When this condition is not satisfied, the flow of the material creates a loss in height of the layer which leads to an accumulated error on the total height of the structure and eventually to its collapse. When the height δ between the nozzle and the sublayers reach a certain threshold, coiling patterns can appear as illustrated in Fig. 7.6.right. Interestingly the coiling patterns observed when extruded cement-based material are very similar to the one observed with other viscous fluid (see [START_REF] Brun | Liquid Ropes: A Geometrical Model for Thin Viscous Jet Instabilities[END_REF] and [START_REF] Yuk | A New 3D Printing Strategy by Harnessing Deformation, Instability, and Fracture of Viscoelastic Inks[END_REF]). [START_REF] Roussel | Numerical simulations of concrete processing: From standard formative casting to additive manufacturing[END_REF]] Fig. 7.8 illustrates the longitudinal tearing of the layers when the printing velocity gets too high over the extrusion velocity. The longitudinal true strain is equal to = ln Vr Ve = ln V * while the true strain rate is equal to = Vr H ln V * [START_REF] Yuk | A New 3D Printing Strategy by Harnessing Deformation, Instability, and Fracture of Viscoelastic Inks[END_REF]. Writing [START_REF] Brun | Liquid Ropes: A Geometrical Model for Thin Viscous Jet Instabilities[END_REF] f the strain at fracture, we can thus highlight a critical dimensionless velocity.

Layer Tearing

V * < V * L = f (7.7)
In the Layer Pressing Strategy, this regime is rarely achieved since the plastic deformation of the fresh material can be high before it tears apart. However, the effect of the longitudinal striction coming from a reduction of the section from nozzle to layer can be observed from the rough surface finish of the layer. For the Infinite Brick Extrusion, numeric simulation has been able to reproduce the tearing of a rectangular shape layer [START_REF] Roussel | Numerical simulations of concrete processing: From standard formative casting to additive manufacturing[END_REF].

Over-Pressing of the Layer

This final printing regime is specific to the Layer Pressing Strategy. Indeed, when the layer height H gets low, a deformation of the sub-layers is observed as the nozzle passes above them. It results in the printed column in a pyramidal section of the linear parts (see Fig. 7.9. The deformation is assumed to be planar so the overall section of each layer remains constant, while the width B increases and the height H decreases. The total height of the structure however is still controlled since the deformation stops when the nozzle has passed, the material is capable to withstand its weight for such small layer heights. The main issues with this phenomenon are the poor control over the final structure thickness (corresponding to the layer width) and the weakening of the material of the substrate each time it is deformed. Indeed, this deformation prevents the linear evolution of the material yield stress, thus threatening the overall structural stability. The measurement of the printing force can help understand better this phenomenon, and identify the principal parameters involved.

Figure 7.9 -Schematic view of the layer over-pressing leading to a pyramidal shape of the overall section (left) and the corresponding 3D Printed example. The area of the layers' cross-section remains constant despite the deformation. Fig. 7.10 illustrates the over-pressing phenomenon on a larger structure. Indeed layers constituting the tower present a periodic deformation that is clearly visible on the top left corner of the picture. These regular beads are not controlled and their effect is increased at the corners of the columns where the material deposition is irregular. In this example, the over-pressing did not lead to a collapse of the structure. One can imagine using this phenomenon for aesthetics purpose provided that it is understood and controlled.

Layer Pressing Characterisation

After identifying the different printing regimes that can be encountered with the Layer Pressing Strategy, the goal of this section is to find the range of parameters that ensures a consistently controlled pressing regime at all times. An experimental protocol is developed to measure the pressing force in action on the sublayers. A similar study with the identification and analysis of the pressing force has been conducted and presented in [START_REF] Reinold | Extrusion Process Simulation and Layer Shape Prediction during 3D-Concrete-Printing Using The Particle Finite Element Method[END_REF]. It proposes a numerical method using Finite Particle Analysis to predict both the layer cross-section geometry and the extrusion force on the sublayers. This experiment aims at identifying the extrusion force applied to the sub-layers in addition to the material weight as a function of the printing parameters and material properties. This test consists of printing layers over a balance to measure directly the vertical forces in action. The balance is directly connected to a computer that records the force applied while the robot is printing (see Fig. 7.11). The toolpath (Fig. 7.12) for the test is a series of straight lines passing over a scale, measuring a mass corresponding to the material weight and the local vertical force due to the extrusion. The toolpath is larger than the balance so the measurement is performed on straight lines only. A plate is installed all around the balance at the same level so the pressing never stops. Without this plate, the pressure in the nozzle would vary when exiting and entering the balance area, leading to a transition phase at the start of the new layer where the force measured is not constant. In the same series, all layers identical, the robot velocity V r , the flow rate V e and the layer height H are constant. This way, the error on the geometric measurement and the recorded force can be assessed. The recording of the balance is shown in Fig. 7.13 for a series of tests with constant material properties. The raw values are retrieved for a complete series of tests. The five layers can clearly be distinguished by the linear uprising parts of the curves. The slope of each part is directly related to the flow rate of the material, while the jumps at the start (upward) and end (downward) of each part show the additional force coming from the extrusion that is of interest here.

Each curve is treated separately to retrieve the force F corresponding to the pressing, from the weight of the material. The experiment is then repeated with different material properties (initial yield stress τ 0 ). Fig. 7.14 show the printing of layers of different heights H over the balance. The vertical force coming from the extrusion is directly extracted from this graph after subtraction of the weight of the material.

The initial yield stress τ 0 of the material is measured before each series using the slug test [Ducoulombier et al., 2020a]. This in-line measurement method is fully automated in our facility and simply requires the same balance used for the pressing test. The initial yield stress is modified by changing the accelerator flow rate in the extrusion head. The values measured for these tests and the corresponding error, defined with a 95% confidence interval on the mass of the slugs, are given in table 7.3. The error is less than 3% for all test materials, except for the material with an 810 Pa yield stress, which might indicate some heterogeneity at the material level. 2.2 2.3 9.9 2.5

Table 7.3 -Initial Yield Stress measurement and error for each series following the protocol from [Ducoulombier et al., 2020a] Figure 7.15 -Rectangle and elliptical section of an extruded layer of height H and width B of the printed layers on the balance, we can plot coefficient β as a function of the dimensionless height H * . Fig. 7.17 shows that β slowly decreases when the layer height increases. For small values of H * , the layer is strongly pressed and its shape is closed to a rectangle, hence β gets close to 1. For larger values of H * , the layer section takes the shape of an ellipse. Similar results from numerical simulations are presented in [START_REF] Comminal | Influence of Processing Parameters on the Layer Geometry in 3D Concrete Printing: Experiments and Modelling[END_REF] with the layer cross-section ranging from an almost rectangular shape to an elliptic one while still pressing the layer, and finally to a circle when the deposition height gets too high. It is noticeable that the aspect ratio is independent of the material properties (given here by the yield stress value).

In practice, a value of β = π 4 ( 0.8) corresponding to a layer with an elliptic section can thus be a satisfactory estimation of the aspect ratio of the lace. This value is represented by the dot line in Fig. 7.17. In that specific case, The relation of mass conservation in the dimensionless form simply becomes:

V * H * B * = 1 (7.9)

Figure 7.16 -Cross-section of a layer constituted of a rectangle and two half-disks.

Finally, an analytical expression of β can be obtained by considering that the layer crosssection takes an oblong shape like presented in Fig. 7.16. In this model, the shape of the cross-section is constituted of a rectangle of width B -H and two half-disks of diameter H. With this model, one can see that the cross-section tends towards a rectangular shape when H decreases, and to a circular one for large values of H. A similar approach was introduced by COMMINAL et al. in [START_REF] Comminal | Numerical modeling of the strand deposition flow in extrusion-based additive manufacturing[END_REF] but with three different analytical relations for an elliptic, an oblong and a rectangular cross-section.

The cross-section area of the oblong layer is:

S = (B -H)H + πH 2 4 (7.10) S = BH + ( π 4 -1)H 2 (7.11)
Introducing the previous expression in the relation of mass conservation S V r = SV e , B can be expressed in function of H:

V * (BH + ( π 4 -1)H 2 ) = πD 2 4 (7.12) B = πD 2 4V * H + (1 - π 4 )H (7.13)
Finally the analytical expression of β defined as β = S BH becomes:

β = 1 + ( π 4 -1)H πD 2 4V * H + (1 -π 4 )H (7.14)
which after simplification yields: This expression of β is plotted on the graph of Fig. 7.17 for different values of the velocity ratio V * . The comparison between the experimental and analytical values of β shows the same tendency to decrease with H * . The influence of the velocity ratio V * in the analytical expression does not perfectly reflect the experimental results with an error up to 20% for the higher value of V * . It gives however a better estimate of the actual layer cross-section than when considering a rectangular shape.

β = 1 1 + ( 4 π -1)V * H * 2

Upper Bound Condition for Pressing Layers

In this part, an upper bound condition for pressing layers is proposed based on a kinematic approach of the extrusion process (see Fig. 7.18). This upper bound is the limit above which no pressing occurs, preventing a good control of the layer geometry. • t c = D/V r is the pressing duration. For a given section of a layer, t c is the total time spent by the nozzle above this section.

• t d = H/V e is the descent time. If the nozzle is immobile and at a distance H from the substrate, t d is the time it takes for the material to go from the nozzle to the substrate.

The material is pressed if it reaches the substrate before the nozzle has completely passed over it. The condition reads: t d < t p ⇔ HV e < DV r (7.16)

The same condition has been found in [START_REF] Yuk | A New 3D Printing Strategy by Harnessing Deformation, Instability, and Fracture of Viscoelastic Inks[END_REF] for additive manufacturing of plastic fluids.

Using the dimensionless parameters, this upper bound condition yields:

H * V * < 1 (7.17)

Self-Weight vs. Pressing Force

In this section, the results of the pressing force are presented. Similarly to the dimensionless material weight parameter ω * introduced in the previous chapter, a dimensionless pressing stress τ * can be created from the printing parameters:

τ * = 4F πD 2 τ 0 √ 3 (7.18)
Similarly to the dimensionless weight ω * , the Von Mises yield criterion is used here, hence the apparition of the term in √ 3.

10 -1 For small layer heights H, the pressing force can get up to ten times higher than the equivalent weight which means that by the time one layer is printed, the sublayer has to withstand the equivalent of ten layers on top of it. This phenomenon is very local in both time and space but is enough to create large deformation that can be an issue for the structural integrity of the printed object. Those results confirm that with the Layer Pressing Strategy, a fine setting of the printing parameters must be performed to avoid deformation of the lower layers.

10 0 10 -2 10 -1 10 0 10 1 H * τ 0 = 490P a ; V * = 2.2 τ * ω * 10 -1 10 0 10 -2 10 -1 10 0 10 1 H * τ 0 = 500P a ; V * = 1.8 τ * ω * 10 -1 10 0 10 -2 10 -1 10 0 10 1 H * τ 0 = 810P a ; V * = 2.2 τ * ω * 10 -1 10 0 10 -2 10 -1 10 0 10 1 H * τ 0 = 1200P a ; V * = 2.38 τ * ω *
The plotted values in the log scale for τ * seem to be aligned in Fig. 7.19. This seems to illustrate a polynomial relation between τ * and H * . Through linear regression, a coefficient of 3 is found which indicates a cubic dependency of the pressing force in 1 H (see Fig. 7.20). the analytical approach in the following section will confirm this tendency.

In Fig. 7.21, the pressing force F for each experiment is directly plotted in function of H * V * . The force drops systematically after the value corresponding to H * V * = 1. This tends to valid the upper bound condition (7.17) for pressing layers.

Analytical Approach

Measuring the pressing force experimentally before each print can be time-consuming, especially without the right equipment. Besides, for structures where the layer geometry varies during the fabrication, estimating the pressing force experimentally becomes impossible. A numerical solution as the one proposed by REINOLD [START_REF] Reinold | Extrusion Process Simulation and Layer Shape Prediction during 3D-Concrete-Printing Using The Particle Finite Element Method[END_REF] can be a good option, but it also requires a model of the entire structure with all the layers. This can lead to heavy computation time.

Finding an analytical solution for the pressing force can provide a simple and efficient way to ensure the printing parameters are correctly set. The analytical formula can thus be integrated Such a relation already exists in the litterature for a layer of material of length L, pressed uniformly between two plates at a speed V low enough so that the viscous effects can be neglected (see the illustration on Fig. 7.22). This relation given in [Coussot, 2005] states:

F = τ 0 LB 2 2H (7.19)
To adapt it to a 3D Printing process, the length L of the plate is replaced by the characteristic length of the problem, namely the nozzle diameter D. The relation becomes:

F = τ 0 DB 2 2H (7.20)
Using the relation of mass conservation (7.4.1), this equation becomes:

F = τ 0 π 2 D 2 32β 2 V * 2 H * 3 (7.21)
The dimensionless pressing stress (7.18) thus becomes in its analytical form:

τ * = π 8βV * 2 H * 3 √ 3 (7.22)
A first result given directly by this relation is the same cubic dependency in 1 H that was found with the experimental measurements.

Figure 7.22 -Uniform pressing of a longitudinal band of material between two plates from [Coussot, 2005]

Fitting Experimental and Analytical Values

The comparison between the experimental data and the analytical formula for the pressing force is illustrated in Fig. 7.23. The graphs show all the forces measured experimentally (circle marks) and the analytical results given by (7.22) as a dashed line. The fitting is satisfactory as long as the layer height H is not too low. When H decreases, the discrepancy between the model and the experimental values may come from a granular effect. The size of the larger particles in the material (≈ 1 mm) is not negligible in regards to the layer height (4 mm).

For common values of the layer height, usually 6mm < H < 15mm, the analytical formula can predict the pressing force applied on the sublayers. The next chapter presents a methodology using this relation to set the printing parameters. The following section introduces a novel method to measure the material initial yield stress τ 0 based on the pressing force measurement experiment.

On the Measurement of the Initial Yield Stress

For material with a high structuration rate (A thix ≥ 10), the experimental measurement of the initial yield stress of the material can be challenging. Indeed, the measurement must instantly provide the required result, otherwise, the material properties evolve too much during the procedure. The use of a rheometer or a slump test is vain since the preparation time of the sample or the filling of the slump cone gives too much time to the material to change. In the best case, the resulting value will be far above the real one and in the worst case, the measuring equipment can be damaged.

The question of finding new rheological tests specific to Concrete 3D Printing and materials with high structuration rates has arisen in the additive manufacturing community. The slug test developed by DUCOULOMBIER [Ducoulombier et al., 2020a] is an inline process that has the specificity to provide yield stress information at the exact moment the material exits the extrusion nozzle. It is used in this dissertation to measure the initial yield stress of the material before each experiment and provide accurate results at a very low cost. Another inline process is proposed in [START_REF] Jacquet | Gravity Driven Tests to Assess Mechanical Properties of Printable Cement-Based Materials at Fresh State[END_REF] for the measurement of the tensile stress. However, this procedure is more adapted to material with high initial yield stress (τ 0 ≥ 1kP a) and a low structuration rate.

The present study of the pressing force during the extrusion can be used to simply provide an estimation of the initial yield stress. With the validation of the analytical formula for the pressing force (7.22), a simple procedure of extruding a layer over a balance with a given height H and width B, and after measuring the pressing force can immediately give back the initial yield stress. This is the reverse problem of what has been studied before. The initial yield stress can be directly given by the analytical formula of the pressing force (7.21):

τ 0 = 32β 2 F V * 2 H * 3 π 2 D 2 (7.23)
Using the analytical expression of β (7.15) in the previous relation yields:

τ 0 = 32F V * 2 H * 3 π 2 D 2 1 + ( 4 π -1)V * H * 2 2 (7.24)
The advantage of this method for yield stress measurement lies in the fact that the process is the same as actually printing an object. There is thus no potential perturbation coming from the test protocol.

Summary and Contributions

This chapter has introduced the different local failure modes that can occur with the Layer Pressing Strategy. A focus has been made on the over-pressing regime that is specific to this printing strategy.

An experimental setup was described leading to key results:

• the shape of the layer's cross-section depends on its height and an analytical formula for its area has been proposed;

• the force applied to the sub-layers during the extrusion can be significantly higher than the weight of a single layer. Again, an analytical formula of this additional force has been proposed and validated by experimental results;

• a new method for the measurement of the initial yield stress has been deduced from the measurement of the extrusion force.

The following chapter will discuss the influence of each parameter on the printing regimes that have been introduced and propose a methodology for the setting of these parameters.

τ t > 2 ρgH √ 3 + 4F πD 2 √ 3 (8.1)
The material structuration is still considered to be linear so τ t = τ 0 + ∆ t × A thix according to [Roussel, 2006]. And the time ∆ t in this expression corresponds to the inter-layer time, or the time it takes to print one layer. It can thus be expressed in function of the layer length L and the printing velocity V r : ∆ t = L Vr . The previous relation becomes:

τ 0 + LA thix V r > 2 ρgH √ 3 + 4F πD 2 √ 3 (8.2)
A new dimensionless parameter is introduced here as a growth velocity ratio:

A * = A thix ∆t τ 0 (8.3)
A * reflects the growth velocity of the structure with regards to the material properties evolution. For instance a value of A * = 0 means that the material does not evolve through time. On the other hand, high values of A * means that the material structuration is high compared to the printing speed, so the structure will be more likely to stand still (at least at the layer scale). If the printing velocity V r is constant, it can be related to the inter-layer time ∆t and the layer length L: ∆t = L Vr . In this case, the structuration parameter becomes:

A * = LA thix τ 0 V r (8.4)
Using the dimensionless weight and pressing stress, relation (8.2) yields:

2ω * + τ * -A * < 1 (8.5)

Printing Parameters Map

The last relation (8.5) can be seen as a lower bound for pressing layer. This finally gives four conditions that are bounding the Controlled Pressing Regime. With the dimensionless form, those four conditions can be expressed in function of the velocity ratio V * and the height ratio H * only:

• Self-supporting layer (ω * < 1):

H * < τ0 √ 3 ρgD • Upper bound condition (H * V * < 1); • Lower bound condition (2ω * + τ * -A * < 1): V * > τ0 2H * 3 (A thix ∆t √ 3-2ρgDH * ) • Tearing boundary (V * < V f );
They can thus be gathered under a single diagram of printing regimes (see Fig. 8.2). The diagram is created for a given nozzle diameter D and material properties τ 0 and A thix . These parameters are usually set first depending on the object geometry, the printing equipment available and the material used.

Each region in the diagram corresponds to a different printing regime whose boundaries are given by the previous conditions: • The target region for the Layer Pressing Strategy is in blue. In this Controlled Pressing region, both the height and the width of the layers are controlled and the extrusion itself does not impact the sub-layers.

• In the purple region of Over Pressing, the extrusion force F creates a deformation of the lower layers leading to poor control of the width B.

• On the right of the figure, the green region corresponds to the case where the material can not withstand its weight. The layer flows on itself thus losing its target height and creating an accumulated vertical error that leads almost certainly to a global failure of the structure.

• On the top, the Tearing region is reached when V * > V f , where V f is a limit velocity above which the layer starts to tear regularly. The parameter V f which has not been studied in this dissertation depends on the material properties.

• Finally, the central region in yellow is where the Infinite Brick Extrusion happens. In this region, the control of both height and width of the layer is correct but due to the absence of pressing of the layers, any small error in the layer height can lead to a vertical drift, causing the same results as the free flow deposition regime. Note that in this yellow region, the equality between the layer height H and the distance between the nozzle and the substrate δ is not valid anymore.

dashed) and the horizontal line defined by V * = 1.

Influence of the Printing Parameters

As stated before, the Printing Regime Diagram is drawn for a given nozzle diameter D and material properties τ 0 and A thix . And the mass conservation curve depends on the layer width B. In this section, the influence of those parameters on the diagram is looked at to understand the leverage a user has when preparing a print. By default and if not stated otherwise, the printing parameters used in all the presented diagrams take the following values:

• Initial yield stress τ 0 = 300P a;

• Structuration rate A thix = 30P a/s;

• Nozzle diameter D = 20mm;

• Layer width B = 25mm.

8.2.1 On the Self-Supporting Condition (ω * < 1)

The vertical line (green) delimiting the boundary of the Free Flow Deposition regime, characterised by the incapacity of the material to withstand its own weight, is entirely controlled by material parameters. Indeed, the condition ω * < 1 depends only on the initial yield stress τ 0 -it is assumed that the density ρ of the material cannot be modified and therefore, is not a parameter in the problem. The last boundary condition of the Controlled Pressing regime is the limit with the Over-Pressing domain. This condition depends on both kinematic and material parameters. Both the weight of the material (ω * ) and the pressing force (τ * ) compete with the structuration parameter A * . It is interesting to look at the influence of A * on the boundary curve. Since the structuration rate A thix appears only in the expression of A * , the displacement of the curve is looked at with regards to the variation of A thix (see Fig. 8.6).

By increasing the structuration rate, the material of the sub-layer hardens faster and is thus more unlikely to deform when extruding the new layer on top of it. This is illustrated on the right diagram of The values taken as an example for A thix in Fig. 8.6 show that the Controlled Pressing regime area can be very sensitive to the structuration rate. It shows indeed a small displacement of the curve when A thix is increased by 50P a, but a major difference when it is decreased by 5P a and then 10P a.

Application to the Slicing Operation

In this section, the use of the Printing Regimes Diagram is illustrated in a case study. The diagram can be used to find the optimum printing parameters that ensure a correct print at a local scale. The different boundary conditions can be integrated within a parametric software such as Grasshopper to explore the influence of each parameter and find the combination that leads to a successful print.

The object for the study is a vertical wall with a trapezoidal shape. Its dimensions are reported in Fig. 8.7. This geometry is purposely chosen to create variations in the layers' length L and height H. The slicing direction is not horizontal but is parallel to the bottom edge of the wall in the lower part and parallel to the top inclined edge in the upper part. In Fig. 8.7, the slicing direction is given by the arrows which are perpendicular to the final layers. That way the layer's length L is always increasing. The actual layers are drawn in black in all the following figures.

The process starts by setting the parameters that are constant during the fabrication process, namely the material properties, the wall thickness and the printing specifications:

• The material density ρ = 2350kg/m3;

• The initial yield stress τ 0 = 160P a;

• The structuration rate Athix = 30P a/s;

• The wall thickness which corresponds to the final layers width B = 30mm;

• The extrusion nozzle diameter D = 20mm;

• The material flow rate Q = 28g/s. The problem consists of finding the parameters H and V r for the layer height and printing velocity. As seen in the previous section, those two parameters are interdependent. Therefore the problem comes down to choosing the total number of layers N of the structure. This way, all layers have the same minimal and maximal height H on the edge of the structure. Choosing N automatically yields the layer geometry which is then used to determine the printing velocity V r at each point of the structure using the relation of mass conservation (7.4.1).

The entire process is done in the parametric environment of Grasshopper. For a given number of layers N , the layer curves are generated. Each curve is then discretised in a high number of points and each point is analysed to find what is its corresponding printing regime. The process is as follows:

1. Retrieve the layer height H at each point.

2. Calculate the printing velocity V r from the relation of mass conservation (7.4.1).

3. Find the printing regime corresponding to each point from the boundary condition of the printing diagram.

In step 2, if the analytical formula for β is used, the expression of V * becomes:

V * H * B * = π 4 1 + 4 π -1 V * H * 2 (8.6) V * = 1 π 4 H * B * -4 π -1 H * 2 (8.7)
To verify the Over Pressing condition (2ω * + τ * -A * = 1) the dimensionless parameter A * must be calculated. In this case, the printing velocity is not constant. Therefore, the inter-layer And it yields:

A * = LA thix mτ 0 m i 1 V i r (8.9)
The structuration parameter A * is thus calculated for every single layer.

The results can be visualised directly on the digital model by using the same colour code as in the printing regime diagram in Fig. 8.2. Fig. 8.10 illustrates the layer pressing analysis of the wall for a number of layers N = 50. It shows an overflow of the layers on the right of the wall, where the layers are thicker. Besides, the bottom part is subject to an over pressing which tends to disappear when the layer length increases. For the four layers marked with black lines on the wall structure, the corresponding diagrams are drawn in Fig. 8.9. Since the layer width B is the same for the entire structure, the red curves representing the relation of mass conservation are identical on the four diagrams. Only the blue curve of the pressing lower bound (the boundary between Controlled Pressing and Over Pressing regimes) changes following the variation of the structuration parameter A * from a layer to another. Finally, the minimal and maximal heights of the layers are drawn by the vertical grey dashed curves on the four diagrams. The analysis of the four diagrams of Fig. 8.9 is as follows: 55. The parameters range for the height of the layers H and the printing velocity V r are given for information in each case.

• For N = 45 (Fig. 8.10-left) the layer height on the right half part of the wall is too important, causing incapacity for the material to withstand its weight (green region). The value of N is low enough so that the average layer height is too high to create an over pressing of the sub-layer.

• For N = 50 (Fig. 8.10-middle) the layer height is lower in average leading to an overpressing regime in the lower part of the wall. The green region has decreased compared to the previous case.

• For N = 55 (Fig. 8.10-right) the green area on the left is now small but almost all the rest of the wall is subjected to an over-pressing due to the low height of the layers.

This example shows that in this case, varying only the number of layers is insufficient to ensure a good printing process on the entire structure. The variation in height H of the layers is too high to remain in the Controlled Pressing regime.

At this point, a few solutions exist to ensure the print to be successful, meaning that the entire structure lies within the Controlled Pressing regime. One strategy is to play on the global printing parameters such as the material properties, or the flow rate. Fig. 8.11 shows a combination of parameters that allows keeping all layers in the Controlled Pressing regime at all times. The initial yield stress τ 0 has been increased to 200P a so layers can withstand their own weight, even for the thicker ones. The structuration rate A thix has also been increased, which leads to a higher value of the dimensionless structuration parameter A * mitigating the chances of over-pressing the sub-layers.

Another solution to print the structure by avoiding local failure can be to modify the slicing strategy. In this example, for instance, a gradient of thickness in the layer can be created, with thicker layers at the bottom by keeping the same slicing direction. This way, the over pressing of the lower layer can be reduced without changing any printing parameter. However, this would Figure 8.11 -Satisfying solution for the slicing where all parameters fits within their admissible range, and the layer pressing is controlled at all times. increase the layer height H, thus putting more layers in the Over Flow region of the diagram (green area). Then all it needs is to increase the initial yield stress τ 0 sufficiently.

It must be noted that with this example, using the classic slicing strategy of creating all layers horizontal is highly inefficient compared to the solution presented before. Fig. 8.12 shows the analysis of the wall structure under horizontal slicing. Even after increasing the structuration rate A thix (from 40P a in the previous working example) to 100P a, the structure is not locally stable. This is due to the very low value of the structuration parameter A * in the upper part of the wall where the length of the layers reduces.

Figure 8.12 -Printing regime analysis on the studied wall after a horizontal slicing. This example helps to understand the benefits of implementing the Printing Regimes Diagram within a parametric environment. Since all boundary conditions are expressed as analytical formulas, the computation time to retrieve the local stability information at each point in the structure is very low, even for a high number of layers. This leaves room for real-time parametric optimisation which is difficult to achieve with numeric simulation using Computational Fluid Dynamics for instance.

Summary and Contributions

In this chapter, the conditions of local stability of the extruded material are transcribed in a printing regime diagram. This diagram offers a clear understanding of the potential failures occurring during the extrusion and what is the influence of each printing parameter.

A case study showed how the information provided by the printing diagram can be implemented into a parametric environment like Grasshopper. This way, a user can directly see the influence of his choice of parameters on the local behaviour of the material.

However, this methodology still has some limitations which can be addressed in future works:

• develop new measurement methods of the structuration rate A thix , especially for highly thixotropic materials;

• evaluate the strain at fracture of the material to give an accurate value of the limit velocity V f ;

• the influence of the inter-layer time on the cohesion between layers is not taken into account in the diagram. Incorporating it would require a preliminary study on the effect of the extrusion force F on the cohesion.

Part V

Conclusion 167

Chapter 9

Conclusion and Perspectives

Summary of results

The objective of this study was first to provide a better understanding of the behaviour, at both global and local scales, of the 3D printed material during the fabrication. It also provides tools and methods to explore the design space of printable geometries, especially those that present ambitious cantilevers that are, by nature, difficult to realise with additive manufacturing. The overall goal is to allow actors from the academy and the industry to have better control of their process. Consequently, new shapes and design can be generated, less failure will occur which means a better productivity with less waste of material and a faster process.

Global Workflow for Concrete 3D Printing

This dissertation started with a comparison between two additive manufacturing process: Masonry and Concrete 3D Printing. The analysis of masonry structures built without temporary supports provided insights on the types of geometries that can potentially be achieved. By looking at the sequential construction of vaults and domes, it appears that the stability during the fabrication is ensured by the formation of arches or compression rings. The other result of this analysis is that the structure being built can be looked at at the scale of the component (brick or layer) and the scale of the structure. By separating the study of the additive manufacturing process according to those two scales, a new workflow for the generation of toolpath for additive manufacturing can be developed. This workflow takes into account the influence of each printing parameter in the process and can, therefore, limit the number of iterations usually necessary with classic workflows. Its main characteristics are:

• the material properties are fully considered at each step of the design process. The slicing operation for instance is informed by the initial yield stress of the material.

• the workflow provides a separate analysis of the global structural behaviour and the local material behaviour. Each step involves specific printing parameters which allows setting them linearly.

• the material flow rate is the only global printing parameter. It provides the velocity of the printing process and guides the structural behaviour (along with the material properties evolution). For large-scale structures, the layer height and robot velocity have no impact on the global structural behaviour.

It must be noted that this workflow is not specific to any printing strategy and can be used for almost any application of additive manufacturing by extrusion of cement-based materials (and even clay). The rest of the study presented in this dissertation followed the two main steps of the workflow: the global structural analysis for the fabrication of vaults and domes and the local characterisation of the extrusion in the case of the Layer Printing Strategy.

Form-Finding of Printable Structures

In the second part of this dissertation, a form-finding algorithm was proposed to generate printable structures with cantilevers. The algorithm relies on the Force Density Method and a yield analysis. The main motivation was to provide in a single process the form-finding algorithm and the stresses analysis when a classic Finite Element model only provides the latter. The approach is similar to the ones used for the design of compression-only masonry structures and specifically HEYMAN's Safe Theorem but it is extended to take into consideration the material properties and its evolution. Indeed, unlike bricks, the printed mortar has very poor strength both in compression and tension in its fresh state. This specification of cement-based additive manufacturing must be considered to explore the design space of such structures.

The form-finding process presented in this dissertation analyses the geometry of the structure and its behaviour following a construction sequence that is guided by the slicing direction. Its main result is the generation of printable geometry. It ensures that it will remain stable at every step of the fabrication and that all internal forces are compressive. Besides, it can evaluate the optimal thickness of the shell structure to ensure that the thrust network remains within the structure envelope at all times. Finally, it sets the minimal material flow rate and consequently the global printing time and this without introducing the robot velocity. The algorithm developed is an alternative to the use of Finite Elements Model that lacks the form-finding aspect and thus requires iteration steps between the geometric model and the structural model.

Characterisation of the Extrusion

In the last part of this dissertation, the analysis of the extrusion process has led to a better understanding of the relation between the printing parameters and the resulting layers. The analysis focuses on the Layer Pressing Strategy, where the extruded material is pressed between the nozzle and the sublayers. This strategy allows printing with some inclination and is thus suited for the generation of cantilevers, just like the inclination of brick beds in masonry vaults and domes follow the surface curvature.

Once again the main objective is to provide users with a way to set the printing parameters so that the control of the layers geometries is ensured and no local failure will occur, and this without going through trial and error process before each new print.

The different local failure modes that can occur have been identified with special attention to the over-pressing case. The pressing of the extruded material leads to a force applied on the sublayers that can deform them. The study of this force has provided interesting insights into the interaction between the material properties, the kinematic parameters and the layer crosssection geometry. This analysis resulted in several analytical formulas that give boundaries to the domain of the Controlled Pressing regime. Therefore, they can easily be implemented in a parametric software and provide immediate feedback to the user regarding the chosen slicing strategy.

Finally, the main lesson in the proposed workflow is that each step of the design process, from the slicing to the structural analysis, of printable structures, must be informed by the material properties. This way, each printing parameter can be set in a logical manner and not experimentally, accelerating then the entire design process of Concrete 3D Printing.

Potential Future Works

The workflow and algorithms presented in this dissertation have been implemented in a computational framework developed by the Build'in Lab of École des Ponts ParisTech. It was coded entirely in C# and Grasshopper components were generated from the core code to provide an interactive environment. From this, the first main work is to develop a fully valid and implemented design tool that can be distributed to the community.

The separation between the global and local analysis of the printing process makes it possible to have two independent tools. This can be interesting if one wants to perform a structural analysis using a Finite Elements Model and still look at the local behaviour of the material to set the layers geometry and robot velocity.

With a tool like that available, future works can then take different directions:

• optimise the form-finding algorithm with more robust computational methods in the solver of the constrained form-fitting problem. One objective would be to fully implement the optimisation guided directly by the membrane stresses, without having to manipulate the force densities explicitly;

• develop new yield criterion specific to fresh mortars used in Concrete 3D Printing. The addition during the process of longitudinal reinforcement, for instance, can lead to the generation of a whole new range of geometries with a significant presence of tensile stresses;

• look at a topology optimisation problem when choosing the slicing strategy. In chapter 4, the possibility to build a dome by creating squinches, that are self-standing structural features, have been mentioned. In the case of large-scale domes and vaults, it can be interesting to look at can be the optimum slicing strategy and the potential division of the geometry in different patches printed separately.

Finally, since the entire workflow and all algorithms presented in this dissertation rely on the material properties and its evolution, new methods to accurately determine those properties must be developed. It is especially important when the structuration of the material is very high (A thix > 10P a/s). The linear model proposed by ROUSSEL [Roussel, 2006] for the evolution of the material yield stress has been used throughout this dissertation but it has not yet been validated for accelerated mortars used in 3D Printing. The ideal method would be an in-line measurement process such as the one proposed by DUCOULOMBIER for the measurement of the initial yield stress [Ducoulombier et al., 2020a], that can correlate the quantity of accelerant added to the mix and the evolution of the structuration.

Conclusion

Concrete 3D Printing is still a rather young technology but the promising results it has shown has already attracted the interest of many universities and industrials. Its development has increased exponentially in the last five years and while the research focused mainly on the formulation of printable materials in the early days, more and more diverse subjects are being treated today, as the addition of reinforcement, the possibility for standardisation or the generation of toolpaths. the evolution of this process is part of the increasing development of automation in the construction sector. New technologies are being developed both for the design of new types of structure and their fabrication that cannot always be performed with traditional construction methods. However, it is interesting to see that even for a "futuristic" technology like 3D Printing, inspiration can be found in ancient building techniques like masonry. The knowledge of construction techniques is critical today when the need for new buildings and infrastructures faces major challenges like climate change and the depletion of natural resources, for instance. To build smarter, more efficiently and with less material, designers, architects and engineers alike, have to comprehend all aspects of the design of structures illustrated by the structural morphology diagram of Fig. 1.11. With the emergence of new technologies like 3D Printing, the interactions between Form, Force, Technology and Material can become less evident. It is then the role of research, like the one presented in this dissertation, to analyse and identify those interactions in order to develop new design tools that can be used to explore geometries and structural systems while perfectly controlling the fabrication process.

List of Figures

1.1 URSCHEL's Machine For Building Walls, patented in 1944, from [Urschel, 1944].

Left: the mechanical rotative arm, right: the extrusion head or end-effector with the cable feeder for longitudinal reinforcement. . . . . . . . . . . . . . . . . . . . 1.2 Functional Diagram of layered construction process, reproduced from [Pegna, 1997] 1.3 Contour Crafting process from [Khoshnevis, 2004] 2.15 Left: In the Infinite Brick Extrusion, the deposited layer geometry is given by the shape of the nozzle, right: in the Layer Pressing Strategy, the nozzle is usually circular and the extruded material is pressed onto the substrate to shape the layer. 2.16 Limitations of the Infinite Brick Extrusion from [START_REF] Bos | Additive manufacturing of concrete in construction: potentials and challenges of 3D concrete printing[END_REF]. Left: local deformation due to cantilevering. Right: twisting of the layer. . . . . . . . . . . . 2.17 Material properties evolution for the two asymptotic cases of extrusion strategies

Figure 1 . 5 -

 15 Figure 1.5 -Examples of geometric freedom in 3D Printing objects (credits: Léo Demont).

Figure 1

 1 Figure 1.8 -Graphic representation of 3D Printing Process as a multi-objectives optimisation problem

Figure 1 .

 1 Figure 1.11 -Concrete 3D Printing through the spectrum of structural morphology. The diagram of interaction is in the centre while specific themes related to additive manufacturing surround it.

2. 1

 1 Form Finding and Structural Design -Between Form and Force This State of the art is conducted by circulating the diagram of structural morphology of Fig. 1.11. It starts with the interaction between the Form and Force aspects with regards to the fabrication of overhangs and continues with the Structure, Material and Technology aspects at the core of the Concrete 3D Printing process.

Figure 2 . 2 -

 22 Figure 2.2 -Clay printing experiment from IAAC. Left: supportless printing of overhang, right: printing of barrel vaults on supporting frame (photo by Peter Magnus).

Figure 2 . 3 -

 23 Figure 2.3 -Left: example of borie from South of France, right: Treasury of Atreus, Mycenes, Greece. Source: Wilhem Lübke, Max Semrau; Plan of Art History. Paul Neff Verlag, Esslingen, 14th edition 1908.

Figure 2 . 5 -

 25 Figure 2.5 -Construction scheme of a Nubian vault

Figure 2 . 6 -

 26 Figure 2.6 -Masonry dome on inclined layers

Figure 2 .

 2 Figure 2.8 -Spiral staircase built by GUASTAVINO with the Catalan vault technique (credits: Avery Collins Archive, Columbia University).

Figure 2 .

 2 Figure 2.10 -The stone-weighted rope device for erecting Gothic vaults (John Fitchen, The Construction of Gothic Cathedrals, 1961, fig. 69, p.182 )

Figure 2 .

 2 Figure 2.13 -Phase change from solid to fluid due to a collision between the 3D Printed structure and the nozzle after completion of the last layer.

Figure 2 .

 2 Figure 2.16 -Limitations of the Infinite Brick Extrusion from [Bos et al., 2016]. Left: local deformation due to cantilevering. Right: twisting of the layer.

Figure 3 . 2 -

 32 Figure 3.2left: masonry curved wall, right: 3d printed curved wall with trapezoidal section for the layers

Figure 3 . 3 -

 33 Figure 3.3 -Effect of the slicing on the geometry of the temporary structure for a spherical half-dome. top: horizontal slicing, bottom: vertical slicing.

Fig. 3 . 5 ,

 35 Figure 3.4 -Illustrations of the slicing of a half-dome by CHOISY in [Choisy, 1883].Left: radial slicing, right: horizontal slicing in the lower part combined with a radial slicing in the upper part.

Figure 3 . 5 -

 35 Figure 3.5 -Left: 3-axis gantry at TU Eindhoven Concrete Printing facility. Right: 9-axis robot at Twente Additive Manufacturing facility.

Fig. 3 .

 3 Fig.3.7 illustrates criteria f for different layer configurations. Vertical structures fall automatically under the f 0 criteria, but without presenting any cantilever.

Figure 3 .

 3 Figure 3.7 -a) vertical wall, b) f 0 shear stress coming for the layer inclination of angle α, c)f 1 combination of bending and shear in the material, d)f 0 bending moment coming from the local cantilever δ

Figure 3 . 8 -

 38 Figure 3.8 -Local cantilever in the case of 3D Printing with corbels. The offset part can be assimilated as vertical layer of height δ to calculate its stability.

Figure 3 .

 3 Figure 3.9 -Comparison between f 0 and f 2 critical angle

Figure 3 .

 3 Figure 3.10 -From left to right: Printing of a cylinder on a vertical surface. A local failure of the bottom part of the cylinder leads to complete tearing of the lower part by pure tensile stress. The failure of the upper part of the cylinder occurs later at the anchoring of the first layer to the plate and is not due to the material itself.

Figure 3 .

 3 Figure 3.11 -Masonry vaults examples. Left: final geometry of a dome guided by a local inclination criterion α, by CHOISY. Right: Persian vault built with squinches, creating piecewise discontinuous layers.

Figure 3 .

 3 Figure 3.12 -Illustration of the layers continuity by comparison of masonry brickworks and equivalent 3d printing path with left: continuous brickwork, middle: patched masonry and right: discontinuous brickwork. The dots mark the discontinuities and the stopping and restarting of the system.

Figure 3 .

 3 Figure 3.13 -Illustration of the different types of supports. The red elements (point, line, surface) defining the positions of the supports.

Figure 3 .

 3 Figure 3.14 -Nubian vault construction by CHOISY from[Choisy, 1883]

Fig. 3 .

 3 Fig. 3.15 categorises different typologies of structures aforementioned and examples of 3D Printed objects based on their behaviour at different scales: the local scale of the extrusion

Figure 3 .

 3 Figure 3.15 -Examples of structure typologies based on their sub-structural system (in place during the fabrication) and the local configuration of the layers.

Figure 3 .

 3 Figure 3.16 -Printing path and local orientation of the extrusion nozzle

Figure 3 .

 3 Figure 3.17 -Decomposition of the fabrication process of a 3D Printed barrel vault with the separation between the the transition to permanent regimes.

Figure 3 .

 3 Figure 3.18 -Relation between masonry parameters s, f and c with regards to the 3D Printing set-up

Figure 3 .

 3 Figure 3.19 -3D Printed arch opening in a vertical wall.

  2.10 from the State of the Art). The tensile stresses are directed towards the walls while the arch edges remain in compression at all times.

Figure 3 .

 3 Figure 3.20 -Toolpath of the 2D crossing. a.3D view, b. front view with distinction between upper and lower parts, c. top view

  Fig. 4.2 represents for several classic arch geometries how they can be entirely divided in arcs of circles of similar or different radii. The red lines show the final geometry of the arch while the dotted lines describe the corresponding circles. An interesting point on these construction is the possibility to provide tangential continuity while changing the circle radius (see Fig. 4.2 c, d and e).

Figure 4 .

 4 Figure 4.1 -Primitive shapes of arches, vaults and domes for additive manufacturing. Triangular, Hemispherical, Segmental, Pointed, Catenary, etc. This figure is reproduced in larger size in Appendix A.

Figure 4 . 2 -

 42 Figure 4.2 -Examples of profiles composed of arcs of different radius. Note the tangential continuity ensured by the alignment of the circles centres and the transition points on the circles

Figure 4 . 3 -

 43 Figure 4.3 -Results of geodesic slicing on a double curve surface from[START_REF] Adiels | Brick patterns on shells using geodesic coordinates[END_REF]

Figure 4 . 4 -

 44 Figure 4.4 -Cusp apparition during the slicing operation of a doubly curved geometry. The colour of the surface corresponds to its mean curvature.

Figure 4 . 5 -

 45 Figure 4.5 -Heat Method applied to a dome with different sources to generate multiple potential strips orientation.

Figure 4 . 6 -

 46 Figure 4.6 -Strips transformation by changing the point sources in the heat method

Figure 4 .

 4 Figure 4.7 -Slicing strategies. a. horizontal slicing, b. radial slicing, c. constant angle slicing

Figure 4 .

 4 Figure 4.8 -Darboux frame from the object surface and the layer direction

Figure 4 .

 4 Figure 4.9 -3D Printed dome with rectangular base (40cm × 80cm). The layers are non planar leading to height variations.

Figure 4 .

 4 Figure 4.10 -Toolpath of the rectangular-base dome before and after addition of mechanism constraints.

Figure 4 .

 4 Figure 4.11 -Iterative Worklow

Figure 4 .

 4 Figure 4.12 -Geometric constraint verification in a CAD environment

Figure 4 .

 4 Figure 4.13 -3D Printed domes with constant layer thickness on the left, and variable layer thickness but constant inter-layer time on the right. The layer width B is the same in both cases and remain unchanged during the print. Note: the dark layer in the centre of the right dome comes a malfunction of the pump at that position, but did not affect the rest of the fabrication.

Figure 4 .

 4 Figure 4.15 -Application of the Linear Workflow to a vertical wall.

Figure 5 .

 5 Figure 5.1 -Global Piecewise structural analysis within the linear workflow for designing 3D Printed structures

  5.2).

Figure 5 . 2 -

 52 Figure 5.2 -Structural failure modes occurring with additive manufacturing process

Figure 5 . 3 -

 53 Figure 5.3 -Thrust line of semi-circular arch under self-weight from[Heyman, 1969]

Figure 5 . 5 -

 55 Figure 5.5 -Global workflow of the constrained form-fitting algorithm.

Figure 5 . 6 -

 56 Figure 5.6 -System of nodes and elements and the corresponding connectivity matrixC = [C f C s ] reproduced from[Schek, 1974].

Figure 5 .

 5 Figure 5.8 -Fitting of the force densities on a hemispherical dome.

Fig. 5 .

 5 Fig.5.8 shows the results of the fitting of the force densities on a spherical dome supported at its base. The values of the force densities are normalised, and by convention negative value corresponds to an element in compression. The model presents an oculus at the top to remove the singularity from the mesh. This prevents bad behaviour of the algorithm at the singularity.

Figure 5 .

 5 Figure 5.9 -Fitting of the force densities on a free-form surface with fixed boundaries.

Figure 5 .

 5 Figure 5.10 -Fitting of the force densities on a S-shaped surface. The green spheres correspond to the amplitude of the residual forces after optimisation.

Figure 5 .

 5 Figure 5.11 -Best-fit optimisation of the hemispherical dome as per Eq. (5.18). Left: result geometry showing compression only in the structure, right: comparison between initial hemispherical shape and the optimised shape

Figure 5 .

 5 Figure 5.12 -Best-fit optimisation of a free-form surface.

Figure 5 .

 5 Figure 5.13 -Best-fit optimisation of a S-shape surface.

Figure 5 .

 5 Figure 5.15 -Minimal thrust line of a hemispherical dome.

Figure 5 .

 5 Figure 5.18 -Comparison between analytical and numerical stresses of a hemispherical dome with an oculus. the meridional N φ and parallel N θ stresses are shown on the graph in function of the angle φ.

  . For hardened concrete, Mohr-Coulomb or Drucker-Prager criteria have proven to be accurate. However, the question of finding the ideal yield criterion for the intermediate state of concrete, specific to additive manufacturing is still open. DI CARLO used a Drucker-Prager criterion for the analysis of the Contour Crafting Process [Di Carlo et al., 2014], WOLFS proposed a numerical simulation method based on Mohr-Coulomb criterion [Wolfs et al., 2018a]. KRUGER et al. and JAYATHILAKAGE work on the development of a new yield criterion customised for concrete printing ([

Figure 6 .

 6 Figure 6.1 -Barrel vault divided into strips. The last strip (in red) is supported by friction only

Figure 6 . 2 -

 62 Figure 6.2 -Section view of a single strip of width w and thickness b at an inclination α

Figure 6 . 3 -

 63 Figure 6.3 -Minimal yield stress necessary to withstand a strip by friction in function of its aspect ratio w/b

Figure 6 . 4 -

 64 Figure 6.4 -Support nodes on a dome and a barrel vault separated into first strip supports (blue) and additional supports (green)

Figure 6 . 5 -

 65 Figure 6.5 -Illustration of the sequencing of the strips of the model. Strip-0 corresponds to the first layer printed on the floor or the supporting area. Each strip is comprised of transverse and longitudinal edges. The halfedge data-structure used to navigate the model is partly represented on strip-1.

  6.7.middle).

Figure 6 . 6 -

 66 Figure 6.6 -Comparison between constrained and unconstrained Laplacian smoothing on a dome.

Figure 6 .

 6 Figure 6.7 -Influence of the triangulation strategy on the final mesh structure.

Figure 6 .

 6 Figure 6.8 -Mesh vertices and their influence area delimited by the dashed lines.

Figure 6 .

 6 Figure 6.11 -Initialisation of the third strip after form-fitting of the first two. The weight of the ghost strip, or next strip is applied on the free node of the current strip (in green). the blue lines indicate the influence area of each node.

Figure 6 .

 6 Figure 6.13 -Initial model geometry (right) and the result of the strip-by-strip form-fitting (middle). the figure on the right shows the optimal flow rates for each strip.

Figure 6 .

 6 Figure 6.14 -Piecewise funicular form-fitting (in black) on an initial hemispherical dome (in red).

Figure 6 .

 6 Figure 6.15 -Strip-by-strip form-fitting of a hemispherical dome for models with different mesh densities. The silhouettes on the right shows the similarity between the three geometries and the comparison with the original dome shape.

Figure 6 .

 6 Figure 6.16 -Application of the strip by strip analysis to a barrel vault. a. Infinite compressive and tensile strength, b. Infinite compressive strength and no tensile strength, c. Limited compressive strength.

Figure 7

 7 Figure 7.1 -Local characterisation of the extrusion seen as the final step in the linear workflow for 3D Printing

Figure 7 . 3 -

 73 Figure 7.3 -3D Printing of a Nubian vault without support. The stability is ensured by the orientation of successive inclined arches.

Figure 7 . 4 -

 74 Figure 7.4 -Series of parametric columns made of ten layers and various heights

Figure 7 . 5 -

 75 Figure 7.5 -Schematic view of the Controlled Pressing strategy (left) and experimental result (right) on a 3D Printed column.

Figure 7 . 6 -

 76 Figure 7.6 -Schematic view of layers flowing out under self-weight and the resulting accumulated error on the total height.

Figure 7 . 7 -

 77 Figure 7.7 -Coiling patterns. Left: coiling patterns with cement-based material extruded at the Navier laboratory, right: coiling patterns with viscous fluid from[START_REF] Brun | Liquid Ropes: A Geometrical Model for Thin Viscous Jet Instabilities[END_REF]

Figure 7

 7 Figure 7.8 -Schematic view of the layer longitudinal tearing (left) and the corresponding 3D Printed example (right).

Figure 7 .

 7 Figure 7.10 -3D Printed tower presenting periodic defects on the corners coming from layer over-pressing (credits: Stefano Borghi)

Figure 7 .

 7 Figure 7.11 -Schematic view of the experiment.

Figure 7 .

 7 Figure 7.12 -Left: Toolpath over the balance, right: Corresponding extrusion of a layer.

Initial

Figure 7 .

 7 Figure 7.17 -Aspect ratio β in function of the dimensionless height H *

Figure 7 .

 7 Figure 7.18 -Schematic view of the Layer Pressing Strategy with the printing parameters

Figure 7 .

 7 Figure 7.19 -Dimensionless pressing stress and dimensionless weight for different layer heights and material properties

Figure 7 .

 7 Figure 7.20 -Dimensionless pressing stress in logarithmic scale in function of the inverse of the layer height. the blue dashed line represent the linear regression from the points.

Figure 7

 7 Figure 7.21 -Illustration of the upper bound condition of the pressing force.

  within a parametric environment to get direct insight into the viability of the process.

Figure 7

 7 Figure 7.23 -Experimental and analytical values of the dimensionless pressing stress for different material properties and layer heights.

Figure 8 . 2 -

 82 Figure 8.2 -Printing Regimes Diagram. The dashed lines correspond to the common printing parameters where V r = V e and H = D.

Figure 8 . 4 -

 84 Figure 8.4 -Influence of the initial yield stress τ 0 on the printing diagram. The dashed lines represent the new boundaries after a variation of dτ 0 = 50P a.

Figure 8 . 5 -

 85 Figure 8.5 -Influence of the layer width B on the relation of mass conservation V * H * B * = π/4β

  Figure 8.6 -Influence of the material structuration A thix on the lower bound condition 2ω * + τ * -A * = 1.

Figure 8

 8 Figure 8.7 -Geometry of the wall and slicing direction for the case study.

Figure 8

 8 Figure 8.8 -Printing regimes on the case study of a trapezoidal wall. The four layers marked 1 to 4 are analysed via the printing regimes diagram in the next figure 8.9.

Figure 8

 8 Figure 8.9 -Printing regimes diagrams corresponding to the four marked layers in the wall illsutrated in the previous figure 8.8.

Figure A. 1 -

 1 Figure A.1 -Primitive shapes from boundary to form and the possible slicing strategies.

  . Left: the extrusion nozzle and the side trowels shaping the layers, right: illustration of 3D Printing of an entire building using a large gantry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Evolution of actors and projects related to Concrete 3D Printing from[START_REF] Buswell | 3D printing using concrete extrusion: A roadmap for research[END_REF] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Examples of geometric freedom in 3D Printing objects (credits: Léo Demont). . . 1.6 Unit cost of a component with regards to the quantity being fabricated for additive manufacturing and traditional casting method, from [Lim et al., 2012]. . . . . . . 1.7 Nailing of layers to enhance cohesion, published in [Perrot et al., 2020]. . . . . . 1.8 Graphic representation of 3D Printing Process as a multi-objectives optimisation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Top left: Krypton column by XtreeE (credits: Lisa Riciotti), top right: topologically optimised post-tensioned bridge by Ghent University (picture from [Vantyghem et al., 2020]), bottom: mechanically optimised reinforced concrete beam (picture from [Maitenaz et al., 2020]). . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Left: Pavilion in Innsbruck 3D Printed by Incremental3d (photo: Rupert Asanger, from [Grasser et al., 2020]), right: artificial reef 3D Printed by XtreeE (from [Gaudillière et al., 2020]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Concrete 3D Printing through the spectrum of structural morphology. The diagram of interaction is in the centre while specific themes related to additive manufacturing surround it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Modification of a dome shape based on the maximum admissible angle α max defined by the angle after which friction cannot support the new layer. . . . . . . . 2.8 Spiral staircase built by GUASTAVINO with the Catalan vault technique (credits: Avery Collins Archive, Columbia University). . . . . . . . . . . . . . . . . . . . . 2.9 Squinches and pendentives with discontinuous layering . . . . . . . . . . . . . . . 2.10 The stone-weighted rope device for erecting Gothic vaults (John Fitchen, The Construction of Gothic Cathedrals, 1961, fig. 69, p.182 ) . . . . . . . . . . . . . . 2.11 Vault built with interlocking bricks from [Loing et al., 2020] . . . . . . . . . . . . 2.12 Left: from 3D geometry to a finite elements model with VoxelPrint (from [Vantyghem et al., 2020]), right: result of a finite elements analysis of a 3D Printing process (from [Wolfs et al., 2018a]) . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Phase change from solid to fluid due to a collision between the 3D Printed structure and the nozzle after completion of the last layer. . . . . . . . . . . . . . . . . . . 2.14 Evolution of the material yield stress with regards to an extrusion-based process by REITER [Reiter et al., 2018]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 1

 1 Masonry wall vs. 3d printed wall at the three different scales . . . . . . . . . . . 3.2 left: masonry curved wall, right: 3d printed curved wall with trapezoidal section for the layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Effect of the slicing on the geometry of the temporary structure for a spherical half-dome. top: horizontal slicing, bottom: vertical slicing. . . . . . . . . . . . . . 3.4 Illustrations of the slicing of a half-dome by CHOISY in [Choisy, 1883].Left: radial slicing, right: horizontal slicing in the lower part combined with a radial slicing in the upper part. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Left: 3-axis gantry at TU Eindhoven Concrete Printing facility. Right: 9-axis robot at Twente Additive Manufacturing facility. . . . . . . . . . . . . . . . . . . 3.6 Shell structure typologies by ISLER [Chilton, 2000] . . . . . . . . . . . . . . . . . 3.7 a) vertical wall, b) f 0 shear stress coming for the layer inclination of angle α, c)f 1 combination of bending and shear in the material, d)f 0 bending moment coming from the local cantilever δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Local cantilever in the case of 3D Printing with corbels. The offset part can be assimilated as vertical layer of height δ to calculate its stability. . . . . . . . . . . 3.9 Comparison between f 0 and f 2 critical angle . . . . . . . . . . . . . . . . . . . . 3.10 From left to right: Printing of a cylinder on a vertical surface. A local failure of the bottom part of the cylinder leads to complete tearing of the lower part by pure tensile stress.The failure of the upper part of the cylinder occurs later at the anchoring of the first layer to the plate and is not due to the material itself. . 3.11 Masonry vaults examples. Left: final geometry of a dome guided by a local inclination criterion α, by CHOISY. Right: Persian vault built with squinches, creating piecewise discontinuous layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Illustration of the initialisation of the strip geometry. Each free node of the current strip is projected onto the free edge of the previous strip (left). The nodes are moved vertically if the projected nodes are higher than the current ones, giving the new strip geometry (the strip free edge is in red). The dashed lines can be seen as the nozzle direction while printing. . . . . . . . . . . . . . . . . . . . . . . 6.11 Initialisation of the third strip after form-fitting of the first two. The weight of the ghost strip, or next strip is applied on the free node of the current strip (in green). the blue lines indicate the influence area of each node. . . . . . . . . . . . 6.12 Strip by strip analysis: a. stresses in the initial configuration. The high values of the force density bounds allow for an unconstrained form-fitting. The red curve on the strip free edge indicates that the maximum stress in the strip exceeds the material yield strength; b. in the middle, the force densities are highly constrained, the strip has deformed and shrunk but is not yet admissible stress-wise; c. after manual fitting of the force density bounds, an admissible geometry for the strip is found, slightly away from the initial geometry but keeping the continuity with the previous strips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Initial model geometry (right) and the result of the strip-by-strip form-fitting (middle). the figure on the right shows the optimal flow rates for each strip. . . . 6.14 Piecewise funicular form-fitting (in black) on an initial hemispherical dome (in red).123 6.15 Strip-by-strip form-fitting of a hemispherical dome for models with different mesh densities. The silhouettes on the right shows the similarity between the three geometries and the comparison with the original dome shape. . . . . . . . . . . . 6.16 Application of the strip by strip analysis to a barrel vault. a. Infinite compressive and tensile strength, b. Infinite compressive strength and no tensile strength, c. Limited compressive strength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Result of the global piecewise form-fitting giving a new geometry at each step of the analysis. The initial dome geometry is shown in the dashed line. . . . . . . . 6.18 Section of a model after piecewise form-fitting with the initial geometry (dashed line) and all temporary geometries corresponding to the activation of each strip. The envelope thickness d can be defined from the point cloud around the node on the initial geometry. The middle of the envelope x * i can then be deduced. . . . . 7.1 Local characterisation of the extrusion seen as the final step in the linear workflow for 3D Printing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 3D Printing of a dome with a gradient of layer heights. The discontinuity in the middle layer is due to a local problem with the pumping equipment. [Carneau et al., 2020b] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 3D Printing of a Nubian vault without support. The stability is ensured by the orientation of successive inclined arches. . . . . . . . . . . . . . . . . . . . . . . . 7.4 Series of parametric columns made of ten layers and various heights . . . . . . . 7.5 Schematic view of the Controlled Pressing strategy (left) and experimental result (right) on a 3D Printed column. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Schematic view of layers flowing out under self-weight and the resulting accumulated error on the total height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Coiling patterns. Left: coiling patterns with cement-based material extruded at the Navier laboratory, right: coiling patterns with viscous fluid from [Brun et al., 2015] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Schematic view of the layer longitudinal tearing (left) and the corresponding 3D Printed example (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Schematic view of the layer over-pressing leading to a pyramidal shape of the overall section (left) and the corresponding 3D Printed example. The area of the layers' cross-section remains constant despite the deformation. . . . . . . . . . . . 7.10 3D Printed tower presenting periodic defects on the corners coming from layer over-pressing (credits: Stefano Borghi) . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Schematic view of the experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12 Left: Toolpath over the balance, right: Corresponding extrusion of a layer. . . . 7.13 Raw data from the balance giving the mass measured for each series. The values given here as an example correspond to a material with an initial yield stress of 500 Pa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14 Layer Pressing for different heights H. . . . . . . . . . . . . . . . . . . . . . . . . 7.15 Rectangle and elliptical section of an extruded layer of height H and width B . . 7.16 Cross-section of a layer constituted of a rectangle and two half-disks. . . . . . . . 7.17 Aspect ratio β in function of the dimensionless height H * . . . . . . . . . . . . . 7.18 Schematic view of the Layer Pressing Strategy with the printing parameters . . . 7.19 Dimensionless pressing stress and dimensionless weight for different layer heights and material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.20 Dimensionless pressing stress in logarithmic scale in function of the inverse of the layer height. the blue dashed line represent the linear regression from the points. 7.21 Illustration of the upper bound condition of the pressing force. . . . . . . . . . . 7.22 Uniform pressing of a longitudinal band of material between two plates from [Coussot, 2005] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.23 Experimental and analytical values of the dimensionless pressing stress for different material properties and layer heights. . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Local scheme of the extrusion process. σ represents the stress applied by the material weight and the extrusion force to the lower part of the first sub-layer (layer n -1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Printing Regimes Diagram. The dashed lines correspond to the common printing parameters where V r = V e and H = D. . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The relation of mass conservation is embedded in the printing regimes diagram in the red dashed line. This indicates the single degree of freedom left in the determination of the remaining printing parameters H and V r when the width B is set beforehand. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Influence of the initial yield stress τ 0 on the printing diagram. The dashed lines represent the new boundaries after a variation of dτ 0 = 50P a. . . . . . . . . . . . 8.5 Influence of the layer width B on the relation of mass conservation V * H * B * = π/4β 8.6 Influence of the material structuration A thix on the lower bound condition 2ω * + τ * -A * = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Geometry of the wall and slicing direction for the case study. . . . . . . . . . . . 8.8 Printing regimes on the case study of a trapezoidal wall. The four layers marked 1 to 4 are analysed via the printing regimes diagram in the next figure 8.9. . . . 8.9 Printing regimes diagrams corresponding to the four marked layers in the wall illsutrated in the previous figure 8.8. . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Variation of the number of layers N on the wall geometry. The colours refer to the printing regimes from the main diagram. . . . . . . . . . . . . . . . . . . . . 8.11 Satisfying solution for the slicing where all parameters fits within their admissible range, and the layer pressing is controlled at all times. . . . . . . . . . . . . . . . 8.12 Printing regime analysis on the studied wall after a horizontal slicing. . . . . . .

8

  Application to Printing Parameters Setting 8.1 Printing Regimes Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Lower Bound Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Printing Parameters Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Printing Regimes and Mass Conservation . . . . . . . . . . . . . . . . . . 8.2 Influence of the Printing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 On the Self-Supporting Condition (ω * < 1) . . . . . . . . . . . . . . . . . 8.2.2 On the Upper Bound Condition (V * H * < 1) . . . . . . . . . . . . . . . . 8.2.3 On the Lower Bound Condition (2ω * + τ * -A * < 1) . . . . . . . . . . . . 8.3 Application to the Slicing Operation . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Summary and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Global Workflow for Concrete 3D Printing . . . . . . . . . . . . . . . . . . 9.1.2 Form-Finding of Printable Structures . . . . . . . . . . . . . . . . . . . . 9.1.3 Characterisation of the Extrusion . . . . . . . . . . . . . . . . . . . . . . . 9.2 Potential Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Table of Primitive Self-Supporting Geometries

	Chapter 1
	V Conclusion
	9 Conclusion and Perspectives
	9.1 Part I
	Introduction

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Acknowledgements

Summary

In this chapter, an algorithm of constrained form-fitting was presented. The use of the Force Density Method ensures that only membrane stresses are involved. The evaluation of the membrane stresses from the force densities allows integrating the yield analysis directly in the form-finding process. Besides, the manual setting of the force densities provides the user with the choice of imposing pure compression (or pure tension) in the entire structure.

The next chapter proposes to use this algorithm to study the design space of printable geometries. The constrained form-fitting is used sequentially to simulate the additive manufacturing process, taking into account the evolution of the material properties. Figure 7.13 -Raw data from the balance giving the mass measured for each series. The values given here as an example correspond to a material with an initial yield stress of 500 Pa.

Results and Analysis

Layer Geometry Control

When printing with a circular nozzle, the extruded layer takes an oval shape, meaning that the corners never form straight angles. Fig. 7.15 shows the layer cross-section embedded in the bounding rectangular shape. The "ideal" rectangular section is achievable when extruding through a rectangular nozzle only. The real section of the layer can be expressed as:

Where β is a corrective factor that can be evaluated. By measuring the exact width and height Chapter 8

Application to Printing Parameters Setting

This chapter aims at understanding the relevant parameter settings for the Layer Pressing Strategy in Concrete 3D Printing. After establishing a lower bound condition for the Controlled Pressing regime, a diagram of all the printing regimes is proposed as a function of the printing parameters.

Printing Regimes Diagram

Lower Bound Condition

The previous analysis of the pressing force applied during the extrusion can be used to evaluate the local stability of the first sub-layer. When the layer n is extruded above layer n -1, the latter has to withstand its own weight, the weight of the new extruded material and the pressing force coming from the extrusion (see Fig. 8.1). Therefore, the stability condition of layer n -1 reads:

Printing Regimes and Mass Conservation

The In practice, the width of the layer B is usually given by the design of the structure to be printed. Therefore it is not a parameter to be set but its value is already known at this stage. In this case, there is only one degree of freedom left in the choice of parameters H * and V * . This degree of freedom is given by the relation of mass conservation (7. There are special cases where additional constraints must be set. For flow-based pultrusion of continuous fibres for instance, where the extruded cement-based material pulls long fibres during the process [START_REF] Demont | Flow-Based Pultrusion of continuous fibers for cement-based composite material and additive manufacturing: rheological and technological requirements[END_REF], the layer cross-section area must equal the nozzle area. According to the relation of mass conservation (7.1), that imposes the printing velocity V r to be equal to the extrusion velocity V e , hence V * = 1. In this case, the layer height is directly given by the horizontal coordinate of the intersection between the mass conservation curve (red Appendix A

Table of Primitive Self-Supporting Geometries

List of Tables