Justin Dirrenberger
email: justin.dirrenberger@ensam.eu

From Architectured Materials to Large-Scale Additive Manufacturing

The classical material-by-design approach has been extensively perfected by materials scientists, while engineers have been optimising structures geometrically for centuries. The purpose of architectured materials is to build bridges across the microscale of materials and the macroscale of engineering structures, to put some geometry in the microstructure. This is a paradigm shift. Materials cannot be considered monolithic anymore. Any set of materials functions, even antagonistic ones, can be envisaged in the future. In this paper, we intend to demonstrate the pertinence of computation for developing architectured materials, and the not-so-incidental outcome which led us to developing large-scale additive manufacturing for architectural applications.

Introduction

Materials are ubiquitous in nature and man-made applications, from electronics to transportation or energy, but also biomedical, defence, architecture or construction. Materials have such a predominance in human existence that prehistoric periods were named after materials use [START_REF] Hummel | Understanding materials science[END_REF]). Technological progress and its impact on everyday's life are intimately linked with our capacity to control matter in the form of industrialised materials. The classical material-by-design approach has been extensively perfected by materials scientists, while civil engineers and architects have been optimising structures geometrically for centuries. The purpose of architectured materials is to build bridges across the microscale of materials and the macroscale of engineering structures, to introduce some geometry within the microstructure. This is a paradigm shift in the sense that materials cannot be considered as monolithic artefacts anymore. Any set of material functions, even antagonistic ones [START_REF] Steeves | Concepts for structurally robust materials that combine low thermal expansion with high stiffness[END_REF], can be envisaged in the future. The development of architectured materials is intrinsically transdisciplinary, on the fringes of physics, chemistry, and mechanical engineering, but also biology, computer science, architecture, design, etc. In this work, we intend to demonstrate the pertinence of a computational approach for developing architectured materials, and the not-so-incidental outcome which led us to developing large-scale additive manufacturing for architectural applications. This paper is an extended version of a preliminary conference abstract submitted for the Next Generation Building workshop held at TU Delft in 2016 [START_REF] Dirrenberger | From architectured materials to the development of large-scale additive manufacturing[END_REF].

The chapter is organised as follows: an introduction to architectured materials is given in Sect. 4.2. An overview of the computational tools relevant for the development of such materials is presented in Sect. 4.3. The translation of architectured materials concepts to the architectural scale corresponds to Sect. 4.4. Finally, conclusions and perspectives are postponed to Sect. 4.5.

Architectured Materials

Many industrial applications require new materials with enhanced specific properties, i.e. performance per unit of mass; this is especially true for the transportation and biomedical sectors. For instance, the automotive industry, with its ever increasing requirements regarding passenger safety and fuel consumption, is an edifying example: nowadays, classical steel-based material solutions are being challenged by new lightweight aluminium alloys and advanced composites. A response from steel manufacturers was the development of advanced high-strength steels (AHSS) for yield strength designed parts in order to reduce both thickness and mass. AHSS with very high strength (1000 MPa and more) commonly exhibit poor sheet formability hence limiting the mass reduction attainable. This is especially true for martensitebased AHSS such as dual-phase (DP), complex phase (CP) and martensitic (MS) steels, depending on their martensite content. The higher the martensite content, the lower the thin-sheet formability. In order to mitigate failure and tearing of thin-sheets during deformation-based forming (stamping, blanking, etc.), a possible solution is to rely on the concept of localised heat treatment in order to soften the material where needed by locally annealing martensite, hence changing the local yield strength/ductility trade-off. Introducing such geometrical discontinuities in terms of material behaviour is characteristic of architectured materials.

Architectured materials are a rising class of materials that bring new possibilities in terms of functional properties, filling the gaps and pushing the limits of Ashby's materials performance maps [START_REF] Ashby | Designing hybrid materials[END_REF]. The term architectured materials encompasses any microstructure designed in a thoughtful fashion, such that some of its materials properties, e.g. yield strength/density, have been improved in comparison to those of its constituents, due to both structure and composite effects, which depend on the multiphase morphology, i.e. the relative topological arrangement between each phase [START_REF] Ashby | Designing hybrid materials[END_REF][START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1: Matériaux aléatoires élastiques et milieux périodiques[END_REF][START_REF] Bréchet | Architectured materials: expanding materials space[END_REF]. Localised material processing methods, such as additive manufacturing, or localised laser heat treatment in the case of AHSS, appear as natural candidates for developing architectured materials.

There are many examples: particulate and fibrous composites, foams, sandwich structures, woven materials, lattice structures, etc. with different objectives. For instance, developing architectured porous materials for structural, acoustic and insulation properties [START_REF] Caty | Fatigue of metal hollow spheres structures[END_REF][START_REF] Fallet | Mechanical behaviour of metallic hollow spheres foam[END_REF], entangled monofilament of perlitic steel [START_REF] Courtois | Mechanical properties of monofilament entangled materials[END_REF], sandwich composite structures [START_REF] Kolopp | Experimental study of sandwich structures as armour against medium-velocity impacts[END_REF], segmented interlocking structures [START_REF] Djumas | Enhanced mechanical performance of bioinspired hybrid structures utilising topological interlocking geometry[END_REF], woven and non-woven textile composites [START_REF] Mezeix | Mechanical behavior of entangled fibers and entangled cross-linked fibers during compression[END_REF][START_REF] Lecampion | Evolution of portland cement mechanical properties exposed to CO2-rich fluids: investigation at different scales[END_REF][START_REF] Dirrenberger | Towards gigantic RVE sizes for stochastic fibrous networks[END_REF], crumpled metallic foils [START_REF] Bouaziz | Compression of crumpled aluminum thin foils and comparison with other cellular materials[END_REF], etc. Architectured materials have been an exciting research topic in recent years, especially regarding the processing of architectured metallic foams [START_REF] Brothers | Density-graded cellular aluminum[END_REF][START_REF] Erk | Titanium with controllable pore fractions by thermoreversible gelcasting of TiH2[END_REF], multiscale architectured ceramics [START_REF] Mirkhalaf | Overcoming the brittleness of glass through bio-inspiration and micro-architecture[END_REF], or architectured metallic sheets [START_REF] Embury | Steel-based composites: driving forces and classifications[END_REF][START_REF] Chéhab | Compositionally graded steels: a strategy for materials development[END_REF]. Apart from processing, most efforts have been focused on the modelling of architectured materials. In particular, the pioneering works done on truss-structures [START_REF] Deshpande | Effective properties of the octet-truss lattice material[END_REF]) and metallic foams [START_REF] Deshpande | Isotropic constitutive models for metallic foams[END_REF] led to useful results for the modelling of architectured materials [START_REF] Fleck | Micro-architectured materials: past, present and future[END_REF]. The mechanical modelling of bio-inspired architectured materials has been pursued successfully, yielding results for multi-layered materials that could be used for modelling architectured metallic sheets [START_REF] Turcaud | An excursion into the design space of biomimetic architectured biphasic actuators[END_REF][START_REF] Stoychev | Shape-programmed folding of stimuli-responsive polymer bilayers[END_REF]. Other works demonstrated the interest for designing recursive or nested material architectures to achieve enhanced specific mechanical properties [START_REF] Ajdari | Hierarchical honeycombs with tailorable properties[END_REF][START_REF] Rayneau-Kirkhope | Ultralight fractal structures from hollow tubes[END_REF]; finally, elastic instabilities have been considered for shape-generation of architectured materials [START_REF] Bertoldi | Negative Poisson's ratio behavior induced by an elastic instability[END_REF][START_REF] Shim | Harnessing instabilities for design of soft reconfigurable auxetic/chiral materials[END_REF].

One can play on many parameters in order to obtain architectured materials, but all of them are related either to the microstructure or the geometry. Parameters related to the microstructure can be optimised for specific needs using a materials-by-design approach, which has been thoroughly developed by chemists, materials scientists and metallurgists. Properties improvements related to microstructural design are intrinsically linked to the synthesis and processing of materials and are therefore due to micro and nanoscale phenomena, taking place at a scale ranging from 1 nm to 10 µm. This scale is below the scope of the present chapter, in terms of topology optimisation, but has been extensively studied in the literature [START_REF] Embury | Steel-based composites: driving forces and classifications[END_REF][START_REF] Olson | Beyond discovery: design for a new material world[END_REF]Freeman 2012).

Until now, most architectured materials in the literature have been obtained empirically. By capitalising on the concept of localised processing of thin structures, our efforts at the PIMM laboratory 1 have been focused on developing systematic tools to determine materials architecturation patterns, for a given set of requirements. For instance, these patterns come as an output from a computational shape-optimisation loop developed around a heuristic generative geometry module, based on cellular automata (CA), and a cost function evaluation module, based on the finite element (FE) method. This cost function has to be minimised for given constraints (Bendsøe and Sigmund 2004). In the case of linear elastic problems, the cost function can be related to the elastic energy density computed by numerical homogenisation at the scale of the structure [START_REF] Allaire | Shape optimization by the homogenization method[END_REF]. The cost function value is then used as a feedback, and an optimised shape is generated accordingly following a specific evolution algorithm. The spatial resolution associated with the shape-optimisation can be chosen to be relevant for localised laser treatment, i.e. 1 mm, corresponding to a representative scale of the underlying microstructure [START_REF] Bironeau | Evaluation of morphological representative sample sizes for nanolayered polymer blends[END_REF]; this allows us to use homogenised behaviour for each grid-cell in the simulation [START_REF] Dirrenberger | Effective properties of architectured materials[END_REF].

From a macroscopic viewpoint, parameters related to the geometry have mainly been the responsibility of structural and civil engineers for centuries: to efficiently distribute materials within structures. An obvious example would be the many different strategies available for building bridges. At the millimetre scale, materials can be considered as structures, i.e. one can enhance the bending stiffness of a component by modifying its geometry while keeping the lineic mass (for beams) or surfacic mass (for plates) unchanged [START_REF] Weaver | The optimal selection of material and section-shape[END_REF]. On the other hand, one might need a lower flexural strength for specific applications, with the same lineic and/or surfacic masses. This can be achieved with strand structures, i.e. by creating topological interfaces in the material. Processing remains the key technological issue for further development of architectured materials, and progress is made every day in this direction at the lab scale, as it was done in [START_REF] Schaedler | Ultralight metallic microlattices[END_REF]) by using a sequence of several processing techniques in order to fabricate ultralight metallic microlattice materials [START_REF] Schaedler | Architected cellular materials[END_REF]. There is still a long way to go for the industry to actually apply architectured materials in product manufacturing.

Architectured materials lie between the microscale and the macroscale. This class of materials involves geometrically engineered distributions of microstructural phases at a scale comparable to the scale of the component [START_REF] Ashby | Designing hybrid materials[END_REF][START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1: Matériaux aléatoires élastiques et milieux périodiques[END_REF][START_REF] Bréchet | Architectured materials: expanding materials space[END_REF], thus calling for enriched models of continuum mechanics in order to determine the effective properties of materials [START_REF] Geers | Multiscale modeling of microstructure-property relations[END_REF][START_REF] Matouš | A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials[END_REF]), e.g. generalised continua theories, in order to describe the behaviour of architectured materials, such as straingradient elasticity [START_REF] Auffray | A complete description of bi-dimensional anisotropic strain-gradient elasticity[END_REF], and strain-gradient plasticity. This topic has been especially fruitful these last few years in the mechanics of materials community [START_REF] Lebée | Homogenization of thick periodic plates: application of the bending-gradient plate theory to a folded core sandwich panel[END_REF][START_REF] Trinh | Evaluation of generalized continuum substitution models for heterogeneous materials[END_REF][START_REF] Chen | Micropolar continuum modeling of bidimensional tetrachiral lattices[END_REF][START_REF] Auffray | A complete description of bi-dimensional anisotropic strain-gradient elasticity[END_REF][START_REF] Placidi | Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second-gradient elasticity[END_REF]Rosi and Auffray 2016;[START_REF] Andreaus | Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity[END_REF][START_REF] Placidi | Identification of two-dimensional pantographic structures with a linear d4 orthotropic second gradient elastic model accounting for external bulk double forces[END_REF][START_REF] Dell'isola | Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives[END_REF]; this results in the availability of versatile models able to describe the various situations encountered with architectured materials. Given mature processing techniques, architectured materials are promised to a bright future in industrial applications due to their enticing customisable specific properties and the opportunity of multifunctionality.

When considering actual applications, one engineering challenge is to predict the effective properties of such materials; computational homogenisation using finite element analysis is a powerful tool to do so. Homogenised behaviour of architectured materials can thus be used in large structural computations, hence enabling the dissemination of architectured materials in the industry. Furthermore, computational homogenisation is the basis for computational topology optimisation [START_REF] Allaire | Shape optimization by the homogenization method[END_REF]Bendsøe and Sigmund 2004;[START_REF] Guest | Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability[END_REF][START_REF] Challis | Design of three dimensional isotropic microstructures for maximized stiffness and conductivity[END_REF]Xu et al. 2016;[START_REF] Vicente | Concurrent topology optimization for minimizing frequency responses of two-level hierarchical structures[END_REF]Xu et al. 2016;[START_REF] Salonitis | A hybrid finite element analysis and evolutionary computation method for the design of lightweight lattice components with optimized strutdiameter[END_REF][START_REF] Asadpoure | Topology optimization of multiphase architected materials for energy dissipation[END_REF][START_REF] Khakalo | Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software[END_REF][START_REF] Wang | Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization[END_REF]) which will give rise to the next generation of architectured materials as it can already be seen in the works of [START_REF] Laszczyk | Design of architectured sandwich core materials using topological optimization methods[END_REF][START_REF] Andreassen | Design of manufacturable 3D extremal elastic microstructure[END_REF][START_REF] Körner | A systematic approach to identify cellular auxetic materials[END_REF][START_REF] Hopkins | Design of nonperiodic microarchitectured materials that achieve graded thermal expansions[END_REF][START_REF] Kotani | Materials inspired by mathematics[END_REF][START_REF] Ghaedizadeh | Tuning the performance of metallic auxetic metamaterials by using buckling and plasticity[END_REF][START_REF] Ren | Anisotropic and dispersive wave propagation within strain-gradient framework[END_REF][START_REF] Liu | Harnessing buckling to design architected materials that exhibit effective negative swelling[END_REF][START_REF] Dalaq | Finite element prediction of effective elastic properties of interpenetrating phase composites with architectured 3D sheet reinforcements[END_REF].

Materials science comes from the following fact: microstructural heterogeneities play a critical role in the macroscopic behaviour of a material [START_REF] Besson | Non-linear mechanics of materials[END_REF][START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1: Matériaux aléatoires élastiques et milieux périodiques[END_REF][START_REF] Jeulin | Mechanics of random and multiscale microstructures[END_REF][START_REF] François | Mechanical behaviour of materials, volume 1: microand macroscopic constitutive behaviour, vol. 180. Solid mechanics and its applications[END_REF][START_REF] Torquato | Random heterogeneous materials[END_REF][START_REF] Ostoja-Starzewski | Microstructural randomness and scaling in mechanics of materials[END_REF]. Constitutive modelling, thanks to an interaction between experiments and simulation, is usually able to describe the response of most materials in use. Such phenomenological models, including little to no information about the microstructure, cannot necessarily account for local fluctuation of properties. In that case, the material is considered as a homogeneous medium. Studying the behaviour of heterogeneous materials involves developing enriched models including morphological information about the microstructure [START_REF] Smith | Computer simulation results for the two-point probability function of composite media[END_REF]Yeong and Torquato 1998;[START_REF] Torquato | Morphology and effective properties of disordered heterogeneous media[END_REF][START_REF] Decker | 3D morphological analysis of the connectivity of a porous medium[END_REF][START_REF] Jeulin | Random texture models for material structures[END_REF][START_REF] Kanit | Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry[END_REF][START_REF] Peyrega | 3D morphological characterization of phonic insulation fibrous media[END_REF][START_REF] Jean | A multiscale microstructure model of carbon black distribution in rubber[END_REF][START_REF] Escoda | Three-dimensional morphological modeling of concrete using multiscale Poisson polyhedra[END_REF]. These models should be robust enough to predict effective properties depending on statistical data (volume fraction, n-point correlation function, etc.) and the physical nature of each phase or constituent. As a matter of fact, advanced models are often restricted to a limited variety of materials. Although isotropic and anisotropic polycrystalline metals, for instance, have been extensively studied by the means of both analytical and computational tools [START_REF] Cailletaud | Some elements of microstructural mechanics[END_REF][START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF][START_REF] Madi | Finite element simulations of the deformation of fused-cast refractories based on X-ray computed tomography[END_REF][START_REF] Berdin | Internal stresses in polycrystalline zirconia: microstructure effects[END_REF][START_REF] Fritzen | Computational homogenization of porous materials of Green type[END_REF][START_REF] Hor | Statistical assessment of multiaxial HCF criteria at the grain scale[END_REF][START_REF] Kowalski | Finite element modeling of periodic polycrystalline aggregates with intergranular cracks[END_REF][START_REF] Amodeo | Atomistically-informed crystal plasticity in MgO polycrystals under pressure[END_REF][START_REF] Schindler | Numerical homogenization of elastic and thermal material properties for metal matrix composites (MMC)[END_REF], architectured materials bring up new challenges regarding the determination of effective properties.

The development of architectured materials is related to the availability of appropriate computational tools for both design and modelling, but also for computerised manufacturing as for the various additive manufacturing techniques considered to produce architectured materials.

Computation for Design, Modelling, and Manufacturing

The merit of additive manufacturing is often summarised as its ability to produce shapes that result from a topology optimisation process. Topology optimisation aims at attaining the most efficient structure for a given set of requirements. It is a longstanding topic of research and development for engineers that can be traced back to the seminal work of [START_REF] Michell | The limit of economy of material in frame structures[END_REF]) on frame structures one century ago, or even 30 years earlier with (Lévy 1874), who gave the first proof for determinacy in statics for single-load trusses. The topic of optimisation has been active ever since. Nowadays, optimality in terms of industrial design is becoming more and more critical due to scarcity of material resources and the need for lightweight structures. This technique has become well-established in the field of structural mechanics, especially when associated with FE simulation. Classical methods (Bendsøe and Kikuchi 1988;[START_REF] Rozvany | Layout optimization of structures[END_REF]Duysinx and Bendsøe 1998), such as SIMP (Bendsøe and Sigmund 2004) (Solid Isotropic Material with Penalisation) rely on node-based values to evaluate and optimise the geometry, i.e. the number of design variables is equal to the number of elements available in the model at initialisation. Then, the optimisation procedure consists in determining at each element if it should either stay a material element or become a void element, i.e. be removed. This technique has been applied to different scales: for instance with regards to the design of efficient building structures [START_REF] Cui | Computational morphogenesis of 3D structures by extended ESO method[END_REF], or as a tool for designing micro-and nanoarchitectured materials [START_REF] Zhou | Design of graded two-phase microstructures for tailored elasticity gradients[END_REF].

Most computational approaches for topology optimisation used in engineering are gradient-based, they are also known as local approaches. In recent years, so-called global approaches emerged, and are currently subject to epistemological controversy in the optimisation community due to the lack of proof for global convergence, as well as inefficiency in comparison with classical gradient methods [START_REF] Sigmund | On the usefulness of non-gradient approaches in topology optimization[END_REF][START_REF] Riche | On global optimization articles in SMO[END_REF]. Although the fact that being based on heuristics has been held against global approaches in structural mechanics, heuristics itself should not be considered a shortcoming but rather an epistemological hypothesis.

The computational framework being developed at PIMM rely on a global topologyoptimisation approach making use of cellular automata, as well as FE for evaluating of the cost function, which has to be minimised for given constraints (Bendsøe and Sigmund 2004). This evaluation step is straightforward since it consists in performing a FE computation on the generated model using predefined local constitutive behaviours, and averaging the response of the structure, to a given set of boundary conditions and applied loads, by the computational homogenisation method [START_REF] Dirrenberger | Homogenization of periodic auxetic materials[END_REF][START_REF] Dirrenberger | Elastoplasticity of auxetic materials[END_REF][START_REF] Dirrenberger | Effective elastic properties of auxetic microstructures: anisotropy and structural applications[END_REF][START_REF] Dirrenberger | Towards gigantic RVE sizes for stochastic fibrous networks[END_REF]. The topology optimisation step is somewhat more difficult as choices have to be made with regards to the many approaches available and the type of problem being dealt with. Several reviews are available on this topic, see for instance (Bendsøe and Sigmund 2004;[START_REF] Rozvany | A critical review of established methods in structural topology optimization[END_REF][START_REF] Eschenauer | Topology optimization of continuum structures: a review[END_REF]. Most developments in topology optimisation dealt with the efficiency of structures, i.e. minimising the mass of materials while optimising the elastic stiffness of a structure under a given load, which means choosing between void and matter for any given point in the design space, either continuous [START_REF] Allaire | Shape optimization by the homogenization method[END_REF] or discrete (Bendsøe and Sigmund 2004).

Architecturation patterns can be either continuous, discrete, periodic or random, therefore cellular automata (CA) seem like natural candidates for generating them. CA correspond to an evolving structure based on a regular lattice, they are characterised by 5 properties: lattice geometry and dimensionality, cellular neighbourhood, cell state, local rule of transition, and boundary conditions. CA have already been used for topological optimisation, but not for solving pattern-type problems [START_REF] Hajela | On the use of energy minimization for CA based analysis in elasticity[END_REF][START_REF] Missoum | Study of a new local update scheme for cellular automata in structural design[END_REF][START_REF] Hopman | Triangular cellular automata for computing two-dimensional elastodynamic response on arbitrary domains[END_REF]). An alternative method, the hybrid cellular automata (HCA) method, was proposed by [START_REF] Tovar | Bone structure adaptation as a cellular automaton optimization process[END_REF]) by taking the best of both CA and SIMP. This method has been developed thoroughly for structural optimisation [START_REF] Tovar | Optimality conditions of the hybrid cellular automata for structural optimization[END_REF]. A critical improvement to the CA approach was the implementation of multigrid methods in order to accelerate the convergence during optimisation [START_REF] Kim | Multigrid accelerated cellular automata for structural design optimization: A 1-D implementation[END_REF][START_REF] Zakhama | Multigrid implementation of cellular automata for topology optimization of continuum structures[END_REF]. The development of genetic algorithms in structural optimisation is known as evolutionary structural optimisation (ESO). It was first proposed by [START_REF] Xie | A simple evolutionary procedure for structural optimization[END_REF][START_REF] Chu | Evolutionary structural optimization for problems with stiffness constraints[END_REF] in the early 1990s, but suffered from various drawbacks (lack of convergence, mesh-dependency) which were partially overcome by the bi-directional evolutionary structural optimisation (BESO) approach developed by the same team a decade later [START_REF] Huang | Convergent and mesh-independent solutions for bi-directional evolutionary structural optimization method[END_REF]. Other developments have been undertaken regarding genetic algorithms, e.g. in terms of multi-criterion optimisation [START_REF] Canyurt | Cellular genetic algorithm technique for the multicriterion design optimization[END_REF].

Therefore, a framework based on CA with a local rule of transition using the BESO approach, along with multigrid implementation appears like an appropriate option to design architectured materials, especially by the ability of such framework to deal with multiple scales of topology optimisation. In order to be fully efficient, the multiscale optimisation scheme must comply with optimising multiple anisotropic materials with nonlinear elastoplastic behaviour, which will yield nonlinearities in structural response of the architectured materials for various sets of requirements/cases of application, in terms of ductile fracture or fatigue properties (Torabian et al. 2016a(Torabian et al. , b, 2017a, b), b).

Generating and modelling shapes for additive manufacturing follows specific rules, coming from both processing constraints, e.g. layer thickness, product dimensions, etc., and the functional properties of the produced part, e.g. mechanical strength, thermal conductivity, etc. An usual and straightforward method for generating an additive manufacturing building path is to use a 3D-to-2D slicing software. It consists in slicing the 3D shape, i.e. computer-aided design (CAD) file, of an object into flat thin layers of constant thickness which can be layered one up onto the other, i.e. computer-aided manufacturing (CAM) file. This results in a cantilever-method strategy. Each layer is then made of a contour line, as well as a filling pattern such as a honeycomb structure or a space-filling curve (Peano curve, Hilbert curve, etc.); the filling density can be adjusted for given requirements. This method is well-established for small-scale additive manufacturing.

Development of the Large-Scale Additive Manufacturing

The industrial applications of small-scale additive manufacturing in production (in opposition to prototyping) concern either geometrically complex products in high value-adding sectors (biomedical, sports, aerospace) and/or parts made of costly materials, typically alloys which include Titanium, Nickel and/or Chrome in their composition. This type of additive manufacturing is usually powder-based, which is intrinsically costly, but enables very good spatial accuracy and microstructural control. Technologies such as direct metal laser sintering (DMLS), selective laser sintering (SLS), electron beam melting (EBM) or direct metal deposition (DMD) are taking off in high-end industries due to its cost-saving possibilities both in terms of reducing the number manufacturing steps and the amount of materials used [START_REF] Guest | Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability[END_REF][START_REF] Allaire | Shape optimization by the homogenization method[END_REF][START_REF] Xu | A comparison of energy consumption in bulk forming, subtractive, and additive processes: review and case study[END_REF][START_REF] Baumers | The cost of additive manufacturing: machine productivity, economies of scale and technology-push. Technological Forecasting & Social Change Bendsøe[END_REF][START_REF] Ford | Additive manufacturing and sustainability: an exploratory study of the advantages and challenges[END_REF].

The DEMOCRITE project (2014-2015, funded by HESAM Université) aimed at transposing such possibilities within an architectural context, i.e. developing largescale additive manufacturing with the same level of material performance than conventional processing methods and comparable accuracy as DMLS, i.e. a spatial resolution circa 0.1% of the building size, about 1 mm for a metre-wide printer. Our group at the PIMM lab explored the potential applications of large-scale additive manufacturing techniques to civil engineering structures. Thanks to this large-scale processing technique, one could apply the concept of architectured materials within an architectural context; multifunctional properties can be achieved for structural elements, by optimising geometry and material composition, as it was done in [START_REF] Gosselin | Large-scale 3D printing of ultra-high performance concrete-a new processing route for architects and builders[END_REF], for an enhanced thermal insulation case study regarding a structural wall element produced by large-scale additive manufacturing. An example of topology optimisation based on the SIMP approach, performed during the DEMOCRITE project is given in Fig. 4.1. The initial optimisation results (Fig. 4.1a) were adapted and refined in order to fit the additive manufacturing constraints as well as the architectural scenario. The rendering shown on Fig. 4.1b actually includes the visible layers of printed concrete resulting from the building process.

Based upon an understanding of the limitations identified in previous projects present in the literature, the DEMOCRITE project dealt with the large-scale additive manufacturing of selective deposition for ultra-high performance concrete (UHPC) [START_REF] Duballet | Additive manufacturing and multi-objective optimization of graded polystyrene aggregate concrete structures Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints[END_REF]. The 3D printing process involved is based on a Fused Deposition Modelling-like technique, in the sense that a material is deposited layer-by-layer through an extrusion printhead. The project also explored the possibilities offered by CAD and optimisation, and their integration within the product design process in the case of large-scale additive manufacturing. Thus, the introduced technology succeeded in solving many of the problems that could be found in the literature. Most notably, the process enabled the production of 3D large-scale complex geometries, without the use of temporary supports, as opposed to 2.5D examples found in the literature for concrete 3D printing [START_REF] Khoshnevis | Automated construction by contour crafting-related robotics and information technologies[END_REF][START_REF] Buswell | Freeform construction: mega-scale rapid manufacturing for construction[END_REF][START_REF] Cesaretti | Building components for an outpost on the Lunar soil by means of a novel 3D printing technology[END_REF].

According to [START_REF] Gosselin | Large-scale 3D printing of ultra-high performance concrete-a new processing route for architects and builders[END_REF], the concept of freeform commonly used in the literature is not adequate nor sufficient for describing concrete 3D printing. For a given printing process and automation complexity, one can attain specific types of topologies within a given time-frame and performance criterion for the material and/or structure to be built. Design conditions for large-scale additive manufacturing depend on many other parameters than just the properties of extruded cementitious materials; parameters such as the printing spatial resolution, overall size of parts to be printed, the environment, the presence of assembling steps, etc. A tentative classification of such relationships between geometrical complexity, processing, and design is proposed in [START_REF] Duballet | Classification of building systems for concrete 3D printing[END_REF]. The cantilever strategy, which is characteristic of small-scale additive manufacturing, is not appropriate for large-scale printing since it does not take into account the processing constraints and their impact on the performance of the printed structure. The building path should be adapted and optimized based on simulation results in order to take into account constraints and to exhibit more robustness for complex geometries.

The processing constraints depend mostly on the fresh material properties in its viscous state, as well as early-age behaviour, in interaction with the building strategy and the stiffness of the structure being built. On the other hand, functional requirements will depend on the properties of the hardened material, its durability [START_REF] Lecampion | Evolution of portland cement mechanical properties exposed to CO2-rich fluids: investigation at different scales[END_REF], as well as the structural geometry for effective stiffness, and other functional properties such as thermal and sound insulation. See [START_REF] Gosselin | Large-scale 3D printing of ultra-high performance concrete-a new processing route for architects and builders[END_REF] for a geometrically induced thermal insulation case study. Both types of constraints have to be considered at the design stage.

Printing path generation is a critical step during the design phase. There are two main approaches to tool-path generation in the context of 3D printing: (1) 3D-to-2D slicing, which is by far the most common method adopted, yields planar layers of equal thickness built on top of each other. This approach is not optimal from a design and structural viewpoint as it will induce cantilevers when two consecutive layers have different sizes and limit the attainable geometries; (2) the tangential continuity method, which has been introduced in [START_REF] Gosselin | Large-scale 3D printing of ultra-high performance concrete-a new processing route for architects and builders[END_REF] in order to optimise the structure being built by creating layers of varying thickness. These layers exhibit a maximised surface area of contact between each other, hence stabilising the overall structure. Moreover, this method is actually exploiting the possibilities Figure 4.3 shows a 3D-printed 4 metre-high post supporting the playground roof of a school in Aix-en-Provence, France, produced by XtreeE. The structure was made of two parts: the 3D-printed envelope, used as a lost formwork, and the core, made of conventional cast concrete. The total printing time was 15 h and 30 min.

A driving force for additive manufacturing is its ability to produce more complex 3D shapes in comparison to casting or subtractive processes. This complexity allows to design optimal structures based on topology optimisation techniques. One of the main current challenges is to modify optimisation algorithms in order to account for the additive manufacturing constraints, especially with regards to the processing parameters and structural stability while printing. A possible answer to these challenges would be to consider the multiphysics phenomenon aspect of 3D printing, which involves the elastic stability of the overall structure being built, the kinetics of hydration, the evolving viscoplasticity of fresh cement, the evolution of temperature within the printing environment, etc. As a matter of fact, all these physical problems, with multiple time and space scales, can be modelled on their own, but coupling them generates complexity and uncertainty regarding the process of 3D printing. Therefore, efforts should be concentrated on understanding and modelling the printing process in its multiple physical aspects, only then optimisation will be fully integrated with the processing, which would virtually change the way 3D printed structures are conceived today. An attempt at multi-objective optimisation for thermal, mechanical and economical criteria, in the context of a hybrid largescale concrete 3D printing and assembling process, was proposed by [START_REF] Duballet | Design of space truss based insulating walls for robotic fabrication in concrete[END_REF]. It consists in fabricating insulating blocks, which would be assembled by robots in order to form a layer of printing support for the mortar to be extruded; at the end, tie columns and ring beams will ensure the masonry confinement. This approach is rather new in comparison with most examples of large-scale concrete 3D printing available [START_REF] Labonnote | Additive construction: state-of-the-art, challenges and opportunities[END_REF][START_REF] Duballet | Classification of building systems for concrete 3D printing[END_REF]. Indeed, the future of large-scale additive manufacturing in architecture and construction might reside in smarter, more parsimonious use of 3D printing on specific parts of the printed object, where it is the most pertinent in order to take advantage of both the material properties and morphology of the structure being built.

Conclusions and Perspectives

An overview was given regarding the concept of architectured materials in general, i.e. highlighting the importance of geometry at multiple scales, as well as the computational tools associated with their development. In fact, this development is made possible by the computerisation of both the designing step, and robotisation of the manufacturing step. This automation of the process theoretically allows for a higher accuracy and infinite customisation of materials and structures according to a given set of requirements. In this context, the top-down approach of architectured materials, i.e. going from the application down to the design of materials, becomes generalisable, hence yielding new opportunities for the built environment sector, e.g. in architectural design, structural engineers' calculations, or the logistics of a construction site, if building information modelling (BIM) is integrated. From an application viewpoint, one of the most critical point is indeed to be able to incorporate robotic actions within the BIM model.

Fig. 4 . 1

 41 Fig. 4.1 An example of implementation of topology optimisation in the DEMOCRITE project: a Boundary conditions and results for the structural optimisation problem; b Rendering of the actual multifunctional wall including both the structural and thermal insulation parts

Fig. 4 . 2

 42 Fig. 4.2 The stand-alone concrete additive manufacturing process developed during the DEM-OCRITE project, currently being implemented commercially by XtreeE

Fig. 4 . 3

 43 Fig. 4.3 Close-up view of a 4 metre-high post in Aix-en-Provence, France. Source www.xtreee. com

Acknowledgements This work is part of the DEMOCRITE (Large-scale additive manufacturing platform) Project PNM-14-SYNG-0002-01, as well as the SCOLASTIC (Systematic Computational Optimisation and Local Laser Processing for Steel-Based Architectured Materials) Project ANR 16-CE08-0009. The author would like to gratefully acknowledge ANR (Agence Nationale de la Recherche), and heSam Université for financial support through its Paris Nouveaux Mondes program.


est destinée au dépôt

Thermographic investigation of heat source in transversely isotropic composites

Benjamin Valès, Victor Alfonso Munoz Cuartas, Hélène Welemane, Marie-Laetitia Pastor, Baptiste Trajin, Marianne Perrin, Arthur Cantarel,

Moussa Karama

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To cite this version:

	Vales, Benjamin and Munoz Cuartas, Victor Alfonso and Welemane,
	Hélène and Pastor, Marie-Laetitia and Trajin, Baptiste and Perrin,
	Marianne and Cantarel, Arthur and Karama, Moussa Thermographic
	investigation of heat source in transversely isotropic composites. In:
	International Conference on Structural Analysis of Advanced Materials
	(ICSAAM 2017), 19 September 2017 -22 September 2017 (Bucarest,
	Romania).

Benjamin Valès, Victor Alfonso Munoz Cuartas, Hélène Welemane, Marie-Laetitia Pastor, Baptiste Trajin, et al.. Thermographic investigation of heat source in transversely isotropic composites. International Conference on Structural Analysis of Advanced Materials (ICSAAM 2017), Sep 2017, Bucarest, Romania. pp.030043-1. hal-01905409 This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 20090

Microporomechanics

Random heterogeneous materials. Microstructure and macroscopic properties

Conduction of heat in solids


G Kolaye
email: kolayegg@gmail.com

I Damakoa

• S Bowong

• R Houe

D Békollè

Theoretical Assessment of the Impact of Climatic Factors in a Vibrio Cholerae Model

Keywords: Mathematical model, V. Cholerae, Climatic factors, Cooperative systems, Stability

A mathematical model for Vibrio Cholerae (V. Cholerae) in a closed environment is considered, with the aim of investigating the impact of climatic factors which exerts a direct influence on the bacterial metabolism and on the bacterial reservoir capacity. We first propose a V. Cholerae mathematical model in a closed environment. A sensitivity analysis using the eFast method was performed to show the most important parameters of the model. After, we extend this V. cholerae model by taking account climatic factors that influence the bacterial reservoir capacity. We present the theoretical analysis of the model. More precisely, we compute equilib-ria and study their stabilities. The stability of equilibria was investigated using the theory of periodic cooperative systems with a concave nonlinearity. Theoretical results are supported by numerical simulations which further suggest the necessity to implement sanitation campaigns of aquatic environments by using suitable prod-ucts against the bacteria during the periods of growth of aquatic reservoirs.

Introduction

Vibrio Cholerae (V. Cholerae) is a Gram-negative, comma-shaped bacterium that causes cholera in humans. Cholera is an acute intestinal infection caused by the ingestion of contaminated foods and water with V. Cholerae bacterium. There are 200 serogroups of V. Cholerae but only the V. Cholerae O1 and O139 are responsible of epidemics (WHO 2017). The etiological agent passes through and survives the gastric acid barrier of the stomach and then penetrates the mucus lining that coats the intestinal epithelial [START_REF] Reidl | Vibirio Cholerae and cholera: out of the water and into the host[END_REF]. Once they colonise the intestinal gut, they produce enterotoxin (which stimulates water and electrolyte secretion by the endothelial cells of the small intestine) that leads to watery diarrhea. Vibrio Cholerae has been reported to be associated with a variety of living organisms, including animals with an exoskeleton of chitin, aquatic plants, protozoa, bivalves, waterbirds, as well as abiotic substrates (e.g. sediments). Most of these are well-known or putative environmental reservoirs for the bacterium, defined as places where the pathogen lives over time, with the potential to be released and to cause human infection. These bacteria are then associated with these environmental reservoirs forming commensal relationships [START_REF] Huq | Influence of water temperature, salinity, and pH on survival and growth of toxigenic Vibrio Cholerae serovar O1 associated with live copepods in a laboratory microcosms[END_REF]. In many mathematical model of cholera [START_REF] Copasso | A model for the 1973 cholera epidemic in the european mediterranean region[END_REF]Cláudia 2001;[START_REF] Pascual | Cholera and climate: revisiting the quantitative evidence[END_REF][START_REF]Hyperinfectivity: a critical element in the ability of V. Cholerae to cause epidemics?[END_REF] and recently [START_REF] Mushayabasa | Is HIV infection associated with an increased risk for cholera? Insights from a mathematical model[END_REF][START_REF] Mwasa | Mathematical analysis of a cholera model with public health interventions[END_REF], bacteria dynamic are generally simplified using one linear differential equation with constant parameters. From the mathematical point of view, the constant parameter assumption has the advantage of simplifying the models and analysis, and facilitating the use of some well-known theory in autonomous dynamical systems. But it is important for mathematical cholera studies to incorporate these seasonal factors that modify the metabolism of bacteria, their density and the number of their environmental reservoirs along a year.

The aim of this paper is to improve our understanding on the impact of climatic factors of the dynamics of V. Cholerae. We first formulate and analyze a mathematical model of V. Cholerae in a closed environment. The sensitivity analysis of the model is carried out to show the most important parameters of the model. After, we extend this model by taking into account the variation of climatic factors. Threshold and equilibria are obtained and global stabilities examined. Numerical simulations are presented to support the theory and to get insight on the role of climatic factors on the dynamics of the population of V. Cholerae.

The rest of the paper is organized as follows. In Sect. 2, we introduced and studied a minimalistic V. Cholerae model. After, this model was extended by considering the seasonality of climatic factors on the bacterial reservoir capacity in Sect. 3. We present a qualitative analysis of the model. More precisely, we compute equilibria and study their stabilities. Numerical simulations are presented to illustrate and confirm theoretical results. We conclude the paper on Sect. 4.

A Minimalistic V. Cholerae Model

The Model

We consider two distinct population of bacteria according to their infectiouness statut: Hyperinfectious bacteria (HI) and latent or Non-Hyperinfectious bacteria (Non-HI) denoted by B H and B L , respectively. The carrying capacity of V. Cholerae in environ- ment is denoted by K. For the population of hyperinfectious bacteria, we assume that the growth rate rB L could limited due to a plausible limitation of shared resources in the medium where they live 1 -B H ∕K , that is, the growth rate in the HI compartment is modeled by the logistic term rB L (1 -B H ∕K) . Hyperinfectious bacteria become Non- Hyperinfectious bacteria at constant rate . Hyperinfectious and Non-Hyperinfectious bacteria die at constant rates H and L , respectively. Then, the dynamics of the population of V. Cholerae in a closed environment can be modeled by the following system of differential equations:

Biological significance and values of parameters of model (1) are given in Table 1.

Theoretical Analysis

We first study the basic properties of the model's solutions (1), which are essential in the proofs of stability results. The right-hand side of model ( 1) is a continuously differentiable map (C 1 ) . Model system (1) is biologically well posed: if the initial data are in ℝ 2 + , then the solution stays in ℝ 2 + : indeed, it is straightforward to show that the compact:

(1)

⎧ ⎪ ⎨ ⎪ ⎩ ḂH = rB L 1 - B H K -( H + )B H , ḂL = B H -L B L .
( To derive some qualitative results from our model (1), we compute equilibria and study their stabilities. Let be a threshold that represents the mean number of Non-HI bacteria produced by one Non-HI bacteria over its lifespan:

) Ω = (B H , B L ) ∈ ℝ 2 + ∶ B H ≤ K, B L ≤ L K , 2
The threshold (3) is sometimes called the basic offspring number. We have the following result.

Lemma 1 : Model system (1) has two possible equilibria:

(i) a trivial equilibrium E0 = (0, 0); (ii) a positive equilibrium E * = (B * H , B * L ) when > 1 defined as follows:
The stability of equilibria of model ( 1) is summarized in Proposition 1.

Proposition 1 Assume that the initial condition (B H (0), B L (0)) ∈ Ω .

1 When < 1 , the trivial equilibrium E0 is globally asymptotically stable (GAS), which implies that the population of V. Cholerae will go until extinction, whatever the initial population. 2 When > 1 , the trivial equilibrium is unstable and the positive equilibrium E * is globally asymptotically stable, that is the population of V. cholerae persists in the environment.

Proof To prove Proposition 1, we will use some results related to cooperative systems [START_REF] Smith | Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems[END_REF]). These kind of systems are very important and well-known in Biology and Ecology, and have been studied extensively, see [START_REF] Smith | Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems[END_REF]. For reader's convenience, we recall the following general definition. Consider a n-dimensional autonomous differential system:

where f is a given vector function, i.e. f = (f ) i , with f i ∶ ℝ n ⟶ ℝ. Then, we recall the following

(3) = r ( H + ) L . (4) B * H = 1 - 1 K and B * L = L 1 - 1 K. (5) ̇x = f (x), x(0) = x 0 , Definition 1 (Smith 2008) System (5) is called cooperative if for every i, j ∈ {1, 2, 3, … , n} such that i ≠ j the function f i (x 1 , … , x n ) is monotone increasing with respect to x j .
It is easy to verify that model ( 1) is a cooperative system in Ω . In the following, we consider the basins of attraction represented as n-dimensional intervals: given

a, b ∈ ℝ n with a ≤ b then [a, b] = {x ∈ ℝ n ∶ a ≤ x ≤ b} .
To show GAS, we will use the following theorem:

Theorem 1 (Anguelov et al. 2012) Let a, b ∈ Ω be such that a < b , ]a;b[⊂ Ω and f (b) ≤ 0 ≤ f (a)
. Then, system (5) defines a (positive) dynamical system on [a, b]. Moreover, if [a, b] contains a unique equilibrium p then p is globally asymptotically stable on [a, b].

We are now be able to prove Proposition 1.

• When ≤ 1 , model (1) has only the trivial equilibrium E0 . Thus, taking a = E0 and b = K, L K we have that f (a) = 0 and f (b) ≤ 0 . It then follows from The- orem 6 [START_REF] Anguelov | Mathematical modeling of sterile insect technology for control of Anopheles mosquito[END_REF], that the trivial E0 is globally asymptotically stable on [a, b], hence on Ω , when < 1 • When > 1 , there exists > 0 such that > 1 + . Let H sufficiently small such that:

H < and L = (1+ ) L H < . Let a = (H , L ) T .
Then, from the right-hand side of model ( 1) and the fact that > 1 + and K ≫ 1 , we can deduce that Hence, it follows from Theorem 6 [START_REF] Anguelov | Mathematical modeling of sterile insect technology for control of Anopheles mosquito[END_REF]) that the nontrivial equilibrium

E * = (B * H , B * L ) is globally asymptotically stable on [a , b]
. Since a can be selected to be smaller than any x > 0 , we have that E * is asymptotically stable on Ω with basin of attraction Ω ′ = Ω∖ E 0 which implies that E 0 is unsta- ble. This concludes the proof. □

Sensitivity Analysis

Let us recall that the basic offspring number is defined as follows:

(6) f (a ) ≥ ⎛ ⎜ ⎜ ⎝ ( + L ) 1 - 1 + H 1 - 1 + H ⎞ ⎟ ⎟ ⎠ ≥ 0. (7) = r ( H + ) L .
The relevance of the basic offspring number is due to the Proposition 1. In fact, it is the key parameter for the existence of a non-trivial equilibrium. It is important to know the relative impact of different parameters of the model (1) on . To do this, we estimate the sensitivity index of with respect to a parameter p is p . Another measure is the elasticity index p [START_REF] Chitnis | Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model[END_REF]) that measures the relative change of with respect to p, defined as follows:

In fact, straightforward computations lead to the following result:

Clearly r and L have the strongest impact on . However, it is slightly different when we focus on the variables B H and B L . Now, we perform the sensitivity analy- sis in order to detect the most sensitive parameters, that is the parameters that most influence the output variable of the model. This can help predict the effect of each parameter on the model and classify them according to their degree of sensitivity. To do so, we use the eFast 1 sensitivity method which is described in [START_REF] Marino | A methodology for performing global uncertainty and sensitivity analysis in systems biology[END_REF]. Figure 1 presents the sensitivity analysis obtained with the eFast method that highlights first-order effects (main effects) and total effects (main and all interaction effects) of the parameters on the Model Outputs, B H and B L . According to the result obtained in Fig. 1, the parameter L has the strongest impact both on B H and B L . This means that elimination of Latent Bacteria (Non-HI) have the most impact bacteria density. These kind of bacteria are usually found in wetlands, rivers, sewage

Υ p = p . p . ( (8)
) Υ r = 1, Υ L = -1, Υ = H H + and Υ H = - H H + 9
. and water sources contaminated by human excrements [START_REF] Xu | Survival and viability of non-culturable Escherichia coli and Vibrio Cholerae in the estuarine and marine environment[END_REF]. Therefore, sensitivity analysis results suggest that effective control strategy to reduce bacteria density would be the implementation of a serious sanitation campaigns of aquatic environments (wetlands, rivers, sewage and water sources contaminated by human excrements) by using suitable products during the periods of growth of aquatic reservoirs and also during cholera epidemic periods. These sanitation actions could be initiated and organized by the public health agents. In Haïti for example, the new solution implemented by United Nation Organization to eradicate Cholera concerns improving access to drinking water and sanitation (Guillaume et al. 2012).

A V. Cholerae Model in a Periodic Environment

The Model

Climatic factors such as water temperature may explain the seasonality of the disease either by exerting a direct influence on the bacterial reservoir capacity (the value of K) or even on parameters affecting their growing (r), their survival ( H and L ) and their metabolism ( ) [START_REF] Colwell | Global climate and infectious disease: the cholera paradigm[END_REF][START_REF] Islam | Association of Vibrio Cholerae O1 with the cyanobacterium, Anabaena sp., elucidated by polymerase chain reaction and transmission electron microscopy[END_REF]. But the bacterial reservoir capacity (K) have an important role on persistence and resurgences of cholera disease. In fact bacteria are strongly associated with both phytoplanktonous and zooplanktonous organisms forming commensal or symbiotic relationships. When V. Cholerae bacteria are associated with such reservoirs, they generally have an advanced survival [START_REF] Xu | Survival and viability of non-culturable Escherichia coli and Vibrio Cholerae in the estuarine and marine environment[END_REF]. Their environmental adaptation will lead to metabolic and phenotypic changes that will condition their survival; This can be compared to a phenomenon of dormancy. Cells are considered "viable but non-culturable" (VNC) because the main effect of this change is the loss of ability live on bacteriological culture media [START_REF] Roszak | Survival strategies of bacteria in the natural environment[END_REF]. Changing for a cultivable state is possibleparticularly when the factors that cause stress become favorable to the development and growth of the bacterial population. The variation of these climatic factors are such all the parameters excepted C must be time-dependent and strictly bounded by positive values [START_REF] Colwell | Global climate and infectious disease: the cholera paradigm[END_REF][START_REF] Islam | Association of Vibrio Cholerae O1 with the cyanobacterium, Anabaena sp., elucidated by polymerase chain reaction and transmission electron microscopy[END_REF][START_REF] Xu | Survival and viability of non-culturable Escherichia coli and Vibrio Cholerae in the estuarine and marine environment[END_REF][START_REF] Roszak | Survival strategies of bacteria in the natural environment[END_REF][START_REF] Huq | Influence of water temperature, salinity, and pH on survival and growth of toxigenic Vibrio Cholerae serovar O1 associated with live copepods in a laboratory microcosms[END_REF]. The model (1) becomes :

where C is a minimal constant bacterial capacity value. Notice that the model in ( 10) belongs to the family of periodic cooperative systems with a concave nonlinearity [START_REF] Smith | Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems[END_REF][START_REF] Jifa | The algebraïc criteria for the asymptotic behavior of cooperative systems with concave nonlinearities[END_REF]. Let us define a periodic cooperative system of n differential equations:

(10)

⎧ ⎪ ⎨ ⎪ ⎩ ḂH = r(t)B L (1 - B H K(t) + C ) -( H (t) + (t))B H , ḂL = (t)B H -L (t)B L , (11) ̇x = f (t, x),
where f ∶ ℝ + × ℝ n ⟶ ℝ n , with concave non-linearities. Let A(t) be a n × n con- tinuous matrix in ℝ , -periodic in t, we denote: and set Then, Set P 0 and Q 0 in ℝ + . For reader's convenience, let us recall Theorem 5.5, p. 230 of [START_REF] Smith | Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems[END_REF].

Theorem 2 Let F(t, x) be continuous in ℝ × [0, P 0 ] × [0, Q 0 ] , -periodic in t for a fixed x and assume D x F(t, x) exists and is continuous in ℝ × [0, P 0 ] × [0, Q 0 ] . Assume also that all solutions are bounded in [0, P 0 ] × [0, Q 0 ] and F(t, 0) = 0 . Assume (a) F i x j ≥ 0, (t, x) ∈ ℝ × [0, P 0 ] × [0, Q 0 ], (b) A(t) = D x F(t, 0) is irreducible for any t ∈ ℝ, (c) if 0 < x < y , then D x F(t, x) > D x F(t, y).
Then 1. If all principal minors of -Ā are non-negative, then lim x⟶∞ x(t) = 0 for every solution of system (11) in [0, P 0 ] × [0, Q 0 ] 2. If -A has at least one negative principal minor, then system (11) possesses a unique positive -periodic solution which attracts all initial conditions in [0,

P 0 ] × [0, Q 0 ].
In this case, we can take

P 0 = Q 0 = K + C . Let x = (B H , B L ) ∈ [0, P 0 ] × [0, Q 0 ] . Without losing to the generality we can suppose B H ≤ K + C and B L ≤ K + C . One has (12) āi,j = max 0≤t≤ a i,j (t)
and a i,j = min 0≤t≤ a i,j (t),

(13) Ā = (ā i,j ) and A = (a i,j ). (14) A ≤ A(t) ≤ Ā for 0 ≤ t ≤ . F(t, x) = ⎛ ⎜ ⎜ ⎝ r(t)B L (1 - B H K(t) + C ) -( H (t) + (t))B H (t)B H -L (t)B L ⎞ ⎟ ⎟ ⎠ .
Clearly, F is continuous and -periodic. We also have F(t, 0) ≡ 0 . Moreover, which implies that D x F(t, x) is continuous and (t, x) . Finally, we have Clearly, A is irreducible for all t. One notes p min ≤ p(t) ≤ p max for all t ≥ 0 and for p = r, , H , L , K . Using the fact that all time-dependent parameters have positive lower and upper bounds, we can deduce that First, according to Theorem 2, we need to study all principal minors of -Ā . Diago- nal terms are positive, and a straightforward computation shows that det(-Ā) ≥ the following inequality is verified From Theorem 2, since all diagonal terms of -A are positive, it remains to verify that det(-A) < 0 . A simple computation gives Then, applying Theorem 2, we have the following result.

F i x j ≥ 0 for all (B H , B L ) ∈ ℝ 2 + . When 0 < x < y , we have obviously D x F(t, y) > D x F

Theorem 3 Let

Now, let us assume all parameters to be time-dependent. Then, Theorem 4 Assume that all parameters are constant, except. Then, (i) If > 1 , model (10) has a unique periodic solution which attracts all initial conditions in Ω. (ii) if ≤ 1 , the solution of model (10) converges to the trivial equilibrium E 0 .

(15) D x F(t, x) = ⎛ ⎜ ⎜ ⎝ -( H (t) + (t)) - r(t)B L K(t) + C r(t) 1 - B H K(t) + C (t) -L (t) ⎞ ⎟ ⎟ ⎠ , ( 16
) A = D x F(t, 0) = -( H (t) + (t)) r(t) (t) -L (t) . (17)
We stress that this result is similar to the one with constant parameters, except that the nontrivial equilibrium is now periodic.

Numerical Simulation

Herein, numerical simulations are presented to illustrate Theorem 4. We used the parameter values given in Table 1. The carrying capacity K of V. Cholerae is 10) using four various initial conditions in ℝ 2 + when r = 1∕40 day -1 (so that = 0.5 ). All other parameter values are given in Table 1 periodic, while all other parameter constant. Several type of functions for K could be chosen. Among them, we choose one of simplest:

(19) (t) = r(t) (t) ( H (t) + (t)) L (t) , (
The function K(t) is a periodic function with a period T = 365 days , K mean = 10 6 cell/ml and k = 0.99 . The Fig. 2 illustrates the model (10) dynamics when < 1 . As expected, the population decays until extinction. However, when > 1 bacteria are maintained in the environment with a periodic density as shown in Fig. 3.

Another simulation (see Fig. 4) illustrates how dynamics depend on climatic factors that affect a direct influence on either the bacteria reservoir capacities (K mean ) or the bacteria mortality ( L ).

Figure 4 shows density variations of bacteria when K is considered periodic and when K mean and L have two different values. The Fig. 4a shows clearly how reser- voirs can play important role on persistence of bacteria in environment. In this work, we proposed a model of V. Cholerae populations growing in a closed environment and qualitatively analyzed their dynamics. The aim was to evaluate the impact of factors on the growth of V. Cholerae populations. We first considered a V. Cholerae model in a closed environment with all parameters constant. The theoretical analysis of the model has been presented. The sensitivity analysis of the model has been also investigated. We found that the best control action would be preventive sanitation. So it is recommended to use suitable products against V. Cholerae during the periods of growth of aquatic reservoirs and also during cholera epidemic periods. After, the climatic variations have been incorporated in this V. cholera model by using periodic functions. Critical periods generally correspond to periods where climatic and environmental factors are favorable to the growth and to the propagation of aquatic reservoirs. We presented a qualitative analysis of the model. More precisely, existence, positivity and boundedness of solutions are presented. Threshold and equilibria are obtained and global stabilities examined. We shown that the model can present one or two biologically meaningful equilibria when the basic offspring number is lower or greater than 1 respectively. Equilibrium stabilities were examined. Numerical simulations have been presented to illustrate and confirm theoretical results. Through numerical simulations, we found that the aquatic reservoirs are playing a significant role among the factors explaining the causes of endemicity of these disease.

Fig. 1

 1 Fig. 1 eFast sensitivity analysis for the a Hyperinfectious and b Non-Hyperinfectious bacteria. White bar: first-order effects; sum of white and grey bars: total effect

  ( Hmin + min ) Lmin . (i) If min > 1 , model (10) has a unique periodic which attracts all initial conditions in Ω. (ii) If max ≤ 1 , the solution of model (10) converges to the trivial equilibrium E 0 . Indeed, the above result is related to the time dependent basic offspring number (t): which satisfies Assume now, that all parameters are constant except K, then -A = -Ā and thus min = max , then we have the following result.

Fig. 2

 2 Fig. 2 Simulation of model (10) using four various initial conditions in ℝ 2+ when r = 1∕40 day -1 (so that = 0.5 ). All other parameter values are given inTable 1

Fig. 3

 3 Fig.3Simulation of model (10) using four various initial conditions in ℝ 2 + when r = 1∕7 day -1 (so that = 2.8571 ). All other parameter values are given inTable 1

Fig. 4

 4 Fig. 4 Simulation of model (10) for two different values of K mean and L . All other parameter values are given inTable 1

Table 1

 1 Numerical values for the parameters of model (1)

	Definition	Symbol	Estimated	Source
	Growth rate of V. Cholerae	r	1∕7 day -1	Rosa et al. (2003)
	Daily death-rate of HI bacteria	H	1∕60 day -1	Kaper and Morris (1995)
	Daily death-rate of Non-HI bacteria	L	1∕30 day -1	Tudor and Strati (1977)
	Transition rate from HI to Non-HI state		1∕30 day -1	Yibeltal (2009)
	V. Cholerae capacity of the environment	K	10 6 cell∕ml	Assumed