Minna Lanz
email: minna.lanz@tut.fi

Matti Majuri
email: matti.majuri@tut.fi

Reijo Tuokko
email: reijo.tuokko@tut.fi

Information flows in Future Advanced Manufacturing Ecosystems

Keywords: knowledge management, semantic information, tacit knowledge, systems thinking, social capital, know-how

Manufacturing is the backbone of each and every society, and in order for society to sustain in long run the manufacturing has to be sustainable as well. The sustainability in the field of manufacturing has traditionally been discussed in a sense of operational efficiency and environmental metrics. Rarely the link between individuals working in the company and the efficiency of operations has been established and discussed deeply. This link is information flow that combines both tacit and formal information in a dynamically changing socio-technical environment. In this paper the information flow between individuals in different levels of company hierarchy is utilized as the observation baseline. This paper discusses the information flow within a company and outlines sociotechnical challenges needed to solve in order to realize future Advanced Manufacturing Ecosystems.

Introduction

Regarding the information flow two major challenges are identified: content of information and dynamic environment where the information is temporally valid. Today's design and operation control systems produce a variety of data. However, the sheer amount of data that lacks the structure, connectivity and intent cannot be utilized efficiently later on. There is ever the greater need to transfer this unstructured mass of bits and bytes to meaningful information and to human understandable knowledge, as it is understood in the industry. However there are major challenges in understanding the actual content of information flow. Today the information flow should connect the different design departments and activities to be part of the life-time information of a product including its processes and services. For example modern computer-aided design (CAD) tools has made it easier and faster to design products made up of more and more complicated and intricate parts. Without sufficient experience or guidance, todays designer runs the risk of developing product designs that are unnecessarily difficult to produce [START_REF] Pulkkinen | PLM state of the practice and future challenges in globally networked manufacturing companies[END_REF], [START_REF] Lohtander | Collecting manufacturing information in a global distributed manufacturing environment[END_REF]. Another example of disruptions in information flow and/or understanding its content is vividly explained by Redman [START_REF] Redman | The impact of poor data quality on the typical enterprise[END_REF]. He stated that quality errors in manufacturing relating to low quality data may raise costs up to 30%. Furthermore, Redmans study indicates that 8-12% revenue losses occur because of data quality issues. Informally, 40-60% of the expenses in service sector may be consumed as a result of poor data.

It is recognized that environment evolves and changes over the time and the system itself evolves, sometimes unpredictably, during its lifecycle. In recent years changes in the environment have become faster, stronger and harder to predict than ever before. Same time the manufacturing is becoming more decentralized causing challenges in the network level operations management. In order to operate in such dynamic and complex environment best of the both worlds; human intelligence and computers computing power are needed to be combined more sufficient manner. As the unpredictable changes occur, detailed simulations of large systems will become unreliable. Algorithms cannot solve the problems that arise, thus applying human intelligence to solve complex problems or choosing optimal scenario becomes highly important. Paradoxically the current operation strategy has forced human to be the processing unit while computer attempts to make intelligent decision over relatively vague input information. This strategy may improve the effectiveness of manufacturing in limited range of situations but on the other hand it definitely reduces the flexibility and agility of the manufacturing system.

Current engineering and production management methodologies and tools have been created for the old markets where the mass-customization was a norm. The IT-tools for these solutions offer proprietary services and interfaces. This paralyses healthy ecosystem growth, emergence of new start-ups, development and neglects the needs of agile and innovative SMEs. Contemporary tools and services, such as cloud-computing, emphasize on technology in favor of actual industry specific business needs. As a consequence ICT-centric solutions tend to help only in communication, but transfer existing (tedious) work practices in the new media. The current solutions are largely developed from the one perspective respecting the constraints from legacy ICT, not from the modern engineering business processes.

The challenges mentioned above are no doubt look very technical issues. However, technology itself cannot solve the issues. It is very likely that the technology-oriented approach will not be sufficient for the future advanced manufacturing ecosystems. This paper discusses of technical information management, tacit knowledge and system interoperation and relations. By understanding the technical possibilities, social know-how and how human-machine systems interoperate the advanced manufacturing ecosystems can emerge.

2 Considering the surrounding systems

Semantics

The main problem is that the models and documents created with different design and operation management systems are that the documents are meaningless from the knowledge point of view for systems other than their authoring system. The second problem is that the Design Intent is a concept that is not captured by the modern design systems. This raises even bigger challenges for the future manufacturing systems that require this information to provide autonomous reasoning. A case study done in Finnish manufacturing sector highlighted the challenge in capturing the design intent and managing the information flow among various stakeholders [START_REF] Jarvenpaa | Studying the Information Sources and Flows in a Company -Support for the Development of New Intelligent Systems[END_REF]. The case study was conducted in a large and globally operating manufacturing company, which possessed large network of subcontractors. The study revealed, that workers personality, human communication, and bureaucracy have a significant effect on what actually gets saved into the information systems, and furthermore what can be retrieved from the system later on. Furthermore the study suggested that the design intent was lost very early from the information content and due to that several important issues were constantly ignored. Different people also have very different ways of doing and understanding things, which may lead to misunderstandings in cooperation.

Knowledge builds on data and/or information that is organized and processed to convey understanding, experience, accumulated learning and expertise. In order to enhance the effectiveness, agility and reactivity of future the manufacturing systems it is needed to look at the detail level; more precisely, where the numbers are coming and how valid those are in given context. On the product design perspective this means that information flow content actually carries the design intent and it can be understood in a context of use. In the manufacturing operations level the simulation results are tied to the given context and are valid in that context. In the recent years industrial standards are being defined in a more computer-readable form, most notably since the emergence of eXtensive Mark-Up Language (XML) -based formats and computing power. XML as a language has number of advantages for developers and implementers. Structural language allows the compilation of specifications, creation of datasets and certain kinds of testing can be performed more easily. However, along the way XML markups have been used as a substitute for modeling the information -a dangerous shortcut that only works in communities that already share a common understanding of the meaning and usage of terms [START_REF] Lanz | Engineering Intelligence -Product-Service Concepts and Requirements in Industry[END_REF], [START_REF] Ray | Manufacturing Interoperability[END_REF]. Knowledge modeling and representation always reflect the view of the modelers on the domain to be modeled, although the ideal should be to initially remain as independent as possible of any purpose or application [START_REF] Spyns | Ontology Engineering and (Digital) Business Ecosystems: a case for a Pragmatic Web[END_REF]. More and more information will be retrieved from diverse sources originating from different circumstances and application. Therefore the resultant data are inherently built on different languages and representations that will put high demand on semantic integration solutions. Also, information will be fetched from diverse locations (over the global networks), which will be grounds for extensive use of web technologies. These two basic requirements (semantic integration, access over web) speaks for the adoption of semantic web technologies to describe and integrate (or perhaps rather inter-link) digital content on built environment. Semantic information enables the creation of built-in intelligence into versatile products. Real intelligent functions will be possible to be created based on semantic rea-soning algorithms. The difference to "glued" intelligence is significant, because the product and the digital services related to products can utilize more detailed and specific information than traditionally [START_REF] Lanz | Engineering Intelligence -Product-Service Concepts and Requirements in Industry[END_REF].

Systems thinking

Centralized systems based on hierarchical data share are giving a way to globally distributed production networks. These networks need to adapt rapidly to the production changes. Due to the resulting larger complexity of heterogeneous interacting components the system can no longer be only guided and controlled based on a human operator. Predicting the future state of dynamic system based on the static and often inaccurate or obsolete models of manufacturing system's state is not reliable or sufficient [START_REF] Lanz | Set-up and first steps on capturing of realistic resource characteristics of an intelligent manufacturing environment[END_REF].

According to International Council of Systems Engineering (INCOSE) there is a growing awareness that the revised systems engineering approach is becoming a key to successfully design, develop and sustain the highly complex systems. However the field suffers from the lack of unified principles and models that support a wide range of domains [START_REF] Anon | International Council on Systems Engineering report[END_REF]. A shift away from a generic definition and application of systems engineering to a more specifically defined and precise application of systems engineering in diverse domains can thus be expected in the future. The future systems engineering environment will also fully support life cycle perspectives. In many respects, the future of systems engineering can be said to be model-based. A key driver will be the continued evolution of complex, intelligent, global systems that exceed the ability of the humans who design them to comprehend and control all aspects of the systems they are creating. The role of modeling will mature to respond to this need.

Social Capital

The role of humans in the Factories of the Future will be defined through the stimulation of working environments and conditions that bring about new forms of collaboration and interaction processes among humans, machines and the manufacturing technologies involved. Sustainability of the human capital, in Europe, is subject to the global economic conditions. Age and employment systems need to be adapted, while the societal impact of manufacturing on local environments ought to be taken into account, in terms of energy demands, life quality, natural resources and safety.

Lately, following to the well-known trends of globalization, transformation from vertical value chains to open value networks, web 2.0 and further developments, crowd-sourcing, dematerialization, virtualization etc., management theory has developed new approaches more coherent with the emergent business and social landscape, leveraging the human factor along with technology enablers. However, as the development advances with steady leaps it also causes pressures towards workforce. This pressure comes from dynamic and fast changes in organizational level, new technical solutions requiring faster learning curves and changes in global economy. In the recent years, analysis on employees wellbeing highlights that, more now than in the past, individuals have to deal with different objectives, desires, expectations and responsibilities, which can be clustered in two main categories of work and life. Bourne et al [START_REF] Bourne | Embracing the whole individual: Advantages of a dual-centric perspective of work and life[END_REF] stated that "dualcentric experience more overall satisfaction, greater work-life balance, and less emotional exhaustion", and that organizations should take care of employees as whole individuals in order to enhance their wellbeing.

According to EFFRA [START_REF]Factories of the Future 2020 Roadmap[END_REF] manufacturing is evolving from being perceived as a production-centric operation to a human-centric business with greater emphasis on workers, suppliers and customers in-the-loop. ActionPlanT [START_REF] Majumdar | ICT FOR MANUFACTURING -The ActionPlanT Vision for Manufacturing 2.0, Factories of the Future 2020 Roadmap[END_REF] envisions workers and managers alike given more opportunity for continuous development of skills and competences through novel knowledge delivery mechanisms. The human-centricity paradigms involve inclusion for young and elderly people Future enterprises will not only be better equipped for transferring skills to a new generation of workers but also proficient in assisting older workers with better user interfaces, intuitive user-experience-driven workflows and other aids, such as mobile and service robots.

Technological Advances

According to European Commission communication [START_REF]Digital Ecosystems -the enabling technologies and paradigms for fostering endogenous local development, local capacity building and knowledge sharing processes providing tailored and personalized ICT services to citizens and business networks[END_REF] a digital ecosystem is a self-organizing digital infrastructure aimed at creating a digital environment for networked organizations that supports the cooperation, the knowledge sharing, the development of open and adaptive technologies and evolutionary business models. Boley and Chang [START_REF] Boley | Digital Ecosystems: Principles and Semantics[END_REF] proposed definition for a digital ecosystem by terms "open, loosely coupled, domain clustered, demand-driven, self-organizing agent environment", where each agent of each species is proactive and responsive regarding its own benefit/profit but is also responsible to its system. The digital ecosystem approach transposes the concepts to the digital world, reproducing the desirable mechanisms of natural ecosystems.

As several interacting natural ecosystems exist, several digital ecosystems exists due to differentiation and the development of endemic product and services tailored to specific local needs. The key enabling technologies developed within the digital ecosystem research aim at providing a knowledge-and serviceoriented infrastructure that supports the spontaneous composition, distribution, evolution and adaptation of ICT-based services. This platform should allow the SME software industry to independently develop (and disseminate on the network) services and software components which will be composed forming complex, evolutive and adopted solutions. These technologies allow the spontaneous development and the cooperative provision of services and solutions, without the need for any keystone player, central coordination or central point of control/failure.

Requirements for the future Advanced Manufacturing Ecosystems

The future advanced manufacturing ecosystems require new and more collaborative approaches, where the technical advantages are understood and accepted, where the social capital is appreciated and utilized and where the interactions of a whole system can be visualized and understood.

As the construction and development of an increasingly artifactual environment continue, little thought has been given to the coherence or compatibility of knowledge between different scientific disciplines. As a result, incompatibilities and contradictions have arisen within that environment, and, most ominously, between it and the surrounding natural environment [START_REF] Yoshikawa | Synthesiology as sustainability science[END_REF]. [START_REF] Yoshikawa | Synthesiology as sustainability science[END_REF] introduced a term synthesiology. Synthesiology is a term for the theory of integration of scientific and technological knowledge from different disciplines with the needs of society.

McDonald (2011)has recommended that management should evolve to foster trust and teamwork; to create a fluid, flexible, customized work community environment; to decompose vertical organizational structures towards inter-intra organizational networks with emergent new roles for facilitators and brokers; to create new exciting, entertaining and challenging workplaces for young people [START_REF] Mcdonald | It's time for management version 2.0: Six forces redefining the future of modern management[END_REF]. [START_REF] Kira | Socially Sustainable Work Organizations: A Chaordic Systems Approach[END_REF] emphasize that socially sustainable work organizations are able to both repeat accustomed and devise innovative solutions, and for this reason it is crucial for them to embrace individual stakeholders complexities and encourage them to cultivate it, while at the same time maintaining a holistic view of the organization [START_REF] Kira | Socially Sustainable Work Organizations: A Chaordic Systems Approach[END_REF]. Furthermore, Kira et al. [START_REF] Kira | Socially Sustainable Work Organizations: A Chaordic Systems Approach[END_REF] suggest that a higher level of collaborative shaping of work (with employee involvement) is highly conducive to sustainable work abilities and development of human resources.

As this paper focuses more on the information flow management in future Manufacturing Ecosystems, the collaboration, motivation and self-organizing interaction between individuals becomes highly important. Fischer [START_REF] Fischer | Supporting the evolution of design artifacts with representations of context and intent[END_REF] summarized well that in interpersonal communication the design intent is mostly casually conveyed and understood based on history, shared experience and circumstances. However, in this modern operation environment each stakeholder has a different context and different understanding of the problem. Communication breakdown occurs when the stakeholders have only little shared context. The message, design intent, gets lost in translation.

Summary

All in all the future interoperability among different stakeholders requires more multi-sectoral approach in order to tackle the arisen challenges. The efficient knowledge capture, processing, utilization and reuse as integral part of future product-services requires holistic approach, that includes understanding from following sector: 1. Social and economic sciences (network economics, community buildings, diffusion of knowledge and practices, legal aspects, business and organizational models), 2. System theory, self-organization of complex systems and epistemology 3. Computer science (mainly network architectures, semantic interoperability and formal languages) [START_REF]Digital Ecosystems -the enabling technologies and paradigms for fostering endogenous local development, local capacity building and knowledge sharing processes providing tailored and personalized ICT services to citizens and business networks[END_REF].

Only by adopting the holistic approach in design, operation management and manufacturing the new Advanced Manufacturing Ecosystems can emerge.


Joseph Gril

Delphine Jullien

David Hunt

Compression set and cupping of painted wooden panels

Factors that need to be considered to explain the observed cupping pattern of panel paintings are reviewed. The anisotropic shrinkage and the action of the paint layer, both a reinforcement and a moisture barrier are the most obvious ones, but the progressive contraction of ancient wood is usually invoked. Although the aging process of wood could be responsible for some of it, several mechanical processes may lead to compression set: mechano-sorption, microbuckling, or even the recovery of locked-in growth strains.

Introduction

Wooden boards used to support historical paintings often exhibit cupping. The origin of this unwanted curvature is not completely elucidated, as sev-eral factors could be involved and the condition of conservation of the pan-els through the centuries is mostly unknown. According to information giv-en by conservators, panel paintings that were not heavily restrained through craddling or similar reinforcement, tend to cup toward the observ-er, meaning that the central part of the board moved frontward relative to the lateral parts (Figure 1). This paper aims at discussing various factors that need to be considered to explain this phenomenon, including the so-called "compression set", a per-manent contraction made evident, for instance, by the gap between an his-torical panel and its original frame.

Anisotropic moisture expansion

Among wood technologists the most well-known cause of board cupping is the anisotropy of moisture shrinkage ratios in the RT plane (Figure 2). There is a simple stress-free solution describing the deformation of an homogeneous wooden piece subjected to a slow humidity change that didn't induce any moisture gradient:

X(x,y) = (1 + α R ).x -(α T -α R ).atan(y/x).y (1) Y(x,y) = (1 + α R ).y + (α T -α R )
.atan(y/x).x where X, Y is the deformed position of a point initially at position x, y and α the moisture expansion having homogeneous polar coordinates centred on the pith (constant values of α R and α T , α RT = 0…). This expression is only valid when the piece does not contain the pith. It can be verified that this solution fulfils all the conditions of a mechanical problem for the fields of stress (σ), strain (ε) and displacement (u): -static equilibrium: σ = 0 => div σ = 0 -kinematic compatibility: (grad u + grad t u)/2 = ε -free boundary conditions -hygro-elastic constitutive equations: ε = α => σ = C(ε-α) = 0 where C is the rigidity tensor. The direction of the cupping depends on the sign of moisture change and ring orientation. We can reasonably assume that when a painter ordered a panel to a craftman, he expected the panel to be flat and air-dry before he applied the ground layer. A difference of humidity between the artist's time and nowadays is the most obvious explanation for the observed cupping. The central deflection d of a board of width 2L aligned parallel to y axis at distance h from the pith can be derived from (1):

d = -(α T -α R ).L.atan(L/h) (2)
the solution being only valid for h exceeding the half-thickness of the board.

As an order of magnitude, a moisture content reduction of 5%, such as resulting from a drop of average RH from 75% to 50% at 20°C, for a poplar flatsawn board 40 cm wide cut 5 cm from the pith, would produce a deflection of 2 mm, independently of the panel thickness (Figure 3). The amount of this effect is compatible with the cupping of ancient panel paintings. How-ever, if it was the only phenomenon involved, the cupping direction should depend on the ring orientation, contrarily to the fact illustrated in Figure 1.

Asymmetry

The next cause of cupping for a panel is the asymmetry between both faces. It is either caused by different boundary conditions or by heterogeneous mechanical behaviour. Both factors occur in panels as soon as they are painted on a single face, since the paint layer reduces the speed of exchanges with the surrounding atmosphere, while contributing mechanically to the whole structure. If the material possessed a simple hygro-elastic behaviour, the panel would become flat again once the new equilibrium conditions are reached -or attain the state described in the previous section if the effect of ring curvature is taken into account. However, the hygro-mechanical behaviour of wood is characterised by a very significant coupling, called mechanosorption. Due to mechano-sorptive coupling, permanent deformations are locked in wood as soon as it subjected to mechanical stress while the moisture content varies. This is one aspect of the "set" that will be discussed in the last section. As a result, a non-zero cupping is generated. In the long term, for a panel subjected to a complex hygrothermal history, both tension set and compression set will be generated in turn. Although they are likely to compensate each other, in some circumstances compression may occur at higher moisture contents than the tension and result in a dominance of the compression part (Hunt and Jeronimidis 2016).

The "paint layer" is a complex composite of various underground layers and paints, sometimes reinforced by canvas. Its physical and mechanical properties may change through the history of the artwork. In addition to its role as moisture barrier, it contributes mechanically to the curvature of the panel by having a different rigidity and moisture expansion than the wood of the panel. According to the findings of [START_REF] Aurand | A re-evaluation of the mechanisms underlying relative humidity induced fatigue damage in panel paintings: the case of 14th century italian preparatory layers[END_REF] and [START_REF] Gauvin | Etude expérimentale et numérique du comportement hygromécanique d'un panneau de bois: application à la conservation des tableaux peints sur bois du patrimoine[END_REF] the elastic modulus of a few typical underground layers was 2 to 10 times higher than that of poplar in T direction. Thus even a thin paint layer may act as significant reinforcement of the wooden panel and prevent to some extent the deformation of the painted face. Moreovover, owing to the viscoleastic and mechanosorptive behaviour of wood, its effective rigidity under long term loading in a variable climate is considerably reduced so that the reinforcing action of the paint layer may even become dominant.

Aging

Thus any process inducing a deformation of the wooden panel transversally to the fibres is likely to induce a cupping. In case of a negative strain, a contraction, the cupping would indeed occur as shown in Figure 1.

A lower level of average humidity in the present storage conditions was evoked as a possible factor in the preceding section. Combined with the asymmetry imposed by the paint layer, it would indeed be able to explain the observed cupping of historical panels. To verify this interpretation, the direction and amount of cupping -and if possible its precise profile -should be recorded on a large number of panel paintings where the ring structure has been documented, so that the quantitative relation between cupping and estimated pith position can be checked. If the hypothesis holds true, the 3rd pattern in Figure 1 should provoke the most cupping.

As an alternative to drier storage conditions, hygroscopic aging could be considered to explain a reduction of average moisture content. [START_REF] Yokoyama | Mechanical characteristics of aged Hinoki wood from Japanese historical buildings[END_REF] observed a reduction by 2% of the equilibrium moisture con-tent in given air-dry conditions, for hinoki wood used for Japanese temples 1500 years ago. Garcia Esteban et al. ( 2005) imposed 5 cycles between 90%RH at 50°C and oven dry state to several hardwood and softwood specimens, and obtained a significant reduction of equilibrium moisture content but at high RH only. In recent years, many attempts to reproduce the features of naturally aged wood through mild -typically, below 150°C -thermal or hygrothermal treatment have been made [START_REF] Kránitz | Effects of aging on wood: a literature review[END_REF]). However Obataya (2016) explained that the observed changes are partly reversible by exposure to high humidity. This suggest that some of the moisture reduction is due to structural changes, such as the formation of permanent hydrogen (hornification), or to the dissolution of cell-wall components such as hemicelluloses: the former being potentially recoverable but not the latter. In any case, in the case of wood used in historical objects, a lower equilibrium moisture content can be expected, compared to recently processes wood. Although dimensions of ancient object cannot be directly compared to their original ones, we can reasonably expect that a lowering of the moisture content results in a contraction. This is at least the case with thermal treated wood, which has many common features with naturally aged wood.

Compression set

Wood is a highly compressible material, especially transversally to the fibre, as evidenced by "Cupido's arrow" (Inoue 1982). Figure 5 shows two aspects of compression set illustrated by this artefact. The arrow's end has been softened in hot water to allow the wood to be considerably compressed without breaking the cell walls (Figure 5b). The compressed shape is fixed by drying the wood under restraint, inducing a considerable mechano-sorptive relaxation so that after release of the restraint the springback is very small. The remaining contraction could be named viscoelastic compression set. Most of it can be recovered by reconditioning the wood through boiling. Figure 5c shows, however, that the recovery is incomplete. Indeed, some damage and permanent folding has been induced in the cell walls. The permanent deformation resulting from such micro-buckling could be named plastic compression set. Transversally to the fibre, viscoelastic tension set can be obtained as well; but plastic tension set is unlikely to occur due to wood brittleness in transverse tension.

Figure 6 shows an experiment by [START_REF] Hoadley | Chemical and physical properties of wood[END_REF] made to evidence the compression set in a board. The board on the right is free to expand, and returns to its initial shape after a full moisture cycle. When prevented from expanding the middle board undergoes a compression set clearly visible after returning to the initial moisture level. On the left the fully restrained board breaks in the final stage. Mazzanti et al (2014) confirmed Hoadley's findings, except for the breaking of the fully restrained board that was never observed. The tangential stress and strain of free, partially restrained and fully restrained poplar specimens were measured. Figure 7 shows the stress of the fully restrained specimen versus the opposite of the strain of the unrestrained specimen, this graph being the closest possible approximation of a stress-strain curve for the restrained specimen. The drop of compressive stress observed during the first wetting cycle (segment JL) can be attributed to some causes discussed above, such as mechano-sorptive relaxation or the occurrence of plastic micro-buckling in the cell walls. In any case, the order of magnitude of -1% for the residual strain is sufficient to explain the observed permanent cupping of panels that would undergo little contraction of their painted face. Hunt and Jeronimidis (2016) studied the effect of compression set on the hygroscopic expansion of poplar wood. They induced the set either by preventing the swelling of specimens suddenly placed in wet condition, or by compressing them in dry state. In both case they were allowed sufficient time to recover in wet conditions. In R direction they obtained compression sets of 0.6% and a increase of the swelling rate of 20% and 9% for wet and dry compression, respectively. In T direction both the final set and the effect on swelling rates were small or not significant.

The recovery of locked-in growth strain should be also considered as one possible mechanism of irreversible strain. Wood formation at stem periphery is accompanied by prestressing. In T direction, the expansion of newly formed wood results in compression, progressively converted into tension with the deposition of newer wood. In R direction the small initial compression applied by the bark is rapidly converted into tension. Hunt and Mazzanti (2016) observed a shift of the relationship between dimensions and weight of specimens exposed during one year to the English outdoor climate, corresponding to a permanent strain of -0.07% and +0.12% in R and T directions, respectively. These values represent a fraction of the typical strain observed on green wood as a result of boiling, explained as the hygrothermal recovery of the viscoelastic strain locked in wood during the process of its formation. As this recovery is usually not fully expressed before drying, it is likely to occur slowly during the following centuries. In R direction at least, a contraction is expected, while in T both a contraction and an expansion can be expected.

Figure 1 :

 1 Figure 1: Why do most unrestrained painted panels cup the same way?

Figure 2 :

 2 Figure 2: Deformation of wooden blocks resulting from T shrinkage exceeding R shrinkage.

Figure 3 :

 3 Figure 3: Central deflection (d) of a flat-sawn board dried from 70 to 50% RH, as a function of distance to pith (h) and half board width (L).

Figure 4

 4 Figure[START_REF] Hoadley | Chemical and physical properties of wood[END_REF] illustrates the cupping resulting from asymmetric moisture diffusion in the ideal case of a fully waterproof paint layer. The transient response is characterised by a peak at a time dependent mostly on the speed of water diffusion and panel thickness.

Figure 4 :

 4 Figure 4: Transient and permanent effect of asymmetric boundary conditions.

Figure 5 :

 5 Figure 5: Cupido's arrow (up) and SEM images of the microstructure (down): (a) uncompressed wood; (b) heavily compressed; (c) after reconditioning (from Inoue 1982).

Figure 6 :

 6 Figure 6: Evidence of compression set according to Hoadley (1995).

Figure 7 :

 7 Figure 7: Stress of fully restrained specimen versus the unloaded strain of a specimen subjected to the same wetting-drying cycles (25°C, 30-80-30%RH): (square) 1st cycle, (triangle) 2nd cycle, (x) 3rd cycle. 0H: compression set; 0K dimensional loss for restrained or partially restrained specimens. J: stress peak, L: end of 1st adsorption; M: end of 1st cycle and start of 2nd (Mazzanti et al 2015)


Denis Serre

The role of the Hilbert metric in a class of singular elliptic boundary value problem in convex domains

, we were led to consider a distance over a bounded open convex domain. It turns out to be the so-called Thompson metric, which is equivalent to the Hilbert metric. It plays a key role in the analysis of existence and uniqueness of solutions to a class of elliptic boundary-value problems that are singular at the boundary.

Introduction

Let Ω be a connected open set in R n . If p ∈ R n , we denote |p| its usual Euclidian norm. The class of boundary value problems that we are interested in is div(a(|∇w|)∇w) + F (|∇w|) w = 0 in Ω, [START_REF] Anderson | Complete minimal varieties in hyperbolic space[END_REF] w > 0 in Ω, [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] w = 0 on ∂Ω.

Hereabove, a is a smooth numerical even function, which satisfies the requirements for ellipticity: [START_REF] Hua | On the Dirichlet problem for minimal graphs[END_REF] a(r) > 0, a(r) + ra ′ (r) > 0, ∀r ≥ 0.

We warn the reader that we do not assume a priori a uniform ellipticity ; it may happen that the ratio a(r) a(r) + ra ′ (r) 1 tends either to 0 or to +∞ as r → +∞. For instance, we allow the principal part to be the minimal surface operator, where a(r) = (1 + r 2 ) -1/2 , for which a(r) a(r) + ra ′ (r) = 1 + r 2 → +∞.

We suppose that F is a smooth, non-negative function, and that (5) F (0) > 0.

The lower order term in [START_REF] Anderson | Complete minimal varieties in hyperbolic space[END_REF] therefore becomes singular at the boundary, where w vanishes.

Notations. In the sequel, we denote b(r) = ra(r), so that a + ra ′ = b ′ . We define a strictly increasing function

G(r) = r 0 sb ′ (s) F (s) ds.
The inverse G -1 will be denoted H.

Data

At first glance, it may look strange that neither the equation, nor the boundary contain some explicit data ; both are "homogeneous". Our data is nothing but the domain itself. The assumption about Ω meets that in other works on non-uniformly elliptic BVPs: it is a bounded convex domain in R n .

Motivations

We came to this class of problems through the analysis of the two-dimensional Riemann problem for the Euler system of a compressible flow, when the gas obeys the so-called Chaplygin equation of state. This problem can be recast as [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF] div ∇w

1 + |∇w| 2 + 2 w 1 + |∇w| 2 = 0,
which is [START_REF] Anderson | Complete minimal varieties in hyperbolic space[END_REF] with

n = 2, a(r) = 1 √ 1 + r 2 , F = 2a.
We first proved the existence and uniqueness (see [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF]) for this problem whenever Ω is uniformly convex, in the sense that the curvature is bounded away from zero along the boundary. Later on, we removed the uniform assumption and proved the existence for every convex bounded planar domain [START_REF] Serre | Gradient estimate in terms of a Hilbert-like distance, for minimal surfaces and Chaplygin gas[END_REF]. This improvement involves an interior Lipschitz estimate of log w in terms of a special metric over Ω, for which the boundary is a horizon. We shall show below that this distance is nothing but the Hilbert metric d H , giving meanwhile a new and rather simple proof of the triangle inequality. It turns out that the very same BVP also governs those graphs x 3 = w(x 1 , x 2 ) that are complete minimal surfaces in the 3-dimensional hyperbolic space H 3 , the upper half-space in R 3 , equipped with the metric

ds 2 = dx 2 1 + dx 2 2 + dx 2 3 x 2 3 ,
of constant negative curvature. The existence of such minimal surfaces was studied by Anderson [START_REF] Anderson | Complete minimal varieties in hyperbolic space[END_REF] in the parametric and the non-parametric cases, the latter involving the graph of w. The non-parametric part of Anderson's paper is however incomplete, in that the author contents himself to establish L ∞ -bounds (by below and above) and claims that it automatically implies regularity estimates in the interior. This claim is not true because the principal part of the PDE, the operator for minimal surfaces, is not a priori uniformly elliptic. Uniform ellipticity requires the knowledge of a prior Lipschitz estimate, which can not be overlooked. The same flaw occurs in Lin's paper [START_REF] Hua | On the Dirichlet problem for minimal graphs[END_REF]. We point out that in both of these motivations, the convexity of Ω is a necessary condition for existence (and therefore a necessary and sufficient one). In the Chaplygin Riemann problem, it is guaranted by the analysis of the propagation of shock waves. If Ω is not convex, a complete minimal surface in H 3 , asymptotic to ∂Ω, exists1 as a current [START_REF] Anderson | Complete minimal varieties in hyperbolic space[END_REF], but it is not a graph over Ω.

Content of this paper

We start by showing in Section 2 the equality between our (not so) new distance and the Hilbert metric in Ω.

Then we turn towards the class of BVPs [START_REF] Anderson | Complete minimal varieties in hyperbolic space[END_REF][START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF][START_REF] Hilbert | Ueber die gerade Linie als kürzeste Verbindung zweier Punkte[END_REF]. We show that essentially the same strategy as the one designed in [START_REF] Serre | Gradient estimate in terms of a Hilbert-like distance, for minimal surfaces and Chaplygin gas[END_REF] works out under the rather mild assumption that ( 7)

+∞ e -G(s) b ′ (s) F (s) ds < ∞.
Our main result is therefore Theorem 1.1 Let a, F be even smooth functions, satisfying [START_REF] Hua | On the Dirichlet problem for minimal graphs[END_REF][START_REF] Marquis | Géométrie de Hilbert. Images des Mathématiques[END_REF][START_REF] Serre | Gradient estimate in terms of a Hilbert-like distance, for minimal surfaces and Chaplygin gas[END_REF]. Then, for every bounded convex domain Ω ⊂ R d , there exists one and only one function

w ∈ C(Ω) ∩ C ∞ (Ω)
solving the BVP (1,2,3). In addition, log w is Lipschitz, with constant 1, with respect to the Hilbert metric d H .

Of course, if a or F has only finite regularity, then w has only finite regularity.

A distance over a bounded convex domain

Let Ω is a non-void, bounded convex open domain in R n . Given two points p, q ∈ Ω, Ωp contains a ball centered at the origin and is therefore absorbing. Thus there exists some λ > 0 such that Ωq ⊂ λ(Ωp). If µ > λ, then also Ωq ⊂ µ(Ωp), by convexity. Likewise, the infimum m(p, q) of all such numbers satisfies the same inclusion, by continuity. Hence the set of these numbers is of the form [m(p, q), +∞). Considering the volumes, we have

|Ω| = |Ω -q| ≤ m(p, q) n |Ω -p| = m(p, q) n |Ω|, which implies (8) m(p, q) ≥ 1.
The equality in (8) stands only if

Ω -q = Ω -p, that is if p = q. If r ∈ Ω is a third point, then Ω -r ⊂ m(q, r)(Ω -q) ⊂ m(q, r)m(p, q)(Ω -p)
and therefore m(p, r) ≤ m(q, r)m(p, q).

All this shows that the logarithm of m is a non-negative function over Ω × Ω, which vanishes only along the diagonal and satisfies the triangle inequality. In other words, the function d Ω (p, q) = log m(p, q) + log m(q, p) is a distance over Ω. In our paper [START_REF] Serre | Gradient estimate in terms of a Hilbert-like distance, for minimal surfaces and Chaplygin gas[END_REF], we used the equivalent metric d ′ Ω (p, q) = max{log m(p, q), log m(q, p)}.

We prove here that d Ω is nothing but the Hilbert distance d H over Ω (see [START_REF] Hilbert | Ueber die gerade Linie als kürzeste Verbindung zweier Punkte[END_REF]). Let us recall the definition of the latter. If p, q ∈ Ω, let r, s be the intersection points of the line L passing through p and q, with the boundary ∂Ω ; we label the points so that r, p, q, s are in this order along L. Then d H is the logarithm of a cross-ratio :

d H (p, q) = log rq • ps rp • qs .
Our first result therefore reads Proposition 2.1 For every non-void, bounded convex open domain Ω ⊂ R n , one has

(9) d Ω ≡ d H .
The equality (9) follows immediately from the Lemma 2.1 With the notation above, there holds (10) m(p, q) = rq rp .

Proof

Let us define θ = 1 u ∈ (0, 1). Because r is a boundary point of Ω, and Ω is convex open, we have (1θ)Ω + θr ⊂ Ω. This is equivalent to writing

Ω -q ⊂ u u -1 (Ω -p),
which is the inequality ≤ in (10).

Conversely, suppose that Ωq ⊂ λ(Ωp). Then we have Ωq ⊂ λ(Ωp) and therefore rq ∈ λ(Ωp). This amounts to writing

1 + u λ p - u λ ∈ Ω, but this is equivalent to v ≤ 1 + u λ ≤ u.
The second inequality gives λ ≥ u u-1 . This implies the inequality ≥ in (10).

Remarks. This characterization of the Hilbert metric is related to the construction of the Hilbert projective metric over the cone

C = {(t, tx) | t > 0 and x ∈ Ω},
see [START_REF] Walsh | Gauge-reversing maps on cones , and Hilbert and Thompson isometries[END_REF]. It provides a much simpler proof of the triangle inequality than the original one. For the classical proof, which involves projetive geometry, see the introductory article in Image des Mathématiques [START_REF] Marquis | Géométrie de Hilbert. Images des Mathématiques[END_REF]. Lemma 2.1 also implies that d ′ Ω is identical to the Thompson metric in Ω.

3 The strategy for existence and uniqueness to the BVP Our first observation is that the PDE (1) is of the quasilinear form i,j

a ij (∇w)∂ i ∂ j w + N(w, ∇w) = 0,
where the principal part is elliptic,

c(p)|ξ| 2 ≤ i,j a ij (p)ξ i ξ j ≤ C(p)|ξ| 2 , 0 < c(p) < C(p) < ∞.
The coefficients a ij involve the gradient of w, but not w itself. Finally the lower order term N is non-increasing in w. Therefore the PDE satisfies the maximum principle (MP).

The MP allows us to compare a sub-solution and a super-solution. A locally Lipschitz function u : Ω → (0, +∞) is a sub-solution of (1) if it satisfies, in the distributional sense,

(11) div(a(|∇u|)∇u) + F (|∇u|) u ≥ 0.
It is a super-solution if it satisfies the opposite inequality ≤ in (11). If in addition u is continuous over Ω, we say that u is a super-solution of the BVP if it is a super-solution of (1), and it satisfies u ≥ 0 on ∂Ω. It is a sub-solution if it satisfies (11), and u ≤ 0 over ∂Ω (but then this means u ≡ 0 on the boundary, because u > 0 in the interior).

If u and v are a sub-solution and a super-solution respectively, of the BVP in some domain Ω, then u ≤ v in Ω. In particular, if w is a solution in Ω, then u ≤ w ≤ v. This immediately implies the uniqueness part of Theorem 1.1.

The method for existence is based on the one hand on a continuation argument, described in Section 6, and on the other hand on a priori estimates. The latter must be robust enough to allow us to pass to the limit in a sequence of solutions. To ensure the boundary condition, we shall use sub-and super-solution respectively to construct barrier functions w ± > 0, with w ± ≡ 0 on the boundary. The fact that w is clamped between w -and w + implies the boundary condition. It also ensures that w is positive and bounded in Ω. In order to pass to the limit in the PDE, we need a precompactness property of ∇w in L ∞ loc (Ω). This will be given by a C 2 loc -regularity estimate and the Ascoli-Arzela theorem. The regularity is a well-known fact (see Gilbarg & Trudinger [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]) whenever the operator

L(p) = i,j a ij (p)∂ i ∂ j
is uniformly elliptic. Since we have not assumed the latter property, it must come as a consequence of the fact that p = ∇w takes its values in a compact subset of R n . In other words, we need an a priori estimate of ∇w in L ∞ loc (Ω). We summarize below the tasks we are going to address:

• Construct a finite upper bound w + of w, continuous up to the boundary, where it satisfies w + ≡ 0.

• Construct a lower bound w -> 0 of w, continuous up to the boundary, where it satisfies w -≡ 0.

• Find a Lipschitz estimate of w in Ω. This estimate may deteriorate near the boundary, but it must be uniform on every compact subset of Ω. This is where the Hilbert metric is at stake.

• Make all these estimates uniform with respect to some approximation.

Of course, the only tool at our disposal is the maximum principle.

The barrier functions

We shall use repeatedly the fact that the PDE ( 1) is invariant under a scaling: if z is a solution in some domain ω, then the function z µ (x) := 1 µ z(µx) is again a solution, in 1 µ ω.

The upper barrier

We write our convex domain as the intersection of slabs

Ω = ν∈S n-1 Π ν , π ν = {x ∈ R n | α -(ν) < x • ν < α + (ν)},
where we have of course α ± (-ν) = -α ∓ (ν). Notice that ν → α ± is continuous.

Our upper bound will be given as the infimum of super-solutions. The building block is the solution of the BVP in the interval (0, 1) : Lemma 4.1 (1-D case.) When n = 1 and the domain is (0, 1), then the BVP admits a unique solution W .

We infer that the BVP in a slab Π ν admits a solution, namely

W ν (x) = (α + (ν) -α -(ν))W x • ν -α -(ν) α + (ν) -α -(ν) .
Because Ω ⊂ Π ν and W ν is non-negative, in particular along ∂Ω, its restriction to Ω is a super-solution of the BVP in Ω. Therefore the solution w satisfies w ≤ W ν . This yields to our upper-bound,

w(x) ≤ w + (x) = inf ν∈S n-1 W ν (x).
The continuity of α ± , plus the uniform continuity of W , imply that w + is continuous over Ω. We point out that, because every y ∈ ∂Ω is a boundary point of some Π ν , w + vanishes on the boundary.

Proof

We already know the uniqueness. Using the reflexion x ↔ 1x, we infer that W must be even: W (1x) = W (x). We anticipate that W is monotonous over (0, 1 2 ) and write the PDE, now an ODE as

(b(W ′ )) ′ + F (W ′ ) W = 0, W (0) = W ′ ( 1 2 ) = 0,
where b(r) := ra(r). We recall that b ′ > 0, from ellipticity.

Let us define z = W ′ • W -1 . Using W ′ = z(W ), we transform the ODE into zb ′ (z) dz dW + F (z) W = 0.
The latter ODE amounts to writing G(z) + log W = cst, from which we obtain z = H(log c W ) for some integrating factor c ∈ R.

Let us make temporarily the choice that c = 1 and consider a maximal solution of the autonomous ODE W ′ = H(-log W ). We have

dW H(-log W ) = dx.
Because of ( 7), we have

0 ds H(-log s) < ∞.
Therefore there exists a unique solution W 0 of the Cauchy problem

W ′ 0 = H(-log W 0 ), W 0 (0) = 0.
This W 0 is increasing. Since the integral

0 e -G(s) b ′ (s) F (s) ds is converging, we have e -G(0) ds H(-log s) < ∞
and therefore W 0 reaches the value e -G(0) at some finite x > 0. Then W ′ 0 (x) = 0. Extending W 0 it by parity, we obtain a solution of the BVP in the interval (0, 2x). Then

W (t) = 1 2x W 0 (2xt)
defines the solution of the BVP over (0, 1).

The lower barrier

The construction of the lower barrier does not make use of the convexity. We begin with a building block: Lemma 4.2 There exists an ǫ > 0 such that the function Z(x) = ǫ 2 (1 -|x| 2 ) be a sub-solution of the BVP in the unit ball B(0; 1).

Proof

Since Z is positive in the ball, it suffices to check that Z satifies (11). This inequality writes

1 2 (ǫ 2 -t 2 )(b(t) + (d -1)a(t)) ≤ F (t), ∀t ∈ [0, ǫ].
Because a and b are non-negative, it is enough to have

ǫ 2 2 (b(t) + (d -1)a(t)) ≤ F (t), ∀t ∈ [0, ǫ].
Let A and B be the upper bounds of a and b over [0, 1] respectively. If ǫ < 1, it is enough to have

ǫ 2 2 (B + (d -1)A) ≤ sup t∈[0,ǫ] F (t),
which is obviously true for ǫ > 0 small enough.

By translation and scaling, we inherit a sub-solution of the BVP in any ball B(x 0 ; ρ) :

Z x 0 ,ρ (x) = ρZ x -x 0 ρ .
If B(x 0 ; ρ) is contained in Ω, then w is a super-solution for the BVP in this ball, and we infer w ≥ Z x 0 ,ρ . This leads us to our lower barrier function

w -(x) = sup{Z x 0 ,ρ | B(x 0 ; ρ) ⊂ Ω}.
We point out that w -is continuous over Ω and is positive in the interior.

The Lipschitz estimate

The main ingredient is the | log w(q) -log w(p)| ≤ max{log m(p, q), log m(q, p)}, ∀p, q ∈ Ω.

Consequently, log w is Lipschitz with constant at most 1, with respect to the Hilbert metric.

Because the restriction to the Hilbert metric to a compact subset K ⊂ Ω is equivalent to the Euclidian distance, we infer a Lipschitz estimate in the classical sense, away from the boundary. Because min K w -> 0 and w + is bounded, this transfers into a local Lipchitz estimate of w: Corollary 5.1 For every compact subset K ⊂ Ω, the restriction w| K enjoys an a priori estimate in the Lipschitz semi-norm sup K |∇w|.

Proof

Given p, q ∈ Ω, the function

x → m(p, q)w x m(p, q) + p
is the solution of the BVP in the domain m(p, q)(Ωp). Since the latter contains Ωq, it is also a super-solution in the domain Ωq. It is therefore larger than or equal to the solution w(x + q) in the latter:

w(x + q) ≤ m(p, q)w x m(p, q) + p , ∀x ∈ Ω -q.
Setting x = 0 in the inequality above, we derive w(q) ≤ m(p, q)w(p).

Exchanging the roles of p and q, we also have w(x) ≤ m(q, p)w(y), whence (12).

The best Lipschitz constant

Lemma 5.1 provides an upper bound for the Lipschitz constant of log w with respect to the Hilbert metric:

c Ω := sup x =y | log w(y) -log w(x)| d H (x, y) ≤ 1.
We may wander whether this bound is accurate or not. Remark that if O is a boundary point and L is a ray emanating from O in Ω, then the restriction of d H to L is logarithmic, in the sense that if x, y ∈ L, then

d H (x, y) = | log t y + log(T L -t x ) -log t x -log(T L -t y )| ∼ | log t y -log t x |,
where t is the affine coordinate along L with origin O, and T L is the coordinate or the other intersection point of L with ∂Ω. If the solution admits a Hölder singularity at a boundary point, of exponent α ∈ (0, 1], we deduce that c Ω ≥ α.

One remarquable application of this principle is the following Proposition 5.1 Let the origin be a conical point of ∂Ω, and denote C the tangent cone at 0. Suppose that the BVP is solvable in the cone C, with a solution V (x) = |x|v x |x| . Then the solution in Ω is asymptotic to V as x → 0. In particular,

c Ω = 1.
The fact that V is homogeneous of degree one is a consequence of the scaling invariance of C (the conic property) and the expected uniqueness.

Proof

Let w be the solution of the BVP in Ω, and recall that for every µ > 0, the function

w µ (x) := 1 µ w(µx)
is the solution of the BVP in the domain 1 µ Ω. Let us list a few properties of the sequence (w µ ) µ>0 :

• For ǫ < η, one has the lower bound (maximum principle)

w ǫ ≥ w η in 1 η Ω. • For ǫ < η, one has the Lipschitz estimate | log w ǫ (x) -log w ǫ (y)| ≤ d ǫ H (x, y) ≤ d η H (x, y), ∀x, y ∈ 1 η Ω,
where we have denoted d η H the Hilbert distance in 1 η Ω. • By the maximum principle, w µ ≤ V for every µ > 0.

The Lipschitz estimates ensures that the PDE remains uniformly elliptic in every compact subdomain of the cone C. Therefore the theory of elliptic regularity applies: every derivative D β w µ remains bounded as µ → 0 + , on every compact subdomain of C. The (monotonic) limit w 0 is again a solution of the PDE. In addition, it satisfies (w 0 ) µ = w 0 , which means that it is homogeneous of degree one. Because of the upper bound w 0 ≤ V , we know that w 0 vanishes along the boundary. All this implies that w 0 is identical to V .

Let us know select two points x, y on the same ray L, close to the origin. The asymptotics above gives

| log w(y) -log w(x)| ∼ | log |y| -log |x|| ∼ d H (x, y). This implies c Ω ≥ 1. With Lemma 5.1, we conclude that c Ω = 1.
Another interesting situation is that of the equation ∆w + 1 w = 0.

When n = 1, and therefore the domain is I = (0, ℓ), the ODE can be integrated by hand and we find u(x) ∼ C ℓ x √ -2 log u. This quasi-Lipschitz behaviour at the boundary implies c I ≥ 1, and therefore c I = 1.

The situation is significantly better for our fundamental example : Proposition 5.2 Consider the BVP for the equation [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF]. When Ω is a disk (hence n = 2), we have c Ω = 1 2 . Proof By scaling, we may work in D = D(0; 1). Then

w(x) = 1 -|x| 2 2 ,
a rare case where the solution is known in close form. In particular, the Hölder singularity of exponent 1 2 implies c Ω ≥ 1 2 . On the other hand d H is given by

d H (x, y) = 2 log 1 -x • y + |y -x| 2 -|x ∧ y| 2 -log(1 -|x| 2 )(1 -|y| 2 ).
There remains the inequality

c Ω ≤ 1 2 . The inequality | log w(x) -log w(y)| ≤ 1 2 d H (x, y) to prove is equivalent to 1 -|x| 2 ≤ 1 -x • y + |y -x| 2 -|x ∧ y| 2 .
It is implied by

(x • (y -x)) 2 + |x ∧ y| 2 ≤ |y -x| 2 ,
which is true because the left-hand side equals (x • (yx)) 2 + |x ∧ (yx)| 2 = |x| 2 |y -x| 2 , and on the other hand |x| < 1.

We now show that the assumption made in Proposition 5.1 is always met in our fundamental example. The cone C is a sector S α of aperture α ∈ (0, π). The self-similar solution is written w α (x) = rA(θ). The boundary condition is A(0) = A(α) = 0. With ∇w α = A e r + A ′ e θ , the ODE satisfied by θ → A(θ) is

A √ 1 + A 2 + A ′2 + A ′ √ 1 + A 2 + A ′2 ′ + 2 A √ 1 + A 2 + A ′2 = 0, that is (13) A(1 + A 2 )(A ′′ + A) + 2(1 + A 2 + A ′2 ) = 0.
The solutions of (13) may not be constant. This equation can therefore be integrated once, into (14)

A 4 (1 + A 2 + A ′2 ) = C(1 + A 2 ) 2 ,
for some positive constant C. This autonomous ODE has the form A ′2 = F C (A) where F C is positive over (0, A * ) with

A * = 1 2 C + √ C 2 + 4C .
The Cauchy problem and we have A ′ C (ℓ) = 0. The maximum principle tells us that at fixed x, the map C → A C (x) is increasing. In particular, C → ℓ is increasing; obviously, it is continuous too.

A ′ = F C (A), A (0)
Let us compute the limits ℓ(0) and ℓ(+∞). We have

ℓ = A * 0 A 2 dA (1 + A 2 )(A * 2 -A 2 )(CA * -2 + A 2 ) = A * 3 1 0 s 2 ds (1 + A * 2 s 2 )(1 -s 2 )(C + A * 4 s 2 )
.

When C → 0 + , one has A * 2 ∼ √ C and therefore ℓ ∼ C 1/4 1 0 s 2 ds √ 1 -s 4 → 0, whence ℓ(0) = 0. When instead C → +∞, we have A * 2 ∼ C and ℓ = 1 0 s 2 ds (s 2 + A * -2 )(1 -s 2 )(s 2 + CA * -4 ) → 1 0 ds √ 1 -s 2 = π 2 .
Extending A C by parity, we obtain a solution A C of (14) vanishing at 0 and 2ℓ, where 2ℓ ranges from 0 to π when C ∈ (0, +∞). Therefore, there exists a unique C for which 2ℓ = α. Then w α = rA C (θ) is the announced solution.

Existence proof

So far, we have proved that if the solution w of the BVP in Ω exists, then it enjoys a finite upper bound w + , a positive lower bound w -, and a Lipschitz estimate over every compact subdomain K ⊂ Ω. This ensures that the linear operator i,j a ij (∇w)∂ i ∂ j is uniformly elliptic on relatively compact subdomains. From regularity theory [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF], we deduce locally uniform estimates of derivatives ∂ β w of every order.

Our existence proof deals first with a modified problem, from which the singularity at the boundary has been removed, and the a priori uniform ellipticity has been restored.

Relaxation of the boundary condition

Let ǫ > 0 be given, we considered the BVP formed by the PDE (1), together with the boundary condition (15) w = ǫ over ∂Ω.

Because of the maximum principle, the solution w ǫ must be unique and satisfy w ǫ > ǫ in Ω. We may therefore replace the singularity 1 w in (1) by a smooth positive, decreasing function g ǫ (w) which coincides with 1 w over (ǫ, +∞). An upper barrier w +,ǫ can be constructed by the following procedure. For every slab Π ν containing Ω, we consider the function

z ǫ,ν (x) = λW x • ν -α λ
with the same W provided by Lemma 4.1. The parameters are chosen so that z = ǫ on the boundary of the slab:

λ = α + (ν) -α -(ν) 1 -2s , α = 1 -s 1 -2s α -(ν) - s 1 -2s α + (ν), s = s(ǫ) := W -1 (ǫ).
Because z ǫ,ν is a solution in Π ν , it is a super-solution in Ω. Therefore our upper barrier is

w +,ǫ = inf ν∈S n-1 z ǫ,ν .
We point out that |∇z ǫ,ν | ≤ W ′ (s(ǫ)). This implies the same bound for w +,ǫ . Let now ā be a smooth numerical function that coincides with a over [0, W ′ (s(ǫ))], such that r → rā(r) is increasing and ā is constant over [1 + W ′ (s(ǫ)), +∞). Then the functions z ǫ,ν are super-solutions of the modified PDE ( 16) div(ā(|∇w|)∇w) + σg ǫ (w)F (|∇w|) = 0 as well, for every σ ∈ [0, 1] ; they are actually solutions when σ = 1. The BVP (16,15) admits therefore the upper barrier function w +,ǫ . Because a solution satisfies ǫ ≤ w ≤ w +,ǫ , and since w +,ǫ = ǫ on the boundary, one infers that the normal derivative ∂ ν w at the boundary is bounded by that of w +,ǫ , that is by W ′ (s(ǫ)). Then, because a PDE of the form above enjoys a maximum principle for derivatives, we find that for any solution of (16,15), one has |∇w| ≤ W ′ (s(ǫ)). All this, together with Theorem 11.3 of [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF], shows that the map T , defined by w → z = T w if div(ā(|∇w|)∇z) + σg ǫ (w)F (|∇w|) = 0, z| ∂Ω = ǫ admits a fixed point w ǫ , which is a classical solution of (16,15). It satisfies the expected bounds (17) ǫ ≤ w ǫ ≤ w +,ǫ .

These bounds ensure that w ǫ is actually a solution of [START_REF] Anderson | Complete minimal varieties in hyperbolic space[END_REF]15). We point out that w ǫ is unique.

Passage to the limit

We now prove that the w ǫ 's satisfy uniform estimates. On the one hand, the same rescaling as before can be used: if p, q ∈ Ω, and if Ωq ⊂ λ(Ωp), then

x → λw ǫ x λ + p solves (16) in Ωq, and is ≥ λǫ ≥ ǫ over ∂(Ωq). By the MP, we deduce w ǫ (x + q) ≤ λw ǫ x λ + p .

Setting x = 0 in the inequality above, we obtain w ǫ (q) ≤ λw ǫ (p).

This shows that w ǫ is Lipschitz with respect to the Hilbert metric, with Lipschitz constant ≤ 1.

On the other hand, the same lower barrier w -applies to the modified BVP, and the upper barrier w +,ǫ converges uniformly towards w + as ǫ → 0+. By regularity theory, we therefore obtain uniform bounds for higher derivatives in every compact subdomain.

By Ascoli-Arzela and a diagonal procedure, we may extract from (w ǫ ) ǫ>0 a subsequence that converges in C 1,β loc (Ω) for some β > 0, to some limit function w. We may pass to the limit in [START_REF] Anderson | Complete minimal varieties in hyperbolic space[END_REF], so that w solves the PDE. On the other hand, passing to the limit in w -≤ w ǫ ≤ w +,ǫ yields w -≤ w ≤ w + . In particular, w ∈ C(Ω) and w satisfies the boundary condition (3). This ends the proof of Theorem 1.1.

Lemma 5 . 1

 51 The solution of the BVP (1,2,3) in a bounded convex open domain Ω satisfies, if it exists (12)

Proposition 5 . 3

 53 The BVP for the fundamental example (6) is solvable in any planar sector S α . Corollary 5.2 Let Ω be a planar open convex domain. Let us restrict to the equation (6). If ∂Ω has a kink, then c Ω = 1. Proof S α := {re iθ | θ ∈ (0, α)}.

  = 0 admits a unique maximal solution A C on an interval [0, ℓ],

This is the parametric part, by far the main one, of Anderson's paper, on which we have no doubt at all.

Acknowledgement

I am indebted to Ludovic Marquis for useful comments about the Hilbert metric and its variants.


L Pinard
email: l.pinard@lma.in2p3.fr

C Michel

B Sassolas

L Balzarini

J Degallaix

V Dolique

R Flaminio

D Forest

M Granata

B Lagrange

N Straniero

J Teillon

G Cagnoli

The mirrors used in the LIGO interferometers for the first detection of gravitational waves

detection of gravitational wave occurred in the LIGO interferometers. These advanced detectors need large fused silica mirrors having optical and mechanical properties never reached up to now.

INTRODUCTION

On September 14, 2015 at 09:50:45 UTC, the two detectors of the Laser Interferometer Gravitational wave Observatory (LIGO) simultaneously observed a transient gravitational-wave signal [START_REF] Abbott | Observation of Gravitational Waves from a Binary Black Hole Merger[END_REF]. The signal matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.

The Laboratoire des Matériaux Avancés (LMA) was in charge of providing the coatings for more than 30 large optics (34/35 cm up to 55 cm diameter and 20 cm thick) for Advanced LIGO and Advanced Virgo gravitational wave interferometers. In particular, LMA has coated the 20 arm cavity mirrors (Input Mirror (IM) and End Mirrors (EM)) of Advanced LIGO and all the mirrors of Advanced Virgo. Furthermore, we have coated the large Advanced Virgo Beam Splitter (Ø55 cm, 6.5 cm in Thickness) which is the largest optical part ever coated by Ion Beam Sputtering IBS) (Fig 1).

The mirror substrates are made in ultra-pure fused silica (Suprasil from Heraeus Quartz Glass) on which a Ti-Ta2O5/SiO2 IBS multilayers is deposited [START_REF] Pinard | Toward a new generation of low-loss mirrors for the advanced gravitational waves interferometers[END_REF]. The increased YAG input laser power in the interferometers implies also to reduce even further the optical absorption of the coating down to subppm level to limit thermal lensing effects (mirror surface deformation induced by the temperature increase). Moreover, in the advanced detectors, the two Fabry-Perot cavities (arms of the Michelson interferometer) must be as identical as possible (same mirror transmission, same finesse). Even if our large IBS coater using RF ion sources is very reproducible, it is almost impossible to reach the symmetry specifications for the High Reflective (HR) coating transmission, especially for the Input Mirrors (Transmission of 1.4+/-0.1% at 1064 nm with ΔT<0.01%). The only solution is to coat the two substrates in the same run using a planetary motion. In addition, the transmission value must be uniform on each mirror on a large area implying a strong improvement of the coating uniformity on a large scale (better than 0.1% over 15 cm diameter). This last requirement is particularly difficult for the HR coating of the EM whose total thickness is close to 6µm.

In the following sections, after having detailed the silica substrate and the coating specifications, we describe how we managed to improve the mirror absorption and the scattering at a level much lower than the first generation of VIRGO mirrors [START_REF] Cimma | IBS sputtering coatings on large substrates: towards an improvement of the mechanical and optical performances[END_REF]. Secondly, the way to improve the multilayers (Ti-Ta2O5/SiO2) thickness uniformity on a large diameter is detailed.

SILICA SUBSTRATES AND COATING SPECIFICATIONS

As an example, the substrates (Ø35 cm, 20 cm thick) chosen for the Advanced Virgo Fabry-Perot cavities mirrors are made of ultra-pure fused silica. It was decided to use the last generation of Suprasil developed by Heraeus (Suprasil 3002) as bulk material for the IM. This choice is mainly driven by the high homogeneity of the refractive index and the extremely low bulk absorption at 1064 nm: 0.2 ppm/cm (part per million). For the EM, crossed by only a small amount of light, the constraints are less stringent and Suprasil 312 was chosen.

Very severe polishing specifications were defined in order to guarantee after coating a low scattering level as well as extremely good flatness (Fig. 2). Indeed, here are below the most stringent specifications asked to the polishing company:

THE ADVANCED VIRGO AND ADVANCED LIGO INPUT AND END MIRRORS A. Thermal noise and absorption

Ten years ago, one important source of noise in the Virgo interferometer was identified to be the coating thermal noise. A lot of work has been done so far [START_REF] Granata | Mechanical loss in state-of-the-art amorphous optical coatings[END_REF] to modify and optimize the high index layers (Ta2O5) which are the main source of mechanical loss (directly proportional to the Ta2O5 total thickness). The best compromise found was to dope Ta2O5 with Ti atoms.

In order to decrease the high index layer total thickness while keeping the mirror properties, alternative mirror coating designs with non-quarter-wave layers (aH (bL cH)x dL, with a,c<1 and b>1) have been studied [START_REF] Pinard | Toward a new generation of low-loss mirrors for the advanced gravitational waves interferometers[END_REF]. Experimental measurements (TNI or Thermal Noise Interferometer, Caltech USA) proved that the coating mechanical losses decreased [START_REF] Villar | Measurement of thermal noise in multilayer coatings with optimized layer thickness[END_REF], compared to a classical design and that the average absorption at 1064 nm is also lower (from our measurements based on the photothermal deflection technique). We applied this recipe to develop the high reflective coating on the IM and EM of Advanced Virgo.

Fig. 5 shows an absorption map realized on an EM HR coating whose transmission is 3 ppm at 1064 nm. The result is remarkable: average value 0.24 ppm on Ø15 cm at 1064 nm.

B. Scattering

To improve the average scattering level after coating, a new wet cleaning machine was developed which allows us to have a better cleaning efficiency on the very small particles.

Moreover, a new procedure was put in place to control of the substrate surface just before closing the coating chamber door (visual observation in the dark with a high power halogen lamp, use of ionized dry air gun). For the 20 Advanced LIGO test masses coated, here is below the average scattering on Ø15 cm at 1064 nm obtained for the two kinds of HR coating (Input mirrors, End mirrors):

• On 10 ITMs : 3.7 +/-1.2 ppm • On 10 ETMs : 4.9 +/-1.5 ppm

•
Best result : 2.3 ppm (Fig. 6) Fig. 6: Average scattering map Ø15 cm on a ITM of Advanced LIGO measured with a CASI scatterometer

C. Antireflective coating

We developed a specific three band antireflective coating (532, 800 and 1064 nm) based on a 16 layers design necessary for the Advanced Virgo interferometer. Due to the stringent specification at 1064 nm (R<100 ppm with a goal of 50 ppm), we tried to find during the design optimization a solution for this wavelength as less sensitive as possible to manufacturing errors. We have done 1000 situations to test the robustness of the design by applying a random error of +/-1% on all layers. On Fig 7, we can see the worst trial and the higher reflectivity value we can have is 100 ppm. The design found has shown experimentally a remarkable robustness to errors. Indeed, 100% of AR coatings made on large optics using this design on more than 20 large substrates have a reflectivity lower than 100 ppm, 80% less than 50 ppm. An example of achievement can be seen on Fig. 8: 13 ppm +/-6 ppm on Ø15 cm.

D. Mirror Flatness

For the first generation of Virgo mirrors made in 2002 and 2009, a large IBS (Ion Beam Sputtering) coating chamber was developed at LMA (Fig. 9). The coating thickness uniformity needed was not so stringent: indeed the mirror flatness should be lower than 3 nm RMS on Ø15 cm. For the Advanced detector, the flatness must be better than 0.5 nm RMS over the same diameter: this is a huge difference. Images LMA Photothèque CNRS 2008 -Hubert Raguet Fig. 9: Large IBS coater Coating thickness control is a crucial point for interference coatings. In the Advanced gravitational waves interferometers, it is required to have coatings with very good thickness uniformity in order to provide constant optical properties over the optics surface. Two Fabry-Perot cavities (made of one Input and one End mirror) must have the same optical characteristics: mirror transmission, finesse, round trip losses. It implies to coat two 35 cm silica substrates (Fig. 9) at the same time (twin mirrors). Indeed, it is unthinkable to get the same mirror transmission at the level required in two successive runs, even if our IBS coater is very reproducible.

To reach the flatness requirement after coating on the Advanced LIGO and Advanced Virgo large mirrors, we developed at LMA a new sample holder with a planetary motion coupled with dedicated masks (between the targets and the substrates). For each material, a mask shape is calculated with an homemade simulation software because the sputtered particles profiles are different. Thus, we are able to coat two 40 kg mirrors in the same run (Fig. 10). The mask shape optimization was made first on monolayer deposited on large fused silica plates (Ø35 cm, 6 mm thick) with spectrophotometric measurements along diameters. Some limitations linked to the metrology device were rapidly observed when the uniformity level became lower than 0.2% on Ø16 cm (unable to well optimize the mask shape, uncertainties).

So we decided to measure the coating uniformity directly on the HR coatings. The uniformity profiles are deducted from reflectivity measurements at large AOI on the edge of the reflectivity band of the multilayer using a CASI scatterometer. The reflectivity variation are converted in centering variation and then in thickness variation. Fig. 11 shows the average uniformity profile obtained on Ø24 cm for a HR coating of an EM (6µn total thickness). The uniformity reached is remarkable: #0.05% on Ø20 cm or 3 nm PV for a HR coating of an EM. The transmission matching of the two IM and the two EM was also a success (average transmission difference between the two IM is 0.002%).

CONCLUSION

LMA was able to reach the required specifications for the arm cavity mirrors of the advanced gravitational wave detectors. This is the first time such performances on coating uniformity, absorption, scattering were achieved on 35 cm mirrors made by IBS. We can in any case say that the first detected gravitational wave (and the future ones….) has passed between two large mirrors coated at LMA.

Fig. 1 :

 1 Fig. 1: Advanced Virgo recycling mirror Ø35 cm (left) and large Beam Splitter Ø55 cm (right)

• Microroughness < 1 Fig. 2 :

 12 Fig. 2: Four Advanced LIGO ITM substrates before coating On Fig. 3, a map of the point defects (Micromap profilometer measurement) and a map of the flatness (Fig. 4) on an EM substrate are shown. The optical performances of these substrates are remarkable and better than expected. The measured microroughness is around 0.6 Angström RMS.Regarding the IBS high reflective coating specifications on the IM and EM, ultimate performances in term of absorption, scattering, flatness are required:

Fig. 3 :Fig. 4 :

 34 Fig. 3: Point defect map Ø15 cm of an Advanced Virgo EM

Fig. 5 :

 5 Fig. 5: Absorption map at 1064 nm on Ø15 cm of a high reflectivity coating of an EM of Advanced Virgo For the 20 Advanced LIGO test masses coated, here is below the average absorption on Ø15 cm obtained for the two kinds of HR coating (Input mirrors, End mirrors): • On the 10 ITMs : 0.22 +/-0.03 ppm (average +/-σ) • On the 10 ETMs : 0.27 +/-0.07 ppm

Fig. 7 :

 7 Fig. 7: Zoom of the AR spectrum around 1064 nm with sensitivity to errors (of random manufacturing errors +/-1% on all layers)

Fig. 8 :

 8 Fig. 8: Reflexion map at 1064 nm on Ø16 cm of an AR coating of an ITM of Advanced LIGO (average value 13 ppm)

Fig. 10 :

 10 Fig. 10: Two Ø35 cm IM installed in the coater sample holder

Fig. 11 :

 11 Fig. 11: Average uniformity profile on Ø24 cm of an EM high reflective coating Fig. 12 shows the flatness achieved on a HR coating of an Advanced Virgo EM (total thickness 6 µm) which satisfies the requirement: 0.37 nm RMS on Ø15 cm.

Fig. 12 :

 12 Fig. 12: Flatness of an Advanced Virgo EM on Ø15 cm (Power removed)


Michael Brundage
email: michael.brundage@stonybrook.edu

Qing Chang
email: qing.chang@stonybrook.edu

Shiyao Wang
email: shiyao.wang@stonybrook.edu

Shaw Feng
email: shaw.feng@nist.gov

Guoxian Xiao
email: guoxian.xiao@gm.com

Jorge Arinez
email: jorge.arinez@gm.com

Energy Savings Opportunities and Energy Efficiency Performance Indicators for a Serial Production Line

Keywords:

Modern manufacturing facilities waste various energy savings opportunities (ESO) and lack proper performance indicators to measure energy efficiency on the production line. This work develops new energy savings opportunity strategies to maximize energy savings for the entire manufacturing facility. The ESO is an opportunity window calculated from on-line production data, such as production count, machine downtime records, buffer levels, and machine idle status, allowing certain machines to be turned off for energy savings without negatively affecting throughput. New energy efficiency performance indicators are presented that use real time production data to identify the least energy efficiency machine on the line. To achieve this goal, a baseline for energy consumption is established for the production line based on a scenario with no downtime events in the system. The energy savings opportunity strategy utilizes the Energy Efficiency Performance Indicators (EEPI) to take the opportunity window for the least energy efficient machine at opportune times, allowing for improvements to be made to the machine, increasing the overall energy efficiency of the line. This strategy takes energy saving opportunities at set increments allowing enough time for the system to recover between each opportunity window. Case studies on a serial production line are performed to validate the conclusions from the paper.

Introduction

With escalating fuel prices and increasing global competition, manufacturing companies are seeking methods to cut costs in any way possible. There are many opportunities to reduce costs in the energy consumption of the facility. These companies are searching for a way to reduce energy cost without sacrificing quality or affecting the yield of their products. The energy consumption in the industrial sector has almost doubled in the past 60 years and accounts for about one-half of the world's total energy consumption [START_REF] Nidumolu | Why Sustainability is Now the Key Driver of Innovation[END_REF]2]. In the US alone, the industrial sector spent over $100 billion in energy costs [START_REF]US Carbon Dioxide Emissions from Energy Sources 2009 Flash Estimate[END_REF] and was responsible for approximately 34% of all energy consumed in 2006. In a typical manufacturing plant, the largest source of energy consumption is the production system where 67% of the total energy cost is attributed to the production process [START_REF] Eia | [END_REF]. Being the center of a manufacturing system, production operation directly impacts energy distribution within the manufacturing environment as a whole. The dynamics of the energy demand, largely determines the total energy cost, since the cost of energy (e.g., electricity) actually varies minute-by-minute depending on demand and peak power.

According to Assessment Study on Sensors and Automation in the Industries of the Future from the US Department of Energy Industrial Technologies Program,"integrated plant-wide control" is projected to achieve about 317 Trillion Btu/yr energy savings, while "real-time control of energy usage" has a projected savings of 280 Trillion Btu/yr [START_REF] Bennet | Assessment Study on Sensors and Automation in the Industries of the Future: Reports on Industrial Controls, Information Processing, Automation, and Robotics[END_REF]. However, most previous research efforts have not focused on the entire production line, instead looking at the individual machine level [6]. At the machine level there can be an 80% reduction in energy consumption if instead of leaving non-bottleneck machines idle, these machines are turned off until needed [START_REF] Mouzon | Operational Methods for Minimization of Energy Consumption of Manufacturing Equipment[END_REF][START_REF] Mouzon | A Framework to Minimize Total Energy Consumption and Total Tardiness on a Single Machine[END_REF]. It has also been discovered that 85% of energy in a manufacturing environment is utilized for functions not related to the production of parts [START_REF] Gutowski | Environmentally Benign Manufacturing: Observations from Japan, Europe and the United States[END_REF].

There are few studies that address factory floor planning while considering energy saving opportunities [START_REF] Chen | Energy Consumption Reduction in Serial Production Lines via Optimal Startup Schedule[END_REF][START_REF] Guerrero | Production System Design to Achieve Energy Savings in an Automotive Paint Shop[END_REF][START_REF] Sun | Energy Consumption Reduction for Sustainable Manufacturing Systems Considering Machines with Multiple Power States[END_REF][START_REF] Fan | A New Approach to Scheduling in Manufacturing for Power Consumption and Carbon Footprint Reduction[END_REF]. Previous work into this topic has been severely limited with most work focusing on maintaining the quality of the product and the desired productivity while neglecting the energy saving potential. These methods treat the energy consumption as an additional cost term for an optimization problem or the consumption is analyzed as a result of high level decision making and scheduling. The energy consumption is considered a byproduct of the production system and not a main driver in the decision process on the factory floor or the control scheme of the overall system. Some existing methods, such as the energy treasure hunt developed at GE [START_REF]Data collection framework on energy consumption in manufacturing[END_REF] focused on developing weekend and daily shutdown plans, and managing the leak tag program. Such program is mainly based on non-operation obvious waste, requires expert knowledge on the part of the inspector, and is a "trial and error" manual procedure. There is still a lack of integrated systematic control methodology to drive overall effective energy savings.

One main obstacle in providing an integrated systematic control scheme is the lack of appropriate performance indicators for the facility. While many companies are able to provide key performance indicators (KPI) for a plant, these indicators do not properly address the problem areas on the floor [START_REF]Rockwell Software Enterprise Energy Management[END_REF][16]. These indicators normally single out the machine with the most energy usage, however this machine may not necessarily be the key issue in terms of energy inefficiency. This is due to the nonlinearity of the production system, which makes it difficult to quantify the impact of individual downtime incidents on the entire operation. The machine center with the most energy usage may not be the least energy efficient machine because of the effects of downtime effects from other machines.

This paper develops and implements new Energy Efficient Performance Indicators (EEPI) that incorporate energy usage from all facets of the manufacturing floor and the facility, and provide energy saving opportunity in real-time production. The EEPI takes into account random downtime events on the manufacturing floor and will allocate the energy usage into two separate categories based on permanent production loss and the lack of synchronization on the floor. This allows the identification of the process that is the most energy inefficient. In addition, the ESO will be applied to save energy and reduce the peak energy consumption so as to reduce overall cost.

The rest of the paper is structured as follows. In section 2 we present the energy savings opportunities for the serial production line. We discuss the energy efficiency performance indicators in Section 3. Section 4 provides simulation studies of the energy opportunities on a serial production line. We dissect the results and provide conclusions and future work in Section 5. e i =(m j , t i , d i ), i = 1,...,n, j = 1,2,...,M denotes a downtime event for station m j at time t i for a duration of d i time units. -E = {e 1 ,e 2 ,...,e n } denotes a sequence of downtime events for the line.

Energy Dynamic Analysis for the Production System

Assumptions and Background

This papers utilizes continuous flow models as seen in Figure [START_REF] Nidumolu | Why Sustainability is Now the Key Driver of Innovation[END_REF] [START_REF] Sevast'yanov | Influence of Storage Bin Capacity on the Average Standstill Time of a Production Line[END_REF][START_REF] Wijngaard | The Effect of Interstage Buffer Storage on Output of Two Unreliable Production Units in Series with Different Production Rates[END_REF][START_REF] Gershwin | Continuous Model of an Unreliable Two-Stage Material Flow System with a Finite Inter-stage Buffer[END_REF]. The continuous flow model will treat the quantity of jobs in the buffer as varying continuously from zero to the capacity of the buffer as opposed to integer steps. This is done for ease of analysis. The actual system dynamics are not affected by this assumption regardless if the system is continuous or discrete [START_REF] Gershwin | Manufacturing Systems Engineering[END_REF][START_REF] Li | Production Systems Engineering[END_REF]. For the serial production line as seen above, we can make the following assumptions: 1. Each station S i has a constant rated speed equal to 1 Tm , where T m is the cycle time of the station. A station will run at its rated speed if it is operational and is neither starved nor blocked. 2. A machine is starved if it is operational and its upstream buffer is empty. 3. A machine is blocked if it is operational and its downstream buffer is full. 4. The first machine, S 1 , is never starved and the last machine, S M , is never blocked. 5. Each Buffer B 2 , B 3 , ..., B M have a finite capacity. B 2 , B 3 , ..., B M denotes the maximum capacity of the buffer. 6. S M * = argmin m=1,...M 1 Tm is unique. 7. W is the actual energy consumption for the production system. 8. The total rated power consumption of the line is P = P 1 + P 2 +, ..., P M .

Energy Savings Opportunities

In this section we introduce the concept of the opportunity window. This opportunity window W m is the longest amount of time machine m can be down without resulting in a permanent production loss for the system [START_REF] Chang | Energy Saving Opportunity Analysis of Automotive Serial Production Systems[END_REF][START_REF] Chang | Transient Analysis of Downtimes and Bottleneck Dynamics in Serial Manufacturing Systems[END_REF][START_REF] Liu | The Costs of Downtime Incidents in Serial Multi-Stage Manufacturing Systems[END_REF]. The definition for the opportunity window can be seen below in equation ( 1):

W m (T d ) = sup{d ≥ 0 : s.t.∃T * (d), T 0 s m (t) dt = T 0 šm (t, e) dt, ∀T ≥ T * (d)} (1)
where T 0 s m (t) dt and T 0 šm (t, e) dt are the production counts of the end of line station, S M at time T with and without the inserted downtime event e i =(m j , t i , d i ) respectively. Utilizing this energy saving opportunity, certain machines can be turned to energy savings mode for an amount of time less than or equal to their respective opportunity window without negatively impacting the normal production.

When there are random downtime events in the system, not every event contributes to a permanent production loss for the production line. It is proven in [START_REF] Chang | Energy Saving Opportunity Analysis of Automotive Serial Production Systems[END_REF][START_REF] Chang | Transient Analysis of Downtimes and Bottleneck Dynamics in Serial Manufacturing Systems[END_REF][START_REF] Liu | The Costs of Downtime Incidents in Serial Multi-Stage Manufacturing Systems[END_REF] that given a realization of the production process subject to a sequence of downtime events E = {e 1 ,e 2 ,...,e n } and supposing that max i=1,...,n {t i + d i } < T, if the slowest machine M * stops for D time units during [0,T ), then for any station m in the production line there exists a T * ≥ T , such that,

T ′ 0 s m (t ′ ) dt ′ - T ′ 0 s m (t ′ , E) dt ′ = D T M * , ∀T ′ > T * (2)
where

T ′ 0 s m (t ′ ) dt ′ and T ′ 0 s m (t ′ , E
) dt ′ are the production volume without and with a sequence of downtime events E respectively. Thus the smallest possible downtime duration d * i for the i th failure event e i =(m j , t i , d i ) is found in equation ( 2):

d * i = inf {d ≥ 0 : s.t.T M * ti+d ti s M * dt ′ = W (m i , b(t i ; E))} (3)
where d * i is the time it takes the buffers between station m and M * to become empty if (m < M * ) or full if (m > M * ). If the actual downtime duration d i > d * i then there is permanent production loss on the line equal to d i -d * i . If the downtime duration d i < d * i is less then the above value than there is no permanent production loss.

Energy Efficiency Performance Indicators

For the development of the Energy Efficiency Performance Indicators, we must first introduce an energy baseline for the factory. The first step in this process if to define the overall production time of the manufacturing line. The entire production line is dictated by the slowest machine in the system, S * m = 1/T * m , where T * m is the cycle time of the slowest machine. If the production count of the line is M , then we can define the overall production time as:

t p = M S m * = M × T m * . (4)
This is the baseline: the shorts possible time that the manufacturing line can produce M products. Knowing that this is the baseline, this means that the actual time it takes to produce M parts will always be:

t r ≥ t p , (5)
where t r is the actual production time to produce M parts. We can now quantify the energy consumption to a dynamic and static portion in the manufacturing process, and categorize the energy consumption in detail (production related and unrelated). The dynamic part is attributed to random disruptions on the line, while the static part is related to synchronization operation. The static part of the energy consumption can be defined as W 1 , which can be seen in equation (6):

W 1 = W × t p t r . (6)
The dynamic portion of the energy consumption due to random downtime events on the line is W 2 and can be defined as:

W 2 = W × t r -t p t r . ( 7
)
As one can see the total energy consumption W 1 + W 2 = W . The next step in the process is to distribute the energy consumption at the machine level to aid in developing the Energy Performance Indicator (EP I) for the entire line. The portion of the energy consumption that is due to normal machine operation can be estimated using the power rating of the individual machines. It is defined as W i,1 :

W i,1 = W 1 × P i P , (8)
where P i is the power consumption of machine i and P is the rated power consumption of the entire line. Next, we develop the portion of energy that is wasted during permanent production loss, which is W i,2 . Permanent production loss occurs when there is a downtime event at machine i d i that is longer than the opportunity window d * i . This will cause the slowest machine to become blocked or starved depending on the location of the down machine. Using this knowledge, the formula for W i,2 becomes:

W i,2 = W 2 × (d i -d * i ) Σ(d i -d * i ) . (9)
We can then use W i,1 and W i,2 to find the energy consumption per part at each machine, ECP P :

ECP P = W i,1 + W i,2 M i , (10)
where M i is the production count of machine i. If we sum the ECP P for every machine, this will give us the performance indicator for the production line, which we will delineate as energy performance indicator (EP I Actual ).

EP I Actual = M i=1 W i,1 + W i,2 M i . ( 11
)
When there is no energy waste the W i,2 term goes to zero, which gives the energy baseline, defined as EP I Baseline :

EP I Baseline = M i=1 W i,1 M i . ( 12
)
As one can see as the production becomes very inefficient and M i becomes small, while W i,1 and W i,2 grow larger the EP I Actual will grow larger than the baseline. The larger the gap between the baseline and the EP I Actual , and the higher the EP I Actual the less efficient the production line is performing.

However, the EP I Actual cannot completely describe the energy efficiency of individual machines. For example, certain machines may have to consume larger energy than other machines, so it cannot be concluded that this machine is energy inefficient. The key is the proportion of the energy consumed in effectively producing products. Therefore, an additional performance indicator is defined as the Energy Efficiency Performance Indicator (EEP I). This performance indicator for an individual machine is equal to:

EEP I i = W i,1 W i,1 + W i,2 . ( 13
)
When there is no energy waste (i.e. no random downtime events) then the W i,2 term will go to 0 making the EEP I i = 1.0. This is the energy baseline. When downtime events cause much permanent production loss with all of the energy consumption being wasted during interruption events, which causes W i,2 to increase making the EEP I i = 0.0.

Case Studies

The production system for the case study is a 5 Machine 4 Buffer system (5M4B) with maximum buffer contents of 18 parts for each buffer. The parameters of the line can be seen in Table 1. The simulation time for this study is 168 hours with the buffers starting at half their maximum capacity. This case study uses a case where the slowest machine is the most efficient with the least energy consumption to illustrate how the EEP I can identify the least energy efficient machine in a situation when it is not the slowest machine.

Case 1: d = 0

The line is first run without any inserted opportunity windows. This will provide us with production data for a line that has no inserted downtime events. This will serve as our base scenario without any energy efficiency control strategy. The EP I Actual for the entire line is calculated using the formula in equation [START_REF] Guerrero | Production System Design to Achieve Energy Savings in an Automotive Paint Shop[END_REF] and can be seen in Figure (2). The solid line indicates the EP I Baseline , which is the energy consumption without any permanent production loss, calculated from equation ( 12). The dashed line represents the actual EP I for the entire manufacturing line. This case will allow us to compare the following cases when we insert downtime events into the production line. If the EP I Actual is greater than this case then the production line is less energy efficient than without any energy opportunity windows. If the gap between the two is smaller than Case 1 then the line is more energy efficient. To find the least energy efficient machine in the line, we utilize equation [START_REF] Fan | A New Approach to Scheduling in Manufacturing for Power Consumption and Carbon Footprint Reduction[END_REF], and plot the results in Figure (3). This indicates the energy efficiency for individual machines since it takes into account the permanent production loss at each machine due to random downtime events. In this case, machine 2 is the least energy efficient machine as indicated by EEP I since downtime events at machine 2 cause the slowest machine, machine 3, to become starved, therefore causing permanent production loss. Machine 3 has an EEP I equal to 1.0 because it has no random downtime events, any time not producing parts is due to the other machines causing it to be blocked or starved. The next case that is run takes into account inserted opportunity windows that are calculated according to equation [START_REF]US Carbon Dioxide Emissions from Energy Sources 2009 Flash Estimate[END_REF]. Using the opportunity window each machine is turned off at various times, allowing sufficient time for the machine to recover before taking the energy savings opportunity again. There is permanent production loss for this case because the random downtime events due to machine inefficiencies cause the buffer levels to not reach their full capacity, therefore decreasing the opportunity window . The permanent production loss is 12.3% compared to case 1.

The production count of machine 3 and machine 5 can be seen in Figure [START_REF] Eia | [END_REF]. Only two machines are shown since each machine except the slowest has the same parameters and would make it difficult to see the inserted opportunity windows in the production count graph if all were shown at once. Therefore, only the slowest machine and the last machine are shown since the production count of the last machine will equal the production count of the entire line. Only a portion of the simulation is shown as well to better illustrate the production count of each machine. As one can see the EP I Actual and the EP I Baseline decreases for the entire line with the insertion of energy savings opportunities, which is due to the small production loss. The last case that is simulated is a more conservative energy saving opportunity strategy. This case has inserted downtime events at each machine that are shorter than the maximum calculated energy opportunity windows for that machine by considering stochastic random downtimes so that less permanent production loss will be observed from case 2. The production count of machine 3 and machine 5 from this case can be seen in Figure [START_REF] Mouzon | Operational Methods for Minimization of Energy Consumption of Manufacturing Equipment[END_REF]. The production count is higher from the previous case during the time interval shown. This is due to the more conservative energy saving opportunity strategy. The EP I Actual and the EP I Baseline are close to those in case 2. This makes sense since the energy consumption is slightly more than in case 2, but there is less of a throughput impact. Since there is less of a production impact in this case, we can conclude that this is the best of the three cases.

Lastly, the EEP I is calculated and shown in Figure [START_REF] Gutowski | Environmentally Benign Manufacturing: Observations from Japan, Europe and the United States[END_REF]. Once again machine 2 is still the least energy efficient machine, which is to be expected since we have not performed any control actions to fix this machine. Theses results demonstrate that EP I Actual and EP I Baseline decreases with inserted ESO. This is due to the fact that although the inserted downtime saves the overall energy consumption per part, too much ESO may have the risk to cause more energy consumed by the idling of the bottleneck machine. EEP I captured the portion of the energy used on actually producing parts for each machine rather than downtime and idling, therefore it is used to identify the least energy efficient machine of the production line and help to find the root cause of energy inefficiency.

Conclusions and Future Work

This paper investigates energy saving opportunities for a serial production line while developing new energy performance indicators for the production line and at the machine level. The performance indicators are tested using simulation studies using three different cases. These studies use different downtime events to prove the concept of the energy opportunity window as well as the energy performance indicators. The indicators are able to correctly identify the machine with the least energy efficiency for each case.

The next step in this research is to develop a control methodology to help alleviate the problem of the least energy efficient machine by utilizing the energy opportunity window or by performing preventative maintenance. Also, a dynamic energy saving opportunity strategy will be developed to optimize cost savings.

2. 1 Nomenclature 1 Tm

 11 Some nomenclature utilized for this paper is as follows:s m (t), m = 1,2,...M denotes the actual processing speed of machine S m at time t. -T 0 s m (t ′ )dt ′ , m= 1,2,...M denotes the production volume of station S m during time interval [0,T). -M * = argmin m=1,...M denotes the slowest machine in the line. -b m (t), m = 2,3,...,M denotes the buffer level of B m at time t.

Fig. 1 :

 1 Fig. 1: A Serial Production Line with M Machines and M-1 Buffers

Fig. 2 :

 2 Fig. 2: EPI for the Entire Line with d = 0

Fig. 3 :

 3 Fig. 3: EEPI for the Each Machine with d = 0

4. 2

 2 Case 2: d = d * i

Fig. 4 :

 4 Fig. 4: Production Count of Machine 3 & 5 for d = d * i

Fig. 5 :

 5 Fig. 5: EPI for the Entire Line with d = d * i

Fig. 6 :

 6 Fig. 6: EEPI for the Each Machine with d = d * i

4. 3

 3 Case 3: d < d * i

Fig. 7 :

 7 Fig. 7: Production Count of Machine 3 & 5 for d < d * i

Fig. 8 :

 8 Fig. 8: EPI for the Entire Line with d < d * i

Fig. 9 :

 9 Fig. 9: EEPI for the Each Machine d < d * i