Stephane Pagano
email: pagano@em-lyon.com

Gilles Neubert
email: gneubert@em-lyon.com

Developing a collaborative framework for mapping and managing key drivers of future value creation based on Intangible Assets

Keywords: Value Creation, Collaborative Networks, Resilient SC, Ecosystem, Intangible assets, Stakeholders

Companies, their suppliers and networks of partners need to be aligned to fulfill the strategy to deliver the value and adapt rapidly whatever challenge arises. To be resilient, companies and their networks have to build ecosystems in a systemic thinking. To help firms achieving this, we propose a methodology based on "value drivers" that allows the mapping, the analysis and the management of intangible assets, and the way they are activated through a multi perspective and a multi stakeholder framework.

Introduction

In a rapidly changing environment, companies have to face many challenges while the competitive and technological environment keeps evolving at a fast pace: Globalization of markets through trade agreements and communication technologies, consumers' maturity and power of influence; strong trend towards a service-focused and dematerialized economy. Today's value chains consist of globally spread stakeholders, production processes, global distribution and retail channels. As part of the procurement process, the purchasing function is a hub upstream the Supply Chain that plays a fundamental role both in term of value creation and adaptation of the business models of the companies.

Our research objective is to construct a dictionary of resources at work in processes, seen here as drivers of performing relationships, by mapping their importance and role in the interactions taking place in the processes. Our applied research proposition is to build tools for the management of value creation in the buying and supply-chain processes integrating multiple stakeholders and multiple perspectives.

The paper is organized as follow: after the introduction, the second part reviews essential concepts underlying the stakes for the sustainability and the performance of firms' supply chain, the third part presents our framework approach and gives some details on our research and propositions. The fourth part concludes on findings, expected contributions and future operational outputs.

Context

Value and value creation

Value is a multi-perspective concept that extends beyond the limits of the firm. With globalization and technological innovation, value is delivered through dynamic networks or chains of interconnected firms or supply chain. This raises the question on how a particular relationship helps a firm creating value in terms of offerings, and what factor(s) are essential to successfully establish a particular relationship [START_REF] Srivastava | Value creation through relationship closeness[END_REF]. The strategic models of Value Creation based on arms-length confrontation with suppliers has led many companies on a short term value creation path, which meant value destruction for many suppliers and their economic ecosystem, impacting their entire value chain and their stakeholders.

Corporate Social Responsibility

There's an increasing pressure to consider the environmental and social aspects in purchasing policies: expectations are rising from stakeholders such as customers, the general public, NGOs, and governments who now hold companies responsible, not only for their own actions, but also all the partners within the supply chain [START_REF] Jiang | The effects of interorganizational governance on supplier's compliance with SCC: An empirical examination of compliant and non-compliant suppliers[END_REF][START_REF] Kovács | Corporate environmental responsibility in the supply chain[END_REF][START_REF] Goebel | The influence of ethical culture on supplier selection in the context of sustainable sourcing[END_REF][START_REF] Igarashi | What is required for greener supplier selection? A literature review and conceptual model development[END_REF]. CSR advocates the idea that companies need to change their business models and value propositions, actually creating value not only for themselves, but also for a larger number of stakeholders. Since 1930, the stakeholder theory poses the question of the definition of the objectives of firms, arguing about shareholders value or about sharing value with partners. Porter and Kramer recently introduced the idea of Shared Value [START_REF] Porter | Creating Shared Value[END_REF]. Over the last decade, many authors [START_REF] Hughes | Simple Rules for Making Alliances Work[END_REF][START_REF] Murphy | Shareholders First? Not So Fast[END_REF][START_REF] Ready | The Power of Collective Ambition[END_REF] called for an urgent renewal of the strategic objectives of firms, towards financial success, but taking into account the actual needs of the communities where they operate or have an impact.

Collaboration and risks mitigation

Usual strategies based on competitive advantage [START_REF] Porter | Competitive advantage: creating and sustaining superior performance: with a new introduction[END_REF][START_REF] Barney | Firm Resources and Sustained Competitive Advantage[END_REF] can't last. To address this issue, firms have investigated other domains: process optimization to save resources, flexibility, changes in relationship with other partners to secure exclusive access to suppliers. These developments impact the complexity and importance of purchasing decisions [START_REF] De Boer | A review of methods supporting supplier selection[END_REF][START_REF] Weele | Purchasing & supply chain management: analysis, strategy, planning and practice[END_REF][START_REF] Oshri | The Handbook of Global Outsourcing and Offshoring[END_REF]. Supplier management is a key issue in supply chain performances and this reflects in a continuing growth of publications [START_REF] Ho | Multi-criteria decision making approaches for supplier evaluation and selection: A literature review[END_REF][START_REF] Cao | Supply chain collaboration: Impact on collaborative advantage and firm performance[END_REF][START_REF] Wu | A literature review of decision-making models and approaches for partner selection in agile supply chains[END_REF][START_REF] Chai | Application of decision-making techniques in supplier selection: A systematic review of literature[END_REF]. Since suppliers can represent up to 90% of firms turnover, the potential impact of inappropriate portfolio management of suppliers can seriously impact the value of a company. Firms also interconnected with their societal, natural and economic environment [START_REF] Porter | The Competitive Advantage of Corporate Philanthropy[END_REF] because it's from these that depend their competitiveness: trained employees, infrastructures, communications, political stability. The state of the global environment therefore impacts the firms activity, operational costs, the availability of the resources, and the value of the firms viewed by the stakeholders. These risks are putting a high pressure on companies that need to adapt rapidly to innovative or disruptive situations.

2.4

The role of intangible assets

The way value is generated lies in the way physical and intellectual resources are managed and activated. Companies are looking for durable and sustainable strategic resource to acquire and maintain competitive advantages. This is why knowledge management should be completely integrated [START_REF] Drucker | Coming of the New Organization[END_REF][START_REF] Barney | Firm Resources and Sustained Competitive Advantage[END_REF][START_REF] Grant | The Knowledge-based View of the Firm: Implications for Management Practice[END_REF]. The importance of knowledge in the success of firms has already been described [START_REF] Drucker | Long-Range Planning-Challenge to Management Science[END_REF][START_REF] Penrose | The theory of the growth of the firm[END_REF] specifically the role of intangible assets in providing sustainable competitive advantage [START_REF] Hall | A framework linking intangible resources and capabiliites to sustainable competitive advantage[END_REF][START_REF] Hall | The strategic analysis of intangible resources[END_REF]. Though there is an abundant literature in economy and management on intangibles assets [START_REF] Sveiby | Methods for Measuring Intangible Assets[END_REF], we think that there is still a great work to do to operationalize intangible assets management, moreover in operations and supply chain management.

Conclusion

We see the complexity of supply chain and value chain configuring when it comes to describe how intangibles assets such as human competencies, organizational knowledge and behavioral characteristics deployed by interacting individuals, departments and organizations achieve each stakeholder's values, expectations and perspective. The "life" of projects and processes such as buyer-supplier relationship can be as complex as a human based interaction between organizations and services can be, because it greatly relies on organizational and personal "qualities" of the actors on both sides of each relation (buyers, vendors, and other stakeholders involved).

As value creation has many meanings, we see the limits of closed-off reasoning and the potential of widening the perspectives of firms through a paradigm shift: stakeholders, environment and future generations no longer being expandable when building sustainable business models and value creation strategies. This requires considering internal organization of the firm, its networks of connections and partners, its tangible and intangible assets and the notions of capital and investment [START_REF] Parung | A metric for collaborative networks[END_REF].

Approach and empirical method development

Firms need to adopt a multi-perspective approach on the supplier selection and management: purchasing process and supply chain play a strategic role, which must go beyond optimization of processes, and take into account the temporal dimensions of the ROI (short, mid and long term) and the management of intangibles.

Measure, management and value creation in the firm are spread over time and must take into account the stakeholders. To ensure the longevity of the firm, the following dimensions and perspective must be included in the processes:

 The time horizon of the firms' strategies;  Value creation/destruction for the firm and its ecosystem;  Actual place of intangible assets as drivers of competitiveness and value creation. To achieve this, we are constructing on an approach by value "enablers" or "drivers" for future value creation [START_REF] Marr | Intellectual capital -defining key performance indicators for organizational knowledge assets[END_REF], mapping intangible assets, processes and stakeholders through events that will generate new value proposition (figure1) and eventually economic value and performance for the firm.

Ground of the research

The global research ground for our projects is based on three companies which offered participating buyers teams for initial explorations. Later on two firms offered a long standing participation for the conduct of the full research, currently ongoing, through long standing projects involving transversal teams.

Connecting the intangibles

To map and understand interactions and flows beyond the specifics of each situation (project, company, process, industry etc.), instead of trying to provide predefined items, we focused on building a generic methodology to analyze and manage value creation in the processes, objectively linking the activation of intangible assets to value creation. The applied output can be used by managers to follow projects and processes along their lifetime, and comprises four main phases: A consistency strategy analysis; the identification of key drivers; the selection of key performance indicators and the construction of dashboards.

The heart of the methodology is to analyze the drivers represented by intangible assets (human capital, relational capital, structural capital) and connect them with the process to achieve value proposition and delivery made to the clients.

Fig. 2. The fuzzy box.

To do so, we need to clarify what happens in what we call a fuzzy box (figure 2) where players activate processes and stakeholders to achieve objectives and create value. Key steps here are to identify helping events occurring during the life of the project to evaluate the role and contribution of intangibles assets, and breaking events to capitalize on experience and construct feedback.

3.3

Mapping value created / destroyed and stakeholders' impact.

We have elaborated an interactive tool that allows the mapping, contextualization and linking of the stakeholders, their interactions and the value created or destroyed for each one of them (figure3). The different stakeholders of a project or process are listed twice: Underneath the time line, each stakeholder has its own timeline to position the source of helping and breaking events. Above the timeline, the same stakeholders are listed with once again their own timeline to position the different impacts of the events: value created or destroyed, assets created. Helping and breaking points of the projects' life are positioned on the timelines. The interaction of the events (a) are visually mapped. When an event is positioned, it is possible to map the value destruction caused by the breaking event (b) and the value created for each helping event (c). Causes and consequences are spread over time and can generate positive or negative impact on different horizons: short, medium and long. Hence, assets created during one project can be identified as a significant source of a helping event in a future project (d), or directly create value (e). This is a capital notion we want to document because assets mobilized in one project are probably the result of a previous investment, just like assets generated in the current project can be drivers of future value, a concept that is at the core of our research work. This can have a significant impact on the affectation of costs and value in industries, because every production line is analyzed through return on investment.

Empirical validation of the method

To validate the method and the tools we have organized a workshop with a team from each of the firms. The objectives of the workshop were to validate the methodology, and to verify the ease of appropriation of the method and the tools, knowing that the concepts and approaches used by the tools were quite far of the everyday operations and scope of the participants.

All the team members found easy to list the stakeholders, to map the helping and breaking events, to connect the values generated, even though these values were outside of direct economic results, and to point value destructions for the different stakeholders. A very significant result is that the method offered the possibility to identify and map the ripple effects of the outcomes of a given project, allowing the affectation of more value to the global outcome of a project due to previously unaccounted dissemination to other projects. Comments were here that the method offered unsuspected outputs: some assets generated are reusable and thus multiplying their effects and value while remaining unaccounted or miss-affected. Value generated by one team could be capitalized by third parties without recognition of the original creators. Some assets generated by one project could be integrated in an innovative Business Models not foreseen before.

The results showed that the buyers and the managers of the purchasing departments easily grab the approach and tools, both to build a reflexive dialogue on past experiences, where they try to identify solutions to achieve better performances, but also to elaborate a decisive argumentation to promote the strategic role of the purchasing function. All participants noted that the method could be used as a knowledge management tool to communicate the history of a project, to illustrate the history of the relationship with a supplier, allowing the understanding of the factors, namely human related, that were decisive along the projects phases.

Conclusion and future works

Our approach to integrate in the same framework different types of value (i.e. economic, monetary but also market, image, competencies, organizational skills etc.) was validated. The method and tools allow the objectivization of the different values created or destroyed, and of the future values that will eventually be generated with the intangible assets activated or created. Very formalized processes can alter the adaptive capacities of companies and networks, building the case for the ability to build transversal qualitative approach of relationships and information sharing, all these heavily relying on the intangible assets of companies.

The ability for firms to adapt, reconfigure and respond, greatly relies on the capacity of the different stakeholders involved to collaborate and mutually adjust to each-others, diffuse information and realign. All this relies almost exclusively on the very own capacities of the actors involved thus illustrating the central importance of intangible assets. This research builds meaningful methods to track, document and manage intangibles assets in response to internal or external pressure on value creation.

Expected outputs are tools to: analyze the strategy of the firm, how it is translated along the supply chain; identify and take into account all the stakeholders; map, contextualize and link the stakeholders, their interactions and the value created or destroyed for each one of them. These deliverables are being developed and refined through field work. They describe, operationalize and value intangible assets, not in the book value of the firms, but in the future value they can generate or destroy. Other outputs are being developed in field now such as knowledge management tools.

Among scientific contributions, we expect this research will contribute to Resource-Based View theory, to Knowledge Management, to Sustainable Supply Chain and to Supply Chain performance.

Fig. 1 .

 1 Fig. 1. Placing intangible assets at the heart of renewed value propositions.

Fig. 3 .

 3 Fig. 3. Visual canvas, mapping events, value and stakeholders' interactions.

Aknowledgements

The authors would like to thank THESAME and the PEAK program: member companies of the program, the F2I, funding structure of the UIMM, the UDIMERA, the Research Division of the Regional Council of Rhône-Alpes, and the General Council of Haute-Savoie.


Alexia De Brauer
email: alexia.de-brauer@math.u-bordeaux1.fr

Angelo Iollo
email: angelo.iollo@math.u-bordeaux1.fr

Thomas Milcent
email: thomas.milcent@u-bordeaux.fr

A Cartesian Scheme for Compressible Multimaterial Models in 3D

Keywords: compressible multimaterial, immersed boundary, Eulerian elasticity

We model the three-dimensional interaction of compressible materials separated by sharp interfaces. We simulate fluid and hyperelastic solid flows in a fully Eulerian framework. The scheme is the same for all materials and can handle large deformations and frictionless contacts. Necessary conditions for hyperbolicity of the hyperelastic neohookean model in three dimensions are proved thanks to an explicit computation of the characteristic speeds. We present stiff multimaterial interactions including air-helium and water-air shock interactions, projectile-shield impacts in air and rebounds.

Introduction

The study of three-dimensional multimaterial phenomena requires efficient numerical modeling and simulation. These problems are insoluble by traditional theoretical and experimental approaches and hazardous to study in the laboratory. Nevertheless, their accurate and efficient simulation is crucial to reduce environmental impact in many industrial domains. Numerical model research in this field is particularly active because of the formidable difficulty in applying classical approaches based on ad hoc models and schemes to multi-physics contexts.

In what follows we focus on phenomena that involve gases, liquids and elastic materials undergoing rapid evolution and large deformations in three space dimensions. For this reason, we privilege a fully Eulerian approach and a monolithic model that describes each material by the same set of conservation equations and an appropriate thermodynamically consistent constitutive law different for each material [START_REF] Godunov | Elements of continuum mechanics[END_REF]. For specific applications, typically in presence of mostly radial phenomena, Lagrangian or Lagrangian-Eulerian approaches may be more pertinent, see [START_REF] Kluth | Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme[END_REF][START_REF] Galera | A two-dimensional unstructured cellcentered multi-material ALE scheme using VOF interface reconstruction[END_REF]6,[START_REF] Maire | A nominally second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids[END_REF].

Fully Eulerian hyperbolic conservative models for continuum mechanics were introduced in [START_REF] Godunov | Elements of continuum mechanics[END_REF][START_REF] Plohr | A conservative eulerian formulation of the equations for elastic flow[END_REF]27] and their numerical analysis started in [START_REF] Fedkiw | A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)[END_REF]24]. In [START_REF] Fedkiw | A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)[END_REF] the authors propose a scheme, the ghost-fluid method, that allows an implicit capturing of the material interfaces. Moreover, they devise a locally non-conservative Godunov solver at the interface that is stable and non oscillatory. This scheme relies on the definition of a ghost medium that requires the storage of additional variables at the material interfaces, where the solution of two Riemann problems, each relative to a different material, is necessary.

A different method was proposed in [24] where a conservative cut-cell technique was developed for hyperelastic and plastic multimaterial simulations. A specific feature of this numerical model is that the deviations of the deformation gradient with respect to the irrotational compatibility condition and to continuity equation are explicitly penalized in the governing equations. This scheme is at the base of many other subsequent works in the literature.

Another approach is introduced in [START_REF] Favrie | Solid-fluid diffuse interface model in cases of extreme deformations[END_REF] for hyperelastic solid and fluid interac-tions. The authors design a conservative non-equilibrium mixture model that relaxes to the desired multimaterial conservation laws. In this approach one compromises on the sharpness of the material interface to avoid oscillations and to apply an HLLC solver previously developed for a single material in [14].

Further developments of this method include plasticity modeling [10] and an hyperbolic sub-system splitting procedure [START_REF] Favrie | A thermodynamically compatible splitting procedure in hyperelasticity[END_REF] where each sub-system has only three waves instead of seven.

The analysis and the simulation of three-dimensional hyperelastic models present specific difficulties due to the seven-wave pattern and to the inherent hyperbolicity constraint. In [4] an approximation of the Riemann problem based on characteristic tracing is employed to simulate the one-dimensional projection of the seven-wave pattern typical of a three-dimensional hyperelastic constitutive law. This approach has further been developed in [3] to deal with actual multimaterial interfaces thanks to a variant of the ghostfluid method, with applications to one-dimensional as well as two-dimensional deformations of multiple hyperelastic sliding materials. A fully conservative cut-cell three-dimensional Eulerian method is then presented in [5] for simulating multiple compressible solid and fluid components where internal boundaries are tracked using level set functions. A different discretization scheme is adopted in [START_REF] Hill | An Eulerian hybrid WENO centered-difference solver for elasticplastic solids[END_REF], where the simulation of elastic and plastic two-dimensional flows is based on a hybrid centered-difference WENO method and a ghostfluid approach at the interface. The authors employ a fourth-order centereddifference stencil away from discontinuities that requires no explicit calculation of the local hyperbolic characteristic information. As a result, the scheme is constructed in a general way to allow for possibly different constitutive relations. In the same spirit, a conventional third-order WENO scheme and a ghost-fluid method are applied in [START_REF] Barton | Eulerian adaptive finitedifference method for high-velocity impact and penetration problems[END_REF] to actual three-dimensional simulations of hyperelastic and plastic impacts and penetration with adaptive mesh refinement. It is known that approximate Riemann solvers for such problems may lead to inaccurate intermediate states as noted in [START_REF] Miller | An iterative Riemann solver for systems of hyperbolic conservation laws, with application to hyperelastic solid mechanics[END_REF]4]. However, these local inaccuracies do not seem to significantly pollute the solution when compared to exact test cases [14,[START_REF] Favrie | Solid-fluid diffuse interface model in cases of extreme deformations[END_REF][START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF].

Hence, the typical features of integration schemes for Eulerian formulations of the hyperelastic conservation laws are: i) tracking of internal boundaries by a level set function; ii) approximate Riemann solver to compute numerical fluxes; iii) ghost-material approach to compute numerical fluxes at the material interface. In this respect, the present study details a twofold contribution to the simulation of complex three-dimensional hyperelastic phenomena.

Firstly, we show under usual assumptions that the hyperelastic neohookean model employed satisfies the necessary conditions for hyperbolicity by explicitly computing the analytic wave speeds of small perturbations. Secondly, we simplify the computations of the numerical fluxes at the material interface such that no ghost material is defined and no mixture model is needed to obtain a non-oscillatory scheme, extending to three space dimensions the method presented in [START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF]. Thanks to this method, we are able to explicitly determine the internal boundary conditions modeling shock-bubble interactions, threedimensional impacts and rebounds of hyperelastic materials immersed in a fluid.

The model

This model was already discussed in [START_REF] Godunov | Elements of continuum mechanics[END_REF][START_REF] Plohr | A conservative eulerian formulation of the equations for elastic flow[END_REF]27,24,7,[START_REF] Favrie | Solid-fluid diffuse interface model in cases of extreme deformations[END_REF]. We follow here the formulation presented in [START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF] where a neohookean model was investigated and extend it to three dimensions.

Forward and backward characteristics

Let Ω 0 ⊂ R 3 be the reference (or initial) configuration of a continuous medium and Ω t ⊂ R 3 the deformed configuration at time t. In order to describe the evolution of this medium in the Lagrangian frame we define the forward characteristics X(ξ, t) as the image at time t in the deformed configuration of a material point ξ belonging to the initial configuration, i.e., X : Ω 0 × [0, T ] -→ Ω t , (ξ, t) → X(ξ, t) (see Fig. 1). The corresponding Eulerian velocity field u is de-

fined as u : Ω t × [0, T ] -→ R 3 , (x, t) → u(x, t) where    X t (ξ, t) = u(X(ξ, t), t) X(ξ, 0) = ξ (1)
To describe the continuous medium in the Eulerian frame, we introduce the deformed configuration, gives the corresponding initial point ξ in the initial configuration, i.e., Y : Ω t × [0, T ] -→ Ω 0 , (x, t) → Y (x, t) (see Fig. 1). Since Y (X(ξ, t), t) = ξ, differentiating with respect to time and space in turn we have:

   Y t + u • ∇ x Y = 0 Y (x, 0) = x (2) and [∇ ξ X(ξ, t)] = [∇ x Y (x, t)] -1 .
(3)

The relation ( 2) is the Eulerian equivalent of the characteristic equation (1).

In addition, equation (3) allows to compute the gradient of the deformation in the Eulerian frame via Y .

Y (x, t) = ξ x = X(ξ, t)
Initial configuration Ω 0 Deformed configuration Ω t Fig. 1. Forward and backward characteristics.

Eulerian model

The local form of the governing equations of mass, momentum and energy conservation in the deformed configuration Ω t can be written as

         ρ t + div x (ρu) = 0 (ρu) t + div x (ρu ⊗ u -σ) = 0 (ρe) t + div x (ρeu -σ T u) = 0 (4)
The physical variables are the density ρ(x, t), the velocity u(x, t), the total energy per unit mass e(x, t) and the Cauchy stress tensor in the physical domain σ(x, t). To close these systems a constitutive law is needed to detail the relationship between the stress tensors and the other physical variables.

Hyperelastic models

The classical results of this section can be found in the textbook [18]. Let us define ε = e -1 2 |u| 2 the internal energy per unit mass. In the hyperelastic context, ε is a function of the deformation tensor ∇ ξ X and the entropy s.

The energy has to be Galilean invariant and we focus in this paper on the isotropic case. It can be proved that the material is Galilean invariant and We assume that ε is the sum of ε vol , a term depending on volume variation and entropy, and ε iso a term accounting for isochoric deformation. Hence the internal energy is given by

ε = ε vol (ρ, s) + ε iso (Tr(B), Tr(B -1 )) (5)
where in three dimensions we have

B(x, t) = [∇ x Y ] -1 [∇ x Y ] -T /J 2 3 (x, t) J(x, t) = det([∇ x Y ]) -1 (6)
Tensor B accounts for isochoric deformations since det(B) = 1. In general the term relative to an isochoric transformation may also depend on entropy.

Here, we will limit the discussion to materials where the isochoric term is independent of the entropy, as in the case of idealized crystals (metals, ceramic).

The Cauchy stress tensor σ(x, t) is then given by (see [18] for the details)

σ(x, t) = -ρ 2 ∂ε vol ∂ρ s (ρ, s)I + 2J -1 σ iso - Tr(σ iso ) 3 I (7)
where

σ iso = ∂ε iso ∂α B - ∂ε iso ∂β B -1 (8)
and α, β denote the first and second argument of ε iso , respectively.

Overall model

Together with equations of mass, momentum and energy conservation (4), the additional equation ( 2) is required in order to record the deformation in the Eulerian frame. However, since σ will directly depend on ∇ x Y (5-7) we take the gradient of (2) as a governing equation and obtain the system in conservative form

                 ρ t + div x (ρu) = 0 (ρu) t + div x (ρu ⊗ u -σ) = 0 (∇ x Y ) t + ∇ x (u • ∇ x Y ) = 0 (ρe) t + div x (ρeu -σ T u) = 0 (9)
where some fluxes of ∇ x Y are zero (see equation ( 13)). The initial density ρ(x, 0), the initial velocity u(x, 0), the initial total energy e(x, 0) and ∇ x Y (x, 0) = I are given together with appropriate boundary conditions. This system is known in the literature as the inverse deformation gradient formulation.

We choose a general constitutive law that models gas, fluids and elastic solids.

The internal energy per unit mass ε = e -1 2 |u| 2 is defined as

ε(ρ, s, ∇ x Y ) = neohookean elastic solid κ(s) γ -1 1 ρ -b 1-γ -aρ van der Waals gas + p ∞ ρ stiffened gas + χ ρ 0 (Tr(B) -3) ( 10
)
where B is defined in (6). We obtain according to ( 7)

σ(ρ, s, ∇ x Y ) = -p(ρ, s)I + 2χJ -1 B - Tr(B) 3 I (11)
where

p(ρ, s) = -p ∞ -aρ 2 + κ(s) 1 ρ -b -γ (12)
Here κ(s) = exp s c v and c v , γ, p ∞ , a, b, χ are positive constants that characterize a given material. Parameters a and b correspond to the van der Waals parameters, p ∞ accounts for fluid or solid materials where intermolecular forces are present (see for example [14,[START_REF] Favrie | Solid-fluid diffuse interface model in cases of extreme deformations[END_REF]). The last term in the energy expression models a neohookean elastic solid where the constant χ is the shear elastic modulus. For very large deformations, more complete models (Mooney-Rivlin, Ogden) take into account an additional invariant in the energy function in three dimensions. Without loosing in generality, here we stick to a neohookean model as it leads to simpler expressions in the following developments.

Numerical Scheme

Let x = (x 1 , x 2 , x 3 ) be the coordinates in the canonical basis of R 3 , u = (u 1 , u 2 , u 3 ) the velocity components, Y i ,j the components of the tensor [∇ x Y ] and σ ij the components of the stress tensor σ. Equations ( 9) become

                                     ρ ρu 1 ρu 2 ρu 3 Y 1 ,1 Y 2 ,1 Y 3 ,1 Y 1 ,2 Y 2 ,2 Y 3 ,2 Y 1 ,3 Y 2 ,3      ρu 1 (ρu 1 ) 2 ρ -σ 11 ρu 1 ρu 2 ρ -σ 21 ρu 1 ρu 3 ρ -σ 31 ρu 1 Y 1 ,1 +ρu 2 Y 1 ,2 +ρu 3 Y 1 ,3 ρ ρu 1 Y 2 ,1 +ρu 2 Y 2 ,2 +ρu 3 Y 2 ,3 ρ ρu 1 Y 3 ,1 +ρu 2 Y 3 ,2 +ρu 3 Y 3 ,3 ρ 0 0 0 0 0 0 ρu 1 ρe-(σ 11 ρu 1 +σ 21 ρu 2 +σ 31 ρu 3 ) ρ      ρu 2 ρu 1 ρu 2 ρ -σ 12 (ρu 2 ) 2 ρ -σ 22 ρu 2 ρu 3 ρ -σ 32 0 0 0 ρu 1 Y 1 ,1 +ρu 2 Y 1 ,2 +ρu 3 Y 1 ,3 ρ ρu 1 Y 2 ,1 +ρu 2 Y 2 ,2 +ρu 3 Y 2 ,3 ρ ρu 1 Y 3 ,1 +ρu 2 Y 3 ,2 +ρu 3 Y 3 ,3 ρ 0 0 0 ρu 2 ρe-(σ 12 ρu 1 +σ 22 ρu 2 +σ 32 ρu 3 ) ρ       ρu 3 ρu 1 ρu 3 ρ -σ 13 ρu 2 ρu 3 ρ -σ 23 (ρu 3 ) 2 ρ -σ 33 0 0 0 0 0 0 ρu 1 Y 1 ,1 +ρu 2 Y 1 ,2 +ρu 3 Y 1 ,3 ρ ρu 1 Y 2 ,1 +ρu 2 Y 2 ,2 +ρu 3 Y 2 ,3 ρ ρu 1 Y 3 ,1 +ρu 2 Y 3 ,2 +ρu 3 Y 3 ,3 ρ ρu 3 ρe-(σ 13 ρu 1 +σ 23 ρu 2 +σ 33 ρu 3 ) ρ      0 0 0 0 0 0 0 0 0 0 0 0 0 0                                      (13)
This system can be written in the compact form

Φ t + (G 1 (Φ)) ,1 + (G 2 (Φ)) ,2 + (G 3 (Φ)) ,3 = 0 (14)
We discretize (14) with a finite volume method on a Cartesian mesh. Let ∆x i be the grid spacing in the x i direction and Ω i,j,k the control volume centered at the node (i∆x 1 , j∆x 2 , k∆x 3 ). The semi-discretization in space of (14) on

Ω i,j,k gives (Φ i,j,k ) t + G 1 i+1/2,j,k -G 1 i-1/2,j,k ∆x 1 + G 2 i,j+1/2,k -G 2 i,j-1/2,k ∆x 2 + G 3 i,j,k+1/2 -G 3 i,j,k-1/2 ∆x 3 = 0 ( 15
)
where Φ i,j,k is the value of the conservative variable integrated on Ω i,j,k . In Fig. 2 are shown sections in two dimensions of the three-dimensional computational cell projected on their respective parallel planes. The fluxes in (15) will be computed by approximate one-dimensional Riemann solvers in the direction orthogonal to the cell sides of the Cartesian mesh.

x y z i∆x 1 j∆x 2 k∆x 3 (i -1)∆x 1 (i + 1)∆x 1 (j -1)∆x 2 (j + 1)∆x 2 (k -1)∆x 3 (k + 1)∆x 3 Φ i-1,j,k Φ i+1,j,k Φ i-1,j,k Φ i+1,j,k Φ i,j-1,k Φ i,j+1,k Φ i,j-1,k Φ i,j+1,k Φ i,j,k-1 Φ i,j,k+1 Φ i,j,k-1 Φ i,j,k+1 G 1 i-1 2 ,j,k G 1 i+ 1 2 ,j,k G 2 i,j-1 2 ,k G 2 i,j+ 1 2 ,k G 3 i,j,k-1 2 G 3 i,j,k+ 1 2 Φ i,j,k
Therefore, we have

G 1 i-1/2,j,k ≈ F(Φ i-1,j,k ; Φ i,j,k ) G 1 i+1/2,j,k ≈ F(Φ i,j,k ; Φ i+1,j,k ) (16)
G 2 i,j-1/2,k ≈ F(Φ i,j-1,k ; Φ i,j,k ) G 2 i,j+1/2,k ≈ F(Φ i,j,k ; Φ i,j+1,k ) (17)
G 3 i,j,k-1/2 ≈ F(Φ i,j,k-1 ; Φ i,j,k ) G 3 i,j,k+1/2 ≈ F(Φ i,j,k ; Φ i,j,k+1 ) (18)
where F(• ; •) is a numerical flux function that will be specified in the following.

The fluxes in [START_REF] Galera | A two-dimensional unstructured cellcentered multi-material ALE scheme using VOF interface reconstruction[END_REF] are the same in the three spatial directions. Let us consider the one-dimensional problem in the x 1 direction

Φ t + (G 1 (Φ)) ,1 = 0 ( 19
)
where we have (Y i ,2 ) t = (Y i ,3 ) t = 0. These equations introduce six characteristics in [START_REF] Jiang | Efficient implementation of Weighted ENO schemes[END_REF] whose speeds are 0. This means also that Y i ,2 and Y i ,3 are just constant parameters in the other equations of the system. Therefore, in the next sections we focus on the wave structure of the following one-dimensional problem

Ψ t + (F (Ψ)) ,1 = 0 ( 20
)
where

Ψ =                              ρ ρu 1 ρu 2 ρu 3 Y 1 ,1 Y 2 ,1 Y 3 ,1 ρe                              F (Ψ) =                              ρu 1 (ρu 1 ) 2 ρ -σ 11 ρu 1 ρu 2 ρ -σ 21 ρu 1 ρu 3 ρ -σ 31 ρu 1 Y 1 ,1 +ρu 2 Y 1 ,2 +ρu 3 Y 1 ,3 ρ ρu 1 Y 2 ,1 +ρu 2 Y 2 ,2 +ρu 3 Y 2 ,3 ρ ρu 1 Y 3 ,1 +ρu 2 Y 3 ,2 +ρu 3 Y 3 ,3 ρ ρu 1 ρe-(σ 11 ρu 1 +σ 21 ρu 2 +σ 31 ρu 3 ) ρ .                             

Characteristic speeds

In this section we extend in three dimensions the criterion given in [START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF] to analytically compute the wave speeds. Also, we prove that when we consider a neohookean constitutive law, the wave speeds are real for any deformation under usual assumptions.

General case

The governing equations ( 20) are closed with the constitutive law (5)-( 7) which defines σ as a non-linear function of the unknowns. In this adiabatic and inviscid model, it can be shown that entropy is just transported along the characteristics for smooth solutions. Since the wave velocities are locally defined by infinitesimal smooth variations of the conservative variables Ψ, the energy equation can be simply replaced by s t + u • ∇s = 0. Thus, instead of ( 20), we study the following quasi-linear system:

                                ρ ρu 1 ρu 2 ρu 3 Y 1 ,1 Y 2 ,1 Y 3 ,1 s                                 ,t +                                 0 1 0 0 0 0 0 0 -(ρu 1 ) 2 ρ 2 2ρu 1 ρ 0 0 -σ 11 ,1 -σ 11 ,2 -σ 11 ,3 -σ 11 ,s -ρu 1 ρu 2 ρ 2 ρu 2 ρ ρu 1 ρ 0 -σ 21 ,1 -σ 21 ,2 -σ 21 ,3 -σ 21 ,s -ρu 1 ρu 3 ρ 2 ρu 3 ρ 0 ρu 1 ρ -σ 31 ,1 -σ 31 ,2 -σ 31 ,3 -σ 31 ,s - ρu 1 Y 1 ,1 +ρu 2 Y 1 ,2 +ρu 3 Y 1 ,3 ρ 2 Y 1 ,1 ρ Y 1 ,2 ρ Y 1 ,3 ρ ρu 1 ρ 0 0 0 - ρu 1 Y 2 ,1 +ρu 2 Y 2 ,2 +ρu 3 Y 2 ,3 ρ 2 Y 2 ,1 ρ Y 2 ,1 ρ Y 2 ,3 ρ 0 ρu 1 ρ 0 0 - ρu 1 Y 3 ,1 +ρu 2 Y 3 ,2 +ρu 3 Y 3 ,3 ρ 2 Y 3 ,1 ρ Y 3 ,1 ρ Y 3 ,3 ρ 0 0 ρu 1 ρ 0 0 0 0 0 0 0 0 ρu 1 ρ                                                                 ρ ρu 1 ρu 2 ρu 3 Y 1 ,1 Y 2 ,1 Y 3 ,1 s                                 ,1 =                                 0 0 0 0 0 0 0 0                                 (21)
where

σ ij ,1 , σ ij ,2 , σ ij ,3
, and σ ij ,s denote the derivative of

σ ij with respect to Y 1 ,1 , Y 2 ,1 , Y 3
,1 and s respectively. We denote by J ac the matrix appearing in [START_REF] Maire | A nominally second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids[END_REF].

We introduce the notation

Σ = [∇σ][∇Y ] :=          σ 11 ,1 σ 11 ,2 σ 11 ,3 σ 21 ,1 σ 21 ,2 σ 21 ,3 σ 31 ,1 σ 31 ,2 σ 31 ,3                   Y 1 ,1 Y 1 ,2 Y 1 ,3 Y 2 ,1 Y 2 ,2 Y 2 ,3 Y 3 ,1 Y 3 ,2 Y 3 ,3          (22)
The characteristic polynomial of J ac can be written under the form

P (λ) = (λ -u 1 ) 2 ρ 3 ((ρ(u 1 -λ) 2 ) 3 + Tr(Σ)(ρ(u 1 -λ) 2 ) 2 + Tr(Cof((Σ))(ρ(u 1 -λ) 2 ) + Det(Σ))
so that the eigenvalues of J ac are

Λ E = u 1 , u 1 , u 1 ± α 1 ρ , u 1 ± α 2 ρ , u 1 ± α 3 ρ ( 23
)
where α 1 , α 2 and α 3 are the roots of the polynomial of third order

X 3 + Tr(Σ)X 2 + Tr(Cof((Σ))X + Det(Σ) = 0 (24)
Therefore, the necessary conditions for system [START_REF] Kluth | Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme[END_REF] to be hyperbolic are α 1 > 0, α 2 > 0 and α 3 > 0. In [START_REF] Ndanou | Criterion of hyperbolicity in hyperelasticity in the case of the stored energy in separable form[END_REF] it is shown that these conditions are also sufficient to ensure that a complete system of eigenvectors of the Jacobian matrix exists.

Roots of third order polynomial

With the change of variables X = Z -Tr(Σ)/3, equation (24) reduces to

Z 3 + PZ + Q = 0 ( 25
)
where

P = Tr(Cof((Σ)) - Tr(Σ) 2 3 Q = 2Tr(Σ) 3 27 - Tr(Σ)Tr(Cof((Σ)) 3 + Det(Σ) (26)
The three solutions are reals if ∆ 3D = 4P 3 + 27Q 2 ≤ 0 (which implies P ≤ 0) and are given by

α k = 2 -P 3 cos   1 3 arccos   -sign(Q) 1 - ∆ 3D 4P 3   + 2kπ 3   - Tr(Σ) 3 k = 1, 2, 3 (27)
Moreover, the α i are positive if Tr(Σ) < 0, Tr(Cof((Σ)) > 0 and Det(Σ) < 0.

In the particular case where ∆ 3D = 0 we obtain a solution of multiplicity two

α 1 = 3Q P - Tr(Σ) 3 α 2 = α 3 = - 3Q 2P - Tr(Σ) 3 (28)

Hyperbolicity of the neohookean model

The internal energy ε with the neohookean model reads

ε = ε iso (ρ, s) + χ ρ 0 (Tr(B) -3) (29)
Let A be

A ij = (-1) i+j (K 1i K 1j + K 2i K 2j + K 3i K 3j ) (30)
where

K 11 = Y 2 ,2 Y 3 ,3 -Y 3 ,2 Y 2 ,3 K 21 = Y 1 ,2 Y 3 ,3 -Y 3 ,2 Y 1 ,3 K 31 = Y 1 ,2 Y 2 ,3 -Y 2 ,2 Y 1 ,3 K 12 = Y 2 ,1 Y 3 ,3 -Y 3 ,1 Y 2 ,3 K 22 = Y 1 ,1 Y 3 ,3 -Y 3 ,1 Y 1 ,3 K 32 = Y 1 ,1 Y 2 ,3 -Y 2 ,1 Y 1 ,3 K 13 = Y 2 ,1 Y 3 ,2 -Y 3 ,1 Y 2 ,2 K 23 = Y 1 ,1 Y 3 ,2 -Y 3 ,1 Y 1 ,2 K 33 = Y 1 ,1 Y 2 ,2 -Y 2 ,1 Y 1 ,2
We can show that J -1 B = AJ

         D 1 D 2 Jβ 2 + 6A 12 D 2 Jβ 3 + 6A 13 6A 12 -3β 2 JA 12 -9A 11 -3β 3 JA 12 6A 13 -3β 2 JA 13 -3β 3 JA 13 -9A 11          -ρc 2          1 0 0 0 0 0 0 0 0          (32)
where

D 1 = -2A 11 -5A 22 -5A 33 D 2 = -2A 11 + A 22 + A 33 β 2 = K 11 Y 1 ,2 -K 21 Y 2 ,2 + K 31 Y 3 ,2 β 3 = K 11 Y 1 ,3 -K 21 Y 2 ,3 + K 31 Y 3 ,3
and we have the relation

β 2 A 12 + β 3 A 13 = 0
Thus, it follows that Tr(Σ) = -ρc 2 + 10 9 χJ

1 3 (4A 11 + A 22 + A 33 ) (33)
Tr(Cof(Σ)) = 4χJ

1 3 A 11 ρc 2 + 1 9 χJ 1 3 (C 1 + 11A 2 11 + 5A 11 (A 22 + A 33 )) (34) det(Σ) = -(2χJ 1 3 ) 2 A 11 A 11 ρc 2 + 2 9 χJ 1 3 C 1 (35)
where

C 1 = 2A 2 11 + (A 12 ) 2 + (A 13 ) 2 + 5(A 11 A 22 -(A 12 ) 2 + A 11 A 33 -(A 13 ) 2 )
We have that C 1 ≥ 0 because

A 11 A 22 -(A 12 ) 2 = (K 12 K 21 -K 11 K 22 ) 2 + (K 12 K 31 -K 11 K 32 ) 2 + (K 22 K 31 -K 21 K 32 ) 2 ≥ 0 A 11 A 33 -(A 13 ) 2 = (K 13 K 21 -K 11 K 23 ) 2 + (K 13 K 31 -K 11 K 33 ) 2 + (K 23 K 31 -K 21 K 33 ) 2 ≥ 0
Since A ii ≥ 0, it follows that Tr(Σ) ≤ 0, Tr(Cof((Σ)) > 0 and Det(Σ) < 0.

The reduced parameters are then given by

P = - 1 3 ρc 2 - 2 9 χJ 1 3 C 2 2 + 4 3 (2χJ 1 3 ) 2 C 3 (36) Q = - 2 27 ρc 2 - 2 9 χJ 1 3 C 2 ρc 2 - 2 9 χJ 1 3 C 2 2 + 2(2χJ 1
3 ) 2 C 3 (37) ∆ 3D = - 1 9 4 3 (2χJ 1
3

) 2 C 3 2 ρc 2 - 2 9 χJ 1 3 C 2 2 + 16 9 (2χJ 1
3

) 2 C 3 ( 38
)
where

C 2 = 7A 11 -5(A 22 + A 33 ) C 3 = (A 12 ) 2 + (A 13 ) 2 ≥ 0
This result shows that the necessary conditions for the neohookean model to be hyperbolic are satisfied, since the wave speeds given by ( 23) and ( 27) are real for any deformation, under the usual assumptions c 2 > 0 and χ ≥ 0.

Small deformations

We investigate the simplifications of the wave speeds for small deformations.

In this case ∇Y ≈ I and A ij = δ ij . Hence, we have

C 1 = 12 C 2 = 3 C 3 = 0 and (33)-(35) reduce to Tr(Σ) = -ρc 2 + 20 3 χ ≤ 0 Tr(Cof(Σ)) = 4χ ρc 2 + 11 3 χ ≥ 0 det(Σ) = -4χ 2 ρc 2 + 8 3 χ ≤ 0
The reduced parameters are given by

P = - 1 3 ρc 2 + 2 3 χ 2 Q = - 2 27 ρc 2 + 2 3 χ 3 ∆ 3D = 0
Then, using ( 23) and ( 28), the waves speeds are given by

Λ E = u 1 , u 1 , u 1 ± c 2 + 4 3 2χ ρ , u 1 ± 2χ ρ , u 1 ± 2χ ρ (39)
Therefore, in the small deformation limit the speeds corresponding to transverse waves coincide, as in the classical theory of linear elasticity.

Numerical flux

In this section we first introduce the HLLC solver employed for a single material. Then we specify the numerical flux for the multimaterial case.

HLLC Solver

We consider equation [START_REF] Kluth | Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme[END_REF] with the initial condition

Ψ(x, t = 0) =          Ψ l if x ≤ 0 Ψ r if x > 0, (40)
The numerical flux function F(Ψ l ; Ψ r ) at the cell interface x = 0 is determined based on the solution of the HLLC [START_REF] Toro | Restoration of the contact surface in the HLL-Riemann solver[END_REF] approximate Riemann problem, similarly to what is done in [14,[START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF]. Even though the exact wave pattern involves seven distinct waves, see [START_REF] Miller | An iterative Riemann solver for systems of hyperbolic conservation laws, with application to hyperelastic solid mechanics[END_REF], the approximate solver approaches the solution using three waves (the contact discontinuity u 1 , the fastest leftward and rightward waves s l and s r ), thus inducing only two intermediate states Ψ -and Ψ + (see Fig. 3). associated fluxes F ± . The numerical flux at the cell interface x = 0 is then given by

x t s r u 1 s l Ψ r Ψ + Ψ - Ψ l
F(Ψ l ; Ψ r ) =                  F (Ψ l ) if 0 ≤ s l F -if s l ≤ 0 ≤ u 1 F + if u 1 ≤ 0 ≤ s r F (Ψ r ) if s r ≤ 0 ( 56
)
where F is defined in [START_REF] Kluth | Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme[END_REF].

Finally, the waves corresponding to the degenerate characteristics in Y i ,2 and Y i ,3 do not contribute to the numerical flux in the direction 1. The variables Y i ,2 and Y i ,3 are updated in time when the fluxes in the directions 2 and 3, respectively, are computed.

Multimaterial solver

The multimaterial solver is detailed in one dimension for sake of clarity and is the same as in [START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF]. In three dimensions we use the same method in all directions. We consider a case where the interface separating materials with different constitutive laws is located between the cell centers k -1 and k. The main idea of the multimaterial solver is that, instead of (56) we take

F l k-1/2 = F - F r k-1/2 = F + (57)
(see Fig. 4, equations ( 41) and ( 42)).

x As for ghost-fluid methods, the scheme is locally non conservative since F -= F + , but it is consistent since F ± are regular enough functions of the states to the left and to the right of the interface and F + = F -when those states are identical. In [START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF] it is shown that the error in conservation is negligible as the number of cell interfaces for which a non-conservative numerical flux is employed is always negligible compared to the total number of mesh cells.

Ψ k-1 Ψ k (k -1)∆x k∆x F k-3/2 (Ψ k-2 , Ψ k-1 ) F l k-1/2 F r k-1/2 F k+1/2 (Ψ k , Ψ k+1 )
A special care is needed at the multimaterial interface. To avoid the singularity due to the discontinuity of Y and to keep the interface sharp, the computation of (Y i ,2 ) ± and (Y i ,3 ) ± is one-sided:

(Y i ,2 ) -= (Y i ,2 ) l (Y i ,2 ) + = (Y i ,2 ) r ( 58
) (Y i ,3 ) -= (Y i ,3 ) l (Y i ,3 ) + = (Y i ,3 ) r ( 59
)
This can be seen as an extension by continuity of Y i ,2 and Y i ,3 relative to each material.

Extension to second-order in each space direction

The scheme is extended to second-order accuracy using a piecewise-linear slope reconstruction in space, with minmod limiter. For the cells separated by the multimaterial interface, the stencil used to compute the slopes is smaller.

We calculate these slopes using the corresponding intermediate state of the multimaterial Riemann problem. For example, if cells k -1 and k do not belong to the same medium, the slope of cell k -1 is computed using Ψ k-2 , Ψ k-1 and the intermediate state to the left of the contact discontinuity. This state is determined thanks to the solution of an approximate Riemann problem between Ψ k-1 and Ψ k , without slope reconstruction.

Interface advection and time integration

Coherently with a fully Eulerian approach, a level set function is used to follow the interface separating different materials. The level set function is transported with the velocity field by the equation :

ϕ t + u • ∇ϕ = 0 ( 60
)
This equation is approximated with a WENO 5 scheme [START_REF] Jiang | Efficient implementation of Weighted ENO schemes[END_REF].

The conservation equations and the interface advection are explicitly integrated in time by a Runge-Kutta 2 scheme. The interface position is advected using the material velocity field. For numerical stability, the integration step is limited by the fastest characteristics over the grid points. Hence, the interface position will belong to the same interval between two grid points for more than one time step. When the physical interface overcomes a grid point, the corresponding conservative variables, say Ψ k , do not correspond anymore to the material present at that grid point before the integration step. When the physical interface moves to the right, then we take Ψ k = Ψ -, whereas if it moves to the left Ψ k = Ψ + . In three dimensions this is done direction by direction taking the resulting average for the conservative variables.

We show in this section three-dimensional shock-bubble interactions and impacts in air. In particular, we simulate elastic rebounds in two and three dimensions. Computation are performed with an MPI parallel code.

Table 1 shows the material properties and initial conditions relative to each test case. Test cases description.

TC Media ρ [kg.m -3 ] u 1 [m.s -1 ] p [Pa] γ a b p ∞ [Pa] χ [Pa]

Shock-bubble interaction

We consider two shock-bubble interaction test cases involving different fluids.

The boundary conditions are fixed at the initial values for inlet and homogeneous Neumann conditions are imposed at outlet. Reflection conditions are imposed for all the other boundaries.

TC1 is the propagation in air of a Mach 1.22 shock through an helium bubble.

The initial configuration and the physical parameters are described in Fig. 5 and in Table 1. The computation is performed on a 1000 × 400 × 400 mesh and lasts for 50h on 300 processors. The zero iso-value of the level set function and schlieren on the horizontal plane through the center of the bubble are presented at different times on Fig. 6. The results are qualitatively similar to those presented in the literature in two dimensions, see for example [START_REF] Quirk | On the dynamics of a shock-bubble interaction[END_REF][START_REF] Fedkiw | A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)[END_REF]22,[START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF].

Air-water shock-bubble interaction TC2 involves a van der Waals gas bubble and a Mach 1.422 shock in water modeled by a stiffened gas constitutive law. The initial configuration and the physical parameters are described in Fig. 7 and in Table 1. This test case is more severe compared to the previous one since it presents larger density ratios. The mesh is 480 × 400 × 400 and the computation lasts 15h on 96 processors. The results are presented in Fig. 8, where we show the zero of the level set function and the schlieren on the vertical plane through the center of the bubble. The bubble is strongly compressed, it changes topology and swirls similarly to what happens in two dimensions, see [START_REF] Allaire | A five-equation model for the simulation of interfaces between compressible fluids[END_REF][START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF].

Impact

We present an impact simulation of a 800m.s -1 projectile on a plate in air (TC3). The initial configuration and the physical parameters are described in Fig. 9 and in Table 1. In this simulation the projectile and the plate are adjacent at initial time. The projectile and the plate are a single material and they are described by the same level set. In other words they will necessarily stick together for the whole simulation. Homogeneous Neumann conditions are imposed at the borders. The computation is performed on a 600 3 mesh and uses 216 processors for 60h. The results are shown on Fig. 10. The elastic material is deformed and oscillates while being displaced rightward. Schlieren images on the planes x = 0.03, y = 0 and z = 0 are presented. Compared to the two dimensional test case [START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF] the deformation is less evident as the mass ratio between the projectile and the plate is smaller. For this test case we present in Tab. 2 the convergence of the mass and energy conservation errors as functions of the grid refinement. The errors are computed by relative L 1 norms in space and time. The simulations last until the fluxes are zero across the domain boundaries for a mesh of 100 3 . This time represents 25% of the total computational time (fourth snapshot in Fig. 10). It can be observed that the conservation errors coherently decrease as the mesh is refined with a rate close to 1, as expected. However, for large elongations of the material, like in the case of rubbers, conservation errors may occur due to the level set discretization. For long filaments, it may occur that when a structure is smaller than a computational cell, the isoline zero of the level set function disappears, and with it, the material contained. Mass and energy conservation L 1 errors in space and time in percentage of the total mass and energy, respectively, for mesh sizes of 100 3 , 200 3 and 400 3 (TC3).

Also, we show in Tab. 3 the relative error between ρ and ρ 0 det(∇ x Y ), and the Frobenius norm of ∇ x × ∇ x Y for several mesh sizes. For these computations, we only consider solid cells situated at a distance of 0.025m from the interface in order to avoid the singularity of the inverse deformation tensor at the border of the elastic solids. The simulations last about half the total computational time (fifth snapshot in Fig. 10). It can be seen that the errors decrease as the mesh is refined, at a rate that is of about 1 for the density, and slightly smaller for ∇ x × ∇ x Y , as expected because of the additional differentiation. In the case where the mesh is 2000 2 , the rebound takes place thanks to a material contact of the disk with the copper layer. An high speed shock and the corresponding jet emerge from the impact. When the mesh size increases to 4000 2 cells, no contact occurs at the rebound and a thin layer of air cells still separates the two materials. The flow and deformation features observed for a 2000 2 mesh are overall retrieved for a 4000 2 mesh.

The pressure through y = 0 for both meshes is presented on Fig. 13. We can observe a significant pressure peak at the contact interface compressing the material and therefore the surrounding fluid that is ejected from the contact region at high speed. TC5 is relative to the impact of a the projectile at 800m.s -1 initially detached from a shield in two dimensions. The initial configuration and the physical parameters of the two-dimensional simulation are described in Fig. 14 and in Table 1. The computation is performed on a 4000 2 mesh and uses 24 2 processors for 6 hours. Interface and schlieren at t = 50µs, 83µs, 99µs, 149µs, 199µs, 298µs, 497µs, 710µs.

Here the disk actually rebounds on an shock layer created in front of the disk. This layer has a thickness of a few grid points. Again a high speed shock forms at the impact. The speed of this shock is of the order of that of the fast waves in the elastic solid. These results show that the overall scheme is robust on very fine meshes for stiff numerical simulations.

Three-dimensional rebound

The initial configuration and the physical parameters of the three-dimensional rebound (TC6) are described in Fig. 16 and in Table 1. The computation is performed on a 400 3 mesh and uses 216 processors for 24 hours. Overall the physics of the interaction is similar to what observed in two dimensions. In the figure the level set of the shield is in blue, whereas as the sphere is red. The level sets are also projected on the visualized planes.

Conclusions and Future Work

We have presented a three-dimensional multimaterial model that describes the dynamics of the interaction between gas, liquids and neohookean materials. This model is hyperbolic and reduces to the well known linear model in the small-deformation limit. The numerical scheme is based on an approximate Riemann solver that is able to model multimaterial interfaces with possibly frictionless contacts. The scheme is simple to implement, stable and robust, as shown by the stiff test cases presented.

The explicit solver proposed is constrained by the time scale of the elastic waves. For problems such as impacts or in general fast-dynamical processes, this is the relevant scale of the physical phenomenon. For physical phenomena that take place on the time scale determined by the material velocity, the stability condition limits the application of such time-dependent approaches. Also, the significantly different time scales present in multimaterial phenomena introduce accuracy and efficiency problems that are well known in low-speed compressible fluids. Future work will include exploration of these issues for compressible multimaterial flows, as well as, enriching the model with plasticity.

  isotropic if, and only if, ε is expressed as a function of the invariants of the right Cauchy-Green tensor C(ξ, t) = [∇ ξ X] T [∇ ξ X] or equivalently of the invariants of the left Cauchy-Green tensor B(ξ, t) = [∇ ξ X][∇ ξ X] T . The three dimensional invariants often considered in the literature are Tr(•), Tr(Cof(•)) = (Tr(•) 2 -Tr(• 2 ))/2 and Det(•).

Fig. 2 .

 2 Fig.2. Discretization on the control volume Ω i,j,k . Two-dimensional sections of the computational cell are projected on planes (xy), (yz) and (zx).

1 3.

 1 Therefore the expression of the stress tensor σ (11) becomes σ = -p(ρ, s)I + 2χJ of the sound speed c 2 (ρ, s)

Fig. 3 .

 3 Fig. 3. HLLC solver wave pattern.

Fig. 4 .

 4 Fig. 4. Fluxes at the material discontinuity.

1 Air

 1

Fig. 5 .

 5 Fig. 5. Sketch of the initial configuration for the shock-bubble interaction TC1. The computational domain is [247.5, 470] × [0, 89] × [0, 89]mm.

Fig. 6 .

 6 Fig. 6. Interaction of a Mach 1.22 shock propagating in air through an helium bubble (TC1). Pictures at t = 62µs, 110µs, 163µs, 264µs, 471µs, 735µs. From left to right, top to bottom.

Fig. 7 .

 7 Fig. 7. Sketch of the initial configuration for shock-bubble interaction TC2. The computational domain is [-0.2, 1] × [0, 1] × [0, 1]m.

Fig. 8 .

 8 Fig. 8. Interaction of a Mach 1.422 water shock and an air bubble (TC2). Interface and schlieren on the plane z = 0.5 at time t = 37µs, 73µs, 110µs, 132µs, 147µs, 195µs. From left to right, top to bottom.

Fig. 9 .

 9 Fig.9. Sketch of the 3D initial configuration for the impact test case TC3 where a projectile impinges on a cylindrical plate with a velocity of 800m.s -1 . The computational domain is [-0.5, 0.5] 3 m.

Fig. 10 .

 10 Fig. 10. Impact of a projectile on a plate (TC3). Interface and schlieren at t = 24µs, 43µs, 88µs, 178µs, 355µs, 710µs. From left to right, top to bottom.

  0 det(∇ x Y )| (%) 0.2300 0.1456 0.0511∇ x × ∇ x Y 0.1716 0.1258 0.0768

Fig. 12 .

 12 Fig. 12. Impact and rebound of a projectile on a layer in two dimensions. Velocity and schlieren at time t = 21µs, t = 450µs and t = 900µs from top to bottom. The mesh is 2000 2 on the left and 4000 2 on the right.

Fig. 13 .

 13 Fig. 13. Pressure on the line y = 0 in the range x ∈ [-0.4, 1] at time t = 450µs.

  Fig. 14. Sketch of the 2D initial configuration for impact and rebound test case TC5 where a projectile impinges a plate at velocity 800m.s -1 . The computational domain is [-0.5, 0.5] 2 m.

Fig. 15 .

 15 Fig. 15. Impact and rebound of a projectile on a plate in two dimensions (TC5).

Fig. 16 .

 16 Fig.[START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF]. Sketch of the initial configuration for the impact test case TC6 where a projectile impinges a shield at velocity 800m.s -1 and rebounds. The computational domain is [-0.5, 0.5] 3 m.

Fig. 17 .

 17 Fig. 17. Impact of a projectile on a shield (TC6). Interface and schlieren on the planes x = 0.025 and y = z = 0 at t = 107µs, 166µs, 206µs, 256µs, 383µs, 710µs. From left to right, top to bottom.

Table 3 L

 3 1 -norm error in space and time of |ρ -ρ 0 det(∇ x Y )| in percentage of ρ and L 1 -Frobenius norm in space and time of ∇ x × (∇ x Y ) for mesh sizes of 100 3 , 200 3 and

	400 3 (TC3).

Acknowledgements

This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the Investments for the future Programme IdEx Bordeaux-CPU (ANR-10-IDEX-03-02). The simulations presented in this paper were carried out using the PLAFRIM experimental parallel testbed, being developed under the Inria PlaFRIM development action with support from LABRI and IMB and other entities: Conseil Régional d'Aquitaine, FeDER, Université de Bordeaux and

Here the stresses (σ 11 ) , (σ 21 ) and (σ 31 ) are independent unknowns.

The HLLC scheme is based on the assumption that every wave is a shock and therefore Rankine-Hugoniot relations give

We introduce the notations

and we denote by Q i l (respectively Q i r ) the i-th component of Q l (respectively Q r ). In the following, except when otherwise explicitly stated, we take u -

The Rankine-Hugoniot relations (43) lead to

The relations for the velocities u ± i and the stresses (σ i1 ) are linked and are described in the three following cases.

(1) In the case where both Ψ l and Ψ r are solid states (χ = 0), all velocity and stress components are continuous across the contact discontinuity.

Hence, it follows that

(2) When at least one of the states Ψ l or Ψ r is fluid (χ = 0) or in the case where both Ψ l and Ψ r are solid states (χ = 0) with a frictionless contact, we have that u 1 and σ 11 are continuous. Hence, equation ( 50) is still verified. However since σ 21 and σ 31 vanish at the interface, the transverse velocities u 2 and u 3 can be discontinuous. It follows that equations (51)-( 52) are replaced by

(3) If Ψ l and Ψ r are solid states (χ = 0) relative to two different objects in frictionless contact, a rebound with material contact takes place when the normal velocities of the states Ψ l and Ψ r are such that u r > u l . In this case σ 11 , σ 21 and σ 31 vanish and the velocities u 1 , u 2 and u 3 are discontinuous. The stress components σ 11 , σ 21 and σ 31 are negligible based on the hypothesis that when u r > u l , a tiny film of gas is instantaneously formed between the two solids at a negligible pressure compared to the stress in the elastic materials. Hence, equations ( 53)-( 54) are still verified and equations ( 50) are replaced by

Similarly to [START_REF] Gorsse | A simple cartesian scheme for compressible multimaterials[END_REF], from equations (43) we consistently obtain

Finally, the robustness of the scheme is strongly influenced by the estimation of s l and s r . We use the estimate presented in [START_REF] Davis | Simplified second-order Godunov-type methods[END_REF] which is a simple way to obtain robust speed estimates :

where λ = max α i ρ .

Equations ( 45)-(55) completely define the intermediate states Ψ ± and their

We now focus on cases similar to impacts except that the projectile is not initially adjacent to the plate and can rebound. Here we do not consider the modeling of the full relevant physics at the scales induced by the arbitrarily small gap between the objects getting in contact. However, in terms of numerical modeling this case may need special care in the solution of the Riemann problem when the projectile and the plate get in material contact. Material contact is defined as a configuration where the level sets pertaining to the projectile and the plate are separated by less than a grid point. We model this phenomenon as a frictionless contact where both the normal and the tangential stresses vanish. This allows a discontinuity in both normal and tangential velocities in the solution of the Riemann problem, see equations ( 53),( 54) and (55).

Two-dimensional rebounds

First, we simulate the solution in two dimensions as the meshes that can be afforded are significantly finer. As depicted on Fig. 11, the first test case is relative to a copper disk moving toward a copper layer on the east boundary.

The surrounding fluid is air. The east border of the computational domain acts as a rigid wall where the displacement vanishes in the normal direction.

The other borders are modeled by homogeneous Neumann conditions.

The gap between the disk and the plate measures 10 -2 m and the disk travels at a speed of 500m.s -1 . A few snapshots of the solution are presented on Fig 12.

The computations are run on 2000 2 and 4000 2 meshes. The computations last 10h and 12h on 144 and 324 processors, respectively.


Senan Doyle

Florence Forbes

Michel Dojat

Automatic multiple sclerosis lesion segmentation with P-LOCUS

P-LOCUS provides automatic quantitative neuroimaging biomarker extraction tools to aid diagnosis, prognosis and follow-up in multiple sclerosis studies. The software performs accurate and precise segmentation of multiple sclerosis lesions in a multi-stage process. In the first step, a weighted Gaussian tissue model is used to perform a robust segmentation. The algorithm avails of complementary information from multiple MR sequences, and includes additional estimated weight variables to account for the relative importance of each voxel. These estimated weights are used to define candidate lesion voxels that are not well described by a normal tissue model. In the second step, the candidate lesion regions are used to populate the weighted Gaussian model and guide convergence to an optimal solution. The segmentation is unsupervised, removing the need for a training dataset, and providing independence from specific scanner type and MRI scanner protocol.

Introduction

MS brain lesion segmentation is important for diagnosis, prognosis, and patient follow-up. Typically, this task is performed manually by a medical expert, however automatic methods are sought to alleviate the tedious, time consuming and subjective nature of manual delineation. Automatic methods are motivated by the demand for large-scale multi-center clinical research studies that require precise, repeatable and cost-e cient analysis. Automatic brain image segmentation remains a challenging task for a number of reasons, including the presence of artefacts, and the heterogeneity of MRI scanner protocol.

Automated or semi-automated MS brain lesion detection methods can be classified according to their use of multiple sequences, a priori knowledge about the structure of normal brain, and the specific tissue segmentation model. In most approaches, normal brain tissue prior probability maps are used to help identify lesion as an outlier.

Existing methods frequently avail of complementary information from multiple sequences. For example, lesion voxels may appear hyperintense in one modality and normal in another. This is implicitly used by neurologists when examining data. In an statistical framework, complementary information from di↵erent sequences can help to better discriminate data generated by di↵erent probabilistic distributions in a multi-dimensional space. Intensity distributions are commonly modeled as multi-dimensional Gaussian distributions. This provides a way to combine the multiple sequences in a single segmentation task but with all the sequences having equal importance. Given that the information content and discriminative power to detect lesions varies between di↵erent MR sequences, we adopt a weighted data model, originally proposed by Forbes et al [START_REF] Forbes | Adaptive weighted fusion of multiple mr sequences for brain lesion segmentation[END_REF], that allows for the identification of atypical lesion voxels and the subsequent inclusion of the lesion class as an additional model component.

Weighted Model

We consider a finite set V of N voxels on a regular 3D grid. The intensities observed at each voxel are denoted by y = {y 1 , . . . , y N } . Each y i = {y i1 , . . . , y iM } is itself a vector of M intensity values corresponding to M di↵erent MR sequences. The goal is to assign each voxel i to one of K classes considering the observed features data y. For brain tissue segmentation, we consider in general 3 tissues plus some possible additional classes to account for lesions in pathological data. We denote the hidden classes by z = {z 1 , . . . , z N }, and the set in which z takes its values by Z. Typically, the z i 's take their values in {1 . . . K}. We consider non-negative weights ! = {! i , i 2 V } in a state space denoted by W and with ! i = {! i1 , . . . , ! iM }. In our general setting the weights are sequence and voxel-specific.

The segmentation task is recast into a missing data framework in which y are observations and z are missing variables. Their joint distribution p(y, z|!; ) is governed by the weights ! 2 W and parameters 2 , which are both unknown and need to be estimated. A prior distribution p(!) is defined on the weights, considered additional missing variables. Denoting the parameters by = { , }, we assume that the joint distribution p(y, z, !; ) is a MRF with the following energy function:

H(y, z, !; ) = H Z (z; ) + HW (!) + X i2V log g(yi|zi, !i; )
where the energy term HW (!) involving only ! does not depend on and the g(yi|zi, !i; )s are probability density functions of y i . For the data term P i2V log g(yi|zi, !i; ) in (1), we consider M-dimensional Gaussian distributions with diagonal covariance matrices. For each class k, t (µ k1 , . . . , µ kM ) is the mean vector and {s k1 , . . . , s kM } the covariance matrix components. When z i = k, then G(yim; µz i m, sz i m) represents the Gaussian distribution with mean µ km and variance s km . The whole set of Gaussian parameters is denoted by = {µ km , s km , k = 1 . . . K, m = 1 . . . M}. Our data term is then defined by setting

g(yi|zi, !i; ) = M Y m=1 G(yim; µz i m, sz i m !im ) ,
which is proportional to

M Q m=1 G(y im ; µ zim , s zim ) !im .
Intuitively, the impact of a larger ! im is to give more importance to the intensity value y im in the model. A weight of one recovers the standard multivariate Gaussian case.

The missing data term H Z (z; ) in ( 1) is set to a standard Potts model, with external field ⇠ and spatial interaction parameter ⌘, and whose energy is

H Z (z; ) = X i2V (⇠iz i + X j2N (i) ⌘ hzi, zji),
where N (i) denotes the voxels neighboring i and hz i , z j i is 1 when z i = z j and 0 otherwise. Parameter = {⇠, ⌘} with ⇠ = { t (⇠ i1 . . . ⇠ iK ), i 2 V } being a set of real-valued K-dimensional vectors and ⌘ a real positive value.

The weights are assumed independent from parameters and independent across modalities. The simplest choice is to define a prior

p(!) = Q M m=1 Q i2V p(! im )
where each p(! im ) is a Gamma distribution with hyperparameters ↵ im (shape) and im (inverse scale). Thus

HW (!) = M X m=1 X i2V ((↵im 1) log !im im !im).
In practice, the set of hyperparameters is fixed so that the modes of each prior p(! im ) are located at some expert weights {! exp im , m = 1 . . . M, i 2 V } accounting for some external knowledge, if available. Formally, we set ↵ im = im ! exp im +1 to achieve this. The expert weights can be chosen according to the specific task. For example, when voxels with typical lesion intensities are not numerous enough to attract a model component, increasing the expert weight for some of them will help in biasing the model toward the identification of a lesion class.

A solution to the model is found using the Expectation-Maximization (EM) framework [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF] combined with a variational approximation for tractability in the presence of Markov dependencies. In particular, the mean field principle provides a deterministic way to deal with intractable MRF models [START_REF] Celeux | EM procedures using mean field-like approximations for Markov model-based image segmentation[END_REF].

Method

Of the four possible input sequences available, the method uses only the unprocessed T1-weighted and Flair. The images are masked, co-registered and corrected for inhomogeneities using the N4 algorithm.

The segmentation process consists of two stages, as detailed in [START_REF] Forbes | A Weighted Multi-Sequence Markov Model For Brain Lesion Segmentation[END_REF]. In the first step, we set K = 3, considering only the three normal tissue classes (with all ! exp im and im set to 1). The ⇠ parameters in the MRF prior are set to ⇠ ik = log f ik where f ik is the normalized value given by a normal tissue atlas. The interaction parameter ⌘ is estimated using a stochastic gradient descent method as specified in [START_REF] Celeux | EM procedures using mean field-like approximations for Markov model-based image segmentation[END_REF]. The estimated weights for the Flair sequence are thresholded at a value of one to identify outlier regions corresponding to candidate lesion regions. This candidate region is refined using additional intensity, location and size constraints, as in [START_REF] Van Leemput | Automated segmentation of multiple sclerosis lesions by model outlier detection[END_REF][START_REF] Freifeld | Lesion detection in noisy MR brain images using constrained GMM and active contours[END_REF][START_REF] Garcia-Lorenzo | Multiple Sclerosis lesion segmentation using an automatic multimodal graph cuts[END_REF]. Retained lesions are hyperintense in Flair, confined to white matter and greater than 5mm 3 .

In the second step, the candidate region is used to specify the parameters of the weight distribution in a K = 4 segmentation setting. We set im according to: im = L for all i 2 L and im = L for all i 6 2 L, where L and L are values to be specified. We set L = 1000 to express our a priori trust in the estimation of the normal brain tissue classes from the preliminary first step, and set L = 10 to allow some flexibility in the weight estimation.

The expert weight is fixed to ! L = 2 and ! L = 1. Large values of ! L make the lesion class more representative and handle the possibility of very small lesions, while a small ! L ensures that the weighting of a large candidate lesion region does not a↵ect the estimation of other classes. A post-processing step removes artefacts based on spatial location.

Conclusion

The adaptive weighting model facilitates accurate and robust MS lesion segmentation from T1-weighted and Flair sequences. The advantage of this approach is that the weights, and therefore the 'outliers' are obtained in a multi-sequence framework that provides a more robust estimation of normal tissue parameters. The method is independent of MRI scanner, and does not require training data.


Nada Matta
email: nada.matta@utt.fr

Jason Xinghang Dai

François Rauscher
email: francois.rauscher@utt.fr

Hassan Atifi
email: hassan.atifi@utt.fr

Guillaume Ducellier
email: guillaume.ducellier@utt.fr

Xinghang Dai
email: xinghang.dai@utt.fr

How to Capture Knowledge from Project Environment?

Keywords: Knowledge representation, design project management, classification, project memory

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

From the beginning, knowledge is a human preoccupation. A lot of questions are still discussed: what is knowledge? How knowledge is built? How is it represented in mind? How can it be kept? How can it be learned? … The notion of Knowledge is defined from the antiquity. Platon, for instance, defined the thought as the intellectual model of objects. Heraclite went towards the definition of the logos as a triangle in which distinguished thought, from expression, from reality. Saussure defined the base of the semiotic as: a representation of knowledge involved in an activity, which is related to a specific symbol [START_REF] Saussure | Cours de linguistique générale[END_REF]. Currently these representations are more and more used to enhance learning from expertise and past experience. So, human who recog-nizes concepts in the reference makes sense. Based on this theory, knowledge engineering approaches provide techniques that help to represent expertise as references and to promote learning from these references, i.e. knowledge representation using semantic networks. For instance, currently, knowledge-sharing (semantic web studies) techniques use ontology as a guide to share a common sense of a concept in a domain [START_REF] Gruber | Toward principles for the design of ontologies used for knowledge sharing?[END_REF]. These techniques are commonly used to represent knowledge in a given domain for a given profession. In our work, we study knowledge produced from a cooperative activity as design projects, in which several actors with different skills and backgrounds work together to reach a given goal. Our challenge is to capture this type of knowledge related to work episodes and to extract and represent the deep knowledge, which belongs to the type of projects and design activities. This paper presents an approach for capturing knowledge from daily design project environment and aggregating this knowledge as classifications.

Design Project knowledge

Capturing knowledge from design projects

Design projects are currently done in cooperative way. Expert interviews, and

Textmining are not adequate to capture the collaborative dimensions of knowledge, which it is produced by interactions between actors. Knowledge can be extracted directly from daily work environment. So, techniques and tools as proposed on CSCW [START_REF] Schmidt | Coordination Mechanisms: Towards a conceptual foundation for CSCW systems design, Computer Supported Cooperative Work[END_REF] are used to support cooperative activities. Information and data will be captured directly from communication, coordination and decision-making support techniques. However, knowledge extracted from daily activity is mainly related to episodic memory. Routines and heuristics rules can then be defined to build semantic memory. By keeping track of knowledge during the realization of each project, the memory is defined as projects cases. These projects will be indexed not only by keywords and the projects' type but also by the main criteria underlined their execution.

This indexation must be linked to a typology of projects and problems. Aggregation and classifications of rules help to extract problem solving strategies related to project's typologies and problems (Erreur ! Source du renvoi introuvable.).

2.2

Representing design knowledge as Project memory

Design project knowledge can be represented as a project memory, which can be described as "the history of a project and the experience gained during the realization of a project" [START_REF] Matta | How to learn from design project knowledge[END_REF]. It must consider mainly:

•

The project organization: participants, competences, sub-teams, tasks, etc.

•

The reference frames used in the project various stages: rules, methods, laws…

•
The realization of the project: solutions evaluation and incidents management.

•

The decision making process: negotiation, argumentation and decision-making.

Often, there are interdependent relations among the various elements of a project memory. Links between these elements emphasize the design rationale in a project.

Project memory architecture Fig. 1.

Traceability of information from design projects

Keeping track of information from design projects mainly consists to extract knowledge from several sources: o Actor work-environment to be aware of activities. In our work, techniques are developed for traceability of decision meetings and e-mails.

Keeping track of decision meetings

Several approaches in CSCW are developed in order to represent design rationale.

We can note mainly IBIS and QOC, in which design rationale, named also as the space of design is represented as issues, options and arguments [START_REF] Shum | Representing Hard-to-Formalise, Contextualised, Multidisciplinary, Organisational Knowledge[END_REF]. Other approaches as DRCS and DIPA link decision-making to other elements of the projects like tasks, results and constraints [START_REF] Matta | How to learn from design project knowledge[END_REF]. To represent cooperative activity, links between elements from the project context and problem solving are needed. Context is important to enhance learning in an organization. The designer needs to match the context of his problem to past ones in order to understand them and solves his problem. Design rationale approaches links decision-making to some aspects of the projects context but it-missed links to project organizations as roles and skills of actors, etc. DYPKM [START_REF] Bekhti | Project memory: An approach of modelling and reusing the context and the design rationale[END_REF] approach recommends keeping track of design rationale from the project context and decision meetings. Structuring information cannot be done directly during the meetings. Also, the meetings animator cannot represent different views of discussions afterward as recommended in several design rationale techniques. Traceability of decision-making has to be done on two steps taking notes during the meetings and structuring notes to define report. The secretary in a meeting has to take notes of discussions in order to keep track of links between these discussions, questions and participants. When writing report, he/she has to distinguish suggestions from arguments and annotate them by criteria. In order to obtain this type of results and integrate traceability during an activity, a tool « Memory Meeting » [START_REF] Matta | How to learn from design project knowledge[END_REF] is defined to support collaborative decision-making traceability (Fig. 2). Results are then linked to other project parts using designers' tools like Product Life cycle management tools.

Example of structured report of decision-making meeting. Information are structured as Fig. 2. design rationale methods adding links to actors organization, skills, …

Keeping track from communication

Several approaches study and analyze e-mails as a specific discourse [START_REF] Rauscher | Discovering Problem-Solving Knowledge in Business Emails, Traceability in Software Design using Computer Mediated Communication[END_REF], for instance, tagging work. Other works use natural language processing in order to identify messages concerning tasks and commitments. Pragmatics analysis of e-mails uses some of these methods like ngrams analysis. However techniques studying e-mails, often do not consider the context of discussions, which is important to identify speaker's intention. In this work, e-mails are extracted from professional projects. So, pragmatics analysis and topic parsing are mixed. The results are linked to project context (skills and roles of messages senders and receivers, project phases, and deliverables, etc.) in order to keep track of speakers' intentions. As pragmatics analysis

shows [START_REF] Atifi | The Effectiveness of Professional Emails: Representations and Communicative Practices[END_REF], there is not only one grid to analyze different types of speech intentions. A grid of request act is built for our study to help the identification problem solving in interactions [START_REF] Rauscher | Discovering Problem-Solving Knowledge in Business Emails, Traceability in Software Design using Computer Mediated Communication[END_REF].

Firstly, messages related to project relevant topics are identified. Then, the volume of messages related to each subject and phases are calculated.

In each message thread (message and answers), are represented:

• Information about organization: authors, to whom, in Copy

• Information about phases: Date and hour of messages and answers

• Information about product: topic and joined files

• Information about message intention: main speech act. The role and skill of messages' senders and receivers help to analyze the role of the message in problem solving and the nature of the content (solution answering a prob-lem, proposition discussions, coordination messages, etc.). In the same way, linking messages to phases helps to identify the main problems in each one. To begin with, the speech act analysis on problem solving is focus on identifying request and solution. So, speech acts that help to localize a request in a message are identified [START_REF] Rauscher | Discovering Problem-Solving Knowledge in Business Emails, Traceability in Software Design using Computer Mediated Communication[END_REF].

Request act grid is identified for this aim. In this grid, there is two request types: direct and indirect. A "Direct" request may use an imperative or a performative form like obligations and want or need statements. An "Indirect" request may use query questions about ability, willingness, and capacity, or may use statements about the willingness (desire) [START_REF] Searle | Speech Acts. An Essay in the Philosophy of Language[END_REF].

Then, we study the organization of related messages thread in order to identify the solution proposed (if it exists) to the request (Fig. 3).

Example of communication analysis results

Knowledge discovery by classification

Low-level data in project memory should be mapped into other forms that might be more compact, more abstract, or more useful [START_REF] Dai | Knowledge Classification for Design project memory[END_REF]. In this section, a classification method to extract knowledge from design project memory will be proposed. In order to generate rules that represent interrelations between concepts or sub-networks, machine-learning techniques are considered. An evaluation of major machine learning techniques (statistical methods, decision tree, rule based method and neural network) is carried out in search for the appropriate algorithm [START_REF] Domingos | A few useful things to know about machine learning[END_REF]. Our intention is to classify project memory into rule-based knowledge, which leads us to choose a rule-based algorithm ITRULE. It can induce an optimal set of rules from a set of examples [START_REF] Dai | Knowledge Classification for Design project memory[END_REF]. 1. Problem-solving view: at a specific project phase, we can classify decision-making process for one particular issue. Solutions that are repetitive will be classified as essential solutions, the solutions that are distinctive will be considered as explorative attempt with its precondition as an explanation.

2. Cooperation view: this classification view allows verifying whether there are parallel tasks that involve cooperative design concerning whole project team. This rule will reveal the influence of concurrent design on project result. i.e. Fig. 4 3. Management view: this classification view will focus on project organization influence on different project memory modules.

Conclusion and perspective

In this paper, we presented our work on traceability and structuring of daily knowledge especially from design projects. Two techniques that help to capture knowledge from decision-making and from communication were described. More techniques in order to capture knowledge from designer's environment (linking to other work on user profiling) are under development. Captured knowledge needs to be classified in order to identify routines. Our classifications rules hypothesis needs to be tested in a large number of examples in order to identify ontology of design projects rules.

o

  Project management to keep organizations and context. o Workflow and documents to capture versioning of results.o E-mails, wikis to obtain interactions relayed to coordination.• Environment:o Meetings to capture negotiation and cooperation.

Fig. 3 .

 3 Knowledge captured like this can be stored as examples of projects. Aggregation of routines must be done in order to extract deep knowledge from these cases. Classification techniques are used for this aim.

  it on enriched XML XML Then project information will be classified according to different views to extract knowledge rules. Here we propose three classification views: Example of classification of Management influence. In 2012: 2 groups (mechanics vs. Fig. 4. Informatics), Decision: 4 databases. In 2013: groups are mix. Decision: 1 database.