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temperature. To account for compressibility, regarding what was done for vis- cous **flows**, various **numerical** methods have been proposed to solve compressible or incompressible problems, for what concerns independent variables, linear sys- tem solvers, and **numerical** stability. Several authors developed in the past unified computational methods for compressible and incompressible viscous **flows** [24–30], showing results for **a** wide range **of** flow speeds, but in two-dimensional simple geometries. Extensions **of** low to vanishing Mach number compressible **flows** to viscoelastic constitutive models have been studied by Webster and co-workers [32, 33, 31] in **a** very comprehensive work, and were compared to experimental results in [34]. However, these studies were devoted to two-dimensional **flows**. Since our objective remains the possibility **of** application **of** this kind **of** models for three- dimensional industrial applications, we have extended **a** previous approach involv- ing weakly compressible **flows** **of** generalized newtonian fluids [35] to viscoelastic constitutive models. We split compressible and viscoelastic effects by considering that each brings its contribution to the stress determination: compressibility was taken into account classically through **a** state equation (the Tait law), whereas viscoelasticity was determined through the evolution **of** the extra-stress tensor. The proposed formulation for the 3D non-steady flow **of** **a** viscoelastic compressible fluid will be presented as follows: in the first part, we outline the material behavior modelling, by drawing the basic features **of** the multi-mode Pom-Pom model, that has been implemented to treat the non-linear viscoelastic behavior, and how does the state law relates density evolution **with** pressure and temperature; in the next section, the computational methods used to solve the viscoelastic compressible flow problem are detailed. Two basic problems need to be treated: the presence **of** convective terms in constitutive equations, and strong non-linearity when we con- sider the coupled problem. **A** splitting **method** allowed separate resolution **of** the evolution equations (through **a** Space-Time Discontinuous Galerkin **finite** **element** **method**) from the flow equations, using mixed **finite** elements. Furthermore, exten- sion to free surface, that is briefly described here, increased the complexity **of** our problem. Validation tests performed on ’benchmark’ geometries (like the contrac- tion or contraction/ expansion flow) are analyzed, and finally, three-dimensional free-surface **flows** are considered in simple and complex geometries.

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1 rue de la Noë, 44321 Nantes, France
SUMMARY
An eXtended Stochastic **Finite** **Element** **Method** has been recently proposed for the **numerical** solution **of** partial dierential equations dened on random domains. This **method** is based on **a** mariage between the eXtended **Finite** **Element** **Method** and spectral stochastic methods. In this paper, we propose an extension **of** this **method** for the **numerical** **simulation** **of** random multi-phased materials. The random geometry **of** material interfaces is described implicitly by using random level-set functions. **A** xed deterministic nite **element** mesh, which is not conforming the random interfaces, is then introduced in order to approximate the geometry and the solution. Classical spectral stochastic nite **element** approximation spaces are not able to capture the irregularities **of** the solution eld **with** respect to spatial and stochastic variables, which leads to **a** deterioration **of** the accuracy and convergence properties **of** the approximate solution. In order to recover optimal convergence properties **of** the approximation, we propose an extension **of** the partition **of** unity **method** to the spectral stochastic framework. This technique allows the enrichment **of** approximation spaces **with** suitable functions based on an **a** priori knowledge **of** the irregularities in the solution. **Numerical** examples illustrate the eciency **of** the proposed **method** and demonstrate the relevance **of** the enrichment procedure. key words: Stochastic partial dierential equations; Random geometry; Random Level-sets; X- FEM; Spectral Stochastic Methods; Partition **of** Unity **Method**

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Fig. 4 – Sketch **of** the rotating drum case.
As stated in step 2 **a** non-conformal mesh matching algo- rithm is used after rotation. Indeed, even if the matching between meshes is conformal at the ﬁrst iteration, it will not be the case at the subsequent ones. Indeed, the rotor mesh will have rotated by an angle which does not necessarily coin- cide **with** the angular discretization **of** the meshes. Hence, in order to solve the governing equations, information has to be passed at the non-conforming interface between the stator and the rotor. This is done thanks to **a** mesh joining algorithm and has to be performed at each time step. The main idea **of** this **method** is to intersect the faces **of** both interfaces and split them so as to build conforming meshes. An illustration **of** this **method** is shown in Fig. **3**. More information on this procedure can be found in EDF (2015).

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Similar results hold on subsets **of** R, provided the discretizations are conforming. Remark **3**. If one chooses another discretization, all results presented hereafter hold provided the estimates (4.20) remain true. For instance, for the RTN [k] **finite**
**element** defined on tetrahedral triangulations **of** R, cf. [14, §2.3.1]. To prove (4.20) in this case, one has simply to apply the results **of** [18, §3.2]. On the other hand, provided that the field q and its divergence are “smooth” in the sense that they belong to PH m+1 (R) for some integer m ≥ 0, using the RTN

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u 0 = 100 m/s. The pressure evolution at the center **of** the plate is shown in Fig. 5(**a**). The results obtained **with** the two geometries are quite similar except for the amplitude **of** the initial peak, which is much higher in the case **of** **a** ﬂat impact surface. This fact has already been discussed in the literature (see e.g. [7]) and could be expected somehow since the impact is less abrupt in the case **of** an hemispheric impactor. In Fig. 5(b) the results obtained **with** an hemispheric impact surface are compared to those derived experimentally by Wilbeck [15] for **a** similar impact speed (117 m /s). The agreement is very good, except for the amplitude **of** the initial peak. Nonetheless, some considerations are in order. First **of** all, the correct amplitude **of** this peak is extremely delicate to evaluate and **a** value **of** 12 times the stagnation pressure is not unphysical, as it can be found elsewhere in the literature both in **numerical** and experimental results (see again [7]). Moreover, in this case the analysis is limited to two dimensions and to an incompressible ﬂuid, while experimental data result, **of** course, from three-dimensional tests, and from the use **of** slightly compressible projectiles.

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Yannick Louvigny and Pierre Duysinx
Figure 1: Nodal displacements (amplified). Figure 2: Twist angle evolution.
isolated from the displacements related to the crankshaft motion. It had been possible because super **element** methods have been used in addition to the **finite** **element** **method** to model the crankshaft. The crankshaft twist angle, measured between the two crankpin, is then determined. Its value is obtained by calculating the difference **of** rotation angle between each crankpin central nodes. Figure 2 illustrates the variations **of** crankshaft twist angle during one complete four-stroke cycle. One notices that the twist angle is proportional to the instant torque produced by the engine as also noted in [**3**] and [4]. To study the dynamics **of** the engine in transient situation, simplified torsional models **of** the crankshaft (including super **element** models and beam **element** models), allowing faster simulations, are developed. To validate the accuracy **of** these crankshaft models, they are used in steady state simulations and the results obtained are compared to the ones obtained **with** the pure **finite** **element** model.

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P. Ciarlet Jr. 1 , L. Giret 1 , 2 , E. Jamelot **3** , * and F.D. Kpadonou 1 , 4
Abstract. We study first the convergence **of** the **finite** **element** approximation **of** the mixed diffusion equations **with** **a** source term, in the case where the solution is **of** low regularity. Such **a** situation commonly arises in the presence **of** three or more intersecting material components **with** different characteristics. Then we focus on the approximation **of** the associated eigenvalue problem. We prove spectral correctness for this problem in the mixed setting. These studies are carried out without, and then **with** **a** domain decomposition **method**. The domain decomposition **method** can be non-matching in the sense that the traces **of** the **finite** **element** spaces may not fit at the interface between subdomains. Finally, **numerical** experiments illustrate the accuracy **of** the **method**.

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The **numerical** **simulation** **of** impacts on dissipative solids has been and is again mainly per- formed **with** the classical **finite** **element** **method** coupled **with** centered differences or Newmark **finite** difference schemes in time [4]. Though this approach has well-known advantages, clas- sical time integrators introduce high frequency noise in the vicinity **of** discontinuities which is hard to remove **with** artificial viscosity without destroying the accuracy **of** the **numerical** so- lution. The **finite** volume **method**, initially developed for the **simulation** **of** gas dynamics [1], has gained recently more and more interest for problems involving impacts on solid media [5, 6, 7, 8]. This **method** shows some advantages to achieve an accurate tracking **of** wavefronts; among others (i) continuity **of** fields is not enforced on the mesh in its cell-centered version, that allows for capturing discontinuous solutions, (ii) the characteristic structure **of** hyperbolic equations can be introduced within the **numerical** solution, either through the explicit solution **of** **a** Riemann problem at cell interfaces, or in **a** implicit way through the construction **of** the **numerical** scheme, (iii) the amount **of** **numerical** viscosity introduced can be controlled locally as **a** function **of** the local regularity **of** the solution, so that to permit the elimination **of** spurious **numerical** oscillations while preserving **a** high order **of** accuracy in more regular zones.

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This work departs from the classical van der Waals type phase change mod- elling which usually requires additional terms related to very small scale effects (see [15]). These are intended to correct the core system intrinsic lack **of** hyperbolicity. **A** drawback **of** this approach is that it requires **numerical** strategies to use very fine discretizing grids. On the contrary, the present system is fully compliant **with** standard **numerical** relaxation techniques for hyperbolic systems although the model equilibrium states are compatible **with** the equilibium Maxwell points for **a** van der Waals law.

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5. Brenner, K., Canc`es, C., and Hilhorst, D. : **Finite** volume approximation for an immiscible two-phase flow in porous media **with** discontinuous capillary pressure. Computational Geosciences 17.3 (2013): 573-597.
6. Brezis, H. : Analyse fonctionnelle: Th´eorie et applications. Vol. 91. Paris: Dunod, 1999.
7. Brinkman, H. C. : **A** calculation **of** the viscous force exerted by **a** flowing fluid on **a** dense swarm **of** particles. Applied Scientific Research 1.1 (1949): 27-34.

In the present study, the proposed **method** aims to overcome this drawback. The main idea is to retain the use **of** the monolithic formulation and coupling it to some additional features that could allow **a** better and accurate resolution, in particularly at the interface between the fluid and solid. Recall that the monolithic resolution, based on the levelset approach consists in considering **a** single grid for both air and solid for which only one set **of** equations need to be solved. Consequently, different subdomains are treated as **a** single fluid **with** variable material properties. One important feature till now is that by solving the whole domain in **a** fully monolithic way there is no need **of** empirical data so as to determine the heat transfer coefficient. The heat exchange at the interface is replaced naturally by solving the convective fluid in the whole domain. Note also that different **numerical** methods introduced in the previous chapters could be used to solve the conjugate and coupled problem without additional efforts. Numerically, the communication between the solid and the fluid is obtained naturally without any further assumption and force modelling. In other words, there is no need for some coupling engines specifically designed to handle data exchange and algorithmic control signals between solid region and fluid region.

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Name and hardening type Swift parameters Back-stress
Swift
Isotropic hardening K=472.19MPa; n=0.171; ε 0 =0.001 C x =51.65; X sat =5.3
Figure 6 illustrates two distinct meshes refinements. Due to the square geometry in XY plane (see Figure 5.**a**) and to benefit the computation time reduction, only half **of** the sheet is modelled. This simplification also can provide **a** similar result as **a** full mesh (Henrard, 2008). The initial refined mesh (reference mesh) is composed by 2048 elements disposed in one layer **of** RESS **finite** **element** in thickness direction. The coarse mesh used **with** adaptive remeshing **method** is modelled by 128 elements on the sheet plane **with** one layer **of** RESS **finite** **element** in thickness direction. However, the nodes at the top layer **of** both meshes define the contact **element** layer at the surface. The contact modelling is based on **a** penalty approach and on **a** Coulomb law (Habraken and Cescotto, 1998). So, both meshes have two layers **of** elements (solid-shell + contact **element**) in thickness direction, and the spherical tool was modelled as **a** rigid body. Finally, Coulomb friction coefficient between the tool and sheet is set to 0.05 (Henrard et al., 2010) and the penalty coefficient is equal to 1000 [N.mm -**3** ].

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The **numerical** scheme proposed in this paper is **a** new cut-cell **method**. As in [16], the immersed boundary is geometrically represented by using the signed algebraic distance to the obstacle boundary. In fluid-cells, that is mesh cells which are far enough from the immersed boundary, classical centered, second-order **finite** volume schemes are used. In our approach, the location **of** the velocity component is, as in [16], adapted to the geometry **of** cut-cells. However, the discrete pressure is placed at the center **of** the cartesian cells for both fluid-cells and cut-cells. In the vicinity **of** the obstacle, second-order interpolations using boundary conditions on the solid boundaries are intro- duced to evaluate the convective fluxes. This results in **a** local first-order approximation **of** the nonlinear terms in cut-cells. **A** pointwise approxima- tion **of** the viscous terms is used in cut-cells. When boundary conditions on the immersed boundary can be used, **a** five-point stencil scheme for the viscous term is employed. Otherwise, **a** six-point first-order approximation is introduced. The resulting linear system is close to the five-point structure symmetric system obtained on cartesian mesh **with** the MAC scheme. **A** direct solver, based on **a** capacitance matrix **method**, is proposed. The effi- ciency **of** the solver is similar to the cartesian grid solver obtained **with** the MAC scheme. The incompressibility **of** the discrete velocity field is enforced up to the computer accuracy. While first-order truncation errors are locally introduced in the scheme in the cut-cells, **a** second-order global accuracy is recovered. Note that **a** similar superconvergence result has been proved by Yamamoto in [19] in the context **of** elliptic equations.

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Albeit these modelization issue, the use **of** state-**of**-the-art industrial **simulation** tools such as FEM usually imply tedious meshing steps in the case **of** large and complex assemblies, **with** complex and manual meshing steps. The idealisation and simplification **of** these structures into **a** mix **of** 2D and 3D **Finite** Elements usually takes significantly more time than the analysis itself. This is **a** major drawback if many complex designs are to be explored: the meshing parametrization may have to be done separately from the CAD parametrization or **with** complex coding **of** case-dependent meshing rules. This is why an alternative and pragmatic path is proposed here for the **simulation** **of** thin structures. This strategy relies on two main ingredients: (i) the eXtended **Finite** **Element** **Method** (X-FEM) [9] and (ii) the Level-Set **method** [10]. The eXtended **Finite** **Element** **Method** specificities are used to explore **a** calculation process that enables **a** simple automation **of** the meshing steps. Even though potentially computationally more expensive, the meshing automation allowed by the X-FEM may lead to drastic time reduction **of** the CAD to mesh process and **a** much tighter link between CAD and calculated assembly. This process may therefore allow easier and faster design explorations in an industrial context.

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where the c i are the unknown values; we have the analogous for the two-dimension framework. So
at each time step, one has to solve **a** non linear matrix system by an iterative **method**.
Then, using the value **of** the normal relative velocity −Dn.∇c at the interfaces between two neighbouring cells (n is the unit normal to the interface), one solves the mixing kinetic energy equation (28). It is done **with** **a** **finite** volume scheme; the only technical point is the advection term (2c − 1)ρKD∇c which is handled **with** an upwind technique which is explicit **with** respect to the time (in each cell, it depends on the sign **of** (2c − 1)n.∇c in the neighbouring cells). **Of** course, the time step has to satisfy **a** classical stabilty criterium for this upwind technique which is **of** the type |(2c − 1)Dn.∇c|δt/δx ≤ 1, but this criterium is generally less strong than the standard one related to the Wilkins scheme v ther δt/δx ≤ 1, where v ther is the sound speed.

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1. Introduction
During the last decades, the room available in car vehicles (e.g. in engine compartment) has plummeted because **of** the rapid development **of** on-board electronics. As **a** result, **a** need for very accurate **numerical** tools for design has appeared in automotive industry. In the meantime, **a** fast computation is necessary so that design duration remains suitable for industry. In this context, ﬂexible pieces represent an outstanding challenge since, unlike most **of** car pieces, they cannot be modeled as rigid body solids in CAD software. This paper focuses on **a** speciﬁc type **of** ﬂexible piece, namely electrical cables. Cables have **a** complex struc- ture. **A** wire is made up **of** copper ﬁlaments wrapped in an elastomer duct. These wires are most **of** the time gathered in bundles which are themselves surrounded by various protections such as tape, PVC tube … Moreover, the full cable is often constituted **of** several drifted cable pieces forming **a** system **with** **a** complex geometry.

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