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• It should be a sufficiently accurate and robust scheme to cope with **unstructured** three- dimensional **grids** composed of tetrahedral and/or hexahedral cells.
Before describing the main features of our finite volume scheme, let us briefly give an overview of the existing cell-centered diffusion scheme on three-dimensional **grids**. The simpler cell- centered finite volume is the so-called two-point flux approximation wherein the normal com- ponent of the heat flux at a cell interface is computed **using** the finite difference of the ad- jacent temperatures. It is well known that this method is consistent if and only if the com- putational grid is orthogonal with respect to the metrics induced by the symmetric positive definite conductivity tensor. This flaw renders this method inoperative for solving anisotropic diffusion problems on three-dimensional **unstructured** **grids**. It has motivated the work of Aavatsmark and his co-authors to develop a class of finite volume schemes based on multi- point flux approximations (MPFA) for solving the elliptic **flow** equation encountered in the context of reservoir simulation, refer to [2, 3]. In this method, the flux is approximated by a multi-point expression based on transmissibility coefficients. These coefficients are computed **using** the pointwise continuity of the normal flux and the temperature across cell interfaces. The link between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes is investigated in [36]. The class of MPFA methods is characterized by cell- centered unknowns and a local stencil. The global diffusion matrix corresponding to this type of schemes on general 3D **unstructured** **grids** is in general non-symmetric. There are many variants of the MPFA methods which differ in the choices of geometrical points and control volumes employed to derive the multi-point flux approximation. For more details about this method and its properties, the interested reader might refer to [4, 5, 17, 30] and the references therein.

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3
with the present contribution. First, Cartesian **grids** are considered instead of **unstructured** ones. Second, exact or approximate local Riemann problem solution is set in mixture cells to enforce interface conditions. In the present contribution, such ingredient is not used, this detail being important when dealing with sophisticated **flow** models, such as multiphase **flow** ones. Last, Ghost Cells in multi-D computations are filled with fluid state normal to the interface in a band (or layer) of cells of finite size. Determination of these cells in the normal direction to the interface may be challenging when dealing with **unstructured** **grids**. In the present contribution this issue is replaced by a simple averaging method. Fluid-fluid and solid-fluid coupling with Level-Set methods have been addressed in the frame of **unstructured** meshes by Farhat et al. (2008, 2012), Wang et al. (2011) and possibly other authors. It seems that similar restrictions as the former lists with Cartesian grid approaches are present:

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One of the current challenges of hydraulic modelling is in the size of the simulations. Larger and larger areas have to be considered to include the real boundaries of the whole hydrodynamic problem or to deal with the complex practical applications submitted to hydraulic engineers. On the other hand, fine **grids** have to be used to obtain a suitable representation of the hydrodynamic fields near the points of interest. Even if **unstructured** meshes or grid refinement allow to decrease the size and the computation time of such simulations, the coupling of different **solvers**, each one used in the part of the simulation where its fundamental characteristics are most suitable, open the door to still unreached modelling possibilities.

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1 Introduction
This work is devoted to the development of a coupling method for fluid-structure interaction in the **compressible** case. We intend to simulate transient dynamics problems, such as the impact of shock waves onto a structure, with possible fracturing causing the ultimate breaking of the structure. An inviscid fluid **flow** model is considered, being convenient for treating such short time scale phenomena. The simulation of fluid-structure interaction problems is often computationally challenging due to the generally different numerical methods used for solids and fluids and the instability that may occur when coupling these methods. Monolithic methods have been employed, **using** an Eulerian formulation for both the solid and the fluid (for instance, the diffusive interface method [16, 1]), or a Lagrangian formulation for both the fluid and the solid (for example, the PFEM method [26]), but in general, most solid **solvers** use Lagrangian formulations and fluid **solvers** use Eulerian formulations. In this paper we consider the coupling of a Lagrangian solid solver with an Eulerian fluid solver.

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Abstract
We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dy- namics equations on **unstructured** meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geomet- ric conservation law (GCL). Fluxes are computed **using** a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conserva- tive for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to as- sess this new scheme. The results obtained for various representative configurations of one and two-dimensional **compressible** fluid flows show the robustness and the accuracy of our new scheme. Key words: Lagrangian hydrodynamics, cell-centered scheme, Generalized Riemann problem, **compressible** **flow**, high-order finite volume methods, **unstructured** mesh

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the Lagrangian frame, we show why classical cell-centered Lagrangian schemes are not able to capture the low-Mach regime except by **using** unreasonably fine meshes. Consequently, we propose a slight modification of the original scheme, which is easy to implement in any scheme **using** an acoustic Godunov solver on **unstructured** mesh, and has a negligible cost in term of CPU time. We demonstrate that this modification cures this flaw. The properties of the orig- inal semi-discrete scheme (consistence, conservation) are preserved. Particular attention is paid to the entropy condition, proving its compatibility with the proposed modification. We assess this new scheme on several low and high- Mach problems, to demonstrate its good behavior in all regimes. Our last test problem is devoted to the study of the growth rate of instability in convergent configurations. It shows that even if the problem is globally very **compressible**, the low-Mach correction can have a significant impact on the solution.

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body motions, while lifting the constraint that mesh nodes remain attached to moving boundaries. The main advantage of this approach is to preserve the mesh connectivity as time evolves, making it well suited for Arbitrary Lagrangian Eulerian (ALE) **flow** **solvers**. By its very definition, the ALE approach combines both the Lagrangian and Eulerian reference frames and allows for a flexible, mov- ing grid. This is helpful in problems with large deformation of boundaries where the grid tracks the fluid or boundary. In addition, an often overlooked issue in moving **grids** is the discretization of the Geometric Conservation Law (GCL) presented in Farhat & Lesoinne (1996). These consist in two equations that state that cell volumes must be bounded by their surfaces (Surface Conservation Law, SCL) and that a volumetric increment of a moving cell must be equal to the sum of changes along the surfaces that enclose the volume (Volume Conservation Law, VCL). These requirements for the time dependent meshes in a finite volume method have been presented in Zhang et al. (1993), Trépanier et al. (1993) and are implicitly verified in the present approach as the grid connectivity is fixed. For example, the continuity equation on a moving grid is

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Project-Team HIEPACS
Research Report n ° 9396 — February 2021 — 16 pages
Abstract: Low-rank compression techniques are very promising for reducing memory footprint and execution time on a large spectrum of linear **solvers**. Sparse direct supernodal approaches are one these techniques. However, despite providing a very good scalability and reducing the memory footprint, they suffer from an important flops overhead in their **unstructured** low-rank updates. As a consequence, the execution time is not improved as expected. In this paper, we study a solution to improve low-rank compression techniques in sparse supernodal **solvers**. The proposed method tackles the overprice of the low-rank updates by identifying the blocks that have poor compression rates. We show that block incomplete LU factorization, thanks to the block fill-in levels, allows to identify most of these non-**compressible** blocks at low cost. This identification enables to postpone the low-rank compression step to trade small extra memory consumption for a better time to solution. The solution is validated within the PaStiX library with a large set of application matrices. It demonstrates sequential and multi-threaded speedup up to 8.5x, for small memory overhead of less than 1.49x with respect to the original version.

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expensive in transitional regimes, in particular for flows in the range of altitude we are interested in. The efficiency of DSMC can be improved by **using** coupling strategies (see [12, 14]) or implicit schemes (see [24, 15]), but these methods are still not very well suited for stationary computations. In contrast, deterministic methods (based on a numerical discretization of the stationary kinetic model) can be more efficient in transitional regimes. Up to our knowledge, there are few deterministic simulation codes specifically designed for steady flows. One of the most advanced ones is the 3D code of Titarev [25] developed for **unstructured** meshes. Another 3D code has been developed by G. Brook [10]. Other codes exist, but they are rather designed for unsteady problems, see for instance [20, 1] or the recent UGKS scheme developed by K. Xu and his collaborators [27, 18], which is an Asymptotic Preserving scheme for unsteady flows.

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is an accurate and robust method, which can produce impressive results, even on **unstructured** polygonal **grids**, see for instance [21].
An alternative to the staggered discretization is to use a conservative cell-centered discretization. This method for Lagrangian gas dynamics in one dimension, has been introduced by Godunov, see [12] and [24]. The multidimensional extension of this method has been performed during the eighties, [2] and [11]. This multidimensional scheme is a cell-centered finite volume scheme on moving structured or **unstructured** meshes. It is constructed by integrating directly the system of conservation laws on each moving cell. The primary variables, density, momentum and total energy are defined in the cells. The flux across the boundary of the cell is computed by solving exactly or approximately a one-dimensional Riemann problem in the direction normal to the boundary. The main problem with this type of method lies in the fact that the node velocity needed to move the mesh cannot be directly calculated. In [2], the node velocity is computed via a special least squares procedure. It consists in minimizing the error between the normal velocity coming from the Riemann solver and the normal projection of the vertex velocity. It turns out that it leads to an artificial grid motion, which requires a very expensive treatment [10]. Moreover, with this approach the flux calculation is not consistent with the node motion.

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Keywords: global stability, **compressible** leading-edge **flow**, direct numerical simulation, Krylov subspace methods, matrix-free implementation, Cayley transformation
R´ esum´ e
L’objet de cette th`ese est l’´etude d’un ´ecoulement **compressible**, impactant un bord d’attaque en fl`eche par une m´ethode de stabilit´e globale. Cette m´ethode, tr`es g´en´erale, permet une meilleure compr´ehension de la stabilit´e de cet ´ecoulement. La d´emarche suivie consiste en deux ´etapes : (i) le d´eveloppement d’un solveur de stabilit´e globale bas´e sur une simulation num´erique directe, et (ii) l’application de ce solveur ` a notre cas d’´etude. Concernant le premier point, nous avons utilis´e une m´ethode combinant des simulations num´eriques directes et des m´ethodes de type Krylov ne n´ecessitant pas la construction explicite de la matrice de stabilit´e (“matrix-free im- plementation”). La flexibilit´e de l’algorithme a ´et´e accrue par l’utilisation d’une transformation de Cayley et sa robustesse am´elior´ee par des techniques de pr´econditionement. La m´ethode ainsi d´evelopp´ee, a ´et´e appliqu´ee ` a l’´etude de la stabilit´e globale de l’´ecoulement **compressible** autour d’un cylindre de section parabolique. La dynamique temporelle des perturbations est caract´eris´ee par le spectre de cet ´ecoulement. Une grande vari´et´e de modes globaux a ´et´e de- couverte : des modes de couche limite, diff´erents types de modes acoustiques et des modes repr´esentant des paquets d’ondes. En particulier, nous avons ´etabli une connection spatiale en- tre les instabilit´es de type “attachment-line” et “crossflow” dans les modes de couche limite. De mˆeme, quelques modes globaux repr´esentant ` a la fois des structures dans la couche limite et des instabilit´es acoustique ont ´et´e trouv´es. Finalement, nous avons pr´esent´e la courbe de stabilit´e marginale des modes de couche limite et des modes acoustiques.

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In this article, we introduce an alternative approach related to X-FEM [24, 14, 22] and CutFEM [4, 17]. We start from surface triangulations generated by a segmentation tool. This surface mesh, which may be of poor quality, is then embedded in a regular and fixed background grid. We now generate signed distance functions for each body, **using** the regular background grid to represent them discretely. As per usual with level-set-based methods, the surfaces of the bodies will be represented by the zero contour line of these functions. Zero contour lines may intersect the background mesh in an arbitrary manner. This allows for a sharp and smooth (continuous and elements-wise linear in this paper) surface representation which are then used simulate contact and compute contact forces between the elastic bodies.

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(LAM P 0 -lim) and P 1 → P lim 1 → P 0 (LAM P 1 -lim) for the vertical displacement at the barycenter
of the plate (left) and the computed numerical dissipation (88) as a function of time (right).
5.3 Finite deformation of a cantilever thick beam
In [40] the authors present a test case involving a finite deformation of a 2D cantilever vertical thick beam of length L having a unit square cross section and initially loaded by a uniform horizontal velocity u 0 = 10 m.s −1 whilst the unit width base is maintained fixed, see figure 5 for a sketch. We consider the initial computational domain ω(t = 0) = [0; 1]×[0; 6] leading to L = 6 m and material characteristics ρ 0 = 1100 kg.m −3 , E = 1.7 · 10 7 Pa and ν = 0.45. Free boundary conditions are considered apart from the fixed-wall bottom of the bar. The mesh is made of N c = 5442 triangles. The simulations are run with the cascade P1 → P lim 1 → P0 . On the left panels of figure 8, we present the pressure distribution along with the deformed shapes at four different output times. The results are qualitatively in agreement with the published ones from the literature. Moreover we observe on the right panels that the yellow cells (unlimited second-order scheme) are massively represented, while only few of them demand dissipation (blue cells). For comparison purposes we also superimpose in black line the shapes obtained with the simpler cascade P1 → P0 from [13]. As can be observed, this latter scheme is genuinely more dissipative, and it numerically justifies the need for **using** a second order limited reconstruction within the cascade.

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In this test case, we projected the video on a two-plane wedge located 800mm away from the camera. First, we re- constructed the 3D wedge with a structured-light technique involving gray code and phase shift [ 18 ] on the full resolu- tion images. Then, another reconstruction was performed with our method by projecting the live band video sequence and **using** a temporal window W of 15 frames. As shown in Fig. 9 , we superimposed both 3D results and computed their differences in millimeters. As can be seen, the maximum er- ror is only 4mm. The average error for both planes is -1.9 mm and -1.3mm while the average angular error is approx- imately 2 degrees for both planes. We can see in Fig. 10 (a) a warped checkerboard projected on the wedge and (b) its projection as seen from an arbitrary point of view.

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sin θ,
where (r, θ) are the polar coordinates corresponding to the Cartesian coordinates (x, y). The corresponding distorted grid obtained setting a0 = 0.25 and n = 8 is plotted in Figure 22(b). Now, we compute the numerical solution of the non-linear test problem on the above distorted grid, **using** two different schemes. These are: the classical five-point scheme and the CCLADS scheme which reduces to a nine-point scheme on quadrangular **grids**. The numerical solution resulting from the five-point scheme is plotted in Figure 23 **using** blue dots. We have displayed the temperatures in all cells as function of the cell center radius versus the reference solution. In this figure, we observe the main flaw of the five-point scheme: in spite of its robustness, it produces a numerical solution wherein the temperature front is aligned with the grid distortion. The corresponding numerical solution is not able to preserve the cylindrical symmetry. In addition, the comparison to the reference solution, shows that the timing of the thermal wave is completely wrong. Let us emphasize that this test case is not a fake problem. It is representative of situations which frequently occur in the framework of plasma physics simulation wherein the heat conduction equation is coupled with a numerical method solving Lagrangian hydrodynamics equations. In this case, grid distortions are induced by fluid motion and thus the use of the five-point scheme to solve the heat conduction equation leads to a very bad result. This weakness of the five-point scheme follows from the fact that its construction is based on a two-point flux approximation 3 , which becomes inaccurate in the

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10: Output(mesh groups); /* Return repartitioned groups */
• Non-intrusive linkage with third-party **solvers**;
• Improvable parallel performances by means of dynamic load balancing.
Open-source software packages are used for every step in the computing chain, from mesh parti- tioning, remeshing, node renumbering, mesh visualization. The remeshing kernel is the sequen- tial Mmg library [DDF14] [Mmg]. Parallelization is performed through Message Passing Interface (MPI) libraries. Partitioning of a centralized input mesh is performed by means of the Metis library [Kar] [Met], and the Scotch library [Sco] is employed for nodes renumbering to reduce cache misses. Finally, mesh files can be saved for visualization in the Medit format (readable by Medit [Fre01] [Med] and Gmsh [GR09]) and in VTK format [Sch+06][Vtk]. Finally, version control is performed with Git and continuous integration testing with Jenkins.

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non structur´ es bidimensionnels
R´ esum´ e : In this paper, we describe a cell-centered Lagrangian scheme devoted to the numeri- cal simulation of solid dynamics on two-dimensional **unstructured** **grids** in planar geometry. This numerical method, utilizes the classical elastic-perfectly plastic material model initially proposed by Wilkins [M.L. Wilkins, Calculation of elastic-plastic **flow**, Meth. Comput. Phys. (1964)]. In this model, the Cauchy stress tensor is decomposed into the sum of its deviatoric part and the thermodynamic pressure which is defined by means of an equation of state. Regarding the devi- atoric stress, its time evolution is governed by a classical constitutive law for isotropic material. The plasticity model employs the von Mises yield criterion and is implemented by means of the radial return algorithm. The numerical scheme relies on a finite volume cell-centered method wherein numerical fluxes are expressed in terms of sub-cell force. The generic form of the sub-cell force is obtained by requiring the scheme to satisfy a semi-discrete dissipation inequality. Sub-cell force and nodal velocity to move the grid are computed consistently with cell volume variation by means of a node-centered solver, which results from total energy conservation. The nominally second-order extension is achieved by developing a two-dimensional extension in the Lagrangian framework of the Generalized Riemann Problem methodology, introduced by Ben-Artzi and Fal- covitz [M. Ben-Artzi and J. Falcovitz, Generalized Riemann Problems in Computational Fluid Dynamics, Cambridge Monographs on Applied and Computational Mathematics, 2003]. Finally, the robustness and the accuracy of the numerical scheme are assessed through the computation of several test cases.

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The performance of both parallel 1D and 2D methods depends on the choice of the values for a number of different parameters. For the 1D case these are the block- column nb on which depends the amount of concurrency and the internal block size ib on which depend the efficiency of elementary BLAS operations and the global amount of flop (this parameter defines how well the staircase structure of each front is ex- ploited). For the parallel 2D STF case these parameters are, the tiles size (mb,nb), the type of panel reduction algorithm set by the bh parameter described in Section 4.2 and the internal block size ib. The choice of these values depends on a number of factors, such as the number of working threads, the size and structure of the matrix, the shape of the elimination tree and of frontal matrices and the features of the underlying archi- tecture. It has to be noted that these parameters may be set to different values for each frontal matrix; moreover, it would be possible to let the software automatically choose values for these parameters. Both these tasks are very difficult and challenging and are out of the scope of this work. Therefore, for our experiments we performed a large number of runs with varying values for all these parameters, **using** the same values for all the fronts in the elimination tree, and selected the best results (shortest running time) among those. For the sequential runs internal block sizes ib={32, 40, 64, 80, 128} were used for a total of five runs per matrix. For the 1D parallel STF case, the used values were (nb,ib)={(128,32), (128,64), (128,128), (160,40), (160,80)} for a total of five runs per matrix. For the 2D case (nb,ib)={(160,32), (160,40), (192, 32), (192,64)}, mb={nb, nb*2, nb*3, nb*4} and bh={4, 8, 12, 16, 20, 24, ∞} for a total of 112 runs per matrix. This choice of values stems from the following observations. For the 1D case, tasks are of relatively coarse grain and, therefore, it is beneficial to choose smaller values for nb with respect to the 2D case in order to improve concurrency. On the other side, in the 2D case concurrency is aplenty and therefore it is beneficial to choose a relatively large nb value in order to achieve a bet- ter BLAS efficiency and keep the runtime overhead small; the internal block size ib, however, has to be relatively small to keep the flop overhead under control.

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Unite´ de recherche INRIA Lorraine, Technoˆpole de Nancy-Brabois, Campus scientifique, 615 rue de Jardin Botanique, BP 101, 54600 VILLERS LE`S NANCY Unite´ de recherche INRIA Rennes, IRI[r]