Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex France Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - [r]

A centered second-order finite volume scheme for the heterogeneous Maxwell equations in three dimensions on arbitrary unstructured meshes Serge Piperno — Malika Remaki — Loula Fezoui.. a[r]

1.3. Explicit versus implicit DGTD methods
From **the** above discussion, it is clear that **the** DGTD method is nowadays **a** very popular numerical method in **the** computational electromagnetics community. **The** works mentioned so far are mostly concerned with time explicit DGTD methods relying on **the** use of **a** single global time step computed so as to ensure stability of **the** simulation. It is however well known that when combined with an explicit time integration method and in **the** presence of an **unstructured** locally refine mesh, **a** high order DGTD method suffers from **a** severe time step size restriction. **A** possible alternative to overcome this limitation is to use smaller time steps, given by **a** local stability criterion, precisely where **the** smallest elements are located. **The** local character of **a** DG formulation is **a** very attractive feature **for** **the** development of explicit local time stepping schemes [20]-[21]-[22]. An alternative approach that has been considered in [23]-[24] is to use **a** hybrid explicit-implicit (or locally implicit) time integration strategy. Such **a** strategy relies on **a** component splitting deduced from **a** partitioning of **the** mesh cells in two sets respectively gathering coarse and fine elements. In these works, **a** second-order explicit leap-frog **scheme** is combined with **a** second-order implicit Crank-Nicolson **scheme** in **the** framework of **a** non-dissipative (**centered** flux based) DG discretization in space. At each time step, **a** large linear system must be solved whose structure is partly diagonal (**for** those rows of **the** system associated to **the** explicit unknowns) and partly sparse (**for** those rows of **the** system associated to **the** implicit unknowns). **The** computational efficiency of this locally implicit DGTD method depends on **the** size of **the** set of fine elements that directly inluences **the** size of **the** sparse part of **the** matrix system. Therefore, an approach **for** reducing **the** size of **the** subsystem of globally coupled (i.e. implicit) unknowns is worth considering if one wants to solver very large-scale problems.

En savoir plus
b Universit´ e Cˆ ote d’Azur, Inria, CNRS, LJAD, France
Abstract
We present **a** time-implicit hybridizable **discontinuous** **Galerkin** (HDG) method **for** numerically solving **the** system of three-dimensional (**3D**) time-domain **Maxwell** equa- tions. This method can be seen as **a** fully implicit variant of classical so-called DGTD (**Discontinuous** **Galerkin** Time-Domain) methods that have been extensively studied during **the** last 10 years **for** **the** simulation of time-domain electromagnetic wave propagation. **The** proposed method has been implemented **for** dealing with general **3D** problems discretized using **unstructured** tetrahedral **meshes**. We provide numer- ical results aiming at assessing its numerical convergence properties by considering **a** model problem on one hand, and its performance when applied to more realis- tic problems. We also include some performance comparisons with **a** **centered** flux time-implicit DGTD method.

En savoir plus
Abstract
In this paper, we present **a** conservative cell-**centered** Lagrangian **Finite** **Volume** **scheme** **for** solving **the** hyperelasticity **equations** on **unstructured** multidimensional grids. **The** starting point of **the** present approach is **the** cell-**centered** FV discretiza- tion named EUCCLHYD and introduced in **the** context of Lagrangian hydrody- namics. Here, it is combined with **the** **a** posteriori Multidimensional Optimal Order Detection (MOOD) limiting strategy to ensure robustness and stability at shock waves with piecewise linear spatial reconstruction. **The** ADER (Arbitrary high or- der schemes using DERivatives) approach is adopted to obtain second-order of ac- curacy in time. This strategy has been successfully tested in an hydrodynamics context and **the** present work aims at extending it to **the** case of hyperelasticity. Here, **the** hyperelasticty **equations** are written in **the** updated Lagrangian frame- work and **the** dedicated Lagrangian numerical **scheme** is derived in terms of nodal solver, Geometrical Conservation Law (GCL) compliance, subcell forces and com- patible discretization. **The** Lagrangian numerical method is implemented in **3D** un- der MPI parallelization framework allowing to handle genuinely large **meshes**. **A** relatively large set of numerical test cases is presented to assess **the** ability of **the** method to achieve effective second order of accuracy on smooth flows, maintaining an essentially non-oscillatory behavior and general robustness across discontinuities and ensuring at least physical admissibility of **the** solution where appropriate. Pure elastic neo-Hookean and non-linear materials are considered **for** our benchmark test problems in 2D and **3D**. These test cases feature material bending, impact, com- pression, non-linear deformation and further bouncing/detaching motions.

En savoir plus
fancy. In practice, **the** non-conformity can result from **a** local refinement of **the** mesh (i.e. h-refinement), of **the** interpolation degree (i.e. p-enrichment) or of both of them (i.e. hp-refinement).
This work is **a** continuation of [16] where **a** hp-like DGTD-P p c :P p f method was introduced **for** solv-
ing **the** two-dimensional time-domain **Maxwell** **equations** on non-conforming triangular **meshes**. It was numerically shown that **the** proposed method has many advantages by comparing it with **a** h-refinement one. **The** main goals of this report are, on one hand, to study numerically **the** convergence of **the** h- and hp-refinement DGTD methods **for** propagation problems in both homogeneous and **heterogeneous** media and using conforming and non-conforming **meshes** and, on **the** other hand, to compare these methods in terms of accuracy and computational costs. **The** rest of this report is organized as follows. In section 2 we recall **the** basic features of our **discontinuous** **Galerkin** time-domain formulation **for** solving **the** first order **Maxwell** **equations** in **the** time domain, based on totally **centered** numerical fluxes and **a** leap-frog time-integration **scheme**. Numerical experiments are presented in section 3 **for** homogeneous media and section 4 **for** **heterogeneous** domain. Finally, conclusions and future works are summarized in section 5.

En savoir plus
• 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.

En savoir plus
Adrien Catella, Victorita Dolean and St´ephane Lanteri
Abstract—Numerical methods **for** solving **the** time-domain **Maxwell** **equations** often rely on cartesian **meshes** and are variants of **the** **finite** difference time-domain (FDTD) method due to Yee [1]. In **the** recent years, there has been an increasing interest in **discontinuous** **Galerkin** time-domain (DGTD) methods dealing with **unstructured** **meshes** since **the** latter are particularly well adapted to **the** discretization of geometrical details that char- acterize applications of practical relevance. However, similarly to Yee’s **finite** difference time-domain method, existing DGTD methods generally rely on explicit time integration schemes and are therefore constrained by **a** stability condition that can be very restrictive on locally refined **unstructured** **meshes**. An implicit time integration **scheme** is **a** possible strategy to overcome this limitation. **The** present study aims at investigating such an implicit DGTD method **for** solving **the** 2D time-domain **Maxwell** **equations** on non-uniform triangular **meshes**.

En savoir plus
In this paper, we study **the** P-SV seismic wave propagation considering an isotropic, linearly elastic medium by solving **the** velocity-stress formulation of **the** elastodynamic **equations**. **For** **the** discretization of this system, we focus on **a** DG method which is **a** **finite** element method allowing discontinuities at **the** interfaces introduced via numerical fluxes as **for** **finite** volumes. Our method is based on **centered** fluxes and **a** leap-frog time-discretization which lead to **a** non-dissipative combination [11]. Moreover, **the** method is suitable **for** complex **unstructured** simplicial **meshes**. **The** extension to higher order in space is realized by Lagrange polynomial functions (of degree 0 to 3 **for** our solver), defined locally on tetrahedra and do not necessitate **the** inversion of **a** global mass matrix since an explicit time **scheme** is used.

En savoir plus
Key words: Maxwell’s **equations**, **discontinuous** **Galerkin** method, leap-frog time **scheme**, sta- bility, convergence, non-conforming **meshes**, high-order accuracy.
1. Introduction
**The** accurate modeling of systems involving electromagnetic waves, in particular through **the** resolution of **the** time-domain **Maxwell** **equations** on space grids, remains of strategic interest **for** many technologies. **The** still prominent **Finite** Difference Time- Domain (FDTD) method proposed by Yee [20] lacks two important features to be fully applied in industrial contexts. First, **the** huge restriction to structured or block- structured grids. Second, **the** efficiency of FDTD methods is limited when fully curvi- linear coordinates are used. Many different types of methods have been proposed in order to handle complex geometries and **heterogeneous** media by dealing with un- structured tetrahedral **meshes**, including, **for** example, mass lumped **Finite** Element Time-Domain (FETD) methods [12,14], mimetic methods [11], or **Finite** **Volume** Time- Domain (FVTD) methods [17], which all fail in being at **the** same time efficient, easily extendible to high orders of accuracy, stable, and energy-conserving.

En savoir plus
characteristics **for** p = 2. In Table I and II, “mesh size” de- notes **the** edge length of **the** tetrahedrons on **the** edge of **the** unit cube and N dof **the** number of degrees of freedom **for**
**the** hybrid variable. We observe that **the** asymptotic conver- gence orders of **the** approximate solutions **for** both E and H are optimal, i.e. of order p + 1 **for** both E and H when using polynomial order p. This convergence rate has been proved **for** **the** Helmholtz equation in 2d [6] but no theoretical result is currently available **for** Maxwell’s **equations** in **3d**.

En savoir plus
SCHWARZ METHODS **FOR** SOLVING **THE** TIME-HARMONIC **MAXWELL** **EQUATIONS**
M. EL BOUAJAJI ˚ , V. DOLEAN : , M.J. GANDER ; AND S. LANTERI ˚ , R. PERRUSSEL ;
Abstract. We show in this paper how to properly discretize optimized Schwarz methods **for** **the** time-harmonic **Maxwell** **equations** using **a** **discontinuous** **Galerkin** (DG) method. Due to **the** multiple traces between elements in **the** DG formulation, it is not clear **a** priori how **the** more sophisticated transmission conditions in optimized Schwarz methods should be discretized, and **the** most natural approach does not lead at convergence of **the** Schwarz method to **the** mono-domain DG discretization, which implies that **for** such discretizations, **the** DG error estimates do not hold when **the** Schwarz method has converged. We present an alternative discretization of **the** transmission conditions in **the** framework of **a** DG weak formulation, and prove that **for** this discretization **the** multidomain and mono-domain solutions **for** **the** Maxwell’s **equations** are **the** same. We illustrate our results with several numerical experiments of propagation problems in homogeneous and **heterogeneous** media.

En savoir plus
Figure 1: Left: Coarse irregular grid (mesh 1). Right: Refined irregular grid (mesh 2).
of these irregular grids is that they include **a** few features of irregular grids generated by **the** geological observations in **the** underground engineering framework. In order to get **a** comparison point, we have compared **the** results obtained with α = 1 and β = 0 (non-constrained **scheme**) using mesh 1, with **the** results obtained with α = 10 −3 and β given by Uzawa’s algorithm (constrained **scheme**) using mesh 1 and mesh 2. Note that **the** choice α = 1 and β = 0 is **the** one which leads to two-point fluxes on rectangular **meshes** and on some regular triangular **meshes** in **the** case of isotropic problems (see [9]). We thus provide in Figure 2 **the** difference between **the** three solutions and **the** analytical solution, computed along **the** line with equation x2 = 0.5. We see that **the** respect of **the** constraints decreases strongly **the** error, which decreases again using **the** finest mesh. This is confirmed by **the** L 2 errors of **the** solution and of its gradient, shown in Table 1, using **a** variety of grids and numerical parameters.

En savoir plus
In **the** present work, we first propose an operator splitting strategy that allows to decouple **the** acoustic and **the** transport phenomenons. **The** approximation algorithm is split into two steps: an acoustic step and **a** trans- port step. **For** one-dimensional problems, this strategy is equivalent to an explicit Lagrange-Projection [15, 13] method, however **the** present splitting does not involve any moving Lagrangian mesh and can be naturally expressed **for** multi-dimensional problems. Following simple lines inspired by [11, 10] we investigate **the** depen- dence of **the** truncation error with respect to **the** Mach number. Let us mention that our study does not involve **a** Taylor expansion in **the** vicinity of **the** zero-Mach limit, nor **a** near-divergence free condition **for** **the** velocity field. Although this analysis is by no mean **a** thorough explanation of **the** low Mach regime behavior of our solver, it is enough to suggest simple means to obtain **a** truncation error with **a** uniform dependence on **the** Mach number **for** M < 1. **The** cure simply relies on modifying **the** pressure terms in **the** flux of **the** acoustic operator that is coherent with **the** correction proposed by [11, 10, 18, 24, 14]. Although this modified **scheme** is based on **a** modified flux definition, one can shows that it can also be rephrased as **a** simple approximate Riemann solver in **the** sense of Harten, Lax and van Leer [17] that is consistent with **the** integral form of **the** gas dynamics equation. This **scheme** is endowed with good stability properties under **a** CFL condition that involves **the** Mach number as **the** time step is still constrained by **the** sound velocity.

En savoir plus
Résumé : Le système des équations de **Maxwell** décrit l’évolution d’un champ électro- magnétique en interaction avec un milieu de propagation. Les différentes propriétés de ce milieu, telles que son caractère isotrope ou anisotrope, homogène ou hétérogène, linéaire ou non-linéaire, sont définies par des lois constitutives qui lient champs et inductions. Dans cette étude, on s’intéresse aux effets non-linéaires et on considère plus particulièrement le cas de milieux non-linéaires de type Kerr. Dans le cas de ce modèle, n’importe quel diélectrique peut se comporter comme un matériau non-linéaire dès lors que l’amplitude du champ électrique se propageant dans le milieu est suffisamment forte. C’est par ex- emple le cas du vide mais l’énergie minimale pour observer des effets non-linéaires doit être similaire à l’énergie totale produite par le soleil en une seconde. Nous considérons néanmoins le vide comme un milieu canididat pour notre étude, au même titre que l’air. Pour ce dernier, l’amplitude minimale du champ électrique pour observer des effets non- linéaires est 10 6 V/m, un niveau qui peut être atteint par les lasers exploités de nos jours.

En savoir plus
where P p i (T i ) denotes **the** space of polynomials {ϕ ij } d i j=1 of total degree at most p i on **the** element T i . **The** space V p (T h ) has **the** dimension d i , **the** local number of degrees of freedom (dof). Note that **the** polynomial degree p i may vary from element to element in **the** mesh and that **a** function v p h ∈ V p (T h ) is **discontinuous** across element interfaces. **For** two distinct triangles T i and T k in T h , **the** intersection T i ∩ T k is an (oriented) edge s ik which we will call interface, with oriented normal vector ~n ik . **For** **the** boundary interfaces, **the** index k corresponds to **a** fictitious element outside **the** domain. By non-conforming interface we mean an interface s ik which has at least one of its two vertices in **a** hanging node or such that p i| sik 6= p k| sik or both of them. Finally, we denote by V i **the** set of indices of **the** elements neighboring T i .

En savoir plus
Fig. 6. Comparison of **the** convergence results between **centered** flux and upwind flux.
It is already known **for** time-domain problems that **the** **centered** flux combined to **a** leap-frog time integration **scheme** results in **a** non-dissipative discontin- uous **Galerkin** method (**a** mandatory feature **for** long time computations, see [6]). As far as time-harmonic problems are concerned, **the** previous results show that **the** upwind flux has better convergence properties. Nevertheless, **the** cen- tered flux remains less expensive both **for** time-domain and time-harmonic problems (arithmetic operations and memory requirements).

En savoir plus
Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex France Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vi[r]

discretize not only **the** gas dynamics **equations** but also **the** vertex motion in order to move **the** mesh. Moreover, **the** numerical fluxes of **the** physical conservation laws must be determined in **a** compatible way with **the** vertex velocity so that **the** geometric conservation law (GCL) is satisfied, namely **the** rate of change of **a** Lagrangian **volume** has to be computed coherently with **the** node motion. This critical requirement is **the** cornerstone of any Lagrangian multi-dimensional **scheme**. **The** most natural way to solve this problem employs **a** staggered discretization in which position, velocity and kinetic energy are **centered** at points, while density, pressure and internal energy are within cells. **The** dissipation of kinetic energy into internal energy through shock waves is ensured by an artificial viscosity term. Since **the** seminal works of von Neumann and Richtmyer [42], and Wilkins [43], many developments have been made in order to improve **the** accuracy and **the** robustness of staggered hydrodynamics [11, 9, 7]. More specifically, **the** construction of **a** compatible staggered discretization leads to **a** **scheme** that conserves total energy in **a** rigorous manner [10, 8]. We note also **the** recent development of **a** variational multi-scale stabilized approach in **finite** element computation of Lagrangian hydrodynamics, where **a** piecewise linear approximation was adopted **for** **the** variables [35, 34]. **The** case of Q1/P0 **finite** element is studied in [36], where **the** kinematic variables are represented using **a** piecewise linear continuous approximation, while **the** thermodynamic variables utilize **a** piecewise constant representation.

En savoir plus
−V dx e −V . On admissible **meshes**, it is possible to write **a** classical two-point **finite** vol- ume **scheme** in order to approximate (1). With **the** Scharfetter-Gummel numerical fluxes, **the** preservation of **the** steady-state is ensured, as **a** discrete counterpart of **the** entropy-dissipation property (see [4]). In this paper, we want to design **a** **scheme** which satisfies **the** same properties but that could be applicable on almost general **meshes** including non-conformal and distorted **meshes**. We propose **a** Discrete Du- ality **Finite** **Volume** **scheme**. Let us mention that **a** robust free energy diminishing **finite** **volume** **scheme** based on **the** VAG **scheme** has already been proposed and analyzed in [3].

En savoir plus