Haut PDF A centered Discontinuous Galerkin Finite Volume scheme for the 3D heterogeneous Maxwell equations on unstructured meshes

A centered Discontinuous Galerkin Finite Volume scheme for the 3D heterogeneous Maxwell equations on unstructured meshes

A centered Discontinuous Galerkin Finite Volume scheme for the 3D heterogeneous Maxwell equations on unstructured meshes

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]

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A Centered Second-Order Finite Volume Scheme for the Heterogeneous Maxwell Equations in Three Dimensions on Arbitrary Unstructured Meshes

A Centered Second-Order Finite Volume Scheme for the Heterogeneous Maxwell Equations in Three Dimensions on Arbitrary Unstructured Meshes

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]

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An implicit hybridized discontinuous Galerkin method for the 3D time-domain Maxwell equations

An implicit hybridized discontinuous Galerkin method for the 3D time-domain Maxwell equations

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.
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An implicit hybridized discontinuous Galerkin method for the 3D time-domain Maxwell equations

An implicit hybridized discontinuous Galerkin method for the 3D time-domain Maxwell equations

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.
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A 3D cell-centered ADER MOOD Finite Volume method for solving updated Lagrangian hyperelasticity on unstructured grids

A 3D cell-centered ADER MOOD Finite Volume method for solving updated Lagrangian hyperelasticity on unstructured grids

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.
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Numerical evaluation of a non-conforming discontinuous Galerkin method on triangular meshes for solving the time-domain Maxwell equations

Numerical evaluation of a non-conforming discontinuous Galerkin method on triangular meshes for solving the time-domain Maxwell equations

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.
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A high-order cell-centered finite volume scheme for simulating three dimensional anisotropic diffusion equations on unstructured grids

A high-order cell-centered finite volume scheme for simulating three dimensional anisotropic diffusion equations on unstructured grids

• 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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An inconditionnally stable discontinuous Galerkin method for solving the 2D time-domain Maxwell equations on unstructured triangular meshes

An inconditionnally stable discontinuous Galerkin method for solving the 2D time-domain Maxwell equations on unstructured triangular meshes

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.
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ANALYSIS OF A DISCONTINUOUS GALERKIN METHOD FOR ELASTODYNAMIC EQUATIONS. APPLICATION TO 3D WAVE PROPAGATION.

ANALYSIS OF A DISCONTINUOUS GALERKIN METHOD FOR ELASTODYNAMIC EQUATIONS. APPLICATION TO 3D WAVE PROPAGATION.

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.
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High-Order Leap-Frog Based Discontinuous Galerkin Method for the Time-Domain Maxwell Equations on Non-Conforming Simplicial Meshes

High-Order Leap-Frog Based Discontinuous Galerkin Method for the Time-Domain Maxwell Equations on Non-Conforming Simplicial Meshes

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.
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A hybridized discontinuous Galerkin method for 3d time-harmonic Maxwell's equations

A hybridized discontinuous Galerkin method for 3d time-harmonic Maxwell's equations

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.
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Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell equations

Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell equations

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.
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A Finite Volume Scheme for Diffusion Problems on General Meshes Applying Monotony Constraints

A Finite Volume Scheme for Diffusion Problems on General Meshes Applying Monotony Constraints

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.
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An all-regime Lagrange-Projection like scheme for the gas dynamics equations on unstructured meshes

An all-regime Lagrange-Projection like scheme for the gas dynamics equations on unstructured meshes

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.
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Discontinuous Galerkin methods for the numerical solution of the nonlinear Maxwell equations in 1d

Discontinuous Galerkin methods for the numerical solution of the nonlinear Maxwell equations in 1d

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.
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Preliminary Investigation of a Nonconforming Discontinuous Galerkin Method for Solving the Time-Domain Maxwell Equations

Preliminary Investigation of a Nonconforming Discontinuous Galerkin Method for Solving the Time-Domain Maxwell Equations

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 .
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Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods

Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods

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).
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An Implicit DGTD Method for Solving the Two-Dimensional Maxwell Equations on Unstructured Triangular Meshes

An Implicit DGTD Method for Solving the Two-Dimensional Maxwell Equations on Unstructured Triangular Meshes

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]

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A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

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.
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A nonlinear Discrete Duality Finite Volume Scheme for convection-diffusion equations

A nonlinear Discrete Duality Finite Volume Scheme for convection-diffusion equations

−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].
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