Research Items (133)
In this work, we address time-dependent wave propagation problems with strong multiscale features (in space and time). Our goal is to design a family of innovative high-performance numerical methods suitable for the simulation of such multiscale problems. Particularly, we extend the Multiscale Hybrid-Mixed finite element method (MHM for short) for the two- and three- dimensional time-dependent Maxwell equations with heterogeneous coefficients. The MHM method arises from the decomposition of the exact electric and magnetic fields in terms of the solutions of locally independent Maxwell problems tied together with a one-field formulation on top of a coarse- mesh skeleton. The multiscale basis functions, which are responsible for upscaling, are driven by local Maxwell problems with the tangential component of the magnetic field prescribed on faces. A high-order discontinuous Galerkin method in space combined with a second-order explicit leap-frog scheme in time discretizes the local problems. This makes the MHM method effective and yields a staggered algorithm within a “divide-and-conquer” framework. Several two-dimensional numerical tests assess the optimal convergence of the MHM method and its capacity to preserve the energy principle, as well as its accuracy for solving heterogeneous media problems on coarse meshes.
- Jan 2018
This volume entitled "European Computational Aerodynamics Research Project (ECARP)" contains the contributions of partners presented in two work shops focused on the following areas: Task 3 on Optimum Design and Task 4.2 on Navier Stokes Flow algorithms on Massively Parallel Processors. ECARP has been supported by the European Union (EU) through the Indus trial and Materials Technology Programme, Area 3 Aeronautics, with the Third Research Framework Programme (1990-1994). Part A of this volume is focused on computational constrained optimization as a follow up of the EU research project" Optimum Design in Aerodynamics" , (AERO-S9-0026) dealing with more viscous flow based real applications. It pro vides the reader with a set of optimization tools and referenced data useful in modern aerodynamic design. Task 3 of the project entitled "Optimum Design" brought together 13 Euro pean partners from the academic and industrial aeronautic oriented community showing state of the art expertise in traditional automated optimization software on current computer technology to improve the capability to optimize aircraft shapes.
- Sep 2017
This article has been published on IEEE Transactions on Antennas and Propagation (http://ieeexplore.ieee.org/document/8038082/). The accurate and efficient simulation of 3D transient multiscale electromagnetic problems is extremely challenging for conventional numerical methods. Assuming a splitting of the underlying tetrahedral mesh in coarse and fine parts and using the Lawson procedure, we derive a family of exponential-based time integration methods for the time-domain Maxwell's equations discretized by a high order discontinuous Galerkin (DG) scheme formulated on locally refined unstructured meshes. These methods remove the stiffness on the time explicit integration of the semi-discrete operator associated to the fine part of the mesh, and allow for the use of high order time explicit scheme for the coarse part operator. They combine excellent stability properties with the ability to obtain very accurate solutions even for very large time step sizes. Here, the explicit time integration of the Lawson-transformed semi-discrete system relies on a Low-Storage Runge-Kutta (LSRK) scheme, leading to a combined Lawson-LSRK scheme. In addition, efficient techniques are also presented to further improve the efficiency of this exponential-based time integration. For the efficient calculation of matrix exponential, we employ the Krylov subspace method. Numerical experiments are presented to assess the stability, verify the accuracy and numerical convergence of the Lawson-LSRK scheme. They also demonstrate that the DGTD methods based on the proposed time integration scheme can be much faster than those based on classical fully explicit time stepping schemes, with the same accuracy and moderate memory usage increase on locally refined unstructured meshes, and are thus very promising for modelling three-dimensional multiscale electromagnetic problems.
- Mar 2017
- 2017 International Applied Computational Electromagnetics Society (ACES) Symposium
In this work, a new family of implicit-explicit (IMEX) schemes based on exponential time integration is developed for the 3D time-domain Maxwell's equations discretized by a high order discontinuous Galerkin (DG) scheme formulated on locally refined unstructured meshes. Numerical experiments demonstrate that the DGTD solver based on the proposed time integration scheme can be much faster than the one based on classical low-storage Runge-Kutta scheme, with the same accuracy and slight memory usage increase.
In this work, we address time dependent wave propagation problems with strong multiscale features (in space and time). Our goal is to design a family of innovative high performance numerical methods suitable to the simulation of such multiscale problems. Particularly, we extend the Multiscale Hybrid-Mixed finite element method (MHM for short) for the two-and three-dimensional time-dependent Maxwell equations with heterogeneous coefficients. The MHM method arises from the decomposition of the exact electric and magnetic fields in terms of the solutions of locally independent Maxwell problems tied together with a one-field formulation on top of a coarse-mesh skeleton. The multiscale basis functions, which are responsible for upscaling, are driven by local Maxwell problems with tangential component of the magnetic field prescribed on faces. A high-order discontinuous Galerkin method in space combined with a second-order explicit leapfrog scheme in time discretizes the local problems. This makes the MHM method effective and yields a staggered algorithm within a " divide-and-conquer " framework. Several numerical tests assess the optimal convergence of the MHM method and its capacity to preserve the energy principle, as well as its accuracy to solving heterogeneous media problems on coarse meshes.
- Nov 2016
This paper is concerned with the approximation of the time domain Maxwell equations in a dispersive propagation medium by a Discontinuous Galerkin Time Domain (DGTD) method. The Debye model is used to describe the dispersive behaviour of the medium. We adapt the locally implicit time integration method from Verwer (2010) and derive a convergence analysis to prove that the locally implicit DGTD method for Maxwell-Debye equations retains its second order convergence.
- Jan 2016
During the last ten years, the discontinuous Galerkin time-domain (DGTD) method has progressively emerged as a viable alternative to well established finite-difference time-domain (FDTD) and finite-element time-domain (FETD) methods for the numerical simulation of electromagnetic wave propagation problems in the time-domain. The method is now actively studied in various application contexts including those requiring to model light/matter interactions on the nanoscale. Several recent works have demonstrated the viability of the DGDT method for nanophotonics. In this paper we further demonstrate the capabilities of the method for the simulation of near-field plasmonic interactions by considering more particularly the possibility of combining the use of a locally refined conforming tetrahedral mesh with a local adaptation of the approximation order.
This paper is concerned with the development of a scalable high order finite element type solver for the numerical modeling of light interaction with nanometer scale structures. From the mathematical modeling point of view, one has to deal with the differential system of Maxwell equations in the time domain, coupled to an appropriate differential model of the behavior of the underlying material (which can be a dielectric and/or a metal) at optical frequencies. For the numerical solution of the resulting system of differential equations, we have designed a high order DGTD (Discontinuous Galerkin Time-Domain) solver that has been adapted to hybrid MIMD/SIMD computing. Here we discuss about this later aspect and report on preliminary performance results on the Curie system of the PRACE research infrastructure.
- Mar 2015
Simulation of wave propagation through complex media relies on proper understanding of the properties of numerical methods when the wavenumber is real and complex. Numerical methods of the Hybrid Discontinuous Galerkin (HDG) type are considered for simulating waves that satisfy the Helmholtz and Maxwell equations. It is shown that these methods, when wrongly used, give rise to singular systems for complex wavenumbers. A sufficient condition on the HDG stabilization parameter for guaranteeing unique solvability of the numerical HDG system, both for Helmholtz and Maxwell systems, is obtained for complex wavenumbers. For real wavenumbers, results from a dispersion analysis are presented. An asymptotic expansion of the dispersion relation, as the number of mesh elements per wave increase, reveal that some choices of the stabilization parameter are better than others. To summarize the findings, there are values of the HDG stabilization parameter that will cause the HDG method to fail for complex wavenumbers. However, this failure is remedied if the real part of the stabilization parameter has the opposite sign of the imaginary part of the wavenumber. When the wavenumber is real, values of the stabilization parameter that asymptotically minimize the HDG wavenumber errors are found on the imaginary axis. Finally, a dispersion analysis of the mixed hybrid Raviart–Thomas method showed that its wavenumber errors are an order smaller than those of the HDG method.
The time-harmonic Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental for the simulation of many modern devices we have become used to in everyday life. The numerical solution of these equations is hampered by two fundamental problems: first, in the high frequency regime, very fine meshes need to be used in order to avoid the pollution effect well known for the Helmholtz equation, and second the large scale systems obtained from the vector valued equations in three spatial dimensions need to be solved by iterative methods, since direct factorizations are not feasible any more at that scale. As for the Helmholtz equation, classical iterative methods applied to discretized Maxwell equations have severe convergence problems. We explain in this paper a family of domain decomposition methods based on well chosen transmission conditions. We show that all transmission conditions proposed so far in the literature, both for the first and second order formulation of Maxwell's equations, can be written and optimized in the common framework of optimized Schwarz methods, independently of the first or second order formulation one uses, and the performance of the corresponding algorithms is identical. We use a decomposition into transverse electric and transverse magnetic fields to describe these algorithms, which greatly simplifies the convergence analysis of the methods. We illustrate the performance of our algorithms with large scale numerical simulations.
We present a high-order discontinuous Galerkin method for the simulation of P-SV seismic wave propagation in heterogeneous media and two dimensions of space. The first-order velocity-stress system is obtained by assuming that the medium is linear, isotropic and viscoelastic, thus considering intrinsic attenuation. The associated stress-strain relation in the time domain being a convolution, which is numerically intractable, we consider the rheology of a generalized Maxwell body replacing the convolution by differential equations. This results in a velocity-stress system which contains additional equations for the anelastic functions including the strain history of the material. Our numerical method, suitable for complex triangular unstructured meshes, is based on a centered numerical flux and a leap-frog time-discretization. The extension to high order in space is realized by Lagrange polynomial functions, defined locally on each element. The inversion of a global mass matrix is avoided since an explicit scheme in time is used. The method is validated through numerical simulations including comparisons with a finite difference scheme.
We present a discontinuous Galerkin method for site effects assessment. The P-SV seismic wave propagation is studied in 2-D space heterogeneous media. The first-order velocity-stress system is obtained by assuming that the medium is linear, isotropic and viscoelastic, thus considering intrinsic attenuation. The associated stress-strain relation in the time domain being a convolution, which is numerically intractable, we consider the rheology of a generalized Maxwell body replacing the convolution by a set of differential equations. This result in a velocity-stress system which contains additional equaltions for the anelastic functions expressing the strain history of the material. Our numerical method, suitable for complex triangular unstructured meshes, is based on centred numerical fluxes and a leap-frog time discretization. The method is validated through numerical simulations including comparisons with a finite-difference scheme. We study the influence of the geological structures of the Nice basin on the surface ground motion through the comparisons of 1-D and 2-D soil response in homogeneous and heterogeneous soil. At last, we compare numerical results with real recordings data. The computed multiple-sediment basin response allows to reproduce the shape of the recorded amplification in the basin. This highlights the importance of knowing the lithological structures of a basin, layers properties and interface geometry.
- May 2014
SUMMARY The great majority of numerical calculations of electric energy deposition in human tissues exposed to microwaves are performed using the finite-difference time-domain (FDTD) method and voxel-based geometrical models of the tissues. The straightforward implementation of the method and its computational efficiency are among the main reasons for FDTD being currently the leading method for numerical assessment of human exposure to electromagnetic waves. However, the rather difficult departure from the commonly used Cartesian grid and cell size limitations regarding the discretization of very detailed structures of human tissues are often recognized as the main weaknesses of the FDTD method in this application context. In particular, interfaces between tissues where sharp gradients of the electromagnetic field may occur are hardly modeled rigorously in these studies (this is generally referred as the staircasing effect). We present here an alternative numerical dosimetry methodology that is based on a DGTD method designed to work with non-conforming cubic-tetrahedral meshes for more flexibility in the discretization process of complex propagation scenes. For example, in the context of head tissues exposure to mobile phone radiation, the computation domain is divided in two regions: an unstructured tetrahedral mesh-based heterogeneous model of head tissues and an orthogonal cartesian mesh of the vacuum part of the propagation domain. We present numerical results comparing this approach with strategies based on fully Cartesian and fully tetrahedral meshes of the whole computational domain. Copyright © 2013 John Wiley & Sons, Ltd.
- Feb 2014
We aim at coupling a method of moments, the Wave Concept Iterative Procedure, and the Hybridizable Discontinuous Galerkin (HDG) method to study electromagnetic susceptibility of innovative planar circuits in 3-D. Hybridizing the Wave Concept Iterative Procedure with volumic methods such as the Frequency-Domain Transmission Line Matrix method, the finite-element method, and the HDG method in 2-D is a first step for the validation of the proposed coupling technique. The considered problem is Maxwell's equations in the frequency domain. Three test cases in 2-D and a preliminary result in 3-D are provided.
- Nov 2013
During the last ten years, the discontinuous Galerkin time-domain (DGTD) method has progressively emerged as a viable alternative to well established finite-difference time-domain (FDTD) and finite-element time-domain (FETD) methods for the numerical simulation of electromagnetic wave propagation problems in the time-domain. In this paper, we review the historical development of the DGTD method and emphasize its recent adoption by the nanophotonic research community. In addition, we discuss about some of our recent efforts aiming at improving the accuracy, flexibility and efficiency of a non-dissipative order DGTD method, and also report on some preliminary works towards its extension to the numerical treatment of physical models and problems that are relevant to nanophotonics.
An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the region of refinement is small relative to the computational domain, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The downside of this approach is having to solve a linear system per time step. But due to the assumed small region of refinement relative to the computational domain, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit–explicit approaches in presence of local refinements.
Purpose ‐ This work is concerned with the development and the numerical investigation of a hybridizable discontinuous Galerkin (HDG) method for the simulation of two-dimensional time-harmonic electromagnetic wave propagation problems. Design/methodology/approach ‐ The proposed HDG method for the discretization of the two-dimensional transverse magnetic Maxwell equations relies on an arbitrary high order nodal interpolation of the electromagnetic field components and is formulated on triangular meshes. In the HDG method, an additional hybrid variable is introduced on the faces of the elements, with which the element-wise (local) solutions can be defined. A so-called conservativity condition is imposed on the numerical flux, which can be defined in terms of the hybrid variable, at the interface between neighbouring elements. The linear system of equations for the unknowns associated with the hybrid variable is solved here using a multifrontal sparse LU method. The formulation is given, and the relationship between the considered HDG method and a standard upwind flux-based DG method is also examined. Findings ‐ The approximate solutions for both electric and magnetic fields converge with the optimal order of p+1 in L2 norm, when the interpolation order on every element and every interface is p and the sought solution is sufficiently regular. The presented numerical results show the effectiveness of the proposed HDG method, especially when compared with a classical upwind flux-based DG method. Originality/value ‐ The work described here is a demonstration of the viability of a HDG formulation for solving the time-harmonic Maxwell equations through a detailed numerical assessment of accuracy properties and computational performances.
This work is about the numerical solution of the time-domain Maxwell's equations in dispersive propagation media by a discontinuous Galerkin time-domain method. The Debye model is used to describe the dispersive behaviour of the media. The resulting system of differential equations is solved using a centred-flux discontinuous Galerkin formulation for the discretization in space and a second-order leapfrog scheme for the integration in time. The numerical treatment of the dispersive model relies on an auxiliary differential equation approach similar to that which is adopted in the finite difference time-domain method. Stability estimates are derived through energy considerations and convergence is proved for both the semidiscrete and the fully discrete schemes.
- Mar 2013
This study is concerned with the numerical solution of 2D electromagnetic wave propagation problems in the frequency domain. We present a high order discontinuous Galerkin method formulated on unstructured triangular meshes for the solution of the system of the time-harmonic Maxwell equations in mixed form. Within each triangle, the approximation of the electromagnetic field relies on a nodal polynomial interpolation and the polynomial degree is allowed to vary across mesh elements. The resulting numerical methodology is applied to the simulation of 2D propagation problems in homogeneous and heterogeneous media as well.
We present a high-order discontinuous Galerkin method for the simulation of P-SV seismic wave propagation in heterogeneous media and two dimensions of space. The first-order velocity-stress system is obtained by assuming that the medium is linear, isotropic and viscoelastic, thus considering intrinsic attenuation. The associated stress-strain relation in the time domain being a convolution, which is numerically intractable, we consider the rheology of a generalized Maxwell body replacing the convolution by differential equations. This results in a velocity-stress system which contains additional equations for the anelastic functions including the strain history of the material. Our numerical method, suitable for complex triangular unstructured meshes, is based on a centered numerical flux and a leap-frog time-discretization. The extension to high order in space is realized by Lagrange polynomial functions, defined locally in each element. The inversion of a global mass matrix is avoided since an explicit scheme in time is used and because of the local nature of the discontinuous Galerkin formulation. The method is validated through numerical simulations including comparisons with a finite difference scheme.
- Jan 2013
The numerical simulation of electromagnetic wave propagation in open domains involving scattering objects or/and inhomogeneous regions naturally raises the question of the appropriate treatment of the artificial truncation of the infinite propagation domain. The main issue is to find a suitable way to efficiently model the far-field propagation region, by limiting the extent of the volume discretization.
- Nov 2012
- International Conference on Domain Decomposition Methods
Transmission conditions between subdomains have a substantial influence on the convergence of iterative domain decomposition algorithms. For Maxwell's equations, transmission conditions which lead to rapidly converging algorithms have been developed both for the curl-curl formulation of Maxwell's equation, see [2, 3, 1], and also for first order formulations, see [7, 6]. These methods have well found their way into applications, see for example [9, 11, 10]. It turns out that good transmission conditions are approximations of transparent boundary conditions. For each form of approximation chosen, one can try to find the best remaining free parameters in the approximation by solving a min-max problem. Usually allowing more free parameters leads to a substantially better solution of the min-max problem, and thus to a much better algorithm. For a particular one parameter family of transmission conditions analyzed in , we investigate in this paper a two parameter counterpart. The analysis, which is substantially more complicated than in the one parameter case, reveals that in one particular asymptotic regime there is only negligible improvement possible using two parameters, compared to the one parameter results. This analysis settles an important open question for this family of transmission conditions, and also suggests a direction for systematically reducing the number of parameters in other optimized transmission conditions.
We study block preconditioning strategies for the solution of large sparse complex coefficients linear systems resulting from the discretization of the time-harmonic Maxwell equations by a high order discontinuous finite element method formulated on unstructured simplicial meshes. The proposed strategies are based on principles from incomplete factorization methods. Moreover, a complex shift is applied to the diagonal entries of the underlying matrices, a technique that has recently been exploited successfully in similar contexts and in particular for the multigrid solution of the scalar Helmholtz equation. Numerical results are presented for 2D and 3D electromagnetic wave propagation problems in homogeneous and heterogeneous media.
We study the numerical solution of 3D time-harmonic Maxwell’s equations by a hybridizable discontinuous Galerkin method. A hybrid term representing the tangential component of the numerical trace of the magnetic field is introduced. The global system to solve only involves the hybrid term as unknown. We show that the reduced system has properties similar to wave equation discretizations and the tangential components of the numerical traces for both electric and magnetic fields are single-valued. On the example of a plane wave propagation in vacuum, the approximate solutions for both electric and magnetic fields have an optimal convergence order.
- Sep 2012
- Electromagnetics in Advanced Applications (ICEAA), 2012 International Conference on
We are interested here in a multielement Discontinuous Galerkin Time Domain method to solve the system of Maxwell equations. The method is formulated on non-conforming and hybrid meshes combining an unstructured triangulation for an accurate discretization of the irregularly shaped objects with a structured (orthogonal) quadrangulation for a gain in CPU time on the rest of the computational domain. We present the discretization schemes and the theoritical aspects in 3D, and we expose the numerical simulations.
- Jul 2012
An hybridization between two numerical methods, the 1d Wave Concept Iterative Procedure (WCIP) and the 2d Finite Element Method (FEM), is developed. Using two examples, comparisons are provided between the new hybrid method and an analytic solution, when available, or the WCIP alone.
A reduced basis method is introduced to deal with a stochastic problem in a numerical dosimetry application in which the field solutions are computed using an iterative solver. More precisely, the computations already performed are used to build an initial guess for the iterative solver. It is shown that this approach significantly reduces the computational cost.
The great majority of numerical calculations of the specific absorption rate (SAR) induced in human tissues exposed to microwaves are performed using the finite difference time-domain (FDTD) method and voxel-based geometrical models. The straightforward implementation of the method and its computational efficiency are among the main reasons for FDTD being currently the leading method for numerical assessment of human exposure to electromagnetic waves. However, the rather difficult departure from the commonly used Cartesian grid and cell size limitations regarding the discretization of very detailed structures of human tissues are often recognized as the main weaknesses of the method in this application context. In particular, interfaces between tissues where sharp gradients of the electromagnetic field may occur are hardly modeled rigorously in these studies. We present here an alternative numerical dosimetry methodology which is based on a high order discontinuous Galerkin time-domain (DGTD) method and adapted geometrical models constructed from unstructured triangulations of tissue interfaces, and discuss its application to the calculation of the SAR induced in head tissues.
- Nov 2011
We report on recent developments aiming at improving the accuracy and the performances of a discontinuous Galerkin time domain method (DGTD) for the simulation of time-domain electromagnetic wave propagation problems involving general domains and heterogeneous media. The common objective of the associated studies is to bring the method to a level of computational efficiency and flexibility that allows to tackle realistic applications of practical interest.
The study of optimized Schwarz methods for Maxwell’s equations started with the Helmholtz equation, see [2–4, 11]. For the rot-rot formulation of Maxwell’s equations, optimized Schwarz methods were developed in , and for the more general form in [9, 10]. An entire hierarchy of families of optimized Schwarz methods was analyzed in , see also  for discontinuous Galerkin discretizations and large scale experiments. We present in this paper a first analysis of optimized Schwarz methods for Maxwell’s equations with non-zero electric conductivity. This is an important case for real applications, and requires a new, and fundamentally different optimization of the transmission conditions. We illustrate our analysis with numerical experiments.
We are interested here in the numerical modeling of time-harmonic electromagnetic wave propagation problems in irregularly shaped domains and heterogeneous media. In this context, we are naturally led to consider volume discretization methods (i.e. finite element method) as opposed to surface discretization methods (i.e. boundary element method). Most of the related existing work deals with the second order form of the time-harmonic Maxwell equations discretized by a conforming finite element method . More recently, discontinuous Galerkin (DG) methods have also been considered for this purpose. While the DG method keeps almost all the advantages of a conforming finite element method (large spectrum of applications, complex geometries, etc.), the DG method has other nice properties which explain the renewed interest it gains in various domains in scientific computing: easy extension to higher order interpolation (one may increase the degree of the polynomials in the whole mesh as easily as for spectral methods and this can also be done locally), no global mass matrix to invert when solving time-domain systems of partial differential equations using an explicit time discretization scheme, easy handling of complex meshes (the mesh may be a classical conforming finite element mesh, a non-conforming one or even a mesh made of various types of elements), natural treatment of discontinuous solutions and coefficient heterogeneities and nice parallelization properties.
- Aug 2011
- Euro-Par 2011: Parallel Processing Workshops
We study the performance of a multi-GPU enabled numerical methodology for the simulation of electromagnetic wave propagation in complex domains and heterogeneous media. For this purpose, the system of time-domain Maxwell equations is discretized by a discontinuous finite element method which is formulated on an unstructured tetrahedral mesh and which relies on a high order interpolation of the electromagnetic field components within a mesh element. The resulting numerical methodology is adapted to parallel computing on a cluster of GPU acceleration cards by adopting a hybrid strategy which combines a coarse grain SPMD programming model for inter-GPU parallelization and a fine grain SIMD programming model for intra-GPU parallelization. The performance improvement resulting from this multiple-GPU algorithmic adaptation is demonstrated through three-dimensional simulations of the propagation of an electromagnetic wave in the head of a mobile phone user.
- Jun 2011
We have recently developed a discontinuous Galerkin frequency domain modelling algorithm for the solution of the 2D transverse magnetic Maxwell equations. This method is formulated on an unstructured triangular discretization of the computational domain and makes use of a high order polynomial interpolation of the electromagnetic field components within each triangular element. The discontinuous nature of the approximation naturally allows for a local definition of the interpolation order that is, in combination with a possibly non-conforming local refinement of the mesh, a key ingredient for obtaining a flexible and accurate discretization method. Moreover, heterogeneity of the propagation media is easily dealt with by assuming element-wise values of the electromagnetic parameters. In this paper, we propose the use of this discontinuous Galerkin frequency domain method as the forward modelling algorithm for solving the inverse problem for the electric permittivity in the 2D case. The inversion process is based on a gradient minimization technique developed by Pratt for seismological applications. Preliminary numerical results are presented for the imaging of a simplified subsurface model with the aim of assessing the performances of the proposed inversion methodology with regards to the number of frequencies, the number of recorded data and the number of sources.
We present a high performance computing methodology for the simulation of electromagnetic wave propagation in biological tissues and its application to the numerical evaluation of radio frequency absorption in head tissues as they are exposed to radiation from a cellular phone. For this purpose, the system of time-domain Maxwell equations is discretized in space by a discontinuous Galerkin method which is formulated on a tetrahedral mesh and which relies on a high order interpolation of the electromagnetic field components within a mesh element. The semi-discretized equations are then time integrated by a second order leap-frog scheme. The resulting numerical methodology is adapted to modern parallel computing systems with multiple GPU acceleration cards by adopting a hybrid strategy that combines a coarse grain SPMD programming model for inter-GPU parallelization and a fine grain SIMD programming model for intra-GPU parallelization. The performance improvement thanks to multiple-GPU acceleration is demonstrated through large-scale simulations that are performed on a cluster of GPUs using realistic heterogeneous models of head tissues built from medical images.
- Mar 2011
Discontinuous Galerkin (DG) methods have been the subject of numerous research activities in the last 15 years and have been successfully developed for various physical contexts modeled by elliptic, mixed hyperbolic-parabolic and hyperbolic systems of PDEs. One major drawback of high order DG methods is their intrinsic cost due to the very large number of globally coupled degrees of freedom as compared to classical high order conforming finite element methods. Different attempts have been made in the recent past to improve this situation and one promising strategy has been recently proposed by Cockburn (Cockburn et al., 2009) in the form of so-called hybridizable DG formulations. The distinctive feature of these methods is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements of the discretization mesh. The present work is concerned with the study of such a hybridizable DG method for the solution of the system of Maxwell equations. In this preliminary investigation, a hybridizable DG method is proposed for the two-dimensional time-domain Maxwell equations time integrated by an implicit scheme.
In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the numerical modeling of electromagnetic wave propagation. Such methods most often rely on explicit time integration schemes which are constrained by a stability condition that can be very restrictive on highly refined meshes. In this paper, we report on some efforts to design a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on locally refined simplicial meshes. The proposed method consists in applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part.
The great majority of numerical calculations of the Specific Absorption Rate (SAR) induced in human tissues exposed to microwaves are performed using the Finite Difference Time Domain (FDTD) method and voxel based geometrical models. The straightforward implementation of the method and its computational efficiency are among the main reasons for FDTD being currently the leading method for numerical assessment of human exposure to electromagnetic waves. However, the rather difficult departure from the commonly used cartesian grid and cell size limitations regarding the discretization of very detailed structures of human tissues are often recognized as the main weaknesses of the method in this application context. We present here an alternative numerical dosimetry methodology combining a high order Discontinuous Galerkin Time Domain (DGTD) method and adapted geometrical models based on unstructured triangulations, and discuss its application to the calculation of the SAR induced in head tissues.
We report on results concerning a discontinuous Galerkin time domain (DGTD) method for the solution of Maxwell equations. This DGTD method is formulated on unstructured simplicial meshes (triangles in 2-D and tetrahedra in 3-D). Within each mesh element, the electromagnetic field components are approximated by an arbitrarily high order nodal polynomial while, in the original formulation of the method, time integration is achieved by a second order Leap-Frog scheme. Here, we discuss about several recent developments aiming at improving the accuracy and the computational efficiency of this DGTD method in view of the simulation of problems involving general domains and heterogeneous media.
- Aug 2010
- EMTS 2010 (ElectroMagnetic Theory, 20th URSI international Symposium on)
This paper is concerned with a preliminary investigation of a Discontinuous Galerkin Time Domain (DGTD) method formulated on hybrid quadrangular-triangular meshes for the solution of the two-dimensional Maxwell equations. The general objective of this study is to enhance the flexibility and the efficiency of DGTD methods for large-scale time domain electromagnetic wave propagation problems with regards to the discretization process of complex propagation scenes and the work discussed here is a first step in this direction. Within each mesh element, the electromagnetic field components are approximated by a high order nodal polynomial and time integration of the associated semi-discrete equations is achieved by a second order Leap-Frog scheme. We study the stability of the resulting DGTD method and present numerical results aiming at the validation of the method on a model problem.
In this paper, we discuss the formulation, stability and validation of a high-order non-dissipative discontinuous Galerkin (DG) method for solving Maxwell’s equations on non-conforming simplex meshes. The proposed method combines a centered approximation for the numerical fluxes at inter element boundaries, with either a second-order or a fourth-order leap-frog time integration scheme. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary-level hanging nodes. The method is proved to be stable and conserves a discrete counterpart of the electromagnetic energy for metallic cavities. Numerical experiments with high-order elements show the potential of the method.
- May 2010
Purpose – The purpose of this paper is to develop a time implicit discontinuous Galerkin method for the simulation of two-dimensional time-domain electromagnetic wave propagation on non-uniform triangular meshes. Design/methodology/approach – The proposed method combines an arbitrary high-order discontinuous Galerkin method for the discretization in space designed on triangular meshes, with a second-order Cranck-Nicolson scheme for time integration. At each time step, a multifrontal sparse LU method is used for solving the linear system resulting from the discretization of the TE Maxwell equations. Findings – Despite the computational overhead of the solution of a linear system at each time step, the resulting implicit discontinuous Galerkin time-domain method allows for a noticeable reduction of the computing time as compared to its explicit counterpart based on a leap-frog time integration scheme. Research limitations/implications – The proposed method is useful if the underlying mesh is non-uniform or locally refined such as when dealing with complex geometric features or with heterogeneous propagation media. Practical implications – The paper is a first step towards the development of an efficient discontinuous Galerkin method for the simulation of three-dimensional time-domain electromagnetic wave propagation on non-uniform tetrahedral meshes. It yields first insights of the capabilities of implicit time stepping through a detailed numerical assessment of accuracy properties and computational performances. Originality/value – In the field of high-frequency computational electromagnetism, the use of implicit time stepping has so far been limited to Cartesian meshes in conjunction with the finite difference time-domain (FDTD) method (e.g. the alternating direction implicit FDTD method). The paper is the first attempt to combine implicit time stepping with a discontinuous Galerkin discretization method designed on simplex meshes.
Nous présentons dans ce rapport une méthode Galerkin discontinue pour la résolution numérique des équations de Maxwell en domaine temporel sur des maillages hybrides. Nous formulons les schémas de discrétisation en trois dimensions d'espace (3D) pour des maillages hybrides alors tétraédriques / hexaédriques, en utilisant une approximation centrée pour approcher les intégrales de surface et un schéma d'intégration en temps de type saute-mouton d'ordre deux. Toujours en 3D, nous étudions la stabilité L2 de cette méthode en montrant qu'elle conserve une énergie discrète et en exhibant une condition suffisante de stabilité de type CFL. Nous réalisons plusieurs simulations numériques en deux dimensions (on utilise dans ce cas des maillages hybrides triangulaire / quadrangulaire) qui visent à valider et réaliser une première évaluation des possibilités de la méthode Galerkin discontinue proposée.
- Jan 2010
In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the solution of the unsteady Maxwell equations modeling electromagnetic wave propagation. One of the main features of DGTD methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems. Such DGTD methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step. A more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part, resulting in an hybrid explicit–implicit (or locally implicit) time integration strategy. In this paper, we report on our recent efforts towards the development of such a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on unstructured simplicial meshes. Numerical experiments for 3D propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes.
Ces dernières années, les méthodes Galerkin discontinues ont fait l'objet de plusieurs travaux visant à leur mise au point pour la résolution numérique des équations de Maxwell instationnaires. Dans la grande majorité de ces travaux, les composantes du champ électromagnétique sont approchées au sein de chaque élément du maillage par une interpolation polynomiale d'ordre élevé. Différentes formes d'interpolation polynomiale ont été considérées mais aucune étude comparative n'a été menée jusqu'ici. On présente dans ce rapport les résultats d'une telle étude réalisée dans le cadre de la résolution numérique des équations de Maxwell 1D.
We report on recent efforts towards the development of a high order, non-conforming, discontinuous Galerkin method for the solution of the system of frequency domain Maxwell's equations in heterogeneous propagation media. This method is an extension of the low order one which was proposed in (http://hal.inria.fr/inria-00155231/en/)
- Aug 2009
- Euro-Par 2009 Parallel Processing, 15th International Euro-Par Conference, Delft, The Netherlands, August 25-28, 2009. Proceedings
Developing highly communicating scientific applications capable of efficiently use computational grids is not a trivial task. Ideally, these applications should consider grid topology 1) during the mesh partitioning, to balance workload among heterogeneous resources and exploit physical neighborhood, and 2) in communications, to lower the impact of latency and reduced bandwidth. Besides, this should not be a complex matter in end-users applications. These are the central concerns of the DiscoGrid project, which promotes the concept of a hierarchical SPMD programming model, along with a grid-aware multi-level mesh partitioning to enable the treatment of grid issues by the underlying runtime, in a seamless way for programmers. In this paper, we present the DiscoGrid project and the work around the GCM/ProActive-based implementation of the DiscoGrid Runtime. Experiments with a non-trivial computational electromagnetics application show that the component-based approach offers a flexible and efficient support and that the proposed programming model can ease the development of such applications.
- Jan 2009
Numerical methods for solving the time domain Maxwell equations often rely on cartesian meshes and are variants of the finite difference time domain method originating in the seminal work of Yee . In the recent years, there has been an increasing interest in discontinuous Galerkin time domain methods dealing with unstructured meshes since the latter are particularly well suited for the discretization of geometrical details that characterize applications of practical relevance. Similarly to Yee's finite difference time domain method, discontinuous Galerkin time domain methods generally rely on explicit time integration schemes and are therefore constrained by a stability condition that can be very restrictive on highly refined or unstructured meshes and when the local approximation relies on high order polynomial interpolation. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable. However, such a time scheme comes at the expense of the inversion of a global linear system at each time step, thus obliterating one of the attractive features of discontinuous Galerkin formulations. In this paper, we report on our recent efforts concerning the design of an implicit time integration scheme in conjunction with a discontinuous Galerkin approximation method for solving the time domain Maxwell equations on unstructured triangular meshes. Despite the memory and computational overheads induced by the inversion of a global linear system at each time step, we demonstrate that an implicit discontinuous Galerkin time domain method is a viable numerical strategy for solving electromagnetic wave propagation problems on locally refined unstructured meshes.
Ce rapport traite de la résolution des équations de Maxwell en régime harmonique par une méthode de type Galerkin discontinu. Il rappelle tout d'abord la formulation du problème et propose quelques éléments de discussion pour le choix du flux numérique utilisé dans la méthode de discrétisation. Il présente ensuite quelques pistes pour la résolution des systèmes linéaires obtenus. En particulier, on étudie la possibilié de résoudre ces systèmes linéaires par des algorithmes multigrille algébriques et des résultats préliminaires sont obtenus dans le cas des équations de Maxwell en deux dimensions d'espace.
We present numerical results concerning the solution of the time-harmonicMaxwellequations discretized by discontinuousGalerkinmethods. In particular, a numerical study of the convergence, which compares different strategies proposed in the literature for the elliptic Maxwellequations, is performed in the two-dimensional case.
This paper is concerned with the design of a high-order discontinuous Galerkin (DG) method for solving the 2-D time-domain Maxwell equations on nonconforming triangular meshes. The proposed DG method allows for using nonconforming meshes with arbitrary-level hanging nodes. This method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements of the mesh, with a leap-frog time integration scheme. Numerical experiments are presented which both validate the theoretical results and provide further insights regarding to the practical performance of the proposed DG method, particulary when nonconforming meshes are employed.
The numerical solution of the three-dimensional time-harmonic Maxwell equations using high order methods such as discontinuous Galerkin formulations require efficient solvers. A domain decomposition strategy is introduced for this purpose. This strategy is based on optimized Schwarz methods applied to the first order form of the Maxwell system and leads to the best possible convergence of these algorithms. The principles are explained for a 2D model problem and numerical simulations confirm the predicted theoretical behavior. The efficiency is further demonstrated on more realistic 3D geometries including a bioelectromagnetism application.
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 (1966). In 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 characterize 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 2-D time-domain Maxwell equations on nonuniform triangular meshes.
We present here a domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by a discontinuous Galerkin method. In order to allow the treatment of irregularly shaped geometries, the discontinuous Galerkin method is formulated on unstructured tetrahedral meshes. The domain decomposition strategy takes the form of a Schwarz-type algorithm where a continuity condition on the incoming characteristic variables is imposed at the interfaces between neighboring subdomains. A multifrontal sparse direct solver is used at the subdomain level. The resulting domain decomposition strategy can be viewed as a hybrid iterative/direct solution method for the large, sparse and complex coefficients algebraic system resulting from the discretization of the time-harmonic Maxwell equations by a discontinuous Galerkin method.
In this work, we discuss the formulation, stability, convergence and numerical validation of a high-order leap-frog based non-dissipative discontinuous Galerkin time-domain (DGTD) method for solving Maxwell's equations on non-conforming simplicial meshes. This DGTD method makes use of a nodal polynomial interpolation method for the approximation of the electromagnetic field within a simplex, and of a centered scheme for the calculation of the numerical flux at an interface between neighboring elements. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes. The method is proved to be stable and conserves a discrete analog of the electromagnetic energy for metallic cavities. The convergence of the semi-discrete approximation to Maxwell's equations is established rigorously and bounds on the global divergence error are provided. Numerical experiments with high-order elements show the potential of the method.
Résumé Aujourd'hui, grâce à Internet, il est possible d'interconnecter des machines du monde entier pour traiter et stocker des masses de données. Cette collection hétérogène et distribuée de ressources de stockage et de calcul a donné naissance à un nouveau concept : les grilles informatiques. L'idée de mutualiser les ressources informatiques vient de plusieurs facteurs, évolution de la recherche en parallélisme qui, après avoir étudié les machines homogènes, s'est attaquée aux environnements hétérogènes puis distribués ; besoins croissants des applications qui nécessitent l'utilisation toujours plus importante de moyens informatiques forcément répartis. La notion de grille peut avoir plusieurs sens suivant le contexte : grappes de grappes, environnements de type GridRPC (appel de procédure à distance sur une grille)., réseaux pair-à-pair, systèmes de calcul sur Internet, etc... Il s'agit d'une manière générale de systèmes dynamiques, hétérogènes et distribués à large échelle. Un grand nombre de problématiques de recherche sont soulevées par les grilles informatiques. Elles touchent plusieurs domaines de l'informatique : algorithmique, programmation, intergiciels, applications, réseaux. L'objectif de GRID'5000 est de construire un instrument pour réaliser des expériences en informatique dans le domaine des systèmes distribués à grande échelle (GRID). Cette plate-forme, ouverte depuis 2006 aux chercheurs de la communauté grille, regroupe un certain nombre de sites répartis sur le territoire national. Chaque site héberge une ou plusieurs grappes de processeurs. Ces grappes sont alors interconnectées via une infrastructure réseau dédiée à 10 Gb/s fournie par RENATER. À ce jour, GRID'5000 est composé de 9 sites: Lille, Rennes, Orsay, Nancy, Bordeaux, Lyon, Grenoble, Toulouse et Nice. Début 2007, GRID'5000 regroupait plus de 2500 processeurs et près de 3500 coeurs.
We describe ongoing research activities at Inria Sophia Antipolis aiming at the construction of efficient and robust unstructured multigrid solvers for complex 2D and 3D flow simulations.
In this paper we discuss the possible advantages and inherent draw-backs of a reduced order model for compressible flows of practical interest. The models considered include Euler and Navier-Stokes equations. Turbulent flows are also investigated, and the model of turbulence evolution is based on the κ − ε formulation with wall laws. The reduced order model is constructed from a set of basis functions determined by proper orthogonal decomposition (POD). The POD basis functions are used to filter, at each time step, the solution obtained from a finite-volume code. Three test cases are discussed: unsteady Euler flow about an oscillating airfoil, laminar vortex shedding from a NACA-0012 airfoil at incidence, and finally turbulent vortex shedding from a square cylinder.
- Jan 2007
Programming non-embarrassingly parallel scientific computing applications such as those involving the numerical resolution of system of PDEs using mesh based methods for grid computing environments is a complex and important issue. This work contributes to this goal by proposing some MPI extensions to let programmers deal with the hierarchical nature of the grid infrastructure thanks to a tree representation of the processes as well as the corresponding extension of collective and point-to-point operations. It leads in particular to support N M communications with transparent data redistribution.
This work is concerned with the design of a hp-like discontinuous Galerkin (DG) method for solving the 2D time-domain Maxwell's equations on non-conforming locally refined triangular meshes. The proposed DG method allows non-conforming meshes with arbitrary-level hanging nodes. This method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements of the mesh, with a leap-frog time integration scheme.
- Nov 2006
Understanding the physics of the rupture process requires very sophisticated and accurate tools in which both the geometry of the fault surface and realistic frictional behaviours could interact during rupture propagation. New formulations have been recently proposed for modelling the dynamic shear rupture of non-planar faults (Ando et al., 2004; Cruz-Atienza &Virieux, 2004; Huang &Costanzo, 2004) providing highly accurate field estimates nearby the crack edges at the expanse of a simple medium description or high computational cost. We propose a new method based on the finite volume formulation to model the dynamic rupture propagation of non-planar faults. After proper transformations of the velocity-stress elastodynamic system of partial differential equations following an explicit conservative law, we construct an unstructured time-domain numerical formulation of the crack problem. As a result, arbitrary non-planar faults can be explicitly represented without extra computational cost. The analysis of the total discrete energy through the fault surface leads us to the specification of dynamic rupture boundary conditions which insure the correct discrete energy time variation and, therefore, the system stability. These boundary conditions are set on stress fluxes and not on stress values, which makes the fracture to have no thickness. Different shapes of cracks are analysed. We present an example of a bidimensional non-planar spontaneous fault growth in heterogeneous media as well as preliminary results of a highly efficient extension to the three dimensional rupture model based on the standard MPI.
Large scale distributed systems such as Grids are difficult to study from theoretical models and simulators only. Most Grids deployed at large scale are production platforms that are inappropriate research tools because of their limited reconfiguration, control and monitoring capabilities. In this paper, we present Grid'5000, a 5000 CPU nation-wide infrastructure for research in Grid computing. Grid'5000 is designed to provide a scientific tool for computer scientists similar to the large-scale instruments used by physicists, astronomers, and biologists. We describe the motivations, design considerations, architecture, control, and monitoring infrastructure of this experimental platform. We present configuration examples and performance results for the reconfiguration subsystem.
- Oct 2006
- Euro-Par'97 Parallel Processing
This paper presents a two-level strategy for the parallelization of a genetic algorithm coupled to a compressible flow solver designed on unstructured meshes. The resulting algorithm is used for the optimum shape design of aerodynamic configurations. Preliminary results are presented for the optimization of two-dimensional airfoils.
- Aug 2006
We discuss the parallel performances of discontinuous Galerkin solvers designed on unstructured tetrahedral meshes for the calculation of three-dimensional heterogeneous electromagnetic and aeroacoustic wave propagation problems. An explicit leap-frog time-scheme along with centered numerical fluxes are used in the proposed discontinuous Galerkin time-domain (DGTD) methods. The schemes introduced are genuinely non-dissipative, in order to achieve a discrete equivalent of the energy conservation. Parallelization of these schemes is based on a standard strategy that combines mesh partitioning and a message passing programming model. The resulting parallel solvers are applied and evaluated on several large-scale, homogeneous and heterogeneous, wave propagation problems.
- Jun 2006
The ever-rising diffusion of cellular phones has brought about an increased concern for the possible consequences of electro-magnetic radiation on human health. Possible thermal effects have been investigated, via experimentation or simulation, by several research projects in the last decade. Concerning numerical modeling, the power absorption in a user's head is generally computed using discretized models built from clinical MRI data. The vast majority of such numerical studies have been conducted using Finite Differences Time Domain methods, although strong limitations of their accuracy are due to heterogeneity, poor definition of the detailed structures of head tissues (staircasing effects), etc. In order to propose numerical modeling using Finite Element or Discontinuous Galerkin Time Domain methods, reliable automated tools for the unstructured discretization of human heads are also needed. Results presented in this article aim at filling the gap between human head MRI images and the accurate numerical modeling of wave propagation in biological tissues and its thermal effects. To cite this article: G.
A general Discontinuous Galerkin framework is introduced for symmetric systems of conservations laws. It is applied to the three-dimensional electromagnetic wave propagation in heterogeneous media, and to the propagation of aeroacoustic perturbations of either uniform or nonuniform, steady solutions of the three-dimensional Euler equations. In all these linear contexts, the time evolution of some quadratic wave energy is given in a balance equation, with a volumic source term for aeroacoustics in a nonuniform flow. An explicit leap-frog time scheme along with centered numerical fluxes are used in the proposed Discontinuous Galerkin Time Domain (DGTD) method, in order to achieve a discrete equivalent of the balance equation for the wave energy. The scheme introduced is genuinely nondissipative. Numerical first-order boundary conditions are developed to bound the domain and stability is proved on arbitrary unstructured meshes and discontinuous finite elements, under some CFL-like stability condition on the time step. Numerical results obtained with a parallel implementation of the method based on mesh partitioning and message passing are presented to show the potential of the method.
- Jan 2006
- 7th International Symposium on Computer Methods in Biomechanics and Biomedical Engineering
Grids raise new challenges for programming, composing, and deploying numerical applications. Heterogeneity, medium to high latency, various underlying systems and protocols call for new paradigms and techniques. Within this framework, the development of high performance numerical methods for the solution of systems of PDEs (Partial Differential Equations), must integrate these factors, representing both new difficulties and new opportunities. In this chapter, we describe an open source middleware for the Grid, ProActive, featuring distributed objects and components. Using ProActive, we demonstrate how to design and implement an object-oriented (OO) time domain finite volume solver on unstructured meshes for the 3D Maxwell’s equations modelling the propagation of electromagnetic waves. We also present some experimental results obtained on an experimental Grid, Grid’5000, running on more than 400 processors.
On s'intéresse à la résolution numérique des équations de Maxwell tridimensionnelles en régime harmonique par des méthodes de type Galerkin Discontinu (GD) en maillages tétraédriques non-structurés où l'on approche les inconnues du problème par des fonctions constantes par morceaux (méthode GD d'ordre 0 ou GDP0) ou linéaires par morceaux (méthode GD d'ordre 1 ou GDP1). De nombreux travaux ces dernières années ont démontré l'intérêt des formulations GD pour la modélisation numérique de phénomènes de propagation d'ondes en milieux hétérogènes. Dans cette étude, on adapte au domaine fréquentiel (ou régime harmonique) des méthodes GD centrées précédemment dévelopées pour les mêmes équations résolues en domaine temporel. Dans ce rapport, on traite essentiellement de la formulation des méthodes en question en 1D et 3D, et de leur analyse théorique (dispersion numérique et caractère bien posé du problème discret). La difficulté principale réside dans le fait que l'on traite d'un problème posé dans le domaine complexe pour lequel des résultats d'inversibilité sont difficiles à prouver en l'absence d'hypothèses supplémentaires. On conclut en présentant une série de résultats numériques préliminaires en 1D et 3D, ces derniers visant essentiellement à valider les méthodes GD proposées. La question de la résolution (itérative ou directe) des systèmes linéaires obtenus sera traitée dans un prochain rapport.
On étudie la stabilité d'une méthode Galerkin discontinu pour la résolution numérique des équations de Maxwell 2D en domaine temporel sur des maillages triangulaires non-conformes. Cette méthode combine l'utilisation d'une approximation centrée pour l'évaluation des flux aux interfaces entre éléments voisins du maillage, á un schéma d'intégration en temps de type saute-mouton. La méthode repose sur une base de fonctions polynomiales nodales Pk et on considère ici les schémas obtenus pour k=0,..3. L'objectif de cette étude est d'exhiber des conditions sous lesquelles les schémas correspondant sont stables, et de comparer ces conditionsá celles obtenues dans le cas de maillages conformes.
Large scale distributed systems like Grids are difficult to study only from theoretical models and simulators. Most Grids deployed at large scale are production platforms that are inappropriate research tools because of their limited reconfiguration, control and monitoring capabilities. In this paper, we present Grid'5000, a 5000 CPUs nation-wide infrastructure for research in Grid computing. Grid'5000 is designed to provide a scientific tool for computer scientists similar to the large-scale instruments used by physicists, astronomers and biologists. We describe the motivations, design, architecture, configuration examples of Grid'5000 and performance results for the reconfiguration subsystem.
- Nov 2005
- 6th IEEE/ACM International Workshop on Grid Computing (GRID 2005)
Grid 2005 held in conjunction with SC'05, the International Conference for High Performance Computing, Networking and Storage
A Discontinuous Galerkin method is used for to the numerical solution of the time-domain Maxwell equations on unstructured meshes. The method relies on the choice of local basis functions, a centered mean approximation for the surface integrals and a second-order leapfrog scheme for advancing in time. The method is proved to be stable for cases with either metallic or absorbing boundary conditions, for a large class of basis functions. A discrete analog of the electromagnetic energy is conserved for metallic cavities. Convergence is proved for P k Discontinuous elements on tetrahedral meshes, as well as a discrete divergence preservation property. Promising numerical examples with low-order elements show the potential of the method.
Existing numerical methods for the solution of the time domain Maxwell equations often rely on explicit time integration schemes and are therefore constrained by a stability condition that can be very restrictive on highly refined or unstructured meshes. The present study aims at investigating the applicability of an implicit time integration scheme in conjunction with a finite volume approximation method on unstructured meshes. If one wants to achieve at least second-order accuracy in time, then the most natural choice for the implicit time integration of the semi-discrete Maxwell equations is the Crank-Nicolson scheme. In the first part of the paper, we study some of the mathematical properties of the resulting Implicit Finite Volume Time Domain (IFVTD) method in the one dimensional case using different boundary conditions (metallic, periodic and absorbing boundary conditions). We prove that the IFVTD method globally conserves a discrete form of the electromagnetic energy if the initial boundary value problem is based on metallic or periodic boundary conditions. If an absorbing boundary condition is used then we show that the discrete electromagnetic energy is decreasing. In this one dimensional study, we also prove that the matrix operator characterizing the IFVTD method is invertible. In the second part of the paper, we extend the results of the one dimensional analysis to the three dimensional Maxwell equations. We conclude with preliminary numerical results on a simple academic test case in order to validate the proposed method.
- Jun 2004
PORFLOW ™ is a CFD software which is dedicated to groundwater flow and nuclear waste management simulations. PORFLOW ™ is developed by the ACRi company located in Bel Air (California, USA) (http://www.acricfd.com) with a subsidiary in Sophia Antipolis (France) (http://www.acri.fr). In this paper, we report on our recent efforts to improve the performances of PORFLOW ™ by focusing our attention on the solution of the large sparse linear systems resulting from the space and time discretizations of flow and transport equations. This work is illustrated by applying PORFLOW ™ to the calculation of the COUPLEX1 and part of the COUPLEX2 problems.
- May 2004
- Parallel and Distributed Processing Symposium, 2004. Proceedings. 18th International
Summary form only given. Within the trend of object-based distributed computing, we present the design and implementation of a numerical simulation for electromagnetic waves propagation. A sequential Java design and implementation is first presented. Further, a distributed and parallel version is derived from the first, using an active object pattern. In addition, benchmarks are presented on this nonembarrassingly parallel application. A first contribution resides in the sequential object-oriented design that proved to be very modular and extensible; the classes and abstractions are designed to allow both element and volume type methods, furthermore, valid on structured, unstructured, or hybrid meshes. Compared to a Fortran version, the performance of this highly modular version proved to be in the same range. It is also shown how smoothly the sequential version can be distributed, keeping the same structuring and object abstractions, allowing to deal with larger data size. Finally, benchmarks on up to 64 processors compare the performances with respect to sequential and parallel versions, putting that in perspective with a comparable Fortran version.
In a previous paper [DL01], we reported on numerical experiments with a non-overlapping domain decomposition method that has been specifically designed for the calculation of steady compressible inviscid flows governed by the two-dimensional Euler equations. In the present work, we study this method from the theoretical point of view. The proposed method relies on the formulation of an additive Schwarz algorithm which involves interface conditions that are Dirichlet conditions for the characteristic variables corresponding to incoming waves (often referred to as natural or classical interface conditions), thus taking into account the hyperbolic nature of the Euler equations. In the first part of this paper, the convergence of the additive Schwarz algorithm is analyzed in the two-and three-dimensional continuous cases by considering the linearized equations and applying a Fourier analysis. We limit ourselves to the cases of two and three-subdomain decompositions with or without overlap and we obtain analytical expressions of the convergence rate of the Schwarz algorithm. Besides the fact that the algorithm is always convergent, surprisingly, there exist flow conditions for which the asymptotic convergence rate is equal to zero. Moreover, this behavior is independent of the space dimension. In the second part, we study the discrete counterpart of the non-overlapping additive Schwarz algorithm based on the implementation adopted in [DL01] but assuming a finite volume formulation on a quadrangular mesh. We find out that the expression of the convergence rate is actually more characteristic of an overlapping additive Schwarz algorithm. We conclude by presenting numerical results that confirm qualitatively the convergence behavior found analytically.
- Apr 2004
We report on our efforts towards the design of efficient parallel hierarchical iterative methods for the solution of sparse and irregularly structured linear systems resulting from CFD applications. The solution strategies considered here share a central numerical kernel which consists in a linear multigrid by volume agglomeration method. Starting from this method, we study two parallel solution strategies. The first variant results from a direct intra-grid parallelization of multigrid operations on coarse grids. The second variant is based on an additive Schwarz domain decomposition algorithm which is formulated at the continuous level through the introduction of specific interface conditions. In this variant, the linear multigrid by volume agglomeration method is used to approximately solve the local systems obtained at each iteration of the Schwarz algorithm. As a result, the proposed hybrid domain decomposition/multigridmethod can be viewed as a particular form of parallelmultigrid in which multigrid acceleration is applied on a subdomain basis, these local calculations being coordinated by an appropriate domain decomposition iteration at the global level. The parallelperformances of these two parallelmultigridmethods are evaluated through numerical experiments that are performed on several clusters of PCs with different computational nodes and interconnection networks.
Because of the accuracy required in a neurosurgical proce- dure, tracking intra-operative deformations is a challenging task. Fur- thermore, the clinical demand for fast non rigid registration will have to be met in a very near future. In this paper, we propose a patient-specific biomechanical model based on block-matching in order to register two MR images of the same patient with a parallel implementation. Com- pared to other intra-operative registration techniques, this method com- bines a viscoelastic mechanical regularization with a correlation-based iconic energy term. We first shortly present the theoretical aspects of our method (more detailed in (9)). Then we describe in more details the parallel implementation of the algorithm. Finally we present a retrospec- tive registration study made of four pre/post operative MRI pairs.
Because of the accuracy required in a neurosurgical procedure, tracking intra-operative deformations is a challenging task. Furthermore, the clinical demand for fast non rigid registration will have to be met in a very near future. In this paper, we propose a patient-specific biomechanical model based on block-matching in order to register two MR images of the same patient with a parallel implementation. Compared to other intra-operative registration techniques, this method combines a viscoelastic mechanical regularization with a correlation-based iconic energy term. We first shortly present the theoretical aspects of our method (more detailed in ). Then we describe in more details the parallel implementation of the algorithm. Finally we present a retrospective registration study made of four pre/post operative MRI pairs.
- Dec 2002
Interface conditions (IC) between subdomains have an important impact on the convergence rate of domain decomposition algorithms. We first recall the Schwarz method which is based on the use of Dirichlet conditions on the boundaries of the subdomains and overlapping subdomains. We explain how it is possible to replace them by more efficient ICs with normal and tangential derivatives so that overlapping is not necessary. It is possible to optimize the coefficients of the IC in order to achieve the best convergence rate. Results are given for the convection—diffusion equation. Then we consider the compressible Euler equations which form a system of equations. We present a new analysis of the use of interface conditions based on the flux splitting. We compute the convergence rate in the Fourier space. We find a dependence of their effectiveness on the Mach number M. For M=⅓, the convergence rate tends to zero as the wavenumber of the error goes to infinity. We stress the differences with the scalar equations. We present numerical results in agreement with the theoretical results. Copyright © 2002 John Wiley & Sons, Ltd.
- Nov 2002
In this work we examine the acceleration of the convergence of a non-overlapping additive Schwarz-type algorithm by modifying the transmission conditions applied to the subdomain interfaces. We have built generalized zero-order interface conditions using the Smith theory of diagonalizing polynomial matrices. The numerical experiments confirmed qualitatively the behaviour in accordance with the theory, but we could not reproduce identically the results obtained in the continuous case. The preliminary results are very encouraging since they lead to a very good convergence rate for certain Mach numbers. Copyright © 2002 John Wiley & Sons, Ltd.
We report on our recent efforts on the formulation and the evaluation of a domain decomposition algorithm for the parallel solution of two-dimensional compressible inviscid flows. The starting point is a flow solver for the Euler equations, which is based on a mixed finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is obtained using a linearized backward Euler implicit scheme. As a result, each pseudo-time step requires the solution of a sparse linear system for the flow variables. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. First, we formulate an additive Schwarz algorithm using appropriate matching conditions at the subdomain interfaces. In accordance with the hyperbolic nature of the Euler equations, these transmission conditions are Dirichlet conditions for the characteristic variables corresponding to incoming waves. Then, we introduce interface operators that allow us to express the domain decomposition algorithm as a Richardson-type iteration on the interface unknowns. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In our approach, the interface unknowns are numerical (normal) fluxes. Copyright © 2001 John Wiley & Sons, Ltd.
- Mar 2001
In this paper we examine how parallel multigrid acceleration can be used to improve the efficiency of two-dimensional compressible steady flow calculations on unstructured meshes. We study two parallel multigrid formulations. The first one is based on the standard approach that relies on domain partitioning for the parallel treatment of the pre- and post-smoothing steps whereas the coarse grid levels are visited sequentially according to predefined cycles (V-cycle, F-cycle or W-cycle). Reducing communication overheads is of crucial importance for parallel multigrid methods. When adopting the standard parallelization technique (i.e., intra-level parallelism based on domain partitioning) the usual drawback is that, as the calculation in a given cycling strategy proceeds from the finest level to the coarsest ones, the ratio between communication and calculation becomes worse resulting in a notable degradation of the parallel efficiency. In order to improve this situation, the second formulation considered in this study makes use of residual and correction filtering techniques allowing a parallel treatment of the various grid levels. This leads to the notion of inter-level parallelism. We propose distributed memory parallel versions of these two multigrid formulations and evaluate them through numerical experiments that are performed on a cluster of Pentium Pro computers interconnected via a 100 Mbit/s FastEthernet switch.
This paper is concerned with the formulation and the evaluation of a hybrid solution method that makes use of domain decomposition and multigrid principles for the calculation of two-dimensional compressible viscous flows on unstructured triangular meshes. More precisely, a non-overlapping additive domain decomposition method is used to coordinate concurrent subdomain solutions with a multigrid method. This hybrid method is developed in the context of a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is performed using a linearized backward Euler implicit scheme. As a result, each pseudo time step requires the solution of a sparse linear system. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically, the Schwarz algorithm is equivalent to a Jacobi iteration on a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In the present approach, the interface unknowns are numerical fluxes. The interface system is solved by means of a full GMRES method. Here, the local system solves that are induced by matrix-vector products with the interface operator, are performed using a multigrid by volume agglomeration method. The resulting hybrid domain decomposition and multigrid solver is applied to the computation of several steady flows around a geometry of NACA0012 airfoil.
We present the ongoing joint work of two research groups to exploit the advantages of hybrid grids for numerical simulation in fluid dynamics. After referring to some general issues of the solution method, the development of an FAS multigrid method is outlined in more detail. Further three alternative attempts for hybrid grid generation are presented.
- Oct 2000
In this paper, we report on our recent efforts concerning the design of parallel linear multigrid algorithms for the acceleration of 3-dimensional compressible flow calculations. The multigrid strategy adopted in this study relies on a volume agglomeration principle for the construction of the coarse grids starting from a fine discretization of the computational domain. In the past, this strategy has mainly been studied in the 2-dimensional case for the solution of the Euler equations (see Lallemand et al. ), the laminar Navier–Stokes equations (see Mavriplis and Venkatakrishnan ) and the turbulent Navier–Stokes equations (see Carr , Mavriplis  and Francescatto and Dervieux ). A first extension to the 3-dimensional case is presented by Mavriplis and Venkatakrishnan in  and more recently in Mavriplis and Pirzadeh . The main contribution of the present work is twofold: on the one hand, we demonstrate the successful extension and application of the multigrid by a volume agglomeration principle to the acceleration of complex 3-dimensional flow calculations on unstructured tetrahedral meshes and, on the other hand, we enhance further the efficiency of the methodology through its adaptation to parallel architectures. Moreover, a nontrivial aspect of this work is that the corresponding software developments are taking place in an existing industrial flow solver.
- Apr 2000
The interest in unstructured meshes for Computational Fluid Dynamics (CFD) applications appears to be increasingly important in the industrial community. Industrial applications require the numerical simulation of complex flows (i.e. the underlying flows exhibit localized high variations of physical quantities) around or within complex geometries. Unstructured meshes are particularly well suited to these kinds of simulation due to their ability in accurately discretizing complex computational domains and, to their flexibility in dynamically refining and derefining, or deforming, in order to match the underlying flow features. Concerning flow solvers, the main question appears to be the lack of efficiency demonstrated by unstructured mesh solvers compared to structured ones. Many efficient methods developed in the structured context are not easily extensible to unstructured meshes and much research work has yet to be done in this direction. During the last ten years, several such works have demonstrated that multigrid principles can yield robust and efficient unstructured mesh solvers (see for example Lallemand et al. , Koobus et al. , Carré , Mavriplis et al. , and ). In this paper, we describe ongoing research activities at Inria Sophia Antipolis aiming at the construction of efficient and robust unstructured multigrid solvers for complex two-dimensional (2D) and three-dimensional (3D) flow simulations. Both academic and industrial aspects are considered.
This paper presents a two-level strategy for the parallelization of a Genetic Algorithm (GA) coupled to a compressible flow solver designed on unstructured triangular meshes. The parallel implementation is based on MPI and makes use of the process group features of this environment. The resulting algorithm is used for the optimum shape design of aerodynamic configurations. Numerical and performance results are presented for the optimization of two-dimensional airfoils for calculations performed on the following systems: an SGI Origin 2000 and an IBM SP-2 MIMD systems; an Pentium Pro (P6/200 MHz) cluster where the interconnection is realized through a FastEthernet (100 Mbits/s) switch.
Fluid flows are very often governed by the dynamics of a mall number of coherent structures, i.e., fluid features which keep their individuality during the evolution of the flow. The purpose of this paper is to study a low order simulation of the Navier–Stokes equations on the basis of the evolution of such coherent structures. One way to extract some basis functions which can be interpreted as coherent structures from flow simulations is by Proper Orthogonal Decomposition (POD). Then, by means of a Galerkin projection, it is possible to find the system of ODEs which approximates the problem in the finite-dimensional space spanned by the POD basis functions. It is found that low order modeling of relatively complex flow simulations, such as laminar vortex shedding from an airfoil at incidence and turbulent vortex shedding from a square cylinder, provides good qualitative results compared with reference computations. In this respect, it is shown that the accuracy of numerical schemes based on simple Galerkin projection is insufficient and numerical stabilization is needed. To conclude, we approach the issue of the optimal selection of the norm, namely the H 1 norm, used in POD for the compressible Navier–Stokes equations by several numerical tests.
Parallelization strategies based on domain partitioning techniques have been widely adopted for parallel finite element computations because of their suitability to distributed memory platforms. In most cases, this parallelization is based on non-overlapping partitions especially for Computational Structural Mechanics applications. However, finite volume (or mixed finite element/finite volume) discretization methods, which are frequently implemented in Computational Fluid Dynamics applications, generally require the use of overlapping mesh partitions to keep the parallelization work simple. Unfortunately, many tools on which the partitioning step relies give poor results when asked for overlapping partitions. In this paper, we describe an efficient method to transform a non-overlapping partition of a domain into an overlapping one. We also propose an optimization strategy for overlapping partitions that mainly aims at reducing the computational load unbalance as well as the size of the interfaces. The new algorithms demonstrate significant improvements as they are applied to generate overlapping partitions in the context of a parallel mixed finite element/finite volume three-dimensional flow solver
The governing equations of viscous flows, the Navier–Stokes equations, are approximated by means of a low order model based on proper orthogonal decomposition (POD). Numerical evidence and analysis of simplified models show that the resulting time-wise semidiscretization is only marginally stable. Here, two methods providing additional stabilization are described: the first is based on a Lax–Wendroff type artificial diffusion term, while the second is a reinterpretation of POD in the frame of the finite element functional least square method.
We present a two-level strategy for parallelization of a genetic algorithm coupled to a compressible flow solver designed on unstructured meshes. The resulting algorithm is used for the optimum shape design of aerodynamic configurations. The basic motivation to use a parallelized genetic algorithm in optimum shape design is very high computational effort needed for accurate evaluation of design configuration. In addition, hard problems need a large population, and this translates directly into higher computational costs. Results are presented for direct optimization problem with the minimization of drag-induced shock on RAE2822 airfoil.
For solving a problem on a fine mesh, the multigrid technology requires the definition of coarse levels, coarse grid operators and integer-grid transfer operators. For non-structured three-dimensional meshes in CFD, two major multigrid techniques have emerged in the last years. The first one relies on the use of non-nested triangulations, while the second technique is associated to finite volume discretization and agglomeration/aggregation techniques. In this paper, we first present some automatic ways to coarsen three-dimensional meshes, and show that these geometrical methods result in efficient multigrid algorithms. Then, we briefly describe the volume agglomeration method and shows an example of its application in a three-dimensional industrial CFD code.
The present paper aims at pointing out the attractivity of a particular type of domain decomposition method as a parallel smoother in a multigrid algorithm. The smoothing ability of standard Schwarz type algorithms have already been studied for elliptic problems such as the Laplace equation. The more recent Restricted Additive Schwarz (RAS) algorithm seems attractive from the parallel efficiency viewpoint. The first part of this paper identifies several formulations of the Schwarz algorithm. In the second part, the Restricted Additive Schwarz algorithm and several variants are analyzed with respect to their smoothing properties, on a model 1D Laplace equation and a 2D advection-diffusion equation. This last model should give a first idea of the behavior of this kind of smoother for compressible fluid flow problems.