Hybrid Explicit-Implicit Time Integration for Grid-Induced Stiffness in a DGTD Method for Time Domain Electromagnetics
02 Nov 2011-pp 163-170
TL;DR: This paper reports on some efforts to design a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on locally refined simplicial meshes.
Abstract: 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.
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01 Feb 2013
TL;DR: This paper provides a comprehensive review of different DGTD schemes, highlighting the fundamental issues arising in each step of constructing a D GTD system, as well as the implementation of different time-stepping schemes.
Abstract: Efficient multiscale electromagnetic simulations require several major challenges that need to be addressed, such as flexible and robust geometric modeling schemes, efficient and stable time-stepping methods, etc. Due to the versatile choices of spatial discretization and temporal integration, discontinuous Galerkin time-domain (DGTD) methods can be very promising in simulating transient multiscale problems. This paper provides a comprehensive review of different DGTD schemes, highlighting the fundamental issues arising in each step of constructing a DGTD system. The issues discussed include the selection of governing equations for transient electromagnetic analysis, different basis functions for spatial discretization, as well as the implementation of different time-stepping schemes. Numerical examples demonstrate the advantages of DGTD for multiscale electromagnetic simulations.
163 citations
Dissertation•
04 Apr 2014
TL;DR: In this paper, the authors discuss the desarrollo de tecnicas numericas eficientes en el analisis de problemas realistas de propagacion, radiacion, dispersion and acoplamiento electromagnetico.
Abstract: En esta tesis se aborda el desarrollo de tecnicas numericas eficientes en el analisis de problemas realistas de propagacion, radiacion, dispersion y acoplamiento electromagnetico. Para este objetivo, se investiga sobre la utilizacion de elementos discontinuos de Galerkin (DG) y su aplicacion en la resolucion de las ecuaciones de Maxwell en el dominio del tiempo. Las principales aportaciones estan basadas en la combinacion del esquema eficiente de integracion temporal leap-frog (LF), junto con un algoritmo de local time-stepping (LTS), con el metodo de discretizacion espacial DG. El algoritmo propuesto se denomina Leap-Frog Discontinuous Galerkin (LFDG). Se ha desarrollado la formulacion espacial DG en su forma semi-discreta. La formulacion se plantea de una forma general, unificando diferentes esquemas de evaluacion de flujos, los cuales han sido aplicados con exito a este metodo. Se ha desarrollado un amplio rango de funcionalidades en el contexto de metodos DG, como las condiciones tipicas de contorno (conductor electrico/magnetico perfecto, condicion de contorno de Silver-Muller de primer orden, interfaces entre materiales con propiedades electricas y/o magneticas diferentes), modelizacion de materiales anisotropos, fuentes electromagneticas (ondas planas, puertos coaxiales o en guia de onda y delta-gap), y condiciones de frontera conformes y uniaxiales perfectamente adaptadas. El esquema de integracion LF se ha aplicado a la formulacion DG semi-discreta, obteniendo el algoritmo LFDG. Ademas se propone un esquema de LTS explicito en combinacion con el algoritmo LFDG. El esquema semidiscreto DG y el algoritmo LFDG se han analizado, y se exploran los limites en cuanto a precision y coste computacional del metodo LFDG. En primer lugar se revisa el problema de los modos espurios en el contexto de DG, y se estudian los espectros numericos de ambos esquemas. Despues, las relaciones numericas de dispersion y disipacion, y la convergencia y anisotropia de los errores de ambos metodos se comparan y analizan. Finalmente, se ha llevado a cabo un analisis en cuanto a coste computacional vs. precision del metodo LFDG, incluyendo una comparacion con el metodo de diferencias finitas en el dominio del tiempo. El algoritmo LFDG se ha implementado de forma paralela y escalable utilizando una tecnica de programacion hibrida OMP-MPI, en la cual se explota la naturaleza paralela del algoritmo propuesto. Se demuestran las capacidades del metodo, siendo capaz de calcular problemas electricamente grandes, manteniendo la precision controlada, y considerando pequenos detalles geometricos gracias a la utilizacion del algoritmo de LTS. El metodo LFDG se ha aplicado a diferentes tipos de problemas electromagneticos (filtros de microondas, antenas, compatibilidad electromagnetica en aeronautica?) comparado con medidas u otras tecnicas numericas. El metodo ha sido aplicado a problemas reales de ingenieria, mostrando importantes propiedades: robustez, estabilidad, versatilidad, eficiencia, escalabilidad y precision.
26 citations
Additional excerpts
...in [76, 77] applied the same approach but locally, proposing a hybrid explicit-implicit timeintegration scheme....
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TL;DR: The Discontinuous Galerkin (DG) method for Maxwell's equations and dedicated techniques for adaptive mesh refinement are presented, which allow for performing transient mesh refinement.
Abstract: [1] A discontinuous Galerkin method (DGM) for Maxwell’s equations in time domain and dedicated techniques for adaptive mesh refinement are presented. Since the DGM is a finite element–type method, it offers two refinement mechanisms: the manipulation of the local mesh step size (h adaptation) and the adaptation of the local approximation order (p adaptation). For both cases, a new approximation is obtained by means of projections between finite element spaces. The projection operators introduced are optimal with respect to the projection error. A reliable estimator for the local smoothness of the solution is presented, which forms the basis for the hp decision, i.e., the choice of the type of adaptation to be performed. The stability and efficiency of the adaptive method are demonstrated, allowing for performing transient mesh refinement, i.e., the continuous adaptation of the mesh according to the current situation.
4 citations
Cites methods from "Hybrid Explicit-Implicit Time Integ..."
...Further efficiency improvements could be made by employing local time stepping methods as introduced by Piperno [2006] or mixed implicit‐explicit time integration as recently proposed by Dolean et al. [2011] ....
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References
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TL;DR: In this paper, a new parallel distributed memory multifrontal approach is described to handle numerical pivoting efficiently, a parallel asynchronous algorithm with dynamic scheduling of the computing tasks has been developed.
940 citations
TL;DR: Convergence is proved for Discontinuous elements on tetrahedral meshes, as well as a discrete divergence preservation property, and a discrete analog of the electromagnetic energy is conserved for metallic cavities.
Abstract: 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 leap-frog 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 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.
284 citations
TL;DR: A discontinuous Galerkin finite element method (DGFEM) along with recently introduced high-order implicit-explicit Runge-Kutta (IMEX-RK) schemes to overcome geometry-induced stiffness in fluid-flow problems.
172 citations
"Hybrid Explicit-Implicit Time Integ..." refers methods in this paper
...In [6], the authors study the application of explicit–implicit Runge–Kutta (so-called IMEX-RK) methods in conjunction with high order discontinuous Galerkin discretizations on unstructured triangular meshes, in the framework of unsteady compressible flow problems (i....
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TL;DR: These schemes have already been developed for N-body mechanical problems and are known as symplectic schemes and are transformed and applied to DGTD methods on wave propagation problems in order to obtain stable and accurate local time-stepping algorithms.
Abstract: The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems. Able to deal with unstructured, possibly locally-refined meshes, they handle
easily complex geometries and remain fully explicit with easy parallelization and extension to high orders of accuracy. Non-dissipative versions exist, where some discrete electromagnetic energy is exactly conserved. However, the stability limit of the methods, related to the smallest elements in the mesh, calls for the construction of local-time stepping algorithms. These schemes have already been developed for N-body mechanical problems and are known as symplectic schemes. They are applied here to DGTD methods on wave propagation problems.
127 citations
TL;DR: Recent efforts towards the development of a hybrid explicit-implicit DGTD method for solving the time domain Maxwell equations on unstructured simplicial meshes are reported on.
120 citations
Additional excerpts
...The proof can be found in [4]....
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