Numerical solution of exterior Maxwell problems by Galerkin BEM and Runge-Kutta convolution quadrature

In this paper we consider time-dependent electromagnetic scattering problems from conducting objects. We discretize the time-domain electric field integral equation using Runge-Kutta convolution quadrature in time and a Galerkin method in space. We analyze the involved operators in the Laplace domain and obtain convergence results for the fully discrete scheme. Numerical experiments indicate the sharpness of the theoretical estimate

A 3D Time Domain CSEM Forward Modeling Code using custEM and FEniCS

A new 3D time domain CSEM forward modeling code TDcustEM built upon the recently published open-source frequency domain code custEM and the open-source finite element toolbox FEniCS is presented. The transformation of the frequency domain data provided by custEM is performed by a Fast Hankel Transform using 80 digital filter coefficients. 3D edge-based tetrahedral meshes generated by TetGen facilitate the calculation of syn- thetic data using topography, arbitrary source geometries and complex subsurface structures. To ensure precision and reliability of the new algorithm, calculated results of different CSEM setups and electromagnetic field components are cross-validated against analytic solutions as well as 1D and 3D time domain modeling codes. Certain modeling studies are conducted regarding possible interpolation and extrapola- tion techniques to reduce the number of necessary frequencies and therefore the compu- tational runtime, which is still an issue of convolutional time domain CSEM approaches. Additional modeling studies showed the importance of precise receiver positioning for measuring the horizontal components of the time derivative of the magnetic field. As the present thesis is embedded in the Collaborative Research Centre 806 – Our Way to Europe, three sedimentary deposits in the East African Rift Valley were subject to mul- tidimensional TEM surveys in the framework of this project. Common 1D (EMUPLUS) as well as laterally and spatially constrained (AarhusInv) in- version techniques were applied to the TEM field data. Sediment thicknesses as well as stratigraphic sequences of the investigated sedimentary basins were derived. An extensive 3D modeling study of one of the target areas representing a volcanically-formed basin including topography was performed using the newly developed TDcustEM code

Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell's equations

In this note we study the temporal convergence of a locally implicit discontinuous Galerkin (DG) method for Maxwell's equations modeling electromagnetic wave propagation. Particularly, we wonder whether the method retains its second-order ordinary differential equation (ODE) convergence under stable simultaneous space-time grid refinement towards the true partial differential equation (PDE) solution. This is not a priori clear due to the component splitting which can introduce order reduction.Dans ce papier nous étudions la convergence temporelle d'une méthode Galerkin discontinue localement implicite pour la résolution des équations de Maxwell modélisant la propagation des ondes électromagnétiques. En particulier, nous nous demandons si pour un raffinement du maillage, simultané et stable en espace-temps, le second ordre de convergence au sens des EDO est conservé pour la solution exacte de l'EDP. Cela n'est pas à priori clair en raison de la décomposition des éléments qui peut introduire une réduction d'ordre

Scalable domain decomposition methods for finite element approximations of transient and electromagnetic problems

The main object of study of this thesis is the development of scalable and robust solvers based on domain decomposition (DD) methods for the linear systems arising from the finite element (FE) discretization of transient and electromagnetic problems. The thesis commences with a theoretical review of the curl-conforming edge (or Nédélec) FEs of the first kind and a comprehensive description of a general implementation strategy for h- and p- adaptive elements of arbitrary order on tetrahedral and hexahedral non-conforming meshes. Then, a novel balancing domain decomposition by constraints (BDDC) preconditioner that is robust for multi-material and/or heterogeneous problems posed in curl-conforming spaces is presented. The new method, in contrast to existent approaches, is based on the definition of the ingredients of the preconditioner according to the physical coefficients of the problem and does not require spectral information. The result is a robust and highly scalable preconditioner that preserves the simplicity of the original BDDC method. When dealing with transient problems, the time direction offers itself an opportunity for further parallelization. Aiming to design scalable space-time solvers, first, parallel-in-time parallel methods for linear and non-linear ordinary differential equations (ODEs) are proposed, based on (non-linear) Schur complement efficient solvers of a multilevel partition of the time interval. Then, these ideas are combined with DD concepts in order to design a two-level preconditioner as an extension to space-time of the BDDC method. The key ingredients for these new methods are defined such that they preserve the time causality, i.e., information only travels from the past to the future. The proposed schemes are weakly scalable in time and space-time, i.e., one can efficiently exploit increasing computational resources to solve more time steps in (approximately) the same time-to-solution. All the developments presented herein are motivated by the driving application of the thesis, the 3D simulation of the low-frequency electromagnetic response of High Temperature Superconductors (HTS). Throughout the document, an exhaustive set of numerical experiments, which includes the simulation of a realistic 3D HTS problem, is performed in order to validate the suitability and assess the parallel performance of the High Performance Computing (HPC) implementation of the proposed algorithms.L’objecte principal d’estudi d’aquesta tesi és el desenvolupament de solucionadors escalables i robustos basats en mètodes de descomposició de dominis (DD) per a sistemes lineals que sorgeixen en la discretització mitjançant elements finits (FE) de problemes transitoris i electromagnètics. La tesi comença amb una revisió teòrica dels FE d’eix (o de Nédélec) de la primera família i una descripció exhaustiva d’una estratègia d’implementació general per a elements h- i p-adaptatius d’ordre arbitrari en malles de tetraedres i hexaedres noconformes. Llavors, es presenta un nou precondicionador de descomposició de dominis balancejats per restricció (BDDC) que és robust per a problemes amb múltiples materials i/o heterogenis definits en espais curl-conformes. El nou mètode, en contrast amb els enfocaments existents, està basat en la definició dels ingredients del precondicionador segons els coeficients físics del problema i no requereix informació espectral. El resultat és un precondicionador robust i escalable que preserva la simplicitat del mètode original BDDC. Quan tractem amb problemes transitoris, la direcció temporal ofereix ella mateixa l’oportunitat de seguir explotant paral·lelisme. Amb l’objectiu de dissenyar precondicionadors en espai-temps, primer, proposem solucionadors paral·lels en temps per equacions diferencials lineals i no-lineals, basats en un solucionador eficient del complement de Schur d’una partició multinivell de l’interval de temps. Seguidament, aquestes idees es combinen amb conceptes de DD amb l’objectiu de dissenyar precondicionadors com a extensió a espai-temps dels mètodes de BDDC. Els ingredients clau d’aquests nous mètodes es defineixen de tal manera que preserven la causalitat del temps, on la informació només viatja de temps passats a temps futurs. Els esquemes proposats són dèbilment escalables en temps i en espai-temps, és a dir, es poden explotar eficientment recursos computacionals creixents per resoldre més passos de temps en (aproximadament) el mateix temps transcorregut de càlcul. Tots els desenvolupaments presentats aquí són motivats pel problema d’aplicació de la tesi, la simulació de la resposta electromagnètica de baixa freqüència dels superconductors d’alta temperatura (HTS) en 3D. Al llarg del document, es realitza un conjunt exhaustiu d’experiments numèrics, els quals inclouen la simulació d’un problema de HTS realista en 3D, per validar la idoneïtat i el rendiment paral·lel de la implementació per a computació d’alt rendiment dels algorismes proposatsPostprint (published version

The Maxwell-Landau-Lifshitz-Gilbert System: Mathematical Theory and Numerical Approximation

This thesis deals with the mathematical theory and numerical approximation of the Landau--Lifshitz--Gilbert equation coupled to the Maxwell equations without artificial boundary conditions. As a starting point, the physical equations are stated on the unbounded three dimensional space and reformulated in a mathematically precise way to a coupled partial differential -- boundary integral system. We derive a weak form of the whole coupled system, state the relation to the strong form and show uniqueness of the Maxwell part of the solution. A numerical algorithm is proposed based on the tangent plane scheme for the LLG part and using a finite element and boundary element coupling as spatial discretization and the backward Euler method and Convolution Quadrature as time discretization for the interior Maxwell part and the boundary, respectively. Under minimal assumptions on the regularity of solutions, we present well-posedness and convergence of the numerical algorithm. For the pure Maxwell equations without the coupling to the LLG equation, we are able to show stronger results than in the coupled case. We derive a weak form for the Maxwell transmission problem and demonstrate existence and uniqueness of the weak solutions as well as equivalence with a strong solution. The proposed algorithm of finite-element/ boundary-element coupling via Convolution Quadrature converges with only minimal assumptions on the regularity of the input data. Again for the full Maxwell--LLG system, we show a-priori error bounds in the situation of a sufficiently regular solution. This is done by a combination of the known linearly implicit backward difference formula time discretizations with higher order non-conforming finite element space discretizations for the LLG equation and the leapfrog and Convolution Quadrature time discretization with higher order discontinuous Galerkin elements and continuous boundary elements for the boundary integral formulation of Maxwell\u27s equations. The precise method of coupling allows us to solve the system at the cost of the individual parts, with the same convergence rates under the same regularity assumptions and the same CFL conditions as for an uncoupled examination. Numerical experiments illustrate and expand on the theoretical results and demonstrate the applicability of the methods. For the formulation of the boundary integral equations, the study of the Laplace transform is inevitable. We collect and extend the properties of the Laplace transform from literature. In the suitable functional analytic setting, we give extensive proofs in a self contained way of all the required properties

A primal-dual finite element approximation for a nonlocal model in plasticity

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

Scalable domain decomposition methods for finite element approximations of transient and electromagnetic problems

The main object of study of this thesis is the development of scalable and robust solvers based on domain decomposition (DD) methods for the linear systems arising from the finite element (FE) discretization of transient and electromagnetic problems. The thesis commences with a theoretical review of the curl-conforming edge (or Nédélec) FEs of the first kind and a comprehensive description of a general implementation strategy for h- and p- adaptive elements of arbitrary order on tetrahedral and hexahedral non-conforming meshes. Then, a novel balancing domain decomposition by constraints (BDDC) preconditioner that is robust for multi-material and/or heterogeneous problems posed in curl-conforming spaces is presented. The new method, in contrast to existent approaches, is based on the definition of the ingredients of the preconditioner according to the physical coefficients of the problem and does not require spectral information. The result is a robust and highly scalable preconditioner that preserves the simplicity of the original BDDC method. When dealing with transient problems, the time direction offers itself an opportunity for further parallelization. Aiming to design scalable space-time solvers, first, parallel-in-time parallel methods for linear and non-linear ordinary differential equations (ODEs) are proposed, based on (non-linear) Schur complement efficient solvers of a multilevel partition of the time interval. Then, these ideas are combined with DD concepts in order to design a two-level preconditioner as an extension to space-time of the BDDC method. The key ingredients for these new methods are defined such that they preserve the time causality, i.e., information only travels from the past to the future. The proposed schemes are weakly scalable in time and space-time, i.e., one can efficiently exploit increasing computational resources to solve more time steps in (approximately) the same time-to-solution. All the developments presented herein are motivated by the driving application of the thesis, the 3D simulation of the low-frequency electromagnetic response of High Temperature Superconductors (HTS). Throughout the document, an exhaustive set of numerical experiments, which includes the simulation of a realistic 3D HTS problem, is performed in order to validate the suitability and assess the parallel performance of the High Performance Computing (HPC) implementation of the proposed algorithms.L’objecte principal d’estudi d’aquesta tesi és el desenvolupament de solucionadors escalables i robustos basats en mètodes de descomposició de dominis (DD) per a sistemes lineals que sorgeixen en la discretització mitjançant elements finits (FE) de problemes transitoris i electromagnètics. La tesi comença amb una revisió teòrica dels FE d’eix (o de Nédélec) de la primera família i una descripció exhaustiva d’una estratègia d’implementació general per a elements h- i p-adaptatius d’ordre arbitrari en malles de tetraedres i hexaedres noconformes. Llavors, es presenta un nou precondicionador de descomposició de dominis balancejats per restricció (BDDC) que és robust per a problemes amb múltiples materials i/o heterogenis definits en espais curl-conformes. El nou mètode, en contrast amb els enfocaments existents, està basat en la definició dels ingredients del precondicionador segons els coeficients físics del problema i no requereix informació espectral. El resultat és un precondicionador robust i escalable que preserva la simplicitat del mètode original BDDC. Quan tractem amb problemes transitoris, la direcció temporal ofereix ella mateixa l’oportunitat de seguir explotant paral·lelisme. Amb l’objectiu de dissenyar precondicionadors en espai-temps, primer, proposem solucionadors paral·lels en temps per equacions diferencials lineals i no-lineals, basats en un solucionador eficient del complement de Schur d’una partició multinivell de l’interval de temps. Seguidament, aquestes idees es combinen amb conceptes de DD amb l’objectiu de dissenyar precondicionadors com a extensió a espai-temps dels mètodes de BDDC. Els ingredients clau d’aquests nous mètodes es defineixen de tal manera que preserven la causalitat del temps, on la informació només viatja de temps passats a temps futurs. Els esquemes proposats són dèbilment escalables en temps i en espai-temps, és a dir, es poden explotar eficientment recursos computacionals creixents per resoldre més passos de temps en (aproximadament) el mateix temps transcorregut de càlcul. Tots els desenvolupaments presentats aquí són motivats pel problema d’aplicació de la tesi, la simulació de la resposta electromagnètica de baixa freqüència dels superconductors d’alta temperatura (HTS) en 3D. Al llarg del document, es realitza un conjunt exhaustiu d’experiments numèrics, els quals inclouen la simulació d’un problema de HTS realista en 3D, per validar la idoneïtat i el rendiment paral·lel de la implementació per a computació d’alt rendiment dels algorismes proposat

Numerical methods for computing Casimir interactions

We review several different approaches for computing Casimir forces and related fluctuation-induced interactions between bodies of arbitrary shapes and materials. The relationships between this problem and well known computational techniques from classical electromagnetism are emphasized. We also review the basic principles of standard computational methods, categorizing them according to three criteria---choice of problem, basis, and solution technique---that can be used to classify proposals for the Casimir problem as well. In this way, mature classical methods can be exploited to model Casimir physics, with a few important modifications.Comment: 46 pages, 142 references, 5 figures. To appear in upcoming Lecture Notes in Physics book on Casimir Physic

Analysis of Isogeometric Non-Symmetric FEM-BEM Couplings for the Simulation of Electromechanical Energy Converters

The main contribution of this thesis consists in providing a rigorous analysis of non-symmetric isogeometric couplings of the Finite Element Method (FEM) and the direct Boundary Element Method (BEM) for some model problems that are relevant for the simulation of electromechanical energy converters. The corresponding (electro)magnetic subsystem of such a multi-physics problem can be modeled by the eddy-current approximation of Maxwell’s equations. We study this type of models in both the static and quasistationary case, which we formulate in terms of the magnetic vector potential in two-dimensional (2D) and three-dimensional (3D) Lipschitz domains with a general topology. We associate FEM with bounded domains that may be filled with non-linear materials, whereas BEM is applied for bounded and unbounded domains that contain linear materials, i.e., for which a fundamental solution is available. Our analysis is based on the framework of strongly monotone and Lipschitz continuous operators, which also incorporates the required physical properties of the considered non-linear materials. To establish well-posedness and stability of the continuous settings, we use either implicit stabilization (in two dimensions) or a formulation in appropriate quotient spaces (in three dimensions) depending on the specific model. Moreover, we show the quasi-optimality of the method with respect to a conforming Galerkin discretization. For the concrete discretization, we consider an isogeometric framework, in particular, we employ conforming B-Spline spaces for the approximation of the solution, and Non-Uniform Rational B-Splines (NURBS) for geometric modelling. This approach facilitates h- and p-refinements, and avoids the introduction of geometrical errors. In this setting, we derive a priori estimates, and discuss the possible improvement of the convergence rates (super-convergence) of the method, when the pointwise error in func- tionals of the solution (more precisely its Cauchy data) is evaluated in the BEM domain. This improvement may double the usual convergence rates under certain circumstances. The theoretical findings are confirmed through several numerical examples. To validate our approach for the complete electromechanical system, we couple the (electro)magnetic and the mechanical subsystems weakly, and compute the needed forces and/or torques by using the Maxwell Stress Tensor (MST) method. For the sake of illustration, time derivatives are discretized by means of a classical implicit Euler scheme. The results of numerical experiments are in agreement with the expectations and the reference solutions