Computational Electromagnetism and Acoustics

It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems

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

Méthode de Galerkin discontinue pour la discrétisation par Éléments finis des équations de maxwell pour la modélisation de problèmes d’électromagnétisme en basses fréquences

RÉSUMÉ: Une discrétisation par éléments finis utilisant la méthode de Galerkin discontinue pour les équations de Maxwell est proposée pour modéliser les problèmes d’électromagnétisme en basses fréquences. L’approximation des équations de Maxwell, dans le régime des basses fréquences, est directement discrétisée avec la méthode de Galerkin discontinue qui a été originellement développée pour les problèmes hyperboliques. On étudie, plus précisément, la modélisation des problèmes sur les supraconducteurs à haute température afin d’évaluer la robustesse de la stratégie numérique proposée. Une analyse dimensionnelle du système d’équations aux dérivées partielles d’ordre un, ainsi qu’un modèle pour les milieux ambiants ayant une conductivité très faible sont aussi proposés. Un problème ayant une solution manufacturée et la propagation d’un front magnétique sont étudiés afin de vérifier la méthodologie numérique proposée. L’induction d’un courant électrique dans un échantillon et dans des câbles électriques supraconducteurs à configuration complexe est étudiée afin de valider le modèle mathématique. De plus, une comparaison sur la capture des forts gradients de la densité de courant et sur la robustesse du schéma de points-fixes est faite entre la stratégie numérique proposée et l’approche numérique populaire au sein de la communauté des ingénieurs électriques en utilisant les problèmes sur les supraconducteurs à haute température.----------ABSTRACT: The discretization of Maxwell’s equations using the discontinuous Galerkin finite element method is proposed for modeling electromagnetism problems in low-frequency regime. The low-frequency approximation to Maxwell’s equations is directly discretized using the discontinuous Galerkin method that was first designed for hyperbolic problems. The modeling of high-temperature superconductors is particularly studied to assess the robustness of the proposed numerical strategy. A dimensional analysis of Maxwell’s equations in low-frequency regime is proposed as well as a model for a medium with very low conductivity, such as air medium. A problem with a manufactured solution and the magnetic front problem are used to verify the proposed numerical strategy. The magnetization of superconducting bulks and wires with a complex structure is used to validate the mathematical model. For hightemperature superconductors modeling, the capture of the sharp gradients of the current density and the robustness of the fixed-point method are studied. The proposed approach is also compared with the popular numerical strategy among the electrical engineering community

Resolution of the Vlasov-Maxwell system by PIC Discontinuous Galerkin method on GPU with OpenCL

International audienceWe present an implementation of a Vlasov-Maxwell solver for multicore processors. The Vlasov equation describes the evolution of charged particles in an electromagnetic field, solution of the Maxwell equations. The Vlasov equation is solved by a Particle-In-Cell method (PIC), while the Maxwell system is computed by a Discontinuous Galerkin method. We use the OpenCL framework, which allows our code to run on multicore processors or recent Graphic Processing Units (GPU). We present several numerical applications to two-dimensional test cases

Reconstruction of electric fields and source distributions in EEG brain imaging

In this thesis, three different approaches are developed for the estimation of focal brain activity using EEG measurements. The proposed approaches have been tested and found feasible using simulated data. First, we develop a robust solver for the recovery of focal dipole sources. The solver uses a weighted dipole strength penalty term (also called weighted L1,2 norm) as prior information in order to ensure that the sources are sparse and focal, and that both the source orientation and depth bias are reduced. The solver is based on the truncated Newton interior point method combined with a logarithmic barrier method for the approximation of the penalty term. In addition, we use a Bayesian framework to derive the depth weights in the prior that are used to reduce the tendency of the solver to favor superficial sources. In the second approach, vector field tomography (VFT) is used for the estimation of underlying electric fields inside the brain from external EEG measurements. The electric field is reconstructed using a set of line integrals. This is the first time that VFT has been used for the recovery of fields when the dipole source lies inside the domain of reconstruction. The benefit of this approach is that we do not need a mathematical model for the sources. The test cases indicated that the approach can accurately localize the source activity. In the last part of the thesis, we show that, by using the Bayesian approximation error approach (AEA), precise knowledge of the tissue conductivities and head geometry are not always needed. We deliberately use a coarse head model and we take the typical variations in the head geometry and tissue conductivities into account statistically in the inverse model. We demonstrate that the AEA results are comparable to those obtained with an accurate head model.Open Acces

Characterisation and Classification of Hidden Conducting Security Threats using Magnetic Polarizability Tensors

The early detection of terrorist threat objects, such as guns and knives, through improved metal detection, has the potential to reduce the number of attacks and improve public safety and security. Walk through metal detectors (WTMDs) are commonly deployed for security screening purposes in applications where these attacks are of particular con-cern such as in airports, transport hubs, government buildings and at concerts. However, there is scope to improve the identification of an object’s shape and its material proper-ties. Using current techniques there is often the requirement for any metallic objects to be inspected or scanned separately before a patron may be determined to pose no threat, making the process slow. This can often lead to build ups of large queues of unscreened people waiting to be screened which becomes another security threat in itself. To improve the current method, there is considerable potential to use the fields applied and measured by a metal detector since, hidden within the field perturbation, is object characterisation information. The magnetic polarizability tensor (MPT) offers an economical characteri-sation of metallic objects and its spectral signature provides additional object character-isation information. The MPT spectral signature can be determined from measurements of the induced voltage over a range of frequencies for a hidden object. With classification in mind, it can also be computed in advance for different threat and non-threat objects, producing a dataset of these objects from which a machine learning (ML) classifier can be trained. There is also potential to generate this dataset synthetically, via the application of a method based on finite elements (FE). This concept of training an ML classifier trained on a synthetic dataset of MPT based characterisations is at the heart of this work.In this thesis, details for the production and use of a first of its kind synthetic dataset of realistic object characterisations are presented. To achieve this, first a review of re-cent developments of MPT object characterisations is provided, motivating the use of MPT spectral signatures. A problem specific, H(curl) based, hp-finite element discreti-sation is presented, which allows for the development of a reduced order model (ROM), using a projection based proper orthogonal decomposition (PODP), that benefits from a-posteriori error estimates. This allows for the rapid production of MPT spectral signatures the accuracy of which is guaranteed. This methodology is then implemented in Python, using the NGSolve finite element package, where other problem specific efficiencies are also included along with a series of additional outputs of interest, this software is then packaged and released as the open source MPT-Calculator. This methodology and software are then extensively tested by application to a series of illustrative examples. Using this software, MPT spectral signatures are then produced for a series of realistic threat and non-threat objects, creating the first of its kind synthetic dataset, which is also released as the open source MPT-Library dataset. Lastly, a series of ML classifiers are documented and applied to several supervised classification problems using this new syn-thetic dataset. A series of challenging numerical examples are included to demonstrate the success of the proposed methodology

Three-dimensional finite-volume time-domain modeling of graphitic fault zones in the Athabasca Basin using unstructured grids

In this thesis, numerical modeling methods for geophysical time-domain electromag- netic (EM) problems and their applications in modeling graphitic faults in the Atha- basca Basin are investigated. A finite-volume time-domain numerical modeling method is developed. The method uses unstructured Delaunay-Voronoï dual meshes. Such unstructured meshes are more flexible and efficient when models containing geological units with complex geometries and topography need to be considered. A model build- ing procedure is established to construct arbitrarily complex models with topography. The procedure locally refines the mesh quality at certain areas such as loop sources and receivers in order to obtain better numerical results. For modeling time-domain EM problems, two approaches are used: the electric field approach and the potential approach. The electric field method directly solves the electric field Helmholtz equation while the potential method solves the Helmholtz equation expressed using vector and scalar potentials. The electric field method is simpler in theory and results in a smaller linear system of equations compared to potential methods. The potential method, on the other hand, is more complex intheory and a larger linear system of equations needs to be solved. However, using the potentials method enables the decomposition of the electric field into galvanic and inductive parts, which is helpful for understanding the physics behind the behaviour of the EM fields in the ground. In addition, the linear system of equations is better conditioned which potentially allows the use of iterative methods to solve it. Both methods are validated by comparing the modeling results with analytic solu- tions for homogeneous half-space models and numerical results for models presented in the literature. The modeling methods developed in this thesis are then applied to the modeling of real EM data collected in the Athabasca Basin. Thin, steeply dip- ping graphitic fault systems, which are linked to the formation of uranium deposits are present in the basin and have a large conductivity contrast with the background host. Because of the close relationship between the graphitic faults and the uranium deposits, time-domain EM surveys are important tools for uranium exploration in the basin. Geological models of the graphitic fault systems are discretized with unstruc- tured grids using the model building procedure developed in this thesis. Two real data sets that were previously collected from the Athabasca Basin are modeled and the modeling results are compared with the real data. The match between the calculated three-component responses and real data is good for models built based on geological information, drilling information, and trial-and-error. These models can help us to infer the complex geometry and conductivity features of the subsurface conductor be- yond the areas targeted by drilling. Therefore, 3D modeling of realistic, complicated real-life conductive targets such as in the uranium exploration in the Athabasca Basin or any other classic mineral exploration for a conductive target with complex shape is an important tool

Derivation and numerical approximation of two-temperature Euler plasma model

International audienceThis paper gives a derivation of the two-temperature Euler plasma system from the two-fluid MHD model. The two-temperature Euler plasma system is proved to be an asymptotic regime for weakly magnetized plasma of the two-fluid MHD model. Our procedure is more general, enabling us to show that assumptions in previous derivations in literature are straightforward consequences of our work. We then propose a finite volume approximation to compute the solution of the two-temperature Euler plasma model in unstructured tessellations used to adequately mesh the toroidal geometry of the tokamak, where flows the plasma. Numerical tests illustrate our method

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