Rapidly converging boundary integral equation solvers in computational electromagnetics

Simon Adrian 1, 2
2 Lab-STICC_IMTA_MOM_PIM
Lab-STICC - Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance
Abstract : The electric field integral equation (EFIE) and the combined field integral equation(CFIE) suffer from the dense-discretization and the low-frequency breakdown: if the average edgelength of the mesh is reduced, or if the frequency is decreased, then the condition number of the system matrix grows. This leads to slowly or non-converging iterative solvers. This dissertation presents new paradigms for rapidly converging integral equation solvers: to overcome the illconditioning, we advance and extend the state of the art both in hierarchical basis and in Calderón preconditioning techniques. For the EFIE, we introduce a hierarchical basis for structured and unstructured meshes based on generalized primal and dual Haar prewavelets. Furthermore, a framework is introduced which renders the hierarchical basis able to efficiently precondition the EFIE in the case that the scatterer is multiply connected. The applicability of hierarchical basis preconditioners to the CFIE is analyzed and an efficient preconditioning scheme is derived. In addition, we present a refinement-free Calderón multiplicative preconditioner (RF-CMP) that yields a system matrix which is Hermitian, positive definite (HPD), and well-conditioned. Different from existing Calderón preconditioners, no dual basis functions and thus no refinement of the mesh is required. Since the matrix is HPD—in contrast to standard discretizations of the EFIE—we can apply the conjugate gradient (CG) method as iterative solver, which guarantees convergence. Eventually, the RF-CMP is extended to the CFIE.
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Simon Adrian. Rapidly converging boundary integral equation solvers in computational electromagnetics. Modeling and Simulation. Ecole nationale supérieure Mines-Télécom Atlantique; Technische Universität (Munich, Allemagne), 2018. English. ⟨NNT : 2018IMTA0074⟩. ⟨tel-02011596⟩

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