Elasmi, Mehdi (2022)
Analysis of Isogeometric Non-Symmetric FEM-BEM Couplings for the Simulation of Electromechanical Energy Converters.
Technische Universität Darmstadt
doi: 10.26083/tuprints-00020329
Dissertation, Erstveröffentlichung, Verlagsversion

Kurzbeschreibung (Abstract)

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.

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