Dissertationsthema: Efficient Parallel Numerical Simulation of Magnetic Field Problems
Bearbeiter: Dipl.-Ing. Klaus Michael Kuhn
Betreuer: O.Univ.-Prof. Dipl.-Ing. Dr. Ulrich Langer
Gutachter: O.Univ.-Prof. Dipl.-Ing. Dr. Heinz W. Engl

This thesis is concerned with the coupling of Finite Element Methods (FEM) and Boundary Element Methods (BEM) by Domain Decomposition (DD) for the solution of 2D and 3D magnetic field problems. We present analytical investigations for 2D and 3D magneto-static problems and discuss practical aspects for the 2-dimensional case. Hereby our aim is to design the formulations of the physical problem such that fast (parallel) numerical solvers of optimal complexity can be applied to the discretized problem.

In particular we consider non-overlapping DD methods. This approach allows naturally the coupling of different discretizations methods as FEM and BEM and is the basis for parallelization. We introduce the tool ADDPre for the DD preprocessing. It can be applied to general 2-dimensional geometries. As a result, a partitioning of the domain into subdomains is obtained in opposite to a simple distribution of a mesh which is not sufficient for DD methods.

We analyze coupled FEM-BEM variational formulations of 2D magnetic field problems. Those formulations are designed such that the discrete problem can be transformed into a symmetric and positive definite (spd) system of equations. Optimal preconditioners for the discrete problem are proposed. The optimality is justified both, theoretically and numerically. Especially we are concerned about the efficient implementation of parallel solvers for coupled FE-BE equations.

We propose a general concept for the formulation of 3D magneto-static problems based on mixed formulations. The approach is such that both, the continuous and the discrete problem are correct representations of the physical problem. Moreover, the concept allows to reduce the mixed problem to an equivalent problem in primal variables which leads finally to a spd system matrix. We introduce a technique for the coupling of vector-valued FEM and scalar BEM similar to the 2-dimensional case. The resulting discrete system can be transformed into a spd system again such that the preconditioned conjugate gradient algorithm can be used as a solver.

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