To BEM, or not to BEM, that is the question

After a brief overview on numerical methods, the presentation focuses on the Boundary Element Method (BEM). The BEM has developed into an important alternative to domain-oriented approaches like Finite Elements, ever since fast implementations are available. The BEM reduces the dimensionality of the problem and can easily take into account unbounded domains, and problems involving motion. This is underlined by use cases such as simulation of electrical machines and accelerator components.

The main part of the talk is about combining BEM with Isogeometric Analysis (IGA). Most Computer-Aided Design (CAD) systems represent geometries in terms of Non-Uniform Rational B-Splines (NURBS). Specifically, CAD systems feature Boundary Representations (B-REP) which are well suited as geometry description for BEM. To live up to the promises of IGA, namely closing the gap between design and analysis, an IGA BEM for electromagnetic scattering and eigenvalue problems has been developed. The B-spline de Rham complex on a boundary manifold will be introduced and a Galerkin discretization of the Electric Field Integral Equation (EFIE) derived, including a convergence result.

Typical challenges of the BEM such as (nearly) singular integrals, fully populated matrices, preconditioning, and the fact that linear eigenvalue (EV) problems in the domain result in non-linear EV problems in BEM will also be addressed briefly, and state-of-the-art solutions reviewed.

Finally, an outlook into modeling and simulation of problems involving motion is given, which links to some aspects of relativity.

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