# Veranstaltung

## Seminar in Numerical Analysis: Stefan Kurz

## Stefan Kurz, TU Darmstadt

**IGA BEM for Maxwell eigenvalue problems**

Abstract:

Superconducting cavities are standard components of particle accelerators. Their design is typically described by parametrized ellipses and determined by mathematical optimization. The simulation model is subject to demanding requirements, such as a relative accuracy of 10^{−9} for the resonance frequency of the accelerating mode. Since the geometry and the electromagnetic fields are smooth, an approach in the gist of isogeometric analysis (IGA) suggests itself. The geometry is modeled by a NURBS mapping, while the electromagnetic fields are discretized by the B-spline de Rahm complex [2]. An IGA finite element method (FEM) for the Maxwell eigenvalue problem was investigated and showed promising results [3]. For the same accuracy, the number of required degrees of freedom was reduced by a factor 3 . . . 9 compared to classical FEM. However, CAD systems feature surface descriptions only, so the volumetric spline model had to be created manually.

To live up to the promises of IGA, namely closing the gap bewteen design and analysis, we suggest an IGA boundary element method (BEM). We will review the state-of-the-art of all relevant building blocks. We will address the B-spline de Rham complex on a boundary manifold, the Galerkin discretization of the electric field integral equation, and present a convergence result. We will discuss a recent contour integral method [1] to solve the resulting non-linear eigenvalue problem. Aspects of integrating so-called ”fast methods” will also be presented, in particular Adaptive Cross Approximation [5] and Calderón preconditioning [4].

[1] W.-J. Beyn. An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl, 436(10):3839–3863, 2012.

[2] A. Buffa, G. Sangalli, and R. Vázquez. Isogeometric analysis in electromagnetics: B-splines approximation. Comput Method Appl M, 199:1143–1152, 2010.

[3] J. Corno, C. de Falco, H. De Gersem, and S. Schöps. Isogeometric simulation of Lorentz detuning in superconducting accelerator cavities. Comput Phys Commun, 201:1–7, February 2016.

[4] J. Li, D. Dault, B. Liu, Y. Tong, and B. Shanker. Subdivision based isogeometric analysis technique for electric field integral equations for simply connected structures. J Comput Phys, 319:145–162, 2016.

[5] B. Marussig, J. Zechner, G. Beer, and T.-P. Fries. Fast isogeometric boundary element method based on independent field approximation. Comput Method Appl M, 284:458–488, 2015.

The work of Stefan Kurz is supported by the ’Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt.

For further information about the seminar, please visit this webpage.