We present first numerical examples of how the framework of isogeometric boundary element methods, in the context of electromagnetism also known as method of moments, can be used to achieve higher accuracies by elevation of the degree of basis functions. Our numerical examples demonstrate the computation of the electric field in the exterior domain.

Spectral methods, cf.

In engineering applications, spectral element methods are rarely considered; the reason simply being that to fully enjoy their convergence properties, meshes with curved elements of increasing orders must be generated. This poses challenges to mesh generation and pre-processing. In contrast, classical

In this document, we present numerical experiments which showcase how the isogeometric framework can be used to obtain an implementation of a spectral boundary element method. We demonstrate the implementation by the solution of electromagnetic scattering problems through

The organisation of the paper is straight forward. We first introduce basic notions of the electric field integral equation, and our

Visualisations of the geometries. The maximum diameter of the boat is smaller than 4.5, the sphere is of radius 1.

Visualisations of the real part of the unknown surface current.

We consider the scattering of an electromagnetic wave under the assumption of constant material coefficients

In a continuous setting, this can be recast as the variational problem of finding an unknown surface current

Numerical examples on the unit sphere. Wave number

Numerical examples for the ship geometry as presented in Fig.

To solve the electric wave equation via a boundary element approach, the unknown is reduced to a vector field on

To apply

Following

For these spaces, it is possible to show existence, uniqueness, and quasi-optimality of the solution

We apply a modified version of the superspace approach as used by e.g.

To showcase the possibility of obtaining an increased accuracy through (mainly)

We excite our model setup by a Hertz-Dipole as defined by

Summarised, we solve for

As a first example, we choose the example of a unit sphere given by 6 patches and define a dipole with

Thus the condition of the system matters little compared to the case in which compression must be applied.

On the sphere, a stable exponential rate of convergence w.r.t.

We present another example. As geometry, we choose the boat depicted in Fig.

The adaptation of isogeometric to spectral element methods is straight forward. Through the use of Bézier extraction, one extracts piecewise smooth parametrisations of geometries. Using known constructions from isogeometric analysis, one can define a global, divergence-conforming basis that consists of tensor product Bernstein polynomials, with supports consisting of one patch each, or in the case of functions identified to achieve normal continuity, multiple patches.
Application of

These results are exceptionally promising since the reduced system size makes the application of direct solvers possible, thus circumventing the need for preconditioners, which are still a challenging topic for boundary element methods for electromagnetic problems.

The code basis, as well as the geometries for these computations are available at

All authors have jointly carried out research and worked together on the manuscript. The numerical tests have been conducted by the last author. All authors read and approved the final manuscript.

Stefan Kurz is also affiliated as Chief Expert with Robert Bosch GmbH.

This article is part of the special issue “Kleinheubacher Berichte 2018”. It is a result of the Kleinheubacher Tagung 2018, Miltenberg, Germany, 24–26 September 2018.

The work of Felix Wolf is supported by the Excellence Initiative of the German Federal and State Governments and the Graduate School of Computational Engineering at TU Darmstadt.

This research has been supported by the DFG (grant no. SCHO1562/3-1) and the DFG (grant no. KU1553/4-1).

This paper was edited by Thomas Eibert and reviewed by three anonymous referees.