Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples

Abstract

We analyze a new approach to three-dimensional electromagnetic scattering problems via fast isogeometric boundary element methods. Starting with an investigation of the theoretical setting around the electric field integral equation within the isogeometric framework, we show existence, uniqueness, and quasi-optimality of the isogeometric approach. For a fast and efficient computation, we then introduce and analyze an interpolation-based fast multipole method tailored to the isogeometric setting, which admits competitive algorithmic and complexity properties. This is followed by a series of numerical examples of industrial scope, together with a detailed presentation and interpretation of the results.

Bembel: The fast isogeometric boundary element C++ library for Laplace, Helmholtz, and electric wave equation

“…For the numerical quadrature of these integrals, we employ regularization techniques as described in [21]. The compression of the resulting densely populated system matrices is based on the embedded fast multipole method (FMM), which is tailored to the framework of isogeometric analysis, see [16,17], and fits into the framework of H 2 -matrices. Its particular advantage is that the matrix compression is directly applied on the reference geometry, that is, the unit square.…”

“…Hence, the employed compression scheme profits from the inherently two-dimensional structure of the problem. The complexity for assembly, storage requirement and matrix-vector multiplication for the system matrix are almost linear in the number of unknowns, see [16,17]. Moreover, this compression technique provably maintains the convergence behaviour for increasingly finer discretizations, cf.…”

“…[16]. An in-depth mathematical analysis of the implemented approach together with numerical studies based on previous versions of Bembel is available in [7,16,17].…”

“…compute ( hmat ) ; 23 VectorXcd rho = gmres . solve ( rhs ) ; The convergence plots were computed around the pictured geometry, with a dipole in direction (0, 0, 1/100) placed at (1, 1, 0) , see [17] and the example programs. The rest of the parameters are chosen as in the listed program.…”

“…With support of B-splines and NURBS for the geometry mappings, the Laplace and Helmholtz code became isogeometric in [7]. Finally, in [17], it has been extended to the electric field integral equation.…”