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

We present 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

Computation of Electromagnetic Fields Scattered From Objects With Uncertain Shapes Using Multilevel Monte Carlo Method

Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (CMLMC) method is used together with a surface integral equation solver. The CMLMC method optimally balances statistical errors due to sampling of the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine. The number of realizations of finer discretizations can be kept low, with most samples computed on coarser discretizations to minimize computational cost. Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.Comment: 25 pages, 10 Figure

Accurate and efficient algorithms for boundary element methods in electromagnetic scattering: a tribute to the work of F. Olyslager

Boundary element methods (BEMs) are an increasingly popular approach to model electromagnetic scattering both by perfect conductors and dielectric objects. Several mathematical, numerical, and computational techniques pullulated from the research into BEMs, enhancing its efficiency and applicability. In designing a viable implementation of the BEM, both theoretical and practical aspects need to be taken into account. Theoretical aspects include the choice of an integral equation for the sought after current densities on the geometry's boundaries and the choice of a discretization strategy (i.e. a finite element space) for this equation. Practical aspects include efficient algorithms to execute the multiplication of the system matrix by a test vector (such as a fast multipole method) and the parallelization of this multiplication algorithm that allows the distribution of the computation and communication requirements between multiple computational nodes. In honor of our former colleague and mentor, F. Olyslager, an overview of the BEMs for large and complex EM problems developed within the Electromagnetics Group at Ghent University is presented. Recent results that ramified from F. Olyslager's scientific endeavors are included in the survey

A new integral representation for quasiperiodic fields and its application to two-dimensional band structure calculations

In this paper, we consider band-structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green's function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green's function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums.Comment: 25 pages, 6 figures, submitted to J. Comput. Phy

Numerical methods for computing Casimir interactions

We review several different approaches for computing Casimir forces and related fluctuation-induced interactions between bodies of arbitrary shapes and materials. The relationships between this problem and well known computational techniques from classical electromagnetism are emphasized. We also review the basic principles of standard computational methods, categorizing them according to three criteria---choice of problem, basis, and solution technique---that can be used to classify proposals for the Casimir problem as well. In this way, mature classical methods can be exploited to model Casimir physics, with a few important modifications.Comment: 46 pages, 142 references, 5 figures. To appear in upcoming Lecture Notes in Physics book on Casimir Physic

Bound on global error of the fast multipole method for Helmholtz equation in 2-D

This paper analyze the global error of the fast multipole method(FMM) for two-dimensional Helmholtz equation. We first propose the global error of the FMM for the discretized boundary integral operator. The error is caused by truncating Graf's addition theorem, according to the limiting forms of Bessel and Neumann functions, we provide sharper and more precise estimates for the truncations of Graf's addition theorem. Finally, using the estimates we derive the explicit bound and convergence rate for the global error of the FMM for Helmholtz equation, numerical experiments show that the results are valid. The method in this paper can also be applied to the FMM for other problems such as potential problems, elastostatic problems, Stokes flow problems and so on