Small Collaboration: Numerical Analysis of Electromagnetic Problems (hybrid meeting)

The classical theory of electromagnetism describes the interaction of electrically charged particles through electromagnetic forces, which are carried by the electric and magnetic fields. The propagation of the electromagnetic fields can be described by Maxwell's equations. Solving Maxwell's equations numerically is a challenging problem which appears in many different technical applications. Difficulties arise for instance from material interfaces or if the geometrical features are much larger than or much smaller than a typical wavelength. The spatial discretization needs to combine good geometrical flexibility with a relatively high order of accuracy. The aim of this small-scale, week-long interactive mini-workshop jointly organized by the University of Duisburg-Essen and the University of Twente, and kindly hosted at the MFO, is to bring together experts in non-standard and mixed finite elements methods with experts in the field of electromagnetism

The FLAME-slab method for electromagnetic wave scattering in aperiodic slabs

The proposed numerical method, "FLAME-slab," solves electromagnetic wave scattering problems for aperiodic slab structures by exploiting short-range regularities in these structures. The computational procedure involves special difference schemes with high accuracy even on coarse grids. These schemes are based on Trefftz approximations, utilizing functions that locally satisfy the governing differential equations, as is done in the Flexible Local Approximation Method (FLAME). Radiation boundary conditions are implemented via Fourier expansions in the air surrounding the slab. When applied to ensembles of slab structures with identical short-range features, such as amorphous or quasicrystalline lattices, the method is significantly more efficient, both in runtime and in memory consumption, than traditional approaches. This efficiency is due to the fact that the Trefftz functions need to be computed only once for the whole ensemble.Comment: Various typos were corrected. Minor inconsistencies throughout the manuscript were fixed. In Section II B. Additional description regarding choice of Trefftz cell, was added. In Section III A. Detailed description about units (used in our calculation) was adde

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

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.Keywords: Imaginary Frequency, Perfectly Match Layer, Casimir Force, Casimir Energy, Perfect Electric ConductorUnited States. Army Research Office (Contract W911NF-07-D-0004)Massachusetts Institute of Technology. Ferry FundUnited States. Defense Advanced Research Projects Agency (Contract N66001-09-1-2070-DOD

The Fourier-Galerkin Method for Band Structure Computations of 2D and 3D Photonic Crystals

In this dissertation we consider the band structure computation of 2D and 3D photonic crystals with the Fourier-Galerkin method. For the 2D Helmholtz equation we discuss the efficient implementation and the convergence. For the three-dimensional problem we use the so-called Harmonic Restarted Arnoldi method for solving the discrete eigenvalue problems without preconditioning

Design optimization of acoustic metamaterials and phononic crystals with a time domain method

A time-dependent adjoint approach for obtaining sensitivity derivatives for shape optimizations of acoustic metamaterials and phononic crystals is presented. The gradient-based design procedure is suitable for large numbers of design variables, and results are shown on achieving effective material properties with a unit cell and the broadband noise reduction with periodic arrays of cylinders. The acoustic wave propagation problem is solved in the time-domain using a Streamline Upwind/Petrov Galerkin formulation. Topology parameterization is accomplished using the homogenization method, and shape optimization is subsequently used afterwards to refine the geometries. Surface parameterization is accomplished using control grids, which are based on a Laplace equation. The combined strategy is compared with penalty-based topology optimization. Furthermore, the proposed topology optimization is also conducted on the design of a broadband acoustic cloaking device