Skip to main content
main-content

Über dieses Buch

This volume consists ofsurvey articles on current topics in computational wave prop­ agation and inverse problems, written by leading experts in their respective fields. The idea to compile such a volume arose in conjunction with the LMS Durham Symposium on Computational Methods for Wave Propagation in Direct Scattering held at the University of Durham from 15th-25th July 2002, which we jointly or­ ganised. The meeting, attended by 70 participants from the UK and overseas, was structured around a number of short, three lecture, survey courses on a range of top­ ics on computational wave propagation and inverse problems beginning at the level of a graduate student. We were delighted to secure the participation of distinguished international researchers to present these lectures. We felt that it would be valuable to record this material for the benefit of a wider audience, and the idea was hatched that the individual lecturers should be invited to contribute a survey article. Fortunately, many of the speakers not only agreed to undertake this arduous task, but produced what we hope you will agree are the high quality contributions found in this volume. Finally, it is a pleasure to thank the Engineering and Physical Sciences Research Council of Great Britain and the London Mathematical Society for providing the generous support that allowed the meeting to take place. Mark Ainsworth Glasgow, 2003 Penny Davies Dugald Duncan Paul Martin Bryan Rynne Contents New Results on Absorbing Layers and Radiation Boundary Conditions Thomas Hagstrom .

Inhaltsverzeichnis

Frontmatter

New Results on Absorbing Layers and Radiation Boundary Conditions

Abstract
Perhaps the defining feature of waves is the fact that they propagate long distances relative to their characteristic dimension, the wavelength. This allows us to use them to probe the world around us — optically, acoustically, and now at a wide range of wavelengths in a variety of media. For numerical simulations, it is precisely this essential characteristic — the radiation of waves to the far field — that leads to the greatest difficulties. One may view this fundamental difficulty as rooted in the existence of (at least) two widely separated spatial scales. The first are the small scales associated with the wavelengths and the scatterer, and the second is the long distance between the scatterer and the observers.
Thomas Hagstrom

Fast, High-Order, High-Frequency Integral Methods for Computational Acoustics and Electromagnetics

Abstract
We review a set of algorithms and methodologies developed recently for the numerical solution of problems of scattering by complex bodies in three-dimensional space. These methods, which are based on integral equations, high-order integration, Fast Fourier Transforms and highly accurate high-frequency integrators, can be used in the solution of problems of electromagnetic and acoustic scattering by surfaces and penetrable scatterers — even in cases in which the scatterers contain geometric singularities such as comers and edges. All of the solvers presented here exhibit high-order convergence, they run on low memories and reduced operation counts, and they result in solutions with a high degree of accuracy. In particular, our approach to direct solution of integral equations results in algorithms that can evaluate accurately in a personal computer scattering from hundred-wavelength-long objects — a goal, otherwise achievable today only by super-computing. The high-order high-frequency methods we present, in turn, are efficient where our direct methods become costly, thus leading to an overall computational methodology which is applicable and accurate throughout the electromagnetic spectrum.
Oscar P. Bruno

Galerkin Boundary Element Methods for Electromagnetic Scattering

Summary
Methods based on boundary integral equations are widely used in die numerical simulation of electromagnetic scattering in the frequency domain. This article examines a particular class of these methods, namely the Galerkin boundary element approach, from a theoretical point of view. Emphasis is put on the fundamental differences between acoustic and electromagnetic scattering. The derivation of various boundary integral equations is presented, properties of their discretised counterparts are discussed, and a-priori convergence estimates for the boundary element solutions are rigorously established.
Annalisa Buffa, Ralf Hiptmair

Computation of resonance frequencies for Maxwell equations in non-smooth domains

Summary
We address the computation by finite elements of the non-zero eigenvalues of the (curl, curl) bilinear form with perfect conductor boundary conditions in a polyhedral cavity. One encounters two main difficulties: (i) The infinite dimensional kernel of this bilinear form (the gradient fields), (ii) The unbounded singularities of the eigen-fields near corners and edges of the cavity. We first list possible variational spaces with their functional properties and provide a short description of the edge and corner singularities. Then we address different formulations using a Galerkin approximation by edge elements or nodal elements.
After a presentation of edge elements, we concentrate on the functional issues connected with the use of nodal elements. In the framework of conforming methods, nodal elements are mandatory if one regularises the bilinear form (curl, curl) in order to get rid of the gradient fields. A plain regularisation with the (div, div) bilinear form converges to a wrong solution if the domain has reentrant edges or corners. But remedies do exist We will present the method of addition of singular functions, and the method of regularisation with weight, where the (div, div) bilinear form is modified by the introduction of a weight which can be taken as the distance to reentrant edges or corners.
Martin Costabel, Monique Dauge

hp-Adaptive Finite Elements for Time-Harmonic Maxwell Equations

Summary
We review the fundamentals of hp-finite element discretisation of Maxwell equations and their numerical implementation, and describe an automatic hp-adaptive scheme for the time-harmonic Maxwell equations.
Leszek Demkowicz

Variational Methods for Time-Dependent Wave Propagation Problems

Abstract
There is an important need for numerical methods for time dependent wave propagation problems and their many applications, for example in acoustics, electromagnetics and geophysics. Although very old, finite difference time domain methods (FDTD methods in the electromagnetics literature) remain very popular and are widely used in wave propagation simulations, and more generally for the resolution of linear hyperbolic systems, among which Maxwell's system is a typical example. These methods allow us to get discrete equations whose unknowns are generally field values at the points of a regular mesh with spatial step h and time step Δt.
Patrick Joly

Some Numerical Techniques for Maxwell’s Equations in Different Types of Geometries

Summary
Almost all the difficulties that arise in finite difference time domain solutions of Maxwell's equations are due to material interfaces (to which we include objects such as antennas, wires, etc.) Different types of difficulties arise if the geometrical features are much larger than or much smaller than a typical wave length. In the former case, the main difficulty has to do with the spatial discretisation, which needs to combine good geometrical flexibility with a relatively high order of accuracy. After discussing some options for this situation, we focus on the tatter case. The main problem here is to find a time stepping method which combines a very low cost per time step with unconditional stability. The first such method was introduced in 1999 and is based on the ADI principle. We will here discuss that method and some subsequent developments in this area.
Bengt Fomberg

On Retarded Potential Boundary Integral Equations and their Discretisation

Summary
The paper deals with the retarded potential boundary integral equations (RPBIE) used in the numerical resolution of transient scattering problems (the so-called time domain boundary element methods). We propose here a review and update of the mathematical analysis of the involved RPBIE. Our approach, via Laplace transform, is described in some details for the classical acoustic scattering problems. The main results are: (i) existence and uniqueness theorems on a functional framework closely linked to the energy of the scattered waves; (ii) space-time variational formulations for the so-called “first kind” RPBIE, with coerciveness obtained by energy estimates. That leads us to advocate choosing these first kind RPBIE and their Galerkin approximations, instead of the second kind RPBIE and the collocation approximations. The actual space-time boundary elements are described in some detail. Examples of numerical experiments that confirm the unconditional stability of our schemes are reported, as well as references for similar results in other related work.
Tuong Ha-Duong

Inverse Scattering Theory for Time-Harmonic Waves

Abstract
First, we will briefly recall the physical models for the (linearised) acoustic and electromagnetic wave propagation and how they reduce to boundary value problems for the Helmholtz equation. For more details we refer to, e.g. [17, 18, 25, 50].
Andreas Kirsch

Herglotz Wave Functions in Inverse Electromagnetic Scattering Theory

Abstract
Ever since the invention of radar during the Second World War, scientists and engineers have strived not only to detect but also to identify unknown objects through the use of electromagnetic waves. Indeed, as pointed out in [19], “Target identification is the great unsolved problem. We detect almost everything; we identify nothing”. A significant step forward in the resolution of this problem occurred in the 1960's with the invention of synthetic aperture radar (SAR) and since that time numerous striking successes have been recorded in imaging by electromagnetic waves using SAR [1], [7]. However, as the demands of randar imaging have increased, the limitations of SAR have become increasingly apparent. These limitations arise from the fact that SAR is based on the “weak scattering” approximation and ignores polarisation effects. Indeed, such incorrect model assumptions have caused some scientists to ask “how (and if) he complications associated with radar based automatic target recognition can be surmounted” ([1], p. 5).
David Colton, Peter Monk

Backmatter

Weitere Informationen