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2007 | Buch

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Boundary Element Analysis

Mathematical Aspects and Applications

herausgegeben von: Prof. Dr.-Ing. Martin Schanz, Prof. Dr. Olaf Steinbach

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Applied and Computational Mechanics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This volume contains eleven state of the art contributions on boundary integral equation and boundary element methods. Beside some historical and more analytical aspects in the formulation and analysis of boundary integral equations also modern fast boundary element methods are described and analyzed from a mathematical point of view. In addition, engineering and industrial applications of those methods are presented showing the ability of state of the art boundary element methods to solve challenging problems from different fields of applications.

This book is addressed to researchers, graduate students and practitioners working on and using boundary element methods. All contributions also show the great achievements of interdisciplinary research between mathematicians and engineers, with direct applications in industry.

Inhaltsverzeichnis

Frontmatter

Some Historical Remarks on the Positivity of Boundary Integral Operators

Abstract

Variational arguments go back a long time in the history of boundary integral equations. Energy methods have shown up very early, then virtually disappeared from the common knowledge and eventually resurfaced in the context of boundary element methods. We focus on some not so well known parts of classical works by well known classical authors and describe the relation of their ideas to modern variational principles in boundary element methods.

Martin Costabel

Averaging Techniques for a Posteriori Error Control in Finite Element and Boundary Element Analysis

Abstract

Averaging techniques for a posteriori error control are established for differential and integral equations within a unifying setting. The reliability and efficiency of the introduced estimator results from two grids T _h and T _H with different polynomial degrees for a smooth exact solution. The proofs are based on first order approximation operators and inverse estimates. For a finer and finer fine mesh T _h, the estimator becomes asymptotically exact. The abstract framework is applicable to a finite element method for the Laplace equation, boundary element methods for Symm’s and the hypersingular integral equation or transmission problems.

Carsten Carstensen, Dirk Praetorius

Coupled Finite and Boundary Element Domain Decomposition Methods

Abstract

The finite element method and the boundary element method often have complementary properties in different situations. The domain decomposition technique allows to use the discretization method which is most appropriate for the subdomain under consideration. The coupling is based on the transmission conditions. The Dirichlet to Neumann (D2N) and Neumann to Dirichlet (N2D) maps are playing a crucial role in representing the transmission conditions. In this paper we study the D2N and N2D maps and their finite and boundary element approximations. Different formulations of the transmission conditions lead to different domain decomposition schemes with different properties. In any case we have to solve large scale systems of coupled finite and boundary element equations. The efficiency of iterative methods heavily depends on the availability of efficient preconditioners. We consider various solution strategies and provide appropriate preconditioners resulting in asymptotically almost optimal solvers.

Ulrich Langer, Olaf Steinbach

The hp-Version of the Boundary Element Method for the Lamé Equation in 3D

Abstract

We analyze the h-p version of the BEM for Dirichlet and Neumann problems of the Lamé equation on open surface pieces. With given regularity of the solution in countably normed spaces we show that the boundary element Galerkin solution of the h-p version converges exponentially fast on geometrically graded meshes. We describe in detail how to use an analytic integration for the computation of the entries of the Galerkin matrix. Numerical benchmarks correspond to our theoretical results.

Matthias Maischak, Ernst P. Stephan

Sparse Convolution Quadrature for Time Domain Boundary Integral Formulations of the Wave Equation by Cutoff and Panel-Clustering

Abstract

We consider the wave equation in a time domain boundary integral formulation. To obtain a stable time discretization, we employ the convolution quadrature method in time, developed by Lubich. In space, a Galerkin boundary element method is considered. The resulting Galerkin matrices are fully populated and the computational complexity is proportional to N log² NM ², where M is the number of spatial unknowns and N is the number of time steps.

We present two ways of reducing these costs. The first is an a priori cutoff strategy, which allows to replace a substantial part of the matrices by 0. The second is a panel clustering approximation, which further reduces the storage and computational cost by approximating subblocks by low rank matrices.

Wolfgang Hackbusch, Wendy Kress, Stefan A. Sauter

Fast Multipole Methods and Applications

Abstract

The symmetric formulation of boundary integral equations and the Galerkin boundary element method are considered to solve mixed boundary value problems of three-dimensional linear elastostatics. Fast boundary element techniques, like the fast multipole method, have to be used to overcome the quadratic complexity of standard boundary element methods. The fast methods provide a data sparse approximation of the fully populated matrices and reduce the computational costs and memory requirements from quadratic order to almost linear ones. Three different approaches to realize the boundary integral operators of linear elastostatics by the fast multipole method are described and numerical examples are given for one of these approaches.

Günther Of

A Fast Boundary Integral Equation Method for Elastodynamics in Time Domain and Its Parallelisation

Abstract

This paper discusses a time domain fast boundary integral equation method for three dimensional elastodynamics and its parallelisation for a large shared memory parallel computer. Some details of the PWTD (Plane Wave Time Domain) approach and an extension of the theory to the anisotropic case are presented. We then examine the parallelisation strategies of the code using OpenMP and MPI-OpenMP hybridisation. In the case of MPI-OpenMP hybrid parallelisation, a numerical example with more than one million spatial DOF is shown. It is concluded that the method is promising, its parallelisation with OpenMP is effective and that larger problem can be analysed with MPI-OpenMP hybrid parallelisation.

Yoshihiro Otani, Toru Takahashi, Naoshi Nishimura

FM-BEM and Topological Derivative Applied to Acoustic Inverse Scattering

Abstract

This study is set in the framework of inverse scattering of scalar (e.g. acoustic) waves. A qualitative probing technique based on the distribution of topological sensitivity of the cost functional associated with the inverse problem with respect to the nucleation of an infinitesimally-small hard obstacle is formulated. The sensitivity distribution is expressed as a bilinear formula involving the free field and an adjoint field associated with the cost function. These fields are computed by means of a boundary element formulation accelerated by the Fast Multipole method. A computationally fast approach for performing a global preliminary search based on the available overspecified boundary data is thus defined. Its usefulness is demonstrated through results of numerical experiments on the qualitative identification of a hard obstacle in a bounded acoustic domain, for configurations featuring O(10⁵) nodal unknowns and O(10⁶) sampling points.

Marc Bonnet, Nicolas Nemitz

Boundary Element Methods for Eddy Current Computation

Abstract

This paper studies numerical methods for time-harmonic eddy current problems in the case of homogeneous, isotropic, and linear materials. It provides a survey of approaches that entirely rely on boundary integral equations and their conforming Galerkin discretization. Starting point are both E- and H-based strong formulation, for which issues of gauging and topological constraints on the existence of potentials are discussed.

Direct boundary integral equations and the so-called symmetric coupling of the integral equations corresponding to the conductor and the non-conducting regions are employed. They give rise to coupled variational problems that are elliptic in suitable trace spaces. This implies quasi-optimal convergence of conforming Galerkin boundary element methods, which make use of divΓ-conforming trial spaces for surface currents.

Ralf Hiptmair

Fast Boundary Element Methods in Computational Electromagnetism

Abstract

When the Boundary Element Method (BEM) is used to analyse electromagnetic problems one is able to achieve an almost linear complexity by applying matrix compression techniques. Beyond this, on symmetrical domains the computational costs can be reduced by significant factors. By using several symmetry considerations (geometry, mesh, kernel, excitation) it will be shown how the combination of the Adaptive Cross Approximation (ACA) and the symmetry exploitation allows an efficient solution of electromagnetic problems. This approach will be demonstrated on the scalar BEM formulation for electrostatics and can also be applied to the vectorial eddy current formulations. The symmetry exploting ACA algorithm not only reduces the problem size due to the symmetry but also possesses an almost linear complexity w.r.t. the number of unknowns.

Stefan Kurz, Oliver Rain, Sergej Rjasanow

BEM-Based Simulations in Engineering Design

Abstract

The simulation of the real-world industrial problems is nowadays faced with a number of the challenging requirements, mainly arising in the daily design praxis of power engineering devices. Complex structures, complex physics, huge dimensions and huge aspect ratio in model dimensions are just some of the critical modelling issues that need to be encountered by the simulation tools. Thanks to the advances achieved in the last several years, BEM become a powerful numerical technique for the simulations of such industrial products. Until recent time this technique has been recognized as a technique offering from one side some excellent features (2D instead of 3D discretization, open-boundary problems, etc.), but from the other side having some serious practical limitations, mostly related to the full-populated, often ill-conditioned matrices. The new, emerging numerical techniques like MBIT (Multipole-Base Integral Technique), ACA (Adaptive Cross-Approximations), DDT (Domain-Decomposition Technique) seems to bridge some of these known bottlenecks, promoting those the BEM in a high-level tool for even daily-design process of the 3D real-world problems.

The aim of this Chapter is to illustrate how this numerical technique can be used for the simulation of both single-physics problems appearing in the Dielectric Design (Electrostatics), and multi-physics problems in Thermal Design (coupling of Electromagnetic-Heat Transfer) and Electro-Mechanical Design (coupling of Electromagnetic-Structural Mechanics) of power engineering devices like power transformers or switchgears.

Zoran Andjelić, Jasmin Smajić, Michael Conry

Backmatter

Titel: Boundary Element Analysis
herausgegeben von: Prof. Dr.-Ing. Martin Schanz
Prof. Dr. Olaf Steinbach
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-47533-0
Print ISBN: 978-3-540-47465-4
DOI: https://doi.org/10.1007/978-3-540-47533-0

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