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This textbook delves into the theory and practical application of boundary integral equation techniques, focusing on their numerical solution for boundary value problems within potential theory and linear elasticity. Drawing parallels between single and double layer potentials in potential theory and their counterparts in elasticity, the book introduces various numerical procedures, namely boundary element methods, where unknown quantities reside on the boundaries of the region of interest. Through the approximation of boundary value problems into systems of algebraic equations, solvable by standard numerical methods, the text elucidates both indirect and direct approaches. Indirect methods involve single or double layer potentials separately while direct methods combine these potentials using Green's or Somigliana's formulas. The two approaches give comparable results for general boundary value problems. Tailored for beginning graduate students, this self-contained textbook offers detailed analytical and numerical derivations for isotropic and anisotropic materials, prioritizing simplicity in presentation while progressively advancing towards more intricate mathematical concepts, particularly focusing on two-dimensional problems within potential theory and linear elasticity.

An authoritative guide to the theory and practice of static and dynamic structures analysis Static and Dynamic Analysis of Engineering Structures examines static and dynamic analysis of engineering structures for methodological and practical purposes. In one volume, the authors – noted engineering experts – provide an overview of the topic and review the applications of modern as well as classic methods of calculation of various structure mechanics problems. They clearly show the analytical and mechanical relationships between classical and modern methods of solving boundary value problems. The first chapter offers solutions to problems using traditional techniques followed by the introduction of the boundary element methods. The book discusses various discrete and continuous systems of analysis. In addition, it offers solutions for more complex systems, such as elastic waves in inhomogeneous media, frequency-dependent damping and membranes of arbitrary shape, among others. Static and Dynamic Analysis of Engineering Structures is filled with illustrative examples to aid in comprehension of the presented material. The book: Illustrates the modern methods of static and dynamic analysis of structures; Provides methods for solving boundary value problems of structural mechanics and soil mechanics; Offers a wide spectrum of applications of modern techniques and methods of calculation of static, dynamic and seismic problems of engineering design; Presents a new foundation model. Written for researchers, design engineers and specialists in the field of structural mechanics, Static and Dynamic Analysis of Engineering Structures provides a guide to analyzing static and dynamic structures, using traditional and advanced approaches with real-world, practical examples.

This book is devoted to the mathematical analysis of the numerical solution of boundary integral equations treating boundary value, transmission and contact problems arising in elasticity, acoustic and electromagnetic scattering. It serves as the mathematical foundation of the boundary element methods (BEM) both for static and dynamic problems. The book presents a systematic approach to the variational methods for boundary integral equations including the treatment with variational inequalities for contact problems. It also features adaptive BEM, hp-version BEM, coupling of finite and boundary element methods – efficient computational tools that have become extremely popular in applications.Familiarizing readers with tools like Mellin transformation and pseudodifferential operators as well as convex and nonsmooth analysis for variational inequalities, it concisely presents efficient, state-of-the-art boundary element approximations and points to up-to-date research.The authors are well known for their fundamental work on boundary elements and related topics, and this book is a major contribution to the modern theory of the BEM (especially for error controlled adaptive methods and for unilateral contact and dynamic problems) and is a valuable resource for applied mathematicians, engineers, scientists and graduate students.

Most texts on computational methods are borne out of research activities at postgraduate study programs, and this is no exception. After being introduced to the boundary element method (BEM) (then referred to as the boundary integral equation method (BIEM)) in 1981 by Prof. Jim Liggett of Cornell University, a number of graduate students and myself under his supervision took active interest in the development of the theory and its application to a wide range of engineering problems. We certainly achieved some amount of success. A personal desire to have a deeper understanding and appreciation of computational methods prompted one to take related courses in fmite deference method, and to undertake a self-instructed study of variational and fmite element methods. These exposures were not only quite instructive but fruitful, and may have provided the motivation for the current research on the Green element method (GEM) - a name coined by Prof. Liggett in 1987 during my visit as Professor to the School of Civil & Environmental Engineering, Cornell University. The main objectives of this text are to serve as an instructional material to senior undergraduate and first year graduate students undertaking a course in computational methods, and as a resource material for research scientists, applied mathematicians, numerical analysts, and engineers who may wish to take these ideas to other frontiers and applications.

At the date of this writing, there is no question that the boundary element method has emerged as one of the major revolutions on the engineering science of computational mechanics. The emergence of the technique from relative obscurity to a cutting edge engineering analysis tool in the short space of basically a ten to fifteen year time span is unparalleled since the advent of the finite element method. At the recent international conference BEM XI, well over one hundred papers were presented and many were pub­ lished in three hard-bound volumes. The exponential increase in interest in the subject is comparable to that shown in the early days of finite elements. The diversity of appli­ cations of BEM, the broad base of interested parties, and the ever-increasing presence of the computer as an engineering tool are probably the reasons for the upsurge in pop­ ularity of BEM among researchers and industrial practitioners. Only in the past few years has the BEM audience become large enough that we have seen the development of specialty books on specific applications of the boundary element method. The present text is one such book. In this work, we have attempted to present a self-contained treatment of the analysis of physical phenomena governed by equations containing biharmonic operators. The biharmonic operator defines a very important class of fourth-order PDE problems which includes deflections of beams and thin plates, and creeping flow of viscous fluids.

1. 1 The Hybrid Displacement Boundary Element Model This work is concerned with the derivation of a numerical model for the solution of boundary-value problems in potential theory and linear elasticity. It is considered a boundary element model because the final integral equation involves some boundary integrals, whose evaluation requires a boundary discretization. Furthermore, all the unknowns are boundary vari­ ables. The model is completely new; it differs from the classical boundary element formulation ·in the way it is generated and consequently in the fi­ nal equations. A generalized variational principle is used as a basis for its derivation, whereas the conventional boundary element formulation is based on Green's formula (potential problems) and on Somigliana's identity (elas­ ticity), or alternatively through the weighted residual technique. 2 The multi-field variational principle which generates the formulation in­ volves three independent variables. For potential problems, these are the potential in the domain and the potential and its normal derivative on the boundary. In the case of elasticity, these variables are displacements in the domain and displacements and tractions on the boundary. For this reason, by analogy with the assumed displacement hybrid finite element model, ini­ tially proposed by Tong [1] in 1970, it can be called a hybrid displacement model. The final system of equations to be solved is similar to that found in a stiffness formulation. The stiffness matrix for this model is symmetric and can be evaluated by only performing integrations along the boundary.

Particle samplers are widely used in workplaces in order to determine the concentration of airborne particles in the atmosphere. They generally operate by drawing air, with the aid of a pump, through one or more orifices in the sampler body and housed within the sampler is a filter through which the air is subsequently drawn. The airborne particles are collected on the filter and their concentration is determined. Various samplers have been designed for this purpose including'static'samplers, which are located in a fixed position in a working environment and determine the dust concentration averaged over a prescribed period of time at that one point, and'personal'samplers which are mounted on a working person near to the breathing zone. The ORB sampler, a static sampler designed by Ogden and Birkett (1978) to have approximately the same entry efficiency, for particles with aerodynamical diameter up to at least 25~m, as a human head equally exposed to all wind directions for wind speeds between 0 and 2. 75m1s, is shown in Fig. l. l and examples of personal samplers are shown in Fig. 1. 2a, b and c and represent a single 4mm hole sampler, a seven hole sampler and a 25mm open face filter holder respectively. These three samplers are some of the most commonly used personal samplers for sampling the total airborne concentrations of workplace dusts in Britain.

Die gut eingeführte Methode der Randelemente baut auf den Grundlagen der Kontinuumsmechnik auf. In diesem Buch wird sie für die Elastodynamik schwingender Strukturen und für die Elastostatik formuliert. Die mathematischen und ingenieurwissenschaftlichen Grundlagen werden bereitgestellt; einfache Beispiele erleichtern das Verständnis.Das Buch liegt nun in einer 2. korrigierten Auflage vor. Es richtet sich an Studierende, Ingenieure und Naturwissenschaftler, die sich Kenntnisse der Randelementmethode erarbeiten wollen.

This text considers the problem of the dynamic fluid-structure interaction between a finite elastic structure and the acoustic field in an unbounded fluid-filled exterior domain. The exterior acoustic field is modelled through a boundary integral equation over the structure surface. However, the classical boundary integral equation formulations of this problem either have no solutions or do not have unique solutions at certain characteristic frequencies (which depend on the surface geometry) and it is necessary to employ modified boundary integral equation formulations which are valid for all frequencies. The particular approach adopted here involves an arbitrary coupling parameter and the effect that this parameter has on the stability and accuracy of the numerical method used to solve the integral equation is examined. The boundary integral analysis of the exterior acoustic problem is coupled with a finite element analysis of the elastic structure in order to investigate the interaction between the dynamic behaviour of the structure and the associated acoustic field. Recently there has been some controversy over whether or not the coupled problem also suffers from the non-uniqueness problems associated with the classical integral equation formulations of the exterior acoustic problem. This question is resolved by demonstrating that.the solution to the coupled problem is not unique at the characteristic frequencies and that it is necessary to employ an integral equation formulation valid for all frequencies.

In three dimensional boundary element analysis, computation of integrals is an important aspect since it governs the accuracy of the analysis and also because it usually takes the major part of the CPU time. The integrals which determine the influence matrices, the internal field and its gradients contain (nearly) singular kernels of order lIr a (0:= 1,2,3,4,.··) where r is the distance between the source point and the integration point on the boundary element. For planar elements, analytical integration may be possible 1,2,6. However, it is becoming increasingly important in practical boundary element codes to use curved elements, such as the isoparametric elements, to model general curved surfaces. Since analytical integration is not possible for general isoparametric curved elements, one has to rely on numerical integration. When the distance d between the source point and the element over which the integration is performed is sufficiently large compared to the element size (d> 1), the standard Gauss-Legendre quadrature formula 1,3 works efficiently. However, when the source is actually on the element (d=O), the kernel 1I~ becomes singular and the straight forward application of the Gauss-Legendre quadrature formula breaks down. These integrals will be called singular integrals. Singular integrals occur when calculating the diagonals of the influence matrices.

This volume contains eight state of the art contributions on mathematical aspects and applications of fast boundary element methods in engineering and industry. This covers the analysis and numerics of boundary integral equations by using differential forms, preconditioning of hp boundary element methods, the application of fast boundary element methods for solving challenging problems in magnetostatics, the simulation of micro electro mechanical systems, and for contact problems in solid mechanics. Other contributions are on recent results on boundary element methods for the solution of transient problems.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 engineering and industry.

The Complex Variable Boundary Element Method (CVBEM) has emerged as a new and effective modeling method in the field of computational mechanics and hydraulics. The CVBEM is a generalization of the Cauchy integral formula into a boundary integral equation method. The model­ ing approach by boundary integration, the use of complex variables for two-dimensional potential problems, and the adaptability to now-popular microcomputers are among the factors that make this technique easy to learn, simple to operate, practical for modeling, and efficient in simulating various physical processes. Many of the CVBEM concepts and notions may be derived from the Analytic Function Method (AFM) presented in van der Veer (1978). The AFM served as the starting point for the generalization of the CVBEM theory which was developed during the first author's research engagement (1979 through 1981) at the University of California, Irvine. The growth and expansion of the CVBEM were subsequently nurtured at the U. S. Geological Survey, where keen interest and much activity in numerical modeling and computational mechanics-and-hydraulics are prevalent. Inclusion of the CVBEM research program in Survey's computational-hydraulics projects, brings the modeling researcher more uniform aspects of numerical mathematics in engineering and scientific problems, not to mention its (CVBEM) practicality and usefulness in the hydrologic investigations. This book is intended to introduce the CVBEM to engineers and scientists with its basic theory, underlying mathematics, computer algorithm, error analysis schemes, model adjustment procedures, and application examples.

This book presents a systematic approach to numerical solution for a wide range of spatial contact problems of geotechnics. On the basis of the boundary element method new techniques and effective computing algorithms are considered. Special attention is given to the formulation and analysis of the spatial contact models for elastic bases. Besides the classical schemes of contact deformation, new contact models are discussed for spatially nonhomogeneous and nonlinearly elastic media properly describing soil properties.

The book comprises 26 contributions by more that 60 leading researchers in Boundary Element Methods (BEM) and other Mesh Reduction Methods (MRM). All contributors are well-known scientists from all over the world. The volume, besides a review chapter by Professor Brebbia on the career and the scientific work of Professor Katsikadelis, is essentially a collection of original articles covering a variety of research topics in the areas of solid mechanics, fluid mechanics, potential theory, inhomogeneous or composite materials, fracture mechanics, damage mechanics, plasticity, heat transfer, dynamics and vibrations, soil-structure interaction. The chapters contained in this volume appear in alphabetical order by first author and most of them are relevant to and reflect the research interests of Professor Katsikadelis.

The fast multipole method is one of the most important algorithms in computing developed in the 20th century. Along with the fast multipole method, the boundary element method (BEM) has also emerged as a powerful method for modeling large-scale problems. BEM models with millions of unknowns on the boundary can now be solved on desktop computers using the fast multipole BEM. This is the first book on the fast multipole BEM, which brings together the classical theories in BEM formulations and the recent development of the fast multipole method. Two- and three-dimensional potential, elastostatic, Stokes flow, and acoustic wave problems are covered, supplemented with exercise problems and computer source codes. Applications in modeling nanocomposite materials, bio-materials, fuel cells, acoustic waves, and image-based simulations are demonstrated to show the potential of the fast multipole BEM. Enables students, researchers, and engineers to learn the BEM and fast multipole method from a single source.

This volume, dedicated to Professor Dimitri Beskos, contains contributions from leading researchers in Europe, the USA, Japan and elsewhere, and addresses the needs of the computational mechanics research community in terms of timely information on boundary integral equation-based methods and techniques applied to a variety of fields. The contributors are well-known scientists, who also happen to be friends, collaborators as past students of Dimitri Beskos. Dimitri is one the BEM pioneers who started his career at the University of Minnesota in Minneapolis, USA, in the 1970s and is now with the University of Patras in Patras, Greece. The book is essentially a collection of both original and review articles on contemporary Boundary Element Methods (BEM) as well as on the newer Mesh Reduction Methods (MRM), covering a variety of research topics. Close to forty contributions compose an over-500 page volume that is rich in detail and wide in terms of breadth of coverage of the subject of integral equation formulations and solutions in both solid and fluid mechanics.

Among numerical methods applied in acoustics, the Finite Element Method (FEM) is normally favored for interior problems whereas the Boundary Element Method (BEM) is quite popular for exterior ones. That is why this valuable reference provides a complete survey of methods for computational acoustics, namely FEM and BEM. It demonstrates that both methods can be effectively used in the complementary cases. The chapters by well-known authors are evenly balanced: 10 chapters on FEM and 10 on BEM. An initial conceptual chapter describes the derivation of the wave equation and supplies a unified approach to FEM and BEM for the harmonic case. A categorization of the remaining chapters and a personal outlook complete this introduction. In what follows, both FEM and BEM are discussed in the context of very different problems. Firstly, this comprises numerical issues, e.g. convergence, multi-frequency solutions and highly efficient methods; and secondly, solutions techniques for the particular difficulties that arise with external problems, e.g. discussion of absorbing boundaries for FEM and treatment of the non-uniqueness problem for BEM. Finally, both parts on FEM and on BEM are completed by chapters on related problems, e.g. formulations for fluid-structure interaction. In addition to time-harmonic problems, transient problems are considered in some chapters. Many theoretical and industrial applications are presented. Overall, this book is a unified review of the state-of-the-art on FEM and BEM for computational acoustics.

This is a sequel to the book “Programming the Boundary Element Method” by G. Beer published by Wiley in 2001. The scope of this book is different however and this is reflected in the title. Whereas the previous book concentrated on explaining the implementation of a limited range of problems into computer code and the emphasis was on programming, in the current book the problems covered are extended, the emphasis is on explaining the theory and computer code is not presented for all topics. The new topics covered range from dynamics to piezo-electricity. However, the main idea, to provide an explanation of the Boundary Element Method (BEM), that is easy for engineers and scientists to follow, is retained. This is achieved by explaining some aspects of the method in an engineering rather than mathematical way. Another new feature of the book is that it deals with the implementation of the method on parallel processing hardware. I. M. Smith, who has been involved in programming the finite element method for decades, illustrates that the BEM is “embarrassingly parallelisable”. It is shown that the conversion of the BEM programs to run efficiently on parallel processing hardware is not too difficult and the results are very impressive, such as solving a 20 000 element problem during a “coffee break”.

Symmetric Galerkin Boundary Element Method presents an introduction as well as recent developments of this accurate, powerful, and versatile method. The formulation possesses the attractive feature of producing a symmetric coefficient matrix. In addition, the Galerkin approximation allows standard continuous elements to be used for evaluation of hypersingular integrals. FEATURES• Written in a form suitable for a graduate level textbook as well as a self-learning tutorial in the field.• Covers applications in two-dimensional and three-dimensional problems of potential theory and elasticity. Additional basic topics involve axisymmetry, multi-zone and interface formulations. More advanced topics include fluid flow (wave breaking over a sloping beach), non-homogeneous media, functionally graded materials (FGMs), anisotropic elasticity, error estimation, adaptivity, and fracture mechanics. • Presents integral equations as a basis for the formulation of general symmetric Galerkin boundary element methods and their corresponding numerical implementation. • Designed to convey effective unified procedures for the treatment of singular and hypersingular integrals that naturally arise in the method. Symbolic codes using Maple® for singular-type integrations are provided and discussed in detail.• The user-friendly adaptive computer code BEAN (Boundary Element ANalysis), fully written in Matlab®, is available as a companion to the text. The complete source code, including the graphical user-interface (GUI), can be downloaded from the web site http://www.ghpaulino.com/SGBEM_book. The source code can be used as the basis for building new applications, and should also function as an effective teaching tool. To facilitate the use of BEAN, a video tutorial and a library of practical examples are provided.

This book is devoted to the mathematical foundation of boundary integral equations. The combination of?nite element analysis on the boundary with these equations has led to very e?cient computational tools, the boundary element methods (see e.g., the authors [139] and Schanz and Steinbach (eds.) [267]). Although we do not deal with the boundary element discretizations in this book, the material presented here gives the mathematical foundation of these methods. In order to avoid over generalization we have con?ned ourselves to the treatment of elliptic boundary value problems. The central idea of eliminating the?eld equations in the domain and - ducing boundary value problems to equivalent equations only on the bou- ary requires the knowledge of corresponding fundamental solutions, and this idea has a long history dating back to the work of Green [107] and Gauss [95, 96]. Today the resulting boundary integral equations still serve as a major tool for the analysis and construction of solutions to boundary value problems.

Although the aim of this book is to give a unified introduction into finite and boundary element methods, the main focus is on the numerical analysis of boundary integral and boundary element methods. Starting from the variational formulation of elliptic boundary value problems boundary integral operators and associated boundary integral equations are introduced and analyzed. By using finite and boundary elements corresponding numerical approximation schemes are considered. This textbook may serve as a basis for an introductory course in particular for boundary element methods including modern trends such as fast boundary element methods and efficient solution methods, as well as the coupling of finite and boundary element methods.

Boundary Element Methods (BEM) play an important role in modern numerical computations in the applied and engineering sciences. These methods turn out to be powerful tools for numerical studies of various physical phenomena which can be described mathematically by partial differential equations. The most prominent example is the potential equation (Laplace equation), which is used to model physical phenomena in electromagnetism, gravitation theory, and in perfect fluids. A further application leading to the Laplace equation is the model of steady state heat flow. One of the most popular applications of the BEM is the system of linear elastostatics, which can be considered in both bounded and unbounded domains. A simple model for a fluid flow, the Stokes system, can also be solved by the use of the BEM. The most important examples for the Helmholtz equation are the acoustic scattering and the sound radiation. The Fast Solution of Boundary Integral Equations provides a detailed description of fast boundary element methods which are based on rigorous mathematical analysis. In particular, a symmetric formulation of boundary integral equations is used, Galerkin discretisation is discussed, and the necessary related stability and error estimates are derived. For the practical use of boundary integral methods, efficient algorithms together with their implementation are needed. The authors therefore describe the Adaptive Cross Approximation Algorithm, starting from the basic ideas and proceeding to their practical realization. Numerous examples representing standard problems are given which underline both theoretical results and the practical relevance of boundary element methods in typical computations.

This course with 6 lecturers intends to present a systematic survey of recent re­ search results of well-known scientists on error-controlled adaptive finite element methods in solid and structural mechanics with emphasis to problem-dependent concepts for adaptivity, error analysis as well as h- and p-adaptive refinement techniques including meshing and remeshing. Challenging applications are of equal importance, including elastic and elastoplastic deformations of solids, con­ tact problems and thin-walled structures. Some major topics should be pointed out, namely: (i) The growing importance of goal-oriented and local error estimates for quan­ tities of interest—in comparison with global error estimates—based on dual finite element solutions; (a) The importance of the p-version of the finite element method in conjunction with parameter-dependent hierarchical approximations of the mathematical model, for example in boundary layers of elastic plates; (Hi) The choice of problem-oriented error measures in suitable norms, consider­ ing residual, averaging and hierarchical error estimates in conjunction with the efficiency of the associated adaptive computations; (iv) The importance of implicit local postprocessing with enhanced test spaces in order to get constant-free, i. e. absolute-not only relative-discretizati- error estimates; (v) The coupling of error-controlled adaptive discretizations and the mathemat­ ical modeling in related subdomains, such as boundary layers. The main goals of adaptivity are reliability and efficiency, combined with in­ sight and access to controls which are independent of the applied discretization methods. By these efforts, new paradigms in Computational Mechanics should be realized, namely verifications and even validations ofengineering models.

The boundary element method is an extremely versatile and powerful tool of computational mechanics which has already become a popular alternative to the well established finite element method. This book presents a comprehensive and up-to-date treatise on the boundary element method (BEM) in its applications to various fields of continuum mechanics such as: elastostatics, elastodynamics, thermoelasticity, micropolar elasticity, elastoplasticity, viscoelasticity, theory of plates and stress analysis by hybrid methods. The fundamental solution of governing differential equations, integral representations of the displacement and temperature fields, regularized integral representations of the stress field and heat flux, boundary integral equations and boundary integro-differential equations are derived. Besides the mathematical foundations of the boundary integral method, the book deals with practical applications of this method. Most of the applications concentrate mainly on the computational problems of fracture mechanics. The method has been found to be very efficient in stress-intensity factor computations. Also included are developments made by the authors in the boundary integral formulation of thermoelasticity, micropolar elasticity, viscoelasticity, plate theory, hybrid method in elasticity and solution of crack problems. The solution of boundary-value problems of thermoelasticity and micropolar thermoelasticity is formulated for the first time as the solution of pure boundary problems. A new unified formulation of general crack problems is presented by integro-differential equations.

This volume reviews and discusses the main numerical methods used today for solving problems in infinite domains. It also presents in detail one very effective method in this class, namely the Dirichlet-to-Neumann (DtN) finite element method. The book is intended to provide the researcher or engineer with the state-of-the-art in numerical solution methods for infinite domain problems, such as the problems encountered in acoustics and structural acoustics, fluid dynamics, meteorology, and many other fields of application. The emphasis is on the fundamentals of the various methods, and on reporting recent progress and forecasting future directions. An appendix at the end of the book provides an introduction to the essentials of the finite element method, and suggests a short list of texts on the subject which are categorized by their level of mathematics.

This Proceedings features a broad range of computational mechanics papers on both solid and fluid mechanics as well as electromagnetics, acoustics, heat transfer and other interdisciplinary problems. Topics covered include theoretical developments, numerical analysis, intelligent and adaptive solution strategies and practical applications.

Boundary Element Techniques in Engineering deals with solutions of two- and three-dimensional problems in elasticity and the potential theory where finite elements are inefficient. The book discusses approximate methods, higher-order elements, elastostatics, time-dependent problems, non-linear problems, and combination of regions. Approximate methods include weighted residual techniques, weak formulations, the inverse formulation, and boundary methods. The text also explains Laplace's equation, indirect formulation, matrix formulation, Poisson's equation, and the Helmholtz equation. It describes how elements with linear variations of u and q (i.e. linear elements) can be developed for two dimensional problems, as well as for quadratic and higher order elements for two-dimensional problems. The text investigates the Dirac delta function as a sum of Eigen functions, including some methods to determine the explicit form of fundamental solutions for recurrent problems. The book also tackles the application of boundary elements to problems with both material and certain types of geometric non-linearities, and also the applications of boundary elements to plasticity problems. The text is ideal for mathematicians, students, and professor of calculus or advanced mathematics.