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Boundary Methods
Elements, Contours, and Nodes




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ISBN 9780824725990
Published March 17, 2005 by CRC Press
248 Pages 85 B/W Illustrations

 
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Book Description

Boundary Methods: Elements, Contours, and Nodes presents the results of cutting-edge research in boundary-based mesh-free methods. These methods combine the dimensionality advantage of the boundary element method with the ease of discretization of mesh-free methods, both of which, for some problems, hold distinct advantages over the finite element method.

After introducing some novel topics related to the boundary element method (BEM), the authors focus on the boundary contour method (BCM)-a variant of the BEM that further reduces the dimensionality of a problem. The final section of the book explores the boundary node method, which combines the BEM with moving least-squares approximants to produce a mesh-free, boundary-only method.

The authors, who are also the primary developers of these methods, clearly introduce and develop each topic. In addition to numerical solutions of boundary value problems in potential theory and linear elasticity, they also discuss topics such as shape sensitivities, shape optimization, and adaptive meshing. Numerical results for selected problems appear throughout the book, as do extensive references.

Table of Contents

INTRODUCTION TO BOUNDARY METHODS

I SELECTED TOPICS IN BOUNDARY ELEMENT
METHODS

BOUNDARY INTEGRAL EQUATIONS
Potential Theory in Three Dimensions
Linear Elasticity in Three Dimensions
Nearly Singular Integrals in Linear Elasticity
Finite Parts of Hypersingular Equations
ERROR ESTIMATION
Linear Operators
Iterated HBIE and Error Estimation
Element-Based Error Indicators
Numerical Examples
THIN FEATURES
Exterior BIE for Potential Theory: MEMS
BIE for Elasticity: Cracks and Thin Shells

II THE BOUNDARY CONTOUR METHOD

LINEAR ELASTICITY
Surface and Boundary Contour Equations
Hypersingular Boundary Integral Equations
Internal Displacements and Stresses
Numerical Results
SHAPE SENSITIVITY ANALYSIS
Sensitivities of Boundary Variables
Sensitivities of Surface Stresses
Sensitivities of Variables at Internal Points
Numerical Results: Hollow Sphere
Numerical Results: Block with a Hole
SHAPE OPTIMIZATION
Shape Optimization Problems
Numerical Results
ERROR ESTIMATION AND ADAPTIVITY
Hypersingular Residuals as Local Error Estimators
Adaptive Meshing Strategy
Numerical Results

III THE BOUNDARY NODE METHOD

SURFACE APPROXIMANTS
Moving Least Squares (MLS) Approximants
Surface Derivatives
Weight Functions
Use of Cartesian Coordinates
POTENTIAL THEORY AND ELASTICITY
Potential Theory in Three Dimensions
Linear Elasticity in Three Dimensions
ADAPTIVITY FOR 3-D POTENTIAL THEORY
Hypersingular and Singular Residuals
Error Estimation and Adaptive Strategy
Progressively Adaptive Solutions: Cube Problem
One-Step Adaptive Cell Refinement
ADAPTIVITY FOR 3-D LINEAR ELASTICITY
Hypersingular and Singular Residuals
Error Estimation and Adaptive Strategy
Progressively Adaptive Solutions: Pulling a Rod
One-Step Adaptive Cell Refinement
Bibliography
Index

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