1st Edition

Boundary Methods Elements, Contours, and Nodes

By Subrata Mukherjee, Yu Xie Mukherjee Copyright 2005
    248 Pages 85 B/W Illustrations
    by CRC Press

    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.

    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

    Biography

    Subrata Mukherjee, Yu Xie Mukherjee