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 |  موضوع: كتاب Boundary Methods Elements Contours and Nodes الأربعاء 13 أغسطس 2014, 1:41 am | |
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أحضرت لكم كتاب
Boundary Methods Elements Contours and Nodes
Subrata Mukherjee
Cornell University
Ithaca, New York, U.S.A.
Yu Xie Mukherjee
Cornell University
Ithaca, New York, U.S.A
ويتناول الموضوعات الأتية :
Contents
Preface v
INTRODUCTION TO BOUNDARY METHODS xiii
I SELECTED TOPICS IN BOUNDARY ELEMENT
METHODS 1
1 BOUNDARY INTEGRAL EQUATIONS 3
1.1 Potential Theory in Three Dimensions . 3
1.1.1 Singular Integral Equations . 3
1.1.2 Hypersingular Integral Equations 5
1.2 Linear Elasticity in Three Dimensions . 6
1.2.1 Singular Integral Equations . 6
1.2.2 Hypersingular Integral Equations 8
1.3 Nearly Singular Integrals in Linear Elasticity . 12
1.3.1 Displacements at Internal Points Close to the Boundary . 12
1.3.2 Stresses at Internal Points Close to the Boundary . 13
1.4 Finite Parts of Hypersingular Equations 14
1.4.1 Finite Part of a Hypersingular Integral Collocated at an
Irregular Boundary Point 14
1.4.2 Gradient BIE for 3-D Laplace’s Equation . 17
1.4.3 Stress BIE for 3-D Elasticity 19
1.4.4 Solution Strategy for a HBIE Collocated at an Irregular
Boundary Point 20
2 ERROR ESTIMATION 23
2.1 Linear Operators . 23
2.2 Iterated HBIE and Error Estimation 25
2.2.1 Problem 1 : Displacement Boundary Conditions . 25
2.2.2 Problem 2 : Traction Boundary Conditions 28
2.2.3 Problem 3 : Mixed Boundary Conditions . 30
2.3 Element-Based Error Indicators . 32
2.4 Numerical Examples . 33
, LLCviii CONTENTS
2.4.1 Example 1: Lam´e’s Problem of a Thick-Walled Cylinder
under Internal Pressure . 34
2.4.2 Example 2: Kirsch’s Problem of an Infinite Plate with a
Circular Cutout . 36
3 THIN FEATURES 39
3.1 Exterior BIE for Potential Theory: MEMS 39
3.1.1 Introduction to MEMS 39
3.1.2 Electric Field BIEs in a Simply Connected Body . 41
3.1.3 BIES in Infinite Region Containing Two Thin Conducting
Plates . 41
3.1.4 Singular and Nearly Singular Integrals . 46
3.1.5 Numerical Results 49
3.1.6 The Model Problem - a Parallel Plate Capacitor . 50
3.2 BIE for Elasticity: Cracks and Thin Shells 54
3.2.1 BIES in LEFM 54
3.2.2 Numerical Implementation of BIES in LEFM . 60
3.2.3 Some Comments on BIEs in LEFM . 61
3.2.4 BIEs for Thin Shells . 62
II THE BOUNDARY CONTOUR METHOD 65
4 LINEAR ELASTICITY 67
4.1 Surface and Boundary Contour Equations . 67
4.1.1 BasicEquations . 67
4.1.2 Interpolation Functions . 68
4.1.3 Boundary Elements . 71
4.1.4 Vector Potentials . 73
4.1.5 Final BCM Equations 74
4.1.6 Global Equations and Unknowns 76
4.1.7 Surface Displacements, Stresses, and Curvatures . 76
4.2 Hypersingular Boundary Integral Equations 78
4.2.1 Regularized Hypersingular BIE . 78
4.2.2 Regularized Hypersingular BCE 78
4.2.3 Collocation of the HBCE at an Irregular Surface Point 80
4.3 Internal Displacements and Stresses 82
4.3.1 Internal Displacements 82
4.3.2 Displacements at Internal Points Close to the Bounding
Surface 82
4.3.3 Internal Stresses . 83
4.3.4 Stresses at Internal Points Close to the Bounding Surface 84
4.4 Numerical Results 85
4.4.1 Surface Displacements from the BCM and the
HBCM . 85
, LLCCONTENTS ix
4.4.2 Surface Stresses 87
4.4.3 Internal Stresses Relatively Far from the Bounding Surface 90
4.4.4 Internal Stresses Very Close to the Bounding Surface . 90
5 SHAPE SENSITIVITY ANALYSIS 93
5.1 Sensitivities of Boundary Variables . 93
5.1.1 Sensitivity of the BIE 93
5.1.2 The Integral Ik 94
5.1.3 The Integral Jk 96
5.1.4 The BCM Sensitivity Equation . 98
5.2 Sensitivities of Surface Stresses . 99
5.2.1 Method One . 100
5.2.2 Method Two . 100
5.2.3 Method Three 100
5.2.4 Method Four . 101
5.3 Sensitivities of Variables at Internal Points 101
5.3.1 Sensitivities of Displacements 101
5.3.2 Sensitivities of Displacement Gradients and Stresses . 103
5.4 Numerical Results: Hollow Sphere . 106
5.4.1 Sensitivities on Sphere Surface . 107
5.4.2 Sensitivities at Internal Points . 108
5.5 Numerical Results: Block with a Hole . 110
5.5.1 Geometry and Mesh . 110
5.5.2 Internal Stresses . 112
5.5.3 Sensitivities of Internal Stresses . 112
6 SHAPE OPTIMIZATION 115
6.1 Shape Optimization Problems 115
6.2 Numerical Results 116
6.2.1 Shape Optimization of a Fillet . 116
6.2.2 Optimal Shapes of Ellipsoidal Cavities Inside Cubes . 118
6.2.3 Remarks 122
7 ERROR ESTIMATION AND ADAPTIVITY 125
7.1 Hypersingular Residuals as Local Error Estimators 125
7.2 Adaptive Meshing Strategy . 126
7.3 Numerical Results 127
7.3.1 Example One - Short Clamped Cylinder under Tension . 127
7.3.2 Example Two - the Lam´e Problem for a Hollow Cylinder 130
III THE BOUNDARY NODE METHOD 133
8 SURFACE APPROXIMANTS 135
8.1 Moving Least Squares (MLS) Approximants . 135
8.2 Surface Derivatives 139
, LLCx CONTENTS
8.3 Weight Functions . 141
8.4 Use of Cartesian Coordinates 142
8.4.1 Hermite Type Approximation 142
8.4.2 Variable Basis Approximation 143
9 POTENTIAL THEORY AND ELASTICITY 151
9.1 Potential Theory in Three Dimensions . 151
9.1.1 BNM: Coupling of BIE with MLS Approximants . 151
9.1.2 HBNM: Coupling of HBIE with MLS Approximants . 155
9.1.3 Numerical Results for Dirichlet Problems on a Sphere 156
9.2 Linear Elasticity in Three Dimensions . 165
9.2.1 BNM: Coupling of BIE with MLS Approximants . 165
9.2.2 HBNM: Coupling of HBIE with MLS Approximants . 167
9.2.3 Numerical Results 168
10 ADAPTIVITY FOR 3-D POTENTIAL THEORY 175
10.1 Hypersingular and Singular Residuals . 175
10.1.1 The Hypersingular Residual . 175
10.1.2 The Singular Residual 176
10.2 Error Estimation and Adaptive Strategy 177
10.2.1 Local Residuals and Errors - Hypersingular Residual Approach . 178
10.2.2 Local Residuals and Errors - Singular Residual Approach 178
10.2.3 Cell Refinement Criterion 179
10.2.4 Global Error Estimation and Stopping Criterion . 179
10.3 Progressively Adaptive Solutions: Cube
Problem 180
10.3.1 Exact Solution 181
10.3.2 Initial Cell Configuration # 1 (54 Surface Cells) . 181
10.3.3 Initial Cell Configuration # 2 (96 Surface Cells) . 182
10.4 One-Step Adaptive Cell Refinement 188
10.4.1 Initial Cell Configuration # 1 (54 Surface Cells) . 190
10.4.2 Initial Cell Configuration # 2 (96 Surface Cells) . 191
11 ADAPTIVITY FOR 3-D LINEAR ELASTICITY 193
11.1 Hypersingular and Singular Residuals . 193
11.1.1 The Hypersingular Residual . 193
11.1.2 The Singular Residual 194
11.2 Error Estimation and Adaptive Strategy 194
11.2.1 Local Residuals and Errors - Hypersingular Residual Approach . 194
11.2.2 Local Residuals and Errors - Singular Residual Approach 195
11.2.3 Cell Refinement Global Error Estimation and Stopping
Criterion . 195
11.3 Progressively Adaptive Solutions: Pulling a Rod . 195
11.3.1 Initial Cell Configuration 197
11.3.2 Adaptivity Results 197
11.4 One-Step Adaptive Cell Refinement 198
Bibliography
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