Boundary Methods

by Mukherjee; Subrata

• ISBN13: 9780824725990
• ISBN10: 0824725999
• Format: Hardcover
• Copyright: 2005-03-17
• Publisher: CRC Press

Summary

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

Preface	v
INTRODUCTION TO BOUNDARY METHODS	xiii
I SELECTED TOPICS IN BOUNDARY ELEMENT METHODS	1	(64)
1 BOUNDARY INTEGRAL EQUATIONS	3	(20)
1.1 Potential Theory in Three Dimensions	3	(3)
1.1.1 Singular Integral Equations	3	(2)
1.1.2 Hypersingular Integral Equations	5	(1)
1.2 Linear Elasticity in Three Dimensions	6	(6)
1.2.1 Singular Integral Equations	6	(2)
1.2.2 Hypersingular Integral Equations	8	(4)
1.3 Nearly Singular Integrals in Linear Elasticity	12	(2)
1.3.1 Displacements at Internal Points Close to the Boundary	12	(1)
1.3.2 Stresses at Internal Points Close to the Boundary	13	(1)
1.4 Finite Parts of Hypersingular Equations	14	(9)
1.4.1 Finite Part of a Hypersingular Integral Collocated at an Irregular Boundary Point	14	(3)
1.4.2 Gradient BIE for 3-D Laplace's Equation	17	(2)
1.4.3 Stress BIE for 3-D Elasticity	19	(1)
1.4.4 Solution Strategy for a HBIE Collocated at an Irregular Boundary Point	20	(3)
2 ERROR ESTIMATION	23	(16)
2.1 Linear Operators	23	(2)
2.2 Iterated HBIE and Error Estimation	25	(7)
2.2.1 Problem 1: Displacement Boundary Conditions	25	(3)
2.2.2 Problem 2: Traction Boundary Conditions	28	(2)
2.2.3 Problem 3: Mixed Boundary Conditions	30	(2)
2.3 Element-Based Error Indicators	32	(1)
2.4 Numerical Examples	33	(6)
2.4.1 Example 1: Lame's Problem of a Thick-Walled Cylinder under Internal Pressure	34	(2)
2.4.2 Example 2: Kirsch's Problem of an Infinite Plate with a Circular Cutout	36	(3)
3 THIN FEATURES	39	(26)
3.1 Exterior BIE for Potential Theory: MEMS	39	(15)
3.1.1 Introduction to MEMS	39	(2)
3.1.2 Electric Field BIEs in a Simply Connected Body	41	(1)
3.1.3 BIES in Infinite Region Containing Two Thin Conducting Plates	41	(5)
3.1.4 Singular and Nearly Singular Integrals	46	(3)
3.1.5 Numerical Results	49	(1)
3.1.6 The Model Problem - a Parallel Plate Capacitor	50	(4)
3.2 BIE for Elasticity: Cracks and Thin Shells	54	(13)
3.2.1 BIES in LEFM	54	(6)
3.2.2 Numerical Implementation of BIES in LEFM	60	(1)
3.2.3 Some Comments on BIEs in LEFM	61	(1)
3.2.4 BIEs for Thin Shells	62	(3)
II THE BOUNDARY CONTOUR METHOD	65	(68)
4 LINEAR ELASTICITY	67	(26)
4.1 Surface and Boundary Contour Equations	67	(11)
4.1.1 Basic Equations	67	(1)
4.1.2 Interpolation Functions	68	(3)
4.1.3 Boundary Elements	71	(2)
4.1.4 Vector Potentials	73	(1)
4.1.5 Final BCM Equations	74	(2)
4.1.6 Global Equations and Unknowns	76	(1)
4.1.7 Surface Displacements, Stresses, and Curvatures	76	(2)
4.2 Hypersingular Boundary Integral Equations	78	(4)
4.2.1 Regularized Hypersingular BIE	78	(1)
4.2.2 Regularized Hypersingular BCE	78	(2)
4.2.3 Collocation of the HBCE at an Irregular Surface Point	80	(2)
4.3 Internal Displacements and Stresses	82	(3)
4.3.1 Internal Displacements	82	(1)
4.3.2 Displacements at Internal Points Close to the Bounding Surface	82	(1)
4.3.3 Internal Stresses	83	(1)
4.3.4 Stresses at Internal Points Close to the Bounding Surface	84	(1)
4.4 Numerical Results	85	(8)
4.4.1 Surface Displacements from the BCM and the HBCM	85	(2)
4.4.2 Surface Stresses	87	(3)
4.4.3 Internal Stresses Relatively Far from the Bounding Surface	90	(1)
4.4.4 Internal Stresses Very Close to the Bounding Surface	90	(3)
5 SHAPE SENSITIVITY ANALYSIS	93	(22)
5.1 Sensitivities of Boundary Variables	93	(6)
5.1.1 Sensitivity of the BIE	93	(1)
5.1.2 The Integral Ik	94	(2)
5.1.3 The Integral Jk	96	(2)
5.1.4 The BCM Sensitivity Equation	98	(1)
5.2 Sensitivities of Surface Stresses	99	(2)
5.2.1 Method One	100	(1)
5.2.2 Method Two	100	(1)
5.2.3 Method Three	100	(1)
5.2.4 Method Four	101	(1)
5.3 Sensitivities of Variables at Internal Points	101	(5)
5.3.1 Sensitivities of Displacements	101	(2)
5.3.2 Sensitivities of Displacement Gradients and Stresses	103	(3)
5.4 Numerical Results: Hollow Sphere	106	(4)
5.4.1 Sensitivities on Sphere Surface	107	(1)
5.4.2 Sensitivities at Internal Points	108	(2)
5.5 Numerical Results: Block with a Hole	110	(5)
5.5.1 Geometry and Mesh	110	(2)
5.5.2 Internal Stresses	112	(1)
5.5.3 Sensitivities of Internal Stresses	112	(3)
6 SHAPE OPTIMIZATION	115	(10)
6.1 Shape Optimization Problems	115	(1)
6.2 Numerical Results	116	(9)
6.2.1 Shape Optimization of a Fillet	116	(2)
6.2.2 Optimal Shapes of Ellipsoidal Cavities Inside Cubes	118	(4)
6.2.3 Remarks	122	(3)
7 ERROR ESTIMATION AND ADAPTIVITY	125	(8)
7.1 Hypersingular Residuals as Local Error Estimators	125	(1)
7.2 Adaptive Meshing Strategy	126	(1)
7.3 Numerical Results	127	(8)
7.3.1 Example One - Short Clamped Cylinder under Tension	127	(3)
7.3.2 Example Two - the Lame Problem for a Hollow Cylinder	130	(3)
III THE BOUNDARY NODE METHOD	133	(70)
8 SURFACE APPROXIMANTS	135	(16)
8.1 Moving Least Squares (MLS) Approximants	135	(4)
8.2 Surface Derivatives	139	(2)
8.3 Weight Functions	141	(1)
8.4 Use of Cartesian Coordinates	142	(9)
8.4.1 Hermite Type Approximation	142	(1)
8.4.2 Variable Basis Approximation	143	(8)
9 POTENTIAL THEORY AND ELASTICITY	151	(24)
9.1 Potential Theory in Three Dimensions	151	(14)
9.1.1 BNM: Coupling of BIE with MLS Approximants	151	(4)
9.1.2 HBNM: Coupling of HBIE with MLS Approximants	155	(1)
9.1.3 Numerical Results for Dirichlet Problems on a Sphere	156	(9)
9.2 Linear Elasticity in Three Dimensions	165	(10)
9.2.1 BNM: Coupling of BIE with MLS Approximants	165	(2)
9.2.2 HBNM: Coupling of HBIE with MLS Approximants	167	(1)
9.2.3 Numerical Results	168	(7)
10 ADAPTIVITY FOR 3-D POTENTIAL THEORY	175	(18)
10.1 Hypersingular and Singular Residuals	175	(2)
10.1.1 The Hypersingular Residual	175	(1)
10.1.2 The Singular Residual	176	(1)
10.2 Error Estimation and Adaptive Strategy	177	(3)
10.2.1 Local Residuals and Errors - Hypersingular Residual Approach	178	(1)
10.2.2 Local Residuals and Errors - Singular Residual Approach	178	(1)
10.2.3 Cell Refinement Criterion	179	(1)
10.2.4 Global Error Estimation and Stopping Criterion	179	(1)
10.3 Progressively Adaptive Solutions: Cube Problem	180	(8)
10.3.1 Exact Solution	181	(1)
10.3.2 Initial Cell Configuration #1 (54 Surface Cells)	181	(1)
10.3.3 Initial Cell Configuration #2 (96 Surface Cells)	182	(6)
10.4 One-Step Adaptive Cell Refinement	188	(5)
10.4.1 Initial Cell Configuration #1 (54 Surface Cells)	190	(1)
10.4.2 Initial Cell Configuration #2 (96 Surface Cells)	191	(2)
11 ADAPTIVITY FOR 3-D LINEAR ELASTICITY	193	(10)
11.1 Hypersingular and Singular Residuals	193	(1)
11.1.1 The Hypersingular Residual	193	(1)
11.1.2 The Singular Residual	194	(1)
11.2 Error Estimation and Adaptive Strategy	194	(1)
11.2.1 Local Residuals and Errors - Hypersingular Residual Approach	194	(1)
11.2.2 Local Residuals and Errors - Singular Residual Approach	195	(1)
11.2.3 Cell Refinement Global Error Estimation and Stopping Criterion	195	(1)
11.3 Progressively Adaptive Solutions: Pulling a Rod	195	(3)
11.3.1 Initial Cell Configuration	197	(1)
11.3.2 Adaptivity Results	197	(1)
11.4 One-Step Adaptive Cell Refinement	198	(5)
Bibliography	203	(16)
Index	219