Boundary Element Programming in Mechanics

by Xiao-Wei Gao , Trevor G. Davies

• ISBN13: 9780521773591
• ISBN10: 0521773598
• Format: Multimedia
• Copyright: 2002-03-11
• Publisher: Cambridge University Press

Summary

This monograph describes the application of boundary element methods (BEM) in solid mechanics, beginning with basic theory and then explaining the numerical implementation of BEM in nonlinear stress analysis. In addition, the authors have developed state-of-the-art BEM source code, available for the first time on a CD-ROM included with the book.

Author Biography

Xiao-Wei Gao is Research Associate, Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe. He was formerly Associate Professor, Institute of Applied Mechanics, Ningxia University, People's Republic of China. Dr. Gao is an authority on nonlinear boundary element analysis and is currently developing BEM techniques for commercial aerospace applications Trevor G. Davies is Senior Lecturer, Glasgow University. Dr. Davies's primary expertise is in numerical analysis of nonlinear and dynamic soil-structure interaction using boundary element methods. He was one of the first researchers to tackle the problem of 3D nonlinear boundary element analysis during the late 1970s. He has published extensively and has co-edited the book Boundary Element Techniques in Geomechanics

Table of Contents

Preface	xiii
Legal Matters	xv
PART I. LINEAR PROBLEMS
Introduction	3	(13)
Introduction	3	(1)
A Note on Programming	3	(2)
Mathematical Preliminaries	5	(7)
Historical Sketch	12	(3)
Approximate Methods	12	(1)
BEM in Solid Mechanics	13	(1)
BEM in Elasticity	14	(1)
BEM in Elasto-Plasticity	14	(1)
Closure	15	(1)
Theory of Elasticity	16	(9)
Introduction	16	(1)
Displacements	16	(1)
Stresses	17	(1)
Stress-Strain Relationships	18	(1)
Navier--Cauchy Equations of Equilibrium	19	(1)
Reduced Forms in Two Dimensions	20	(3)
Plane Strain	21	(1)
Plane Stress	22	(1)
Axisymmetry	22	(1)
Anisotropic Materials	23	(1)
Closure	24	(1)
Boundary Integral Equations for Elasticity	25	(9)
Introduction	25	(1)
The Kelvin Fundamental Solution	25	(2)
Betti's Reciprocal Work Theorem	27	(1)
Somigliana Identity	28	(2)
Boundary Integral Equations	30	(1)
Internal Stresses	31	(2)
Closure	33	(1)
Numerical Implementation	34	(33)
Introduction	34	(1)
Boundary Discretization	34	(3)
Interpolation of Field Quantities	37	(1)
Discretized Boundary Integral Equations	38	(1)
Adaptive Integration	39	(8)
Singular Integration	47	(6)
Weakly Singular Integrals in Three Dimensions	47	(2)
Weakly Singular Integrals in Two Dimensions	49	(2)
Strongly Singular Integrals	51	(2)
Evaluation of Boundary Stresses	53	(4)
Symmetry	57	(3)
Corners and Edges	60	(3)
Multiple Regions	63	(1)
System Equation Solution	64	(2)
Closure	66	(1)
The Elastic Program Code	67	(40)
Introduction	67	(1)
Scope of the Program	67	(1)
Program Structure	67	(1)
Global Variables	68	(7)
Global Variables in Module Program Units	68	(7)
Global Variables Passed through Argument Lists	75	(1)
Main Program: BEMECH	75	(1)
Subroutine INPUT-CTR	76	(1)
Subroutine BLOCK-DATA	77	(1)
Subroutine INPUT-EL	78	(1)
Subroutine TREAT-T	79	(1)
Subroutine AXES-COS	80	(1)
Subroutine SHAPEF	81	(1)
Subroutine DSHAPE and DSHAP3D	82	(2)
Subroutine EL-COEFS	84	(1)
Subroutine ADAPTINT	85	(2)
Subroutine CHOSEGP	87	(2)
Subroutines SETGAS and GAUSSV	89	(1)
Subroutines MINDIST and IVSNR 123	90	(1)
Subroutine INT-HG	91	(1)
Subroutine EVAL-HG	92	(1)
Subroutine FORM-HG	93	(1)
Subroutine SINGUGH	94	(1)
Subroutine SIN2DHG	95	(1)
Subroutine SETDSUB	96	(1)
Subroutine BDSTRS	97	(1)
Subroutine BSCOEF	98	(1)
Subroutine SYMTRY	99	(1)
Subroutine HGTOEQS	100	(1)
Subroutine INNERPS	101	(1)
Subroutine EL-SOLVE	102	(1)
Subroutine INVSOLVR	103	(1)
Subroutines OUTPUT and SIGTITL	104	(2)
Closure	106	(1)
Linear Applications	107	(14)
Introduction	107	(1)
Thick-Walled Cylinder under Internal Pressure	107	(4)
Circular Rigid Foundation on a Semi-Infinite Medium	111	(1)
A Three-Dimensional Machine Component	112	(5)
Closure	117	(4)
PART II. NONLINEAR PROBLEMS
Rate-Independent Plasticity Theory	121	(17)
Introduction	121	(1)
Isotropic Yield Criteria	121	(5)
Stress Invariants and Principal Stresses	122	(1)
The Tresca Criterion	123	(1)
The Von Mises Criterion	124	(1)
The Mohr--Coulomb Criterion	124	(1)
The Drucker--Prager Criterion	125	(1)
Principles of Elasto--Plastic Flow	126	(2)
Constitutive Relationships	128	(2)
Isotropic Hardening Materials	130	(4)
Equivalent Uniaxial Yield Criteria	131	(1)
Equivalent Plastic Strain	131	(1)
Explicit Derivations	132	(2)
Kinematic Hardening Materials	134	(1)
Mixed Hardening Materials	135	(1)
Closure	136	(2)
Boundary Integral Equations in Elasto-Plasticity	138	(8)
Introduction	138	(1)
Boundary Integral Equations	138	(2)
Internal Stress Integral Equations	140	(3)
Integration of Strongly Singular Domain Integrals	143	(2)
Closure	145	(1)
Numerical Implementation	146	(16)
Introduction	146	(1)
Domain Discretization	146	(2)
Weakly Singular Domain Integrals	148	(1)
Strongly Singular Domain Integrals	149	(1)
Boundary Stresses - Traction-Recovery Method	150	(3)
System Equations	153	(3)
Initial Stress Representation	153	(1)
Plastic Multiplier Representation	154	(2)
System Equation Solution	156	(4)
Newton-Raphson Method	156	(1)
Transition from Elastic to Elasto-Plastic States	157	(1)
Automatic Incrementation of Boundary Loading	158	(1)
Summary	159	(1)
Closure	160	(2)
The Elasto-Plastic Program Code	162	(25)
Introduction	162	(1)
Scope of the Program	162	(1)
Program Structure	162	(2)
Global Variables	164	(1)
Subroutine INPUT_NL	165	(1)
Subroutine NL_COEFS	166	(1)
Subroutine INT_CELL	167	(1)
Subroutine SIN_CELL	168	(2)
Subroutine CELL_BOUND	170	(1)
Subroutine EVAL_KE	171	(2)
Subroutine INTSUBC	173	(1)
Subroutine PBSCOEF	173	(1)
Subroutine NL_SOLVE	174	(2)
Subroutine PL_FLOW	176	(1)
Subroutine DF_DSIG	177	(1)
Subroutine P_M_ITER	178	(3)
Subroutine MATRICES	181	(1)
Subroutine SIG_SCALE	182	(1)
Subroutine SIGCROSS	183	(1)
Subroutine DF_MATRX	184	(1)
Subroutine UPDATEV	185	(1)
Closure	186	(1)
Nonlinear Applications	187	(16)
Introduction	187	(1)
A Cube Subjected to Uniaxial Tension	187	(4)
A Thick-Walled Cylinder Subjected to Internal Pressure	191	(2)
A Rigid Punch under Plane Strain	193	(3)
A Flexible Square Footing	196	(3)
Multiplanar Tubular DX-Joint	199	(2)
Closure	201	(2)
Epilogue	203	(6)
Review	203	(1)
The Way Forward	203	(6)
Automatic Integration	203	(1)
Computation of Boundary Stresses	204	(1)
Stress-Return Algorithm	204	(1)
System Equation Solver	205	(1)
Local Boundary Conditions	205	(1)
Nonlinear Hardening	206	(1)
Advanced Yield Functions	206	(1)
Finite Strain Elasto-Plasticity	206	(1)
Infinite Boundary Elements	207	(1)
Multiple Regions	207	(2)
Appendix A. Derivation of Kernel Functions	209	(4)
A.1. Derivation of the Strain Kernel	209	(1)
A.2. Derivation of the Stress Kernel	210	(1)
A.3. Derivation of the Traction Kernel	211	(1)
A.4. Kernel Functions for Plane Strain and Plane Stress	211	(2)
Appendix B. Shape Functions	213	(4)
B.1. One-Dimensional Shape Functions	213	(1)
B.2. Two-Dimensional Shape Functions	214	(1)
B.3. Three-Dimensional Shape Functions	215	(2)
Appendix C. Degenerate Elements: Singular Mapping	217	(4)
Appendix D. Elasto-Plastic Flow Theory	221	(7)
D.1. Derivation of the Plastic Flow Rule and Plastic Loading Rule	221	(3)
D.2. Derivations for Kinematic Hardening Materials	224	(1)
D.3. Derivations for Mixed Hardening Materials	225	(1)
D.4. Derivation of the Deformation State Function Γ	226	(2)
Appendix E. Domain Integral Formulations	228	(4)
E.1. Boundary Integral Equations: Initial Strain Formulation	228	(1)
E.2. Analytical Integration of the Strongly Singular Volume Integral	228	(2)
E.3. Interior Stress Equation: Initial Strain Formulation	230	(1)
E.4. Analytical Integration of Eijkl in Two Dimensions	230	(1)
E.5. Analytical Integration: Initial Strain Formulation	231	(1)
Appendix F. Solution of the Nonlinear System Equations	232	(7)
F.1. The Newton-Raphson Iterative Algorithm	232	(1)
F.2. System Equation Solution Strategies	233	(6)
F.2.1. The Initial Stress Iteration Technique	234	(1)
F.2.2. The Implicit Solution Technique	235	(1)
F.2.3. The Variable Stiffness Technique	236	(1)
F.2.4. The Mixed Representation Technique	237	(2)
Appendix G. Elements of Elasto-Plasticity	239	(3)
Appendix H. Description of Input Data	242	(5)
References	247	(6)
Index	253