Methods of Fundamental Solutions in Solid Mechanics

__Methods of Fundamental Solutions in Solid Mechanics__ presents the fundamentals of continuum mechanics, the foundational concepts of the MFS, and methodologies and applications to various engineering problems. Eight chapters give an overview of meshless methods, the mechanics of solids and structures, the basics of fundamental solutions and radical basis functions, meshless analysis for thin beam bending, thin plate bending, two-dimensional elastic, plane piezoelectric problems, and heat transfer in heterogeneous media. The book presents a working knowledge of the MFS that is aimed at solving real-world engineering problems through an understanding of the physical and mathematical characteristics of the MFS and its applications. Cover 1 Methods of Fundamental Solutions in Solid Mechanics 2 Copyright 3 Dedication 4 About the authors 215 Preface 7 Acknowledgments 284 List of abbreviations 9 Part I: Fundamentals of meshless methods 10 1 - Overview of meshless methods 11 1.1 Why we need meshless methods 11 1.2 Review of meshless methods 14 1.3 Basic ideas of the method of fundamental solutions 16 1.3.1 Weighted residual method 16 1.3.2 Method of fundamental solutions 18 1.4 Application to the two-dimensional Laplace problem 21 1.4.1 Problem description 21 1.4.2 MFS formulation 22 1.4.3 Program structure and source code 26 1.4.3.1 Input data 27 1.4.3.2 Computation of coefficient matrix 27 1.4.3.3 Solving the resulting system of linear equations 27 1.4.3.4 Source code 28 1.4.4 Numerical experiments 32 1.4.4.1 Circular disk 32 1.4.4.2 Interior region surrounded by a complex curve 37 1.4.4.3 Biased hollow circle 41 1.5 Some limitations for implementing the method of fundamental solutions 42 1.5.1 Dependence of fundamental solutions 45 1.5.2 Location of source points 45 1.5.3 Ill-conditioning treatments 47 1.5.3.1 Tikhonov regularization method 47 1.5.3.2 Singular value decomposition 47 1.5.4 Inhomogeneous problems 49 1.5.5 Multiple domain problems 49 1.6 Extended method of fundamental solutions 51 1.7 Outline of the book 55 References 56 2 - Mechanics of solids and structures 131 2.1 Introduction 60 2.2 Basic physical quantities 61 2.2.1 Displacement components 61 2.2.2 Stress components 61 2.2.3 Strain components 64 2.3 Equations for three-dimensional solids 64 2.3.1 Strain-displacement relation 64 2.3.2 Equilibrium equations 65 2.3.3 Constitutive equations 66 2.3.4 Boundary conditions 67 2.4 Equations for plane solids 69 2.4.1 Plane stress and plane strain 69 2.4.2 Governing equations 71 2.4.3 Boundary conditions 72 2.5 Equations for Euler–Bernoulli beams 72 2.5.1 Deformation mode 73 2.5.2 Governing equations 75 2.5.3 Boundary conditions 78 2.5.4 Continuity requirements 79 2.6 Equations for thin plates 80 2.6.1 Deformation mode 80 2.6.2 Governing equations 82 2.6.3 Boundary conditions 84 2.7 Equations for piezoelectricity 87 2.7.1 Governing equations 88 2.7.2 Boundary conditions 95 2.8 Remarks 95 References 96 3 - Basics of fundamental solutions and radial basis functions 11 3.1 Introduction 98 3.2 Basic concept of fundamental solutions 98 3.2.1 Partial differential operators 98 3.2.2 Fundamental solutions 101 3.3 Radial basis function interpolation 104 3.3.1 Radial basis functions 104 3.3.2 Radial basis function interpolation 108 3.4 Remarks 111 References 111 Part II: Applications of the meshless method 113 4 - Meshless analysis for thin beam bending problems 114 4.1 Introduction 114 4.2 Solution procedures 115 4.2.1 Homogeneous solution 116 4.2.2 Particular solution 116 4.2.3 Approximated full solution 117 4.2.4 Construction of solving equations 118 4.2.5 Treatment of discontinuous loading 119 4.3 Results and discussion 121 4.3.1 Statically indeterminate beam under uniformly distributed loading 121 4.3.2 Statically indeterminate beam under middle-concentrated load 124 4.3.3 Cantilever beam with end-concentrated load 124 4.4 Remarks 129 References 129 5 - Meshless analysis for thin plate bending problems 131 5.1 Introduction 131 5.2 Fundamental solutions for thin plate bending 132 5.3 Solutions procedure for thin plate bending 134 5.3.1 Particular solution 135 5.3.2 Homogeneous solution 136 5.3.3 Approximated full solution 137 5.3.4 Construction of solving equations 138 5.4 Results and discussion 139 5.4.1 Square plate with simple-supported edges 139 5.4.2 Square plate on a winkler elastic foundation 141 5.5 Remarks 145 References 145 6 - Meshless analysis for two-dimensional elastic problems 284 6.1 Introduction 148 6.2 Fundamental solutions for two-dimensional elasticity 151 6.3 Solution procedure for homogeneous elasticity 154 6.3.1 Solution procedure 154 6.3.2 Program structure and source code 158 6.3.2.1 Input data 159 6.3.2.1.1 First DOF 159 6.3.2.1.2 Second DOF 160 6.3.2.2 Computation of coefficient matrix 160 6.3.2.3 Solving the resulting system of linear equations 160 6.3.2.4 Source code 161 6.3.3 Results and discussion 166 6.3.3.1 Thick-walled cylinder under internal pressure 166 6.3.3.2 Infinite domain with circular hole subjected to a far-field remote tensile 173 6.4 Solution procedure for inhomogeneous elasticity 176 6.4.1 Particular solution 179 6.4.2 Homogeneous solution 184 6.4.3 Approximated full solution 186 6.4.4 Results and discussion 186 6.4.4.1 Rotating disk with high speed 186 6.4.4.2 Symmetric thermoelastic problem in a long cylinder 189 6.5 Further analysis for functionally graded solids 192 6.5.1 Concept of functionally graded material 192 6.5.2 Thermomechanical systems in FGMs 195 6.5.2.1 Strain-displacement relationship 196 6.5.2.2 Constitutive equations 197 6.5.2.3 Static equilibrium equations 198 6.5.2.4 Boundary conditions 199 6.5.3 Solution procedure for FGMs 199 6.5.3.1 Analog equation method 199 6.5.3.2 Particular solution 200 6.5.3.3 Homogeneous solution 201 6.5.3.4 Approximated full solution 202 6.5.3.5 Construction of solving equations 203 6.5.4 Numerical experiments 204 6.5.4.1 Functionally graded hollow circular plate under radial internal pressure 204 6.5.4.2 Functionally graded elastic beam under sinusoidal transverse load 208 6.5.4.3 Symmetrical thermoelastic problem in a long functionally graded cylinder 209 6.6 Remarks 212 References 213 7 - Meshless analysis for plane piezoelectric problems 215 7.1 Introduction 215 7.2 Fundamental solutions for plane piezoelectricity 216 7.3 Solution procedure for plane piezoelectricity 223 7.4 Results and discussion 226 7.4.1 Simple tension of a piezoelectric prism 226 7.4.2 An infinite piezoelectric plane with a circular hole under remote tension 229 7.4.3 An infinite piezoelectric plane with a circular hole subject to internal pressure 234 7.5 Remarks 72 References 238 8 - Meshless analysis of heat transfer in heterogeneous media 215 8.1 Introduction 241 8.2 Basics of heat transfer 241 8.2.1 Energy balance equation 241 8.2.2 Fourier's law 244 8.2.3 Governing equation 244 8.2.4 Boundary conditions 245 8.2.5 Thermal conductivity matrix 245 8.3 Solution procedure of general steady-state heat transfer 247 8.3.1 Solution procedure 247 8.3.1.1 Analog equation method 248 8.3.1.2 Particular solution 248 8.3.1.3 Homogeneous solution 250 8.3.1.4 Approximated full solution 250 8.3.1.5 Construction of solving equations 251 8.3.2 Results and discussion 251 8.3.2.1 Isotropic heterogeneous square plate 251 8.3.2.2 Isotropic heterogeneous circular disc 255 8.3.2.3 Anisotropic homogeneous circular disc 257 8.3.2.4 Anisotropic heterogeneous hollow ellipse 258 8.4 Solution procedure of transient heat transfer 263 8.4.1 Solution procedure 264 8.4.1.1 Time marching scheme 264 8.4.1.2 Approximated full solution 265 8.4.1.3 Construction of solving equations 266 8.4.2 Results and discussion 266 8.4.2.1 Isotropic homogeneous square plate with sudden temperature jump 267 8.4.2.2 Isotropic homogeneous square plate with nonzero initial condition 270 8.4.2.3 Isotropic homogeneous square plate with cone-shaped solution 273 8.4.2.3.1 Isotropic functionally graded finite strip 273 8.5 Remarks 278 References 278 Appendix A - Derivatives of functions in terms of radial variable r 280 Appendix B - Transformations 284 B.1 Coordinate transformation 284 B.2 Vector transformation 285 B.3 Stress transformation 286 Appendix C - Derivatives of approximated particular solutions in inhomogeneous plane elasticity 288 C.1 Power spline (PS) function 288 C.2 Thin plate spline (TPS) function 289 Index 291 A 98 B 291 C 134 D 291 E 292 F 292 G 292 H 292 I 293 K 293 L 293 M 293 N 293 O 293 P 293 Q 294 R 294 S 294 T 294 U 295 V 295 W 295 Y 295 Back Cover 296 Cover......Page 1 Methods of Fundamental Solutions in Solid Mechanics......Page 2 Copyright......Page 3 Dedication......Page 4 8 - Meshless analysis of heat transfer in heterogeneous media......Page 240 Preface......Page 7 B.1 Coordinate transformation......Page 284 List of abbreviations......Page 9 Part I: Fundamentals of meshless methods ......Page 10 3 - Basics of fundamental solutions and radial basis functions......Page 11 1.2 Review of meshless methods......Page 14 1.3.1 Weighted residual method......Page 16 1.3.2 Method of fundamental solutions......Page 18 1.4.1 Problem description......Page 21 1.4.2 MFS formulation......Page 22 1.4.3 Program structure and source code......Page 26 1.4.3.3 Solving the resulting system of linear equations......Page 27 1.4.3.4 Source code......Page 28 1.4.4.1 Circular disk......Page 32 1.4.4.2 Interior region surrounded by a complex curve......Page 37 1.4.4.3 Biased hollow circle......Page 41 1.5 Some limitations for implementing the method of fundamental solutions......Page 42 1.5.2 Location of source points......Page 45 1.5.3.2 Singular value decomposition......Page 47 1.5.5 Multiple domain problems......Page 49 1.6 Extended method of fundamental solutions......Page 51 1.7 Outline of the book......Page 55 References......Page 56 5 - Meshless analysis for thin plate bending problems......Page 60 2.2.2 Stress components......Page 61 2.3.1 Strain-displacement relation......Page 64 2.3.2 Equilibrium equations......Page 65 2.3.3 Constitutive equations......Page 66 2.3.4 Boundary conditions......Page 67 2.4.1 Plane stress and plane strain......Page 69 2.4.2 Governing equations......Page 71 7.5 Remarks......Page 72 2.5.1 Deformation mode......Page 73 2.5.2 Governing equations......Page 75 2.5.3 Boundary conditions......Page 78 2.5.4 Continuity requirements......Page 79 2.6.1 Deformation mode......Page 80 2.6.2 Governing equations......Page 82 2.6.3 Boundary conditions......Page 84 2.7 Equations for piezoelectricity......Page 87 2.7.1 Governing equations......Page 88 2.8 Remarks......Page 95 References......Page 96 A......Page 98 3.2.2 Fundamental solutions......Page 101 3.3.1 Radial basis functions......Page 104 3.3.2 Radial basis function interpolation......Page 108 References......Page 111 Part II: Applications of the meshless method ......Page 113 4.1 Introduction......Page 114 4.2 Solution procedures......Page 115 4.2.2 Particular solution......Page 116 4.2.3 Approximated full solution......Page 117 4.2.4 Construction of solving equations......Page 118 4.2.5 Treatment of discontinuous loading......Page 119 4.3.1 Statically indeterminate beam under uniformly distributed loading......Page 121 4.3.3 Cantilever beam with end-concentrated load......Page 124 References......Page 129 5.1 Introduction......Page 131 5.2 Fundamental solutions for thin plate bending......Page 132 5.3 Solutions procedure for thin plate bending......Page 134 5.3.1 Particular solution......Page 135 5.3.2 Homogeneous solution......Page 136 5.3.3 Approximated full solution......Page 137 5.3.4 Construction of solving equations......Page 138 5.4 Results and discussion......Page 139 8.2.5 Thermal conductivity matrix......Page 245 5.4.2 Square plate on a winkler elastic foundation......Page 141 References......Page 145 6.1 Introduction......Page 148 6.2 Fundamental solutions for two-dimensional elasticity......Page 151 6.3.1 Solution procedure......Page 154 6.3.2 Program structure and source code......Page 158 6.3.2.1.1 First DOF......Page 159 6.3.2.3 Solving the resulting system of linear equations......Page 160 6.3.2.4 Source code......Page 161 6.3.3.1 Thick-walled cylinder under internal pressure......Page 166 6.3.3.2 Infinite domain with circular hole subjected to a far-field remote tensile......Page 173 6.4 Solution procedure for inhomogeneous elasticity......Page 176 6.4.1 Particular solution......Page 179 6.4.2 Homogeneous solution......Page 184 6.4.4.1 Rotating disk with high speed......Page 186 6.4.4.2 Symmetric thermoelastic problem in a long cylinder......Page 189 6.5.1 Concept of functionally graded material......Page 192 6.5.2 Thermomechanical systems in FGMs......Page 195 6.5.2.1 Strain-displacement relationship......Page 196 6.5.2.2 Constitutive equations......Page 197 6.5.2.3 Static equilibrium equations......Page 198 6.5.3.1 Analog equation method......Page 199 6.5.3.2 Particular solution......Page 200 6.5.3.3 Homogeneous solution......Page 201 6.5.3.4 Approximated full solution......Page 202 6.5.3.5 Construction of solving equations......Page 203 6.5.4.1 Functionally graded hollow circular plate under radial internal pressure......Page 204 6.5.4.2 Functionally graded elastic beam under sinusoidal transverse load......Page 208 6.5.4.3 Symmetrical thermoelastic problem in a long functionally graded cylinder......Page 209 6.6 Remarks......Page 212 References......Page 213 7.1 Introduction......Page 215 7.2 Fundamental solutions for plane piezoelectricity......Page 216 7.3 Solution procedure for plane piezoelectricity......Page 223 7.4.1 Simple tension of a piezoelectric prism......Page 226 7.4.2 An infinite piezoelectric plane with a circular hole under remote tension......Page 229 7.4.3 An infinite piezoelectric plane with a circular hole subject to internal pressure......Page 234 References......Page 238 8.2.1 Energy balance equation......Page 241 8.2.3 Governing equation......Page 244 8.3.1 Solution procedure......Page 247 8.3.1.2 Particular solution......Page 248 8.3.1.4 Approximated full solution......Page 250 8.3.2.1 Isotropic heterogeneous square plate......Page 251 8.3.2.2 Isotropic heterogeneous circular disc......Page 255 8.3.2.3 Anisotropic homogeneous circular disc......Page 257 8.3.2.4 Anisotropic heterogeneous hollow ellipse......Page 258 8.4 Solution procedure of transient heat transfer......Page 263 8.4.1.1 Time marching scheme......Page 264 8.4.1.2 Approximated full solution......Page 265 8.4.2 Results and discussion......Page 266 8.4.2.1 Isotropic homogeneous square plate with sudden temperature jump......Page 267 8.4.2.2 Isotropic homogeneous square plate with nonzero initial condition......Page 270 8.4.2.3.1 Isotropic functionally graded finite strip......Page 273 References......Page 278 Appendix A - Derivatives of functions in terms of radial variable r......Page 280 B.2 Vector transformation......Page 285 B.3 Stress transformation......Page 286 C.1 Power spline (PS) function......Page 288 C.2 Thin plate spline (TPS) function......Page 289 D......Page 291 H......Page 292 P......Page 293 T......Page 294 Y......Page 295 Back Cover......Page 296 Methods of Fundamental Solutions in Solid Mechanics presents the fundamentals of continuum mechanics, the foundational concepts of the MFS, and methodologies and applications to various engineering problems. Eight chapters give an overview of meshless methods, the mechanics of solids and structures, the basics of fundamental solutions and radical basis functions, meshless analysis for thin beam bending, thin plate bending, two-dimensional elastic, plane piezoelectric problems, and heat transfer in heterogeneous media. The book presents a working knowledge of the MFS that is aimed at solving real-world engineering problems through an understanding of the physical and mathematical characteristics of the MFS and its applications. Explains foundational concepts for the method of fundamental solutions (MFS) for the advanced numerical analysis of solid mechanics and heat transfer Extends the application of the MFS for use with complex problems Considers the majority of engineering problems, including beam bending, plate bending, elasticity, piezoelectricity and heat transfer Gives detailed solution procedures for engineering problems Offers a practical guide, complete with engineering examples, for the application of the MFS to real-world physical and engineering challenges Since the basic concept behind the method of fundamental solutions (MFS) was developed primarily by V. D. Kupradze and M. A. Alexidze in 1964, the meshless MFS has become an effective tool for the solution of a large variety of physical and engineering problems, such as potential problems, elastic problems, crack problems, fluid problems, piezoelectric problems, antiplane problems, inverse problems, and free-boundary problems. More recently, it has been extended to deal with inhomogeneous partial differential equations, partial differential equations with variable coefficients, and time-dependent problems, by introducing radial basis function interpolation (RBF) for particular solutions caused by inhomogeneous terms.