Methods of Fundamental Solutions in Solid Mechanics

Chapter 1 Overview of meshless methods1.1 Why we need meshless methods1.2 Review of meshless Methods1.3 Basic ideas of the method of fundamental solutions1.3.1 Weighted residual method1.3.2 Method of fundamental solutions1.4 Application to two-dimensional Laplace problem1.4.1 Problem description1.4.2 MFS formulation1.4.3 Program structure and source code1.4.3.1 Input data1.4.3.2 Computation of coefficient matrix1.4.3.3 Solving the resulting system of linear equations1.4.3.4 Source code1.4.4 Numerical experiments1.4.4.1 Circular disk1.4.4.2 Interior region surrounded by a complex curve1.4.4.3 Biased hollow circle1.5 Some limitations for implementing the method of fundamental solutions1.5.1 Dependence of fundamental solutions1.5.2 Location of source points1.5.3 Ill-conditioning treatment1.5.4 Inhomogeneous problems1.5.5 Multiple domain problems1.6 Extended method of fundamental solutions1.7 Outline of the bookReferencesChapter 2 Mechanics of solids and structures2.1 Introduction2.2 Basic physical quantities2.2.1 Displacement components2.2.2 Stress components2.2.3 Strain components2.3 Equations for three-dimensional solids2.3.1 Strain-displacement relation2.3.2 Equilibrium equations2.3.3 Constitutive equations2.3.4 Boundary conditions2.4 Equations for plane solids2.4.1 Plane stress and plane strain2.4.2 Governing equations2.4.3 Boundary conditions2.5 Equations for Euler-Bernoulli beams2.5.1 Deformation mode2.5.2 Governing equations2.5.3 Boundary conditions2.5.4 Continuity requirements2.6 Equations for thin plates2.6.1 Deformation mode2.6.2 Governing equations2.6.3 Boundary conditions2.7 Equations for piezoelectricity2.7.1 Governing equations2.7.2 Boundary conditions2.8 RemarksReferencesChapter 3 Basics of fundamental solutions and radial basis functions3.1 Introduction3.2 Basic concept of fundamental solutions3.2.1 Partial differential operator3.2.2 Fundamental solutions3.3 Radial basis function interpolation3.3.1 Radial basis functions3.3.2 Radial basis function interpolation3.4 RemarksReferencesChapter 4 Meshless analysis for thin beam bending problems4.1 Introduction4.2 Solution procedure4.2.1 Homogeneous solution4.2.2 Particular solution4.2.3 Approximated full solution4.2.4 Construction of solving equations4.2.5 Treatment of discontinue loading4.3 Results and discussions4.3.1 Statically indeterminate beam under uniformly distributed loading4.3.2 Statically indeterminate beam under middle concentrated load4.3.3 Cantilever beam with end concentrated load4.4 RemarksReferencesChapter 5 Meshless analysis for thin plate bending problems5.1 Introduction5.2 Fundamental solutions for thin plate bending5.3 Solution procedure for thin plate bending5.3.1 Particular solution5.3.2 Homogeneous solution5.3.3 Approximated full solution5.3.4 Construction of solving equations5.4 Results and discussion5.4.1 Square plate with simple-supported edges5.4.2 Square plate on a Winkler elastic foundation5.5 RemarksReferencesChapter 6 Meshless analysis for two-dimensional elastic problems6.1 Introduction6.2 Fundamental solutions for two-dimensional elasticity6.3 Solution procedure for homogeneous elasticity6.3.1 Solution procedure6.3.2 Program structure and source code6.3.2.1 Input data6.3.2.2 Computation of coefficient matrix6.3.2.3 Solving the resulting system of linear equations6.3.2.4 Source code6.3.3 Results and discussion6.3.3.1 Thick-walled cylinder under internal pressure6.3.3.2 Infinite domain with circular hole subjected to a far-field remote tensile6.4 Solution procedure for inhomogeneous elasticity6.4.1 Particular solution6.4.2 Homogeneous solution6.4.3 Approximated full solution6.4.4 Results and discussion6.4.4.1 Rotating disk with high speed6.4.4.2 Symmetric thermoelastic problem in a long cylinder6.5 Further analysis for functionally graded solids6.5.1 Concept of functionally graded material6.5.2 Thermo-mechanical systems in FGMs6.5.1.1 Strain-displacement relationship6.5.1.2 Constitutive equations6.5.1.3 Static equilibrium equations6.5.1.4 Boundary conditions6.5.3 Solution procedure for FGMs6.5.3.1 Analog equation method6.5.3.2 Particular solution6.5.3.3 Homogeneous solution6.5.3.4 Approximated full solution6.5.3.5 Construction of solving equations6.5.4 Numerical experiments6.5.4.1 Functionally graded hollow circular plate under radial internal pressure6.5.4.2 Functionally graded elastic beam under sinusoidal transverse load6.5.4.3 Symmetrical thermoelastic problem in a long functionally graded cylinder6.6 RemarksReferencesChapter 7 Meshless analysis for plane piezoelectric problems7.1 Introduction7.2 Fundamental solutions for plane piezoelectricity7.3 Solution procedure for plane piezoelectricity7.4 Results and discussion7.4.1 Simple tension of a piezoelectric prism7.4.2 An infinite piezoelectric plane with a circular hole under remote tension7.4.3 An infinite piezoelectric plane with a circular hole subject to internal pressure7.5 RemarksReferencesChapter 8 Meshless analysis for heat transfer in heterogeneous media8.1 Introduction8.2 Basics of heat transfer8.2.1 Energy balance equation8.2.2 Fourier’s law8.2.3 Governing equation8.2.4 Boundary conditions8.2.5 Thermal conductivity matrix8.3 Solution procedure for general steady-state heat transfer8.3.1 Solution procedure8.3.1.1 Analog equation method8.3.1.2 Particular solution8.3.1.3 Homogeneous solution8.3.1.4 Approximated full solution8.3.1.5 Construction of solving equations8.3.2 Results and discussion8.3.2.1 Isotropic heterogeneous square plate8.3.2.2 Isotropic heterogeneous circular disc8.3.2.3 Anisotropic homogeneous circular disc8.3.2.4 Anisotropic heterogeneous hollow ellipse8.4 Solution procedure of transient heat transfer8.4.1 Solution procedure8.4.1.1 Time marching scheme8.4.1.2 Approximated full solution8.4.1.3 Construction of solving equations8.4.2 Results and discussion8.4.2.1 Isotropic homogeneous square plate with sudden temperature jump8.4.2.2 Isotropic homogeneous square plate with nonzero initial condition8.4.2.3 Isotropic homogeneous square plate with cone-shaped solution8.4.2.4 Isotropic functionally graded finite strip8.5 RemarksReferencesAppendix A Derivatives of function in terms of radial variable rAppendix B TransformationsB.1 Coordinate transformationB.2 Vector transformationB.3 Stress transformationAppendix C Derivatives of approximated particular solutions in inhomogeneous plane elasticity