Methods of Fundamental Solutions in Solid Mechanics

Methods of Fundamental Solutions in Solid Mechanics
Copyright
Dedication
About the authors
Preface
Acknowledgments
List of abbreviations
1 -
Overview of meshless methods
	1.1 Why we need meshless methods
	1.2 Review of meshless methods
	1.3 Basic ideas of the method of fundamental solutions
		1.3.1 Weighted residual method
		1.3.2 Method of fundamental solutions
	1.4 Application to the two-dimensional Laplace problem
		1.4.1 Problem description
		1.4.2 MFS formulation
		1.4.3 Program structure and source code
			1.4.3.1 Input data
			1.4.3.2 Computation of coefficient matrix
			1.4.3.3 Solving the resulting system of linear equations
			1.4.3.4 Source code
		1.4.4 Numerical experiments
			1.4.4.1 Circular disk
			1.4.4.2 Interior region surrounded by a complex curve
			1.4.4.3 Biased hollow circle
	1.5 Some limitations for implementing the method of fundamental solutions
		1.5.1 Dependence of fundamental solutions
		1.5.2 Location of source points
		1.5.3 Ill-conditioning treatments
			1.5.3.1 Tikhonov regularization method
			1.5.3.2 Singular value decomposition
		1.5.4 Inhomogeneous problems
		1.5.5 Multiple domain problems
	1.6 Extended method of fundamental solutions
	1.7 Outline of the book
	References
2 - Mechanics of solids and structures
	2.1 Introduction
	2.2 Basic physical quantities
		2.2.1 Displacement components
		2.2.2 Stress components
		2.2.3 Strain components
	2.3 Equations for three-dimensional solids
		2.3.1 Strain-displacement relation
		2.3.2 Equilibrium equations
		2.3.3 Constitutive equations
		2.3.4 Boundary conditions
	2.4 Equations for plane solids
		2.4.1 Plane stress and plane strain
		2.4.2 Governing equations
		2.4.3 Boundary conditions
	2.5 Equations for Euler–Bernoulli beams
		2.5.1 Deformation mode
		2.5.2 Governing equations
		2.5.3 Boundary conditions
		2.5.4 Continuity requirements
	2.6 Equations for thin plates
		2.6.1 Deformation mode
		2.6.2 Governing equations
		2.6.3 Boundary conditions
	2.7 Equations for piezoelectricity
		2.7.1 Governing equations
		2.7.2 Boundary conditions
	2.8 Remarks
	References
3 -
Basics of fundamental solutions and radial basis functions
	3.1 Introduction
	3.2 Basic concept of fundamental solutions
		3.2.1 Partial differential operators
		3.2.2 Fundamental solutions
	3.3 Radial basis function interpolation
		3.3.1 Radial basis functions
		3.3.2 Radial basis function interpolation
	3.4 Remarks
	References
4 -
Meshless analysis for thin beam bending problems
	4.1 Introduction
	4.2 Solution procedures
		4.2.1 Homogeneous solution
		4.2.2 Particular solution
		4.2.3 Approximated full solution
		4.2.4 Construction of solving equations
		4.2.5 Treatment of discontinuous loading
	4.3 Results and discussion
		4.3.1 Statically indeterminate beam under uniformly distributed loading
		4.3.2 Statically indeterminate beam under middle-concentrated load
		4.3.3 Cantilever beam with end-concentrated load
	4.4 Remarks
	References
5 -
Meshless analysis for thin plate bending problems
	5.1 Introduction
	5.2 Fundamental solutions for thin plate bending
	5.3 Solutions procedure for thin plate bending
		5.3.1 Particular solution
		5.3.2 Homogeneous solution
		5.3.3 Approximated full solution
		5.3.4 Construction of solving equations
	5.4 Results and discussion
		5.4.1 Square plate with simple-supported edges
		5.4.2 Square plate on a winkler elastic foundation
	5.5 Remarks
	References
6 -
Meshless analysis for two-dimensional elastic problems
	6.1 Introduction
	6.2 Fundamental solutions for two-dimensional elasticity
	6.3 Solution procedure for homogeneous elasticity
		6.3.1 Solution procedure
		6.3.2 Program structure and source code
			6.3.2.1 Input data
				6.3.2.1.1 First DOF
				6.3.2.1.2 Second DOF
			6.3.2.2 Computation of coefficient matrix
			6.3.2.3 Solving the resulting system of linear equations
			6.3.2.4 Source code
		6.3.3 Results and discussion
			6.3.3.1 Thick-walled cylinder under internal pressure
			6.3.3.2 Infinite domain with circular hole subjected to a far-field remote tensile
	6.4 Solution procedure for inhomogeneous elasticity
		6.4.1 Particular solution
		6.4.2 Homogeneous solution
		6.4.3 Approximated full solution
		6.4.4 Results and discussion
			6.4.4.1 Rotating disk with high speed
			6.4.4.2 Symmetric thermoelastic problem in a long cylinder
	6.5 Further analysis for functionally graded solids
		6.5.1 Concept of functionally graded material
		6.5.2 Thermomechanical systems in FGMs
			6.5.2.1 Strain-displacement relationship
			6.5.2.2 Constitutive equations
			6.5.2.3 Static equilibrium equations
			6.5.2.4 Boundary conditions
		6.5.3 Solution procedure for FGMs
			6.5.3.1 Analog equation method
			6.5.3.2 Particular solution
			6.5.3.3 Homogeneous solution
			6.5.3.4 Approximated full solution
			6.5.3.5 Construction of solving equations
		6.5.4 Numerical experiments
			6.5.4.1 Functionally graded hollow circular plate under radial internal pressure
			6.5.4.2 Functionally graded elastic beam under sinusoidal transverse load
			6.5.4.3 Symmetrical thermoelastic problem in a long functionally graded cylinder
	6.6 Remarks
	References
7 -
Meshless analysis for plane piezoelectric problems
	7.1 Introduction
	7.2 Fundamental solutions for plane piezoelectricity
	7.3 Solution procedure for plane piezoelectricity
	7.4 Results and discussion
		7.4.1 Simple tension of a piezoelectric prism
		7.4.2 An infinite piezoelectric plane with a circular hole under remote tension
		7.4.3 An infinite piezoelectric plane with a circular hole subject to internal pressure
	7.5 Remarks
	References
8 -
Meshless analysis of heat transfer in heterogeneous media
	8.1 Introduction
	8.2 Basics of heat transfer
		8.2.1 Energy balance equation
		8.2.2 Fourier's law
		8.2.3 Governing equation
		8.2.4 Boundary conditions
		8.2.5 Thermal conductivity matrix
	8.3 Solution procedure of general steady-state heat transfer
		8.3.1 Solution procedure
			8.3.1.1 Analog equation method
			8.3.1.2 Particular solution
			8.3.1.3 Homogeneous solution
			8.3.1.4 Approximated full solution
			8.3.1.5 Construction of solving equations
		8.3.2 Results and discussion
			8.3.2.1 Isotropic heterogeneous square plate
			8.3.2.2 Isotropic heterogeneous circular disc
			8.3.2.3 Anisotropic homogeneous circular disc
			8.3.2.4 Anisotropic heterogeneous hollow ellipse
	8.4 Solution procedure of transient heat transfer
		8.4.1 Solution procedure
			8.4.1.1 Time marching scheme
			8.4.1.2 Approximated full solution
			8.4.1.3 Construction of solving equations
		8.4.2 Results and discussion
			8.4.2.1 Isotropic homogeneous square plate with sudden temperature jump
			8.4.2.2 Isotropic homogeneous square plate with nonzero initial condition
			8.4.2.3 Isotropic homogeneous square plate with cone-shaped solution
				8.4.2.3.1 Isotropic functionally graded finite strip
	8.5 Remarks
	References
A -
Derivatives of functions in terms of radial variable r
B -
Transformations
	B.1 Coordinate transformation
	B.2 Vector transformation
	B.3 Stress transformation
C -
Derivatives of approximated particular solutions in inhomogeneous plane elasticity
	C.1 Power spline (PS) function
	C.2 Thin plate spline (TPS) function
Index
	A
	B
	C
	D
	E
	F
	G
	H
	I
	K
	L
	M
	N
	O
	P
	Q
	R
	S
	T
	U
	V
	W
	Y