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.

• 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

Engineers and scientists in mechanical engineering, civil engineering, applied mechanics, materials science, computational mechanics, aerospace engineering; research students, and researchers in advanced numerical analysis of solid mechanics and heat transfer in applied mathematics, mechanics, and material engineering

Chapter 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 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 treatment
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
Chapter 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
Chapter 3 Basics of fundamental solutions and radial basis functions
3.1 Introduction
3.2 Basic concept of fundamental solutions
3.2.1 Partial differential operator
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
Chapter 4 Meshless analysis for thin beam bending problems
4.1 Introduction
4.2 Solution procedure
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 discontinue loading
4.3 Results and discussions
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
Chapter 5 Meshless analysis for thin plate bending problems
5.1 Introduction
5.2 Fundamental solutions for thin plate bending
5.3 Solution 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
Chapter 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
6.3.2.2 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 Thermo-mechanical systems in FGMs
6.5.1.1 Strain-displacement relationship
6.5.1.2 Constitutive equations
6.5.1.3 Static equilibrium equations
6.5.1.4 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
Chapter 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
Chapter 8 Meshless analysis for 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 for 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
8.4.2.4 Isotropic functionally graded finite strip
8.5 Remarks
References
Appendix A Derivatives of function in terms of radial variable r
Appendix B Transformations
B.1 Coordinate transformation
B.2 Vector transformation
B.3 Stress transformation
Appendix C Derivatives of approximated particular solutions in inhomogeneous plane elasticity