Methods of Fundamental Solutions in Solid Mechanics

Methods of Fundamental Solutions in Solid Mechanics

1st Edition - June 6, 2019

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  • Authors: Hui Wang, Qing-Hua Qin
  • eBook ISBN: 9780128182840
  • Paperback ISBN: 9780128182833

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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.

Key Features

  • 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

Table of Contents

  • 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 Input data Computation of coefficient matrix Solving the resulting system of linear equations Source code
    1.4.4 Numerical experiments Circular disk Interior region surrounded by a complex curve 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
    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
    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
    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
    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
    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 Input data Computation of coefficient matrix Solving the resulting system of linear equations Source code
    6.3.3 Results and discussion Thick-walled cylinder under internal pressure 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 Rotating disk with high speed 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 Strain-displacement relationship Constitutive equations Static equilibrium equations Boundary conditions
    6.5.3 Solution procedure for FGMs Analog equation method Particular solution Homogeneous solution Approximated full solution Construction of solving equations
    6.5.4 Numerical experiments Functionally graded hollow circular plate under radial internal pressure Functionally graded elastic beam under sinusoidal transverse load Symmetrical thermoelastic problem in a long functionally graded cylinder
    6.6 Remarks
    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
    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 Analog equation method Particular solution Homogeneous solution Approximated full solution Construction of solving equations
    8.3.2 Results and discussion Isotropic heterogeneous square plate Isotropic heterogeneous circular disc Anisotropic homogeneous circular disc Anisotropic heterogeneous hollow ellipse
    8.4 Solution procedure of transient heat transfer
    8.4.1 Solution procedure Time marching scheme Approximated full solution Construction of solving equations
    8.4.2 Results and discussion Isotropic homogeneous square plate with sudden temperature jump Isotropic homogeneous square plate with nonzero initial condition Isotropic homogeneous square plate with cone-shaped solution Isotropic functionally graded finite strip
    8.5 Remarks
    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

Product details

  • No. of pages: 312
  • Language: English
  • Copyright: © Elsevier 2019
  • Published: June 6, 2019
  • Imprint: Elsevier
  • eBook ISBN: 9780128182840
  • Paperback ISBN: 9780128182833

About the Authors

Hui Wang

Dr. HUI WANG was born in Luoyang City of China in 1976. He received his Bachelor degree in Theoretical and Applied Mechanics from Lanzhou University, China in 1999. Subsequently he joined the College of Science as an assistant lecturer at Zhongyuan University of Technology (ZYUT) and spent two years teaching at ZYUT. He earned his Master degree from Dalian University of Technology in 2004 and Doctoral degree from Tianjin University in 2007, both of which are in Solid Mechanics. Since 2007, he has worked at College of Civil Engineering and Architecture, Henan University of Technology as a lecturer. He was promoted to Associate Professor in 2009 and Professor in 2015. From August 2014 to August 2015, he worked at Australian National University (ANU) as a visiting scholar, and then from February 2016 to February 2017, he joined the ANU as a Research Fellow. His research interests include computational mechanics, meshless methods, hybrid finite element method and mechanics of composites. So far, he has authored three academic books by CRC Press and Tsinghua University Press respectively, 7 book chapters and 62 academic journal papers (47 indexed by SCI and 8 indexed by EI). In 2010, He was awarded the Australia Endeavour Award.

Affiliations and Expertise

Professor, Department of Engineering Mechanics, Henan University of Technology, Zhengzhou, China

Qing-Hua Qin

Dr. QINGHUA QIN born in Guilin City of China in 1958 received his Bachelor of Engineering degree in mechanical engineering from Chang An University, China in 1982, earning his Master of Science degree in 1984 and Doctoral degree in 1990 from Huazhong University of Science and Technology (HUST) in China. Both degrees are in applied mechanics. He joined the Department of Mechanics as an associate lecturer at HUST in 1984, and was promoted to lecturer in mechanics in 1987 during his PhD candidature period. After spending 10 years lecturing at HUST, he was awarded the DAAD/K.C. Wong research fellowship in 1994, which enabled him to work at the University of Stuttgart in Germany for nine months. In 1995 he left HUST to take up a postdoctoral research fellowship at Tsinghua University, China, where he worked until 1997. He was awarded a Queen Elizabeth II fellowship in 1997 and a Professorial fellowship in 2002 at the University of Sydney and stayed there till December 2003, both by the Australian Research Council, and is currently working as a professor in the Research School of Engineering (RSE) at the Australian National University, Australia. He was appointed a guest professor at HUST in 2000. He was also awarded a Cheung Kong Professorship at Tianjin University in 2001 from the Ministry of Education of China. He has published more than 300 journal papers and 7 books in the field of applied mechanics.

Affiliations and Expertise

Guest Professor, Research School of Engineering, Australian National University, Acton, Australia

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