An Introduction to Meshfree Methods and Their Programming

The finite difference method (FDM) hasbeen used tosolve differential equation systems for centuries. The FDM works well for problems of simple geometry and was widely used before the invention of the much more efficient, robust finite element method (FEM). FEM is now widely used in handling problems with complex geometry. Currently, we are using and developing even more powerful numerical techniques aiming to obtain more accurate approximate solutions in a more convenient manner for even more complex systems. The meshfree or meshless method is one such phenomenal development in the past decade, and is the subject of this book. There are many MFree methods proposed so far for different applications. Currently, three monographs on MFree methods have been published. Mesh Free Methods, Moving Beyond the Finite Element Method d by GR Liu (2002) provides a systematic discussion on basic theories, fundamentals for MFree methods, especially on MFree weak-form methods. It provides a comprehensive record of well-known MFree methods and the wide coverage of applications of MFree methods to problems of solids mechanics (solids, beams, plates, shells, etc.) as well as fluid mechanics. The Meshless Local Petrov-Galerkin (MLPG) Method d by Atluri and Shen (2002) provides detailed discussions of the meshfree local Petrov-Galerkin (MLPG) method and itsvariations. Formulations and applications of MLPG are well addressed in their book.

1 Fundamentals: 1.1 Numerical simulation; 1.2 Basics of mechanics for solids; 1.3 Strong-forms and weak-forms; 1.4 Weighted residual method; 1.5 Global weak-form for solids; 1.6 Local weak-form for solids ;1.7 Discussions and remarks. 2 Overview of meshfree methods: 2.1 Why MeshFree methods; 2.2 Definition of MeshFree methods; 2.3 Solution procedure of MFree methods; 2.4 Categories of Meshfree methods; 2.5 Future development. 3 Meshfree shape function construction: 3.1 Introduction; 3.2 Point interpolation methods; 3.3 Moving least squares shape functions; 3.5 Remarks; Appendix; Computer programs. 4 MFree methods based on global weak-forms: 4.1 Introduction; 4.2 Meshfree radial point interpolation method ; 4.3 Element Free Galerkin method; 4.4 Source code; 4.5 Example for two-dimensional solids – a cantilever beam; 4.6 Example for 3D solids; 4.7 Examples for geometrically nonlinear problems; 4.8 MFree2DÓ ; 4.9 Remarks; Appendix; Computer programs. 5. MFree methods based on local weak-forms: 5.1 Introduction; 5.2 Local radial point interpolation method; 5.3 Meshless Local Petrov-Galerkin method ; 5.4 Source code; 5.5 Examples for two dimensional solids – a cantilever beam; 5.6 Remarks ; Appendix; Computer programs. 6 Meshfree collocation methods: 6.1 Introduction; 6.2 Techniques for handling derivative boundary conditions; 6.3 Polynomial point collocation method for 1D problems; 6.4 Stabilization in convection-diffusion problems using MFree methods; 6.5 Polynomial point collocation method for 2D problems; 6.6 Radial point collocation method for 2D problems; 6.7 RPCM for 2D solids; 6.8 Remarks. 7 MFree methods based on local weak form and collocation : 7.1 Introduction; 7.2 Meshfree collocation and local weak-form methods; 7.3 Formulation for 2-D statics; 7.4 Source code; 7.5 Examples for testing the code; 7.6 Numerical examples for 2D elastostatics; 7.7 Dynamic analysis for 2-D solids; 7.8 Analysis for incompressible flow problems; 7.9 Remarks;Appendix; Computer programs.Reference. Index.