An Introduction to Meshfree Methods And Their Programming

by Liu, G. R.; Gu, Y. T.

• ISBN13: 9781402032288
• ISBN10: 1402032285
• Format: Hardcover
• Copyright: 2005-07-30
• Publisher: Springer Verlag

Summary

This book aims to present meshfree methods in a friendly and straightforward manner, so that beginners can very easily understand, comprehend, program, implement, apply and extend these methods. It provides first the fundamentals of numerical analysis that are particularly important to meshfree methods. Typical meshfree methods, such as EFG, RPIM, MLPG, LRPIM, MWS and collocation methods are then introduced systematically detailing the formulation, numerical implementation and programming. Many well-tested computer source codes developed by the authors are attached with useful descriptions. The application of the codes can be readily performed using the examples with input and output files given in table form. These codes consist of most of the basic meshfree techniques, and can be easily extended to other variations of more complex procedures of meshfree methods. Readers can easily practice with the codes provided to effective learn and comprehend the basics of meshfree methods.

Table of Contents

Preface	xiii
Authors	xix
1 Fundamentals	1	(36)
1.1 Numerical simulation	1	(2)
1.2 Basics of mechanics for solids	3	(11)
1.2.1 Equations for three-dimensional solids	4	(5)
1.2.1.1 Stress components	4	(1)
1.2.1.2 Strain-displacement equations	5	(1)
1.2.1.3 Constitutive equations	6	(1)
1.2.1.4 Equilibrium equations	7	(1)
1.2.1.5 Boundary conditions and initial conditions	8	(1)
1.2.2 Equations for two-dimensional solids	9	(4)
1.2.2.1 Stress components	9	(1)
1.2.2.2 Strain-displacement equation	10	(1)
1.2.2.3 Constitutive equations	11	(1)
1.2.2.4 Equilibrium equations	12	(1)
1.2.2.5 Boundary conditions and initial conditions	12	(1)
1.3 Strong-forms and weak-forms	13	(1)
1.4 Weighted residual method	14	(13)
1.4.1 Collocation method	17	(1)
1.4.2 Subdomain method	18	(1)
1.4.3 Least squares method	19	(1)
1.4.4 Moment method	20	(1)
1.4.5 Galerkin method	20	(1)
1.4.6 Examples	21	(23)
1.4.6.1 Use of the collocation method	23	(1)
1.4.6.2 Use of the subdomain method	23	(1)
1.4.6.3 Use of the least squares method	24	(1)
1.4.6.4 Use of the moment method	24	(1)
1.4.6.5 Use of the Galerkin method	25	(1)
1.4.6.6 Use of more terms in the approximate solution	26	(1)
1.5 Global weak-form for solids	27	(7)
1.6 Local weak-form for solids	34	(2)
1.7 Discussions and remarks	36	(1)
2 Overview of meshfree methods	37	(17)
2.1 Why Meshfree methods	37	(2)
2.2 Definition of Meshfree methods	39	(1)
2.3 Solution procedure of MFree methods	40	(4)
2.4 Categories of Meshfree methods	44	(7)
2.4.1 Classification according to the formulation procedures	45	(2)
2.4.1.1 Meshfree methods based on weak-forms	45	(1)
2.4.1.2 Meshfree methods based on collocation techniques	46	(1)
2.4.1.3 Meshfree methods based on the combination of weak-form and collocation techniques	47	(1)
2.4.2 Classification according to the function approximation schemes	47	(2)
2.4.2.1 Meshfree methods based on the moving least squares approximation	48	(1)
2.4.2.2 Meshfree methods based on the integral representation method for the function approximation	48	(1)
2.4.2.3 Meshfree methods based on the point interpolation method	49	(1)
2.4.2.4 Meshfree methods based on the other meshfree interpolation schemes	49	(1)
2.4.3 Classification according to the domain representation	49	(5)
2.4.3.1 Domain-type meshfree methods.	50	(1)
2.4.3.2 Boundary-type meshfree methods.	50	(1)
2.5 Future development	51	(3)
3 Meshfree shape function construction	54	(91)
3.1 Introduction	54	(6)
3.1.1 Meshfree interpolation/approximation techniques	55	(3)
3.1.2 Support domain	58	(1)
3.1.3 Determination of the average nodal spacing	58	(2)
3.2 Point interpolation methods	60	(37)
3.2.1 Polynomial PIM shape functions	61	(13)
3.2.1.1 Conventional polynomial PIM	61	(6)
3.2.1.2 Weighted least square (WLS) approximation	67	(2)
3.2.1.3 Weighted least square approximation of Hermite-type	69	(5)
3.2.2 Radial point interpolation shape functions	74	(12)
3.2.2.1 Conventional RPIM	74	(7)
3.2.2.2 Hermite-type RPIM	81	(5)
3.2.3 Source code for the conventional RPIM shape functions	86	(11)
3.2.3.1 Implementation issues	86	(2)
3.2.3.2 Program and data structure	88	(2)
3.2.3.3 Examples of RPIM shape functions	90	(7)
3.3 Moving least squares shape functions	97	(17)
3.3.1 Formulation of MLS shape functions	97	(5)
3.3.2 Choice of the weight function	102	(4)
3.3.3 Properties of MLS shape functions	106	(2)
3.3.4 Source code for the MLS shape function	108	(6)
3.3.4.1 Implementation issues	108	(3)
3.3.4.2 Program and data structure	111	(1)
3.3.4.3 Examples of MLS shape functions	111	(3)
3.4 Interpolation error using Meshfree shape functions	114	(8)
3.4.1 Fitting of a planar surface	118	(1)
3.4.2 Fitting of a complicated surface	118	(4)
3.5 Remarks	122	(2)
Appendix	124	(7)
Computer programs	131	(14)
4 Meshfree methods based on global weak-forms	145	(92)
4.1 Introduction	145	(3)
4.2 Meshfree radial point interpolation method	148	(13)
4.2.1 RPIM formulation	148	(7)
4.2.2 Numerical implementation	155	(6)
4.2.2.1 Numerical integration	155	(2)
4.2.2.2 Properties of the stiffness matrix	157	(1)
4.2.2.3 Enforcement of essential boundary conditions	158	(2)
4.2.2.4 Conformability of RPIM	160	(1)
4.3 Element Free Galerkin method	161	(6)
4.3.1 EFG formulation	161	(2)
4.3.2 Lagrange multiplier method for essential boundary conditions	163	(4)
4.4 Source code	167	(10)
4.4.1 Implementation issues	167	(4)
4.4.1.1 Support domain and the influence domain	167	(2)
4.4.1.2 Background cells	169	(1)
4.4.1.3 Method to enforce essential boundary conditions	169	(1)
4.4.1.4 Shape parameters used in RBFs	169	(2)
4.4.2 Program description and data structures	171	(6)
4.5 Example for two-dimensional solids – a cantilever beam	177	(19)
4.5.1 Using MFree_Global.f90	179	(7)
4.5.2 Effects of parameters	186	(7)
4.5.2.1 Parameter effects on RPIM method	187	(4)
4.5.2.2 Parameter effects on EFG method	191	(2)
4.5.3 Comparison of convergence	193	(1)
4.5.4 Comparison of efficiency	194	(2)
4.6 Example for 3D solids	196	(2)
4.7 Examples for geometrically nonlinear problems	198	(3)
4.7.1 Simulation of upsetting of a billet	199	(1)
4.7.2 Simulation of large deflection of a cantilever beam	200	(1)
4.7.3 Simulation of large deflection of a fixed-fixed beam	201	(1)
4.8 MFree2D©	201	(3)
4.9 Remarks	204	(1)
Appendix	205	(14)
Computer programs	219	(18)
5 Meshfree methods based on local weak-forms	237	(73)
5.1 Introduction	237	(2)
5.2 Local radial point interpolation method	239	(11)
5.2.1 LRPIM formulation	239	(7)
5.2.2 Numerical implementation	246	(4)
5.2.2.1 Type of local domains	246	(1)
5.2.2.2 Property of the stiffness matrix	247	(1)
5.2.2.3 Test (weight) function	248	(1)
5.2.2.4 Numerical integration	248	(2)
5.3 Meshless Local Petrov-Galerkin method	250	(4)
5.3.1 MLPG formulation	250	(2)
5.3.2 Enforcement of essential boundary conditions	252	(1)
5.3.3 Commons on the efficiency of MLPG and LRPIM	253	(1)
5.3.3.1 Comparison with FEM	254	(1)
5.3.3.2 Comparison with MFree global weak-form methods	254	(1)
5.4 Source code	254	(8)
5.4.1 Implementation issues	254	(2)
5.4.2 Program description and data structures	256	(6)
5.5 Examples for two dimensional solids – a cantilever beam	262	(17)
5.5.1 The use of the MFree_local.f90	262	(5)
5.5.2 Studies on the effects of parameters	267	(9)
5.5.2.1 Parameters effects on LRPIM	268	(6)
5.5.2.2 Parameter effects on MLPG	274	(2)
5.5.3 Comparison of convergence	276	(2)
5.5.4 Comparison of efficiency	278	(1)
5.6 Remarks	279	(2)
Appendix	281	(11)
Computer programs	292	(18)
6 Meshfree collocation methods	310	(70)
6.1 Introduction	310	(1)
6.2 Techniques for handling derivative boundary conditions	311	(2)
6.3 Polynomial point collocation method for 1D problems	313	(22)
6.3.1 Collocation equations for 1D system equations	313	(10)
6.3.1.1 Problem description	313	(1)
6.3.1.2 Function approximation using MFree shape functions	314	(1)
6.3.1.3 System equation discretization	315	(1)
6.3.1.4 Discretization of Dirichlet boundary condition	316	(1)
6.3.1.5 Discretized system equation with only Dirichlet boundary conditions	316	(1)
6.3.1.6 Discretized system equations with DBCs	317	(6)
6.3.2 Numerical examples for 1D problems	323	(12)
6.4 Stabilization in convection-diffusion problems using MFree methods	335	(8)
6.4.1 Nodal refinement	338	(1)
6.4.2 Enlargement of the local support domain	338	(1)
6.4.3 Total upwind support domain	339	(2)
6.4.4 Adaptive upwind support domain	341	(1)
6.4.5 Biased support domain	342	(1)
6.5 Polynomial point collocation method for 2D problems	343	(9)
6.5.1 PPCM formulation for 2D problems	344	(2)
6.5.2 Numerical examples	346	(6)
6.6 Radial point collocation method for 2D problems	352	(12)
6.6.1 RPCM formulation	352	(1)
6.6.2 RPCM for 2D Poisson equations	352	(2)
6.6.3 RPCM for 2D convection-diffusion problems	354	(10)
6.6.3.1 Steady state convection-diffusion problem	354	(5)
6.6.3.2 Linear dynamic convection-diffusion equations	359	(5)
6.7 RPCM for 2D solids	364	(14)
6.7.1 Hermite-type RPCM	364	(7)
6.7.2 Use of regular grid (RG)	371	(7)
6.8 Remarks	378	(2)
7 Meshfree methods based on local weak form and collocation	380	(74)
7.1 Introduction	380	(1)
7.2 Meshfree collocation and local weak-form methods	381	(3)
7.2.1 Meshfree collocation method	381	(1)
7.2.2 Meshfree weak-form method	382	(1)
7.2.3 Comparisons of meshfree collocation and weak-form methods	383	(1)
7.3 Formulation for 2-D statics	384	(7)
7.3.1 The idea	384	(2)
7.3.2 Local weak-form	386	(1)
7.3.3 Discretized system equations	387	(3)
7.3.4 Numerical implementation	390	(1)
7.3.4.1 Property of stiffness matrix	390	(1)
7.3.4.2 Type of local domains	391	(1)
7.3.4.3 Numerical integration	391	(1)
7.4 Source code	391	(2)
7.4.1 Implementation issues	392	(1)
7.4.2 Program description	392	(1)
7.5 Examples for testing the code	393	(7)
7.6 Numerical examples for 2D elastostatics	400	(10)
7.6.1 1D truss member with derivative boundary conditions	400	(1)
7.6.2 Standard patch test	401	(2)
7.6.3 Higher-order patch test	403	(4)
7.6.4 Cantilever beam	407	(3)
7.6.5 Hole in an infinite plate	410	(1)
7.7 Dynamic analysis for 2-D solids	410	(13)
7.7.1 Strong-form of dynamic analysis	412	(1)
7.7.2 Local weak-form for the dynamic analysis	412	(1)
7.7.3 Discretized formulations for dynamic analysis	413	(3)
7.7.3.1 Free vibration analysis	414	(1)
7.7.3.2 Direct analysis of forced vibration	415	(1)
7.7.4 Numerical examples	416	(7)
7.7.4.1 Free vibration analysis	417	(1)
7.7.4.2 Forced vibration analysis	417	(6)
7.8 Analysis for incompressible flow problems	423	(20)
7.8.1 Simulation of natural convection in an enclosed domain	423	(11)
7.8.1.1 Governing equations and boundary conditions	423	(1)
7.8.1.2 Discretized system equations	424	(3)
7.8.1.3 Numerical results for the problem of natural convection	427	(7)
7.8.2 Simulation of the flow around a cylinder	434	(9)
7.8.2.1 Governing equation and boundary condition	434	(3)
7.8.2.2 Computation procedure	437	(1)
7.8.2.3 Results and discussion	437	(6)
7.9 Remarks	443	(2)
Appendix	445	(5)
Computer programs	450	(4)
Reference	454	(19)
Index	473