Development of strong form methods with applications in computational mechanics

... performance of strong form methods; To formulate strong form schemes for complex problems of practical applications; To develop powerful and versatile commercial software packages of strong form. .. mappings In contrast, the formulation procedure of the strong form of meshfree methods is relatively simple and straightforward, compared with the meshfree weak form methods The meshfree strong form. .. development of strong form methods is rather sluggish Available literatures for the strong form methods are still limited Therefore, the strong form methods are now in great demand Strong form methods

Founded 1905 DEVELOPMENT OF STRONG FORM METHODS WITH APPLICATIONS IN COMPUTATIONAL MECHANICS ZHANG JIAN (M. Eng., National University of Singapore, Singapore) (B. Eng., Dalian University of Technology, P. R. China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2008 Acknowledgements I would like to express my sincerest gratitude and appreciation to my supervisors, Professor Liu Gui-Rong, Professor Lam Khin Yong and Assistant Professor Li Hua. Professor Liu’s sharp thinking has always saved me from going into wrong directions. I would like to thank him for his dedicated support, invaluable guidance and continuous encouragement throughout the duration of this thesis. His influence on me is far beyond this thesis and will benefit me in my whole life. Also, I would like to thank Professor Lam and Assistant Professor Li for their sage advice, great patience and support in the entire candidature. Their dedication to research and vast knowledge inspire me in my future work. Many thanks are conveyed to my fellow colleagues and friends, Dr. Kee Buck Tong, Bernard, Mr. Xu Xiangguo, George, Dr. Zhang Guiyong, Dr. Deng Bin, Mr. Song Chengxiang, Mr. Zhou Chengen, Dr. Dai Keyang, Dr. Zhao Xin, Dr. Gu Yuantong, Dr. Wu Tianyun, Dr. Huynh Dinh Bao Phuong, Mr. Nguyen Thoi Trung, Mr. Khin Zaw, Mr. Li Zirui and Dr. Cheng Yuan. The constructive suggestions, helpful discussions and valuable perspectives among our group definitely help to improve the quality of my research work. Most importantly, these guys have made my life during my Ph.D. candidature a more meaningful one. To my family, I appreciate their warm care and strong support. Especially to my beloved wife, Ms. Sun Guoyuan, without her endless encouragement, support and understanding, and sacrifice of all her time to take care of me, it is impossible for me i to finish this thesis. This piece of work is also a present for our daughter, Zhang Mingjia Elysia, who was born on 26 January 2008. Last not the least, I am very grateful to the National University of Singapore for granting me the Research Scholarship and other support throughout my Ph.D. candidature. Many thanks are also conveyed to Centre for Advanced Computations in Engineering Science (ACES) and Department of Mechanical Engineering for their material support to every aspect of this work. ii Table of Contents Acknowledgements ....................................................................................................... i Table of Contents ....................................................................................................... iii Summary ...................................................................................................................... ix Nomenclature ........................................................................................................... xiii List of Figures ............................................................................................................ xvi List of Tables............................................................................................................. xxv Chapter 1 Introduction ................................................................................................ 1 1.1 Background .......................................................................................................... 1 1.2 Literature Review ................................................................................................. 4 1.2.1 The classification of meshfree methods ........................................................ 5 1.2.2 Meshfree methods based on weak forms ...................................................... 6 1.2.3 Meshfree methods based on strong forms .................................................... 7 1.3 Objectives ........................................................................................................... 10 1.4 Organization of the Thesis .................................................................................. 14 Chapter 2 Radial Point Interpolation Based Finite Difference Method ............... 17 2.1 Introduction ........................................................................................................ 17 2.2 Function Approximation..................................................................................... 19 2.2.1 Smoothed particle hydrodynamics (SPH) approximation .......................... 19 2.2.2 Reproducing kernel particle method (RKPM) approximation.................... 20 2.2.3 Moving least squares (MLS) approximation .............................................. 20 iii 2.2.4 Partition of unity methods ........................................................................... 22 2.2.5 Polynomial point interpolation ................................................................... 22 2.2.6 Radial point interpolation ........................................................................... 27 2.3 Radial Point Collocation Method (RPCM) ........................................................ 35 2.3.1 Formulation ................................................................................................. 35 2.3.2 Issues in RPCM........................................................................................... 37 2.4 Radial Point Interpolation Based Finite Difference Method .............................. 39 2.5 Numerical Examples .......................................................................................... 42 2.5.1 Poisson’s equation ...................................................................................... 43 2.5.2 Internal pressurized hollow cylinder ........................................................... 45 2.5.3 Infinite plate with a circular hole ................................................................ 46 2.5.4 Bridge pier .................................................................................................. 47 2.5.5 Triangle dam of complicated shape ............................................................ 48 2.6 Parameter Study.................................................................................................. 49 2.6.1 Number of local supporting nodes .............................................................. 49 2.6.2 Relations between the numbers of grid points and field nodes .................. 50 2.7 Remarks .............................................................................................................. 51 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation .............. 74 3.1 Introduction ........................................................................................................ 74 3.2 Gradient Smoothing Method (GSM) .................................................................. 75 3.2.1 Gradient smoothing ..................................................................................... 76 3.2.2 Smoothing domains .................................................................................... 78 iv 3.2.3 Discretization schemes................................................................................ 79 3.2.4 Formulae for derivative approximation ...................................................... 80 3.2.4.1 Two-point quadrature schemes .............................................................. 81 3.2.4.2 One-point quadrature schemes .............................................................. 83 3.2.4.3 Directional correction ............................................................................ 84 3.3 Analyses of Discretization Stencil ...................................................................... 85 3.3.1 Basic principles for stencil assessment ....................................................... 86 3.3.2 Stencils for approximated gradients............................................................ 87 3.3.2.1 Uniform Cartesian mesh ........................................................................ 87 3.3.2.2 Equilateral triangular mesh ................................................................... 88 3.3.3 Stencils for approximated Laplace operator ............................................... 88 3.3.3.1 Uniform Cartesian mesh ........................................................................ 88 3.3.3.2 Equilateral triangular mesh ................................................................... 89 3.3.4 Truncation errors ......................................................................................... 90 3.4 Application and Validation of GSM ................................................................... 90 3.4.1 The governing equations ............................................................................. 91 3.4.2 Evaluation of numerical errors.................................................................... 93 3.4.3 Types of mesh ............................................................................................. 94 3.4.4 The role of directional correction ............................................................... 94 3.4.5 Comparison among four favorable schemes ............................................... 94 3.4.6 Robustness to irregularity of meshes .......................................................... 97 3.5 Remarks .............................................................................................................. 98 v Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems......... 113 4.1 Introduction ...................................................................................................... 113 4.2 Convergence Study of the GSM ....................................................................... 114 4.3 Numerical Examples ........................................................................................ 116 4.3.1 Cantilever beam ........................................................................................ 116 4.3.2 Infinite plate with a circular hole .............................................................. 118 4.3.3 Bridge pier ................................................................................................ 119 4.3.4 An automotive part: connecting rod ......................................................... 120 4.4 Remarks ............................................................................................................ 121 Chapter 5 Adaptive Analyses for Solids using the GSM ...................................... 138 5.1 Introduction ...................................................................................................... 138 5.2 Adaptive Strategy ............................................................................................. 141 5.2.1 Error indicator ........................................................................................... 141 5.2.2 Refinement procedure and stopping criterion ........................................... 142 5.3 Numerical Examples ........................................................................................ 143 5.3.1 Patch test ................................................................................................... 143 5.3.2 Poisson’s equation with a sharp peak ....................................................... 144 5.3.3 Infinite plate with a circular hole .............................................................. 147 5.3.4 Short cantilever plate ................................................................................ 148 5.3.5 L -shaped plate ........................................................................................ 151 5.3.6 Mode-I crack problem............................................................................... 152 5.3.7 Singular loading problem .......................................................................... 154 vi 5.4 Remarks ............................................................................................................ 155 Chapter 6 Vibration Analyses of 2-D Solids using the GSM ................................ 185 6.1 Introduction ...................................................................................................... 185 6.2 The Governing Equations of 2-D Elastodynamics ........................................... 185 6.3 Free Vibration Analysis .................................................................................... 186 6.3.1 Strong form formulation ........................................................................... 186 6.3.2 Numerical results ...................................................................................... 188 6.3.2.1 A cantilever beam ................................................................................ 188 6.3.2.2 A variable cross-section beam ............................................................. 189 6.3.2.3 A shear wall ......................................................................................... 189 6.4 Forced Vibration Analysis ................................................................................ 189 6.4.1 Direct analysis of forced vibration ............................................................ 190 6.4.2 Numerical results ...................................................................................... 192 6.5 Remarks ............................................................................................................ 193 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction..202 7.1 Introduction ...................................................................................................... 202 7.2 Linearly Weighted Gradient Smoothing Method (LWGSM) ........................... 202 7.2.1 Gradient smoothing functions ................................................................... 203 7.2.2 Determination of coefficients ................................................................... 204 7.2.3 Approximation of spatial derivatives ........................................................ 206 7.2.3.1 Approximation of 1st-order derivatives (gradients) ............................ 206 7.2.3.2 Approximation of 2nd-order derivatives (Laplace operator) .............. 207 vii 7.3 Relations between GSM and LWGSM............................................................. 208 7.3.1 The formulation ........................................................................................ 208 7.3.2 Treatment of boundary conditions ............................................................ 210 7.4 Numerical Tests ................................................................................................ 211 7.4.1 Full model ................................................................................................. 211 7.4.2 Half model ................................................................................................ 212 7.5 Remarks ............................................................................................................ 213 Chapter 8 Conclusions ............................................................................................. 220 8.1 Concluding Remarks ........................................................................................ 220 8.2 Recommendations for Future Research ........................................................... 223 References ................................................................................................................. 226 Publications Arising From Thesis ........................................................................... 239 viii Summary Meshfree methods have been actively studied and many techniques are developed aiming to overcome some drawbacks in the conventional numerical methods, such as the finite difference method (FDM) and the finite element method (FEM). Among the meshfree methods, the strong form methods using local nodes possess good potential to become popular alternative numerical methods and most attractive feature to facilitate the implementation for adaptive analysis. This is because the concept of the strong form methods is very simple, and its formulation procedure is straightforward. Neither formulation procedure nor construction of shape function requires numerical integration. However, the development of reliable strong form methods using local nodes remains much challenging, mainly due to the stability issues. Currently, most of the reliable strong form methods are still restricted for structured grids and regular domains. The instability is a crucial issue that limits the applications of strong form methods, especially in the adaptive analysis. The solution with strong form method is usually not stable and hence often less accurate than solution using weak form method. The primary objective of the present work is, therefore, to develop new strong form methods that are stable, so that the features of strong form methods, such as simplicity, stability and accuracy, can be realized for adaptive and dynamic analyses in various problems of computational mechanics. As the first part of this work, a novel radial point interpolation based finite difference method (RFDM) is proposed, in which the radial point interpolation using ix local irregular nodes is used together with the conventional finite difference procedure to achieve both the adaptivity to irregular domain and the stability in the solution that is often encountered in the collocation methods. Several numerical examples are presented to demonstrate the accuracy and stability of the RFDM for problems with complex shapes and regular and extremely irregular nodes. Also, a numerical study on the effects of the parameters for RFDM is conducted. In the second part of this work, as the main achievement of this thesis, a gradient smoothing method (GSM) is developed and applied systematically in computational mechanics. The theoretical aspects of the gradient smoothing method are first exploited with focus on the principle of gradient smoothing and its numerical procedure to solve partial differential equations. Stencil analyses of different types of discretization schemes for spatial partial differential terms are carried out from points of views of both efficiency and accuracy. The compactness of stencil and positivity of the coefficients of supporting nodes are concerned in the analyses. The gradient smoothing method has been successfully explored in the following aspects: • GSM for static analyses of solid mechanics The GSM is applied to static analyses of solid mechanics problems. The gradient smoothing operations are utilized to develop the first- and second-order derivative approximations by successively computing the weights for a set of nodal points surrounding a node of interest. Using the approximated derivatives, the strong form of governing system equations can be simply collocated at each scattered node in the problem domain. The x computational accuracy, efficiency and stability of the present method with regular and irregular nodes are demonstrated through extensive numerical examples. In comparison with other well-established numerical approaches such as the finite element method (FEM), the proposed GSM produces encouraging results. • GSM for adaptive analyses of computational mechanics The GSM is further developed for the adaptive analyses. It can effectively overcome the instability issue while retaining the strong form feature of simplicity in formulation procedures which is particularly suitable for adaptive analysis. In this thesis, a posteriori error indicator based on residual of the governing equation is adopted. By evaluating the residual of the governing equation for each triangular cell in the problem domain, error indicator effectively identifies the necessary regions to be refined. Simple refinement procedure using Delaunay diagram is adopted in the adaptive process. Compared with the well-known finite element method, the GSM for adaptive procedure demonstrates good reliability and performs well in several solid mechanics problems including singularities and concentrated loading. • GSM for dynamic analyses of solids and structures The free and forced vibrations analyses of two-dimensional solids and structures are also conducted using the GSM. The governing equations of elastodynamics are discretized with the strong form of GSM. The validity, accuracy and stability of the present GSM for dynamic analyses are well xi demonstrated through intensive numerical investigations. • Linearly weighted gradient smoothing method --- a further step from the GSM Moving beyond the GSM, a linearly weighted gradient smoothing method (LWGSM) has been devised with piecewise linear smoothing functions for gradient smoothing operation. The relations between GSM and LWGSM are derived theoretically and numerically. It is very interesting to find that LWGSM and GSM (Scheme VIII) have resulted in the identical solutions. Some numerical tests are conducted to show the properties of different schemes within the LWGSM. xii Nomenclature a Coefficient vector A Area of the domain b Body force vector c Damping coefficient C Damping matrix dc Characteristic length (average nodal spacing) ee Error norm in energy eu Error norm for variable u e∂u ∂x Error norm for derivative ∂u ∂x E Young’s modulus f ,F Force vector G Shear modulus I Moment of inertia of section K Stiffness matrix L(), B() Differential operator m Mass density M Mass matrix n Unit outward normal vector n Number of nodes in a local support domain P Loading xiii p(x ) Polynomial basis function Pm Polynomial moment matrix q Parameter of MQ radial basis function q Eigenvector r Distance R (x ) Radial basis function RQ Moment matrix of radial basis function t Specified traction vector U Displacement vector Us Displacement vector of local support domain u Field variable uc Field variable at the centoid of a triangular cell um Field variable at the midpoint of a mesh-edge uh Approximation of field function u u Specified displacement vector ) w Weight function x , y ,x Spatial coordinates αc Parameter of MQ radial basis function δ Kronecker delta function ∇ Divergence operator ΔS x , ΔS y Length component of a domain edge Δt Time step xiv ε Strain tensor ε con Convergence error index ηg Estimated global residual norm ηj Estimated local residual norm η mg Maximum value of η g throughout the adaptation κg Tolerant coefficient of the estimated global residual norm κl Local refinement coefficient Γ Problem boundary Γt Natural (Neumann) boundary Γu Essential (Dirichlet) boundary γ Mesh irregularity Ο() Truncation error φi Shape function component Φ Shape function vector υ Poisson’s ratio θ Rotation σ Stress tensor ω Frequency Ω Problem domain Ωi Smoothing domain for field node i () ( L ) , () ( R ) Pointers to the left- and right-side domain faces with an edge xv List of Figures Fig. 2.1 Pascal’s triangles of monomials for two-dimensional space .......................... 55 Fig. 2.2 Local support domains used in meshfree methods ......................................... 55 Fig. 2.3 A problem governed by PDEs in domain Ω . ................................................ 56 Fig. 2.4 Background grids for finite difference used in the RFDM ............................ 56 Fig. 2.5 100 regular field nodes ( • ) and 441 finite difference grid points ( × ) ........... 57 Fig. 2.6 Distribution of 121 extremely irregular nodes for Poisson’s equation........... 57 Fig. 2.7 Result along the line of x = 0.5 for Poisson’s equation ............................... 58 Fig. 2.8 Result along the line of y = 0.5 for Poisson’s equation .............................. 58 Fig. 2.9 Node distributions: (a) 50; (b) 100; (c) 200 and (d) 400 nodes...................... 59 Fig. 2.10 Error norms of solution for Poisson’s equation ............................................ 59 Fig. 2.11 Hollow cylinder subjected to internal pressure ............................................ 60 Fig. 2.12 Node distribution for the hollow cylinder .................................................... 60 Fig. 2.13 Radial displacement u r along the line of y = x in the hollow cylinder .. 61 Fig. 2.14 Circumferential stress σ θ along the line of y = x in the hollow cylinder ..................................................................................................................... 61 Fig. 2.15 Radial stress σ r along the line of y = x in the hollow cylinder ............. 62 Fig. 2.16 Node distributions in the hollow cylinder: (a) 200; (b) 400; (c) 800 nodes. 62 Fig. 2.17 Error norms of of radial displacement ur for hollow cylinder .................. 63 Fig. 2.18 Quarter model of the infinite plate with a circular hole subjected to a unidirectional tensile load .......................................................................... 63 xvi Fig. 2.19 Node distribution: 366 nodes ( • ) & 987 points (intersections of the dashed) ...................................................................................................................... 64 Fig. 2.20 Normal stress σ xx along the edge of x = 0 in the plate. .......................... 64 Fig. 2.21 A bridge subjected to a uniformly distributed pressure on the top. .............. 65 Fig. 2.22 Nodal distribution in the bridge model: 386 field nodes (dots) and 995 grid points (intersections of dashed lines). .......................................................... 65 Fig. 2.23 Distribution of normal stress σ yy in the bridge: (a) RFDM; (b) ANSYS. . 66 Fig. 2.24 A triangle dam subjected to uniformly distributed pressure on the surface. 67 Fig. 2.25 Node distributions in the triangle dam: (a) 334 and (b) 4462 field nodes.... 68 Fig. 2.26 Displacements along the line of x = 8 : (a) x -direction; (b) y -direction. 69 Fig. 2.27 Normal stress σ yy distribution: (a) RFDM; (b) ANSYS; (c) Reference .... 70 Fig. 2.28 Distribution of a set of 100 randomly scattered nodes in a square domain.. 71 Fig. 2.29 Error norms of the field variable u computed by RFDM based on RBFs using different numbers of local supporting nodes...................................... 71 Fig. 2.30 CPU time required by RFDM based on RBFs using different numbers of local supporting nodes ................................................................................ 72 Fig. 2.31 Condition numbers of the coefficient matrix of the RFDM based on RBFs using different numbers of local supporting nodes ..................................... 72 Fig. 2.32 Optimal grid points for Poisson’s equation .................................................. 73 Fig. 2.33 Optimal grid points for internal pressurized hollow cylinder ...................... 73 Fig. 3.1 Illustration of triangle cells and gradient smoothing domains defined in GSM ...................................................................................................................... 104 xvii Fig. 3.2 Stencils for approximated gradients ( ∂u i ∂u i , ) on uniform Cartesian mesh ∂x ∂y ..................................................................................................................... 104 Fig. 3.3 The stencil for approximated gradients ( ∂u i ∂u i , ) on equilateral triangular ∂x ∂y mesh (Identical for I, II, III, IV, V, VI, VII and VIII). ................................. 105 Fig. 3.4 Stencils for the approximated Laplace operator ( ∂ 2 ui ∂ 2ui + 2 ) on uniform ∂x 2 ∂y Cartesian mesh. ........................................................................................... 105 ∂ 2 ui ∂ 2ui Fig. 3.5 Stencils for a Laplace operator ( 2 + 2 ) discretized onto equilateral ∂x ∂y triangles. ...................................................................................................... 106 Fig. 3.6 Contour plots of exact solutions to the two Poisson’s problems .................. 106 Fig. 3.7 Profile plot of convergence history .............................................................. 107 Fig. 3.8 Representative meshes under investigation.................................................. 107 Fig. 3.9 Contours of relative errors on Cartesian mesh in the first Poisson’s problem ..................................................................................................................... 108 Fig. 3.10 Profiles of computational accuracy based on uniform Cartesian mesh...... 108 Fig. 3.11 Profiles of computational accuracy based on right triangular mesh ........... 109 Fig. 3.12 Profile of computational accuracy based on regular triangular mesh ........ 110 Fig. 3.13 Triangular meshes with various irregularity............................................... 111 Fig. 3.14 Overlapped cells in the computational domain ( γ = 0.16). ........................ 111 Fig. 3.15 Contours of solutions to Poisson’s equations discretized onto irregular meshes. ..................................................................................................... 112 xviii Fig. 3.16 Numerical errors in the GSM solutions (Scheme II and VII) to the second Poisson problem with respect to irregularity of meshes ............................ 112 Fig. 4.1 Node distribution of Poisson’s equation: (a) 50; (b) 200; (c) 882 and (d) 3528 elements........................................................................................................ 124 Fig. 4.2 Comparison of convergence rate and accuracy between GSM and FEM for Poisson’s equation with regular nodes: (a) eu and (b) e∂u / ∂x ................... 125 Fig. 4.3 Irregular nodes of Poisson’s equation: (a) 58; (b) 222; (c) 894 and (d) 3632 elements ....................................................................................................... 126 Fig. 4.4 Comparison of convergence rate and accuracy between GSM and FEM for Poisson’s equation with irregular nodes: (a) eu and (b) e∂u / ∂x ................ 127 Fig. 4.5 Cantilever beam subjected to a parabolic load at the free end ..................... 128 Fig. 4.6 Domain discretization of cantilever beam: (a) nodes and (b) elements ....... 128 Fig. 4.7 Deflection of cantilever beam along the line y = 0 computed using the same mesh (480 triangular elements) for GSM and FEM ........................... 129 Fig. 4.8 Normal stress σ xx along the line x = L / 2 computed using GSM and FEM ...................................................................................................................... 129 Fig. 4.9 Shear stress τ xy along the line x = L / 2 computed using GSM and FEM ..................................................................................................................... 130 Fig. 4.10 Quarter model of the infinite plate with a circular hole subjected to a unidirectional tensile load ........................................................................ 130 Fig. 4.11 Quarter model of the infinite plate: (a) nodes and (b) elements ................. 131 Fig. 4.12 Normal stress σ xx along the edge of x = 0 in a plate with a central hole xix subjected to a unidirectional tensile load ................................................... 131 Fig. 4.13 A bridge pier subjected to a uniformly distributed pressure on the top ..... 132 Fig. 4.14 Half model of the bridge pier: (a) nodes and (b) element .......................... 132 Fig. 4.15 Displacement in y -direction along the line x = 0 ................................... 133 Fig. 4.16 Displacement in y -direction along the line y = 30 ................................ 133 Fig. 4.17 Displacement in y -direction along the line y = 15 ................................. 134 Fig. 4.18 Normal stress σ yy along the line y = 15 ................................................ 134 Fig. 4.19 An automotive part: the connecting rod ..................................................... 135 Fig. 4.20 Half model of the connecting rod: (a) node distribution and (b) element distribution ................................................................................................ 136 Fig. 4.21 Displacement in x -direction along the line y = 0 ................................... 136 Fig. 4.22 Distribution of normal stresses along the line y = 0 : (a) σ xx and (b) σ yy (GSM uses 2877 triangular elements while ANSYS adopts a very fine triangular mesh to get the reference solution) ........................................... 137 Fig. 5.1 Residual evaluated at the center ( ) of a triangular cell. ............................. 160 Fig. 5.2 Illustration of the refinement procedure ....................................................... 160 Fig. 5.3 Patches of five nodes in the essential-patch-test .......................................... 161 Fig. 5.4 (a) Patches for the natural-patch-test: a uniform axial traction along the right end of the patch; (b) Patch with 35 regular nodes; (c) Patch with 35 irregular nodes ............................................................................................................ 162 Fig. 5.5 Three-dimensional plots of the exact solution to the Poisson’s equation with a sharp peak: (a) u ; (b) ∂u ∂u and (c) . .................................................... 163 ∂x ∂y xx Fig. 5.6 Node distributions of uniform refinement for Poisson’s equation with a sharp peak at the center. ......................................................................................... 164 Fig. 5.7 Adaptive nodes from the 2nd to 5th step for solving Poisson’s equation .... 165 Fig. 5.8 Estimated global residual at each adaptive step for Poisson’s equation....... 165 Fig. 5.9 Comparison of error and convergence rate between uniform and adaptive refinements for solving Poisson’s equation with a sharp peak ................... 166 Fig. 5.10 Approximated values of field function u along the line y = 0.5 at the first and fifth steps .................................................................................... 166 Fig. 5.11 The three-dimensional plots of adaptive GSM solutions for Poisson’s equation with a sharp peak at the final adaptive step: (a) u ; (b) ∂u ; (c) ∂x ∂u ............................................................................................................ 167 ∂y Fig. 5.12 Quarter model of the infinite plate with a circular hole ............................. 168 Fig. 5.13 Nodes of uniform refinement for infinite plate: from 39 to 1513 nodes. ... 168 Fig. 5.14 Node distributions of adaptive refinement at the 3rd and 6th steps for the quarter model of infinite plate with a circular hole ................................... 169 Fig. 5.15 Estimated global residual at each adaptive step for the infinite plate ........ 169 Fig. 5.16 Comparison of error norm of displacement u x between uniform and adaptive refinements for infinite plate with a circular hole ..................... 170 Fig. 5.17 Normal stress σ xx along x = 0 at the 3rd and 6th steps ........................ 170 Fig. 5.18 A short cantilever plate subjected to a uniformly distributed pressure ...... 171 Fig. 5.19 Node distributions of ‘uniform’ refinement for short cantilever plate ....... 171 Fig. 5.20 Node distributions of adaptive refinement at the 3rd and 6th steps for short xxi cantilever plate ........................................................................................... 172 Fig. 5.21 Estimated global residual at each adaptive step for short cantilever plate. 172 Fig. 5.22 Comparison of displacement u y (1,0) for short cantilever plate between GSM and FEM with uniform and adaptive refinements .......................... 173 Fig. 5.23 Comparison of computed strain enegy for short cantilever plate between GSM and FEM with uniform and adaptive refinements .......................... 173 Fig. 5.24 Comparison of error and convergence in energy norm for short cantilever plate between GSM and FEM with uniform and adaptive refinements. ... 174 Fig. 5.25 Comparison of condition number of coefficient matrix for short cantilever plate between GSM and FEM with uniform and adaptive refinements .... 174 Fig. 5.26 L -shaped plate subjected to a tensile load in the horizontal direction ..... 175 Fig. 5.27 Selected node distributions of uniform refinement for L -shaped plate.... 175 Fig. 5.28 Node distributions of adaptive refinement at the 3rd and 5th steps for L -shaped plate ......................................................................................... 176 Fig. 5.29 Comparison of computed strain energy between uniform and adaptive refinement for L -shaped plate................................................................ 176 Fig. 5.30 Comparison of error and convergence rate in energy norm between uniform and adaptive refinements for L -shaped plate. .......................................... 177 Fig. 5.31 Mode-I crack problem: (a) geometry; (b) half model with boundary conditions................................................................................................. 177 Fig. 5.32 Selected node distributions of uniform refinement for Mode-I crack ........ 178 Fig. 5.33 Node distributions of adaptive refinement at the 3rd, 5th, 7th and 9th steps xxii for Mode-I crack problem .......................................................................... 179 Fig. 5.34 Comparison of error in displacement in y -direction between uniform and adaptive refinements for Mode-I crack problem ....................................... 180 Fig. 5.35 Comparison of strain energy between uniform and adaptive refinement... 180 Fig. 5.36 Comparison of error and convergence rate in energy norm between uniform and adaptive refinements for Mode-I crack problem................................. 181 Fig. 5.37 A square solid subjected to a singular loading at the center of the top edge ................................................................................................................... 181 Fig. 5.38 Node distributions of uniform refinement for singular loading problem ... 182 Fig. 5.39 Node distributions of adaptive refinement at the 6th and 11th steps for singular loading problem .......................................................................... 182 Fig. 5.40 Displacement u x ( x,5) between uniform and adaptive refinements......... 183 Fig. 5.41 Displacement u y ( x,5) between uniform and adaptive refinements........... 183 Fig. 5.42 Comparison of computed strain energy between uniform and adaptive refinement for singular loading problem ................................................. 184 Fig. 6.1 A cantilever beam ......................................................................................... 197 Fig. 6.2 Nodal distribution of the beam: (a) 63 nodes and (b) 306 nodes ................. 197 Fig. 6.3 Eigenmodes for the cantilever beam obtained using the GSM .................... 198 Fig. 6.4 A cantilever beam with variable cross-sections (a) and its mesh (b) ........... 199 Fig. 6.5 A shear wall with four openings ................................................................... 200 Fig. 6.6 Displacement u y at the middle point of free end using different time steps ..................................................................................................................... 200 xxiii Fig. 6.7 Transient displacement u y at the middle point of the free end of the beam using the Newmark method ( δ = 0.5 and β = 0.25 ) ............................... 201 Fig. 7.1 Linearly weighted smoothing functions for different types of gradient smoothing domains: mGSD, cGSD and nGSD ......................................... 218 Fig. 7.2 The schematic of a linearly weighted smoothing domain and its contained sub-triangles ................................................................................................ 218 Fig. 7.3 Schematic of treatment of natural boundary conditions............................... 219 Fig. 7.4 The half model of a Poisson’s equation........................................................ 219 Fig. 7.5 Contour plots of relative errors using linear interpolation (LI) and gradient smoothing (GS) in the LWGSM ................................................................. 219 xxiv List of Tables Table 2.1 Typical radial basis functions with dimensionless shape parameters. ......... 53 Table 2.2 Computed results of Poisson’s equation. ..................................................... 53 Table 2.3 Error norms of solution for Poisson’s equation ........................................... 53 Table 2.4 Error norms of solution for internal pressurized hollow cylinder ............... 54 Table 2.5 Optimal grid points for Poisson’s equation ................................................. 54 Table 2.6 Optimal grid points for internal pressurized hollow cylinder. ..................... 54 Table 3.1 Spatial discretization schemes for approximating derivatives .................. 100 Table 3.2 Truncation errors in the approximation of first-derivatives in the GSM ... 101 Table 3.3 Truncation errors in the approximation of the Laplace operator in the GSM .................................................................................................................. 101 Table 3.4 Comparison of accuracy by Schemes I and II in the first Poisson’s problem ................................................................................................................... 102 Table 3.5 Comparison of numerical errors approximated on regular triangular mesh for the first Poisson problem with favorable schemes ............................. 102 Table 3.6 Comparison of allowable maximum time step and numerical error for irregular triangular meshes. ..................................................................... 103 Table 4.1 Relative errors of Poisson’s equation with Dirichlet boundary conditions computed using the same sets of regularly distributed nodes for GSM and FEM ......................................................................................................... 123 Table 4.2 Relative errors of Poisson’s equation with Neumann boundary conditions xxv computed using the same sets of irregularly distributed nodes for GSM and FEM .......................................................................................................... 123 Table 4.3 Comparison of the CPU time computed using GSM and FEM ................ 123 Table 5.1 Error norms of displacements for essential-patch-test .............................. 157 Table 5.2 Error norms of displacements for natural-patch-test ................................. 157 Table 5.3 Error norms of uniform refinement for Poisson’s equation with a sharp peak ................................................................................................................... 157 Table 5.4 Error norms of adaptive refinement for Poisson’s equation with a sharp peak .......................................................................................................... 157 Table 5.5 Error norms of uniform refinement for infinite plate with a circular hole.157 Table 5.6 Error norms of adaptive refinement for infinite plate with a circular hole 157 Table 5.7 Error norms of uniform refinement for short cantilever plate. .................. 158 Table 5.8 Error norms of adaptive refinement for short cantilever plate using GSM .................................................................................................................. 158 Table 5.9 Error norms of adaptive refinement for short cantilever plate using FEM 158 Table 5.10 Error norms of uniform refinement for L -shaped plate......................... 158 Table 5.11 Error norms of adaptive refinement for L -shaped plate ........................ 159 Table 5.12 Error norms of uniform refinement for Mode-I crack problem .............. 159 Table 5.13 Error norms of adaptive refinement for Mode-I crack problem .............. 159 Table 6.1 Natural frequencies (Hz) of a cantilever beam with different nodal distribution .............................................................................................. 195 Table 6.2 Natural frequencies of a variable cross-section cantilever beam .............. 196 xxvi Table 6.3 Natural frequencies of a shear wall ........................................................... 196 Table 7.1 Comparisons between the GSM and the LWGSM .................................... 215 Table 7.2 The L2 -norm error of Poisson’s equation using different approaches of LWGSM with different distributions of right triangles ........................... 215 Table 7.3 The L2 -norm error of Poisson’s equation using the GSM (Scheme VII and VIII) with different distributions of right triangles ................................... 215 Table 7.4 The L2 -norm error of Poisson’s equation using different approaches of LWGSM with different distributions of irregular triangular cells. .......... 216 Table 7.5 The L2 -norm error of Poisson’s equation using the GSM (Scheme VII and VIII) with different distributions of irregular triangular cells .................. 216 Table 7.6 The L2 -norm error of half model using different approaches of LWGSM with different distributions of irregular triangular cells. .......................... 217 Table 7.7 The L2 -norm error of half model using the GSM Scheme VIII with different distributions of irregular triangular cells. ................................... 217 xxvii Chapter 1 Introduction Chapter 1 Introduction 1.1 Background With the rapid development of computer technology in the past few decades, a broad range of numerical methods have been developed for different types of problems and achieved great success, for example, the finite difference method (FDM), finite element method (FEM), finite volume method (FVM) and recently the meshfree methods (MM). FDM is one of the oldest methods, which can be traced back to the early 1910s. It is widely adopted in numerical simulations mainly because of its simplicity and efficiency. FEM is one of the most successful and dominant numerical methods in the last century. It is extensively used in modeling and simulation of engineering and science due to its versatility for complex geometries of solids and structures and its flexibility for many types of non-linear problems. Most practical engineering problems related to solids and structures are currently solved using FEM packages. The finite volume method is used to discretize an integral form of the partial differential equation (PDE) for a physical law, e.g., conservations of mass, momentum, or energy. The FVM is now well developed for solving fluid flow problems and implemented widely in commercial computational fluid dynamics (CFD) software. More recently, many meshfree methods have been proposed to get rid of the elements and meshes which are necessary for FEM, and to avoid the inherent 1 Chapter 1 Introduction shortcomings and difficulties of FEM when dealing with certain classes of problems. The existing numerical methods may generally be classified into two major categories according to their formulation procedures of discretizing the governing equations: (1) the methods based on a variational principle or a weak form of system equations (short for weak form method), and (2) the methods based on the strong form of governing equations (short for strong form method). Among these developed weak form methods, the finite element method is most well established. Relying on meshes or elements that are connected to each other by the nodes to model the problem domain, the FEM has encountered several limitations, including high computational cost in generating meshes, low accuracy in the derivatives of primary field variables, difficulties in the implementation for adaptive analysis, no allowance for large distortion of element and simulation of failure process (e.g., dynamic crack growth with arbitrary paths, breakage of structures or components with a large number of fragments), etc. Therefore, the idea of getting rid of the elements and meshes is naturally evolving. A new class of numerical methods, meshfree methods, has been devised. The meshfree methods have achieved remarkable progress over the past few years. Currently, the meshfree weak form method is most widely used due to its excellent stability. It includes the element free Galerkin (EFG) method, reproducing kernel particle method (RKPM), meshless local Petrov-Galerkin (MLPG) method, and point interpolation methods (PIM). The use of global or local integrations to establish the discrete equations is a common feature of the meshfree weak form methods. The 2 Chapter 1 Introduction integrations have significant effects on computational stability, accuracy and convergence. However, the formulation procedures are relatively more complicated and more difficult to be implemented due to the background integrations and variable mappings. In contrast, the formulation procedure of the strong form of meshfree methods is relatively simple and straightforward, compared with the meshfree weak form methods. The meshfree strong form method is regarded as a truly meshfree method as no mesh is required for field variable approximation or integration. With such distinct features, the strong form of meshfree methods is very efficient and easy to be implemented for adaptive analyses and simulations, even for the problems difficult to be solved by the traditional FEM. Smoothed particle hydrodynamics (SPH) and the generalized finite difference method (GFDM) may be under this category. Radial point collocation method (RPCM) is also a meshfree strong-form method formulated using radial basis functions and nodes in local supporting domains. However, the instability of the meshfree strong form methods has been a main challenge that limits the application of meshfree strong form methods that use local nodes. Researchers have introduced several stabilization schemes, in which stabilization factors need to be determined. Currently, most of the ‘full-proof’ strong form methods are still relying very much on the structured grid and restricted regular domain. Although methods like generalized finite difference method (GFDM) can be used for irregular domain and unstructured grid, a proper stencil (node selection) is somehow still needed for function approximation. Such inconvenience procedures give difficulties to the strong form methods for extensive applications. Compared with 3 Chapter 1 Introduction the well-established weak form methods, the development of strong form methods is rather sluggish. Available literatures for the strong form methods are still limited. Therefore, the strong form methods are now in great demand. Strong form methods demonstrate very good potential to become powerful numerical tools. However, there are still some technical problems that need to be solved before they become efficient and practical for engineering applications. The major challenges to the researchers and scientists working on strong form methods are given as follows: 1. To stabilize strong form formulations using irregular local nodes; 2. To improve the accuracy, efficiency and performance of strong form methods; 3. To formulate strong form schemes for complex problems of practical applications; 4. To develop powerful and versatile commercial software packages of strong form methods. Hence, further research work is very necessary to establish strong form methods as powerful numerical tools. 1.2 Literature Review As the problems of computational mechanics become more and more challenging, the conventional numerical methods, for instance, FDM, FEM and FVM, are no longer well suited. The demand of a new class of numerical methods formulated without reliance on mesh or element grows more prominent. Originated about thirty 4 Chapter 1 Introduction years ago, meshfree methods were well established and discussed as one of the hottest research topics in the area of computational mechanics. The earliest meshfree method is the smoothed particle hydrodynamics (SPH) (Lucy, 1977) which was used to study the astrophysical phenomena without boundaries such as exploding stars and dust clouds. Most early research studies on SPH were reflected in the publications of Monaghan and his co-workers (Gingold and Monaghan, 1977; Monaghan and Lattanzio, 1985; Monaghan, 1992). A comprehensive survey of the recent research works of SPH can be found in the book by Liu and Liu (2003). Besides the SPH method, the collocation methods also have great influence on the development of the meshfree methods. As early as 1980s, to get rid of the regular grids in the FDM formulation, many research works were devoted to establish a collocation method based on arbitrarily scattered nodes. Generalized finite difference method was therefore proposed and well discussed by many researchers (Girault, 1974; Perrone and Kao, 1975; Liszka and Orkisz, 1977, 1980). 1.2.1 The classification of meshfree methods With the progressively development of meshfree methods, it is very important to categorize mehfree methods into different classes for better understanding. There are many different ways to classify the meshfree methods. In this section, various types of classification will be briefly introduced. The first type of classification categorizes the meshfree methods according to the interpolation or approximation function. The most popular approximations include SPH approximation (Lucy, 1977; Gingold and Monaghan, 1977), MLS approximation 5 Chapter 1 Introduction (Nayroles et al., 1992; Belytschko et al., 1994), RKPM approximation (Liu et al., 1995, 1997; Liu and Jun, 1998), partition of unity methods (Melenk and Babuska, 1996; Babuska and Melenk, 1997), PIM approximation (Liu and Gu, 2001a; Liu and Zhang, 2008), RPIM approximation (Wang and Liu, 2002a; Liu et al., 2006), etc. Another type of classification uses the domain representation to categorize the meshfree methods, which include domain-type and boundary-type of meshfree methods. In the domain-type methods, both problem domain and boundary are represented by field nodes. Examples of this type of meshfree methods include element-free Galerkin (EFG) method (Belytschko et al., 1994), point interpolation method (PIM) (Liu and Gu, 2001a), local radial point interpolation method (LRPIM) (Liu and Gu, 2001b), SPH method (Lucy, 1977), etc. On the contrary, only boundary is represented by field nodes in the boundary-type meshfree methods, for example, boundary node method (BNM) (Mukherjee and Mukherjee, 1997), boundary point interpolation method (BPIM) (Gu and Liu, 2002), boundary radial point interpolation method (BRPIM) (Gu and Liu, 2003). In this thesis, the classification according to the formulation procedure is adopted. Meshfree methods may mainly be categorized into methods based on strong forms of partial differential equations (PDEs) and methods based on weak forms of system equations. There are exceptions to this classification because some meshfree methods can be used in both strong form and weak form (Gu, 2002; Liu and Gu, 2005). 1.2.2 Meshfree methods based on weak forms The meshfree methods based on Galerkin weak forms (variational principles) are 6 Chapter 1 Introduction relatively young. From the early 1990s, due to the successful application of variational principles in the FEM, more and more research efforts have been devoted to the study of meshfree methods based on Galerkin weak forms. Several landmark papers were published in this period of time. The first landmark paper was published by Nayroles et al. (1992), who proposed the diffuse element method (DEM). Belytschko et al. (1994) published another landmark paper to propose the element free Galerkin (EFG) method based on the DEM. After this publication, the meshfree methods based on the Galerkin weak forms developed very fast. It is reflected by a large number of new meshfree methods proposed, including the reproducing kernel particle method (RKPM) (Liu et al., 1995), the meshless local Petrov-Galerkin (MLPG) method (Atluri and Zhu, 1998), the point interpolation method (PIM) (Liu and Gu, 2001a), the local radial point interpolation method (LRPIM) (Liu and Gu, 2001b), the radial point interpolation method (RPIM) (Wang and Liu, 2002a), the linear conforming point interpolation method (LC-PIM) (Zhang et al., 2008) and the linear conforming radial point interpolation method (LC-RPIM) (Liu et al., 2006; Li et al., 2007). Several review papers (Belytschko et al., 1996; Liu et al., 1996; Li and Liu, 2002) and two special issues (Computer Methods in Applied Mechanics and Engineering, Vol. 139, 1996; Computational Mechanics, Vol. 25, 2000) are also devoted to the development of meshfree methods. More details on meshfree weak form methods can be found in books by Liu (2002) and Liu and Gu (2005). 1.2.3 Meshfree methods based on strong forms Compared with meshfree weak form methods, strong form meshfree methods 7 Chapter 1 Introduction have a longer history of development. To approximate the strong form of a PDE using meshfree methods, the PDE and boundary conditions are usually discretized by a specific collocation technique. One of the most famous meshfree methods based on the strong form is the method of smoothed particle hydrodynamics (SPH). SPH was first invented to solve astrophysical problems in three-dimensional open space, in particular polytropes (Lucy, 1977; Gingold and Monaghan, 1977). The basic idea of SPH is that the state of a system can be discretized by arbitrarily distributed particles. The earliest applications of SPH were mainly focused on astrophysical problems and fluid dynamics related areas, such as the simulation of binary stars and stellar collisions (Benz, 1988; Monaghan, 1992), gravity currents (Monaghan, 1995), heat transfer (Cleary, 1998), and so on. Recently, the SPH method has been applied for the simulations of high (or hyper) velocity impact (HVI) problems. Libersky and his co-workers have made outstanding contributions in the application of SPH to impact problems (Libersky and Petscheck, 1991; Libersky et al., 1995; Randles and Libersky, 1996). The main shortcomings of the SPH methods (Li and Liu, 2002) include tensile instability, lack of interpolation consistency, zero-energy mode, and difficulty in enforcing essential boundary condition. Some improvements and modifications of the SPH method have been developed (Monaghan and Lattanzio, 1985; Swegle et al., 1995; Morris, 1996). The generalized finite difference method (GFDM) is also considered as the category of meshfree strong form methods, which directly discretizes the governing equations. The bases of the GFDM were published in the early seventies. The early 8 Chapter 1 Introduction contributors to the GFDM include Jensen (1972), Perrone and Kao (1975), etc. Jensen (1972) was the first to introduce fully arbitrary mesh. He considered Taylor series expansions interpolated on six-node stars in order to derive the finite difference formulae approximating derivatives of up to the second order. While he used that approach to the solution of boundary value problems given in local formulation, Nay and Utku (1973) extended it to the analysis of problems posed in the variational (energy) form. However, these very early GFDM formulations were later essentially improved and extended by many other authors, but the most robust of the methods was developed by Liszka and Orkisz (Liszka and Orkisz, 1980; Liszka, 1984), and the most advanced version was given by Orkisz (1998). The explicit finite difference formulae used in the GFDM, as well as the influence of the main parameters involved, was studied by later investigators (Benito et al., 2001; Gavete et al., 2003). Nevertheless, this category of strong form methods received much less attention. One possible reason might be that the discrete equations yielded by these methods do not have the favorable properties such as symmetric, positive definite, well-conditioned and so on. Radial point collocation method (RPCM) is the first meshfree method of strong form (Liu et al., 2002, 2003, 2005) formulated using radial basis functions and nodes in local supporting domains. Like other strong form methods, the RPCM suffers from problems of instability (Liu and Gu, 2005). Poor accuracy and instability issues often arise, especially when Neumann boundary conditions exist. This is particularly true for solid mechanics problems with force boundary conditions. The system equations 9 Chapter 1 Introduction behave like ill-posed inversed problems (Liu and Han, 2003). Several techniques have been proposed to overcome these shortcomings in the meshfree strong form methods. Examples include the finite point method (Onate et al., 1996, 2001), Hermite-type collocation method (Liu et al., 2002, 2003), fictitious point approach (Liu et al., 2005), stabilized least-squares radial point collocation method (LS-RPCM) (Liu et al., 2006; Kee et al., 2007), and meshfree weak-strong (MWS) form method (Liu and Gu, 2003a; Liu et al., 2004). There are other meshfree methods (particle methods) developed based on the strong forms, such as the vortex method (Chorin, 1973; Bernard, 1995), Hp-cloud method (Liszka et al., 1996), the meshfree collocation method (Zhang et al., 2000), and so on. Detailed discussion of these methods can be referred to the relevant papers and books (Liu, 2002; Liu and Gu, 2005). 1.3 Objectives The meshfree methods in computational mechanics have been actively proposed and increasingly developed in order to overcome some drawbacks in the conventional numerical methods, e.g., finite difference method (FDM) and finite element method (FEM). Among the meshfree methods, the strong form methods possess good potential to become popular alternative numerical methods and most attractive feature to facilitate the implementation for adaptive analysis. The concept of the strong form methods is very simple, and its formulation procedure is straightforward. Neither formulation procedure nor construction of shape function requires numerical 10 Chapter 1 Introduction integration. The truly meshfree feature of strong form methods eases the refining or coarsening procedure in adaptive analysis. Nodes can be quite freely inserted or deleted without worrying too much about the connectivities. Unlike traditional numerical methods relying on meshes or elements, strong form meshfree methods can efficiently eliminate the costly and troublesome remeshing procedure. Nevertheless, the development of strong form methods remains very challenging. Currently, most of the reliable strong form methods are still restricted for structured grids and regular domains. FDM is regarded as the earliest, classical and reliable method of strong form (Richtmyer, 1957; Richtmyer and Morton, 1967). However, while dealing with more geometrically complex and practical problems, FDM relying on the structure grids has encountered great difficulty. The strong form methods formulated without relying on the structure grids are therefore very attractive. Although the methods like generalized finite difference method (GFDM) (Girault, 1974; Liszka and Orkisz, 1977; Liszka and Orkisz, 1980; Orkisz, 1998) can be used for irregular domains and unstructured grids, due to practical reasons, as well as for the purpose of generation of well-conditioned finite difference schemes, implementations of such methods using arbitrary irregular grids may sometimes be required to satisfy certain requirements, e.g., regularity in subdomains with guaranteed smooth transition, mesh with varying element topology and distribution of nodes with topological restrictions. Also, to consider finite difference (FD) operator generation at a node, one of the star selection criteria used in these methods and considered the best one (Kleiber, 1998), termed the Voronoi neighborhood criterion, is 11 Chapter 1 Introduction relatively more complicated and more difficult to be implemented practically. Such inconvenience procedures give difficulties to the strong form methods in the adaptive process as nodal distribution during the adaptation can become highly irregular and hence a ‘proper’ stencil can be costly and difficult to form. In addition, instability is a crucial issue that limits the applications of strong form methods, especially in the adaptive analysis. The solution with strong form method is usually not stable and less accurate than solution using weak form method. Without effective stabilization techniques, it is impossible to use such strong form methods in adaptive analysis. Although researchers have provided several alternative schemes, such as adding derivatives to primary field variables (Zhang et al., 2000), introducing auxiliary collocation points (Zhang et al., 2001), coupling strong formulation with weak form (Liu and Gu, 2002, 2003; Gu and Liu, 2005), and augmenting additional terms to the original governing equations (Onate et al., 2001), the stabilization effect is not yet to be satisfactory, and the implementation of these procedures can be very complicated for adaptive analysis. Compared with the weak form methods, the development of the strong form methods is relatively sluggish. The literature for the strong form meshfree methods in adaptive analysis is very little. As the instability issue is still the fatal shortcoming of strong form methods, it is impossible to extend the strong form methods to adaptive analysis without an effective measure to stabilize the solution. In this light, the primary objective of the present work is: 1) to propose and evaluate new strong form methods to obtain the stability of solution; 2) to utilize the features of strong form 12 Chapter 1 Introduction methods, such as simplicity, stability and accuracy; and 3) to facilitate an easier implementation for adaptive and dynamic analyses in computational mechanics. The main achievements from this thesis include: 1. A radial point interpolation based finite difference method (RFDM) is proposed as an alternative meshfree strong form method. The point interpolation using radial basis functions (RBFs) and nodes in local support domain is incorporated into classical finite difference method (FDM) for stable solutions to partial differential equations. 2. A novel gradient smoothing method (GSM) based on the strong form formulation is proposed, in which gradient smoothing technique is utilized to construct first- and second-order derivative approximations by systematically computing weights for a set of nodal points surrounding a node of interest. The basic principles are introduced in details. The formulation procedure and theoretical analysis are presented thoroughly. Different schemes and techniques are investigated in detail and compared to generate the optimal solutions. 3. The gradient smoothing method (GSM) is validated and examined through extensive numerical investigations of static solid mechanics problems. The computational accuracy, efficiency and stability are well demonstrated even with extremely irregular nodes. The GSM is also compared with other established numerical methods, such as the finite element method (FEM). 4. The gradient smoothing method (GSM) is further developed for adaptive 13 Chapter 1 Introduction analysis in solid mechanics. An effective and robust residual based error indicator and a simple refinement procedure using Delaunay diagram are implemented in the GSM for adaptive procedures. The GSM demonstrates very good reliability and performance in the adaptive analyses of several solid mechanics problems including singularities and concentrated loading. 5. The proposed gradient smoothing method (GSM) is extended and formulated for elasto-dynamic analyses of two-dimensional solids and structures. As an efficient, accurate and stable method of strong form, the present GSM has a good potential for solving linear and non-linear time dependent problems. 6. A linearly weighted gradient smoothing method (LWGSM) is devised, in which a piecewise linear polynomial is adopted as the smoothing function. The theoretical principles are investigated in detail. Some numerical tests are conducted to examine the accuracy, efficiency and stability of the proposed LWGSM. 1.4 Organization of the Thesis This thesis consists of eight chapters and is organized as follows. Chapter 2 presents a novel radial point interpolation based finite difference method (RFDM) in which radial point interpolation using local irregular nodes is used together with the conventional finite difference procedure to achieve both the adaptivity to irregular domain and the stability in the solution that is often encountered in the collocation methods. Several numerical examples are presented to 14 Chapter 1 Introduction demonstrate the accuracy and stability of the RFDM for problems with complex shapes and regular and extremely irregular nodes. Also, a numerical study on the effects of the parameters for RFDM is conducted. Chapter 3 exploits the theoretical aspects of gradient smoothing method. It focuses on elucidating the principle of gradient smoothing and its numerical procedure to solve partial differential equations. Stencil analyses of different types of discretization schemes for spatial partial differential terms are carried out from points of views of both efficiency and accuracy. The compactness of stencil and positivity of the coefficients of supporting nodes are concerned in the analyses. The favorable schemes are selected for further study. Chapter 4 applies the proposed gradient smoothing method (GSM) to solid mechanics problems. The gradient smoothing operation is adopted to develop the first- and second-order derivatives for a node of interest by calculating weights for a set of surrounding field nodes. A simple collocation procedure is then applied to the governing system equations of strong form at each node scattered in the problem domain using the approximated derivatives. Several numerical examples are presented to demonstrate the computational accuracy and stability of the present method with regular and irregular nodes. The proposed GSM is examined in detail by comparison with other established numerical approaches such as the finite element method, producing the convincing results. Chapter 5 develops the adaptive analyses of solid mechanics using the proposed gradient smoothing method (GSM). The present method is found very stable and can 15 Chapter 1 Introduction be easily applied to arbitrarily irregular triangular meshes for complex geometry. Unlike other strong form methods, the present method has excellent stability that is of great importance for adaptive analysis. The reliability and performance of the proposed GSM for adaptive procedure are demonstrated in several solid mechanics problems, including the singularities and concentrated loading, compared with the well-known finite element method. Chapter 6 further employs the GSM to study elasto-dynamic problems in 2-D solids and structures. Strong form formulations for 2-D elastodynamic governing equations are developed. Numerical examples have demonstrated the validity, accuracy and stability of the present GSM for free and forced vibration analyses, compared with other well-established numerical approaches. Chapter 7 establishes the linearly weighted gradient smoothing method (LWGSM) with a piecewise linear smoothing function, moving beyond the gradient smoothing method (GSM). The theoretical basis and formulation procedure are investigated in the same way as that for GSM. Numerical and comparative examples and results are shown to validate the accuracy, efficiency and stability of the developed LWGSM. Chapter 8 ends the thesis by concluding several remarks and recommendations for further studies. 16 Chapter 2 Radial Point Interpolation Based Finite Difference Method Chapter 2 Radial Point Interpolation Based Finite Difference Method 2.1 Introduction In a meshfree method, the problem domain is represented by a number of scattered nodes. In order to seek for the numerical solution to a problem governed by PDEs and given boundary conditions, an efficient interpolation technique is required to construct trial function from these scattered data nodes before the discretized system equations can be formed. The quality of the numerical solution highly depends on the interpolation for function approximation. Several interpolation techniques have been developed and used in meshfree methods, such as smooth particle hydrodynamics (SPH) approximation, reproducing kernel particle method (RKPM) approximation, moving least squares (MLS) approximation, partition of unity methods, polynomial point interpolation method (PPIM) and radial point interpolation method (RPIM). As SPH, RKPM, MLS and partition of unity approximations are not used in this study, only a brief introduction to them is given. Details for their formulations can be referred to the respective references. In this thesis, PPIM and RPIM approximations are mostly used and hence more comprehensive details are provided. The well-known radial basis functions (RBFs) were first introduced in early 90s. Kansa (1990) was one of the pioneers who used RBFs in the meshfree strong form 17 Chapter 2 Radial Point Interpolation Based Finite Difference Method collocation method. As RBFs can be used to interpolate scattered data, such feature provides great flexibility in term of nodal distribution and hence eliminates regular grids. Recently, Liu et al. (2002, 2003, 2005) have revised the conventional scheme of RBFs and proposed a novel scheme with RBFs using local nodes. Instead of using all the nodes in the domain for function approximation, only local nodes, the neighbouring nodes of a point of interest, are selected to approximate the field function and its partial derivatives. In this work, the strong form collocation method using local RBFs and collocation technique is called radial point collocation method (RPCM). As mentioned in Chapter 1, the instability problem has concerned many researchers for strong form meshfree methods, like RPCM. In this work, a radial point interpolation based finite difference method (RFDM) is developed as an alternative meshfree strong form method. In this novel method, the point interpolation using radial basis functions and nodes in local support domain is incorporated into classical finite difference method (FDM) for stable solutions to partial differential equations defined in a domain that is represented by a set of irregularly distributed nodes. A least-square technique is adopted to acquire a system matrix of good properties including symmetry and positive definiteness, which helps greatly in solving the resulting set of algebraic system equations more efficiently and accurately by using standard solver such as the Cholesky solver. The proposed RFDM can effectively avoid the instability in conventional collocation methods, while retaining the feature of simplicity in formulation procedures with little additional computational cost. 18 Chapter 2 Radial Point Interpolation Based Finite Difference Method Section 2.2 briefs the interpolation techniques for function approximation. In section 2.3, the standard radial point collocation method is introduced. Section 2.4 gives theoretical formulation of the RFDM. Several numerical examples are presented in section 2.5. A numerical study on the effects of the parameters for RFDM is conducted in section 2.6. Some conclusions are drawn in section 2.7. 2.2 Function Approximation One of the central and most important issues in meshfree methods is the development of meshfree approximation functions. Several meshfree approximation formulations have been proposed and developed. 2.2.1 Smoothed particle hydrodynamics (SPH) approximation The smoothed particle hydrodynamics (SPH) method (Lucy, 1977; Gingold and Monaghan, 1977) is always regarded as one of the earliest developed meshfree methods. The SPH shape function is represented in an integral form of Kernel interpolation. A field function u at a point x is approximated by u h (x ) = ) ∫ u(ξ )w(x − ξ , h ) dΩξ Ωξ (2.1) ) where w is known as a kernel or weight function, h is a measure of the size of the support, and Ωξ is the influence domain in SPH approximation. Equation (2.1) is often called a smoothing function in SPH. Monaghan (1982) listed the five conditions that the kernel should satisfy. The kernel functions often used in SPH can be found in the references of Monaghan (1982, 1992), Belytschko et al. (1996) and Liu and Liu 19 Chapter 2 Radial Point Interpolation Based Finite Difference Method (2003). 2.2.2 Reproducing kernel particle method (RKPM) approximation Reproducing kernel particle method (RKPM) is another well-know meshfree method proposed by Liu et al. (1995, 1997). Liu’s advance is achieved by adding a correction function to the kernel in equation (2.1). This correction function is particularly useful in improving the SPH approximation near the boundaries as well as to make it linearly or C 1 consistent. The integral representation of a field function u with correction function can be given as ) u h (x ) = ∫ u (ξ ) C (x, ξ )w(x − ξ , h ) dΩξ Ωξ (2.2) where C (x, ξ ) is the correction function. An example of the correction function in one dimension is C (x, ξ ) = c1 (x ) + c2 (x )(ξ − x ) (2.3) where c1 (x ) and c2 (x ) are the coefficients, which can be obtained by enforcing the corrected kernel to reproduce the field function required (Liu et al., 1995). 2.2.3 Moving least squares (MLS) approximation The moving least squares (MLS) approximation originated from data fitting and surface construction in the mathematical community (Mclain, 1974; Gordon and Wixom, 1978). An excellent description of MLS can be found in the landmark paper by Lancaster and Salkausdas (1981). The MLS approximation is now widely used in meshfree methods for constructing meshfree shape functions. Nayroles et al. (1992) 20 Chapter 2 Radial Point Interpolation Based Finite Difference Method were the pioneers who adopted MLS approximation in the meshfree methods. Other meshfree methods that use MLS approximation include EFG (Belytschko et al., 1994), FPM (Onate et al., 2001), etc. In the MLS approximation, a field function u at any point of interest x can be approximated in the following form: m u h (x ) = ∑ p j (x ) a j (x ) = p T (x ) a(x ) (2.4) j =1 where m is the number of basis used in the approximation, p j ( x ) is the basis function of the space coordinates and a j (x ) is the corresponding coefficient. a(x ) is obtained at the point x by minimizing a weighted discrete L2 norm of the residual [ n ) J = ∑ w(x − x i ) pT (x i )a(x ) − ui 2 ] (2.5) i =1 ) where n is the number of nodes in the support domain of x and wi is the weight function. One should note that the number of nodes in the support domain is equal or greater than the number of basis in the approximation, i.e., n ≥ m . The major advantages of MLS approximation is that its continuity is mainly related to the continuity of the chosen weight function. In other words, a low order polynomial basis, e.g., a linear basis, may be used to generate higher continuous approximations by choosing an appropriate weight function. Also, the MLS approximation is very flexible in the nodal selection. Due to the least-squares procedure, the major disadvantages of MLS are that the MLS shape functions lack the Kronecker delta function properties, which can cause difficulties in the imposing of Dirichlet boundary conditions, and it is computationally expensive. 21 Chapter 2 Radial Point Interpolation Based Finite Difference Method 2.2.4 Partition of unity methods The partition of unity methods were proposed and developed by Duarte and Oden (1996) and Babuska and Melenk (1997). Melenk and Babuska (1996) proposed the following approximation technique called the partition of unity finite element method (PUFEM): u h (x ) = ∑ Φ 0I (x )∑ β jI p j (x ) I j (2.6) where β jI are the unknowns and p j is the basis which typically includes a certain degree of monomial terms and possibly some enhancement functions. Φ 0 is a function that satisfies conditions of the partition of unity (Duarte and Oden, 1996). It can be constructed from an MLS shape function. Duarte and Oden (1996) have proposed a slightly more general partition of unity method, called the hp method. In hp approximation, MLS shape functions of order k are employed instead of the partition of unity functions of PUFEM. The approximation is ⎛ ⎞ u h (x ) = ∑ Φ kI (x )⎜ u I + ∑ biI qiI (x )⎟ I i ⎝ ⎠ (2.7) The function qiI (x ) are either high-order monomials or enhancement functions for a node i . A major advantage of this approximation is that it allows the basis q to vary from node to node and thus makes p-adaptivity easy. 2.2.5 Polynomial point interpolation Polynomial function is one of the earliest basis functions used in the interpolation scheme. As its name implies, polynomial function is used as a basis function in the 22 Chapter 2 Radial Point Interpolation Based Finite Difference Method polynomial point interpolation approximation. Considering a smooth and continuous field function u in a problem domain Ω , the approximated field function u h at any point of interest x can be represented in the following form m u h (x ) = ∑ pi (x ) ai = PT (x ) a (2.8) i =1 where pi (x ) is the monomial of the polynomial function in the Euclidean space and ai is the corresponding coefficient. Completed polynomial basis function is usually preferred in the approximation of polynomial point interpolation. For example, completed polynomial basis functions used in one-dimensional and two-dimensional spaces are PT (x ) = [1 x ], m = 2, p = 1 [ (2.9) ] PT (x ) = 1 x y x 2 xy y 2 , m = 6, p = 2 (2.10) where p is the order of polynomial and m is the number of monomials. Pascal’s triangles (Zienkiewicz and Taylor, 1992) can be utilized to determine the basis of the approximation. Fig. 2.1 shows the Pascal’s triangles for two-dimensional space. To obtain the undetermined coefficient ai , the approximation function in Eq. (2.8) is enforced to pass through the field value at each supporting node, which can be expressed in the following matrix form U s = Pm a (2.11) where U s denotes the vector of field value at supporting nodes, U s = { u1 u2 u3 L u n } T (2.12) Pm denotes the moment matrix, 23 Chapter 2 Radial Point Interpolation Based Finite Difference Method ⎡1 ⎢1 ⎢ Pm = ⎢1 ⎢ ⎢M ⎢⎣1 x1 x2 y1 y2 x3 y3 M M xn yn x1 y1 L pn (x1 )⎤ x 2 y 2 L pn (x 2 )⎥ ⎥ x3 y 3 L pn (x 3 )⎥ ⎥ M O M ⎥ x n y n L pn (x n )⎥⎦ (2.13) and a denotes the coefficient vector of monomials a = { a1 a2 a3 L a n } T (2.14) The unknown coefficients can be easily obtained as −1 a = Pm U s if Pm −1 (2.15) is not singular. One should note that, to form a square moment matrix Pm , the number of supporting nodes must be equal to the number of monomials in the polynomial function, i.e., n = m . Substituting the Eq. (2.15) into Eq. (2.8) yields the following expression n u h ( x ) = P T (x ) Pm U s = ∑ φi (x ) ui = Φ(x ) U s −1 (2.16) i =1 where Φ(x ) is a vector of shape function defined as Φ(x ) = P T (x ) Pm −1 (x ) = {φ1 (x ) φ2 (x ) φ3 (x ) L φn (x )} (2.17) and φi (x ) is the shape function of polynomial point interpolation for supporting node i . The derivatives of the approximated field function can be easily obtained in terms of the derivatives of shape functions. It is because the shape function is in the form of polynomial function. For instance, the first-order derivative of the approximated field function with respect to x can be expressed as u, x ( x ) = h ∂ T −1 P (x ) Pm (x )U s = Φ, x (x ) U s ∂x (2.18) Apparently, construction of polynomial point interpolation shape function is very simple and straightforward. The shape function possesses numerous attractive 24 Chapter 2 Radial Point Interpolation Based Finite Difference Method properties (Liu, 2002; Liu and Gu, 2001a, 2005): • Kronecker Delta property The shape function possesses Kronecker Delta property ⎧1 if i = j, ⎩0 if i ≠ j, i, j = 1,2, L , n i, j = 1,2, L , n φi (x j ) = ⎨ (2.19) because the polynomial point interpolation shape function is obtained by enforcing the approximation to pass through the field value at each supporting node. With this important property, Dirichlet boundary conditions can be handled very easily. • Reproducibility of polynomial function As the shape function is in the form of polynomial function, it can reproduce any polynomial function that is included in the basis functions. • Partition of unity The shape function also has the property of partition of unity n ∑ φ (x ) = 1 (2.20) i i =1 This can be easily proved by assuming all nodal values are equal to a constant, U s = { c c c L c} T (2.21) and the polynomial point approximation can be shown as n n i =1 i =1 u h (x ) = ∑ φi (x ) ui = c ⋅ ∑ φi (x ) = c (2.22) which leads to the conclusion of Eq. (2.20). • Compact support As the shape function is only constructed using the vicinity nodes of a point of 25 Chapter 2 Radial Point Interpolation Based Finite Difference Method interest, it is considered as a compact support shape function. • No weight function Unlike other meshfree approximations, for example, MLS, no weight function or kernel function is used in the construction of shape function of polynomial point interpolation. Although the shape function possesses many attractive properties, the invertibility of moment matrix is still the most challenging issue to be resolved (Liu, 2002; Liu and Gu, 2003b, 2005). The construction of shape function is not flexible and robust. As mentioned above, the number of supporting nodes must be equal to the number of monomials in the polynomial used in the approximation. Such strict criterion causes the nodal selection in great difficulty. Furthermore, the inappropriate selection of monomials or supporting nodes will result in non-invertible moment matrix. To overcome the singular moment matrix Pm , Liu and Gu (2003b) have proposed a matrix triangularization algorithm (MTA) to efficiently select the proper enclosure of nodes and monomials. The MTA provides a versatile procedure to exclude the nodes and removes the monomials that cause singular moment matrix Pm . As singularity problem is overcome through the MTA procedure, invertible moment matrix can be formed and hence shape function of polynomial point interpolation can be constructed. Besides the MTA procedure, weighted least-squares (WLS) method is also a very common technique to avoid the singular moment matrix Pm in the procedure of forming the shape function. As mentioned previously, in the formulation of MLS approximation, weight function is introduced to an overdetermined moment matrix Pm 26 Chapter 2 Radial Point Interpolation Based Finite Difference Method and the shape functions are constructed through seeking the minimal of the residual. Of course, augmenting the polynomial basis with radial basis functions (RBFs) in the approximation is also a very good idea to construct shape function using arbitrary field nodes. The details of such approximation are given in the following section. 2.2.6 Radial point interpolation Radial basis functions (RBFs) are very well-known for its excellent performance in surface fitting based on arbitrarily scattered data (Franke, 1982; Hardy, 1990; Powell, 1992). It has been widely used in the mathematic community since many decades ago. The intensive reviews of the RBFs can be found in (Powell, 1992; Liu, 2002; Liu and Gu, 2005). Consider a field function u (x ) defined in a problem domain Ω which is represented by a set of arbitrarily distributed nodes. A local support domain of a point of interest x determines the vicinity nodes that are used for approximation or interpolation of function value at x . A support domain can have different shapes and its dimension and shape can be different from point to point, as shown in Fig. 2.2. Most often used shapes are circular or rectangular. In this study, the number of field nodes (both regular and irregular) in the local support domain is predefined, i.e., n . According to the different distances between the field nodes and the point of interest x , the n nodes which are the nearest to the point of interest are adopted in the support domain. Then, the value of the field function u (x ) at any interest point x can be approximated by interpolating the values of the field function at the vicinity nodes in the local support domain: 27 Chapter 2 Radial Point Interpolation Based Finite Difference Method u h (x ) = ∑ ai Ri ( x − x i ) + ∑ b j p j (x ) = R (x ) a + P(x ) b n m i =1 j =1 (2.23) where n is the total number of supporting nodes selected from the surrounding of the point of interest x for approximation, m is the number of monomials in the polynomial function, R (x ) is the radial basis function and P(x ) is the monomial in polynomial function for augmentation. a and b are the undetermined coefficients of radial basis functions and monomials of polynomial function, respectively. The vectors in Eq. (2.23) are defined as R (x ) = [R1 ( x − x1 ) R2 ( x − x 2 P(x ) = [1 x a = { a1 ) K Rn ( x − x n )] y L pm (x )] (2.24) (2.25) a2 L an } (2.26) b = { b1 b2 L bm } (2.27) T T Enforcing the interpolation in Eq. (2.23) passing through field value at all nodes in the supporting domain leads to the following expression U s = R Q a + Pm b (2.28) where U s is the vector of function values at supporting nodes, U s = { u1 u2 L un } T (2.29) the moment matrix of radial basis functions is ⎡ R1 ( x1 − x1 ) R2 ( x1 − x 2 ) L Rn ( x1 − x n ⎢ R ( x − x1 ) R2 ( x 2 − x 2 ) L Rn ( x 2 − x n RQ = ⎢ 1 2 ⎢ M M O M ⎢ ⎣ R1 ( x n − x1 ) R2 ( x n − x 2 ) L Rn ( x n − x n )⎤ ⎥ )⎥ ⎥ ⎥ )⎦ ( n×n ) (2.30) and the polynomial moment matrix is 28 Chapter 2 Radial Point Interpolation Based Finite Difference Method ⎡1 x1 ⎢1 x 2 Pm = ⎢ ⎢M M ⎢ ⎣1 xn y1 L pm ( x1 ) ⎤ y 2 L pm ( x 2 ) ⎥ ⎥ M O M ⎥ ⎥ y n L pm ( x n )⎦ ( n×m ) (2.31) In order to guarantee a unique approximation, the polynomial terms here have to satisfy an additional orthogonal condition (Hardy, 1990; Kansa and Hon, 2000), PmT a = 0 (2.32) The combination of Eqs. (2.28) and (2.32) yields to the following expression ⎧U s ⎫ ⎡ R Q ⎨ ⎬=⎢ T ⎩ 0 ⎭ ⎣ Pm Pm ⎤ ⎧ a ⎫ ⎧a ⎫ ⎨ ⎬ = G⎨ ⎬ ⎥ 0 ⎦ ⎩b ⎭ ⎩b ⎭ (2.33) One should note that the moment matrix corresponding to the RBFs, R Q , is symmetric. As a result, the matrix G is also a symmetric matrix. A unique solution to the unknown vectors of coefficients a and b can be obtained as ⎧a ⎫ −1 ⎧U s ⎫ ⎨ ⎬=G ⎨ ⎬ ⎩b ⎭ ⎩0⎭ (2.34) if the inverse of G exists. If the order of polynomial function m used in Eq. (2.23) is much lower than the number of radial basis n , G −1 in Eq. (2.34) is hardly to be singular (Liu, 2002; Liu and Gu, 2005). Substituting Eq. (2.34) into Eq. (2.23) leads to ⎧U ⎫ ~ ⎧U ⎫ u h ( x ) = { R ( x ) P( x )}G −1 ⎨ s ⎬ = Φ( x ) ⎨ s ⎬ ⎩0⎭ ⎩0⎭ (2.35) ~ Φ( x ) = { R ( x ) P( x )}G −1 = {φ1 ( x ) φ2 ( x ) L φn ( x ) φn +1 ( x ) L φn +m ( x )} (2.36) where Finally, the radial point interpolation shape functions corresponding to the nodal displacements vector Φ(x) are obtained as 29 Chapter 2 Radial Point Interpolation Based Finite Difference Method Φ( x ) = {φ1 ( x ) φ2 ( x ) L φn ( x )} (2.37) Eq. (2.35) can be rewritten as n u h ( x ) = Φ( x )U s = ∑ φi ui (2.38) i =1 The derivative of the unknown variable function can be evaluated easily by differentiating Eq. (2.38) as u,hl ( x ) = Φ, l ( x )U s (2.39) where l denotes the coordinates either x or y . A comma designates a partial differentiation with respect to the indicated spatial coordinate that follows. Several research works (Powell, 1992; Schaback, 1994) already showed the inverse of R Q usually existed. As the order of the polynomial function is much lower than the number of radial basis, the singularity problem of G matrix is therefore not encountered. However, the condition of R Q can be ill if too many supporting nodes are used for approximation (Kansa and Hon, 2000; Liu and Gu, 2005). Radial basis functions (RBFs) are widely known and used in many meshfree methods, because not only RBFs are very flexible for interpolating scattered data, but also the shape functions constructed using RBFs possess many distinguished properties as well. These properties are well studied and examined in the Liu’s book (Liu, 2002; Liu and Gu, 2005) and many research papers (Wang and Liu, 2002a, b). Some important properties of radial point interpolation shape function are introduced as follows: • Kronecker Delta property 30 Chapter 2 Radial Point Interpolation Based Finite Difference Method The shape function possesses Kronecker Delta property. With such unique property, the imposition of Dirichlet boundary conditions in meshfree methods becomes very straightforward. No special treatment such as penalty technique has to be applied. • Partition of unity The shape function possesses partition of unity as given in Eq. (2.20) if the linear polynomial terms ( m = 3 ) or higher terms are included in the approximation shown in Eq. (2.23). • Reproducibility of polynomial function The shape function of radial point interpolation can ensure the reproducibility of the polynomial function. If a k -order polynomial function is augmented with RBFs in the approximation given in the Eq. (2.23), the shape function can reproduce the same order of polynomial. One should note that without the polynomial term, pure RBF approximation can not reproduce even linear polynomial function. • High continuity Because of the high continuity of the RBFs, the shape function can also obtain high-order derivatives. • Compact support With only surrounding nodes of the point of interest selected for local approximation, the radial point interpolation shape function is considered as a kind of compact support shape function. 31 Chapter 2 • Radial Point Interpolation Based Finite Difference Method No weight function Similar to the polynomial point interpolation shape function, no weight function is required in the derivation of the shape function of radial point interpolation. Plenty types of radial basis functions (RBFs) are available and widely used in the mathematical community. The characteristics of the RBFs have been well studied in many literatures (Hardy, 1990; Kansa, 1990; Powell, 1992). Some of the most commonly used RBFs in meshfree methods are listed in Table 2.1. In this thesis, a very classical type of RBFs, Multi-Quadrics (MQ) (Hardy, 1990), with dimensionless shape parameters is adopted in radial point interpolation approximation. It can be expressed in the following form [ Ri (x ) = ri 2 + (α c d c ) ] 2 q (2.40) where ri is the Euclidean norm of the point of interest x and node i , ri = x − x i 2 (2.41) and d c is the characteristic length, α c and q are the dimensionless shape parameters of MQ-RBF. The characteristic length is also known as an “average” nodal spacing in the local domain. For instance, in the two-dimensional space, it is known as dc = As n −1 (2.42) where As is the area of the support domain and n is the number of supporting nodes in the support domain. There are several implementation issues of radial point interpolation shape function to be noted. 32 Chapter 2 Radial Point Interpolation Based Finite Difference Method 1. Singularity of moment matrix Unlike polynomial point interpolation, there is usually no singularity problem in the approximation of radial point interpolation. Mathematicians have already shown the moment matrix of radial basis function R Q in Eq. (2.30) is usually invertible in their works (Powell, 1992; Schaback, 1994; Wendland, 1995). If the order of polynomial function m used in Eq. (2.23) is much lower than the number of radial basis n , the inverse of moment matrix G in Eq. (2.34) is hardly to be singular (Liu, 2002; Liu and Gu, 2005). 2. Augmentation of polynomial functions In the radial point interpolation, the polynomial terms used in the approximation play a very important role. As mentioned above, pure RBF approximation can not reproduce polynomial function. Introducing additional polynomial terms to the approximation has brought several advantages and resulted in favourable properties of shape functions (Liu, 2002; Wang and Liu, 2002b; Liu and Gu, 2005). The use of polynomial function not only reduces the effects of dimensionless shape parameters on the approximation but also provides better stability. In general, the accuracy of the numerical solution is also improved or at least no undesirable effect. In this thesis, polynomial with completed second order, m = 6 , is adopted in the radial point interpolation approximation. 3. Values of dimensionless shape parameters of RBFs Determination of appropriate shape parameters is very important for RBFs. The shape parameters definitely have certain significant effects on the 33 Chapter 2 Radial Point Interpolation Based Finite Difference Method numerical solution. Therefore, determining the optimal shape parameters for RBFs is always the primary task for radial point interpolation approximation. However, the focus of the thesis is not on the study of RBFs and their optimal dimensionless shape parameters. As thorough studies of dimensionless shape parameters have been conducted (Liu, 2002; Wang and Liu, 2002a, b; Liu and Gu, 2005), recommended values for shape parameters of MQ-RBF: α c = 4.0 and q = 1.03 are used throughout this work. In this work, radial point interpolation approximation is adopted in various meshfree methods of strong form. Compared with polynomial point interpolation approximation, the reasons are very obvious as listed in the following: • Flexibility in nodal and basis selection In the radial point interpolation formulation, the selection of supporting nodes and basis function are much flexible. Unlike the polynomial point interpolation, the number of supporting nodes in the radial point interpolation can be any number, as long as the number of radial basis is much more than the number of monomials, n > m , as shown in Eq. (2.23). • Non-singular moment matrix Singularity problem is the most critical problem in the construction of polynomial point interpolation shape function. Inappropriate selected nodes and basis can cause singular moment matrix easily. This is one of the fatal shortcomings, which prohibits polynomial point interpolation shape functions from being used in the adaptive analysis. As moment matrix in the radial point 34 Chapter 2 Radial Point Interpolation Based Finite Difference Method interpolation formulation is always invertible, it makes radial point interpolation approximation more robust and suitable for arbitrary scattered nodes. • Less sensitive to nodal distribution In the polynomial point interpolation, unfavourable nodal distribution can easily cause the moment matrix singular. However, the shape function of radial point interpolation is more robust to the nodal distribution. Such important feature benefits the meshfree methods in the adaptive analysis where nodal distribution can be severely scattered throughout the domain sometimes. • Good accuracy A vast number of research works has shown that radial point interpolation is a very good approximation (Liu, 2002; Wang and Liu, 2002a, b; Liu and Gu, 2005; Liu et al., 2005). Radial point interpolation shape function can be used in the surface fitting to approximate the field function and their derivatives in very high accuracy. 2.3 Radial Point Collocation Method (RPCM) 2.3.1 Formulation Consider a partial differential governing equation defined in a domain Ω shown in Fig. 2.3: Lu = f in domain Ω (2.43) with Neumann boundary conditions 35 Chapter 2 Radial Point Interpolation Based Finite Difference Method Bu = g on boundary Γt (2.44) and Dirichlet boundary conditions u = u on boundary Γu where L( ) , B ( ) (2.45) are the differential operators and u is the primary field variable. Assume that the above equations, Eqs. (2.43)-(2.45), can be collocated at the field nodes inside the domain and on the boundaries, respectively. The discretized governing system equations can be shown as follows L(ui ) = f i in Ω (2.46) with Neumann boundary conditions B (ui ) = g i on Γt (2.47) and Dirichlet boundary conditions ui = ui on Γu (2.48) where subscript “ i ” denotes the collocation point. In the radial point collocation method (RPCM), the radial point interpolation is used to approximate the field function using local nodes. As shown in Fig. 2.3, the local support domain is formed by the surrounding nodes of the collocation point. The resultant algebraic equations can then be assembled and expressed in the following matrix form KU = F (2.49) where K is the stiffness matrix, U is the vector of unknown variables at all nodes in the problem domain and F is the nodal force vector. The vector of unknown nodal values can then be easily solved as 36 Chapter 2 Radial Point Interpolation Based Finite Difference Method U = K −1F (2.50) if K is not singular and well-conditioned. It is often found that K behave far from well, partially because the radial point collocation method uses local nodes for interpolation, which leads to possible instability in the solutions. It should be noted that the stiffness matrix K is generally not symmetric in the collocation methods. 2.3.2 Issues in RPCM Meshfree strong form methods possess many attractive and distinguished features. However, it is not easy to construct shape functions using arbitrary scattered nodes. Radial point interpolation is a good candidate to be used for constructing shape functions in the strong form collocation method. Besides great flexibility for irregular grids, the radial point interpolation shape function also possesses the Kronecker Delta property. Dirichlet boundary condition can be imposed directly. Furthermore, as compared to polynomial point interpolation, radial point interpolation is more robust as the moment matrix is always invertible. All those properties of radial point interpolation shown in section 2.2 make radial point collocation method (RPCM) a very promising meshfree strong method. However, through the study of RPCM, some crucial issues found have prevented RPCM to be a good meshfree strong form method. First, in the conventional RBF scheme (Kansa, 1990; Kansa and Hon, 2000), all nodes are used to approximate any point of interest in the domain. It means that a full coefficient matrix is formed and hence limits its application to the large scale problems. Furthermore, the condition of full coefficient matrix of the meshfree collocation method based on global RBFs is 37 Chapter 2 Radial Point Interpolation Based Finite Difference Method often ill (Kansa and Hon, 2000). The idea of local RBFs is a very natural choice and it has been proposed by Liu et al. (Liu and Gu, 2001b; Wang and Liu, 2002a; Liu at al., 2005) to avoid the undesired properties caused by the global RBFs. In the local scheme, the coefficient matrix is sparse matrix rather than full matrix. The computational cost is drastically reduced. Furthermore, ill-conditioned coefficient matrix caused by the global scheme is also avoided. The idea of local RBF scheme has been devoted to many research works in meshfree methods and significant results have been obtained (Liu and Gu, 2001b; Liu at al., 2005, 2006b; Kee et al., 2007). Stability is another key issue to be concerned in the RPCM. In our studies (Liu, 2002; Liu and Gu, 2005; Kee et al., 2007), the RPCM solution is found unstable while dealing with the Neumann boundary conditions in higher dimensional space. In one-dimensional space, the RPCM has demonstrated good numerical performance. However, the RPCM does not always work in the two-dimensional space. From our numerical experiments, RPCM is still able to provide excellence results without the presence of Neumann boundary conditions. However, the solution of RPCM becomes unstable whenever Neumann boundary conditions take place. Some strategies or techniques have been proposed to deal with the Neumann boundary conditions, such as special grids arrangement on the Neumann boundaries (Liszka et al., 1996), adding fictitious nodes (Kansa and Hon, 2000), special discretization scheme (Onate, 2001; Zhang et al., 2000), coupling with weak formulation (Liu and Gu, 2003a), etc. Nevertheless, there is still room for improvement. Most of the proposed strategies or techniques are either not practical to be extended to adaptive analysis or not 38 Chapter 2 Radial Point Interpolation Based Finite Difference Method effectively restore the stability. Hence, an effective and practical stabilization scheme for meshfree strong form method is desired. 2.4 Radial Point Interpolation Based Finite Difference Method In this section, the radial point interpolation based finite difference method (RFDM) is formulated. A general frame for constructing difference schemes is first proposed. Then a least-square technique is adopted to solve the differential equations. Consider now a problem with a field variable U governed by a second-order partial differential equation as: A ∂ 2U ∂ 2U ∂ 2U ∂U ∂U B C + + +D +E + FU + G = 0 2 2 ∂x∂y ∂x ∂y ∂x ∂y in Ω (2.51) with boundary conditions a ∂U + bU = f ∂x on Γ (2.52) where A, B, C , D, E , F , G, a, b and f are given constants or functions of x and y , Ω is the problem domain, Γ is the boundary of domain Ω . As shown in Fig. 2.4, the dashed lines are background grids for regular rectangular mesh. The circles are field nodes and the black dots are finite difference (FD) grid points. There are M field nodes that carry the field variable U , and N FD grid points in the problem domain. We require N > M . The classical central finite difference formulas are as follows (Kleiber, 1998) ∂U ∂x = i, j 1 (U i+1, j − U i −1, j ) 2h (2.53) 39 Chapter 2 Radial Point Interpolation Based Finite Difference Method ∂U ∂y 1 (U i, j +1 − U i, j −1 ) 2k (2.54) = 1 (U i −1, j − 2U i, j + U i+1, j ) h2 (2.55) = 1 (U i, j −1 − 2U i, j + U i, j +1 ) k2 (2.56) 1 (U i −1, j −1 − U i −1, j +1 + U i+1, j +1 − U i+1, j −1 ) 4hk (2.57) ∂ 2U ∂x 2 ∂ 2U ∂y 2 ∂ 2U ∂x∂y = i, j = i, j i, j i, j Substituting Eqs. (2.53)-(2.57) into Eq. (2.51), we have c1U i −1, j −1 + c 2U i −1, j + c3U i −1, j +1 + c 4U i , j −1 + c5U i , j + c 6U i , j +1 + c 7U i +1, j −1 + c8U i +1, j + c9U i +1, j +1 = −G (2.58) where c1 = c9 = −c3 = −c7 = B A D C E , c2 = 2 − , c4 = 2 − , 4hk 2h 2k h k C E A D ⎛ A C⎞ c5 = −2⎜ 2 + 2 ⎟ + F , c6 = 2 + , c8 = 2 + 2k 2h k ⎠ k h ⎝h (2.59) Here, h is assumed to be equal to k . When the grid points approach the boundary Γ , the difference schemes in Eqs. (2.53)-(2.57) will not work. In such cases, we adopt, accordingly, backward difference, forward difference or both methods. In addition, when geometry of the boundary is very complex, we use simply the RPCM for the nodes on and near the boundary. As described in section 2.2, the value of field function U ( x, y ) at grid point (x , y ) i j can be approximated by interpolating the values of the field function at the vicinity field nodes in the local support domain. From Eq. (2.38), we then have n U ( xi , y j ) = ∑ φlU l (2.60) l =1 40 Chapter 2 Radial Point Interpolation Based Finite Difference Method where φl is the value of the radial point interpolation shape function at the local supporting field node, and U l is the value of field function at the field node in the supporting domain. Similarly, the values of U ( x, y ) at other eight grid points in Eq. (2.58) are obtained. Thus, after incorporating the values of field function at the grid points into the differential equation (2.58), a set of discretized governing equations can be obtained. Following the procedures given by Eqs. (2.46)-(2.47), Neumann boundary conditions (2.52) can also be discretized at the nodes on Neumann boundary Γt . Finally, a set of N algebraic equations can be obtained and expressed in the matrix form K ( N ×M ) U ( M ×1) = F( N ×1) (2.61) where K ( N ×M ) denotes the stiffness matrix, U ( M ×1) is the vector of unknown variables at all nodes in the problem domain Ω and F( N ×1) is the nodal force vector. To solve the discretized equations (2.61), a least-square technique is utilized K T( M × N ) K ( N ×M ) U ( M ×1) = K T( M × N ) F( N ×1) (2.62) K ( M ×M ) U ( M ×1) = F( M ×1) (2.63) or where K ( M ×M ) = K T( M × N ) K ( N ×M ) is the modified system matrix, and F( M ×1) = K T( M × N ) F( N ×1) is the modified nodal “force” vector. Dirichlet boundary conditions (2.52) (when a = 0 ) are directly imposed at the nodes on the Dirichlet boundary Γu in the final stage. Thus, the final expression of the discretized system equations can be written as follows 41 Chapter 2 Radial Point Interpolation Based Finite Difference Method ˆ ˆ K ( M × M ) U ( M ×1) = F( M ×1) (2.64) ˆ ˆ where K ( M × M ) is the final system matrix, and F( M ×1) is the final nodal “force” vector. It should be noted that the Neumann boundary conditions are imposed in the process of forming the stiffness matrix K ( N ×M ) and nodal force vector F( N ×1) in Eq. (2.61) before implementing the least-square procedure. Dirichlet boundary conditions are imposed in the final stage only after the modified system matrix K ( M ×M ) and modified nodal “force” vector F( M ×1) are formed through the least-square procedure. It is clear that the system matrix K ( M ×M ) is symmetric and positive definite through the least-square procedure. Stable results can be then obtained using a standard linear equation solver, such as the Cholesky solver. 2.5 Numerical Examples To examine the proposed RFDM, intensive numerical studies are carried out. The RFDM is first applied to a Poisson’s equation problem that has exact solution for validity. Then an internal pressurized hollow cylinder is further investigated. As the third example, an infinite plate with a circular hole subjected to a unidirectional tensile load is considered. A bridge pier subjected to a uniformly distributed pressure on the top is studied in the example 4. To demonstrate the robustness of the proposed method to all the problem domains with irregular shapes, a relatively complicated triangle dam is tested as the last example. In this work, the MQ radial basis function augmented with quadratic polynomial function is used in computing the shape 42 Chapter 2 Radial Point Interpolation Based Finite Difference Method functions of radial point interpolation. The dimensionless parameters (see, Table 2.1) for the MQ radial basis function are taken as α c = 4.0 and q = 1.03 . In the numerical studies, an error indicator ( L2 -norm) is defined as follows ∑ (u e= numerical − u exact ) 2 ∑ (u ) exact 2 (2.65) 2.5.1 Poisson’s equation The proposed RFDM is first examined through solving a two-dimensional Poisson’s equation: ∂ 2u ∂ 2u + = sin (π x ) sin (π y ) ∂x 2 ∂y 2 (2.66) The problem domain is Ω = {( x, y ) ∈ [0, 1; 0, 1]}. The exact solution is u ( x, y ) = − 1 2π 2 sin (π x )sin (π y ) (2.67) Dirichlet boundary conditions and Neumann boundary conditions are considered in the following studies. To validate the present RFDM, we start with regular distribution of 10 × 10 field nodes, as shown in Fig. 2.5. The background grid for finite difference is 21× 21 regular rectangular mesh. A Dirichlet boundary is considered here, that is, the essential boundary conditions are imposed on all edges as u=0 along x = 0, x = 1, y = 0 and y = 1 (2.68) Both RFDM and RPCM use twenty vicinity nodes in the local support domain for interpolation. Classical FDM is also used to solve this problem with the regular background grid. The comparisons at some selected nodes are listed in Table 2.2. It 43 Chapter 2 Radial Point Interpolation Based Finite Difference Method can be found that the results obtained by the present RFDM are more accurate than those by both RPCM and FDM. To compare the present RFDM with RPCM, a critically irregular distribution of field nodes is employed and shown in Fig. 2.6, where there are 121 irregular nodes in the domain. More than thirty nodes concentrate in one corner of the domain. The background grid is still 21× 21 regular mesh. Twenty vicinity field nodes are used as the local supporting nodes. The results along x = 0.5 and y = 0.5 are plotted, respectively, in Fig. 2.7 and Fig. 2.8. Results obtained by the RPCM are relatively inaccurate. However, the RFDM is able to provide results quite close to the exact solutions using even such an extremely irregular node distribution. The use of finite difference grids has clearly and significantly improved the stability of the RPCM. In the RFDM, finite difference schemes based on regular grids are used to discretize the governing equations, which have been proven stable in general. The radial point interpolation shape functions created using nodes in local support domains can be a source of instability. However, the shape functions of radial point interpolation are used for function interpolation only, no derivatives of RPIM shape functions are used, and hence magnification of error by differentiation is avoided (Liu and Han, 2003). Therefore, the stability is significantly improved. The convergence studies are conducted using the same boundary conditions as Eq. (2.68). In Fig. 2.9, four distributions of irregular nodes are shown. They are 50, 100, 200 and 400 field nodes, respectively. Twenty vicinity nodes are used for creating shape functions. The overall error norm of field variable u is obviously improved 44 Chapter 2 Radial Point Interpolation Based Finite Difference Method from 0.11% to 0.0095% , as shown in Table 2.3 and Fig. 2.10. 2.5.2 Internal pressurized hollow cylinder A hollow cylinder under internal pressure shown in Fig. 2.11 is now used in the benchmark study. The parameters are taken as internal pressure p = 100 Pa , shear modulus G = 8000 Pa and Poisson’s ratio υ = 0.25 . This problem was studied by several other researchers (Brebbia, 1978; Gu and Liu, 2002) as a benchmark problem, since the analytical solution is available. The exact solutions of radial and circumferential stresses are σr = a2 p b2 − a2 ⎛ b2 ⎜⎜1 − 2 ⎝ r ⎞ ⎟⎟ ⎠ (2.69) σθ = a2 p b2 − a2 ⎛ b2 ⎜⎜1 + 2 ⎝ r ⎞ ⎟⎟ ⎠ (2.70) The radial displacement for plane strain problem is υ (1 − υ 2 ) r ⎛ ⎞ σr ⎟ ur = ⎜σ θ − 2(1 + υ )G ⎝ 1 −υ ⎠ (2.71) where r is the radial coordinate, a is the inner radius and b is the outer radius. Due to the symmetry of the problem, only one quarter of the cylinder needs to be modelled. As shown in Fig. 2.12, there are 95 nodes irregularly distributed in this problem domain. In the RFDM, 24 vicinity nodes are used in the support domain. The RFDM results are compared with the FEM results and analytical solutions. In this study, commercial FEM software, ANSYS, is used to compute the FEM results. In the FEM model, the same set of nodes in Fig. 2.12 is used. Triangle element is adopted in the FEM computation. The radial displacement, circumferential stress and radial 45 Chapter 2 Radial Point Interpolation Based Finite Difference Method stress along the line of y = x are plotted in Fig. 2.13 to Fig. 2.15, respectively. It can be found that the RFDM results are in very good agreement with the exact solutions. In comparison with FEM results, the radial displacement and circumferential stress by RFDM are generally more accurate than those of FEM. The convergence studies are conducted using three different nodal densities (200, 400 and 800 irregular nodes), as shown in Fig. 2.16. Twenty-one vicinity nodes are used for interpolation. The overall error norm of radial displacement u r has been improved a lot from 1.66 % to 0.45 % , as shown in Table 2.4 and Fig. 2.17. A very linear steady convergence is observed. 2.5.3 Infinite plate with a circular hole To validate the RFDM in simulating stress concentration, we consider an infinite plate with a central circular hole subjected to a unidirectional tensile load of 1.0 in the x direction. Due to the symmetry, only the upper right quadrant of the plate is modelled, as shown in Fig. 2.18. The plane strain problem is considered, and the geometries and material parameters used are a = 1 , b = 5 , Young’s modulus E = 1.0 × 10 3 and Poisson’s ratio υ = 0.3 . Symmetry conditions are imposed on the left and bottom edges, and the inner boundary of the hole is traction free. The exact solutions for the stresses in the plate are given in the polar coordinate (Timoshenko and Goodier, 1970): 46 Chapter 2 Radial Point Interpolation Based Finite Difference Method σ xx = 1 − a2 r2 ⎛3 ⎞ 3a cos 4θ ⎜ cos 2θ + cos 4θ ⎟ + 4 ⎝2 ⎠ 2r σ xy = − a2 r2 4 ⎛1 ⎞ 3a sin 4θ ⎜ sin 2θ + sin 4θ ⎟ + 4 ⎝2 ⎠ 2r σ yy 4 (2.72) 4 a2 ⎛ 1 ⎞ 3a cos 4θ = − 2 ⎜ cos 2θ − cos 4θ ⎟ − 4 r ⎝2 ⎠ 2r where (r , θ ) are the polar coordinates and θ is measured counterclockwise from the positive x -axis. Traction boundary conditions given by the exact solution (2.72) are imposed on the right ( x = 5) and top ( y = 5) edges. Fig. 2.19 shows the node distribution in the problem domain, in which there are 366 field nodes and 987 background grid points. Twenty-three vicinity nodes are used in the support domain. The distribution of stress σ xx at x = 0 obtained using the RFDM is shown in Fig. 2.20. It can be observed from this figure that the RFDM yields satisfactory results for the problem. 2.5.4 Bridge pier In this example, the RFDM is used for the stress analysis of a bridge pier subjected to a uniformly distributed pressure on the top, as shown in Fig. 2.21. The problem is solved as a plain strain case with material properties E = 4 × 1010 Pa , υ = 0.15 and loading P = 10 5 Pa . Due to the symmetry, only right half of the bridge is modelled as shown in Fig. 2.22 where there are 386 field nodes and 995 background grid points in the model and 30 vicinity nodes are used in the support domain. In this study, the FEM results are computed with ANSYS using quadratic element and the same set of nodes in Fig. 2.22. 47 Chapter 2 Radial Point Interpolation Based Finite Difference Method Comparison of the stress distribution σ yy computed by the RFDM and the FEM are shown in Fig. 2.23. The results computed by the RFDM are in good agreement with the results computed by the FEM. 2.5.5 Triangle dam of complicated shape As the last example, to generalize the present RFDM to all problem domains with irregular shapes, a triangle dam with complicated geometry subjected to a uniformly distributed pressure on the surface is studied, as shown in Fig. 2.24(a). The problem is treated as the plane strain case with the same material properties as in the bridge pier mentioned above. Due to symmetry, only the right half of the dam is simulated. The geometry of the triangle dam is shown in Fig. 2.24(b). Fig. 2.25(a) shows the node distribution of 334 irregular field nodes (dots), where there are 742 background grid points (intersections of dashed lines) in the problem domain and 24 vicinity nodes in the local supporting domain for the present RFDM. For comparison, commercial FEM software, ANSYS, is used to compute the FEM results with quadratic element and same set of nodes in Fig. 2.25(a). Since no analytical solution is available for this problem, a reference solution is obtained by ANSYS using a very fine mesh of 4462 irregular nodes, as shown in Fig. 2.25(b). Fig. 2.26 shows the displacement components along the line of x = 8 . It can be found that the RFDM results are more accurate than those of FEM, according to the FEM reference solutions. The stress distribution σ yy by RFDM is plotted in Fig. 2.27(a). The result obtained using ANSYS with the same nodes as RFDM is shown in Fig. 2.27(b). Fig. 2.27(c) is the reference result for σ yy . It can be concluded that the 48 Chapter 2 Radial Point Interpolation Based Finite Difference Method RFDM results are accurate enough for general engineering requirement. 2.6 Parameter Study There are two important parameters used in the present RFDM: the numbers of local supporting nodes and finite difference grid points. In this work, the effects of the number of local nodes in support domain are investigated using one simple example. The detailed description on the effects of local supporting nodes can be found in the paper by Kee et al. (2008). Also, the relations between the numbers of finite difference grid points and field nodes are discussed in details. 2.6.1 Number of local supporting nodes In this study, a simple Poisson problem with the same governing equation and exact solution in Eq. (2.66) and (2.67) is considered. The following mixed boundary conditions are considered in problem domain Ω , where Neumann boundary conditions are ∂u ∂x =− x =0 ∂u ∂x 1 sin(π y ) , 2π = x =1 1 sin(π y ) 2π (2.73) and Dirichlet boundary conditions are u=0 along y = 0 and y = 1 (2.74) This problem is modeled by a set of 100 irregularly scattered nodes, as shown in Fig. 2.28. The error norms of the numerical solution obtained by the RFDM based on the local RBFs with different numbers of local nodes are plotted in Fig. 2.29. One can observe that the accuracy of the numerical solutions provided by the RFDM based on 49 Chapter 2 Radial Point Interpolation Based Finite Difference Method local RBFs can be as good as the RFDM based on global RBFs. However, the computational time required for the RFDM based on local RBFs is much lesser than the RFDM based on global RBFs, as shown in Fig. 2.30. The CPU time required by the RFDM using 16 local nodes is 60 times less than the time using global RBFs. The computational time is increasing exponentially as the number of local supporting nodes increases. It is also observed that the condition number of the coefficient (stiffness) matrix K is increasing greatly while more local supporting nodes are used for function approximation as illustrated in Fig. 2.31. The condition number of K formulated using 16 local supporting nodes is less than half of the condition number of K formulated using global supporting nodes. The condition of K will get worse and adversely affect the accuracy of the numerical solution if the field nodes become more and randomly distributed. 2.6.2 Relations between the numbers of grid points and field nodes In this study, two aforementioned numerical examples are presented: Poisson’s equation and internal pressurized hollow cylinder. Twenty vicinity nodes are fixed as the local supporting nodes for interpolation. In the Poisson’s problem, the boundary conditions are the same as Eqs. (2.73) and (2.74). Three distributions of regular field nodes are used in the investigation: 15 × 15 , 21 × 21 and 30 × 30 . The “optimal” numbers of grid points ( N ) with respect to the corresponding field nodes ( M ) for Poisson’s equation are shown in Table 2.5 and Fig. 2.32. It is found that the ratio of N M should be between 2 and 3. In Eq. (2.61), there are M unknown variables and N algebraic equations. To get the solutions, 50 Chapter 2 Radial Point Interpolation Based Finite Difference Method N should not be less than M . On the other hand, if N becomes too large, the M unknown variables will be over smoothed through the least-square procedure. Namely, the solutions will be inaccurate. In the internal pressurized hollow cylinder, as shown in (2.16), three distributions of irregular field nodes are investigated. The “optimal” numbers of grid points ( N ) with respect to the corresponding field nodes ( M ) for hollow cylinder are shown in Table 2.6 and Fig. 2.33. It is also found that the ratio of N M is between 2 and 3. According to the above parameter investigations made for both Poisson’s equation and internal pressurized hollow cylinder, the relationship between M and N is proposed as follows N ∈ [2M , 3M ] (2.75) 2.7 Remarks In this work, a radial point interpolation based finite difference method (RFDM) has been presented for solving partial differential equations, with an emphasis on solid mechanics problems. By incorporating the radial point interpolation into the classical finite difference approach, the proposed RFDM overcomes the instability of radial point collocation method. The use of a least-square technique helps further to obtain a system matrix with good properties and the resulting set of algebraic equations can be solved more efficiently and accurately by using standard solver such as Cholesky solver. A number of numerical examples are studied and some important parameters are investigated in detail. 51 Chapter 2 Radial Point Interpolation Based Finite Difference Method From the research work conducted, the following conclusions can be drawn: 1. Shape function generated using RBFs augmented with polynomials possess the Delta property, which allows the essential boundary conditions to be enforced directly. 2. From the comparison studies with the radial point collocation method, it is found that the RFDM has good stability due to the use of finite difference grids to get the discrete system equations. Shape functions constructed using local supporting nodes are used for function interpolation only (not the derivatives). 3. Based on the study of examples in this paper, the relationship between field nodes M and the corresponding finite difference grid points N is recommended as N = (2 ~ 3) M for the RFDM. 4. From the stress analyses of several numerical examples, the RFDM provides very good results compared to even the well-established finite element method. 5. As demonstrated in this work, the RFDM can be applied with good performance to problem domains of irregular shapes. In summary, it is concluded that the RFDM is a stable, robust and reliable numerical method based on strong form formulation for mechanics problems. However, the computational cost has been increased slightly. From the next chapter, more efficient strong form methods will be developed. 52 Chapter 2 Radial Point Interpolation Based Finite Difference Method Type Expression [ Dimensionless Parameters ] Ri (x ) = ri 2 + (α c d c ) Multi-quadrics (MQ) Gaussian (EXP) αc ≥ 0 , q 2 q Ri ( x ) = exp ⎛ r ⎞ −α c ⎜ i ⎟ ⎝ dc ⎠ 2 αc Thin plate spline (TPS) Ri (x ) = riη η Logarithmic Ri (x ) = riη log ri η Table 2.1 Typical radial basis functions with dimensionless shape parameters. Points RPCM FDM RFDM Exact A(0.5,0.5) -0.05088411 -0.05076489 -0.05071812 -0.05066059 B(0.95,0.5) -0.00786775 -0.00794138 -0.00791603 -0.00792506 C(0.5,0.3) -0.04112709 -0.04106966 -0.04099582 -0.04098528 Table 2.2 Computed results of Poisson’s equation. No. of field nodes 50 100 200 400 Error norm 1.1142E-003 5.8513E-004 3.0903E-004 9.5384E-005 Table 2.3 Error norms of solution for Poisson’s equation. 53 Chapter 2 Radial Point Interpolation Based Finite Difference Method No. of field nodes 200 400 800 Error norm 1.6565E-002 8.5961E-003 4.4910E-003 Table 2.4 Error norms of solution for internal pressurized hollow cylinder. No. of field nodes ( M ) 225 441 900 No. of optimal grid points ( N ) 576 1024 2209 Ratio ( N M ) 2.56 2.32 2.45 Table 2.5 Optimal grid points for Poisson’s equation. No. of field nodes ( M ) 200 400 800 No. of optimal grid points ( N ) 498 970 1881 Ratio ( N M ) 2.49 2.43 2.35 Table 2.6 Optimal grid points for internal pressurized hollow cylinder. 54 Chapter 2 Radial Point Interpolation Based Finite Difference Method Fig. 2.1 Pascal’s triangles of monomials for two-dimensional space. × × Local support domain × Problem domain × : point of interest : field nodes Fig. 2.2 Local support domains used in meshfree methods. 55 Chapter 2 Radial Point Interpolation Based Finite Difference Method Ω : Global domain Ω I : Local support domain Γu : Dirichlet boundary Γt : Neumann boundary Fig. 2.3 A problem governed by PDEs in domain Ω . y Background grids for finite difference Field nodes y j +1 yj y j −1 Problem domain Ω FD Grid points Γ k k xi −1 xi xi +1 h x h Fig. 2.4 Background grids for finite difference used in the RFDM. 56 Chapter 2 Radial Point Interpolation Based Finite Difference Method 1 Location y 0.8 0.6 B A 0.4 C 0.2 0 0 0.2 0.4 0.6 Location x 0.8 1 Fig. 2.5 100 regular field nodes ( • ) and 441 finite difference grid points ( × ). 1 Location y 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 Location x 0.8 1 Fig. 2.6 Distribution of 121 extremely irregular nodes for Poisson’s equation. 57 Chapter 2 Radial Point Interpolation Based Finite Difference Method 0 -0.01 RFDM RPCM Exact u(0.5,y) -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 0 0.2 0.4 0.6 Location y 0.8 1 0.8 1 Fig. 2.7 Result along the line of x = 0.5 for Poisson’s equation. 0 -0.01 RFDM RPCM Exact u(x,0.5) -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 0 0.2 0.4 0.6 Location x Fig. 2.8 Result along the line of y = 0.5 for Poisson’s equation. 58 Chapter 2 Radial Point Interpolation Based Finite Difference Method 1 1 0.5 0.5 0 0 0.5 (a) 0 1 0 1 1 0.5 0.5 0 0 0.5 (c) 0 1 0 0.5 (b) 1 0.5 (d) 1 Fig. 2.9 Node distributions: (a) 50; (b) 100; (c) 200 and (d) 400 nodes. -3 Error norm 10 10 -4 2 10 Number of field nodes Fig. 2.10 Error norms of solution for Poisson’s equation. 59 Chapter 2 Radial Point Interpolation Based Finite Difference Method p 20 15 15 Fig. 2.11 Hollow cylinder subjected to internal pressure. Number of field nodes = 95 25 Location y 20 15 10 5 0 0 5 10 15 Location x 20 25 Fig. 2.12 Node distribution for the hollow cylinder. 60 Chapter 2 Radial Point Interpolation Based Finite Difference Method 0.085 0.08 RFDM FEM Exact Radial displacement ur 0.075 0.07 0.065 0.06 0.055 0.05 0.045 0.04 10 15 Radius r 20 25 Fig. 2.13 Radial displacement u r along the line of y = x in the hollow cylinder. 160 RFDM FEM Exact Circumferential stress σ θ 140 120 100 80 60 40 20 10 15 20 25 Radius r Fig. 2.14 Circumferential stress σ θ along the line of y = x in the hollow cylinder. 61 Chapter 2 Radial Point Interpolation Based Finite Difference Method 0 -10 -20 Radial stress σr -30 -40 -50 -60 -70 RFDM FEM Exact -80 -90 -100 10 15 20 Radius r 25 Fig. 2.15 Radial stress σ r along the line of y = x in the hollow cylinder. 25 25 20 20 15 15 10 10 5 5 0 0 5 10 15 (a) 5 10 20 25 0 0 5 10 15 (b) 20 25 25 20 15 10 5 0 0 15 20 25 (c) Fig. 2.16 Node distributions in the hollow cylinder: (a) 200; (b) 400; (c) 800 nodes. 62 Error norm Chapter 2 Radial Point Interpolation Based Finite Difference Method 10 -2 10 Number of field nodes 3 Fig. 2.17 Error norms of of radial displacement ur for hollow cylinder. y b p a x Fig. 2.18 Quarter model of the infinite plate with a circular hole subjected to a unidirectional tensile load. 63 Chapter 2 Radial Point Interpolation Based Finite Difference Method 5 Location y 4 3 2 1 0 0 1 2 3 Location x 4 5 Fig. 2.19 Node distribution: 366 nodes ( • ) & 987 points (intersections of the dashed). 3.5 RFDM Analytical 3 Stress σxx 2.5 2 1.5 1 0.5 1 2 3 Location y 4 5 Fig. 2.20 Normal stress σ xx along the edge of x = 0 in the plate. 64 Chapter 2 Radial Point Interpolation Based Finite Difference Method Fig. 2.21 A bridge subjected to a uniformly distributed pressure on the top. 30 25 20 15 10 5 0 0 5 10 15 20 Fig. 2.22 Nodal distribution in the bridge model: 386 field nodes (dots) and 995 grid points (intersections of dashed lines). 65 Chapter 2 Radial Point Interpolation Based Finite Difference Method Fig. 2.23 Distribution of normal stress σ yy in the bridge: (a) RFDM; (b) ANSYS. 66 Chapter 2 Radial Point Interpolation Based Finite Difference Method (a) (b) Fig. 2.24 A triangle dam subjected to uniformly distributed pressure on the surface. 67 Chapter 2 Radial Point Interpolation Based Finite Difference Method (a) (b) Fig. 2.25 Node distributions in the triangle dam: (a) 334 and (b) 4462 field nodes. 68 Chapter 2 Radial Point Interpolation Based Finite Difference Method (a) (b) Fig. 2.26 Displacements along the line of x = 8 : (a) x -direction; (b) y -direction. 69 Chapter 2 Radial Point Interpolation Based Finite Difference Method Fig. 2.27 Normal stress σ yy distribution: (a) RFDM; (b) ANSYS; (c) Reference. 70 Chapter 2 Radial Point Interpolation Based Finite Difference Method Fig. 2.28 Distribution of a set of 100 randomly scattered nodes in a square domain. Fig. 2.29 Error norms of the field variable u computed by RFDM based on RBFs using different numbers of local supporting nodes. 71 Chapter 2 Radial Point Interpolation Based Finite Difference Method Fig. 2.30 CPU time required by RFDM based on RBFs using different numbers of local supporting nodes. Fig. 2.31 Condition numbers of the coefficient matrix of the RFDM based on RBFs using different numbers of local supporting nodes. 72 Number of grid points Chapter 2 Radial Point Interpolation Based Finite Difference Method 10 3 Number of grid points Number of field nodes Fig. 2.32 Optimal grid points for Poisson’s equation. 10 10 3 3 Number of field nodes Fig. 2.33 Optimal grid points for internal pressurized hollow cylinder. 10 3 73 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation Chapter 3 Gradient Smoothing Method: The Theoretical Formulation 3.1 Introduction Meshfree methods have been developed recently and implemented on nodes at first place instead of meshes as done in traditional methods, e.g., finite element method. A survey paper written by Babuska et al. (2002) provides the mathematical foundation of various meshfree methods. Overview of computational and implemental issues related to meshfree methods of both weak and strong forms can be found in the monographs by Liu (Liu, 2002; Liu and Gu, 2005). By removing some restrictions due to the use of cells, meshfree methods are more flexible and more suitable for adaptive analysis. They have been successfully applied to problems where cell-based methods are often difficult to give satisfactory results. For example, meshfree methods are very attractive for capturing crack and shock in structure mechanics, and interface between multi-phases in fluid flow problems. In general, many meshfree methods used on weak form are comprehensively studied. When governing equations in strong form are treated, most of these meshfree methods exhibit instability issues and thus special techniques are required to resolve it (Liu et al., 2006; Kee et al., 2007). In meshfree methods, gradient smoothing is often used. It is a key technique in the widely used smoothed particle method (SPH) (Lucy, 1977; Liu and Liu, 2003) and is of 74 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation great importance in stabilizing the nodal integrations and installing linear conformability (Chen et al., 2001; Liu et al, 2005, 2006) for methods based on Galerkin weak form. This thesis presents a novel and efficient numerical scheme called gradient smoothing method (GSM) based on strong form governing equations. In GSM, the partial differential equations (PDEs) are directly discretized at nodes with the help of relevant gradient smoothing domains. Since the governing equations are directly discretized at nodes in the physical domain, the implementation procedure is as simple as the traditional finite difference method. However, GSM can be applied for arbitrary geometries. In the following sections, the theory of GSM is first introduced. The GSM approximations to the gradients (first-order derivatives) and Laplace operator (second-order derivatives) of a field variable are presented in detail. Stencil analyses on coefficients of influence corresponding to various schemes of GSM are then conducted. Important features of stencils for the discretized Laplace operator are discussed. Numerical solutions to Poisson’s equations are obtained using four recommended GSM schemes and investigated in details to reveal the properties on convergence and stability. The computational efficiency, accuracy in results, and robustness to the mesh irregularity for GSM are also intensively examined. 3.2 Gradient Smoothing Method (GSM) In the GSM, derivatives at various locations, including nodes, centroids of cells and midpoints of edges of cells, can be approximated on relevant gradient smoothing 75 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation domains with gradient smoothing operations. The details about the principles of the GSM and its approximations to derivatives are introduced in the following subsections. 3.2.1 Gradient smoothing A two-dimensional elastostatic problem is governed by the following equilibrium equation in the domain Ω : σ ij , j + bi = 0 in Ω (3.1) where σ ij is the stress tensor and bi is the body force. Boundary conditions are given as follows: ui = ui on Γu (3.2) σ ij n j − ti = 0 on Γt (3.3) where ui denotes the prescribed boundary displacement on Dirichlet boundary Γu ; ti is the traction on Neumann boundary Γt and ni is the unit outward normal vector. In the present method, the problem domain Ω is discretized by triangular cells as shown in Fig. 3.1. For the i th node, a smoothing domain Ω i is generated by sequentially connecting the centroids with mid-edge points of surrounding triangular cells. Γi is the boundary of the smoothing cell Ω i . A smooth operation to the gradient of field variable u is adopted as follows (Chen et al., 2001; Liu et al., 2005, 2007): ∇ h u ( x i ) = ∫ ∇ h u ( x ) Φ ( x − x i ) dΩ i Ωi (3.4) 76 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation Integration by parts for Eq. (3.4) leads to ∇ h u ( x i ) = ∫ u h ( x ) n ( x ) Φ ( x − x i ) d Γ − ∫ u h ( x ) ∇ Φ ( x − x i ) dΩ Γi Ωi (3.5) where Φ is a smoothing function. n denotes the unit normal vector of any one of domain faces, as shown in Fig. 3.1. Consider a weighted Shepard function (Shepard, 1968) as the smoothing function Φ(x − x i ) = φ (x − x i ) M ∑ φ (x − x j =1 j ) Aj (3.6) where Ai = ∫ dΩ is the area (or volume) of the representative domain (smoothing Ωi domain) of the i th field node obtained from the diagram in Fig. 3.1. M is the number of surrounding triangular cells for the i th node. The weighted Shepard function in Eq. (3.6) meets the following weighted partition of unity: M ∑ Φ (x − x j =1 j ) Aj = 1 (3.7) For simplicity, a piecewise constant function φ is used in the present method: ⎧1 ⎩0 φ (x − x i ) = ⎨ x ∈ Ωi x ∉ Ωi (3.8) Consequently the smoothing function is ⎧1 Ai x ∈ Ω i Φ (x − x i ) = ⎨ x ∉ Ωi ⎩0 (3.9) Substituting Eq. (3.9) into Eq. (3.5), the smoothed gradient of field function u is obtained: ∇ h u ( x i ) = ∫ u h ( x ) n( x ) Φ ( x − x i ) d Γ = Γi 1 u h ( x ) n( x ) d Γ ∫ Ai Γi (3.10) Note that the choice of constant Φ makes the second term on the right-hand side of 77 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation Eq. (3.5) vanish, and the area integration becomes line integration along the edges of smoothing cell. Similarly, by successively using the smoothing procedures described in Eqs. (3.4)-(3.10), the second-order gradients (derivatives) at the same location can be evaluated easily by differentiating Eq. (3.10) as ∇ 2 u(x i ) = 1 ∇ h u ( x ) n( x ) d Γ ∫ Ai Γi (3.11) Hence, spatial derivatives at any node of interest can be approximated using Eqs. (3.10) and (3.11) together with proper smoothing domains that will be discussed in the following sections. 3.2.2 Smoothing domains In the GSM, problem domain is first divided into a set of meshes (regular or irregular) formed by connecting nodes, and the values of field functions are stored at nodes and a set of system equations is formed by approximating derivatives at nodes.. Based on these meshes, a smooth domain can be constructed for any point of interest. Depending on the location of a point of interest, different types of smoothing domains are used based on the compact principle. As shown in Fig. 3.1, three types of gradient smoothing domains, which are used for approximation of spatial derivatives, are constituted on the basis of primitive unstructured triangular meshes. The first is node-associated gradient smoothing domain (nGSD) for the approximation of derivatives at any field node of interest. It is formed by connecting relevant centroids of the triangles with midpoints of the corresponding connecting edges. The second is a triangular cell itself formed by primitive mesh, which is employed for 78 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation approximating derivatives at the centroid of the cell. It is called centroid-associated gradient smoothing domain (cGSD) here. The third is named midpoint-associated gradient smoothing domain (mGSD) used for the calculation of gradients at the midpoint of an edge of interest. The favorable mGSD is formed by connecting the end-nodes of an edge of interest with the centroids of the triangles on the both sides of the edge, as shown in Fig. 3.1. For approximation of the first-order derivative at any field node, only nGSD is used. For the second-order derivatives at field nodes, the values of the first-order derivative at the centroid of the triangles and midpoint of the connecting edges surrounding the node of interest are first needed. They are calculated using the cGSD and mGSD respectively. The same nGSD is also used for the approximation of the second-order derivatives at corresponding field node. To calculate the gradients at midpoints of edges and centroids of cells with mGSD and cGSD, Eqs. (3.10) and (3.11) can also be used for approximation in the analogous manner. This novel combination of use of the three types of domains provides stability and ensures the accuracy of the solution. Different schemes can be devised for this purpose, and the details will be elucidated further in the next sections. 3.2.3 Discretization schemes We now need to accurately evaluate the integrals along the boundaries of various types of GSDs. In the current work, both the one-point quadrature (rectangular rule) and two-point quadrature (trapezoidal rule) are used and examined for the approximation of derivatives at nodes. Only two-point quadrature is used for the 79 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation approximation of gradients at both midpoints and centroids. As listed in Table 3.1, eight discretization schemes for spatial differential terms are developed, using different types of quadrature and method of approximation. In the one-point quadrature schemes (I, II and VII), the integration over any smoothing domain edge is approximated with the rectangular rule where the integrand is evaluated only at the midpoint of a mesh-edge of interest. In the two-point quadrature schemes (III, IV, V, VI and VIII), the integration is evaluated with the trapezoidal rule, where values of the field variable and its gradients at the two end-nodes of each smoothing domain edge (the midpoint of the edge of interest and the centroid of a triangular cell) are used. In this work, both the first- and second-order derivatives at a node of interest are always approximated with gradient smoothing operation detailed in Section 3.2.1. The gradients at midpoints of mesh-edges can be calculated in two ways: either by simple interpolation using gradients at the both end-nodes of the edge (I, II, III, IV, V and VI) or by gradient smoothing using Eq. (3.10) based on the mGSD (VII and VIII). Similarly, the gradients at centroids can be obtained either by simple interpolation using gradients at the nodes of the cGSD (III and IV) or by gradient smoothing based on the cGSD (V, VI and VIII). Note that when one-point quadrature schemes are used, there is no need to approximate the gradients at centroids, since the integrands in Eqs. (3.10) and (3.11) are evaluated only at the midpoints of the mesh-edges. 3.2.4 Formulae for derivative approximation 80 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation 3.2.4.1 Two-point quadrature schemes • First-order derivatives at nodes Using Eq. (3.10), the first-order derivatives of the field variable u are given by ∂ui 1 = ∂x Ai ⎧1 ∑ ⎨⎩ 2 (ΔS )( ) [(u ) + (uc )ijk + ∂u i 1 = Ai ∂y ⎧1 ∑ ⎨⎩ 2 (ΔS )( ) [(u ) + (u c )ijk + ni k =1 ni k =1 L x ijk L y ij k (L ) m ijk (L ) m ijk ] [ ] 1 (ΔS x )ij(Rk ) (um )ijk + (uc )ij(Rk ) ⎫⎬ 2 ⎭ ] 12 (ΔS )( ) [(u R y ij k ) m ijk ] (R ) ⎫ + (u c )ijk ⎬ ⎭ (3.12) (3.13) where (ΔS x )ij(L ) = ΔS ij(L ) (n x )ij(L ) (3.14) (ΔS )( ) = ΔS ( ) (n )( ) (3.15) (ΔS x )ij(R ) = ΔSij(R ) (n x )ij(R ) (3.16) (ΔS )( ) = ΔS ( ) (n )( ) (3.17) k L y ij k k R y ij k k L ijk k R ijk k L y ij k k R y ij k In equations above, u , um and uc denote values of the field variable u at nodes, midpoints of mesh-edges and centroids of triangular cells, respectively. ΔS x and ΔS y are the two components of the length of a domain edge. n x and n y represent the two components of the unit normal vector of a domain edge. i denotes the node of interest and jk is the other end-node of the edge linked to node i (see Fig. 3.1). Superscripts ( L) and ( R) are pointers to the two domain faces (left-side and right-side) associated with the edge of interest, ijk . The total number of supporting nodes within the stencil of node i is denoted by ni . These geometrical parameters are computed and stored before the intensive calculation is started. The values of the field variable u at non-storage locations, i.e. um and uc , are computed by arithmetic averaging of function values at related nodes, respectively, in the fashion of 81 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation (um )ij k = ui + u jk 2 ⎧ (u + u k + u jk +1 ) 3 , k = 1,2, L , ni − 1 k = ni jni + u j1 ) 3 , ⎩ i (3.19) ⎧(u + u k + u jk −1 ) 3 , k = 2,3, L , ni k =1 j1 + u jni ) 3 , ⎩ i (3.20) (uc )ij(L ) = ⎨(u i + u j k (uc )ij(R ) = ⎨(ui + u j k • (3.18) First-order derivatives at midpoints and centroids Analogous to the discretization at field nodes described above, the first-order derivatives at midpoints of connecting edges ( (∇u m )ijk ) and centroids of the triangles ( (∇u c )ijk (L ) and (∇u c )ijk ) can also be approximated with the gradient smoothing (R ) technique using Eqs. (3.10), but based on the related mGSD and cGSD, respectively. This applies for gradients at midpoints in Schemes VII and VIII, and gradients at centroids in Schemes V, VI and VIII. Similarly, the geometrical parameters including the areas, edge vectors and normal vectors of mGSDs and cGSDs should be predetermined and stored for solving the algebraic equations. Alternatively, the gradients at non-storage positions can be approximated by simple interpolation of gradients at relevant nodes. The discretized gradients in this manner take the same formulae as in Eqs. (3.18)-(3.20), but with the substitution of the gradients (∇u ) for field variable (u ) . In Schemes I, II, III, IV, V and VI, the gradients at midpoints are approximated in this manner. So are the gradients at centroids in Schemes III and IV. • Second-order derivatives at nodes The second-order derivatives with two-point quadrature schemes are obtained using Eq. (3.11) and given in the following form 82 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation ⎧1 (ΔS x )ij(Lk ) ⎡⎢ ∂ (um )ijk + ∂ (uc )ij(Lk ) ⎤⎥ ⎫⎪ ⎪ ∂x ∂ ui 1 ⎪2 ⎣ ∂x ⎦ ⎪ = ∑⎨ ⎬ 2 ∂x Ai k =1 ⎪ 1 ∂ (R ) ⎡ ∂ (R ) ⎤ + (ΔS x )ijk ⎢ (um )ijk + (uc )ijk ⎥ ⎪ ⎪⎩ 2 ∂x ⎦ ⎪⎭ ⎣ ∂x (3.21) ⎧1 ∂ (L ) ⎡ ∂ (L ) ⎤ ⎫ ⎪ (ΔS y )ijk ⎢ (um )ijk + (uc )ijk ⎥ ⎪ ∂y ∂ ui 1 ⎪2 ⎣ ∂y ⎦ ⎪ = ∑⎨ ⎬ 2 Ai k =1 ⎪ 1 ∂y ∂ (R ) ⎡ ∂ (R ) ⎤ ⎪ + (ΔS y )ij ⎢ (um )ijk + (uc )ijk ⎥ ⎪ 2 k ∂y ⎣ ∂y ⎦ ⎪⎭ ⎩ (3.22) ⎧1 (ΔS y )ij(Lk ) ⎡⎢ ∂ (um )ijk + ∂ (uc )ij(Lk ) ⎤⎥ ⎫⎪ ⎪ ∂x ∂ ui 1 ⎪2 ⎣ ∂x ⎦ ⎪ = ∑⎨ ⎬ ∂x∂y Ai k =1 ⎪ 1 ∂ (R ) ⎡ ∂ (R ) ⎤ + (ΔS y )ij ⎢ (um )ijk + (uc )ijk ⎥ ⎪ k ⎪⎩ 2 ∂x ⎦ ⎪⎭ ⎣ ∂x (3.23) 2 2 2 ni ni ni All the first-order derivatives used in Eq. (3.21)-(3.23) are approximated as described in previous two subsections. 3.2.4.2 One-point quadrature schemes In the one-point quadrature schemes (I, II and VII), it is assumed that (uc )ij(L ) = (uc )ij(R ) = (um )ij k k (3.24) k and (∇uc )ij(L ) = (∇uc )ij(R ) = (∇um )ij k k k (3.25) As a result, Eqs. (3.12), (3.13) and (3.21)-(3.23) can be simplified as ∂ui 1 = ∂x Ai ∑ (ΔS ) (u ) (3.26) ∂ui 1 = ∂y Ai ∑ (ΔS ) (u ) (3.27) ni k =1 x ijk m ijk ni k =1 y ij k m ijk and ∂ 2 ui 1 = 2 Ai ∂x ni ∑ (ΔS ) k =1 x ijk ∂ (um )ijk ∂x (3.28) 83 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation ∂ 2 ui 1 = 2 Ai ∂y ∑ (ΔS ) ∂ (um )ijk ∂y (3.29) ∂ 2 ui 1 = ∂x∂y Ai ∑ (ΔS ) ∂ (um )ijk ∂x (3.30) ni k =1 y ij k ni k =1 y ij k where (ΔS x )ij = (ΔS x )ijk + (ΔS x )ijk (3.31) (ΔS ) = (ΔS y )ij + (ΔS y )ij (3.32) k y ij k (L ) (R ) (L ) (R ) k k As shown in Eqs. (3.26)-(3.30), in one-point quadrature schemes, only values for the field variable and its gradients at midpoints are needed in the approximations. Thus, the domain edge vectors for any pair of domain edges connected with the mesh-edge ijk can be lumped together, which in return reduces the storage space for geometrical parameters. It should be noted that in Scheme VII, the gradients at midpoints in Eqs. (3.28)-(3.30) are approximated with gradient smoothing technique based on mGSDs, while they are approximated using simple interpolation approach in Schemes I and II. The one-point quadrature schemes are clearly simpler and much more cost-effective. Schemes based on two-point quadrature impose extra requirements in computation and storage of values at centroids and face vectors for cGSDs. When mGSDs are used for approximation of gradients at midpoints, such demands become a bit higher. However, schemes based on two-point quadrature can give more accurate results. This will be verified and discussed later in the section on numerical examples. 3.2.4.3 Directional correction It is found that decoupling solutions are predicted when the gradients at midpoints, which are approximated using simple interpolation approach, are used for 84 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation approximating the second-order derivatives. To circumvent such a problem, the approximated gradients at midpoints are required to be remedied with the directional correction technique proposed by Crumpton et al. (1997). As in Schemes II, IV and VI, the directional correction takes the form of (∇u~m )ij k ⎡ r ⎛ ∂u ⎞ ⎤ r = (∇um )ijk − ⎢(∇um )ijk ⋅ tijk − ⎜ ⎟ ⎥ tijk ⎝ ∂l ⎠ ijk ⎥⎦ ⎢⎣ (3.33) where (∇um )ij k = 1 (∇ui + ∇u jk ) 2 (3.34) u j − ui ⎛ ∂u ⎞ ⎜ ⎟ ≈ k Δlijk ⎝ ∂l ⎠ ijk (3.35) r rijk r tijk = Δlijk (3.36) r rijK = X jK − X i (3.37) ΔlijK = X jK − X i (3.38) Here X i and X jK denotes the spatial positions of node i and jk , respectively. Details about the role of directional correction will be addressed in the following section on stencil analyses. 3.3 Analyses of Discretization Stencil Before conducting intensive numerical investigations, careful studies of the stencils of supporting nodes for various schemes proposed for the GSM are carried out. The coefficients of influence for the node where derivatives are approximated are derived and analyzed. The objectives for stencil analyses are to select the most suitable schemes that satisfy the basic principles of numerical discretization. For 85 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation clarity, the stencils for approximating the Laplace operator using both uniform Cartesian and equilateral triangular meshes are focused in current study. 3.3.1 Basic principles for stencil assessment In the stencil analyses, the following five basic rules are considered to assess the quality of a stencil resulting from a discretization scheme: (a) Consistency at each domain face; (b) Positivity of coefficients of influence; (c) Negative-slope linearization of the source term; (d) Sum of the neighbor coefficients; (e) The compactness of the stencil. The first four rules are summarized by Patankar (1980) with consideration of solutions with physically realistic behavior and overall balance. To satisfy Rule (a), it requires that the same expression of approximation must be used on the interface of two adjacent GSDs, so that when the gradient smoothing technique is applied to the GSDs, the local conservation of quantities is automatically ensured and so for the global conservation. Rule (b) requires that the coefficient for the node of interest and the coefficients of influence must be positive, once the discretization equation is ni written in the form of aii ui = ∑ aijk u jk + bi . Rule (c) relates to the treatment of the k =1 source terms. As addressed by Patankar (1980), it is essential to keep the slope of linearization to be negative, since a positive slope can lead to computational ni instabilities and physically unrealistic solutions. Rule (d) tells aii = ∑ aijk . k =1 86 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation Besides, Barth (2003) proposed a few lemmas to address the necessity of positivity of coefficients to satisfy a discrete maximum principle that is a key tool in the design and analysis of numerical schemes suitable for non-oscillatory discontinuity (for example, shock). At steady state, non-negativity of the coefficients becomes sufficient to satisfy a discrete maximum principle that can be applied successively to obtain global maximum principle and stable results. His statements reiterate the importance of Rule (b) as mentioned by Patankar (1980). In addition, as commented by Barth (2003), the very first layer of nodes surrounding the node of interest should be included in its stencil. Moreover, as the stencil becomes larger, not only the computational cost increases, but eventually the accuracy decreases as less valid data from further away is brought into approximation. Thus, for the concerns about numerical accuracy and efficiency, Rule (e) on the compactness of the stencil is adopted as additional factor for the assessment of discretization schemes. 3.3.2 Stencils for approximated gradients The stencils for gradient approximation using the eight types of discretization schemes are derived and the coefficients are shown in Fig. 3.2 and Fig. 3.3. 3.3.2.1 Uniform Cartesian mesh It is found that the three one-point quadrature schemes (I, II and VII) give the same stencil when uniform Cartesian mesh is used, as shown in Fig. 3.2(a). This stencil is also identical to that of 2-point based central-differencing scheme in the FDM. It is also observed that the stencil for all two-point quadrature schemes is the same too, as shown in Fig. 3.2(b). This stencil is identical to that of 6-point based 87 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation central-differencing scheme in the FDM. This finding confirms that when uniform Cartesian meshes are used, the GSM is identical to the FDM. The GSM schemes, however, works for irregular meshes. 3.3.2.2 Equilateral triangular mesh It is interesting to know that the stencil for approximated gradients on equilateral triangular mesh is the same regardless of the GSM schemes, as shown in Fig. 3.3. This stencil is identical to that of interpolation method using six surrounding nodes. Note that for irregular triangular meshes, the interpolation method will fail as addressed by Liu (2002), but our GSM still performs well, as will be demonstrated in the section on numerical examples. This is due to the crucial stability provided by the smoothing operation. 3.3.3 Stencils for approximated Laplace operator 3.3.3.1 Uniform Cartesian mesh The stencils for the approximated Laplace operator with GSM schemes on uniform Cartesian meshes are derived and listed in Fig. 3.4. Three schemes I, III and V, given in Fig. 3.4(a), (c) and (e), result in wide stencils with unfavorable weighting coefficients (zeros and negatives) on uniform Cartesian meshes. With such kinds of stencils, unexpected decoupling solutions may be produced (Blazek, 2001). This is also confirmed in current study and will be illustrated in the following section on numerical examples. As mentioned by Monier (1999), such kinds of stencils can not damp out high frequency numerical errors. Therefore, Schemes I, III and V are now labeled as unfavorable. 88 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation It is found that such unfavorable stencils are resulted from the simple interpolation that is used to approximate the gradients at the midpoints in the three schemes. In Schemes II and VI where the directional correction to the approximated gradients at midpoints is made, relatively compact stencils with favorable coefficients are obtained, as depicted in Fig. 3.4(b) and Fig. 3.4(f). Scheme II is a 5-point based stencil and Scheme VI corresponds to 9-node based compact stencil. However, as shown in Fig. 3.4(d), even with directional correction, unfavorable stencil can still occur as in Scheme IV. Therefore, Schemes II and VI are found to be favorable and Scheme IV is also labeled as unfavorable. The compact and favorable stencils are obtained using Schemes VII and VIII where the gradients at midpoints are approximated also by applying gradient smoothing operation to the mGSDs, as seen in Fig. 3.4(b) and (f). This implies that using gradient smoothing technique on mGSDs to approximate the gradients at midpoints is a good alternative to using simple interpolation with directional correction technique. From the point of views of the consistency in the approximation of derivatives at different locations, Schemes VII and VIII are superior to Schemes II and VI. 3.3.3.2 Equilateral triangular mesh The analyses are also conducted on equilateral triangular mesh and resulted stencils are shown in Fig. 3.5. It is found again that Schemes I and III result in an unfavorable stencil, because the coefficients for the first layer of supporting nodes are negative as seen in Fig. 3.5(a), which violates the basic Rule (b). Schemes IV and V, 89 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation and Schemes II, VI, VII and VIII produce two sets of stencils with favorable coefficients, as shown in Fig. 3.5(b) and (c), respectively. The stencil for Schemes IV and V consists of the two layers of surrounding points around the node of interest. The stencil for Schemes II, VI, VII and VIII uses only the first layer of neighboring nodes. According to Rule (e), schemes II, VI, VII and VIII are more favorable than Schemes IV and V, because of their relatively compact stencils. In summary, based on the stencil analyses, four schemes, i.e. II, VI, VII and VIII, are selected as favorable schemes to be further examined and used in the GSM, because they produce compact stencils with favorable coefficients on the both types of regular meshes of concern (uniform Cartesian and equilateral triangular meshes). 3.3.4 Truncation errors We next conduct analyses on truncation errors in the approximation of first- and second-order derivatives with the four recommended schemes in the GSM and the results are summarized in Table 3.2 and Table 3.3. They are derived based on uniform Cartesian mesh and equilateral triangular mesh, respectively. It is clear that all these schemes are of second-order accuracy. The truncation errors for Scheme VII are identical to those for Scheme II, and Schemes VIII and VI have the same truncation errors. All these theoretical findings will be further conformed when these schemes are used to numerical investigations in the following section. 3.4 Application and Validation of GSM Through the stencil analyses for all the schemes, four schemes (II, VI, VII and 90 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation VIII) are selected to use in the GSM because of their compact stencils with favorable coefficients. Numerical investigations for solution to two-dimensional Poisson’s equations are consequently conducted using the GSM. Different spatial discretization schemes are tested and compared with one another in terms of numerical accuracy and computational efficiency. The roles of directional correction and gradient smoothing technique used for approximation of gradients at midpoints are numerically verified. In addition, the effects of the type, density and irregularity of meshes upon the accuracy and stability are intensively investigated. 3.4.1 The governing equations Poisson’s equations for a square domain are solved with our GSM code. Poisson’s equations govern many physical problems, such as the heat conduction problems with sources. Two problems with variations in source and boundary conditions are studied. In current study, the Dirichlet conditions are applied on the boundaries. The pseudo-transient approach was adopted for pursuing steady-state solutions. The governing equations under investigation take the following form: ∂u ∂ 2 u ∂ 2 u = + − f ( x, y , t ) ∂t ∂x 2 ∂y 2 (0 ≤ x ≤ 1, 0 ≤ y ≤ 1) (3.39) For the first problem, the source and initial conditions are prescribed as f ( x, y , t ) = 13 exp( −2 x + 3 y ) u( x, y ,0) = 0 , 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 (3.40) As plotted in Fig. 3.6(a), the analytical solution to this problem is known as uˆ ( x, y ) = e −2 x +3 y (0 ≤ x ≤ 1, 0 ≤ y ≤ 1) (3.41) For the second Poisson problem, the source, initial conditions and analytical solution 91 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation are as follows: f ( x, y , t ) = sin(π x ) sin(π y ) u( x, y ,0) = 0 uˆ ( x, y ) = − 1 2π 2 , sin(π x ) sin(π y ) 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 (0 ≤ x ≤ 1, 0 ≤ y ≤ 1) (3.42) (3.43) The contour plot of the analytical solution to the second problem is shown in Fig. 3.6(b). Boundary conditions adopted in simulations for the both problems are consistent with the corresponding analytical solutions. These analytical solutions are used for the evaluation of numerical errors in the GSM solutions. For a time-dependent problem, the governing equation can be rewritten in the form of ∂u = −R ∂t (3.44) where R represents the residual that is a function of the field variable and its derivatives. In current study, the temporal term ( ∂u ) is discretized with the explicit ∂t five-stage Runge-Kutta (RK5) method: ui(1) ui( 2 ) ui( 3) ui( 4 ) ui( 5) ui( 0 ) = uin = ui( 0 ) − α1ΔtRi( 0 ) = ui( 0 ) − α 2 ΔtRi(1) = ui( 0 ) − α 3 ΔtRi( 2 ) = ui( 0 ) − α 4 ΔtRi( 3) = ui( 0 ) − α 5 ΔtRi( 4 ) (3.45) where the residual Ri( k ) is evaluated with the values of the field function and its derivatives approximated with the k th-step RK solution at node i for every time step. Δ t denotes the time step, and the coefficients adopted in current study are α1 = 0.0695, α 2 = 0.1602, α 3 = 0.2898, α 4 = 0.5060 and α 5 = 1.000. With the RK5 method, only the 0th and 5th-stage solutions at nodes should be stored in memory. 92 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation The RK5 method has been widely used in the simulations of many transient problems, because of its satisfactory efficiency and stability. 3.4.2 Evaluation of numerical errors Three types of numerical errors are evaluated in this study. The convergence error index, ε con , takes the form of ε con = ∑ (u nnode i =1 ( n +1) i − ui( n ) ) 2 ∑ (u nnode i =1 (1) i − ui( 0 ) ) 2 (3.46) where ui( n ) denotes the predicted value of the field variable at the node i at the n th iteration. nnode is the total number of nodes in the domain. The value of ε con is monitored during iterations and used to terminate the iterative process. In most simulations, in order to exclude the effect due to the temporal discretization, computations are not stopped until ε con becomes stabilized, as indicated in Fig. 3.7. The numerical error in a GSM solution for the overall field is defined using L2 -norm of error, which is evaluated using e= nnode ∑ (ui − uˆi )2 i =1 nnode ∑ uˆ i =1 2 i (3.47) where ui and uˆi are predicted and analytical solutions at node i , respectively. This type of error is used to compare the accuracy among different schemes. The third type of error is the node-wise relative error, which is estimated in the fashion of rerrori = ui − uˆi uˆi (3.48) The node-wise relative errors distributed over the computational domain are used to identify problematic regions in simulations. 93 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation • Types of mesh Four types of meshes, i.e. uniform Cartesian, unstructured right triangular mesh, regular triangular and irregular triangular meshes (bold line), as shown in Fig. 3.8, are used in the GSM and investigated in current study. The irregular meshes are designed for the robustness study of the GSM to the irregularity of meshes. The numerical results for various types of meshes with different spatial discretization schemes are discussed in details below. 3.4.4 The role of directional correction As shown in Fig. 3.9(a), the decoupled solution is predicted using Scheme I when it is applied onto uniform Cartesian meshes in solving the first Poisson problem. The saw-toothed numerical errors (checkerboard problem) are generated and can not be dampened out. With the directional correction in Scheme II, such a problem is overcome, as shown in Fig. 3.9(b). This is consistent with the findings in stencil analyses. Note that with directional correction, the overall numerical error is significantly reduced, as shown in Table 3.4. Comparing to Scheme I, the overall errors with Scheme II decrease by 5 times. However, with Scheme II, smaller time step is needed for stability requirement, which results in more computations compared to Scheme I. This becomes more obvious when finer meshes are used in simulation. 3.4.5 Comparison among four favorable schemes The four favorable schemes are comprehensively tested on different types of 94 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation meshes with various densities for solutions to Poisson’s equations. Profiles about computational accuracy for the first Poisson problem are plotted in Fig. 3.10, Fig. 3.11 and Fig. 3.12 together with fitted lines. The errors in these plots are evaluated using Eq. (3.47). It is apparent that the numerical errors decrease as the number of nodes increases. • Uniform Cartesian mesh As shown in Fig. 3.10(a)-(b), on a set of uniform Cartesian mesh, schemes VII is as accurate as scheme II, because the two schemes result in the same discretization stencils, as already highlighted in the stencil analyses. This is also true for the schemes VI and VIII. In addition, the two-point quadrature schemes (VI and VIII) give relatively lower accuracy than the one-point quadrature schemes (II and VII). They also result in higher computational costs than the one-point quadrature schemes. Therefore, based on uniform Cartesian mesh, Schemes II and VII are equivalent and they are superior to Schemes VI and VIII. The slope coefficients of trendlines confirm that these four schemes are of second-order accuracy, which is consistent with our findings in the analyses on truncation errors. • Right triangular mesh Profiles of computational accuracy based on right triangles are shown in Fig. 3.11(a)-(b). Results reveal that Scheme VI gives slightly more accurate approximation than Scheme II. It implies that the scheme VI may result in better accuracy on irregular meshes. Besides, it is interesting to find that based on right triangles, Scheme 95 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation VII is as accurate as Scheme VIII. • Regular triangular mesh When regular triangular mesh is used, the both two-point quadrature schemes (VI and VIII) produce slightly more accurate approximation than one-point quadrature schemes on regular triangles, as shown in Fig. 3.12(a)-(b). Such a finding is consistent with what is observed when right triangular mesh is used. Table 3.5 summarizes the numerical errors ( L2 -norm) for different discretization schemes when regular triangular mesh is used. It is clear that Scheme VII is slightly more accurate than Scheme II, and Scheme VIII is more accurate than Scheme VI. This is also true when right triangular meshes are used in simulations. Such discrepancy in accuracy is related to the approximation of gradient at boundary nodes. In Schemes II and VI, it is necessary to approximate the gradients at boundary nodes, which are used for approximation of gradients at midpoints of internal mesh-edges associated with boundary nodes. Thus, additional errors are introduced due to the inaccurate approximation. Comparatively, in Schemes VII and VIII, subjected to Dirichlet boundary conditions, approximations to gradients at boundary nodes are entirely avoided due to the gradient smoothing techniques applied to the mGSDs. In general, in terms of computational cost, Scheme VII and Scheme II are almost the same, and Scheme VIII and Scheme VI are very close to each other. Schemes VI and VIII need roughly twice computational time as Schemes II and VII, and they are more accurate than Schemes II and VII. However, the improvement in accuracy is not significant. Therefore, to balance the numerical accuracy and computational efficiency, 96 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation the two one-point quadrature schemes (II and VII) are preferred to be used in practice. 3.4.6 Robustness to irregularity of meshes Similar findings described above are also observed in the solutions to the second Poisson’s equation. For the second Poisson problem, additional studies on effects of irregularity of triangular meshes are carried out. It is well-known that triangular meshes have the best adaptivity to complex geometries and can be generated automatically and efficiently. Therefore, it is very desirable to use triangular mesh in the GSM so that it can become a very robust method for engineering problems of complicated geometries. Our objective is thus to further identify the sensitivity of the GSM to mesh quality. To study this in a systematic manner, we first define the irregularity of mesh, γ , using the following formula: ( ai − bi ) 2 + (bi − ci ) 2 + ( ci − ai ) 2 ∑ ai2 + bi2 + ci2 i =1 γ = ne ne (3.49) where ai , bi and ci , respectively, denote the lengths of mesh-edges of a triangular cell, and ne stands for the total number of cells in the overall domain. Using Eq. (3.49), the irregularity vanishes for equilateral triangles and positive for all other shapes including isosceles triangles. Fig. 3.13 shows six sets of triangular meshes with various irregularities, but with the same number of field nodes. It is obvious that as the irregularity increases, the mesh is highly skewed. Note that when the irregularity is larger than 0.152, overlapped cells are found in the domain, as shown in Fig. 3.14. Such a change cannot 97 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation be accurately reflected in the value of irregularity using Eq. (3.49). Convergent results for all irregular meshes are obtained using the GSM with Scheme VII. Contour plots of the approximated field variable are selectively shown in Fig. 3.15. It is clear that on all sets of meshes without overlapped cells, predicted results are reasonably accurate. Note that as irregularity of mesh increases, the time step (Δt) has to be reduced so as to guarantee the stable and convergent results, as shown in Table 3.6. It is interesting to find that for meshes without overlapped cells, the numerical errors predicted with the GSM do not vary so much, as the irregularity of meshes increases (see Fig. 3.16). Once overlapped cells occur in the domain, sudden jumps in numerical errors are noticed. However, for the both Schemes II and VII, stable results are still obtained. Besides, Scheme VII shows much better stability and accuracy than Scheme II amongst all irregular meshes examined here. This means that the GSM with Scheme VII is very robust and insensitive to mesh irregularity. In other words, with the proposed GSM, reasonably stable and accurate results can be obtained even with a mesh of very low quality. Such an attractive feature can be attributed to the use of smoothing techniques in Scheme VII, which provides the crucial stability and robustness to the GSM. 3.5 Remarks In current study, a novel gradient smoothing method (GSM) formulated based on the strong form of governing equations is developed. The GSM valid for both regular 98 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation and irregular meshes can be applied for general PDEs with domains of arbitrary geometry. Intensive studies have been carried out for eight schemes of GSM. It is found that • Schemes II, VI, VII and VIII are favorable, because of their compact stencils with positive coefficients of influence. The GSM solutions to Poisson’s equations have confirmed that all the four schemes indeed give stable and accurate results. • Schemes VII and VIII that use gradient smoothing technique on mGSDs outperform in terms of robustness, stability and accuracy. • Schemes VI and VIII that use two-point quadrature rule give slightly more accurate approximation but at higher computational cost. The one-point quadrature based schemes (Schemes II and VII) have well-balanced performance in terms of both efficiency and accuracy. • Scheme VII is superior to Scheme II in terms of robustness against irregularity of meshes. Therefore, the GSM with Scheme VII is more preferable in practice. 99 Chapter 3 Scheme Gradient Smoothing Method: The Theoretical Formulation Quadrature Type of GSDs Approximation Approximation of gradients at of gradients at midpoints centroids Directional correction I One-point nGSD Interpolation - No II One-point nGSD Interpolation - Yes III Two-point nGSD Interpolation Interpolation No IV Two-point nGSD Interpolation Interpolation Yes V Two-point VI Two-point VII One-point nGSD, cGSD nGSD, cGSD Two-point Interpolation nGSD, Gradient mGSD smoothing nGSD, VIII Interpolation mGSD, cGSD Gradient smoothing Gradient smoothing - Gradient Gradient smoothing smoothing No Yes No No Table 3.1 Spatial discretization schemes for approximating derivatives. 100 Chapter 3 Scheme Gradient Smoothing Method: The Theoretical Formulation Type of mesh Truncation error 3 h 2 ∂ U ij Ο x (h ) = − + Ο( h3 ) 3 6 ∂x 2 II and VII Uniform Cartesian 3 h 2 ∂ U ij Ο y (h ) = − + Ο( h 3 ) 6 ∂y 3 2 VI and VIII II, VI, VII and VIII Ο x (h 2 ) = −h 2 ( 3 3 5 ∂ U ij 1 ∂ U ij ) + Ο( h 3 ) + 24 ∂x 3 2 ∂x∂y 2 Ο y (h 2 ) = −h 2 ( 3 3 5 ∂ U ij 1 ∂ U ij ) + Ο( h3 ) + 24 ∂y 3 2 ∂x 2 ∂y Ο x (h 2 ) = −h 2 ( 1 ∂ 3U i 1 ∂ 3U i + ) + Ο( h 3 ) 24 ∂x3 8 ∂x∂y 2 Ο y (h 2 ) = −h 2 ( 1 ∂ 3U i 1 ∂ 3U i + ) + Ο( h3 ) 24 ∂y 3 8 ∂x 2 ∂y Uniform Cartesian Equilateral Triangular Table 3.2 Truncation errors in the approximation of first-derivatives in the GSM. Scheme II and VII Type of mesh Truncation error Ο( h 2 ) = − Uniform Cartesian VI and VIII Uniform Cartesian II, VI, Equilateral VII and VIII Triangular 4 4 h 2 ∂ U ij ∂ U ij ( 4 + ) + Ο( h 3 ) 4 12 ∂x ∂y 4 ∂ 4U ij ∂ 4U ij h 2 ∂ U ij Ο( h ) = − ( 4 + 3 2 2 + ) + Ο( h 3 ) 4 ∂x ∂y ∂y 12 ∂x 2 Ο( h 2 ) = − ∂ 4U ∂ 4U h 2 ∂ 4U i ( 4 + 2 2 i 2 + 4 i ) + Ο( h 3 ) ∂x ∂y ∂y 16 ∂x Table 3.3 Truncation errors in the approximation of the Laplace operator in the GSM. 101 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation Scheme I No. of nodes Scheme II e iteration e iteration 36 1.96e-2 20 5.07e-3 24 121 8.58e-3 89 1.66e-3 82 441 2.63e-3 202 4.87e-4 303 1681 7.16e-4 728 1.33e-4 1379 6561 1.86e-4 2679 3.38e-5 4299 Table 3.4 Comparison of accuracy by Schemes I and II in the first Poisson’s problem. No. of nodes Δt e Scheme II Scheme VII Scheme VI Scheme VIII 131 0.008 1.95e-3 1.65e-3 1.68e-3 1.55e-3 478 0.001 7.02e-4 6.53e-4 6.46e-4 6.20e-4 1887 0.0005 1.89e-4 1.70e-4 1.74e-4 1.65e-4 7457 0.0001 4.38e-5 3.98e-5 4.12e-5 3.94e-5 29629 0.00003 1.22e-5 1.10e-5 1.11e-5 1.05e-5 Table 3.5 Comparison of numerical errors approximated on regular triangular mesh for the first Poisson problem with favorable schemes. 102 Chapter 3 Mesh Gradient Smoothing Method: The Theoretical Formulation irregularity e Maximum time step (Δt) Scheme VII Scheme II (a) 0.021 0.01 0.0163 0.0172 (b) 0.028 0.01 0.0169 0.0177 (c) 0.048 0.009 0.0179 0.0188 (d) 0.079 0.0075 0.0194 0.0206 (e) 0.118 0.004 0.0214 0.0231 (f) 0.152 0.0005 0.0234 0.0259 Table 3.6 Comparison of allowable maximum time step and numerical error for irregular triangular meshes. 103 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation jk (n) ij( Lk ) (n) ij( Rk ) i jk −1 field node centroid of triangular cell midpoint of connecting edge nGSD cGSD mGSD Fig. 3.1 Illustration of triangle cells and gradient smoothing domains defined in GSM. (0, (− 1 ) 2 1 , 0) 2 (0, − (− ( 1 , 0) 2 1 ) 2 (a) I, II and VII Fig. 3.2 Stencils for approximated gradients ( 1 1 1 1 , ) (0, 3 ) ( , ) 16 16 8 16 16 3 (− , 0) 8 1 1 (− , − ) 16 16 3 (0, − ) 8 3 ( , 0) 8 1 1 ( ,− ) 16 16 (b) III, IV,V, VI and VIII ∂u i ∂u i , ) on uniform Cartesian mesh. ∂x ∂y 104 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation 1 3 1 3 (− , ) ( , ) 6 6 6 6 1 ( , 0) 3 1 (− , 0) 3 3 1 3 1 (− , − ) ( , − ) 6 6 6 6 Fig. 3.3 The stencil for approximated gradients ( ∂u i ∂u i , ) on equilateral triangular ∂x ∂y mesh (Identical for I, II, III, IV, V, VI, VII and VIII). 1 4 1 4 4 1 1 1 4 (a) I 1 128 1 32 3 64 1 32 1 128 1 32 3 64 1 32 7 16 7 16 1 32 3 64 1 128 − − 3 32 − 1 32 1 128 1 32 3 64 1 32 1 128 (d) IV 1 64 1 64 3 32 1 64 1 4 1 − 32 1 4 3 32 1 1 32 4 11 1 − 8 32 1 1 − 32 4 1 1 3 64 64 32 (e) V 3 64 9 64 9 64 3 64 1 128 (c) III 1 64 − 1 128 3 64 3 32 3 64 (b) II and VII 7 16 71 32 7 16 9 64 3 64 1 3 64 3 32 19 3 − 32 32 3 64 1 1 1 4 9 64 3 64 1 128 1 64 3 32 1 64 1 4 1 2 1 4 1 2 3 1 2 1 4 1 2 1 4 (f) VI and VIII ∂ 2 ui ∂ 2ui Fig. 3.4 Stencils for the approximated Laplace operator ( 2 + 2 ) on uniform ∂x ∂y Cartesian mesh. 105 Chapter 3 1 9 1 − 9 1 9 1 9 Gradient Smoothing Method: The Theoretical Formulation − 1 9 − 1 9 1 9 2 3 − 1 9 1 9 1 9 − − 1 9 1 9 1 9 1 9 1 18 1 9 1 9 1 9 1 9 1 18 1 18 1 18 5 18 5 18 1 18 5 18 1 18 (a) I and III 7 3 5 18 1 18 1 18 5 18 5 18 2 3 1 18 2 3 1 18 1 18 2 3 4 2 3 1 18 (b) IV and V Fig. 3.5 Stencils for a Laplace operator ( 2 3 2 3 (c) II, VI, VII and VIII ∂ 2 ui ∂ 2ui + 2 ) discretized onto equilateral ∂x 2 ∂y triangles. u: -0.05 -0.045 -0.04 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0 -0.01 u -0.02 -0.03 -0.04 -0.05 0 0 0.2 0.2 0.4 y 0.4 0.6 0.6 0.8 1 (a) f ( x, y , t ) = 13 exp( −2 x + 3 y ) x 0.8 1 (b) f ( x, y , t ) = sin (π x )sin (π y ) Fig. 3.6 Contour plots of exact solutions to the two Poisson’s problems 106 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation Fig. 3.7 Profile plot of convergence history. (a) Uniform Cartesian (b) Right triangular (c) Regular triangular (d) Irregular triangular Fig. 3.8 Representative meshes under investigation. 107 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation rerror rerror 0.008 0.0075 0.007 0.0065 0.006 0.0055 0.005 0.0045 0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 0.0032 0.003 0.0028 0.0026 0.0024 0.0022 0.002 0.0018 0.0016 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 (a) Scheme I (b) Scheme II Fig. 3.9 Contours of relative errors on Cartesian mesh in the first Poisson’s problem. (a) (b) Fig. 3.10 Profiles of computational accuracy based on uniform Cartesian mesh. 108 Chapter 3 0 log10(e) -1 Gradient Smoothing Method: The Theoretical Formulation Scheme II (right triangular mesh) Scheme VI (right triangular mesh) Trendline for Scheme II Trendline for Scheme VI -2 y = 1.9088x - 0.8821 -3 -4 y = 1.8873x - 0.8712 -5 -6 -2.5 -2 -1.5 -1 -0.5 log10(h) (a) Scheme VII (right triangular mesh) 0 log10(e) -1 -2 -3 Scheme VIII (right triangular mesh) Trendline for Schemes VII and VIII y = 1.9151x - 0.8871 -4 -5 -6 -2.5 -2 -1.5 -1 -0.5 log10(h) (b) Fig. 3.11 Profiles of computational accuracy based on right triangular mesh. 109 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation 0 log10(e) -1 -2 -3 Scheme II (regular triangular mesh) Scheme VI (regular triangular mesh) Trendline for Scheme II Trendline for Scheme VI y = 1.8282x - 0.8324 -4 y = 1.8459x - 0.7703 -5 -6 -2.5 -2 -1.5 -1 -0.5 log10(h) (a) 0 log10(e) -1 -2 -3 Scheme VII (regular triangular mesh) Scheme VIII (regular triangular mesh) Trendline for Scheme VII Trendline for Scheme VIII y = 1.8206x - 0.8762 -4 y = 1.8311x - 0.8415 -5 -6 -2.5 -2 -1.5 -1 -0.5 log10(h) (b) Fig. 3.12 Profile of computational accuracy based on regular triangular mesh. 110 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation (a) γ = 0.021 (b) γ = 0.028 (c) γ = 0.048 (d) γ = 0.079 (e) γ = 0.118 (f) γ = 0.152 Fig. 3.13 Triangular meshes with various irregularity. Fig. 3.14 Overlapped cells in the computational domain ( γ = 0.16). 111 Chapter 3 Gradient Smoothing Method: The Theoretical Formulation -0.05 -0.04 -0.03 -0.02 -0.01 -0.05 -0.04 -0.03 -0.02 -0.01 -0.05 -0.04 -0.03 -0.02 -0.01 (a) γ = 0.021 (b) γ = 0.028 (c) γ = 0.048 -0.05 -0.04 -0.03 -0.02 -0.01 -0.05 -0.04 -0.03 -0.02 -0.01 -0.05 -0.04 -0.03 -0.02 -0.01 (d) γ = 0.079 (e) γ = 0.118 (f) γ = 0.152 Fig. 3.15 Contours of solutions to Poisson’s equations discretized onto irregular meshes. 0.060 Scheme VII Scheme II Error 0.040 0.020 0.000 0.00 0.05 0.10 0.15 0.20 Irregularity Fig. 3.16 Numerical errors in the GSM solutions (Scheme II and VII) to the second Poisson problem with respect to irregularity of meshes. 112 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems 4.1 Introduction In Chapter 3, the theoretical aspects of the gradient smoothing method (GSM) have been discussed in detail. A gradient smoothing method (GSM) of one-point quadrature scheme (Scheme VII) has been proposed for practical use with excellent balance of accuracy and efficiency. In the GSM, gradient smoothing technique is utilized to construct first- and second-order derivative approximations by systematically computing weights for a set of nodal points surrounding a node of interest. A simple collocation procedure is then applied to the governing system equations at each node scattered in the problem domain using the approximated derivatives. In contrast with the conventional finite difference and generalized finite difference methods with topological restrictions, the GSM can be easily applied to arbitrarily irregular meshes for complex geometry. In current study, several numerical examples are presented to demonstrate the computational accuracy and stability of the GSM for solid mechanics problems with regular and irregular nodes. The GSM is examined in detail by comparison with other established numerical approaches such as the finite element method, producing convincing results. Section 4.2 gives a simple convergence study of the GSM using Poisson problems. 113 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems Several numerical examples are presented in section 4.3. Some conclusions are drawn in section 4.4. 4.2 Convergence Study of the GSM In this study, the GSM is examined through solving a two-dimensional Poisson’s equation as in Chapter 2: ∂ 2u ∂ 2u + = sin(π x ) sin(π y ) ∂x 2 ∂y 2 (4.1) with problem domain Ω = {( x, y ) ∈ [0,1; 0,1]}. The corresponding exact solution is u ( x, y ) = − 1 2π 2 sin (π x )sin (π y ) (4.2) Dirichlet and Neumann boundary conditions are considered for regularly and irregularly distributed nodes, respectively. In the numerical studies, a L2 -norm error indicator is also used as eu = ∑ (u numerical ∑ (u − u exact ) ) 2 (4.3) exact 2 Analogously, the error norm for the first-order derivative is e∂u ∂x = ⎡⎛ ∂u ⎞ numerical ⎛ ∂u ⎞ exact ⎤ −⎜ ⎟ ⎥ ∑ ⎢⎜⎝ ∂x ⎟⎠ ⎝ ∂x ⎠ ⎥⎦ ⎢⎣ ⎡⎛ ∂u ⎞ exact ⎤ ∑ ⎢⎜⎝ ∂x ⎟⎠ ⎥ ⎣⎢ ⎦⎥ 2 2 (4.4) We start with the four regular distributions of 6 × 6 , 11 × 11 , 22 × 22 and 43 × 43 field nodes, as shown in Fig. 4.1. A Dirichlet boundary is considered, in which the essential boundary conditions are imposed on all edges as 114 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems along x = 0, x = 1, y = 0 and y = 1 u=0 The relative errors in u and (4.5) ∂u of GSM are compared with three-node linear finite ∂x elements in Table 4.1 and Fig. 4.2. The convergence rates are also demonstrated in Fig. 4.2, where h is the averaged element size. As shown in Fig. 4.2a, the GSM achieves a little higher convergence rate for field variable u compared with the linear FEM. As for the first-order derivative ∂u in Fig. 4.2b, the GSM is more accurate than ∂x FEM. Further, the four distributions of irregular field nodes presented in Fig. 4.3 are investigated. They are 40, 132, 488 and 1897 nodes, respectively. The mixed boundary conditions are considered here in problem domain Ω , where Neumann boundary conditions are ∂u ∂x =− x =0 ∂u ∂x 1 sin(π y ) , 2π = x =1 1 sin(π y ) 2π (4.6) and Dirichlet boundary conditions are u=0 The relative errors in u and along y = 0 and y = 1 (4.7) ∂u for GSM and FEM are presented in Table 4.2 and ∂x Fig. 4.4. As shown in Fig. 4.4a, the GSM not only achieves a higher convergence rate but also obtains more accurate results than FEM. With the increase of irregular nodes, it appears that GSM are more and more accurate than FEM. Similarly, it can be seen from Fig. 4.4b that the GSM is more accurate than FEM in the computation of the first-order derivatives of variable u . This is because the GSM directly discretizes the governing equations based on the gradient smoothing technique which guarantees the first-order continuity. However, in terms of the first-order derivatives (e.g., stresses 115 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems and strains), the FEM suffers from discontinuity problems and requires the use of post processing to produce better results. It can be observed from this numerical study that the GSM is quite stable even with the Neumann boundary conditions and yields very accurate results for both the regular and irregular field node distributions. 4.3 Numerical Examples 4.3.1 Cantilever beam A 2-D cantilever beam with length L and height D subjected to a parabolic traction at the free end is studied as a benchmark problem here, as shown in Fig. 4.5. Assume the beam has a unit thickness so that the problem is simplified into plane stress case. The analytical solution is available by Timoshenko and Goodier (1970): ux = − uy = P 6 EI Py 6 EI ⎡ D2 ⎤ 2 L x x y − + + − ( 6 3 ) ( 2 ν )( )⎥ ⎢ 4 ⎣ ⎦ ⎡ ⎤ D2 x 2 − + + + ( 3L − x ) x 2 ⎥ 3 ν y ( L x ) ( 4 5 ν ) ⎢ 4 ⎣ ⎦ σ xx = − τ xy = P( L − x) y I (4.8) (4.9) (4.10) σ yy = 0 (4.11) P D2 ( − y2 ) 2I 4 (4.12) where the moment of inertia I for a beam with rectangular cross section and unit thickness is given by I = D 3 12 . The geometries and material properties are taken as L = 48 m, D = 12 m, Young’s modulus E = 3 × 107 N/m2, Poisson’s ratio υ = 0.3 , 116 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems loading (integration of the distributed traction) P = −1000 N. The governing equations of this problem are given by Eqs. (3.1)-(3.3), which are also used for the following numerical investigations. In this study, cantilever beam is simulated by 273 regularly distributed nodes and 480 triangular elements, as shown in Fig. 4.6 (a) and (b). To validate the present method, the GSM results are compared with the FEM and analytical solutions, respectively. The same set of nodes and elements are used for modeling of cantilever beam by the GSM and FEM. In the FEM, three-node linear element and 3 gauss integration points are used in the numerical integration scheme. The computing results of the deflection along the line y = 0 are plotted in Fig. 4.7. From this figure, it is observed that the GSM is able to provide the results as accurate as the FEM for deflection of the cantilever beam as shown in Fig. 4.7. In terms of stresses, the FEM requires post processing procedures to provide better results as it suffers from discontinuity in stresses. In contrast, the GSM does not encounter discontinuity problem in stresses. As shown in Fig. 4.8, the normal stress σ xx computed by the GSM is smooth rather than discontinue like the FEM. Also, the shear stress τ xy plotted in Fig. 4.9 is more accurate than that of FEM. From this point of view, the GSM does perform better than the FEM for computing the stresses. Table 4.3 shows the comparison of the computational efficiency between GSM and FEM using the same set of meshes of 273, 527, 1127, 2275 and 3825 regularly distributed nodes. It is found that GSM uses less CPU time than FEM when a small number of nodes are used. This is because of that when node number is small the 117 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems CPU time is largely controlled by the overhead operations in creating the algebraic system equations. As GSM creates the system equations by discretizing directly (by collocation) the governing equation and doesn’t need any integration that is on the other hand necessary for FEM, the GSM is therefore more efficient than FEM when a small number of nodes are used. This is clearly demonstrated in Table 4.3. When a large number of nodes are used, however, the CPU time is mainly determined by solving the algebraic system equations. In this case FEM is faster than GSM, but is only about twice faster. This can be examined simply by the complicity analysis of the equation solvers used in the FEM and GSM. We know that the bandwidth of the system matrix generated by GSM is the same as the FEM, but the matrix in GSM is not symmetric and a solver for asymmetric system equations needs to be used. In the FEM, however, the matrix is symmetric and hence a solver for symmetric system equations can be used. The complexity of a symmetric solver is about twice faster than an asymmetric solver for matrices of the same dimension and bandwidth. This analysis is confirmed numerically as shown in Table 4.3. Our conclusion is: 1) for small systems, GSM is more efficient than FEM, and gives more accurate results in terms of stresses; 2) for large systems, FEM is about twice as faster as GSM, but GSM gives more accurate results in terms of stresses. 4.3.2 Infinite plate with a circular hole To validate the GSM in simulating stress concentration, we consider an infinite plate with a central circular hole subjected to a unidirectional tensile load p = 1.0 in the x -direction. Due to the symmetry, only the upper right quadrant of the plate is 118 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems modelled, as shown in Fig. 4.10, in which the plane strain problem is considered, and the geometries and material parameters used are a = 1 , b = 5 , Young’s modulus E = 1.0 × 10 3 and Poisson’s ratio υ = 0.3 . Symmetry conditions are imposed on the left and bottom edges, and the inner boundary of the hole is traction free. The corresponding exact solutions for the stresses in the plate are given in the polar coordinate (Timoshenko and Goodier, 1970) in Eq. (2.72). The traction boundary conditions given by the exact solutions (2.72) are imposed on the right ( x = 5) and top ( y = 5) edges. Fig. 4.11a shows the distribution of 261 irregular nodes in the problem domain, in which there are 465 triangular elements (see Fig. 4.11b). The distribution of normal stress σ xx along the line x = 0 obtained using the GSM is shown in Fig. 4.12. It can be observed from this figure that the GSM yields very satisfactory results for the stress concentration problem. 4.3.3 Bridge pier In this example, the GSM is used for the stress analysis of a bridge pier subjected to a uniformly distributed pressure on the top, as shown in Fig. 4.13. The problem is solved as a plain strain case with material properties E = 4 × 1010 Pa , υ = 0.15 and loading P = 10 5 Pa . Due to the symmetry, only right half of the bridge is modelled as shown in Fig. 4.14 where there are 590 field nodes (see Fig. 4.14a) in the model and 1077 triangular elements (see Fig. 4.14b). As there are no analytical solutions available for this problem, a reference solution of displacements and stresses are computed with 119 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems commercial software ANSYS using very fine triangular mesh for purpose of validation. The displacements in y -direction along the lines x = 0 , y = 30 and y = 15 are plotted in Fig. 4.15, Fig. 4.16 and Fig. 4.17 respectively. The solutions obtained by GSM are in good agreement with the reference (ANSYS) solutions. Also, comparison of the stress distribution σ yy along the line y = 15 computed by the GSM and ANSYS is shown in Fig. 4.18. It can be concluded from the figure that the GSM results are accurate enough for general engineering requirement. 4.3.4 An automotive part: connecting rod As the last numerical example, to generalize the present GSM to all problem domains with irregular shapes, a connecting rod as an automotive part with complicated geometry is studied as a plane stress solid mechanics problem, as shown in Fig. 4.19a. The material properties are given as Young’s modulus E = 3 × 10 7 Pa and Poisson’s ratio υ = 0.3 . The edge of the hole 1 is fixed and the right edge of the hole 2 is subjected to a constant pressure P = 200Pa (see Fig. 4.19b). Due to symmetry, only upper half of the connecting rod is simulated and shown in Fig. 4.19b. Symmetric conditions are imposed along the bottom edge of the half connecting rod. Fig. 4.20a shows the node distribution of 1634 irregular field nodes, in which there are 2877 triangular elements in the problem domain, as shown in Fig. 4.20b. Since no analytical solution is available for this problem, commercial FEM software, ANSYS, is also used to compute the reference solutions with a very fine mesh of triangular elements for purpose of comparison. 120 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems Fig. 4.21 shows the displacement in x -direction along the line y = 0 . The stress distributions σ xx and σ yy along the line y = 0 by GSM are plotted in Fig. 4.22a and Fig. 4.22b. It can be found that the GSM results are very accurate compared with the ANSYS reference solutions. 4.4 Remarks In this work, a gradient smoothing method (GSM) has been presented for solving partial differential equations, with emphases on solid mechanics problems. By adopting the gradient smoothing technique, the first- and second-order derivative approximations can be obtained with a favourable weight distribution for a set of field nodes surrounding a node of interest. Unlike the traditional finite difference method with structured and orthogonal grids or the generalized finite difference methods with some topological requirements, the GSM is flexible to perform the use of pre-existing meshes which are originally created for finite element or finite difference methods, regardless of their topology. The selected star of the GSM is simply generated by sequentially connecting the centroids with mid-edge points of surrounding elements for the node of interest, compared with the Voronoi neighborhood criterion. Since the GSM directly discretizes the governing equations using the gradient smoothing technique, the first-order continuity can be obtained which leads to the better results in the computations of stresses and strains for solid mechanics problems compared with the finite element method. By comparison with FEM (ANSYS) or analytical solution via several numerical examples, it can be concluded that the proposed GSM 121 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems method achieves very accurate and stable solutions using arbitrary and irregular computational meshes. Compared with the FEM, GSM is more efficient than FEM, and gives more accurate results in terms of stresses when a small number of nodes are used. For large systems, FEM is about twice as faster as GSM, but GSM give more accurate results in terms of stresses. The GSM can be easily applied to adaptive analysis, which will be discussed in the following chapter. 122 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems No. of field nodes 36 121 484 1849 h 0.2 0.1 0.0476 0.0238 GSM 5.2167E-2 1.2647E-2 2.8468E-3 7.1060E-4 FEM 4.6600E-2 1.1600E-2 2.6000E-3 6.5940E-4 GSM 0.20343 7.7494E-2 2.3482E-2 8.4210E-3 FEM 0.22220 8.1900E-2 2.4400E-2 8.6000E-3 eu e∂u / ∂x Table 4.1 Relative errors of Poisson’s equation with Dirichlet boundary conditions computed using the same sets of regularly distributed nodes for GSM and FEM. No. of field nodes 40 132 488 1897 h 0.1878 0.09534 0.04741 0.02350 GSM 6.7903E-2 1.3605E-2 2.7761E-3 6.0463E-4 FEM 6.7900E-2 1.4800E-2 3.3000E-3 7.7356E-4 GSM 0.15545 5.3777E-2 1.9578E-2 6.8747E-3 FEM 0.1881 6.4200E-2 2.1700E-2 7.2000E-3 eu e∂u / ∂x Table 4.2 Relative errors of Poisson’s equation with Neumann boundary conditions computed using the same sets of irregularly distributed nodes for GSM and FEM. No. of field nodes CPU time of GSM (s) CPU time of FEM (s) 273 0.73 1.67 527 1.83 3.14 1127 5.67 6.53 2275 26.78 16.34 3825 84.76 44.25 Table 4.3 Comparison of the CPU time computed using GSM and FEM. 123 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems (a) (b) (c) (d) Fig. 4.1 Node distribution of Poisson’s equation: (a) 50; (b) 200; (c) 882 and (d) 3528 elements. 124 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems (a) (b) Fig. 4.2 Comparison of convergence rate and accuracy between GSM and FEM for Poisson’s equation with regular nodes: (a) eu and (b) e∂u / ∂x . 125 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems (a) (b) (c) (d) Fig. 4.3 Irregular nodes of Poisson’s equation: (a) 58; (b) 222; (c) 894 and (d) 3632 elements. 126 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems (a) (b) Fig. 4.4 Comparison of convergence rate and accuracy between GSM and FEM for Poisson’s equation with irregular nodes: (a) eu and (b) e∂u / ∂x . 127 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems D y x P L Fig. 4.5 Cantilever beam subjected to a parabolic load at the free end. (a) (b) Fig. 4.6 Domain discretization of cantilever beam: (a) nodes and (b) elements. 128 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems Fig. 4.7 Deflection of cantilever beam along the line y = 0 computed using the same mesh (480 triangular elements) for GSM and FEM. Fig. 4.8 Normal stress σ xx along the line x = L / 2 computed using GSM and FEM. 129 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems Fig. 4.9 Shear stress τ xy along the line x = L / 2 computed using GSM and FEM. y b p a x Fig. 4.10 Quarter model of the infinite plate with a circular hole subjected to a unidirectional tensile load. 130 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems (a) (b) Fig. 4.11 Quarter model of the infinite plate: (a) nodes and (b) elements. Fig. 4.12 Normal stress σ xx along the edge of x = 0 in a plate with a central hole subjected to a unidirectional tensile load. 131 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems P 20 30 R10 Fig. 4.13 A bridge pier subjected to a uniformly distributed pressure on the top. (a) (b) Fig. 4.14 Half model of the bridge pier: (a) nodes and (b) element. 132 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems Fig. 4.15 Displacement in y -direction along the line x = 0 . Fig. 4.16 Displacement in y -direction along the line y = 30 . 133 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems Fig. 4.17 Displacement in y -direction along the line y = 15 . Fig. 4.18 Normal stress σ yy along the line y = 15 . 134 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems (a) (b) Fig. 4.19 An automotive part: the connecting rod. 135 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems (a) (b) Fig. 4.20 Half model of the connecting rod: (a) node distribution and (b) element distribution. Fig. 4.21 Displacement in x -direction along the line y = 0 . 136 Chapter 4 A Gradient Smoothing Method for Solid Mechanics Problems (a) (b) Fig. 4.22 Distribution of normal stresses along the line y = 0 : (a) σ xx and (b) σ yy (GSM uses 2877 triangular elements while ANSYS adopts a very fine triangular mesh to get the reference solution). 137 Chapter 5 Adaptive Analyses for Solids using the GSM Chapter 5 Adaptive Analyses for Solids using the GSM 5.1 Introduction In advanced design of products of high precision, adaptive analysis is becoming an important tool in practical numerical computations (Huerta et al., 1999). It is a fundamental tool to obtain numerical solutions with a desired accuracy. In an adaptive procedure, there are three essential ingredients: (1) an effective and stable numerical method for arbitrary problem domains and irregular meshes; (2) a tool for estimating the error of the numerical solution; and (3) an algorithm to refine the problem domain (Liu and Tu, 2002). The first ingredient is a prerequisite, without which an adaptive process will break down. The error estimator is crucial in assessing the local and global errors in the numerical solution at a stage of analysis, whereby a decision can be made on whether a refinement is required. The third is performed according to the error information provided by the error estimate. The effectiveness and efficiency of all these three pieces of techniques are critical to the performance of an adaptive analysis. To conduct a posteriori error estimation, two values of a quantity – a computed value and a reference value – are usually required. The first is the raw data of the numerical solution while the second is derived from the raw data via postprocessing (smoothing or projection). In FEM, it is well known that the raw stresses (or derivatives) do not possess inter-element continuity and have a low 138 Chapter 5 Adaptive Analyses for Solids using the GSM accuracy at nodes and element boundaries. The improved values are obtained by smoothing the inter-element discontinuity. The difference between the raw and improved values forms a basis for error estimation in FEM solution. Detailed descriptions of this approach can be found in FEM literatures, e.g., by Zienkiewicz and Taylor (1992). To establish an adaptive finite element procedure, one of the most important components is a robust automatic mesh generation scheme. However, to develop and implement automatic mesh generators with good control of element size and shape is not an easy task. During the last decade, many research efforts have been devoted to this area (Shephard and Weatherill, 1991; Lo, 1997) and yet it still remains an active research topic in computational mechanics and geometry. Currently, automatic mesh generators of triangular elements for complex geometry are available. Unfortunately, the triangular elements used in FEM are known to be ‘too stiff’ and inaccurate (Liu and Quek, 2003). Compared with the finite element method, the meshfree methods enjoy much more flexibility in model generation since they can approximate field variables entirely based on a group of discrete nodes and require no predefined node connectivity. For meshfree methods that require background cells, triangular cells can be used, which will not affect the accuracy in the solutions. Nodes used in many meshfree methods can be irregular or unstructured. Nodes can be quite freely inserted or deleted without worrying too much about the connectivities. Therefore, the meshfree methods are particularly attractive for the development of adaptive strategies. Several adaptive procedures and error estimates for meshfree methods have 139 Chapter 5 Adaptive Analyses for Solids using the GSM been proposed. Duarte and Oden (1996) derived an error estimator for the h-p cloud methods that involves only the computation of interior residuals and residuals where Neumann boundary conditions are prescribed. Chung et al. (2000) and Lee and Zhou (2004) proposed adaptive refinement procedures for the element-free Galerkin (EFG) method. Liu and Tu (2002) developed an adaptive scheme based on triangular background mesh for meshfree methods. As addressed in Chapter 1, meshfree strong form methods possess many good features for adaptive analysis due to its simplicity. Unlike weak form methods, strong form methods need no integration and hence no mapping is needed. However, the instability problem has been a key factor that limits the application of meshfree strong form methods that use local nodes. Researchers have made lots of efforts to this aspect (Onate, 2001; Liu et al., 2006; Kee et al., 2007). However, it is still an open problem to be solved. In this work, a residual based error indicator is adopted in the GSM for adaptive analyses. The proposed GSM can effectively overcome the instability issue, while retaining the strong form feature of simplicity in formulation procedures which is particularly suitable for adaptive analysis. By evaluating the residual of the governing equation for each triangular cell in the domain, error indicator effectively identifies the necessary regions to be refined. Simple refinement procedure using Delaunay diagram is adopted in the adaptive scheme. Additional nodes can be inserted into the domain easily without worrying about the nodal connectivity and remeshing the domain. 140 Chapter 5 Adaptive Analyses for Solids using the GSM Section 5.2 provides a brief description of a posteriori error indicator based on residual of the governing equation. Section 5.3 illustrates the capabilities of the present GSM through some numerical examples including different levels of stress concentration. The performance is also assessed by comparing the convergence rate obtained with those by uniform refinement. Conclusions are stated in section 5.4. 5.2 Adaptive Strategy A good error indicator is of great importance in the adaptive analysis. In this work, a robust error indicator based on residual of the governing equations (Kee et al., 2007) is adopted. The residual based error indicator provides a good measurement for the quality of the local approximation and the global accuracy of the solution. The brief descriptions of the error indicator and refinement procedures are given as follows. 5.2.1 Error indicator In this adaptive scheme, the same set of triangular cells used for GSDs is used. The error indicator for a triangular cell is computed by evaluating the residual of the strong form governing equations at the center of the triangular cell, as shown in Fig. 5.1. In this work, we use two types of error indicators: local and global error indicators. The local indicator is used to determine the cells that need to be refined, and the global error indicator is used to control the iterations of refinement. The local error indicator is defined as η j = ∫ Lu − f L2 dΩ ≈ 1 A j L(u j ) − f j 3 (5.1) L2 where A j is the area of the j th cell, and Lu j − f j L2 is the L2 -norm of the 141 Chapter 5 Adaptive Analyses for Solids using the GSM residual for the governing equation evaluated at the center of corresponding cell by simple interpolation using the nodal values of the displacements. With the above definition of the local error indicator, the global error indicator is estimated using the global residual norm that can be easily obtained as ⎡1 ⎤ η g = ∫ ( Lu − f L ) dΩ ≈ ∑ ⎢ A j (L(u j ) − f j )⎥ 2 ⎦ j =1 ⎣ 3 nc 2 2 (5.2) where nc is the total number of the triangular cells. 5.2.2 Refinement procedure and stopping criterion The refinement criterion for the j th cell in the adaptive scheme is that when η j ≥ κ l max(η i ) 1≤i ≤ nc (5.3) cell j is refined, where κ l is a local refinement coefficient defined by the analyst. Eq. (5.3) simply leads that a certain percentage of cells that have maximum errors are refined. In advance, the triangular cells are classified into two groups: interior cells and edge cells. An interior cell is a cell that has no edge on the boundaries of the problem domain, and an edge cell is a cell which has at least one edge on the boundaries. For example, cell a and cell b are interior and edge cells respectively, as shown in Fig. 5.2. Then if this interior cell needs to be refined, a new node will be added at the centroid of the triangle; for an edge cell, two new nodes will be added at the centroid and the midpoint of the edge which is on the boundaries (see Fig. 5.2). Finally, the formation of the new mesh will be performed using the Delaunay technique based on the new nodes, as sketched in Fig. 5.2. The estimated global residual norm defined in Eq. (5.2) is used as an indicator for 142 Chapter 5 Adaptive Analyses for Solids using the GSM termination criterion of the adaptive process. The stopping criterion is that when η g ≤ κ g η mg (5.4) is met the adaptive process will be terminated, where κ g is the global residual tolerant coefficient and η mg is the allowable maximum value of global residual error throughout the adaptive process. 5.3 Numerical Examples 5.3.1 Patch test In the first example, both standard patch test with maximum (full) essential (Dirichlet) boundaries called essential-patch-test here and a patch test with maximum natural (Neumann) boundaries called natural-patch-test here are conducted using the present GSM. For essential-patch-test, six different patches of a solid are first examined as shown in Fig. 5.3. All patches have only five field nodes: four corner nodes and one inner node whose location varies inside the domain. The dimension of the patch is 1 by 1, and the material properties are taken as Young’s modulus E = 1.0 and Poisson’s ratio υ = 0.25 . The displacements are prescribed on the outside boundaries using a linear function of x and y : ux = x + y and uy = x − y (5.5) To satisfy the patch test, it requires that the displacements of any interior nodes should be given by the same linear functions in the patch test. As shown in Table 5.1, all six patches have passed the standard patch test to machine accuracy. It is found that all patches have the same level of accuracy regardless of the extreme irregularity of the 143 Chapter 5 Adaptive Analyses for Solids using the GSM cells in the patches. This demonstrates the good stability of the proposed GSM. In natural-patch-test, two different patches are subjected to a uniform axial traction of unit intensity along the right end of the cantilever beam, as shown in Fig. 5.4. There are 35 nodes regularly distributed in the first patch (see Fig. 5.4(b)) and irregularly distributed in the second patch (see Fig. 5.4(c)). The dimension of the cantilever beam is 3.0 by 6.0. The material properties are also taken as E = 1.0 and υ = 0.25 . The exact solutions of the displacements for this problem are ux = x and uy = − y 4 (5.6) It is observed that the two patches of both regular and irregular node distributions pass the higher-order patch test to machine accuracy, as shown in Table 5.2. This shows again that the GSM has excellent stability. It is known that the essential-patch-test is more critical to methods based on global weak forms, and the natural-patch-test is, on the other hand, more critical to methods based on strong forms (Liu and Gu, 2005). Our GSM passes both, which proves numerically that GSM is capable of producing linear fields regardless of types of boundary condition, and hence the GSM solution will converge to any high order continuous fields. More details of conducting the patch tests can be found in Refs. (Zienkiewicz and Taylor, 1992; Liu, 2002; Liu and Gu, 2005). 5.3.2 Poisson’s equation with a sharp peak In the second example, we test further the stability, accuracy, and the peak capturing ability of the GSM using adaptive scheme, and study a Poisson’s problem whose solution has a very sharp peak. Such a Poisson’s equation is defined as 144 Chapter 5 [ Adaptive Analyses for Solids using the GSM ∇ u = − 400 + (200 x − 100) + (200 y − 100) 2 2 2 ] − 100( x − 1 ) 2 − 100( y − 1 ) 2 2 2 e (5.7) in the domain of Ω : [0,1] × [0,1], with Neumann boundary conditions, ∂u = 0 along Γt : x = 0 and y = 0 ∂n (5.8) and Dirichlet boundary conditions, u = 0 along Γu : x = 1 and y = 1 (5.9) The analytical solution for this problem is given as − 100( x − 1 ) 2 − 100( y − 1 ) 2 2 2 u=e (5.10) Three-dimensional plots for the analytical solutions of field function u and its first-order derivatives are shown in Fig. 5.5. Because the analytical solution is available for this problem, the true error in the numerical solution of GSM can be examined. The Poisson’s equation is first solved using our GSM with the six regular distributions of 11 × 11 , 16 × 16 , 21 × 21 , 30 × 30 , 46 × 46 and 61 × 61 (= 3721) field nodes. Four selected node distributions are shown in Fig. 5.6. The overall error norm of the field variable u is reduced from 33.54% to 0.51% as the mesh is refined uniformly, as shown in Table 5.3. This shows that the present GSM is very stable and accurate. The Poisson’s equation is now studied again using our GSM, but with adaptive analysis. The initial mesh has 121 regularly distributed nodes (see Fig. 5.6). The adaptive procedure ends up at 5th iteration step with 1107 irregularly distributed nodes in the problem domain. The local predefined refinement coefficient is κ l = 0.05 and the global residual tolerant coefficient is set as κ g = 0.1 . Due to the 145 Chapter 5 Adaptive Analyses for Solids using the GSM presence of sharp peak, most of the nodes are inserted automatically into the high gradient region as shown in Fig. 5.7. This demonstrates the fact that our GSM is capable of capturing the ‘peak’. From Fig. 5.8, one can observe that the estimated global residual is reduced steadily. This shows the excellent stability of the present GSM, even for extremely irregularly distributed nodes. While estimated global residual norm is reduced in the adaptive process, the true error norm of field function u is significantly reduced from 33.54% to 0.56% as shown in Table 5.4. Compared with the uniform refinement with 3721 regular nodes, the similar accuracy can be obtained using the adaptive refinement with only 1107 nodes. The comparison of convergence rate ( R ) between uniform and adaptive refinements is plotted in Fig. 5.9, where h is the averaged cell size. The convergence rate for the uniform refinement is found to be about 2.07 that conforms the theoretical prediction of 2.0 as shown in Chapter 3. The adaptive refinement using GSM achieves a convergence rate of about 3.14 for field function u which is much higher compared with the uniform refinement. The GSM solutions for field function u along the line y = 0.5 at the first and fifth (final) steps are plotted with analytical solution in Fig. 5.10. It is clear that this adaptive scheme is effective to improve in an automatically manner the accuracy of the solution for field function u . The three-dimensional plots of the approximated field function and its derivatives at the final step are provided in Fig. 5.11. It shows not only the approximated field function but also field function derivatives are in very good agreement with the analytical solutions shown in the Fig. 5.5. 146 Chapter 5 Adaptive Analyses for Solids using the GSM 5.3.3 Infinite plate with a circular hole This numerical example is a stress analysis of an infinite plate with a central circular hole subjected to a unidirectional tensile load p = 1.0 in the x -direction. A plane strain problem is considered. The problem has stress concentration near the hole, and hence is a good test of our adaptive GSM for stress concentration capturing. Due to the symmetry, only the upper right quadrant of the plate is modelled, as shown in Fig. 5.12. The geometry and material parameters used are a = 1 , b = 5 , Young’s modulus E = 1.0 × 10 3 and Poisson’s ratio υ = 0.3 . Because the analytical solution is also available for this problem, the true error in the numerical solution of GSM can be examined. Using the present GSM, we start this benchmark problem from approximately uniform refinement with 39, 98, 199, 403, 826 and 1513 field nodes. Also, four selected node distributions are shown in Fig. 5.13. As shown in Table 5.5, the error norm of the displacement u x is significantly and steadily reduced from 90.12% to 0.98%. This shows again that the GSM has very good stability and accuracy. The adaptive analysis using GSM starts with 39 nodes ‘evenly’ distributed in the quarter model (see Fig. 5.13). The local refinement coefficient is set as κ l = 0.05 and the global residual tolerant coefficient is predefined as κ g = 0.05 . As shown in Fig. 5.14, the adaptive analysis ends at the 6th step with 567 nodes irregularly distributed in the problem domain. The estimated global residual at each adaptive step is plotted in Fig. 5.15. One can observe that the global residual norm is gradually reduced at each adaptive step. It demonstrates again the excellent stability of the GSM even when irregular nodes are 147 Chapter 5 Adaptive Analyses for Solids using the GSM used. The error norms of the displacement u x are shown in Table 5.6 and Fig. 5.16. As shown in Fig. 5.16, the GSM with uniform refinement can only bring the error norm down to 0.98% with 1513 nodes while the error can be brought down to 0.48% using the GSM for adaptive refinement with 567 nodes. As the node increases further, the error norm reduces very slowly for the ‘uniform’ refinement case due to the stress concentration that is not uniform but only near the hole. However, for the GSM with the adaptive refinement, the error norm is dramatically reduced at a steady rate, because of its ability to capture the stress concentration. For validation purpose, the distributions of normal stress σ xx along the line x = 0 at the 3rd and 6th steps are plotted in Fig. 5.17. It is very clear that the accuracy of both displacement and stress has been greatly improved through the effective adaptive scheme using the GSM for the stress concentration problem. 5.3.4 Short cantilever plate In this example, the GSM is used for the stress analysis of a short cantilever plate subjected to a uniformly distributed pressure on the top, as shown in Fig. 5.18. The problem is solved as a plain strain case with material properties E = 1.0 , υ = 0.3 and loading p = 1.0 . This problem has stress singularity near the two left corners, and hence is very good for testing the adaptive GSM for stress singularity capturing. As analyzed in the work of Johnson and Hansbo (1991) and Steeb et al. (2002), the exact solution of energy norm u is defined as u = (∫ σ ε dΩ) T Ω 12 (5.11) and found to be 1.379745 . Since the analytical solution for the displacements is not 148 Chapter 5 Adaptive Analyses for Solids using the GSM known, a reference solution is obtained using a very fine mesh of 58060 degrees of freedom. The calculated energy norm u h using this very fine mesh is 1.3794663 , which is almost the exact energy of 1.379745. The calculated value of displacement in y -direction at the tip node A (1,0) is − 2.875323 . We assume this value as reference ‘exact’ solution. The uniform refinement for this example uses six regular meshes with 73, 214, 488, 755, 1376 and 2498 evenly distributed nodes. Four such distributions of nodes are shown in Fig. 5.19. The GSM solutions of displacement u y ( A) , energy norm and their error norms are presented in Table 5.7. In this work, the relative error in energy norm is defined as ee = uh − u u (5.12) As the uniform refinement advances, both the displacement u y (1,0) and energy norm approach the ‘exact’ (reference) solutions gradually. The adaptive refinement starts with the same coarse mesh of 73 field nodes as shown in Fig. 5.19. The local refinement coefficient is predetermined as κ l = 0.05 and the global residual tolerant coefficient is set as κ g = 0.05 . As shown in Fig. 5.20, the adaptive analysis ends at the 6th step with 1889 nodes distributed irregularly in the whole plate domain. Due to the stress concentration in this problem, more nodes are added into the two corner areas at the left side of the plate (see Fig. 5.20). It can be observed from Fig. 5.21 that the estimated global residual is monotonically reduced as nodes increase. Table 5.8 shows displacement at point A , energy and their error norms for each adaptive step. Compared with the results of uniform refinement, the 149 Chapter 5 Adaptive Analyses for Solids using the GSM GSM with adaptive scheme is clearly more effective. It leads to a very fine accuracy using much less nodes, as shown in Fig. 5.22 and Fig. 5.23. The comparison of convergence rate in energy norm between uniform and adaptive refinements is demonstrated in Fig. 5.24. The convergence rate obtained using the adaptive scheme with GSM is much higher than that of uniform refinement. For comparison purpose, we now study the present problem using linear FEM with both uniform and adaptive models with the same set of initial nodes as that used in the GSM. The adaptive procedure used in the FEM is also the same as that used in adaptive GSM with the same tolerant coefficients. Table 5.9 shows displacement at point A , energy and their error norms for each adaptive step. The comparisons of GSM and FEM with both uniform and adaptive refinements are plotted in Fig. 5.22 and Fig. 5.23. The energy errors of the numerical results calculated using Eq. (5.12) are plotted in Fig. 5.24 with respect to h . Here, h is taken as the average nodal spacing for different nodal configurations. The results show that the adaptive models for both GSM and FEM have obtained higher convergence rate than uniform refinements. This demonstrates the effectiveness of the present adaptive procedure. Second, compared with FEM, the GSM achieves much higher convergence rate for adaptive refinement. Fig. 5.25 presents the comparison of condition numbers of the coefficient matrix for both uniform and adaptive refinements. For uniform refinement, GSM has almost the same condition numbers as those of FEM. However in adaptive procedure, GSM produces much smaller condition numbers than FEM, which demonstrates the 150 Chapter 5 Adaptive Analyses for Solids using the GSM excellent stability of our present GSM. 5.3.5 L -shaped plate Fig. 5.26 shows an L -shaped plate subjected to a tensile force p = 10 in the horizontal direction. This is a classical problem to examine adaptive refinement schemes (Zienkiewicz and Taylor, 1992; Liu et al., 2006). Since there is a singular point at the concave corner, an adaptive scheme is again required to identify the point of singularity and to refine the region around the node. This example is investigated as a plain stress problem. The geometry and material parameters used are a = 5 , Young’s modulus E = 3.0 × 10 7 and Poisson’s ratio υ = 0.3 . The boundary conditions are imposed as demonstrated in Fig. 5.26. As the exact solution for total strain energy is not available, a reference solution is obtained using linear FEM with a very fine mesh of 8732 nodes. The computed energy norm is 3.220292E-2, which is assumed as the reference solution. We also start this example from ‘uniform’ refinement with 108, 369, 710, 1430, 2194 and 2948 nodes, respectively. Four selected distributions of nodes are shown in Fig. 5.27. The values of strain energy and its error norm are shown in Table 5.10. The adaptive analysis starts with an initial mesh of 108 uniformly distributed nodes (see Fig. 5.27). The local refinement coefficient is predefined as κ l = 0.025 , and the global residual tolerant coefficient is κ g = 0.05 . The adaptive refinement ends at the 5th step with 1489 irregular nodes in the problem domain. The node distributions of 3rd and 5th steps are plotted in Fig. 5.28, which shows that the adaptive scheme is able to detect the singular point and refine the surrounding area accordingly. The 151 Chapter 5 Adaptive Analyses for Solids using the GSM calculated strain energy and its error norm for each adaptive step are presented in Table 5.11. Fig. 5.29 shows the comparison of energy norm between uniform and adaptive refinements. It is observed that the adaptive scheme converges to the reference solution much faster. The comparison of convergence rate in energy norm is shown in Fig. 5.30. Since the adaptive scheme can automatically refine the high stress region near the concave corner, it significantly accelerates the process of convergence, and hence significantly improves the accuracy. 5.3.6 Mode-I crack problem In this example, a Mode-I crack problem is considered for adaptive analysis. A square plate with sides of length 2a and a crack of length a is used, as shown in Fig. 5.31(a). The exact displacement and stress solutions in the crack tip neighborhood are given by (Perrone et al., 1978; Anderson, 1991): ux = KI 2μ r ⎛ θ ⎞⎡ ⎛ θ ⎞⎤ cos⎜ ⎟ ⎢κ − 1 + 2 sin 2 ⎜ ⎟⎥ 2π ⎝ 2 ⎠⎣ ⎝ 2 ⎠⎦ (5.13) uy = KI 2μ r ⎛ θ ⎞⎡ ⎛ θ ⎞⎤ sin⎜ ⎟ ⎢κ + 1 − 2 cos 2 ⎜ ⎟⎥ 2π ⎝ 2 ⎠⎣ ⎝ 2 ⎠⎦ (5.14) θ⎛ θ 3θ ⎞ cos ⎜1 − sin sin ⎟ 2⎝ 2 2 ⎠ 2π r (5.15) θ⎛ θ 3θ ⎞ cos ⎜1 + sin sin ⎟ 2⎝ 2 2 ⎠ 2π r (5.16) σ xx = KI σ yy = KI σ xy = KI 2π r sin θ θ 3θ cos cos 2 2 2 (5.17) where r is the distance from the crack tip and θ is the angle measured from the line of the crack. The stress intensity factor is prescribed as K I = 2π . μ is the 152 Chapter 5 Adaptive Analyses for Solids using the GSM shear modulus, and κ is defined as 3 −υ 1+υ Plane stress κ = 3 − 4υ Plane strain κ= (5.18) The problem is solved as a plain strain case with geometry and material parameters a = 1 , E = 3 × 10 7 , υ = 0.3 . Due to the symmetry, only upper half of the plate is modeled, as shown in Fig. 5.31(b). To extend the above solutions to the whole studied domain, we impose on the square plate boundary (the upper, left and right edges) the exact traction. Essential boundary conditions are applied as demonstrated in Fig. 5.31(b). Six distributions of uniform nodes, 76, 252, 542, 848, 1481 and 2717, are first investigated. Fig. 5.32 shows the selected node distributions. Similarly, the reference solution of strain energy, 3.146553E-4, is obtained using FEM (Gauss integration) with a very fine mesh of 9600 nodes based on analytical solutions of stress components. The computed displacement error in y -direction ( u y ), strain energy and its error norm are shown in Table 5.12. It can be observed that both the displacement u y and energy norm approach the exact (reference) solutions as nodes increase. However, the displacement error reduces very slowly due to the stress singularity near the crack tip. The error is still bigger than 10 percent with 2717 evenly distributed field nodes. As shown in Fig. 5.32, the distribution of 76 field nodes is started for adaptive refinement. The local refinement coefficient is prescribed as κ l = 0.05 and the global residual tolerant coefficient is set as κ g = 0.05 . As shown in Fig. 5.33, the adaptive analysis ends at the 9th step with 1144 extremely irregular nodes in the half 153 Chapter 5 Adaptive Analyses for Solids using the GSM plate domain. Due to stress concentration at the crack tip, a large number of nodes are inserted into the crack tip neighborhood (see Fig. 5.33). Table 5.13 shows displacement error in y -direction, energy and its error norm for each adaptive step. Compared with uniform refinement, the adaptive scheme is much more effective. As shown in Fig. 5.34, the displacement error of u y reduces tremendously as adaptive refinement goes. The comparisons of strain energy and its convergence rate between uniform and adaptive refinements are presented in Fig. 5.35 and Fig. 5.36, respectively. It can be easily found that the GSM using adaptive scheme converges more than two times faster than uniform refinement. 5.3.7 Singular loading problem To further examine the capability of our strong form GSM, a square solid subjected to a singular loading P = 1 at the center of the top edge is studied, as shown in Fig. 5.37. The solid is constrained in x and y directions along the left, right and bottom sides respectively. This problem is solved as a plain strain case with geometry and material parameters as a = 10 , E = 10 7 , υ = 0.3 . This singular loading case is studied using the GSM with both uniform and adaptive models. For the uniform refinement, the problem domain is presented using 121, 625, 1681 and 3721 nodes respectively, as shown in Fig. 5.38. For the adaptive procedure, 11 steps of adaptive refinement are performed with κ l = 0.1 and the nodal configuration at the 6th and 11th steps is shown in Fig. 5.39. The figure shows that the present adaptive GSM can accurately catch the steep gradient of stresses and the occurrence of refinement properly concentrates around the point with singular 154 Chapter 5 Adaptive Analyses for Solids using the GSM loading. The displacements in x and y directions along the line y = 5 for both uniform and adaptive refinements are plotted in Fig. 5.40 and Fig. 5.41, respectively. For this problem, the reference solutions are obtained using adaptive FEM with a very fine mesh of 7431 nodes. In Fig. 5.42, the values of strain energy are presented for the results of uniform and adaptive refinements. The reference solution of the strain energy is 4.2812E-3. It can be concluded that the present GSM with adaptive procedure can effectively produce reliable results for problems with high stress concentration, singular points and even singular loading. 5.4 Remarks In the current work, a novel gradient smoothing method (GSM) based on a strong form of governing equations is developed for solid mechanics problems: 1) In the present method, different types of smoothing domains are devised in a novel manner and used for approximation of derivatives; 2) The GSM can be used for mechanics problems with any arbitrarily irregular domains, singularities, and singular loading, which is very difficult for a strong form method. Both stability and accuracy have been demonstrated in comparison with the widely-used FEM. The GSM has exhibited even much better than the FEM for adaptive analysis judging from the condition numbers. 3) Due to the excellent stability, the GSM is further extended to adaptive analysis and found effective. A simple yet robust residual based error indicator is adopted in our adaptive procedure. By approximating the residual of the governing equation in the domain, this error indicator can efficiently capture the region to be 155 Chapter 5 Adaptive Analyses for Solids using the GSM refined. From intensive numerical studies carried out on several benchmark problems with and without singularities, the following conclusions can be drawn: 1. The GSM can reproduce linear fields regardless of the types of boundary conditions (essential or natural). Hence, the solution will converge to any higher-order continuous fields as the field mesh is refined. 2. The study of numerical examples shows that the proposed GSM not only can obtain accurate and stable results but also is successful in the implementation for adaptive analysis with steady convergences. 3. For problems without singularity, even though there is no significant improvement in the convergence rate compared with the uniform refinement, our adaptive GSM can lead to solutions with much higher accuracy. 4. For problems where singular points exist, nearly optimal nodal distributions are generated automatically in the process of adaptive analysis. As a result, far less degrees of freedom are needed to achieve the desired accuracy compared with uniform refinement. 5. In summary, we conclude that the GSM is a stable, robust and reliable numerical method based on strong form formulation for adaptive analysis of solid mechanics problems. 156 Chapter 5 Adaptive Analyses for Solids using the GSM Error norm of u y Error norm of u x Patch Patch a 6.2944E-16 Patch b 1.6149E-16 Patch c 2.1959E-16 Patch d 1.6785E-16 Patch e 1.8025E-16 Patch f 9.8661E-16 Table 5.1 Error norms of displacements for essential-patch-test. Error norm of u y Error norm of u x Patch 6.3624E-16 2.7057E-16 3.4147E-16 5.3408E-16 3.2913E-16 2.3039E-16 Regular nodes 1.1019E-14 Irregular nodes 8.5544E-14 Table 5.2 Error norms of displacements for natural-patch-test. 7.9795E-14 2.7266E-12 No. of field nodes 121 256 441 900 2116 3721 Error norm (%) 33.54 9.02 4.84 2.22 0.91 0.51 Table 5.3 Error norms of uniform refinement for Poisson’s equation with a sharp peak. Step 1 2 3 4 5 No. of field nodes 121 183 313 584 1107 Error norm (%) 33.54 8.66 3.09 1.10 0.56 Table 5.4 Error norms of adaptive refinement for Poisson’s equation with a sharp peak. No. of field nodes 39 98 199 403 826 1513 Error norm of u x (%) 90.12 39.68 6.59 1.67 1.12 0.98 Table 5.5 Error norms of uniform refinement for infinite plate with a circular hole. Step No. of field nodes 1 39 2 99 3 158 4 197 5 395 6 567 Error norm of u x (%) 90.12 8.88 3.01 2.18 0.78 0.48 Table 5.6 Error norms of adaptive refinement for infinite plate with a circular hole. 157 Chapter 5 Adaptive Analyses for Solids using the GSM No. of field nodes 73 214 488 755 1376 Value -2.5170 -2.6748 -2.7397 -2.7592 -2.7901 GSM Error 12.46 6.98 4.72 4.04 2.96 u y (1,0) norm (%) Value -2.7581 -2.8277 -2.8503 -2.8575 -2.8643 FEM Error 4.08 1.65 0.87 0.62 0.38 norm (%) Value 1.2867 1.3344 1.3521 1.3566 1.3642 GSM Error 6.74 3.29 2.00 1.68 1.13 norm (%) Energy Value 1.3492 1.3668 1.3727 1.3746 1.3764 FEM Error 2.21 0.94 0.51 0.37 0.24 norm (%) Table 5.7 Error norms of uniform refinement for short cantilever plate. 2498 -2.8094 2.29 -2.8685 0.24 1.3683 0.83 1.3776 0.16 Step 1 2 3 4 5 6 No. of field nodes 73 193 400 657 1109 1889 Value -2.5170 -2.7557 -2.8304 -2.8405 -2.8561 -2.8620 u y (1,0) Error norm (%) 12.46 4.16 1.56 1.21 0.67 0.47 Value 1.2867 1.3496 1.3647 1.3712 1.3751 1.3768 Energy Error norm (%) 6.74 2.18 1.09 0.62 0.34 0.21 Table 5.8 Error norms of adaptive refinement for short cantilever plate using GSM. Step 1 2 3 4 5 6 7 No. of field nodes 73 127 209 370 541 1016 1614 Value -2.7581 -2.8039 -2.8222 -2.8439 -2.8532 -2.8606 -2.8643 u y (1,0) Error 4.08 2.48 1.85 1.09 0.77 0.51 0.38 norm (%) Value 1.3492 1.3620 1.3664 1.3726 1.3747 1.3762 1.3770 Energy Error 2.21 1.29 0.97 0.52 0.36 0.26 0.20 norm (%) Table 5.9 Error norms of adaptive refinement for short cantilever plate using FEM. No. of field 108 369 710 1430 nodes Strain energy 3.5135E-2 3.3590E-2 3.3234E-2 3.2895E-2 Error norm (%) 9.11 4.31 3.20 2.15 Table 5.10 Error norms of uniform refinement for L -shaped plate. 2194 2948 3.2787E-2 3.2726E-2 1.81 1.62 158 Chapter 5 Adaptive Analyses for Solids using the GSM Step 1 2 3 4 No. of field nodes 108 242 538 817 Strain energy 3.5135E-2 3.3378E-2 3.2824E-2 3.2515E-2 Error norm (%) 9.11 3.65 1.93 0.97 Table 5.11 Error norms of adaptive refinement for L -shaped plate. 5 1489 3.2346E-2 0.44 No. of field nodes 76 252 542 848 1481 2717 Error norm of u y (%) 49.68 29.03 21.13 17.37 13.85 10.64 Value 2.8491E-4 2.9301E-4 2.9798E-4 3.0067E-4 3.0333E-4 Error norm 9.45% 6.88% 5.30% 4.44% 3.60% Table 5.12 Error norms of uniform refinement for Mode-I crack problem. 3.0588E-4 2.79% Energy Step No. of field nodes 1 76 2 115 3 224 4 275 5 364 Error norm of u y (%) 49.68 30.18 15.74 13.15 12.92 3.0395E-4 3.40 3.0474E-4 3.15 3.0774E-4 2.20 Energy Value Error norm (%) 2.8491E-4 2.9463E-4 9.45 6.36 Step No. of field nodes 6 595 7 710 8 832 9 1144 Error norm of u y (%) 7.63 6.27 4.53 2.18 Value 3.1021E-4 3.1148E-4 3.1188E-4 3.1258E-4 Error norm (%) 1.41 1.01 0.88 0.66 Table 5.13 Error norms of adaptive refinement for Mode-I crack problem. Energy 159 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.1 Residual evaluated at the center ( ) of a triangular cell. Cell a Cell b Old node Fig. 5.2 Illustration of the refinement procedure. New node 160 Chapter 5 Adaptive Analyses for Solids using the GSM (a) (b) (c) (d) (e) (f) Fig. 5.3 Patches of five nodes in the essential-patch-test. 161 Chapter 5 Adaptive Analyses for Solids using the GSM y x 3 6 (a) (b) (c) Fig. 5.4 (a) Patches for the natural-patch-test: a uniform axial traction along the right end of the patch; (b) Patch with 35 regular nodes; (c) Patch with 35 irregular nodes. 162 Chapter 5 Adaptive Analyses for Solids using the GSM (a) (b) (c) Fig. 5.5 Three-dimensional plots of the exact solution to the Poisson’s equation with a sharp peak: (a) u ; (b) ∂u ∂u and (c) . ∂x ∂y 163 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.6 Node distributions of uniform refinement for Poisson’s equation with a sharp peak at the center. 164 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.7 Adaptive nodes from the 2nd to 5th step for solving Poisson’s equation. Fig. 5.8 Estimated global residual at each adaptive step for Poisson’s equation. 165 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.9 Comparison of error and convergence rate between uniform and adaptive refinements for solving Poisson’s equation with a sharp peak. Fig. 5.10 Approximated values of field function u along the line y = 0.5 at the first and fifth steps. 166 Chapter 5 Adaptive Analyses for Solids using the GSM (a) (c) (b) Fig. 5.11 The three-dimensional plots of adaptive GSM solutions for Poisson’s equation with a sharp peak at the final adaptive step: (a) u ; (b) ∂u ∂u ; (c) . ∂x ∂y 167 Chapter 5 Adaptive Analyses for Solids using the GSM y b p a x Fig. 5.12 Quarter model of the infinite plate with a circular hole. Fig. 5.13 Nodes of uniform refinement for infinite plate: from 39 to 1513 nodes. 168 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.14 Node distributions of adaptive refinement at the 3rd and 6th steps for the quarter model of infinite plate with a circular hole. Fig. 5.15 Estimated global residual at each adaptive step for the infinite plate. 169 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.16 Comparison of error norm of displacement u x between uniform and adaptive refinements for infinite plate with a circular hole. Fig. 5.17 Normal stress σ xx along x = 0 at the 3rd and 6th steps. 170 Chapter 5 Adaptive Analyses for Solids using the GSM y p=1.0 1.0 A 1.0 x Fig. 5.18 A short cantilever plate subjected to a uniformly distributed pressure. Fig. 5.19 Node distributions of ‘uniform’ refinement for short cantilever plate. 171 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.20 Node distributions of adaptive refinement at the 3rd and 6th steps for short cantilever plate. Fig. 5.21 Estimated global residual at each adaptive step for short cantilever plate. 172 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.22 Comparison of displacement u y (1,0) for short cantilever plate between GSM and FEM with uniform and adaptive refinements. Fig. 5.23 Comparison of computed strain enegy for short cantilever plate between GSM and FEM with uniform and adaptive refinements. 173 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.24 Comparison of error and convergence in energy norm for short cantilever plate between GSM and FEM with uniform and adaptive refinements. Fig. 5.25 Comparison of condition number of coefficient matrix for short cantilever plate between GSM and FEM with uniform and adaptive refinements. 174 Chapter 5 Adaptive Analyses for Solids using the GSM y a p a x a a Fig. 5.26 L -shaped plate subjected to a tensile load in the horizontal direction. Fig. 5.27 Selected node distributions of uniform refinement for L -shaped plate. 175 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.28 Node distributions of adaptive refinement at the 3rd and 5th steps for L -shaped plate. Fig. 5.29 Comparison of computed strain energy between uniform and adaptive refinement for L -shaped plate. 176 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.30 Comparison of error and convergence rate in energy norm between uniform and adaptive refinements for L -shaped plate. y ( r, θ ) 2a x a 2a a 2a (a) (b) Fig. 5.31 Mode-I crack problem: (a) geometry; (b) half model with boundary conditions. 177 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.32 Selected node distributions of uniform refinement for Mode-I crack. 178 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.33 Node distributions of adaptive refinement at the 3rd, 5th, 7th and 9th steps for Mode-I crack problem. 179 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.34 Comparison of error in displacement in y -direction between uniform and adaptive refinements for Mode-I crack problem. Fig. 5.35 Comparison of strain energy between uniform and adaptive refinement. 180 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.36 Comparison of error and convergence rate in energy norm between uniform and adaptive refinements for Mode-I crack problem. Fig. 5.37 A square solid subjected to a singular loading at the center of the top edge. 181 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.38 Node distributions of uniform refinement for singular loading problem. Fig. 5.39 Node distributions of adaptive refinement at the 6th and 11th steps for singular loading problem. 182 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.40 Displacement u x ( x,5) between uniform and adaptive refinements. Fig. 5.41 Displacement u y ( x,5) between uniform and adaptive refinements. 183 Chapter 5 Adaptive Analyses for Solids using the GSM Fig. 5.42 Comparison of computed strain energy between uniform and adaptive refinement for singular loading problem. 184 Chapter 6 Vibration Analyses of 2-D Solids using the GSM Chapter 6 Vibration Analyses of 2-D Solids using the GSM 6.1 Introduction In this chapter, the gradient smoothing method (GSM) is further extended to formulate the free and forced vibration analyses of two-dimensional solids. In the free vibration analysis, frequencies and eigenmodes are obtained by solving the linear eigenvalue equation. In the forced vibration analysis, both the explicit time integration method (the central difference method) and the implicit time integration method (the Newmark method) are used to solve the forced vibration system equation. Section 6.2 gives the basic equations of elastodynamics. The GSM is then developed for free and forced vibration analyses in sections 6.3 and 6.4, respectively. A brief remark is presented in section 6.5. 6.2 The Governing Equations of 2-D Elastodynamics The strong form of two-dimensional small displacement elastodynamics problem of solid mechanics in the domain Ω bounded by Γ is as follows: σ ij , j + bi = mu&&i + cu&i where m is the mass density, c is the damping coefficient, u&&i = acceleration, u&i = (6.1) ∂ 2 ui is the ∂t 2 ∂ui the velocity, σ ij the stress tensor, which corresponds to the ∂t 185 Chapter 6 Vibration Analyses of 2-D Solids using the GSM displacement filed ui , bi the body force tensor, and ( ), j denotes ∂ . The ∂x j auxiliary conditions are given as follows: Essential boundary condition: ui = ui on Γu (6.2) Natural boundary condition: σ ij n j = ti on Γt (6.3) Displacement initial condition: u( x, t0 ) = u 0 ( x ) x ∈Ω (6.4) Velocity initial condition: u& ( x, t0 ) = v 0 ( x ) x ∈Ω (6.5) in which the ui , ti , u0 and v 0 denote the prescribed displacements, tractions, initial displacements and velocities, respectively, and n j is the unit outward normal to the domain Ω . Equation (6.1) for isotropic materials can be written as the following standard strong form E 1 −υ 2 E 1 −υ 2 ⎛ ∂ 2u 1 − υ ∂ 2u ∂ 2 v ⎞ ∂ 2u ∂u ⎟ ⎜⎜ 2 + + − −c =0 + b m x 2 2 ⎟ 2 ∂y ∂t ∂t ∂x∂y ⎠ ⎝ ∂x ⎛ ∂ 2 v 1 − υ ∂ 2v ∂ 2u ⎞ ∂ 2v ∂v ⎜⎜ 2 + ⎟ + + − −c =0 b m y 2 2 ⎟ 2 ∂x ∂x∂y ⎠ ∂t ∂t ⎝ ∂y where E and υ (6.6) are Young’s modulus and Poisson ratio, u and v are displacements at x and y direction. The GSM is used directly to discretize Eq. (6.6). 6.3 Free Vibration Analysis 6.3.1 Strong form formulation The governing equation for no damping free vibration is as follows: 186 Chapter 6 Vibration Analyses of 2-D Solids using the GSM σ ij , j = mu&&i (6.7) The boundary conditions are usually the same form of Eqs. (6.2) and (6.3), but the traction t = 0 . In the free vibration analysis, u( x, t ) can be written as u(x, t ) = u( x ) sin(ω t + ϕ ) (6.8) where ω is the frequency. Substituting Eq. (6.8) into Eq. (6.7) leads to the following equations: σ ij , j + ω 2 m ui = 0 (6.9) It should be noted that the stresses, σ , and displacements, u , in Eq. (6.9) are only the function of coordinator x . Equation (6.9) for isotropic materials can be written as E 1 −υ 2 E 1 −υ 2 ⎛ ∂ 2u 1 − υ ∂ 2u ∂ 2 v ⎞ ⎟⎟ + mω 2 u = 0 ⎜⎜ 2 + + 2 2 ∂ x ∂ y ∂ x ∂ y ⎠ ⎝ ⎛ ∂ 2 v 1 − υ ∂ 2 v ∂ 2u ⎞ ⎜⎜ 2 + ⎟⎟ + mω 2 v = 0 + 2 2 ∂x ∂x∂y ⎠ ⎝ ∂y (6.10) Applying the GSM formulation into the strong form (6.10) for all nodes leads to the following discrete system equations K u − ω 2 Mu = 0 (6.11) Where K is the stiffness matrix, M is the mass matrix. For free vibration analysis, Eq. (6.11) can also be written as: (K − ω 2 M) q = 0 (6.12) where q is the eigenvector. Equation (6.12) is the GSM strong formulation for free vibration analysis. In order to determine the frequencies, ω , and free vibration modes, it is necessary to solve the linear eigenvalue equation. However, it remains the boundary conditions in Eqs. (6.2) and (6.3) need to be satisfied. 187 Chapter 6 Vibration Analyses of 2-D Solids using the GSM 6.3.2 Numerical results The GSM is used for free vibration analysis of 2-D structures. Except specially mentioned, the units are taken as standard international (SI) units in the following examples. 6.3.2.1 A cantilever beam The GSM is first applied to analyze free vibration of a cantilever beam as shown in Fig. 6.1. The problem has been analyzed by Nagashima (1999) using node-by-node meshless (NBNM) method, which is based on a global weak form. A plane stress problem is considered. The parameters are taken as length L =100mm, height D =10 mm, thickness t =1.0mm, Young’s modulus E = 2.1 × 10 4 kgf/mm 2 , Poisson ratio υ = 0.3 , mass density m = 8.0 × 10 −10 kgfs 2 /mm 4 . Fig. 6.2 shows two kinds of nodal arrangements, coarse arrangement (63 nodes) and fine arrangement (306 nodes). Frequency results of these two nodal arrangements obtained by the GSM are listed in Table 6.1. The results obtained by FEM software, ANSYS, and NBNM method (Nagashima, 1999) are also listed in the same table. From this table, one can observe that the results by the GSM are in good agreement with those obtained using FEM and NBNM. The convergence of the present method is also demonstrated in this table. As the number of nodes increases, results obtained by the GSM approach the FEM reference results, which are computed using 9616 degrees of freedom (DOF). The first ten eigenmodes obtained by the GSM with fine arrangement are plotted in Fig. 6.3. Compared with FEM results and Nagashima’s (1999) results, the GSM obtains almost identical results. 188 Chapter 6 Vibration Analyses of 2-D Solids using the GSM 6.3.2.2 A variable cross-section beam In this example, the present GSM is used for free vibration analysis of a cantilever beam with variable cross-sections, shown in Fig. 6.4(a). Results are obtained for following numerical parameters: the length L = 10 , the height H (0) = 5 , H ( L) = 3 , the thickness t = 1.0 , E = 3.0 × 107 , υ = 0.3 and m = 1.0 . The nodal arrangement is shown in Fig. 6.4(b). Results obtained by the presented GSM, the MLPG method (Gu and Liu, 2001) and the FEM software, ABAQUS, are listed and compared in Table 6.2. The obtained results are in very good agreement. 6.3.2.3 A shear wall Fig. 6.5 shows a shear wall with four openings, which has been solved using boundary element method by some researchers (Brebbia et al., 1984). The problem is solved for the plane stress case with E = 1000 , υ = 0.2 , t = 1.0 and m = 1.0 . A total of 574 uniformed nodes are used to discretize the problem domain. The problem is also analyzed using FEM software ABAQUS. Natural frequencies of the first 8 modes are calculated and listed in Table 6.3. Results obtained by BEM, FEM and MLPG are listed in the same table. Results obtained by the present GSM are in good agreement with those obtained using BEM, FEM and MLPG. 6.4 Forced Vibration Analysis In the forced vibration analysis, u is a function of both space coordinates and time. Only space domain is discretized. The discrete form of governing equation (6.1) for forced vibration can be written as 189 Chapter 6 Vibration Analyses of 2-D Solids using the GSM && (t ) + CU & (t ) + KU(t ) = F(t ) MU (6.13) & and U && are vectors of displacements, velocities and accelerations for where U , U all nodes in the entire problem domain. M is the mass matrix, C is the damping matrix, K is the stiffness matrix and F is the force matrix. 6.4.1 Direct analysis of forced vibration The methods of solving Eq. (6.13) can be largely divided into two categories: the modal analysis and the direct analysis. The direct analysis methods are utilized in this chapter. Several direct analysis methods have been developed to solve the dynamic equation (6.13), such as central difference method (CDM) and Newmark method (Petyt, 1990). The CDM and Newmark method are used in this chapter. (a) The central difference method The central difference method (CDM) consists of expressing the velocity and acceleration at time t in terms of the displacement at time t − Δt , t and t + Δt using central finite difference formulation: 1 (u(t − Δt ) − 2u(t ) + u(t + Δt ) ) Δt 2 1 (− u(t − Δt ) + u(t + Δt ) ) u& (t ) = 2Δt &&(t ) = u (6.14) (6.15) where Δt is the time step. The response at time t + Δt is obtained by evaluating the equation of motion at time t . The CDM is, therefore, an explicit method. The CDM is conditionally stable. The stable critical time step for CDM can be obtained from the maximum frequencies based on the dispersion relation using (Belytschko et al., 2000) 190 Chapter 6 Vibration Analyses of 2-D Solids using the GSM Δt crit = max i 2 ωi (ξ 2 i + 1 − ξi ) (6.16) where ωi is the frequency and ξ i the fraction of critical damping in this mode. For non-uniform arrangements of the nodes, the critical time step can be obtained by the eigenvalue inequality: Δt crit = min 2 max Q λ Qmax (6.17) where λ Qmax is the maximum eigenvalue at the quadrature point x Q . The value of λ Qmax depends on the size of the local quadrature cell and the size of the support domain (Belytschko et al., 2000). (b) The Newmark method The Newmark method is a generalization of the linear acceleration method. This latter method assumes that the acceleration varies linearly within the interval (t, t + Δt ) . This gives && = u && t + u 1 (u&& t + Δt − u&& t )τ Δt (6.18) and && t + δ u t + Δt ] Δt u& t + Δt = u& t + [(1 − δ )u (6.19) ⎡⎛ 1 ⎤ 2 ⎞ && && u t + Δt = u t + u& Δt + ⎢⎜ − β ⎟u t + β u t + Δ t ⎥ Δt ⎠ ⎣⎝ 2 ⎦ (6.20) The response at time t + Δt is obtained by evaluating the equation of motion at time t + Δt . The Newmark method is, therefore, an implicit method. The Newmark method is unconditionally stable provided that δ ≥ 0.5 and β ≥ 1 (δ + 0.5)2 4 (6.21) One can find that δ = 0.5 and β = 0.25 lead to acceptable results for most of 191 Chapter 6 Vibration Analyses of 2-D Solids using the GSM the problems considered. Therefore, δ = 0.5 and β = 0.25 are always used in this chapter for simplification. 6.4.2 Numerical results The forced vibration for a 2-D structure, a cantilever beam, as shown in Fig. 4.5, is analyzed. In this numerical example for the forced vibration analysis, the beam is subjected to a parabolic traction at its free end, P = 1000 g (t ) . g (t ) is a function of time. As shown in Fig. 4.6, 273 uniformly distributed nodes are used to discretize the problem domain. For simplification, m = 1.0 is considered. Displacements and stresses for all nodes are obtained. Detailed results of vertical displacement u y on the middle node of the free end of the beam are presented. For comparison, solutions for this problem are also obtained using the FEM software package, ABAQUS/Explicit. A simple harmonic loading with g (t ) = sin(ω f t ) is first considered, where the frequency of dynamic load ω f = 27 is used in this example. In order to investigate the properties of two different direct time integration methods, CDM and Newmark method, results of different time steps are obtained and plotted in Fig. 6.6. From this figure, it can be found that when the time step is smaller (e.g., Δt = 1 × 10 −4 ) both methods obtain results in good agreement with FEM. When Δt ≥ Δt crit (from Eq. (6.17), Δt crit ≈ 1 × 10 −3 in this example), CDM will become unstable. However, the Newmark method is always stable for any time step. It demonstrates that CDM is a conditionally stable method and Newmark method is an unconditionally stable method. A bigger time step can be used in the Newmark method. Even when 192 Chapter 6 Vibration Analyses of 2-D Solids using the GSM Δt = 1 × 10 −3 or Δt = 1 × 10 −2 is used, very good results can also be obtained using Newmark method. However, it should be noted that the computational error would increase with the increase of time step in the Newmark method. The accuracy of the Newmark method would become unacceptable when the time step is too big (e.g., Δt = 5 × 10 −2 ). The unconditionally stability of the Newmark method is very useful for structural forced vibration analysis in engineering applications, especially when responses for a longer time are needed. A big time step can be used in the Newmark method, thus considerable computations can be saved. Many time steps are also calculated to check the stability of the present GSM, with which very stable results are obtained. The transient response of the beam subjected to an impulse (suddenly loaded and suddenly vanished) force P = 1000 g (t ) is considered. The function g (t ) is taken as ⎧1 g (t ) = ⎨ ⎩0 t ≤ 0.5s t > 0.5s (6.22) The present GSM is used to obtain the transient reponse without damping ( c = 0 ). The Newmark method is utilized in this analysis. Vertical displacements on the middle point of the free end of the beam are plotted in Fig. 6.7. For comparison, the result obtained by the FEM software package, ABAQUS/Explicit, is shown in the same figure. Results obtained by the present GSM are in good agreement with those obtained using FEM. Many time steps are also calculated to check the stability of the GSM formulation. 6.5 Remarks 193 Chapter 6 Vibration Analyses of 2-D Solids using the GSM GSM formulations for free vibration and forced vibration analyses of two-dimensional solids and structures are presented in this chapter. Strong forms are developed from the dynamic partial differential equations. Direct time integration methods, the central difference method and Newmark method, are utilized and compared for the forced vibration analyses. Programs of the present GSM have been developed, and a number of numerical examples for free vibration and forced vibration analyses are presented to demonstrate the validity and efficiency of the present method. The results presented are encouraging. It is demonstrated that the GSM is easy to implement, and very flexible for free vibration and forced vibration analyses of solids and structures. 194 Chapter 6 Vibration Analyses of 2-D Solids using the GSM 63 nodes 306 nodes Nagashima FEM (1999) (ANSYS) Reference Nagashima FEM (1999) (ANSYS) ANSYS (9616 DOF) Mode GSM 1 815.62 926.10 983.27 831.82 844.19 853.80 823.75 2 4958.93 5484.11 5918.8 4948. 19 5051.21 5122.0 4941.7 3 12824.48 12831.88 12841 12824.58 12827.60 12828 12824 4 13455.86 14201.32 15629 13055.56 13258.21 13495 13017 5 23826.69 25290.04 28500 23629.15 23992.82 24541 23656 6 36489.65 37350.18 38588 36092.19 36432.15 37478 36081 7 38645.42 38320.59 43714 38428.28 38436.43 38472 38445 8 49678.60 50818.64 60724 49698.37 49937.19 51704 49684 9 63117.79 63283.70 64534 63016.54 63901.16 64072 63979 10 64077.81 63994.48 79276 64217.31 64085.90 66832 64061 GSM Table 6.1 Natural frequencies (Hz) of a cantilever beam with different nodal distributions. 195 Chapter 6 Vibration Analyses of 2-D Solids using the GSM ω (rd/s) Modes 1 2 3 4 5 GSM 262.35 917.16 952.58 1845.50 2570.55 MLPG (Gu and Liu, 2001) 263.21 923.03 953.45 1855.14 2589.78 FEM (ABAQUS) 262.09 918.93 951.86 1850.92 2578.63 Table 6.2 Natural frequencies of a variable cross-section cantilever beam. ω (rd/s) Mode GSM MLPG (Gu and Liu, 2001) FEM (ABAQUS) Brebbia et al. (1984) 1 2.077 2.069 2.073 2.079 2 7.052 7.154 7.096 7.181 3 7.645 7.742 7.625 7.644 4 11.693 12.163 11.938 11.833 5 15.028 15.587 15.341 15.947 6 17.833 18.731 18.345 18.644 7 19.281 20.573 19.876 20.268 8 21.246 23.081 22.210 22.765 Table 6.3 Natural frequencies of a shear wall. 196 Chapter 6 Vibration Analyses of 2-D Solids using the GSM D y x L Fig. 6.1 A cantilever beam. (a) 63 nodes (b) 306 nodes Fig. 6.2 Nodal distribution of the beam: (a) 63 nodes and (b) 306 nodes. 197 Chapter 6 Vibration Analyses of 2-D Solids using the GSM Mode 1 Mode 3 Mode 5 Mode 7 Mode 9 Mode 2 Mode 4 Mode 6 Mode 8 Mode 10 Fig. 6.3 Eigenmodes for the cantilever beam obtained using the GSM. 198 Chapter 6 Vibration Analyses of 2-D Solids using the GSM y H(x) x L (a) (b) Fig. 6.4 A cantilever beam with variable cross-sections (a) and its mesh (b). 199 Chapter 6 Vibration Analyses of 2-D Solids using the GSM Fig. 6.5 A shear wall with four openings. Fig. 6.6 Displacement u y at the middle point of free end using different time steps. 200 Chapter 6 Vibration Analyses of 2-D Solids using the GSM Fig. 6.7 Transient displacement u y at the middle point of the free end of the beam using the Newmark method ( δ = 0.5 and β = 0.25 ). 201 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction 7.1 Introduction Gradient smoothing method (GSM) has been well developed and thoroughly investigated from theoretical aspects, static analyses and adaptive analyses to dynamic analyses. Excellent stability, efficiency and accuracy have been fully demonstrated through extensive numerical examples. Based on the strong form formulation, the GSM is a stable, robust and reliable numerical method for various applications in solid mechanics. In the GSM formulation, a piecewise constant smoothing function is adopted, as in Eq. (3.9). However, in this chapter, a piecewise linear smoothing function is adopted as an alternative to further develop the so-called linearly weighted gradient smoothing method (LWGSM). Theoretical aspects of the LWGSM are provided in detail in the following sections. The relations between GSM and LWGSM are explored also in this chapter. Some numerical tests are conducted to demonstrate the properties of the LWGSM. 7.2 Linearly Weighted Gradient Smoothing Method (LWGSM) 202 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction Similar to the GSM, in the LWGSM, derivatives at various locations, including nodes, centroids of cells and midpoints of cell-edges, are approximated over relevant gradient smoothing domains using a linearly weighted gradient smoothing operation. The details about the theory, principle and implementation procedure of the LWGSM are introduced in this section with a focus on the approximation of spatial derivatives. 7.2.1 Gradient smoothing functions In the GSM as presented in Chapter 3, a piecewise constant smoothing function is adopted. In this work, at a node of interest, i , the following piecewise linear smoothing function is adopted: ⎧ a + a1 ( x − x i ) Φ (x − x i ) = ⎨ 0 ⎩0 x ∈ Ωi x ∉ Ωi (7.1) where a0 and a1 are matrices of coefficients that are dependent on the geometry of each sub-triangle within the smoothing domain Ω i . Fig. 7.1 shows the spatial distribution of piecewise linear smoothing functions over a smoothing domain for an inner node. In addition, the smoothing function is assumed to satisfy the following weighted partition of unity: 2 Ni ∑ ∫ Φ (x − x ) dA = 1 k =1 Ω ( k ) i i (7.2) where N i represents the number of edges connected with the node of interest i . As shown in Fig. 7.2, the linear smoothing function is piecewise continuous on the 2 N i sub-triangles contained in the smoothing domain Ω i . The smoothing functions vanish at the boundary ( ∂ Ω i ) of the smoothing domain Ω i , meaning that 203 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction Φ(x − x i ) = 0 x ∈ ∂Ω i (7.3) Analogously, with the gradient smoothing operations presented in Chapter 3, the firstand second-order derivatives can be approximated in the forms of ∇ h u ( x i ) = − ∫ u h ( x )∇Φ ( x − x i )dΩ (7.4) ∇ 2 u ( x i ) = − ∫ ∇u h ( x )∇Φ ( x − x i )dΩ (7.5) Ωi Ωi It is apparent that within the smoothing domain for the node of interest i , the piecewise weighting function Φ is continuous and its derivatives, ∇Φ , are discontinuous. 7.2.2 Determination of coefficients For a two-dimensional case, the piecewise linear smoothing function can be rewritten as Φ ( x − x i ) = a i + b i ( x − xi ) + c i ( y − y i ) (7.6) The coefficients in matrices a, b and c are dependent on the geometry of each sub-triangle within the smoothing domain. In detail, the matrices of coefficients are in the form of ⎡ ai ,1 ⎤ ⎡ bi ,1 ⎤ ⎡ ci ,1 ⎤ ⎢ a ⎥ ⎢ b ⎥ ⎢ c ⎥ ⎢ i ,2 ⎥ ⎢ i,2 ⎥ ⎢ i ,2 ⎥ ai = ⎢ M ⎥ , bi = ⎢ M ⎥ , c i = ⎢ M ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ai , 2 N I −1 ⎥ ⎢bi ,2 N I −1 ⎥ ⎢ci , 2 N I −1 ⎥ ⎢⎣ ai ,2 N i ⎥⎦ ⎢⎣ bi , 2 N i ⎥⎦ ⎢⎣ ci , 2 N i ⎥⎦ (7.7) As shown in Fig. 7.2, there are totally 2 N i sub-triangles that form the smoothing domain for an internal node of interest. For a boundary node, there are ( 2 N i − 2) sub-triangles, correspondingly. Furthermore, Eq. (7.6) gives 204 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction ∂Φ ( x − x i ) = bi , ∂x ∂Φ ( x − x i ) = ci ∂y (7.8) Let us only concern about the discretization of equations on an inner node. The smoothing function is required to satisfy the weighted partition of unity: ⎧ [ai , 2 k −1 + bi ,2 k −1 ( x − xi ) + ci , 2 k −1 ( y − yi )]dA + ⎫ ⎪⎪Ωi( 2∫k −1) ⎪⎪ ⎨ ⎬ =1 ∑ k =1 ⎪ [ ] + − + − ( ) ( ) a b x x c y y dA ⎪ i i ,2k i ∫ i ,2k i,2k ⎪⎩Ωi( 2 k ) ⎪⎭ Ni (7.9) Besides, for a sub-triangle Δimk ck , it is further assumed that Φ mk = Φ ( x mk − x i ) = 0 (7.10) Φ ck = Φ ( x ck − x i ) = 0 (7.11) With Eq. (7.9), it is easily obtained ai , 2 k = Φ ( x i − x i ) = 3 Ai (7.12) where is the total area of the smoothing domain for the node of interest i . This formulation implies that all coefficients in the matrix a i are constant, regardless of the sub-triangles involved. The parameters bi , 2 k and ci , 2 k for each sub-triangle can then be obtained as the solutions to Eqs. (7.10) and (7.11), in the form of bi , 2 k = − ai , 2 k ci , 2 k = ai , 2 k y ck − y mk ( x mk − xi )( y ck − yi ) − ( x ck − xi )( y mk − yi ) xc k − xmk ( xmk − xi )( yc k − yi ) − ( xc k − xi )( ymk − yi ) (7.13) (7.14) It is also noticed that the denominator in Eqs. (7.13) and (7.14) relates to the area of the sub-triangle Δimk ck , which is 205 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction Ai , 2 k = ( x mk − xi )( y ck − yi ) − ( x ck − xi )( y mk − y i ) /2 (7.15) Therefore, the following relations hold: χ >0 ⎧⎪− ai , 2 k ( y ck − y mk ) / 2 ⎪⎩ai , 2 k ( y ck − y mk ) / 2 ξ i , 2 k = bi , 2 k Ai , 2 k = ⎨ χ 0 , we have ξ i , 2 k = bi , 2 k Ai , 2 k = −ai ,2 k ( y c − y m ) / 2 k k ηi , 2 k = ci , 2 k Ai , 2 k = ai , 2 k ( xc − xm ) / 2 k (7.18) k Analogously, for the ( 2k − 1) th sub-triangle, Δimk ck −1 , the relevant coefficients are ai , 2 k −1 = 3 Ai ξ i , 2 k −1 = bi , 2 k −1 Ai , 2 k −1 = −ai , 2 k −1 ( y c − y m ) / 2 k −1 (7.19) k ηi , 2 k −1 = ci , 2 k −1 Ai ,2 k −1 = ai ,2 k −1 ( xc − xm ) / 2 k −1 k where Ai , 2 k −1 = ( x mk − xi )( y ck −1 − yi ) − ( x ck −1 − xi )( y mk − yi ) . 7.2.3 Approximation of spatial derivatives 7.2.3.1 Approximation of 1st-order derivatives (gradients) Eq. (7.4) can be written as 206 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction ∇ h u( x i ) = − ∫ u h ( x )∇Φ ( x − x i )dΩ Ωi Ni ⎡ ⎤ = − ∑ ⎢ ∫ u h ( x )∇Φ ( x − x i )dΩ + ∫ u h ( x )∇Φ ( x − x i )dΩ⎥ k =1 ⎢ ⎥⎦ Δimk ck −1 ⎣ Δimk ck (7.20) Thus, Eq. (7.20) gives the approximation to the gradients as Ni ⎡ ⎤ ∂ui = − ∑ ⎢bi , 2 k ∫ u h ( x )dΩ + bi , 2 k −1 ∫ u h ( x )dΩ⎥ ∂x k =1 ⎢ ⎥⎦ Δimk ck Δimk ck −1 ⎣ Ni ⎡ ⎤ ∂ui = − ∑ ⎢ci , 2 k ∫ u h ( x )dΩ + ci , 2 k −1 ∫ u h ( x )dΩ⎥ ∂y k =1 ⎢ ⎥⎦ Δimk ck Δimk ck −1 ⎣ (7.21) (7.22) The integral for function u( x ) over each sub-triangle can be simply approximated in the fashion of ∫u h ( x )dΩ = Δimk ck ∫u h ( x )dΩ = Δimk ck −1 [ ] 1 u( x i ) + u( x mk ) + u( x ck ) Ai , 2 k 3 [ ] 1 u( x i ) + u( x mk ) + u( x ck −1 ) Ai , 2 k −1 3 (7.23) Substitute Eq. (7.23) into Eqs. (7.21) and (7.22) and we have [ ] ⎫ ⎧1 u( x i ) + u( x mk ) + u( x ck ) ξ i ,2 k Ni ⎪ ⎪⎪ ∂ui ⎪3 = −∑ ⎨ ⎬ ∂x k =1 ⎪ 1 ⎪ u ( x ) u ( x ) u ( x ) ξ + + + i mk ck −1 i , 2 k −1 ⎪ ⎪⎩ 3 ⎭ [ [ ] ] ⎫ ⎧1 u( x i ) + u( x mk ) + u( x ck ) ηi , 2 k ⎪⎪ ⎪ ∂ui ⎪3 = −∑ ⎨ ⎬ ∂y k =1 ⎪ 1 ⎪ ⎪⎩+ 3 u( x i ) + u( x mk ) + u( x ck −1 ) ηi , 2 k −1 ⎪⎭ Ni (7.24) [ ] (7.25) The function values at midpoints of edges and centroids of triangles can be simply approximated with linear interpolation of values at constitutive nodes. 7.2.3.2 Approximation of 2nd-order derivatives (Laplace operator) Analogously, with the help of linear interpolation in integral calculation, the 207 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction 2nd-order derivatives can be approximated as follows: ⎧ 1 ⎡ ∂u( x i ) ∂u( x mk ) ∂u( x ck ) ⎤ ⎫ + + ξi,2k ⎪ ⎪ ⎢ ⎥ Ni ∂x ∂x ⎦ ∂ 2 ui ⎪ 3 ⎣ ∂x ⎪ = −∑ ⎨ ⎬ 2 ∂x ∂u( x mk ) ∂u( x ck −1 ) ⎤ k =1 ⎪ 1 ⎡ ∂u ( x i ) + + + ξ i , 2 k −1 ⎪⎪ ⎥ ⎪ 3 ⎢ ∂x ∂x ∂x ⎦ ⎣ ⎩ ⎭ (7.26) ⎧ 1 ⎡ ∂u( x i ) ∂u( x mk ) ∂u( x ck ) ⎤ ⎫ + + ηi , 2 k ⎪ ⎪ ⎢ ⎥ Ni ∂y ∂y ⎦ ∂ 2 ui ⎪ 3 ⎣ ∂y ⎪ = −∑ ⎨ ⎬ 2 ∂y ∂u( x mk ) ∂u( x ck −1 ) ⎤ k =1 ⎪ 1 ⎡ ∂u ( x i ) ⎪ + + + ⎥ ηi , 2 k −1 ⎪ ⎪ 3 ⎢ ∂y ∂ ∂ y y ⎣ ⎦ ⎩ ⎭ (7.27) ⎧ 1 ⎡ ∂u( x i ) ∂u( x mk ) ∂u( x ck ) ⎤ ⎫ + + η ⎪ ⎪ , 2 i k ⎢ Ni ∂x ∂x ⎥⎦ ∂ 2 ui ⎪ 3 ⎣ ∂x ⎪ = −∑ ⎨ ⎬ ∂x∂y ∂u( x mk ) ∂u( x ck −1 ) ⎤ k =1 ⎪ 1 ⎡ ∂u ( x i ) + + + ηi , 2 k −1 ⎪⎪ ⎥ ⎪ 3 ⎢ ∂x ∂x ∂x ⎦ ⎣ ⎩ ⎭ (7.28) In above equations, the gradients at midpoints of edges and centroids of triangles can be approximated using either linear interpolation or gradient smoothing operations, as done in GSM with piecewise constant smoothing function. As for the approximation with gradient smoothing operation, both piecewise constant and linear smoothing functions can be used. Thus, the following three approaches are proposed and tested in current study: 1. simple average of gradients at constitutive nodes using linear interpolation (LI); 2. Gradient smoothing (GS) using piecewise constant smoothing functions (PCGS); 3. Gradient smoothing (GS) using piecewise linear smoothing functions (PLGS). 7.3 Relations between GSM and LWGSM 7.3.1 The formulation 208 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction As described in previous section, the LWGSM adopts the piecewise linear smoothing functions while GSM uses the piecewise constant smoothing functions. As a result, the partial differential derivatives in the LWGSM are approximated based on the integration over the surface of a gradient smoothing domain. However, they are approximated based on the integration over the bounding edges of the smoothing domain in the GSM. These variations are summarized in Table 7.1. Apparently, the two methods are different from each other. However, once the simple linear approximation to the surface integral is adopted in the LWGSM for computing the derivatives, the relations between the two methods can be identified. The 1st-order derivatives are approximated as follows: {[ ] [ ] ∂ui 1 Ni = − ∑ u ( x i ) + u( x mk ) + u( x ck ) ξ i , 2 k + u( x i ) + u( x mk ) + u( x ck −1 ) ξ i , 2 k −1 ∂x 3 k =1 } ⎧ ⎫ ⎡ u( x i ) u( x mk ) + u( x ck ) ⎤ + y ck − y mk ⎢ ⎪ ⎪ ⎥ 2 1 Ni ⎪ ⎪ ⎣ 2 ⎦ = ∑⎨ ⎬ Ai k =1 ⎪ ⎡ u( x i ) u( x mk ) + u( x ck −1 ) ⎤ ⎪ + y mk − y ck −1 ⎢ + ⎥⎪ ⎪ 2 2 ⎣ ⎦⎭ ⎩ u( x mk ) + u( x ck ) u( x mk ) + u( x ck −1 ) ⎤ 1 Ni ⎡ = ∑ ⎢ y ck − y mk + y mk − y ck −1 ⎥ Ai k =1 ⎣ 2 2 ⎦ ( ) ( ( ) ) ( {[ ] ) [ ] ∂ui 1 Ni = − ∑ u( x i ) + u( x mk ) + u( x ck ) ηi , 2 k + u( x i ) + u( x mk ) + u( x ck −1 ) ηi , 2 k −1 ∂y 3 k =1 = 1 Ai = 1 Ai ⎧ ⎫ ⎡ u( x i ) u( x mk ) + u( x ck ) ⎤ + xmk − xck ⎢ ⎪ ⎪ ⎥ Ni 2 ⎪ ⎪ ⎣ 2 ⎦ ⎨ ⎬ ∑ ⎡ u( x i ) u( x mk ) + u( x ck −1 ) ⎤ ⎪ k =1 ⎪ + xck −1 − xmk ⎢ + ⎥⎪ ⎪ 2 2 ⎣ ⎦⎭ ⎩ Ni u( x mk ) + u( x ck ) u( x mk ) + u( x ck −1 ) ⎤ ⎡ + xck −1 − xmk ∑ ⎢ x mk − xck ⎥ 2 2 k =1 ⎣ ⎦ ( } ) ( ( (7.29) ) ) ( (7.30) ) Note that the normal vectors for edges mk ck and ck −1mk are 209 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction ( n x ) mk ck = y ck − y mk , ( n y ) mk ck = xmk − x ck ( n x ) ck −1mk = y mk − y ck −1 , ( n y ) ck −1mk = xck −1 − x mk (7.31) Therefore, the LWGSM results in the same formulations as what we obtained with the GSM Scheme VIII, as presented in Chapter 3. Similarly, we can derive that the approximation to the 2nd-order derivative (Laplace operator) is also the same as that in Scheme VIII of GSM. Thus, it is expected to obtain the identical results using the LWGSM as predicted with GSM Scheme VIII. 7.3.2 Treatment of boundary conditions For Dirichlet boundary conditions, since the values of variables on boundaries are prescribed as occurred in the following testing case of full model, no calculations are needed at boundary nodes with the help of gradient smoothing operation. Once the natural (Neumann) boundary conditions are imposed at boundaries, the values of functions at boundary nodes are needed to be numerically solved. In the current implementation, the variables at such boundaries are required to satisfy the boundary conditions only. For example in the following half-model case, the first-order derivative with respect to the x -axis is zero (see Fig. 7.3), which is directly used in the approximation of discretized boundary conditions at relevant boundary nodes. In this case, the gradients at the symmetrical boundary for the scalar u are approximated as ∂ui =0 ∂x (7.32) 210 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction 3 [u ( x ) + u ( x 4 [u ( x ) + u ( x ∂ui i mk ) + u ( x ck )]η i , 2 k i mk ) + u ( x ck −1 )]η i , 2 k −1 = −∑ −∑ 3 3 ∂y k =1 k =2 1 = Ai u ( x mk ) + u ( x ck ) ⎤ 1 ⎡ ∑ ⎢ x mk − x ck ⎥+ A 2 k =1 ⎣ i ⎦ 3 ( ) ∑ (x 4 k =2 ck −1 − xmk ) u ( x mk ) + u ( x ck −1 ) (7.33) 2 It can be found that the boundary conditions are imposed also in the same way as that in the GSM (Scheme VIII). 7.4 Numerical Tests In this section, some numerical tests are conducted to validate the relations between LWGSM and GSM. Also, three different approaches for gradient smoothing operations are utilized in the LWGSM. For simplicity and comparison, a Poisson’s equation, as presented in Chapter 3, is solved with the proposed LWGSM. The governing equation and analytical solutions are the same as Eqs. (3.39) and (3.43). Numerical errors are defined in the section 3.4.2. In the following study, two different cases are covered: a full model of Poisson’s equation with all Dirichlet boundary conditions and a half model of Poisson’s equation with Dirichlet and Neumann boundary conditions. 7.4.1 Full model In this model, the boundary conditions are presented in Eq. (3.42). Similarly, both right triangular cells and irregular triangles are computed for LWGSM and GSM. Table 7.2 shows the L2 -norm error using different approaches of LWGSM with different distributions of right triangles: 11 × 11 , 21 × 21 , 41 × 41 , 81× 81 and 161× 161 . To approximate the gradients at the midpoints of edges and centroids of 211 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction triangles, linear interpolation generates much worse results than those by gradient smoothing operation, as shown in Table 7.2. However, the identical results are obtained using piecewise constant and piecewise linear smoothing functions. In Table 7.3, the L2 -norm errors using GSM Scheme VII and VIII with different distributions of right triangles are presented. It is found that GSM Scheme VII and VIII generate the same results, which are indentical to those in Table 7.2 by LWGSM with piecewise constant and piecewise linear smoothing functions. This validates the theoretical formulation in Section 7.3. Table 7.4 summarizes the L2 -norm errors for different approaches of LWGSM when irregular triangular meshes are used. It is clear that piecewise constant and piecewise linear smoothing functions generate more accurate results than those by linear interpolation. This is also true when right triangular meshes are used in simulations. It is also found that piecewise contstant and piecewise linear smoothing functions generate the same results in the LWGSM. In Table 7.5, the L2 -norm errors using GSM Scheme VII and VIII with different irregular triangles are presented. It is also found that GSM Scheme VIII generate the same results as those in Table 7.4 by LWGSM with piecewise constant and piecewise linear smoothing functions. This validates the theoretical formulation in Section 7.3 once more. Also, it is observed that the results by GSM Scheme VII are more accurate than those by Scheme VIII. This observation is in accordance with the previous results for GSM in Chapter 3. 7.4.2 Half model The boundary conditions and analytical solution are shown in Fig. 7.4 for the half 212 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction model of the above Poisson’s equation. Table 7.6 gives the L2 -norm errors for different approaches of LWGSM when irregular triangles are used in the half model Poisson’s equation with Neumann boundary conditions. Also, it is obvious that the results by piecewise constant and piecewise linear smoothing functions are more accurate than those by linear interpolation. Again, piecewise contstant and piecewise linear smoothing functions have the indentical effect in the LWGSM. Fig. 7.5 shows the contours of the relative errors, which are defined in Eq. (3.48). In Fig. 7.5(a), it is observed that the relative errors concentrate on the areas around nodes. This is due to the gradient interpolations across the nodes. However, when gradient smoothing operation is adopted, the relative errors are reduced significantly, as shown in Fig. 7.5b. In Table 7.7, the L2 -norm errors using GSM Scheme VIII with different irregular triangles are summarized. GSM Scheme VIII produces the identical results to those in Table 7.6 by LWGSM with gradient smoothing operations. This confirms that the LWGSM and GSM Scheme VIII generate the same results, not only theoretically but also numerically. 7.5 Remarks In this study, a linearly weighted gradient smoothing method (LWGSM) is briefly introduced as an alternative to the gradient smoothing method (GSM). The theoretical formulation and discretization are provided comprehensively. Intensive studies have been carried out for different schemes of LWGSM. Some conclusions are drawn as follows: 213 Chapter 7 • Linearly Weighted Gradient Smoothing Method: An Introduction LWGSM with gradient smoothing operation (piecewise constant and piecewise linear) provides much better results than LWGSM with linear interpolation for derivatives at midpoints of edges and centroids of cells. However, piecewise constant smoothing and piecewise linear smoothing generate the identical result. • LWGSM is identical to the GSM Scheme VIII, which has been validated from both theoretical aspects and numerical tests. • It is really interesting to find that through different formulation procedures, the LWGSM and GSM have the same property. LWGSM can be viewed as one of the GSM schemes, which may be constructed in a variety of ways and from different fundamentals. This implies that lots of efforts can be made to further develop the gradient smoothing methods (GSM). 214 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction Smoothing domain Smoothing functions GSM Connection between midpoints and centroids ∇ h u(x i ) = Approximation of derivatives x ∈ Ωi ⎧1 A i Φ=⎨ ⎩0 ∇ 2 u(x i ) = 1 Ai 1 Ai ⎧a + b i ( x − xi ) + c i ( y − yi ) x ∈ Ω i Φ=⎨ i x ∉ Ωi ⎩0 x ∉ Ωi ∫u h LWGSM Connection between midpoints and centroids ( x ) n( x ) d Γ ∂Ωi ∇ h u ( x i ) = − ∫ u h ( x )∇Φ ( x − x i )dΩ Ωi ∫ ∇ u ( x ) n( x ) d Γ h ∂Ωi ∇ 2 u ( x i ) = − ∫ ∇u h ( x )∇Φ ( x − x i )dΩ Ωi Table 7.1 Comparisons between the GSM and the LWGSM. Nodes 11 X 11 21 X 21 41 X 41 81 X 81 161 X 161 Linear interpolation 7.30e-2 1.78e-2 4.44e-3 1.11e-3 2.78e-4 Error GS (piecewise constant) 8.27e-3 2.06e-3 5.14e-4 1.29e-4 3.21e-5 GS (piecewise linear) 8.27e-3 2.06e-3 5.14e-4 1.29e-4 3.21e-5 Table 7.2 The L2 -norm error of Poisson’s equation using different approaches of LWGSM with different distributions of right triangles. Nodes 11 X 11 21 X 21 41 X 41 81 X 81 161 X 161 Error Scheme VII 8.27e-3 2.06e-3 5.14e-4 1.29e-4 3.21e-5 Scheme VIII 8.27e-3 2.06e-3 5.14e-4 1.29e-4 3.21e-5 Table 7.3 The L2 -norm error of Poisson’s equation using the GSM (Scheme VII and VIII) with different distributions of right triangles. 215 Chapter 7 Nodes 131 478 1887 7457 29629 Linearly Weighted Gradient Smoothing Method: An Introduction Linear interpolation 6.93e-2 1.58e-2 3.32e-3 8.16e-4 1.97e-04 Error GS (piecewise constant) 9.96e-3 3.31e-3 7.90e-4 1.95e-4 4.95e-05 GS (piecewise linear) 9.96e-3 3.31e-3 7.90e-4 1.95e-4 4.95e-05 Table 7.4 The L2 -norm error of Poisson’s equation using different approaches of LWGSM with different distributions of irregular triangular cells. Nodes 131 478 1887 7457 29629 Error Scheme VII 7.50e-3 3.33e-3 7.94e-4 1.97e-4 4.96e-05 Scheme VIII 9.96e-3 3.31e-3 7.90e-4 1.95e-4 4.95e-05 Table 7.5 The L2 -norm error of Poisson’s equation using the GSM (Scheme VII and VIII) with different distributions of irregular triangular cells. 216 Chapter 7 Nodes 21 72 244 963 3741 Linearly Weighted Gradient Smoothing Method: An Introduction Linear interpolation 0.28 6.0e-2 1.46e-2 4.06e-3 8.21e-4 Error GS (piecewise constant) 7.93e-2 1.58e-2 3.85e-3 8.21e-4 1.99e-4 GS (piecewise linear) 7.93e-2 1.58e-2 3.85e-3 8.21e-4 1.99e-4 Table 7.6 The L2 -norm error of half model using different approaches of LWGSM with different distributions of irregular triangular cells. Nodes 21 72 244 963 3741 Scheme VIII Time-step 0.0075 0.001 0.0005 0.000125 0.000035 Error 7.93e-2 1.58e-2 3.85e-3 8.21e-4 1.99e-4 Table 7.7 The L2 -norm error of half model using the GSM Scheme VIII with different distributions of irregular triangular cells. 217 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction Fig. 7.1 Linearly weighted smoothing functions for different types of gradient smoothing domains: mGSD, cGSD and nGSD. jk+1 jNi mk＋1 mNi (2Ni-1) (2k) (2Ni) cNi i m1 ck (1) (2) mk jk (2k-1) ck-1 c1 j1 Fig. 7.2 The schematic of a linearly weighted smoothing domain and its contained sub-triangles. 218 Chapter 7 Linearly Weighted Gradient Smoothing Method: An Introduction j1 j2 c1 m2 c2 j3 m1 i m3 c3 m4 j4 Fig. 7.3 Schematic of treatment of natural boundary conditions. U(x,y) = 0 ∂U =0 ∂x ∂U ∂ 2U ∂ 2U = 2 + 2 − f ( x, y ) ∂t ∂x ∂y (a) Half model and its boundary conditions Fig. 7.4 The half model of a Poisson’s equation. (b) Contour of the solution (a) LI (b) GS Fig. 7.5 Contour plots of relative errors using linear interpolation (LI) and gradient smoothing (GS) in the LWGSM. 219 Chapter 8 Conclusions Chapter 8 Conclusions 8.1 Concluding Remarks This study has focused on the development of numerical methods based on strong form governing equations for solids and structures. Several strong form methods have been originally proposed, developed and applied in this thesis. Through studies, the following conclusions are drawn: 1. A novel meshfree strong form method, radial point interpolation based finite difference method (RFDM), has been proposed and developed. Radial basis functions (RBFs) are very accurate and efficient for function interpolation with many distinguished characteristics. Radial point interpolation using RBFs and nodes in local support domains is adopted together with the classical finite difference method to achieve both the adaptivity to irregular geometry and the stability in the solution that is often encountered in the strong form methods, while retaining the feature of simplicity in formulation procedures. A least-square technique is utilized to acquire a system matrix of good properties including symmetry and positive definiteness, which helps greatly in solving the resulting set of algebraic system equations more efficiently and accurately by standard solver such as the Cholesky solver. Numerical examples are presented to demonstrate the accuracy and stability of the RFDM for problems 220 Chapter 8 Conclusions with complex shapes and extremely irregular nodes. In parametric studies, it has been found that the RFDM based on local RBFs can provide as good accuracy as the RFDM based on global RBFs while it is much more efficient than the RFDM based on global RBFs. It is also found that the optimal number of background grid points ( N ) with respect to the corresponding field node ( M ) should be 2 M ≤ N ≤ 3M . 2. The theoretical aspects of gradient smoothing method (GSM) have been firstly exploited. The principles of gradient smoothing and its numerical procedures to discretize partial differential equations are elucidated in detail. The approximations to the gradients (first-order derivatives) and Laplace operator (second-order derivatives) of a field variable are presented with a variety of GSM schemes. Stencil analyses of different types of discretization schemes for spatial partial differential terms are carried out from points of views of both efficiency and accuracy. The compactness of stencil and positivity of the coefficients of supporting nodes are the main concern in the analyses. Numerical solutions to Poisson’s equations are obtained using four favorable GSM schemes and investigated thoroughly to reveal the properties on accuracy, convergence and stability. The computational efficiency and robustness to the mesh irregularity for different GSM schemes are also intensively examined. The GSM Scheme VII is preferred in the following studies. 3. The proposed gradient smoothing method (GSM) has been applied to static 221 Chapter 8 Conclusions analyses of solid mechanics problems. The gradient smoothing operations are utilized to develop the first- and second-order derivative approximations by systematically computing the weights for a set of nodal points surrounding a node of interest. Using the approximated derivatives, the strong form governing system equations can be simply collocated at each scattered node in the problem domain. The computational accuracy, efficiency and stability of the present method with regular and irregular nodes are demonstrated through extensive numerical examples. In comparison with other well-established numerical approaches such as the finite element method (FEM), the proposed GSM produces encouraging results. 4. The gradient smoothing method has been further developed for the adaptive analyses in solid mechanics. The proposed GSM can effectively overcome the instability issue while retaining the strong form feature of simplicity in formulation procedures which is particularly suitable for adaptive analysis. In this thesis, a posteriori error indicator based on residual of the governing equation is adopted. By evaluating the residual of the governing equation for each triangular cell in the problem domain, error indicator effectively identifies the necessary regions to be refined. Simple refinement procedure using Delaunay diagram is adopted in the adaptive process. Compared with the well-known finite element method, the GSM for adaptive procedure demonstrates good reliability and performance in several solid mechanics problems including singularities and concentrated loading. 222 Chapter 8 Conclusions 5. The GSM has been employed further to the elasto-dynamic analyses of two-dimensional solids and structures. For free vibration analysis, frequencies and eigenmodes are obtained by solving the linear eigenvalue equation. In the forced vibration analysis, both the explicit time integration method (the central difference method) and the implicit time integration method (the Newmark method) are used to solve the forced vibration system equation. Numerical examples have demonstrated the validity, accuracy and stability of the present GSM for dynamic analyses. 6. Moving beyond the gradient smoothing method, a linearly weighted gradient smoothing method (LWGSM) has been devised with piecewise linear smoothing functions for gradient smoothing operation. The theoretical aspects and formulation procedures are examined in the same way as that for GSM. The relations between GSM and LWGSM are derived both theoretically and numerically. It is very interesting to find that LWGSM and GSM (Scheme VIII) have resulted in the identical solutions, while they are developed from different principles. Some numerical tests are conducted to show the properties of different schemes within the LWGSM. 8.2 Recommendations for Future Research Based on the presented work in this thesis, some suggestions on further research are given below: 1. The present GSM has not yet been formulated for solving the volumetric 223 Chapter 8 Conclusions locking problem, for which special techniques are needed. A large number of such techniques have been developed by many researchers for weak form methods, one of which is the so-called “selective” formulation. If the selective formulation is used in the GSM with proper design of smoothing domains, the GSM should also be able to solve this type of locking problems. However, formulation towards this direction needs a lot more careful consideration and intensive investigation. 2. In this thesis, the formulations and numerical examples are focused on two-dimensional problems in solid mechanics. However, the idea and general procedure of the GSM can be generalized to three-dimensional cases. The challenging work will be in the construction of 3D smoothing domains and coding. On the other hand, the proposed schemes can be easily applied to other classes of problems, e.g., heat transfer, fluid flow, etc. The validity and effectiveness for fluid dynamics problems have been well demonstrated through the research works (Liu and Xu, 2008) by another colleague in Liu’s group. 3. It is also possible to extend the adaptive analyses to non-linear and dynamic problems. 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(1991) Special issue on adaptive meshing, International Journal for Numerical Methods in Engineering 32. Steeb H, Maute A and Ramm E (2002) Goal-oriented error estimation in solid mechanics, in: E. Stein (Ed.), Error-controlled Adaptive Finite Elements in Solid Mechanics, John Wiley & Sons, England. Swegle JW, Hicks DL and Attaway SW (1995) Smoothed particle hydrodynamics stability analysis, Journal of Computational Physics 116, 123-134. Timoshenko SP and Goodier JN (1970) Theory of Elasticity (3rd edn), McGraw-Hill, New York. Wang JG and Liu GR (2002a) A point interpolation meshless method based on radial 237 References basis functions, International Journal for Numerical Methods in Engineering 54, 1623-1648. Wang JG and Liu GR (2002b) On the optimal shape parameters of radial basis function, Computer Methods in Applied Mechanics and Engineering 191, 21-26. Wendland H (1995) Piecewise polynomial positive definite and compactly supported radial basis functions of minimal degree, Advances in Computational mathematics 4, 389-396. Zhang GY, Liu GR, Kee BBT and Wang YY (2008) An adaptive analysis procedure using the linearly conforming point interpolation method (LC-PIM), International Journal of Solids and Structures (submitted). Zhang X, Song KZ, Lu MW and Liu X (2000) Meshless methods based on collocation with radial basis functions, Computational Mechanics 26, 333-343. Zhang X, Liu XH, Song KZ and Lu MW (2001) Least-squares collocation meshless method, International Journal for Numerical Methods in Engineering 51, 1089-1100. Zienkiewicz OC and Taylor RL (1992) The Finite Element Method, McGraw-Hill, New York, USA. 238 Publications Arising From Thesis 1. Liu GR, Zhang Jian, Li Hua, Lam KY and Kee BBT (2006) Radial point interpolation based finite difference method for mechanics problems, International Journal for Numerical Methods in Engineering 68, 728-754. 2. Zhang Jian, Liu GR, Lam KY and Li Hua (2007) A gradient smoothing method (GSM) for adaptive stress analysis in solids, in: Abstract Book International Conference on Computational Methods (ICCM), Hiroshima, Japan, 4-6 April 2007, pp. 104. 3. Liu GR, Zhang Jian, Lam KY, Li Hua, Xu G, Zhong ZH, Li GY and Han X (2008) A gradient smoothing method (GSM) with directional correction for solid mechanics problems, Computational Mechanics 41, 457-472. 4. Zhang Jian, Liu GR, Lam KY, Li Hua and Xu G (2008) A gradient smoothing method (GSM) based on strong form governing equation for adaptive analysis of solid mechanics problems, Finite Elements in Analysis and Design (In press). 5. Kee BBT, Liu GR, Song CX, Zhang Jian and Zhang GY (2008) A study on the effects of the number of local nodes used in meshfree methods based on radial basis functions, Mathematics and Mechanics of Solids (Submitted). 239 [...]... approximation Such inconvenience procedures give difficulties to the strong form methods for extensive applications Compared with 3 Chapter 1 Introduction the well-established weak form methods, the development of strong form methods is rather sluggish Available literatures for the strong form methods are still limited Therefore, the strong form methods are now in great demand Strong form methods demonstrate... (Computer Methods in Applied Mechanics and Engineering, Vol 139, 1996; Computational Mechanics, Vol 25, 2000) are also devoted to the development of meshfree methods More details on meshfree weak form methods can be found in books by Liu (2002) and Liu and Gu (2005) 1.2.3 Meshfree methods based on strong forms Compared with meshfree weak form methods, strong form meshfree methods 7 Chapter 1 Introduction... efficient and practical for engineering applications The major challenges to the researchers and scientists working on strong form methods are given as follows: 1 To stabilize strong form formulations using irregular local nodes; 2 To improve the accuracy, efficiency and performance of strong form methods; 3 To formulate strong form schemes for complex problems of practical applications; 4 To develop powerful... background integrations and variable mappings In contrast, the formulation procedure of the strong form of meshfree methods is relatively simple and straightforward, compared with the meshfree weak form methods The meshfree strong form method is regarded as a truly meshfree method as no mesh is required for field variable approximation or integration With such distinct features, the strong form of meshfree methods. .. finite difference method (GFDM) may be under this category Radial point collocation method (RPCM) is also a meshfree strong- form method formulated using radial basis functions and nodes in local supporting domains However, the instability of the meshfree strong form methods has been a main challenge that limits the application of meshfree strong form methods that use local nodes Researchers have introduced... according to their formulation procedures of discretizing the governing equations: (1) the methods based on a variational principle or a weak form of system equations (short for weak form method), and (2) the methods based on the strong form of governing equations (short for strong form method) Among these developed weak form methods, the finite element method is most well established Relying on meshes... software packages of strong form methods Hence, further research work is very necessary to establish strong form methods as powerful numerical tools 1.2 Literature Review As the problems of computational mechanics become more and more challenging, the conventional numerical methods, for instance, FDM, FEM and FVM, are no longer well suited The demand of a new class of numerical methods formulated without... uniform refinement for Poisson’s equation with a sharp peak 157 Table 5.4 Error norms of adaptive refinement for Poisson’s equation with a sharp peak 157 Table 5.5 Error norms of uniform refinement for infinite plate with a circular hole.157 Table 5.6 Error norms of adaptive refinement for infinite plate with a circular hole 157 Table 5.7 Error norms of uniform refinement for... to categorize the meshfree methods, which include domain-type and boundary-type of meshfree methods In the domain-type methods, both problem domain and boundary are represented by field nodes Examples of this type of meshfree methods include element-free Galerkin (EFG) method (Belytschko et al., 1994), point interpolation method (PIM) (Liu and Gu, 2001a), local radial point interpolation method (LRPIM)... effectively overcome the instability issue while retaining the strong form feature of simplicity in formulation procedures which is particularly suitable for adaptive analysis In this thesis, a posteriori error indicator based on residual of the governing equation is adopted By evaluating the residual of the governing equation for each triangular cell in the problem domain, error indicator effectively