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Meshfree methods

About: Meshfree methods is a(n) research topic. Over the lifetime, 2216 publication(s) have been published within this topic receiving 69596 citation(s).
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Abstract: An element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.

4,955 citations

Journal ArticleDOI
TL;DR: A new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time-frequency and space-wave number localization of a window function is developed and is called the reproducingkernel particle method (RKPM).
Abstract: A new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time-frequency and space-wave number localization of a window function is developed. This method is motivated by the theory of wavelets and also has the desirable attributes of the recently proposed smooth particle hydrodynamics (SPH) methods, moving least squares methods (MLSM), diffuse element methods (DEM) and element-free Galerkin methods (EFGM). The proposed method maintains the advantages of the free Lagrange or SPH methods; however, because of the addition of a correction function, it gives much more accurate results. Therefore it is called the reproducing kernel particle method (RKPM). In computer implementation RKPM is shown to be more efficient than DEM and EFGM. Moreover, if the window function is C∞, the solution and its derivatives are also C∞ in the entire domain. Theoretical analysis and numerical experiments on the 1D diffusion equation reveal the stability conditions and the effect of the dilation parameter on the unusually high convergence rates of the proposed method. Two-dimensional examples of advection-diffusion equations and compressible Euler equations are also presented together with 2D multiple-scale decompositions.

2,482 citations

Journal ArticleDOI
Abstract: A local symmetric weak form (LSWF) for linear potential problems is developed, and a truly meshless method, based on the LSWF and the moving least squares approximation, is presented for solving potential problems with high accuracy. The essential boundary conditions in the present formulation are imposed by a penalty method. The present method does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the “energy”. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. No post-smoothing technique is required for computing the derivatives of the unknown variable, since the original solution, using the moving least squares approximation, is already smooth enough. Several numerical examples are presented in the paper. In the example problems dealing with Laplace & Poisson's equations, high rates of convergence with mesh refinement for the Sobolev norms ||·||0 and ||·||1 have been found, and the values of the unknown variable and its derivatives are quite accurate. In essence, the present meshless method based on the LSWF is found to be a simple, efficient, and attractive method with a great potential in engineering applications.

2,195 citations

29 Jul 2002
Abstract: Preliminaries Physical Problems in Engineering Solid Mechanics: A Fundamental Engineering Problem Numerical Techniques: Practical Solution Tools Defining Meshfree Methods Need for Meshfree Methods The Ideas of Meshfree Methods Basic Techniques for Meshfree Methods Outline of the Book Some Notations and Default Conventions Remarks Meshfree Shape Function Construction Basic Issues for Shape Function Construction Smoothed Particle Hydrodynamics Approach Reproducing Kernel Particle Method Moving Least Squares Approximation Point Interpolation Method Radial PIM Radial PIM with Polynomial Reproduction Weighted Least Square (WLS) Approximation Polynomial PIM with Rotational Coordinate Transformation Comparison Study via Examples Compatibility Issues: An Analysis Other Methods Function Spaces for Meshfree Methods Function Spaces Useful Spaces in Weak Formulation G Spaces: Definition G1h Spaces: Basic Properties Error Estimation Concluding Remarks Strain Field Construction Why Construct Strain Field? Historical Notes How to Construct? Admissible Conditions for Constructed Strain Fields Strain Construction Techniques Concluding Remarks Weak and Weakened Weak Formulations Introduction to Strong and Weak Forms Weighted Residual Method A Weak Formulation: Galerkin A Weakened Weak Formulation: GS-Galerkin The Hu-Washizu Principle The Hellinger-Reissner Principle The Modified Hellinger-Reissner Principle Single-Field Hellinger-Reissner Principle The Principle of Minimum Complementary Energy The Principle of Minimum Potential Energy Hamilton's Principle Hamilton's Principle with Constraints Galerkin Weak Form Galerkin Weak Form with Constraints A Weakened Weak Formulation: SC-Galerkin Parameterized Mixed Weak Form Concluding Remarks Element Free Galerkin Method EFG Formulation with Lagrange Multipliers EFG with Penalty Method Summary Meshless Local Petrov-Galerkin Method MLPG Formulation MLPG for Dynamic Problems Concluding Remarks Point Interpolation Methods Node-Based Smoothed Point Interpolation Method (NS-PIM) NS-PIM Using Radial Basis Functions (NS-RPIM) Upper Bound Properties of NS-PIM and NS-RPIM Edge-Based Smoothed Point Interpolation Methods (ES-PIMs) A Combined ES/NS Point Interpolation Methods (ES/NS-PIM) Strain-Constructed Point Interpolation Method (SC-PIM) A Comparison Study Summary Meshfree Methods for Fluid Dynamics Problem Introduction Navier-Stokes Equations Smoothed Particle Hydrodynamics Method Gradient Smoothing Method (GSM) Adaptive Gradient Smoothing Method (A-GSM) A Discussion on GSM for Incompressible Flows Other Improvements on GSM Meshfree Methods for Beams PIM Shape Function for Thin Beams Strong Form Equations Weak Formulation: Galerkin Formulation A Weakened Weak Formulation: GS-Galerkin Three Models Formulation for NS-PIM for Thin Beams Formulation for Dynamic Problems Numerical Examples for Static Analysis Numerical Examples: Upper Bound Solution Numerical Examples for Free Vibration Analysis Concluding Remarks Meshfree Methods for Plates Mechanics for Plates EFG Method for Thin Plates EFG Method for Thin Composite Laminates EFG Method for Thick Plates ES-PIM for Plates Meshfree Methods for Shells EFG Method for Spatial Thin Shells EFG Method for Thick Shells ES-PIM for Thick Shells Summary Boundary Meshfree Methods RPIM Using Polynomial Basis RPIM Using Radial Function Basis Remarks Meshfree Methods Coupled with Other Methods Coupled EFG/BEM Coupled EFG and Hybrid BEM Remarks Meshfree Methods for Adaptive Analysis Triangular Mesh and Integration Cells Node Numbering: A Simple Approach Bucket Algorithm for Node Searching Relay Model for Domains with Irregular Boundaries Techniques for Adaptive Analysis Concluding Remarks MFree2D(c) Overview Techniques Used in MFree2D Preprocessing in MFree2D Postprocessing in MFree2D Index References appear at the end of each chapter.

1,738 citations

08 Jun 2005
TL;DR: This book provides first the fundamentals of numerical analysis that are particularly important to meshfree methods, and provides most of the basic meshfree techniques, and can be easily extended to other variations of more complex procedures of mesh free methods.
Abstract: This book aims to present meshfree methods in a friendly and straightforward manner, so that beginners can very easily understand, comprehend, program, implement, apply and extend these methods. It provides first the fundamentals of numerical analysis that are particularly important to meshfree methods. Typical meshfree methods, such as EFG, RPIM, MLPG, LRPIM, MWS and collocation methods are then introduced systematically detailing the formulation, numerical implementation and programming. Many well-tested computer source codes developed by the authors are attached with useful descriptions. The application of the codes can be readily performed using the examples with input and output files given in table form. These codes consist of most of the basic meshfree techniques, and can be easily extended to other variations of more complex procedures of meshfree methods. Readers can easily practice with the codes provided to effective learn and comprehend the basics of meshfree methods.

1,054 citations

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