Showing papers on "Meshfree methods published in 2002"

Mesh Free Methods: Moving Beyond the Finite Element Method

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.

The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element Methods

Abstract: A comparison study of the efficiency and ac- curacy of a variety of meshless trial and test functions is presented in this paper, based on the general concept of the meshless local Petrov-Galerkin (MLPG) method. 5 types of trial functions, and 6 types of test functions are explored. Different test functions result in different MLPG methods, and six such MLPG methods are pre- sented in this paper. In all these six MLPG methods, absolutely no meshes are needed either for the interpo- lation of the trial and test functions, or for the integration of the weak-form; while other meshless methods require background cells. Because complicated shape functions for the trial function are inevitable at the present stage, in order to develop a fast and robust meshless method, we explore ways to avoid the use of a domain integral in the weak-form, by choosing an appropriate test function. The MLPG5 method (wherein the local, nodal-based test function, over a local sub-domain Ω s (or Ω te) centered at a node, is the Heaviside step function) avoids the need for both a domain integral in the attendant symmetric weak-form as well as a singular integral. Convergence studies in the numerical examples show that all of the MLPG methods possess excellent rates of convergence, for both the unknown variables and their derivatives. An analysis of computational costs shows that the MLPG5 method is less expensive, both in computational costs as well as definitely in human-labor costs, than the FEM, or BEM. Thus, due to its speed, accuracy and robustness, the MLPG5 method may be expected to replace the FEM, in the near future.

A Meshless Local Petrov-Galerkin Method for Solving the Bending Problem of a Thin Plate

Abstract: Meshless methods have been extensively popularized in literature in recent years, due to their flex- ibility in solving boundary value problems. The mesh- less local Petrov-Galerkin(MLPG) method for solving the bending problem of the thin plate is presented and discussed in the present paper. The method uses the moving least-squares approximation to interpolate the solution variables, and employs a local symmetric weak form. The present method is a truly meshless one as it does not need a mesh, either for the purpose of inter- polation of the solution or for the integration of the en- ergy. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions are enforced by the penalty method. Sev- eral numerical examples are presented to illustrate the implementation and performance of the present method. The numerical examples presented in the paper show that high accuracy can be achieved for arbitrary nodal distri- butions for clamped and simply-supported edge condi- tions. No post processing procedure is required to com- pute the strain and stress, since the original solution from the present method, using the moving least squares ap- proximation, is of C 2 type.

A hybrid boundary node method

Abstract: A new variational formulation for boundary node method (BNM) using a hybrid displacement functional is presented here. The formulation is expressed in terms of domain and boundary variables, and the domain variables are interpolated by classical fundamental solution; while the boundary variables are interpolated by moving least squares (MLS). The main idea is to retain the dimensionality advantages of the BNM, and get a truly meshless method, which does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regular shaped domains (in general, semi-sphere in the 3-D problem) and their boundaries. Numerical examples presented in this paper for the solution of Laplace's equation in 2-D show that high rates of convergence with mesh refinement are achievable, and the computational results for unknown variables are most accurate. No further integrations are required to compute the unknown variables inside the domain as in the conventional BEM and BNM. Copyright © 2001 John Wiley & Sons, Ltd.

Numerical comparisons of two meshless methods using radial basis functions

Abstract: The recent advance in the development of various kinds of meshless methods for solving partial differential equations has drawn attention of many researchers in science and engineering. One of the domain-type meshless methods is obtained by simply applying the radial basis functions (RBFs) as a direct collocation, which has shown to be effective in solving complicated physical problems with irregular domains. More recently, a boundary-type meshless method that combines the method of fundamental solutions and the dual reciprocity method with the RBFs has been developed. In this paper, the performances of these two meshless methods are compared and evaluated. Numerical results indicate that these two methods provide a similar optimal accuracy in solving both 2D Poisson's and parabolic equations.

Numerical method for shape optimization using meshfree method

Abstract: A numerical method for continuum-based shape design sensitivity analysis and optimization using the meshfree method is proposed. The reproducing kernel particle method is used for domain discretization in conjunction with the Gauss integration method. Special features of the meshfree method from a sensitivity analysis viewpoint are discussed, including the treatment of essential boundary conditions, and the dependence of the shape function on the design variation. It is shown that the mesh distortion that exists in the finite element–based design approach is effectively resolved for large shape changing design problems through 2-D and 3-D numerical examples. The number of design iterations is reduced because of the accurate sensitivity information.

Comparisons of two meshfree local point interpolation methods for structural analyses

Abstract: As truly meshless methods, the local point interpolation method (LPIM) and the local radial point interpolation method (LR-PIM), are based on the point interpolations and local weak forms integrated in a local domain of very simple shape. LPIM and LR-PIM are examined and compared with each other. They are also compared with the established FEM and the meshless local Petrov-Galerkin (MLPG) method. The numerical implementations of these two methods are discussed in detail. Parameters that influence the performance of them are detailedly studied. The convergence and efficiency of them are thoroughly investigated. LPIM and LR-PIM formulations are developed for structural analyses of 2-D elasto-dynamic problems and 1-D Timoshenko beam problems in the first time. It is found that LPIM and LR-PIM are very easy to implement, and very efficient obtaining numerical solutions to problems of computational mechanics.

An analysis of the linear advection–diffusion equation using mesh-free and mesh-dependent methods

Abstract: The numerical solution of advection – diffusion equations has been a long standing problem and many numerical methods that attempt to find stable and accurate solutions have to resort to artificial methods to stabilize the solution. In this paper, we present a meshless method based on thin plate radial basis functions (RBF). The efficiency of the method in terms of computational processing time, accuracy and stability is discussed. The results are compared with the findings from the dual reciprocity/boundary element and finite difference methods as well as the analytical solution. Our analysis shows that the RBFs method, with its simple implementation, generates excellent results and speeds up the computational processing time, independent of the shape of the domain and irrespective of the dimension of the problem. q 2002 Elsevier Science Ltd. All rights reserved.

Meshless approach to shape optimization of linear thermoelastic solids

Abstract: This paper presents a formulation for shape optimization in thermoelasticity using a meshless method, namely the element-free Galerkin method. Two examples are treated in detail and comparisons with previously published finite element analysis results demonstrate the excellent opportunities the EFG offers for solving these types of problems. Smoother stresses, no remeshing, and better accuracy than finite element solutions, permit answers to shape optimization problems in thermoelasticity that are practically unattainable with the classical FEM without remeshing. For the thermal fin example, the EFG finds finger shapes that are missed by the FEM analysis, and the objective value is greatly improved compared to the FEM solution. A study of the influence of the number of design parameters is performed and it is observed that the EFG can give better results with a smaller number of design parameters than is possible with traditional methods. Copyright © 2001 John Wiley & Sons, Ltd.

The Architecture of SAGE – A Meshfree System Based on RFM

Abstract: In a meshfree system, a geometric model of a domain neither conforms to, nor is restricted by a spatial discretization Such systems for engineering analysis offer numerous advantages over the systems that are based on traditional mesh-based methods, but they also require radical approaches to enforcing boundary conditions and novel computational tools for differentiation, integration, and visualization of fields and solutions We show that all of these challenges can be overcome, and describe SAGE (Semi-Analytic Geometry Engine) – a successful system specifically intended for meshfree engineering analysis Our approach and individual modules are based on Rvachev’s Function Method (RFM) but the described techniques, algorithms, and software are applicable to all mesh-based and meshfree methods and have broad use beyond solutions of boundary value problems

Probabilistic fracture mechanics by Galerkin meshless methods - part II: reliability analysis

Abstract: This is the second in a series of two papers generated from a study on probabilistic meshless analysis of cracks. In this paper, a stochastic meshless method is presented for probabilistic fracture-mechanics analysis of linear-elastic cracked structures. The method involves an element-free Galerkin method for calculating fracture response characteristics; statistical models of uncertainties in load, material properties, and crack geometry; and the first-order reliability method for predicting probabilistic fracture response and reliability of cracked structures. The sensitivity of fracture parameters with respect to crack size, required for probabilistic analysis, is calculated using a virtual crack extension technique described in the companion paper [1]. Numerical examples based on mode-I and mixed-mode problems are presented to illustrate the proposed method. The results show that the predicted probability of fracture initiation based on the proposed formulation of the sensitivity of fracture parameter is accurate in comparison with the Monte Carlo simulation results. Since all gradients are calculated analytically, reliability analysis of cracks can be performed efficiently using meshless methods.

Probabilistic fracture mechanics by Galerkin meshless methods – part I: rates of stress intensity factors

Abstract: This is the first in a series of two papers generated from a study on probabilistic meshless analysis of cracks. In this paper (Part I), a Galerkin-based meshless method is presented for predicting first-order derivatives of stress-intensity factors with respect to the crack size in a linear-elastic structure containing a single crack. The method involves meshless discretization of cracked structure, domain integral representation of the fracture integral parameter, and sensitivity analysis in conjunction with a virtual crack extension technique. Unlike existing finite-element methods, the proposed method does not require any second-order variation of the stiffness matrix to predict first-order sensitivities, and is, consequently, simpler than existing methods. The method developed herein can also be extended to obtain higher-order derivatives if desired. Several numerical examples related to mode-I and mixed-mode problems are presented to illustrate the proposed method. The results show that first-order derivatives of stress-intensity factors using the proposed method agree very well with reference solutions obtained from either analytical (mode I) or finite-difference (mixed mode) methods for the structural and crack geometries considered in this study. For mixed-mode problems, the maximum difference between the results of proposed method and finite-difference method is less than 7 . Since the rates of stress-intensity factors are calculated analytically, the subsequent fracture reliability analysis can be performed efficiently and accurately.

Modelling three‐dimensional piece‐wise homogeneous domains using the α‐shape‐based natural element method

Abstract: In this paper, the application of the natural element method (NEM) to the numerical analysis of two- and three-dimensional piece-wise homogeneous domains is presented. The NEM differs from other meshless methods in its capability to accurately reproduce essential boundary conditions along convex boundaries. The α-shape-based extension of this method (α-NEM) generalizes this behaviour to non-convex domains, enables us to construct models entirely in terms of the initial cloud of points and allows us to simulate material discontinuities in a straightforward manner. In the following sections, simple and effective algorithms are presented for the construction of α-shapes in domains composed of various materials. Examples are presented in two- and three-dimensional cases in the context of linear elastostatics showing good performance even with the simple numerical quadrature used. Copyright © 2002 John Wiley & Sons, Ltd.

An Introduction to Moving Least Squares Meshfree Methods

Abstract: We deal here with some fundamental aspects of a category of meshfree methods based on Moving Least Squares (MLS) approximation and interpolation. These include EFG, RKPM and Diffuse Elements. In this introductory text, we discuss different formulations of the MLS from the point of view of numerical precision and stability. We talk about the issues of both “diffuse” and “full” derivation and we give proof of convergence of both approaches. We propose different algorithms for the computation of MLS based shape functions and we give their explicit forms in 1D, 2D and 3D. The topics of weight functions, the interpolation property with or without singular weights, the domain decomposition and the numerical integration are also discussed. We formulate the integration constraint, necessary for a method to satisfy the linear patch test. Finally, we develop a custom integration scheme, which satisfies this integration constraint.

Meshfree analysis and design sensitivity analysis for shell structures

Abstract: A unified design sensitivity analysis method for a meshfree shell structure with respect to size, shape, and configuration design variables is presented in this paper. A shear deformable shell formulation is characterized by a CAD connection, thickness degeneration, meshfree discretization, and nodal integration. Because of a strong connection to the CAD tool, the design variable is selected from the CAD parameters, and a consistent design velocity field is then computed by perturbing the surface geometric matrix. The material derivative concept is utilized in order to obtain a design sensitivity equation in the parametric domain. Numerical examples show the accuracy and efficiency of the proposed design sensitivity analysis method compared to the analytical solution and the finite difference solution. Copyright © 2002 John Wiley & Sons, Ltd.

Moving particle finite element method

Abstract: This paper presents the fundamental concepts behind the moving particle finite element method, which combines salient features of finite element and meshfree methods. The proposed method alleviates certain problems that plague meshfree techniques, such as essential boundary condition enforcement and the use of a separate background mesh to integrate the weak form. The method is illustrated via two-dimensional linear elastic problems. Numerical examples are provided to show the capability of the method in benchmark problems. Copyright © 2001 John Wiley & Sons, Ltd.

A Meshfree Method For Incompressible Fluid Flows with Incorporated Surface Tension

Abstract: A meshfree particle method is used to simulate free surface flows. This is a Lagrangian method. Flows are modeled by the incompressible Navier-Stokes equations. The particle projection method is used to solve the Navier-Stokes equations. The spatial derivatives are approximated by the weighted least squares method (WLS). The pressure Poisson equation is solved by a local iterative procedure with the help of WLS. Numerical experiments are presented for two dimensional cases. In the case of breaking dam problem the numerical result is compared with the experimental result. The surface tension effects are studied in different shapes of drops and Laplace's law is verified. Finally, the collisions of two drops are simulated.

Alternative total Lagrangian formulations for corrected smooth particle hydrodynamics (CSPH) methods in large strain dynamics problems

Abstract: This paper discusses alternative Lagrangian formulations for smooth particle hydrodynamics method. These Lagrangian formulations are here employed in solving large strain problems that involve elasto-plastic and hyperelastic materials. It has previously been shown in the literature that the Lagrangian formulation for continuum eliminates the problem of tension instability which is generally coupled with Eulerian continuum formulation of smooth particle hydrodynamics and other meshless methods. This paper presents the details of the methodologies used in formulating Lagrangian smooth particle hydrodynamics method and their characteristics.