Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

Meshless Methods for Numerically Solving Boundary Value Problems of Elliptic Type Partial Differential Equations

Abstract: MESHLESS METHODS FOR NUMERICALLY SOLVING BOUNDARY VALUE PROBLEMS OF ELLIPTIC TYPE PARTIAL DIFFERENTIAL EQUATIONS by Minhwa Choi Dr. Xin Li, Examination Committee Chair Associate Professor of Mathematics University of Nevada, Las Vegas, USA In this dissertation we propose and examine numerical methods for solving the boundary value problems of partial differential equations (PDEs) by meshless methods. Typically, such a problem is described as Lu(x) = f(x), x ∈ Ω, (0.1) Bu(x) = g(x), x ∈ ∂Ω, (0.2) where Ω is a domain in R, s ≥ 2, L a linear partial differential operator, and B a linear operator for the boundary conditions. First we aim at getting approximate particular solutions up of a nonhomogeneous equation (0.1) by radial basis methods. For instance, the collocation method by radial basis functions for finding particular solutions up of (0.1) is very popular in the literature. Now the particular solutions of certain important PDEs by RBF approximation are available, with the order of convergence to the exact solutions provided. Here we explore and examine the numerical performances of these particular solutions in various examples. Once up is available, we then consider and solve the following boundary value problems

An error estimator and adaptivity based on basis coefficient-vector field of element-free Galerkin method

Abstract: In this paper, an error estimator for element-free Galerkin (EFG) method has been proposed. Since meshfree methods do not require a structured mesh or a sense of nodal belongingness, the methods offer the advantage of insertion, deletion, and redistribution of nodes adaptively in the problem domain. The trial function of the field variable is constructed entirely in terms of consistent basis functions and its associated coefficient. The proposed error estimator is based on the nodal coefficient-vector of the basis functions that are used to construct the trial function. After obtaining the nodal coefficient-vector from EFG solution, an attempt is made to recover the best nodal coefficient-vector based on the reduced domain of influence [Chung and Belytschko (1998)], which is sufficient enough to maintain the regularity of the EFG moment matrix and also ensuring that sufficient influencing nodes are present in all the four quadrants defined at the sample node. The vertices of the Voronoi polygon of the critical error nodes are considered as potential neighborhood and new nodes are inserted at the vertices. Numerical studies have been carried out to illustrate the performance of the proposed methodology of error estimator and adaptivity.

Hybrid Methods: Review of particle-based numerical methods and their coupling to other continuum methods.

Abstract: This report reviews the current state of meshless methods, summarises some commonly used examples, and describes methods for coupling meshless methods to finite element methods. The aim of the report is to provide an explanation of why meshless methods are useful and to identify which meshless methods and coupling methods should be investigated further.

Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods

Abstract: For both linear and nonlinear analysis, finite element method (FEM) software packages, whether commercial or in-house, have contributed significantly to ease the analysis of simple and complex structures with various working conditions. However, the literature offers other discretization techniques equally accurate, which show a higher meshing flexibility, such as meshless methods. Thus, in this work, the radial point interpolation meshless method (RPIM) is used to obtain the required variable fields for a nonlinear elastostatic analysis. This work focuses its attention on the nonlinear analysis of two benchmark plate-bending problems. The plate is analysed as a 3D solid and, in order to obtain the nonlinear solution, modified versions of the Newton–Raphson method are revisited and applied. The material elastoplastic behaviour is predicted assuming the von Mises yield surface and isotropic hardening. The nonlinear algorithm is discussed in detail. The analysis of the two benchmark plate examples allows us to understand that the RPIM version explored is accurate and allows to achieve smooth variable fields, being a solid alternative to FEM.

Dispersive Divergence-Free Vector Meshless Method for Time-Domain Analysis of Frequency-Dependent Media

Abstract: The dispersive meshless method with scalar basis function has been successfully applied for analysis of frequencydependent media. However, as scalar-based meshless methods are not always divergence-free in the absence of source, inaccurate or even wrong solutions may be found in the results. To overcome this problem, in this paper, a new dispersive formulation of meshless method with vector basis function is proposed for analysis of dispersive materials. The divergencefree property of this method improves the accuracy of simulation by eliminating the artificial charges generated due to the numerical analysis. Moreover, the frequency behavior of the dispersive medium is modeled using auxiliary-differentialequations (ADE) method. The efficiency and divergence-free property of the proposed method are verified and compared with dispersive meshless method with scalar basis function by a numerical example.