Meshfree methods

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

Dispersion and pollution of the improved meshless weighted least-square (IMWLS) solution for the Helmholtz equation

Abstract: The meshless weighted least-square (MWLS) method is a meshless method based on the moving least-square (MLS) approximation. Compared with the Galerkin based meshless methods, the MWLS avoids numerical integrations, which improves the computational efficiency significantly. The MLS may form ill-conditioned system of equations, an accurate solution of which is difficult to obtain. In this paper, by using the weighted orthogonal basis function to construct the improved moving least-square (IMLS) approximation and the Lagrange multiplier method to enforce the Dirichlet boundary condition, we derive the formulas and perform the dispersion analysis for an improved meshless weighted least-square (IMWLS) method for two-dimensional (2D) Helmholtz problems. Results demonstrated that the IMWLS is more accurate and has advantages in handling dispersion. A 2D industrial model problem illustrated that the proposed method can easily reach higher frequency without losing accuracy.

Analysis of meshless weak and strong formulations for boundary value problems

Abstract: This paper introduces a weak meshless procedure combined with a multi-resolution numerical integration and its comparison with a strong local meshless formulation for approximating displacement and strain modeled in the form of Elliptic Boundary Value Problems (EBVPs) in one- and two-dimensional spaces. Assets and losses of both strong and weak meshless approaches are considered in detail. The meshless weak formulation considered in the current paper is the well-known Element Free Galerkin (EFG) method whereas the Local Radial Basis Functions Collocation Method (LRBFCM) is taken as a strong formulation. First aspect of the current work is implementation of the new numerical integration techniques introduced in Siraj-ul-Islam et al. (2010) and Aziz et al. (2011) [1,2] in the EFG method and its comparison with numerical integration based on standard Gaussian quadrature, adaptive integration and stabilized nodal integration techniques used in the context of EFG and other allied weak meshless formulations. Second aspect of the current work is analysis of comparative performance of the localized versions of strong and weak meshless formulations. Standard numerical tests are conducted to validate performance of both the approaches.

Localized Method of Approximate Particular Solutions with Cole-Hopf Transformation for Multi-Dimensional Burgers Equations

Abstract: The Burgers equations depict propagating wave with quadratic nonlinearity, it can be used to describe nonlinear wave propagation and shock wave, where the nonlinear characteristics cause difficulties for numerical analysis. Although the solution approximation can be executed through iterative methods, direct methods with finite sequence of operation in time can solve the nonlinearity more efficiently. The resolution for nonlinearity of Burgers equations can be resolved by the Cole–Hopf transformation. This article applies the Cole–Hopf transformation to transform the system of Burgers equations into a partial differential equation satisfying the diffusion equation, and uses a combination of finite difference and the localized method of approximate particular solution (FD-LMAPS) for temporal and spatial discretization, respectively. The Burgers equations with behaviors of propagating wave, diffusive N-wave or within multi-dimensional irregular domain have been verified in this paper. Effectiveness of the FD-LMAPS has also been further examined in some experiments, and all the numerical solutions prove that the FD-LMAPS is a promising numerical tool for solving the multi-dimensional Burgers equations.

Numerical solution of time-dependent stochastic partial differential equations using RBF partition of unity collocation method based on finite difference

Abstract: Meshfree methods based on radial basis functions (RBFs) are popular tools for the numerical solution of stochastic partial differential equations (SPDEs) due to their nice properties. However, the RBF collocation methods in global view have some disadvantages for the numerical solution of time-dependent SPDEs. Calculation of matrix condition number in the resulting dense linear systems indicates that the meshless method using global RBFs may be unstable at each realization to solve SPDEs. In order to avoid numerical instabilities in global RBF methods, we are interested in the use of RBF methods in local view for the numerical solution of time-dependent SPDEs. In this paper, the RBF partition of unity collocation method based on a finite difference scheme for the Gaussian random field (RBF-PU-FD) as a localized RBF approximation presented to deal with these issues. For this purpose, we simulate the Gaussian field with spatial covariance structure at a finite collection of predetermined collocation points. The matrices formed during the RBF-PU-FD method will be sparse and, hence, will not suffer from ill-conditioning and high computational cost. We will show that the method is viable through analyzing its numerical accuracy, CPU time, stability and sparsity structure. For the test problems, we perform 1000 realizations and statistical criterions such as mean, standard deviation, lower bound and upper bound of prediction are computed and evaluated using the Monte-Carlo method.