Meshfree methods

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

Contaminant Transport Modelling Through Landfill Liners

Abstract: In the framework of meshfree methods, in this study a methodology is developed based on radial point interpolation method (RPIM). This methodology is applied to a one-dimensional (1D) contaminant transport in the saturated porous media. The 1D form of advection-dispersion equation involving reactive contaminant is considered. The Galerkin weak form of the governing equation is formulated using 1D meshfree shape functions constructed using thin plate spline radial basis functions. MATLAB code is developed to obtain the numerical solution. Numerical examples are presented to illustrate the applicability of the proposed RPIM and the results are compared with those obtained from the analytical and finite element solutions. The RPIM has generated results with no oscillations and they are insensitive to Peclet constraints. In order to test the practical applicability and performance of the RPIM, a case study of contaminant transport through landfill liner is presented. A good agreement is obtained between the results of the RPIM and the field investigation data.

An Implicit Gradient Meshfree Formulation for Convection-Dominated Problems

Abstract: Meshfree approximations are ideal for the gradient-type stabilized Petrov–Galerkin methods used for solving Eulerian conservation laws due to their ability to achieve arbitrary smoothness, however, the gradient terms are computationally demanding for meshfree methods. To address this issue, a stabilization technique that avoids high order differentiation of meshfree shape functions is introduced by employing implicit gradients under the reproducing kernel approximation framework. The modification to the standard approximation introduces virtually no additional computational cost, and its implementation is simple. The effectiveness of the proposed method is demonstrated in several benchmark problems.

Meshless formulation and parameters study for an elastic bar problem

Abstract: A recent strong interest in computational methods is in developing and exploring Mesh less Methods as an alternative numerical technique to other grid based methods like Finite Element Method. Meshfree methods are expected to be more adaptive and flexible in solving fracture mechanics and large deformation problems where Finite Element Method (FEM) is not suitable. Meshless Local Petrov Galerkin (MLPG) method is one of the popular meshfree technique based on a local weak form of the governing differential equation and employs shape functions derived from Moving Least Square (MLS) approximants. Present work demonstrates application of MLPG to an elastic bar problem with linear body force and investigates effect of some significant parameters on solution accuracy in MLPG formulation. These parameters include order of monomial basis function and weight function selection while constructing MLS shape functions, number of nodes in the problem domain (uniform and non-uniform distribution), size of support domain and sub-domain and different Gauss quadrature integration schemes. Axially loaded elastic bar problem is selected for clarity and unambiguous presentation. MLPG results are compared and validated with exact solution and FEM. It is also shown that MLPG stress results are continuous in the domain while FEM stress results are discontinuous at element boundary.

Numerical Investigation of Preconditioning for Iterative Methods in Linear Systems Obtained by Extended Element-Free Galerkin Method

Abstract: In an extended element-free Galerkin method (X-EFG), the essential and natural boundary conditions can be imposed by the collocation based method. However, a coefficient matrix of linear systems obtained by X-EFG become asymmetric, although a symmetric structure exists in a part of the coefficient matrix. In fact, the structure of the coefficient matrices almost becomes symmetric when the size of linear systems is large. Hence, efficient effects may be obtained by using preconditioning for symmetric matrices. The purpose of the present study is to investigate effects of preconditioning for symmetric matrices to linear systems obtained by X-EFG. To this end, the incomplete Cholesky factorization (IC) is applied to the linear systems by regarding the coefficient matrix as symmetric one. In numerical experiments, it is found that the linear systems obtained by X-EFG can be efficiently solved by using IC as preconditioning for GMRES(m) and Bi-CGSTAB. In some cases, the efficiency of IC is superior than that of the incomplete LDU factorization as preconditioning for these iterative methods.

A Hybrid Multiscale Finite Cloud Method and Finite Volume Method in Solving High Gradient Problem

Abstract: Meshfree methods are always the alternatives to be considered besides the finite element method in dealing with complex engineering problems. Among these complex problems, there are problems where discontinuities or high gradients may only emerge at a small region of a domain and the remaining parts of the domain are smooth and continuous. In this case, adding nodes to the critical region is essential for the better accuracy of solutions. It would be effective if a meshfree method is applied to the critical region and a mesh-based method is employed to solve the remaining part of the domain since node refinement is easier for a meshfree method whereas a mesh-based method is simple to develop and capable of producing accurate solutions for smooth regions. In this study, the finite cloud method (FCM) is coupled with the finite volume method (FVM) such that node refinement only takes place at the region where the FCM is applied. A handshaking approach is proposed to enforce the continuity near the interface where both methods are applied. Numerical examples and computation methods are presented in this paper and the numerical results are examined.