Meshfree methods

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

The numerical solution of Cahn–Hilliard (CH) equation in one, two and three-dimensions via globally radial basis functions (GRBFs) and RBFs-differential quadrature (RBFs-DQ) methods

Abstract: The present paper is devoted to the numerical solution of the Cahn–Hilliard (CH) equation in one, two and three-dimensions. We will apply two different meshless methods based on radial basis functions (RBFs). The first method is globally radial basis functions (GRBFs) and the second method is based on radial basis functions differential quadrature (RBFs-DQ) idea. In RBFs-DQ, the derivative value of function with respect to a point is directly approximated by a linear combination of all functional values in the global domain. The main aim of this method is the determination of weight coefficients. GRBFs replace the function approximation into the partial differential equation directly. Also, the coefficients matrix which arises from GRBFs is very ill-conditioned. The use of RBFs-DQ leads to the improvement of the ill-conditioning of interpolation matrix RBFs. The boundary conditions of the mentioned problem are Neumann. Thus, we use DQ method directly on the boundary conditions, which easily implements RBFs-DQ on the irregular points and regions. Here, we concentrate on Multiquadrics ( MQ ) as a radial function for approximating the solution of the mentioned equation. As we know this radial function depends on a constant parameter called shape parameter. The RBFs-DQ can be implemented in a parallel environment to reduce the computational time. Moreover, to obtain the error of two techniques with respect to the spatial domain, a predictor–corrector scheme will be applied. Finally, the numerical results show that the proposed methods are appropriate to solve the one, two and three-dimensional Cahn-Hilliard (CH) equations.

Generalized finite difference method for solving two-dimensional non-linear obstacle problems

Abstract: In this study, the obstacle problems, also known as the non-linear free boundary problems, are analyzed by the generalized finite difference method (GFDM) and the fictitious time integration method (FTIM). The GFDM, one of the newly-developed domain-type meshless methods, is adopted in this study for spatial discretization. Using GFDM can avoid the tasks of mesh generation and numerical integration and also retain the high accuracy of numerical results. The obstacle problem is extremely difficult to be solved by any numerical scheme, since two different types of governing equations are imposed on the computational domain and the interfaces between these two regions are unknown. The obstacle problem will be mathematically formulated as the non-linear complementarity problems (NCPs) and then a system of non-linear algebraic equations (NAEs) will be formed by using the GFDM and the Fischer–Burmeister NCP-function. Then, the FTIM, a simple and powerful solver for NAEs, is used solve the system of NAEs. The FTIM is free from calculating the inverse of Jacobian matrix. Three numerical examples are provided to validate the simplicity and accuracy of the proposed meshless numerical scheme for dealing with two-dimensional obstacle problems.

J-integral evaluation for 2D mixed-mode crack problems employing a meshfree stabilized conforming nodal integration method

Abstract: Two-dimensional (2D) in-plane mixed-mode fracture mechanics problems are analyzed employing an efficient meshfree Galerkin method based on stabilized conforming nodal integration (SCNI). In this setting, the reproducing kernel function as meshfree interpolant is taken, while employing the SCNI for numerical integration of stiffness matrix in the Galerkin formulation. The strain components are smoothed and stabilized employing Gauss divergence theorem. The path-independent integral (J-integral) is solved based on the nodal integration by summing the smoothed physical quantities and the segments of the contour integrals. In addition, mixed-mode stress intensity factors (SIFs) are extracted from the J-integral by decomposing the displacement and stress fields into symmetric and antisymmetric parts. The advantages and features of the present formulation and discretization in evaluation of the J-integral of in-plane 2D fracture problems are demonstrated through several representative numerical examples. The mixed-mode SIFs are evaluated and compared with reference solutions. The obtained results reveal high accuracy and good performance of the proposed meshfree method in the analysis of 2D fracture problems.