The Meshless Local Petrov-Galerkin Method for Large Deformation Analysis of Hyperelastic Materials

Abstract

Nonlinear formulations of the meshless local Petrov-Galerkin method (MLPG) are presented for the large deformation analysis of hyperelastic materials which are considered to be incompressible or nearly incompressible. The MLPG method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties. In this paper, a simple Heaviside test function is chosen for reducing the computational effort by simplifying domain integrals for hyperelasticity problems. Trial functions are constructed using the radial basis function (RBF) coupled with a polynomial basis function. The plane stress hypothesis and a pressure projection method are employed to overcome the incompressibility or nearly incompressibility in the plane stress and plane strain problems, respectively. Effects of the sizes of local subdomain and interpolation domain on the performance of the present MLPG method are investigated. The behaviour of shape parameters of multiquadrics (MQ) function has been studied. Numerical results for several examples show that the present method is effective in dealing with large deformation hyperelastic materials problems.

On Solving Nonlinear Elasticity Problems Using a Boundary-Elements-Based Solution Method

“…The constants c and q are known as the shape parameters. The reader is reminded that these parameters are known to affect the accuracy of the solution in methods such as the LRPIM [27,28], the analog equation method [12], and even the MLPG [40]. Accordingly, results from these methods are usually presented with the associated optimal values of these shape parameters.…”

An implicit Updated Lagrangian Fragile Points Method with a support domain refinement scheme for solving large deformation problems

“…Another group of meshless methods is based on the Galerkin weak form, such as Element Free Galerkin (EFG) 17 and the Meshless Local Petrov‐Galerkin (MLPG), 18 and thus are more stable compared to SPH. Though both EFG and MLPG are capable of solving large deformation problems, 19,20 the high‐order rational trial functions have to be used in most approaches to satisfy the continuity requirement which results in the difficulties in solving the integrals. For this reason, EFG and MLPG are more difficult to implement and more computational expensive compared to FEM and SPH, and this drawback is more significant for complex large deformation problems.…”

A Locally-Continuous Meshless Local Petrov-Galerkin Method Applied to a Two-point Boundary Value Problem

“…The MLPG has already been used to solve various types of boundary value problems (Amini et al, 2018;Han and Atluri, 2004;Hu and Sun, 2011;Kamranian et al, 2017;Liu et al, 2011;Sheikhi et al, 2019;Zhang et al, 2006). However, in developing those formulations, the authors broke the underlying consistency with the Moving Least-square assumptions, which, in our view, led to shape functions that have unduly complex forms, making their computation and their derivatives' computation quite costly (Liu, 2009;Mirzaei and Schaback, 2013).…”