An adaptive finite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems

Abstract

'An adaptive nite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems.', Computer methods in applied mechanics and engineering., 267 . pp. 111-132. Further information on publisher's website:http://dx.doi.org/10.1016/j.cma.2013.07.018Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be re ected in this document. Changes may have been made to this work since it was submitted for publication. A de nitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, 267, 2013, 10.1016/j.cma.2013.07.018. Additional information:Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractIn this paper, an automatic adaptive coupling procedure is proposed for the finite element method (FEM) and the element-free Galerkin method (EFGM) for linear elasticity and for problems with both material and geometrical nonlinearities. In this new procedure, initially the whole of the problem domain is modelled using the FEM. During an analysis, those finite elements which violate a predefined error measure are automatically converted to an EFG zone. This EFG zone can be further refined by adding nodes, thus avoiding computationally expensive FE remeshing. Local maximum entropy shape functions are used in the EFG zone of the problem domain for two reasons: their weak Kronecker delta property at the boundaries allows straightforward imposition of essential boundary conditions and also provides a natural way to couple the EFG and FE regions compared to the use of moving least squares basis functions. The Zienkiewicz & Zhu error estimation procedure with the superconvergent patch method for strains and stresses recovery is used in the FE region of the problem domain, while the Chung & Belytschko error estimation procedure is used in the EFG region.

The effective interface approach for coupling of the FE and meshless FD methods and applying essential boundary conditions

“…In the simplest case, the basis functions for test function approximation may be the same as the ones applied for the FEM approach. Therefore, those shape functions m as well as their derivatives, may be found directly (as polynomials) like in the FEM, or, to preserve the consistently meshless notation, by means of the above-given meshless formulas (10) in which the number of nodes m in the MFD star is equal to the number of nodes in the integration cell (3 for triangle, 4 for rectangle, 4 for tetrahedron etc.) and, consequently, the number of unknown interpolation coefficients s. It should be noted that the result of (10) is independent from the selection of the weight functions as they have no influence on the approximation schemes in case when m = s is assumed.…”

“…• reproductivity conditions in transition zone [8,19,23], • local maximum entropy shape functions in the meshless domain [10,27].…”

“…It belongs to the constant repertoire of boundary value problems analysis done by means of discrete methods e.g. [1][2][3][4][5][6][7][8][9][10] and is applied mainly to minimize or to eliminate drawbacks of the coupled methods by associating one method with another. In this paper, two computational methods are considered, namely the Finite Element Method (FEM, e.g.…”

A Meshless Method of Lines for Numerical Solution of Some Coupled Nonlinear Evolution Equations

“…A theoretical foundation to the RBF method for data interpolation and solving partial differential equations was provided (see [9,19,17,37,35,7,8]). Due to simple applicability, Kansa's method is recently extended to solve various types of ordinary and partial differential equations including the 1-D nonlinear Burgers' equation with the shock wave, heat transfer, shallow water equation for tide and current simulations, the free boundary-value problems, a class of KdV equations, RLW equation, Schrodinger equation and recently a system of nonlinear PDEs (see [12,11,6,10,28,15,36,34,22,27,1,29,14,21,26,33,30] and the references therein). Some most commonly used radial basis functions are as follows:…”

Efficient implementation of Galerkin meshfree methods for large-scale problems with an emphasis on maximum entropy approximants

“…Because the LME basis functions do not satisfy the Kronecker-delta property at nodes, these schemes are referred to as approximants instead of interpolants. The capabilities of LME approximants have been examined in a variety of computational mechanics applications, such as linear and nonlinear elasticity [25,26], plate [27] and thin-shell analysis [28,29], convection-di↵usion problems [30,31], and phase-field models of biomembranes [32,33] and fracture mechanics [34,35,36].…”