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An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics
Author(s) -
Hillman Michael,
Chen JiunShyan
Publication year - 2015
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5183
Subject(s) - quadrature (astronomy) , meshfree methods , galerkin method , nonlinear system , mathematics , convergence (economics) , numerical integration , gaussian quadrature , kernel (algebra) , stability (learning theory) , mathematical optimization , finite element method , mathematical analysis , computer science , nyström method , integral equation , structural engineering , physics , quantum mechanics , engineering , combinatorics , machine learning , economic growth , electrical engineering , economics
Summary Convergent and stable domain integration that is also computationally efficient remains a challenge for Galerkin meshfree methods. High order quadrature can achieve stability and optimal convergence, but it is prohibitively expensive for practical use. On the other hand, low order quadrature consumes much less CPU but can yield non‐convergent, unstable solutions. In this work, an accelerated, convergent, and stable nodal integration is developed for the reproducing kernel particle method. A stabilization scheme for nodal integration is proposed based on implicit gradients of the strains at the nodes that offers a computational cost similar to direct nodal integration. The method is also formulated in a variationally consistent manner, so that optimal convergence is achieved. A significant efficiency enhancement over a comparable stable and convergent nodal integration scheme is demonstrated in a complexity analysis and in CPU time studies. A stability analysis is also given, and several examples are provided to demonstrate the effectiveness of the proposed method for both linear and nonlinear problems. Copyright © 2015 John Wiley & Sons, Ltd.