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for The Role of Quadrature in Meshfree Method: Variational Consistency in Galerkin Weak Form and Collocation in Strong Form

The Role of Quadrature in Meshfree Method: Variational Consistency in Galerkin Weak Form and Collocation in Strong Form

J. S. Chen
Chancellor’s Professor
Department of Civil & Environmental Engineering Department
University of California, Los Angeles

The approximation functions such as moving least-squares (MLS) and reproducing kernel (RK) used in meshfree methods are typically rational functions with overlapping supports. These approximation functions are capable of reproducing monomials of arbitrary order in arbitrary discretizations, however, this completeness property does not guarantee optimal rate of convergence in the Galerkin solution of PDEs if the domain integration is not sufficiently accurate. Variational consistency (VC) conditions offer a paradigm for formulating the quadrature rules and test functions to achieve optimal convergence for a given PDE in hand. We show that Gauss quadrature rules and nodal integrations are not variationally consistent with MLS and RK. The VC conditions can be used to correct integration methods that are variationally inconsistent via modification of test functions, and several variationally consistent methods are derived under a unified framework. VC conditions reveal that the numerical integration by parts conditions should hold for the inner product between test function and the differential operator acting on the monomials. Solving PDE by a direct collocation on the strong form without the integration by parts procedure avoids the constraint of meeting VC conditions. We show how strong form collocation is compared to the weak form with VC quadrature rules in convergence and efficiency. 

Biographical Sketch
J. S. Chen had his undergraduate education in National Central University (Taiwan), and received MS and PhD from Northwestern University. He is currently the Chancellor’s Professor of Civil & Environmental Engineering Department at UCLA. He is also Professor of Mechanical & Aerospace Engineering Department and Professor of Mathematics Department at UCLA. Prior to joining UCLA in 2001, he was the Assistant/Associate Professor of the Mechanical Engineering Department at the University of Iowa. His research interests are in computational solid mechanics and multiscale materials modeling, with particular emphasis on meshfree methods. He has been the department chair of Civil & Environmental Engineering Department of UCLA (July 2007- June 2012), and past President of US Association for Computational Mechanics (2010–2012). He has received numerous awards, including GenCorp Technology Achievement Award; James Lightners Faculty Fellowship and The Faculty Scholar Award from The University of Iowa; UCLA Chancellor’s Professor; Fellow of US Association for Computational Mechanics; Fellow of International Association for Computational Mechanics; Fellow of International Chinese Association for Computational Mechanics; Outstanding Alumnus of National Central University (Taiwan); Tongji Chair of Tongji University (China); the ICACM Award from International Chinese Association for Computational Mechanics; the Computational Mechanics Award from International Association for Computational Mechanics, among others. He is serving as the Editor-in-Chief, Associate Editor, or Editorial Board member for nine international journals.

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