A General Framework of Domain Integration in Meshfree Methods
Tuesday, December 13, 3:30PM – 5PM
Jiun-Shyan (JS) Chen
The rate of convergence in the Galerkin based numerical methods for solving PDEs is determined by the order of completeness in the approximation functions and the order of accuracy in the domain integration of the weak form. In this work, we first show that the integration constraint required to achieve quadratic rate of convergence in L2 norm in the meshfree approximation with linear completeness can be expressed as a divergence condition, and this condition can be met under the assumed gradient framework with a built-in divergence operator. We then obtain the conditions for higher order exactness in the Galerkin approximation (thus higher order convergence rates), and a correction is proposed for cases where the condition is violated. Essentially, the proposed correction attempts to account for the lack of domain integration exactness. The correction allows for arbitrarily high order exactness in the Galerkin approximation for arbitrary types of integration, and it is demonstrated numerically that convergence rates are restored and the error is significantly reduced when these conditions are met.
Biographical Sketch J. S. Chen had his undergraduate education in Natural Central University, Taiwan, and received MS and PhD from Northwestern University. He worked at GenCorp Research Division in 1989-1994, and thereafter joined the faculty of the Mechanical Engineering Department of The University of Iowa in 1994. He moved to UCLA in 2001 and is currently the Department Chair of Civil & Environmental Engineering Department. His research interests are in computational solid mechanics and multiscale materials modeling. 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, Outstanding Alumnus of National Central University, Taiwan, Tongji Chair of Tongji University, China, and The ICACM Award from International Chinese Association for Computational Mechanics. He is currently the President of US Association for Computational Mechanics.
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