A new partition of unity finite element free from the linear dependence problem and possessing the delta property

Cai, Yongchang; Zhuang, Xiaoying; Augarde, Charles

Abstract

Partition-of-unity based finite element methods (PUFEMs) have appealing capabilities for p-adaptivity and local refinement with minimal or even no remeshing of the problem domain. However, PUFEMs suffer from a number of problems that practically limit their application, namely the linear dependence (LD) problem, which leads to a singular global stiffness matrix, and the difficulty with which essential boundary conditions can be imposed due to the lack of the Kronecker delta property. In this paper we develop a new PU-based triangular element using a dual local approximation scheme by treating boundary and interior nodes separately. The present method is free from the LD problem and essential boundary conditions can be applied directly as in the FEM. The formulation uses triangular elements, however the essential idea is readily extendable to other types of meshed or meshless formulation based on a PU approximation. The computational cost of the present method is comparable to other PUFEM elements described in the literature. The proposed method can be simply understood as a PUFEM with composite shape functions possessing the delta property and appropriate compatibility.

Citation

Cai, Y., Zhuang, X., & Augarde, C. (2010). A new partition of unity finite element free from the linear dependence problem and possessing the delta property. Computer Methods in Applied Mechanics and Engineering, 199(17-20), 1036-1043. https://doi.org/10.1016/j.cma.2009.11.019

• Journal Article Type: Article
• Publication Date: Jan 1, 2010
• Deposit Date: May 6, 2010
• Publicly Available Date: May 7, 2010
• Journal: Computer Methods in Applied Mechanics and Engineering
• Print ISSN: 0045-7825
• Electronic ISSN: 1879-2138
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 199
• Issue: 17-20
• Pages: 1036-1043
• DOI: https://doi.org/10.1016/j.cma.2009.11.019
• Keywords: Partition of unity, PUFEM, Meshless,Linear dependence, Interpolation, Delta property, Dual local approximation.
• Public URL: https://durham-repository.worktribe.com/output/1546106