TY - JOUR

T1 - A new partition of unity finite element free from the linear dependence problem and possessing the delta property

AU - Cai, Yongchang

AU - Zhuang, Xiaoying

AU - Augarde, Charles

N1 - Funding information: The authors gratefully acknowledge the support of National Natural Science of China, NSFC ( 10972161 ). The second author is supported by a Dorothy Hodgkin Postgraduate Award from UK EPSRC at Durham University.

PY - 2010/11/29

Y1 - 2010/11/29

N2 - 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.

AB - 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.

KW - Delta property

KW - Dual local approximation

KW - Interpolation

KW - Linear dependence

KW - Meshless

KW - Partition of unity

KW - PUFEM

UR - http://www.scopus.com/inward/record.url?scp=78650678986&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2009.11.019

DO - 10.1016/j.cma.2009.11.019

M3 - Article

AN - SCOPUS:78650678986

VL - 199

SP - 1036

EP - 1043

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

IS - 17-20

ER -