A weak form quadrature element method for plane elasticity problems

A weak form quadrature element method is proposed and applied to analysis of plane elasticity problems. A variational formulation of plane elasticity problems is established and the differential quadrature analog of the derivatives in the functional is introduced. Several typical plane elasticity problems are studied to verify the proposed method. Results show that the method is highly efficient and promising. It is applied to the analysis of nearly incompressible materials and shown to be robust against volumetric locking. Similarities and dissimilarities, advantages and disadvantages as compared with other numerical methods, typically the p-version finite element method are discussed.

[1]  Alfred G. Striz,et al.  High-accuracy plane stress and plate elements in the quadrature element method , 1995 .

[2]  A finite difference variational method for bending of plates , 1980 .

[3]  J. Z. Zhu,et al.  The finite element method , 1977 .

[4]  S. Dey,et al.  Isoparametric finite difference energy method for plate bending problems , 1983 .

[5]  R. Cook,et al.  Concepts and Applications of Finite Element Analysis , 1974 .

[6]  Alfred G. Striz,et al.  Static analysis of structures by the quadrature element method(QEM) , 1994 .

[7]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[8]  C. Shu.  Differential Quadrature and Its Application in Engineering , 2000 .

[9]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[10]  J. Hutchinson,et al.  Buckling of Bars, Plates and Shells , 1975 .

[11]  H. Zhong.  Triangular differential quadrature , 2000 .

[13]  Hongzhi Zhong,et al.  Triangular differential quadrature and its application to elastostatic analysis of Reissner plates , 2001 .

[14]  S. Dey,et al.  Variational finite difference method for free vibration of sector plates , 1990 .

[15]  K. M. Liew,et al.  Static Analysis of Reissner-Mindlin Plates by Differential Quadrature Element Method , 1998 .

[16]  K. Bathe.  Finite Element Procedures , 1995 .

[17]  S. Fenves.  Numerical and computer methods in structural mechanics , 1973 .

[18]  R. Varga,et al.  Numerical Solution of Plane Elasticity Problems , 1963 .

[19]  Xinwei Wang,et al.  Static and free vibrational analysis of rectangular plates by the differential quadrature element method , 1998 .

[20]  David Bushnell.  Finite-Difference Energy Models versus Finite-Element Models: Two Variational Approaches in One Computer Program , 1973 .

[21]  N. Khot,et al.  An automated, energy-based, finite-difference procedure for the elastic collapse of rectangular plates and panels , 1980 .

[22]  A note on incorporation of domain decomposition into the differential quadrature method , 2003 .

[23]  Hongzhi Zhong,et al.  Solution of poisson and laplace equations by quadrilateral quadrature element , 1998 .

[24]  C. Bert,et al.  Differential Quadrature Method in Computational Mechanics: A Review , 1996 .

[25]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE AND LONG-TERM INTEGRATION , 1971 .

[26]  Chuei-Tin Chang,et al.  New insights in solving distributed system equations by the quadrature method—I. Analysis , 1989 .

[27]  Yuqiu Long,et al.  Generalized conforming triangular membrane element with vertex rigid rotational freedoms , 1994 .

[28]  Charles W. Bert,et al.  Free Vibration of Plates by the High Accuracy Quadrature Element Method , 1997 .

[29]  I. Babuska,et al.  Finite Element Analysis , 2021 .