Modified and Trefftz unsymmetric finite element models
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Modified and Trefftz unsymmetric finite element models Q. Xie • K. Y. Sze • Y. X. Zhou
Received: 5 September 2014 / Accepted: 25 November 2014 Ó Springer Science+Business Media Dordrecht 2014
Abstract The unsymmetric finite element method employs compatible test functions but incompatible trial functions. The pertinent 8-node quadrilateral and 20-node hexahedron unsymmetric elements possess exceptional immunity to mesh distortion. It was noted later that they are not invariant and the proposed remedy is to formulate the element stiffness matrix in a local frame and then transform the matrix back to the global frame. In this paper, a more efficient approach will be proposed to secure the invariance. To our best knowledge, unsymmetric 4-node quadrilateral and 8-node hexahedron do not exist. They will be devised by using the Trefftz functions as the trial function. Numerical examples show that the two elements also possess exceptional immunity to mesh distortion with respect to other advanced elements of the same nodal configurations. Keywords Unsymmetric Finite element method Petrov–Galerkin Trefftz 4-node 8-node
Q. Xie K. Y. Sze (&) Y. X. Zhou Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong e-mail: [email protected] Y. X. Zhou Faculty of Civil Engineering & Mechanics, Jiangsu University, Zhenjiang 212013, People’s Republic of China
1 Introduction Tremendous efforts have been put on developing finite element (FE) models with excellent accuracy and low susceptibility to mesh distortion. In this regard, advanced FE techniques such as hybrid/mixed method (Pian and Sumihara 1984; Pian and Tong 1986; Yuan et al. 1993; Sze 2000; Qin 2003; Sze et al. 2004, 2010; Cen et al. 2011; Freitas and Moldovan 2011; Cao et al. 2012), incompatible displacement/enhanced assumed strain modes (Taylor et al. 1976; Simo and Rifai 1990; Liu and Sze 2010), reduced integration and stabilization (Hughes 1980; Bachrach 1987; Sze et al. 2004), assumed strain formulation (Macneal 1982; Kim et al. 2003; El-Abbasi and Meguid 2000; Cardoso et al. 2008) and discrete shear gap method (Bletzinger et al. 2000) have been developed. Many of them have yielded FE models with excellent accuracy when the mesh is regular. However, their accuracy often drops considerably when the mesh is distorted. Rajendran et al. (Rajendran and Liew 2003; Ooi et al. 2004; Liew et al. 2006; Ooi et al. 2008) proposed the unsymmetric FE method (US-FEM), which belongs to the Petrov–Galerkin formulation. The incompatible metric interpolants expressed in the metric or Cartesian coordinates are employed as the trial functions to satisfy the quadratic completeness for the unsymmetric 8-node quadrilateral plane element (UQ8) and 20-node hexahedral element (UH20). On the other hand, the test functions are the conventional compatible parametric interpolants. UQ8 and UH20 possess exceptional
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immunity to mesh distortion. It was noted later that they are not invariant, i.e., the element predictions change when the inc
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