Allman's triangle, rotational DOF and partition of unity

A simplification of the 1984 Allman triangle (one of historically most important elements with rotational dofs) is presented. It is found that this old element takes a typical form of the partition of unity approximation. The Allman's rotation presented in the partition of unity form offers merits and convenience in formulation and practical applications. The stiffness matrix of the 1984 Allman triangle, which is originally computed from the linear strain triangular element (the 6 nodes quadratic triangle), can be obtained instead in a cheaper way from that of the constant strain triangular element. The constraint of the rotational terms during essential boundary treatment, which remains equivocal and ambiguous, is understood to be mandatory. The partition of unity notion enables a straightforward extension of the Allman's rotational dof to meshfree approximations. In numerical examples, we discuss suppression of spurious zero-energy modes and patch tests. Standard benchmarks are carried out to assess performance of the newly formulated triangle and a meshfree approximation with the rotational dofs. Copyright © 2006 John Wiley & Sons, Ltd.

[1]  I. Babuska,et al.  The generalized finite element method , 2001 .

[2]  Genki Yagawa,et al.  Generalized nodes and high‐performance elements , 2005 .

[3]  Genki Yagawa,et al.  Linear dependence problems of partition of unity-based generalized FEMs , 2006 .

[4]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[5]  K. Y. Sze,et al.  Hybrid stress tetrahedral elements with Allman's rotational D.O.F.s , 2000 .

[6]  E. Oñate,et al.  A hierarchical finite element method based on the partition of unity , 1998 .

[7]  Carlos A. Felippa,et al.  A triangular membrane element with rotational degrees of freedom , 1985 .

[8]  I. Babuska,et al.  The design and analysis of the Generalized Finite Element Method , 2000 .

[9]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[10]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[11]  Kenjiro Terada,et al.  Finite cover method for linear and non‐linear analyses of heterogeneous solids , 2003 .

[12]  Marc Alexander Schweitzer,et al.  Partition of Unity Method , 2003 .

[13]  D. J. Allman,et al.  Evaluation of the constant strain triangle with drilling rotations , 1988 .

[14]  T. Liszka,et al.  A generalized finite element method for the simulation of three-dimensional dynamic crack propagation , 2001 .

[15]  T. P. Pawlak,et al.  Solid elements with rotational degress of freedom. II Tetrahedron elements , 1991 .

[16]  Ivo Babuška,et al.  Generalized finite element methods for three-dimensional structural mechanics problems , 2000 .

[17]  F. Brezzi,et al.  On drilling degrees of freedom , 1989 .

[18]  Genki Yagawa,et al.  FREE MESH METHOD USING TETRAHEDRAL ELEMENT INCLUDING THE VERTEX ROTATIONS , 2004 .

[19]  Hitoshi Matsubara,et al.  Advanced 4‐node tetrahedrons , 2006 .

[20]  Jeen-Shang Lin,et al.  A mesh-based partition of unity method for discontinuity modeling , 2003 .

[21]  Robert D. Cook,et al.  Solid elements with rotational degrees of freedom: Part II—tetrahedron elements , 1991 .

[22]  Guangqi Chen,et al.  DEVELOPMENT OF HIGH-ORDER MANIFOLD METHOD , 1998 .

[23]  Gen-Hua Shi,et al.  Manifold Method of Material Analysis , 1992 .

[24]  Robert D. Cook,et al.  On the Allman triangle and a related quadrilateral element , 1986 .

[25]  C. Felippa.  A study of optimal membrane triangles with drilling freedoms , 2003 .

[26]  T. Strouboulis,et al.  The generalized finite element method: an example of its implementation and illustration of its performance , 2000 .

[27]  O. C. Zienkiewicz,et al.  A new cloud-based hp finite element method , 1998 .

[28]  D. Allman.  A compatible triangular element including vertex rotations for plane elasticity analysis , 1984 .