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Locking‐free continuum displacement finite elements with nodal integration
Author(s) -
Krysl P.,
Zhu B.
Publication year - 2008
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2354
Subject(s) - quadrilateral , hexahedron , finite element method , mathematics , displacement (psychology) , tetrahedron , node (physics) , kinematics , mathematical analysis , geometry , residual , physics , structural engineering , classical mechanics , algorithm , engineering , psychology , psychotherapist
An assumed‐strain finite element technique is presented for linear, elastic small‐deformation models. Weighted residual method (reminiscent of the strain–displacement functional) is used to weakly enforce the balance equation with the natural boundary condition and the kinematic equation (the strain–displacement relationship). A priori satisfaction of the kinematic weighted residual serves as a condition from which strain–displacement operators are derived via nodal integration. A variety of element shapes is treated: linear triangles, quadrilaterals, tetrahedra, hexahedra, and quadratic (six‐node) triangles and (27‐node) hexahedra. The degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials. A straightforward constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Furthermore, the numerical inf–sup test is applied in select problems and several variants of the proposed formulations (linear triangles, quadrilaterals, tetrahedra, hexahedra, and 27‐node hexahedra) pass the test. Examples are used to illustrate the performance with respect to sensitivity to shape distortion and the ability to resist volumetric locking. Copyright © 2008 John Wiley & Sons, Ltd.