Locking-free continuum displacement finite elements with nodal integration.

*(English)*Zbl 1195.74182Summary: 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.

##### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

##### Keywords:

linear elasticity; finite element; distortion sensitivity; volumetric locking; assumed strain; weighted residual; nodal quadrature
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\textit{P. Krysl} and \textit{B. Zhu}, Int. J. Numer. Methods Eng. 76, No. 7, 1020--1043 (2008; Zbl 1195.74182)

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