Locking‐free continuum displacement finite elements with nodal integration

Abstract

SUMMARYAn 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.

Maximum-entropy meshfree method for compressible and near-incompressible elasticity

“…(17), Ω is the union of all the elements attached to node a, i.e., Ω = ∪Ω e a . Even though our approach shares common features with the method proposed by Krysl and Zhu [48], there exist notable differences. We use averages of strain matrices from the elements attached to a particular node to satisfy the near-incompressibility constraint in the weak form (10), whereas in Ref.…”

“…We use averages of strain matrices from the elements attached to a particular node to satisfy the near-incompressibility constraint in the weak form (10), whereas in Ref. [48] the averages are used to obtain a strain field that satisfies a a kinematic constraint in a displacement-based weak form within a nodal integration scheme. A reference mesh for our proposed method is illustrated in Fig.…”

“…In the past few years, there has been renewed efforts to remedy the poor performance of low-order triangular and tetrahedral finite elements in the near-incompressible regime [41,42,43,44,45,46,47,48]. These approaches are broadly based on the idea of reducing the number of incompressibility constraints by defining nodal-averaged pressures or strains.…”

“…In Ref. [48], a special form of a nodal strain matrix was computed from the elements attached to the node, and it led to a locking-free displacement-based formulation. Although these nodal methods tend not to lock, several authors have reported pressure oscillations for highly constrained problems [45,46].…”

“…Our approach for the latter uses the notion of nodal-averaged pressure and leads to a displacementformulation as in Krysl and Zhu [48]; however, we differ in that the starting point of our method is the displacement/pressure mixed formulation, and numerical integration is tailored for meshfree basis functions using Gauss quadrature. Since numerical errors are prevalent when standard tensorproduct Gauss quadrature is used in meshfree methods, we appeal to assumed strain methods [49] and nodal integration techniques [20,21,22] to define a modified strain tensor.…”