An unsymmetric 8-node hexahedral element with high distortion tolerance

Among all 3D 8-node hexahedral solid elements in current finite element library, the ‘best’ one can produce good results for bending problems using coarse regular meshes. However, once the mesh is distorted, the accuracy will drop dramatically. And how to solve this problem is still a challenge that remains outstanding. This paper develops an 8-node, 24-DOF (three conventional DOFs per node) hexahedral element based on the virtual work principle, in which two different sets of displacement fields are employed simultaneously to formulate an unsymmetric element stiffness matrix. The first set simply utilizes the formulations of the traditional 8-node trilinear isoparametric element, while the second set mainly employs the analytical trial functions in terms of 3D oblique coordinates (R, S, T). The resulting element, denoted by US-ATFH8, contains no adjustable factor and can be used for both isotropic and anisotropic cases. Numerical examples show it can strictly pass both the first-order (constant stress/strain) patch test and the second-order patch test for pure bending, remove the volume locking, and provide the invariance for coordinate rotation. Especially, it is insensitive to various severe mesh distortions

A 14-node brick element, PN5X1, exactly representing linear stress fields

We consider Papcovitch–Neuber (PN) solution to the Navier equation, comprising only the vector potential, and develop a new displacement-based 14-node brick element. We assume PN solution in polynomial form. We impose constraints on unknown coefficients of the polynomials such that the element correctly represents linear stress fields. To validate the performance of the new element which we call PN5X1, we conduct several pathological tests available in the literature. PN5X1 predicts, as anticipated, both stresses and displacements accurately at every point inside the elastic continuum for linear stress fields

A 14-node brick element, PN5X1, for plates and shells

We use the Papcovitch–Neuber (PN) solution to the Navier equation and develop a new 14-node brick element PN5X1, to represent linear stress fields exactly. We perform patch tests, to assess its ability to converge in plate and shell problems. We also test the new element in rectangular and circular plates with simply supported and clamped edges and subject them to point and uniformly distributed loads. We also use it in locking tests and in low-energy deformation mode tests of rectangular plates. PN5X1 passes the patch tests and predicts, in the limit of discretization, theoretical values in almost all the plate and shell problems

Eight-node brick, PN340, represents constant stress fields exactly

We develop a new eight-node brick element, PN340, whose interpolation functions exactly satisfy the Navier equation. We begin with the Papcovitch–Neuber solutions in polynomial form, to the Navier equation and consider all the terms necessary to represent cubic displacement fields. We derive constraints on the unknown polynomial coefficients to make the eight-node brick element represent exactly every constant stress field. We provide explanations for the occurrence of kinematic modes. Based on this understanding, we develop a systematic procedure to identify the maximum independent degrees of freedom which the cubic displacement field will have while satisfying the Navier equation. Kinematic modes will never occur if the newly identified dof are used. The newly developed element PN340, based on our present procedure, predicts both stresses and displacements accurately at every point in the element in all the constant stress fields. In tests involving higher order stress fields the element is assured to converge in the limit of discretisation