A new three-dimensional variable-node solid element with rotational degrees of freedom is presented. The developed element is established by introducing variable nodes to the basic 8-node brick type solid element for an effective connection between the refined mesh region and the coarse mesh region with a possible minimum number of degrees of freedom. By introducing the variable node solid elements, some difficulties associated with connecting the different layer patters in the common adaptive h-refinement on hexahedral mesh can be easily overcome. In this paper, the improvement of variable-node solid element is achieved by adding rotational degrees of freedom to the existing variable-node hexahedral element and by addition of non-conforming modes. Rotational degrees of freedom are viewed as particularly advantageous in structural analysis. For example, if spatial beams and shell elements are combined with solid element, each node will possess six degrees of freedom, i.e., three displacements and three rotations. Thus the developed element permits an easy connection to other structural elements which have rotational degrees of freedom.
The derivation of the presented element in this paper is based on the variational principles in which the rotations are introduced as independent variables. In these variational principles, skew-symmetric components of stresses are introduced as Lagrange multipliers to enforce the equality of the independent rotations with the skew-symmetric components of the displacement gradient. The proposed elements are built on a special hierarchical shape functions to introduce nodal rotations for variable node solid element. In addition to special hierarchical shape functions, edge- tangential non-conforming modes and bubble-mode-like non-conforming modes are selectively added to displacement fields for the purpose of improving element behavior. Among these modes, the addition of edge-tangential non-conforming modes is verified as a good remedy for the Poisson*s ratio locking phenomena.
To show the performance of the proposed elements and the applicability to practical three-dimensional problems, several numerical tests were performed. It was verified through numerical tests that the element shows good performance and can be effectively used in practical three-dimensional problems.