Mid-Node Admissible Space for 3D Quadratic Tetrahedral Finite Elements

The Mid-Node Admissible Spaces (MAS) [1,2] for two-dimensional quadratic triangular finite elements are extended to three-dimensional quadratic tetrahedral finite elements (3DQTE). The MAS concept for 3DQTE postulates a bounded region within which a mid-side node of a curved edge of the 3DQTE can be placed to ensure maintaining a specified minimum and maximum Jacobian determinant value at any point of the element. The theorems that form the basis of the MAS and their mathematical proofs, followed by the procedure to construct the MAS for 3DQTE, are presented. Based on the MAS developments, a robust element quality metric for 3DQTE is developed. The metric is based on the Jacobian determinant over the entire element without requiring that it actually be computed everywhere on the element. The metric is relatively inexpensive to compute, especially for mildly distorted elements. It is shown to be able to detect elements of poor quality that other distortion metrics fail to detect. It also approves good quality elements regardless of the extent to which they may appear to be geometrically distorted.