Reciprocal mass matrices and a feasible time step estimator for finite elements with Allman's rotations

Abstract

Finite elements with Allman's rotations provide good computational efficiency for explicit codes exhibiting less locking than linear elements and lower computational cost than quadratic finite elements. One way to further raise their efficiency is to increase the feasible time step or increase the accuracy of the lowest eigenfrequencies via reciprocal mass matrices. This article presents a formulation for variationally scaled reciprocal mass matrices and an efficient estimator for the feasible time step for finite elements with Allman's rotations. These developments take special care of two core features of such elements: existence of spurious zero-energy rotation modes implying the incompleteness of the ansatz spaces, and the presence of mixed-dimensional degrees of freedom. The former feature excludes construction of dual bases used in the standard variational derivation of reciprocal mass matrices. The latter feature destroys the efficiency of the existing nodal-based time step estimators stemming from the Gershgorin's eigenvalue bound. Finally, the developments are tested for standard benchmarks and triangular, quadrilateral, and tetrahedral finite elements with Allman's rotations.