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.

Higher‐order inverse mass matrices for the explicit transient analysis of heterogeneous solids

“…The direct construction of a sparse RMM for FEM via general variational formulation has been presented in Reference 16 and then in Reference 17 including selective mass scaling. These works have been extended to reciprocal mass matrices with a feasible time step estimator for finite elements with Allman's rotations, see Reference 18. The direct mass matrix inversion for discontinuous Galerkin isogeometric analysis has been analyzed in Reference 19.…”

Partitioned formulation of contact‐impact problems with stabilized contact constraints and reciprocal mass matrices

“…By using this inverse, only a cheap sparse matrix‐vector multiplication is required to explicitly compute accelerations and advance the integration in time. The key idea of a RMM was introduced by Tkachuk and Bischoff 30,31 and it is basically to exploit Hamilton's principle, discretizing both displacement and momentum fields with biorthogonal shape functions, to produce a RMM in the form of a second‐order tensor product where only a diagonal projection operator needs to be inverted. This basic procedure also requires some cumbersome modifications of the RMM for the application of boundary conditions, a difficulty that can be circumvented by using localized Lagrange multipliers 32 .…”