Time step estimates for explicit dynamics with reciprocal mass matrices

Abstract

In this contribution, a novel local, node-based time step estimate for explicit dynamics with reciprocal mass matrices is presented. Reciprocal mass matrices are sparse matrices used in explicit dynamics that allow computation of nodal acceleration from the total force vector. They can be built algebraically and variationally and aim for higher critical time steps or/and accuracy than the lumped mass matrix approximation. Since the element eigenvalue inequality by Fried is not valid for reciprocal mass matrices, element-based estimates may be not conservative and they are consequently inadequate. Therefore, the nodal time step estimate for diagonally lumped mass matrices based on Gershgorin's theorem is further developed for application to reciprocal mass matrices. Additionally, simplifications of the proposed time step estimate that improve computational efficiency, especially for contact problems with the penalty method, are discussed. These simplifications use subadditivity and submultiplicativity properties of expressions used in the estimate. Finally, efficiency of the proposed estimate is illustrated by a numerical example.