Large‐step explicit time integration via mass matrix tailoring

Abstract

Summary A method for tailoring mass matrices that allows large time‐step explicit transient analysis is presented. It is shown that the accuracy of the present tailored mass matrix preserves the low‐frequency contents while effectively replacing the unwanted higher mesh frequencies by a user‐desired cutoff frequency. The proposed mass tailoring methods are applicable to elemental, substructural as well as global systems, requiring no modifications of finite element generation routines. It becomes most computationally attractive when used in conjunction with partitioned formulation as the number of higher (or lower) modes to be filtered out (or retained) are significantly reduced. Numerical experiments with the proposed method demonstrate that they are effective in filtering out higher modes in bars, beams, plain stress, and plate bending problems while preserving the dominant low‐frequency contents.

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

“…Although this seems a modest improvement, note that combined with BPM we are doubling the critical time‐step corresponding to the case without penalization. This gain may be further increased by adopting partition‐by‐partition SMS strategies 48 With BPM and scaling the RMM it is possible to make computations with larger time‐steps than the critical time‐step for a diagonal lumped mass matrix with better accuracy.…”

“…We describe here a simple method to reduce the highest frequencies of the system that can be combined with the proposed RMM for better efficiency. The mass scaling process is applied element by element although it could also be applied in a partitioned manner 48 . We start examining the original mode shapes of an element with n DOFs.…”

“…In summary, the present study addresses in multiple fronts the explicit solution process of contact‐impact problems with the objective of improving its applicability to real‐life applications by integrating the following four algorithmic components: The contact constraints are modeled by a bipenalty contact‐impact formulation 18 that incorporates stiffness and mass penalties preserving the stability limit of contact‐free problems for efficient explicit time integration; A method of localized Lagrange multipliers 47 is employed, which facilitates the partitioned governing equations for each substructure along with the completely localized contact penalty forces pertaining to each free substructure; A method for the direct construction of sparse inverse mass matrices of the free bodies in contact 32 is combined with the localized Lagrange multipliers approach; An element‐by‐element mass matrix scaling technique 48 that allows the extension of the time integration step is adopted to improve the overall performance of the algorithm. …”

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

“…Algebraic constructions of the reciprocal mass matrix are introduced in papers. [27][28][29][30] These formulations apply to solid finite elements with displacement degrees of freedom and plate finite elements with rotations. The starting point of the formulation is a diagonal mass matrix and a stiffness matrix.…”

“…They can be assembled from elemental matrices and their usage does not require large changes of the finite element code. Algebraic constructions of the reciprocal mass matrix are introduced in papers 27‐30 . These formulations apply to solid finite elements with displacement degrees of freedom and plate finite elements with rotations.…”