Direct and sparse construction of consistent inverse mass matrices: general variational formulation and application to selective mass scaling

• Published in: International journal for numerical methods in engineering Vol. 101; no. 6; pp. 435 - 469
• Main Authors: Tkachuk, A., Bischoff, M.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 10.02.2015; Wiley Subscription Services, Inc
• Subjects: biorthogonal basis; Construction; explicit dynamics; Finite element method; Hamilton's principle; Inverse; Mass matrix; Mathematical analysis; Mathematical models; mixed finite elements; penalized Hamilton's principle; reciprocal mass matrix; selective mass scaling; Vectors (mathematics)
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.4805

SummaryClassical explicit finite element formulations rely on lumped mass matrices. A diagonalized mass matrix enables a trivial computation of the acceleration vector from the force vector. Recently, non‐diagonal mass matrices for explicit finite element analysis (FEA) have received attention due to the selective mass scaling (SMS) technique. SMS allows larger time step sizes without substantial loss of accuracy. However, an expensive solution for accelerations is required at each time step. In the present study, this problem is solved by directly constructing the inverse mass matrix. First, a consistent and sparse inverse mass matrix is built from the modified Hamiltons principle with independent displacement and momentum variables. Usage of biorthogonal bases for momentum allows elimination of momentum unknowns without matrix inversions and directly yields the inverse mass matrix denoted here as reciprocal mass matrix (RMM). Secondly, a variational mass scaling technique is applied to the RMM. It is based on the penalized Hamiltons principle with an additional velocity variable and a free parameter. Using element‐wise bases for velocity and a local elimination yields variationally scaled RMM. Thirdly, examples illustrating the efficiency of the proposed method for simplex elements are presented and discussed. Copyright © 2014 John Wiley & Sons, Ltd.