Lumped mass finite element implementation of continuum theories with micro‐inertia

Abstract

SUMMARYContinuum theories can be equipped with additional inertia terms to make them capable of capturing wave dispersion effects observed in micro-structured materials. Such terms, often called micro-inertia, are convenient and straightforward extensions of classical continuum theories. Furthermore, the critical time step is usually increased via the inclusion of micro-inertia. However, the drawback exists that standard finite element discretisation leads to mass matrices that cannot be lumped without losing the micro-inertia terms. In this paper, we will develop a solution algorithm based on Neumann expansions by which this disadvantage is avoided altogether. The micro-inertia terms are translated into modifications of the residual force vector, so that the system matrix is the usual lumped mass matrix and all advantages of explicit time integration are maintained. The numerical stability of the algorithm and its effect on the dispersive properties of the model are studied in detail. Numerical examples are used to illustrate the various aspects of the algorithm.

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

“…Thus, Y can be easily inverted. Elimination of V from (18) leads to an equation of motion in the form 8 <…”

“…For the eigenmodes and eigenvalue analysis, a free motion is regarded, that is, F ext D 0 and P O U d D 0. This assumption simplifies the equation of motion (18) to…”

“…In order to be competitive with the LMM, such an inverse of the mass matrix must yield a larger critical time step than the LMM. An example of such a construction is given in [18]. This approach combines micro-inertia continua with additional length scales for mass scaling together with Neumann expansion for an approximation of the inverse of a matrix.In this paper, a variational construction is followed.…”

Large‐step explicit time integration via mass matrix tailoring

“…[2][3][4][5][6] To alleviate the stepsize limitations imposed by the mesh frequencies whose response components contribute very little when low modes dominate the transient responses, various mass matrix tailoring have been introduced by altering the mass matrices to reduce/filter out the high frequencies of the dynamical system without affecting the low-mid frequencies. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] Most of the existing methods cited above require either replacing existing elements by tailored elements and/or adopt element component-dependent time stepping procedures, leading to either elemental and/or global approaches, depending on how the modification of the mass matrix is made.…”

Time step estimates for explicit dynamics with reciprocal mass matrices

“…Recently, reciprocal mass matrices (RMM), i.e. directly assembled inverse mass matrices, were constructed by Lombardo and Askes [1], Tkachuk and Bischoff [2] and Gonzalez et al [3] as an efficient alternative to diagonally lumped mass matrices (LMM) in explicit dynamics. The critical time step ∆t crit and the number of time steps n step are directly connected to the highest angular eigenfrequency of the system ω max through the stability condition of the central difference method…”