Selective mass scaling and critical time-step estimate for explicit dynamics analyses with solid-shell elements

Explicit time integration is a popular method to simulate the dynamical behaviour of a system. Unfortunately, explicit time integration is only conditionally stable: the time step must be chosen not larger than the so-called "critical time step", otherwise the numerical solution may become unstable. To reduce the CPU time needed to carry out simulations, it is desirable to explore methods that increase the critical time step, which is the main objective of our paper. To do this, first we discuss and compare three approaches to increase the critical time step: micro-inertia formulations from continuum mechanics, inertia penalties which are used in computational mechanics, and mass scaling techniques that are mainly used in structural dynamics. As it turns out, the similarities between these methods are significant, and in fact they are identical in 1D if linear finite elements are used. This facilitates interpretation of the additional parameters in the various methods. Next, we derive, for a few simple finite element types, closed-form expressions for the critical time step with micro-structural magnification factors. Finally, we discuss computational overheads and some implementational details.

Increasing the critical time step: micro-inertia, inertia penalties and mass scaling

Summary
Classical 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.

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

Explicit predictor–multicorrector time discontinuous galerkin methods for non-linear dynamics

Summary
In this work, a new, unconditionally stable, time marching procedure for dynamic analyses is presented. The scheme is derived from the standard central difference approximation, with stabilization being provided by a consistent perturbation of the original problem. Because the method only involves constitutive variables that are already available from computations at previous time steps, iterative procedures are not required to establish equilibrium when nonlinear models are focused, allowing more efficient analyses to be obtained. The theoretical properties of the proposed scheme are discussed taking into account standard stability and accuracy analyses, indicating the excellent performance of the new technique. At the end of the contribution, representative nonlinear numerical examples are studied, further illustrating the effectiveness of the new technique. Numerical results obtained by the standard central difference procedure and the implicit constant average acceleration method are also presented along the text for comparison. Copyright © 2015 John Wiley & Sons, Ltd.

A stabilized central difference scheme for dynamic analysis

The applicability of explicit time integration in structural dynamics is often limited by a restrictively small solution time increment. This paper presents a mass penalty technique whereby the highest natural frequencies of a dynamic system can be preferentially lowered with only a minimal effect on the structural modes. Because the critical time increment is inversely proportional to the highest natural frequency, a dramatic increase in the solution time increment may be achieved. An expression is developed for the frequency shift and a number of useful properties of this technique are derived analytically for linear systems. Its utility in non-linear analyses is demonstrated with several examples.

Brian H. Aubert

Papers

A mass penalty technique to control the critical time increment in explicit dynamic finite element analyses