Studies in Numerical Stability and Critical Time Step Estimation by Wave Dispersion Analysis Versus Eigenvalue Computation

Abstract eng:
The critical Courant number limiting the length of time step in explicit integration schemes is inversely proportional to the maximum natural frequency of a finite element mesh. The well known recommendation Co = 1 for linear finite elements is deemed to be best. In fact, for some configurations this choice may dangerously overestimate the critical time step. It was shown in an earlier paper that by increasing the number of elements in the finite element mesh one will paradoxically improve the mesh’s stability towards its theoretical limit. The present paper refines some details, presenting small scale numerical tests. The first test involves a long truss/bar consisting of one row of elements whose critical Courant number changes as elements are added one after another. Since this increases the critical number one may pick up a time step such that it is supercritical to a certain mesh but becomes subcritical by merely adding one element. In a similar fashion, a square area is tested in the second example, using different arrangements of edge supports. It turns out that the numerical solutions to wave propagation may be strongly influenced by small variation of distant boundary conditions, which should normally be physically insignificant.

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