Stability of semidiscrete formulations for elastodynamics at small time steps

Abstract

Solutions of direct time-integration schemes for elastodynamics that converge in time to conventional semidiscrete formulations may be polluted at small time steps by spurious oscillations that violate the principle of causality, for example by arising before wave fronts. This degradation is not an artifact of the time-marching scheme, but rather a property of the solution of the semidiscrete formulation itself. An analogy to singularly perturbed elliptic problems provides an upper bound on the time step for the onset of these oscillations. A simple procedure of spatial stabilization is proposed to remove this pathology from implicit time-integration schemes, without affecting unconditional temporal stability. Spatially stabilized implicit time-integration methods are free of noncausal oscillations at small time steps.