Influence of periodically fluctuating material parameters on the stability of explicit high-order spectral element methods

This paper aims at studying the influence of material heterogeneity on the stability of explicit time marching schemes for the high-order spectral element discretisation of wave propagation problems. A periodic fluctuation of the density and stiffness parameters is considered, where the period is related to the characteristic element size of the mesh. A new stability criterion is derived analytically for quadratic and cubic one-dimensional spectral elements in heterogeneous materials by using a standard Von Neumann analysis. The analysis presented illustrates the effect of material heterogeneity on the stability limit and also reveals the origin of instabilities that are often observed when the stability limit derived for homogeneous materials is adapted by simply changing the velocity of the wave to account for the material heterogeneity. Several extensions of the results derived for quadratic and cubic one-dimensional spectral elements are discussed, including higher order approximations, different periodicity of the material parameters and higher dimensions. Extensive numerical results demonstrate the validity of the new stability limits derived for heterogeneous materials with periodic fluctuation. Finally numerical examples of the stability for randomly fluctuating material properties are also presented, discussing the applicability of the theoretical limits derived for material properties with periodic fluctuation.