Highly accurate stability-preserving optimization of the Zener viscoelastic model, with application to wave propagation in the presence of strong attenuation

Abstract

This article concerns the numerical modeling of time-domain mechanical waves in vis-coelastic media based on a generalized Zener model. To do so, classically in the literature relaxation mechanisms are introduced, resulting in a set of so-called memory variables and thus in large computational arrays that need to be stored. A challenge is thus to accurately mimic a given attenuation law using a minimal set of relaxation mechanisms. For this purpose, we replace the classical linear approach of Emmerich & Korn (1987) with a nonlinear optimization approach with constraints of positivity. We show that this technique is significantly more accurate than the linear approach. Moreover it ensures that physically-meaningful relaxation times that always honor the constraint of decay of total energy with time are obtained. As a result these relaxation times can always be used in a stable way in a modeling algorithm, even in the case of very strong attenuation for which the classical linear approach may provide some negative and thus unusable coefficients.