Determining kernels in linear viscoelasticity

TL;DR

This paper addresses the inverse problem of identifying kernel functions in a nonlocal viscoelastic wave equation, using PDE-constrained optimization and sensitivity analysis, with numerical validation in 3D.

Contribution

It introduces a rigorous approach for determining kernels in complex viscoelastic models through well-posedness analysis and sensitivity derivation, supported by numerical experiments.

Findings

Successful kernel identification in 3D simulations

Established well-posedness of the inverse problem

Derived first-order sensitivities for optimization

Abstract

In this work, we investigate the inverse problem of determining the kernel functions that best describe the mechanical behavior of a complex medium modeled by a general nonlocal viscoelastic wave equation. To this end, we minimize a tracking-type data misfit function under this PDE constraint. We perform the well-posedness analysis of the state and adjoint problems and, using these results, rigorously derive the first-order sensitivities. Numerical experiments in a three-dimensional setting illustrate the method.