Numerical Methods for the Simulation of Wave Propagation Phenomena in Linear Viscoelastic Kelvin-Voigt Materials

Numerical simulation of wave propagation phenomena is a relevant subject in many different fields of science and engineering, such as, among others, geophysics, ocean acoustics, materials science and biomedical engineering. In particular, we develop a mathematical and numerical strategy to accurately simulate acoustic waves, as this could be a critical element in defining effective Early Warning Systems aiming at reducing the possibility of harm or loss due to extreme natural disasters. An effective modelling of wave propagation phenomena in real media requires to take into account several non-trivial but fundamental physical aspects, such as energy dissipation due to viscous effects and heterogeneous spatial distribution of material properties, which pose additional challenges from both a mathematical modelling and numerical point of view. In this thesis we consider a linear viscoelastic material behavior described by means of the well-known Kelvin-Voigt rheology. Material heterogeneity has been taken into account by dividing the domain into multiple layers, each of them being associated with a distinct material and therefore characterised by a particular set of coefficients that define its physical properties. The system of Partial Differential Equations obtained combining the Cauchy momentum equations with the Kelvin-Voigt constitutive model has been reduced from second order to first order in time by doubling the solution variables, that are the displacement and velocity fields. Then, existence, uniqueness and regularity results for the solution of the resulting initial-boundary value problem have been obtained by relying on the standard theory of linear evolution equations in Hilbert spaces. The weak formulation of the problem has been discretised in space by means of both the linear Lagrangian Finite Element Method and the Spectral/Spectral Element Method. The latter approach is based on the partition of the domain into a set of material subdomains and the approximation of the unknown displacement and velocity fields by means of linear combinations of either Legendre or Chebyshev polynomials defined within each material subdomain. Two different strategies have been considered for enforcing the continuity of the solution across the material interfaces. The first approach consists of using boundary-adapted polynomials defined in such a way they match at the interfaces, whereas the second strategy relies on a Mortar-like approach, thus requiring L^2 weak continuity of the basis functions at the material interfaces. The discretisation in space of the weak formulation yields a system of Ordinary Differential Equations that are solved either by means of standard numerical time-stepping schemes or by exact integration via eigenvalue decomposition. The spatial convergence of the implemented numerical schemes has been investigated by solving several problems with manufactured solution. Finally, numerical simulations of wave propagation phenomena in homogeneous and heterogeneous materials due to the action of a point source are also carried out to validate the proposed numerical strategy.