Wave propagation in Kelvin–Voigt heterogeneous materials: A spectral-Galerkin approach

In this work we present a numerical algorithm for the three-dimensional simulation of wave propagation phenomena in viscoelastic heterogeneous materials of Kelvin-Voigt type. The Partial Differential Equations defining the model are obtained by combining the Cauchy equations with the Kelvin-Voigt stress-strain constitutive relations, which defines the stress tensor as the sum of an elastic and a viscous contribution. The numerical approach that we propose is based on a spectral-Galerkin method, whereby the solution of the Partial Differential Equation is approximated by using a linear combination of suitable basis functions defined on the whole computational domain. In particular, we have used basis functions depending on cylindrical coordinates (r, θ, z) and obtained by multiplication of linear combination of Chebyshev polynomials of the first kind (for the radial r and axial z coordinates) and trigonometric functions (for the angle θ). Moreover, material heterogeneity is taken into account by partitioning the domain into several subdomains, each one associated to a different material, and by defining a different set of basis functions on each of them. Then, weak L^2 continuity of the basis functions is imposed at the material interfaces. The resulting system of Ordinary Differential Equations is solved by means of the Crank-Nicolson time integration scheme or by exact integration of the forcing term. A set of numerical examples aiming at demonstrating the capabilities of the present approach is finally presented.