Parameter recovery and transmission problems in poroelastic media

Abstract

In this work we use a finite difference scheme to solve relevant boundary value problems modeling ultrasound propagation through a cancellous bone specimen. In one boundary value problem we consider the bone as an orthotropic medium. In the second boundary value problem the bone is considered to be an isotropic medium. After the boundary value problems are solved numerically we perform various numerical tests to show convergence of the schemes. After convergence is exhibited we deem the problem suitable for performing an inverse problem where certain parameters of interest are tried to be recovered. In addition we develop a new model where we consider the bone specimen to be circular and we also add a layer of muscle to the configuration. For this model, we assume scattering of an infinite cylinder by an incoming plane compressional wave. Analytical solutions are derived using a Helmholtz type decomposition. In deriving the analytical solutions in this model, the solutions are assumed to be time harmonic. We recover time dependence by performing a numerical inverse fast Fourier transform. Plots of the radial displacements will be provided to suggest that the model is physically acceptable. Finally we propose a generalization of the circular problem by assuming the specimen contains a layer of cortical bone. Analytical solutions can still be derived in the time harmonic case. A method for generating these solutions is discussed using a generalization of a method proposed by Vekua.