Time Dependent Solutions for the Biot Equations

Abstract

In this paper we show that the Biot model for the ultrasound interrogation of bone rigidity, under certain restrictions, can be shown to lead to a unique solution. More precisely, we consider the classical experimental method for measuring bone parameters, that is where a bone sample in a water bath and the bone sample interrogated with an ultrasound devise. This procedure leads to an inverse problem where the ultrasound signal is measured in various positions in the water tank. In order to solve the inverse problem an accurate forward solver is necessary. It is shown that the forward problem may be formulated in terms of a boundary element method. To this end, the Biot system of equations describing the acoustic interaction with a porous material is written in a convenient, compact form. The system, and the transition conditions between the porous material, are rewritten in a Laplace transformed space. The transformed problem is reformulated as a nonlocal boundary problem. Using a variational approach it is shown that the variational problem is equivalent to a nonlocal problem and the solution is shown to be unique. We then use Lubisch’s approach to find estimates in the time domain without recourse to using the inverse Laplace transformation.