Topological derivative for an acoustic-elastic coupled system based on two-phase material model

Abstract

This letter presents an explicit formulation for the topological derivative of a two-dimensional acoustic-elastic coupled system, expressed with a two-phase material model based on Biot's theory. First, we briefly explain the two-phase material model in which the objective functional is assumed to be a domain integral of a certain function of velocity potential. The shape derivative of the objective functional is obtained using the usual Lagrangian formulation and we then construct the adjoint equation. Since it is known that the limit value of a shape derivative is equal to the topological derivative, asymptotic behavior for the boundary value problem for the state and adjoint variables is searched for and, based on the solutions, an explicit formulation of the topological derivative is thereby obtained. With the objective functional defined as the squared norm of the acoustic pressure, the topological derivatives for equivalent acoustic and elastic material domains are numerically compared with the numerical difference when a hole domain with a finite radius appears, using the FEM. The provided numerical examples demonstrate the validity of our topological derivative formulation and the procedure for calculating topological derivatives.