A level-set approach based on reaction–diffusion equation applied to inversion problems in acoustic wave propagation

Abstract

Abstract In this work, a topology optimization procedure based on the level-set method is applied to the solution of inverse problems for acoustic wave propagation in the time-domain. In this class of inverse problems the presence of obstacles in a background medium must be identified. Obstacles and background are defined by means of a level-set function that evolves by following the solution of a reaction–diffusion equation. Within this approach, no initial guess for the topology nor level-set reinitialization procedures are necessary, contrary to what is commonly observed when the Hamilton–Jacobi equation is used. The objective function is defined as the domain and time integration of the squared difference between experimental and simulation pressure signals. The finite element method is used for the spatial and level-set function discretizations and a time-marching procedure (Newmark scheme) is used to solve the wave propagation problem, as well as the adjoint problem for the sensitivity analysis. Both procedures provide the information needed to define the velocity field for the level set evolution. Results show that the proposed technique is capable to find the location and shape of obstacles within a background medium. Systematic tests show that, as expected, the distribution of sources and receivers shows to have a major influence on the final solution. Results also reproduce known difficulties; when the so called inverse crime is avoided, the identification procedure worsens its performance. Filters and smoothing are among different features that deserve further investigation. Although the formulation presented here focuses on the acoustic wave propagation problem, its extension to wave propagation in elastic media is straightforward.