Identifying the wavenumber for the inverse Helmholtz problem using an enriched finite element formulation

Abstract

We investigate the inverse problem of identifying the wavenumber for the Helmholtz equation. The problem solution is based on measurements taken at few points from inside the computational domain or on its boundary. A novel iterative approach is proposed based on coupling the secant and the descent methods with the partition of unity method. Starting from an initial guess for the unknown wavenumber the forward problem is solved using the partition of unity method. Then the secant/descent methods are used to improve the initial guess by minimizing a predefined objective function based on the difference between the solution and a set of data points. In the next round of iterations the improved wavenumber estimate is used for the forward problem solution and the partition of unity approximation is improved by adding more enrichment functions. The iterative process is terminated when the objective function has converged and a set of two predefined tolerances are met. To evaluate the estimate accuracy we propose to utilize extra data points. To validate the approach and test its efficiency two wave applications with known analytical solutions are studied. The results show that the proposed approach can achieve high accuracy for the studied applications even when the considered data is contaminated with noise. Despite the clear advantages that were previously shown in the literature for solving the forward Helmholtz problem, this work presents a first attempt to solve the inverse Helmholtz problem with an enriched finite element approach.