Detection of cavities in Helmholtz-type equations using the boundary element method

Helmholtz-type equations arise naturally in many physical applications related to wave propagation, vibration phenomena and heat transfer. These equations are often used to describe the vibration of a structure, the acoustic cavity problem, the radiation wave, the scattering of a wave and heat conduction in fins. In this paper, the numerical recovery of a single and two circular cavities in Helmholtz-type equations from boundary data is investigated. The boundary element method (BEM), in conjunction with a constrained least-squares minimisation, is used to solve this inverse geometric problem. The accuracy and stability of the proposed numerical method with respect to the distance between the cavities and the outer boundary of the solution domain, the location and size of the cavities, and the distance between the cavities are also analysed. Unique and stable numerical solutions are obtained.