High-order topological expansions for Helmholtz problems in 2d

Methods of topological analysis are inherently related to singular perturbations. For topology variation, a trial geometric object put in a test domain is examined by reducing the object size from a finite to an infinitesimal one. Based on the singular perturbation of the forward Helmholtz problem, a topology optimization approach, which is a direct one, is described for the inverse problem of object identification from boundary measurements. Relying on the 2d setting in a bounded domain, the high-order asymptotic result is proved rigorously for the Neumann, Dirichlet, and Robin type conditions stated at the object boundary. In particular, this implies the first-order asymptotic term called a topological derivative. For identifying arbitrary test objects, a variable parameter of the surface impedance is successful. The necessary optimality condition of minimum of the objective function with respect to trial geometric variables is discussed and realized for finding the center of the test object.