International Conference
"Modern Problems of Mathematics, Informatics and Bioinformatics",
devoted to the 100th anniversary of professor Alexei A. Lyapunov

Novosibirsk, Russia, 2011, October 11-14

Алексеев Г.В.  

Existence of 3D cloaking shells for the model of anisotropic acoustics

Recently a number of papers devoted to describing and constructing of so called cloaking shells were published. The existence of objects which do not scatter some incident electromagnetic or acoustic waves was proved in these papers. These objects play the role of invisibility zone which renders any object in the interior of the shell free of acoustic scattering. Corresponding phenomenon has obtained a name of “electromagnetic or acoustic cloaking” and the mentioned shells have obtained the name of “electromagnetic or acoustic cloaking shells”.
In this paper we study the mentioned problem of constructing cloaking shells for the model of anisotropic acoustics using methods of inverse problems. Based on the optimization technique of solving inverse problems we reduce our inverse problem to the problem of minimization of a suitable tracking type cost functional. We deduce the optimality system for the general control problem which describes the first-order necessary optimality conditions. Using the optimality system analysis we shall establish the sufficient conditions for the data which provide a local stability and uniqueness of solutions of control problems under consideration in the case of concrete tracking-type cost functionals. Then the efficient numerical algorithm of solving our control problem based on Newton's method of solving nonlinear equations and finite element method of solution of Helmholtz boundary value problems is developed. The convergence of this algorithm is studied and some results of numerical experiments are presented.