Numerical Solution of the Optimization Issue with Restrictions for the Processes of Acoustic Wave’s Diffraction

Abstract

The research is devoted to solving the optimal control problem for the equations of diffraction of acoustic waves by three-dimensional inclusion.It minimizes the deviation of the pressure field in the inclusion from a specific interval, due to sound sourceschanges in the external environment.
External boundary value problems for partial differential equations are the mathematical model of the process. In our research, we offer an algorithm for the numerical solution of the problem subject to limited set of controls. Namely, it imposes a restriction on the power of sources with which one can control the acoustic field.
The solution to the problem can be represented as a linear combination of the solutions of direct diffraction problems and unknown coefficients. We use the coordinate descent to find the required coefficients.
If we solve direct problems by potential theory, the diffraction problem reduces to a mixed system of weakly singular boundary integral Fredholm equations of the first and second kind on the inclusion surface. The approximation of integral equations by a system of linear algebraic equations is carried out by dividing the unit on the surface, associated with a system of nodal points, and also consistent with the discretization order of the method of averaging weakly singular kernels of integral operators. Multiple integrals arising during discretization are calculated analytically. This allows one to obtain explicit formulas for approximating boundary integral operators with singularities in kernels and use them to calculate the coefficients of systems of linear algebraic equations. At the same time, preliminary triangulation of the surface is not required.
We ran numerical experiments and mathematical modeling of the diffraction process of acoustic waves. When solving a problem on a computer, parallel computations are used.