A strategy for implementing shape-measure method in designing damp source of inhomogeneous damped wave equation

in this article, the shape-measure method in polar coordinates is employed to determine the optimal shape of the damping source and obtaining the optimal damping amount for an inhomogeneous damped wave system in a given time. Using Green’s function and the properties of Sobolov spaces, the damped wave system and its related constraints are cast in the form of a vibrational problem. Next, by introducing the energy function as the target function, the resulting optimization problem is presented in polar coordinates. After representing the problem in an optimal control one, by done Defining two unknown measures, the solution space is extended to a subspace of positive Radon measures, which allowing the presentation of the problem as a linear optimization problem. Subsequently, two steps of approximation on the number of constraints and the solution space beside the discretization scheme, satisfactorily approximate the original optimization problem with a finite linear programming one. Ultimately, the solution of this problem yields the control function and the optimal path, which is the optimal shape. A numerical example is also presented based on the mentioned strategy.

Keywords:

Optimal shape design , Damped wave equation , Shape-measure method