An approach to find the optimal design of 3-D damping set for a damped wave equation in spherical coordinates

Document Type : Research Article

Authors

Hajar Alimorad ¹

Alireza Fakharzadeh Jahromi ²

Mina Goodarzi ²

¹ Department of Mathematics, Jahrom University, P.O. Box 74135-111, Jahrom, Iran.

² Department of Mathematics, Shiraz University of Technology, Shiraz, Iran.

10.22067/ijnao.2026.96965.1802

Abstract

Damped waves play an important role in various aspects of modern life, such as analyzing ocean waves, storage tanks, automotive suspension systems, designing earthquake damping systems, and MRI machines in medicine. Furthermore, analyzing some of these systems is more naturally and efficiently performed in spherical coordinates. Considering these two important points, in this paper, we model the problem of finding the optimal shape of the damping set to minimize the energy of damped waves within a given end time defined in spherical coordinates. The basic idea is to describe the unknown 3D shape using two positive measures in spherical coordinates. This helps us to change the original problem into a linear optimization task that can be solved easily and globally. After explaining and modeling the problem and determining the weak solution in spherical coordinates, the measure-theoretical embedding process is also extended for the problem in spherical coordinates. First, for a fixed region, the problem is formulated as an optimal control one in variational form and then converted into an infinite-dimensional linear optimization problem. To determine the nearly optimal shape of the damping set, we use an appropriate finite linear programming problem through an iterative method based on the particle swarm optimization algorithm. The advantages of this method over other methods are that it provides a linear treatment for our nonlinear optimization problem, has a global minimizer, and is independent of the initial shape and mesh design. To demonstrate the sufficiency of this method, some applicable numerical examples are also provided.

Keywords

Damping set

Measure theory

Optimal shape

PSO algorithm

Wave equation

Subjects

• Control and Optimization

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