Abstract:
The purpose of the research is to develop a numerical scheme using iterative methods for solving systems of equations for solving bulk acoustic problems with inhomogeneous refraction index. The paper presents a formulation of the acoustic wave propagation problem in the form of a volume integral Fredholm equation of the second kind. A structured volume rectangular mesh is used to discretize the problem for the purpose of subsequent numerical solution. Using discretization, the problem formulation is reduced to a discretized operator in the form of a system of equations with a large number of variables and an operator matrix of high dimensionality. Taking into account the peculiarities of the integral kernels of the Helmholtz equation in integral form, numerical methods for solving the systems of equations using modifications of the matrix-vector multiplication of Toeplitz matrices by a vector based on the fast discrete Fourier transform are given. Numerical results of a set of programs for modeling propagation realizations of a plane wave model in a volumetric medium with inhomogeneous refraction index are demonstrated. Special attention in this paper is paid to the possibility of fast solution of mathematical physics problems on a structured grid of high dimensionality, which will allow us to consider the features of the solution on complex inhomogeneous boundaries, as well as to simplify the approximation of the solution. Finally, conclusions will be drawn about the quality of the obtained solutions on different examples of inhomogeneities of the considered volume domain.
Keywords:volume integral equations, acoustic problem, Fredholm equation, iterative methods, fast Fourier transform, Helmholtz equation.
UDC:519.642+51-73 BBK:
22.19
Received: February 17, 2025 Published: May 31, 2025
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