Geodesic
Using of the finite element method for solving the problem of diffraction of acoustic wave on a set of elastic inhomogeneous bodies
Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 244-255

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The paper considers the direct problem of diffraction of a harmonic sound wave by a set of linearly elastic bodies. The statement of the problem of the diffraction of a plane acoustic wave propagating in an ideal fluid by a given set of inhomogeneous anisotropic elastic bodies is presented. The problem statement is two-dimensional. As a method for solving the problem, a modification of the finite element method is proposed. Both the general idea of the method as applied to diffraction problems and the algorithm for solving this problem are described. For discretization in the space surrounding elastic bodies, in the two-dimensional case, a region bounded by a circle is selected. The area is divided into elements: in this work, it is proposed to use triangular elements of the first order. For each triangular element, a local matrix is built, the structure of which is based on the Helmholtz equation (for liquid elements) or general equations of motion of a continuous medium and Hooke's law (for elastic elements), as well as boundary conditions. Local matrices of elements make it possible to form a sparse global matrix for a system of linear algebraic equations, the solution of which determines the required values of pressure and displacements at the grid nodes. The interpolation procedure makes it possible to calculate the pressure and displacements at an arbitrary point inside the region, and the boundary conditions determine the scattered wave at points outside the region.
Keywords: finite element method, harmonic wave, plane wave, ideal fluid, elastic body. 
@article{CHEB_2023_24_5_a18,
     author = {D. R. Biryukov},
     title = {Using of the finite element method for solving the problem of diffraction of acoustic wave on a set of elastic inhomogeneous bodies},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {244--255},
     publisher = {mathdoc},
     volume = {24},
     number = {5},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a18/}
}
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D. R. Biryukov. Using of the finite element method for solving the problem of diffraction of acoustic wave on a set of elastic inhomogeneous bodies. Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 244-255. http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a18/