A numerical method for solving the boundary integral equations of the three-dimensional scalar diffraction problem

2019 ◽  
Author(s):  
А.А. Каширин ◽  
С.И. Смагин
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Diffraction Problem ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Scalar Diffraction

Описывается метод численного решения трехмерной стационарной задачи дифракции акустических волн на однородном включении. Исходная задача сводится к граничным слабо сингулярным интегральным уравнениям Фредгольма первого рода с одной неизвестной функцией, имеющим дискретный спектр. Эти уравнения решаются численно. Для поиска решения исходной задачи на спектре предложен метод интерполяции решения. Приводятся результаты численных экспериментов.

scholarly journals Algorithm for Numerical Solution of Diffraction Problem on the Joint of Two Open Three-Layer Waveguides

2018 ◽  
Vol 186 ◽  
pp. 01010
Author(s):  
Dmitriy Divakov ◽  
Anastasiia Tiutiunnik ◽  
Anton Sevastianov
Keyword(s):  
Galerkin Method ◽  
Numerical Solution ◽  
Dimensional Space ◽  
Diffraction Problem ◽  
Liouville Problem ◽  
Planar Waveguides ◽  
Problem Solution ◽  
Scalar Diffraction ◽  
Open Waveguide ◽  
The Galerkin Method

This paper describes the algorithm for the numerical solution of the diffraction problem of waveguide modes at the joint of two open planar waveguides. For planar structures under consideration, we can formulate a scalar diffraction problem, which is a boundary value problem for the Helmholtz equation with a variable coefficient in two-dimensional space. The eigenmode problem for an open three-layer waveguide is the Sturm-Liouville problem for a second-order operator with piecewise constant potential on the axis, where the potential is proportional to the refractive index. The described problem is singular and has a mixed spectrum and therefore the Galerkin method can not be used in this definition. One way to adapt the Galerkin method for the problem solution is to artificially limit the area, which is equivalent to placing the open waveguide in question in a hollow closed waveguide whose boundaries are remote from the real boundaries of the waveguide layer of the open waveguide. Thus, we obtain a diffraction problem on a finite interval and with a discrete spectrum, which can be solved by the projection method, as done in this paper.

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