The method of hypersingular integral equations in the problem of electromagnetic wave diffraction by a dielectric body with a partial perfectly conducting coating

2017 ◽  
Vol 32 (6) ◽  
Author(s):  
Alexey V. Setukha ◽  
Elizaveta N. Bezobrazova
Keyword(s):  
Electromagnetic Wave ◽  
Integral Equations ◽  
Numerical Scheme ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
The Body ◽  
Dielectric Body ◽  
Piecewise Constant ◽  
Conducting Coating ◽  
Monochromatic Electromagnetic Wave

AbstractA problem of scattering of a monochromatic electromagnetic wave by a homogeneous dielectric body is considered. A part of the boundary of the body is a perfectly conducting thin surface. The problem is reduced to a system of the boundary integral equations containing integrals with strong singularity, the integrals are understood in terms of Hadamard final part. A numerical solution scheme is constructed on the base of approximate solution of these equations using the methods of piecewise-constant approximations and collocation. The constructed numerical scheme is tested on a model example.

Nystrom method for the Muller boundary integral equations on a dielectric body of revolution: axially symmetric problem

2015 ◽  
Vol 9 (11) ◽  
pp. 1186-1192 ◽  
Author(s):  
Vitaliy S. Bulygin ◽  
Yuriy V. Gandel ◽  
Ana Vukovic ◽  
Trevor M. Benson ◽  
Phillip Sewell ◽  
...  
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Body Of Revolution ◽  
Symmetric Problem ◽  
Axially Symmetric ◽  
Dielectric Body ◽  
Nyström Method ◽  
Nystrom Method

scholarly journals Peculiarities of the boundary integral equation method in the problem of electromagnetic wave scattering on ideally conducting bodies of small thickness

2016 ◽  
pp. 460-473
Author(s):  
А.В. Сетуха ◽  
С.Н. Фетисов
Keyword(s):  
Electromagnetic Wave ◽  
Integral Equations ◽  
Wave Scattering ◽  
Boundary Integral ◽  
The Body ◽  
Fine Grid ◽  
Small Thickness ◽  
Electromagnetic Wave Scattering ◽  
Linear Algebraic Equations ◽  
Conducting Bodies

Для численного решения классической задачи дифракции электромагнитной волны на идеально проводящих объектах используется метод граничных интегральных уравнений с гиперсингулярными интегралами, к которым применяются метод кусочно-постоянных аппроксимаций и метод коллокации. В результате задача сводится к системе линейных уравнений, коэффициециенты которой выражаются через интегралы по ячейкам разбиения с сильной степенной особенностью. Для вычисления этих интегралов применяется развитый ранее подход, основанный на выделении в явном виде членов с сильной особенностью, вычисляемых аналитически. В рамках этого подхода в настоящей статье протестирована численная схема, в которой вычисление оставшихся членов со слабосингулярными интегралами по ячейкам разбиения осуществляется путем построения более мелкой сетки второго уровня с домножением подынтегрального выражения на сглаживающий множитель. На примере задачи дифракции на теле в форме прямоугольного крыла показано, что такая схема, в частности, позволяет решать задачи дифракции на телах малой толщины. При этом толщина тела может быть даже меньше диаметра ячеек основного разбиения, но при условии, что диаметр ячеек сетки второго уровня существенно меньше, чем толщина тела. The method of boundary integral equations with hypersingular integrals is used for the numerical solution of the classical problem of electromagnetic wave scattering on ideally conducting bodies. The corresponding integral equations are solved by the methods of piecewise constant approximations and collocation. As a result, the problem is reduced to a system of linear algebraic equations whose coefficients are expressed in terms of integrals over partition cells with a strong power singularity. These integrals are evaluated using the previously developed approach based on the extraction of terms with a strong singularity calculated analytically. The proposed numerical scheme based on the calculation of the remaining terms with weakly singular integrals over partition cells is performed by constructing a fine grid of second level with multiplication of the integrands on a smoothing factor is tested. By the example of scattering on a rectangular it is shown, in particular, that this scheme allows one to solve the scattering problem on bodies of small thickness. In this case, the thickness of a body may be less then the diameter of the first level cells. However, the diameter of the second level cells must be much less than the thickness of the body.

scholarly journals Piecewise constant collocation for first-kind boundary integral equations

1994 ◽  
Vol 35 (3) ◽  
pp. 396-398
Author(s):  
I. G. Graham ◽  
Y. Yan
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Parameter Space ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Piecewise Constant ◽  
Break Points ◽  
Image Position ◽  
A Minor ◽  
Closed Polygon

We wish to correct a minor error in the recent paper [2]. That paper was concerned with an integral equation defined on a closed polygon Γ with r corners at the points x0, x2, …, x2r = x0. We parameterized Γ using a mapping γ:[−π,π] → Γ defined as follows. For each l, introduce the mid-point x2l−1 of the side joining x2l—2 to x2l. Then introduce 2r + 1 points in parameter spacewith the property that for each j = 1, …, 2rwhere mj are integers and . Then γ(s) is defined byfor j = 1, …, 2r. The {Sj} are then the preimages of the {xj} under γ. Moreover, in view of (1), a family of uniform meshes can be constructed on [−π, π] which include {Sj} as the break-points. Then γ maps these to meshes which are uniform on each segment joining xj−1 to xj (which we denote Γj). These meshes are used to discretize the integral equation.

scholarly journals Piecewise-constant collocation for first-kind boundary integral equations

1991 ◽  
Vol 33 (1) ◽  
pp. 39-64 ◽  
Author(s):  
I. G. Graham ◽  
Y. Yan
Keyword(s):  
Linear Systems ◽  
Integral Equations ◽  
Collocation Method ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Piecewise Constant ◽  
Collocation Points

AbstractWe examine the piecewise-constant collocation method, with collocation points the mid-points of subintervals, for first-kind integral equations with logarithmic kernels on polygonal boundaries. Previously this method had been shown to converge subject to certain restrictions on the angles at the corners of the polygon. Here, by considering a slightly modified collocation method, we are able to remove any restrictions on these angles, and to generalise slightly the meshes which may be used. Moreover, the modification leads to new results on the convergence of preconditioned two-(or multi-) grid methods for solving the resultant linear systems.

scholarly journals A numerical boundary integral equation method for elastodynamics. I

1978 ◽  
Vol 68 (5) ◽  
pp. 1331-1357
Author(s):  
David M. Cole ◽  
Dan D. Kosloff ◽  
J. Bernard Minster
Keyword(s):  
Integral Equations ◽  
Initial Value Problems ◽  
Numerical Scheme ◽  
Boundary Integral Equations ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Unknown Boundary ◽  
Boundary Values ◽  
Initial Value ◽  
Time Stepping

abstract The boundary initial value problems of elastodynamics are formulated as boundary integral equations. It is shown that these integral equations may be solved by time-stepping numerical methods for the unknown boundary values. A specific numerical scheme is presented for antiplane strain problems and a numerical example is given.

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