A Generalized Overrelaxation Method for Solving Singular Volume Integral Equations in Low-Frequency Scattering Problems

2005 ◽  
Vol 41 (9) ◽  
pp. 1262-1266 ◽  
Author(s):  
N. V. Budko ◽  
A. B. Samokhin ◽  
A. A. Samokhin
Keyword(s):  
Integral Equations ◽  
Low Frequency ◽  
Volume Integral ◽  
Scattering Problems

scholarly journals Versatile Solver of Nonconformal Volume Integral Equation Based on SWG Basis Function

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Chunbei Luo ◽  
Mingjie Pang ◽  
Hai Lin
Keyword(s):  
Integral Equation ◽  
Basis Function ◽  
High Efficiency ◽  
Low Frequency ◽  
Piecewise Constant Function ◽  
Volume Integral ◽  
Scattering Problems ◽  
Volume Integral Equation ◽  
Inhomogeneous Dielectric ◽  
Brute Force Method

In this paper, a versatile solver of a nonconformal volume integral equation based on the Schaubert-Wilton-Glisson (SWG) basis function is presented. Instead of using a piecewise constant function, the robust conventional SWG basis function is chosen and used directly for discontinuous boundaries. A new map method technique is proposed for constructing SWG pairs, which reduces the complexity from ON2 to ONlogN compared with a brute-force method. The integral equation is solved by the method of moments (MoM) and further accelerated by the multilevel fast multipole algorithm (MLFMA). What’s more, the hybrid scheme of MLFMA and adaptive cross approximation (ACA) is developed to resolve the low-frequency (LF) breakdown when dealing with over-dense mesh objects. Numerical results show that when in analysis of radiation or scattering problems from inhomogeneous dielectric objects or in LF conditions, the proposed solver shows high efficiency without loss of accuracy, which demonstrates the versatile performance of the proposed method.

Compressing H2 Matrices for Translationally Invariant Kernels

2021 ◽  
Vol 35 (11) ◽  
pp. 1392-1393
Author(s):  
R. Adams ◽  
J. Young ◽  
S. Gedney
Keyword(s):  
Integral Equations ◽  
Low Frequency ◽  
Volume Integral ◽  
Translational Invariance ◽  
Numerical Examples ◽  
Large Memory ◽  
Sparse Samples ◽  
Invariant Kernels

H2 matrices provide compressed representations of the matrices obtained when discretizing surface and volume integral equations. The memory costs associated with storing H2 matrices for static and low-frequency applications are O(N). However, when the H2 representation is constructed using sparse samples of the underlying matrix, the translation matrices in the H2 representation do not preserve any translational invariance present in the underlying kernel. In some cases, this can result in an H2 representation with relatively large memory requirements. This paper outlines a method to compress an existing H2 matrix by constructing a translationally invariant H2 matrix from it. Numerical examples demonstrate that the resulting representation can provide significant memory savings.

scholarly journals Integral Equations for Problems on Wave Propagation in Near-Earth Plasma

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1395
Author(s):  
Danila Kostarev ◽  
Dmitri Klimushkin ◽  
Pavel Mager
Keyword(s):  
Magnetic Field ◽  
Integral Equations ◽  
Field Potential ◽  
Low Frequency ◽  
Fredholm Equation ◽  
Compression Waves ◽  
Parallel Component ◽  
Electric Field Potential ◽  
The Magnetic Field ◽  
Low Frequency Waves

We consider the solutions of two integrodifferential equations in this work. These equations describe the ultra-low frequency waves in the dipol-like model of the magnetosphere in the gyrokinetic framework. The first one is reduced to the homogeneous, second kind Fredholm equation. This equation describes the structure of the parallel component of the magnetic field of drift-compression waves along the Earth’s magnetic field. The second equation is reduced to the inhomogeneous, second kind Fredholm equation. This equation describes the field-aligned structure of the parallel electric field potential of Alfvén waves. Both integral equations are solved numerically.

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