scholarly journals Image-source method and truncation of a series expansion of the integral solution—Case of an angular sector in two dimensions

2006 ◽  
Vol 120 (2) ◽  
pp. 597-610 ◽  
Author(s):  
Vincent Martin ◽  
Thomas Guignard
Keyword(s):  
Series Expansion ◽  
Two Dimensions ◽  
Integral Solution ◽  
Image Source ◽  
Source Method ◽  
Angular Sector

scholarly journals Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

2021 ◽  
pp. 1-26
Author(s):  
ELENA CHERKAEV ◽  
MINWOO KIM ◽  
MIKYOUNG LIM
Keyword(s):  
Composite Materials ◽  
Series Expansion ◽  
Function Theory ◽  
Effective Conductivity ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Geometric Function Theory ◽  
Boundary Integral Formulation ◽  
Geometric Function ◽  
Series Expression

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

scholarly journals A Green’s Function for Acoustic Problems in Pekeris Waveguide Using a Rigorous Image Source Method

2021 ◽  
Vol 11 (6) ◽  
pp. 2722
Author(s):  
Zhiwen Qian ◽  
Dejiang Shang ◽  
Yuan Hu ◽  
Xinyang Xu ◽  
Haihan Zhao ◽  
...  
Keyword(s):  
Green’S Function ◽  
Shallow Water ◽  
Green's Function ◽  
Plane Waves ◽  
Spherical Wave ◽  
Sound Field ◽  
Efficient Computation ◽  
Image Source ◽  
Source Method ◽  
Pekeris Waveguide

The Green’s function (GF) directly eases the efficient computation for acoustic radiation problems in shallow water with the use of the Helmholtz integral equation. The difficulty in solving the GF in shallow water lies in the need to consider the boundary effects. In this paper, a rigorous theoretical model of interactions between the spherical wave and the liquid boundary is established by Fourier transform. The accurate and adaptive GF for the acoustic problems in the Pekeris waveguide with lossy seabed is derived, which is based on the image source method (ISM) and wave acoustics. First, the spherical wave is decomposed into plane waves in different incident angles. Second, each plane wave is multiplied by the corresponding reflection coefficient to obtain the reflected sound field, and the field is superposed to obtain the reflected sound field of the spherical wave. Then, the sound field of all image sources and the physical source are summed to obtain the GF in the Pekeris waveguide. The results computed by this method are compared with the standard wavenumber integration method, which verifies the accuracy of the GF for the near- and far-field acoustic problems. The influence of seabed attenuation on modal interference patterns is analyzed.

Combination of acoustical radiosity and the image source method

2013 ◽  
Vol 133 (6) ◽  
pp. 3963-3974 ◽  
Author(s):  
Georgios I. Koutsouris ◽  
Jonas Brunskog ◽  
Cheol-Ho Jeong ◽  
Finn Jacobsen
Keyword(s):  
Image Source ◽  
Source Method
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