A Green's Function for the Domain Bounded by Nonconcentric Spheres

Chen, Jeng‐Tzong; Lee, Jia-Wei; Shieh, Hung-Chih

doi:10.1115/1.4007071

Cited by 5 publications

(3 citation statements)

References 7 publications

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“…Note also that an explicit form of the Dirichlet Green function for two nonconcentric spheres was derived by using bispherical coordinates in [81].…”

Section: Interior Problem For Two Concentric Spheresmentioning

confidence: 99%

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

Grebenkov

Traytak

2019

Journal of Computational Physics

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We apply the generalized method of separation of variables (GMSV) to solve boundary value problems for the Laplace operator in three-dimensional domains with disconnected spherical boundaries (i.e., an arbitrary configuration of non-overlapping partially reactive spherical sinks or obstacles). We consider both exterior and interior problems and all most common boundary conditions: Dirichlet, Neumann, Robin, and conjugate one. Using the translational addition theorems for solid harmonics to switch between the local spherical coordinates, we obtain a semi-analytical expression of the Green function as a linear combination of partial solutions whose coefficients are fixed by boundary conditions. Although the numerical computation of the coefficients involves series truncation and matrix inversion, the use of the solid harmonics as basis functions naturally adapted to the intrinsic symmetries of the problem makes the GMSV particularly efficient, especially for exterior problems. The obtained Green function is the key ingredient to solve boundary value problems and to determine various characteristics of stationary diffusion such as reaction rate, escape probability, harmonic measure, residence time, and mean first passage time, to name but a few. The relevant aspects of the numerical implementation and potential applications in chemical physics, heat transfer, electrostatics, and hydrodynamics are discussed.

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“…Note also that an explicit form of the Dirichlet Green function for two nonconcentric spheres was derived by using bispherical coordinates in [81].…”

Section: Interior Problem For Two Concentric Spheresmentioning

confidence: 99%

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

Grebenkov

Traytak

2019

Journal of Computational Physics

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“…As a result, G ( x 0 , x ) is proportional to − log | x − x 0 | as x → x 0 . Writing it as a sum of singular and regular parts, and expressing both in bipolar coordinates [ 5 , 28 ], Heyda was able to find a series expression for Green’s function with absorbing boundaries. A different approach to the same problem [ 29 ], because the transformed domain is rectangular, is to expand Green’s function in trigonometric eigenfunctions.…”

Section: Introductionmentioning

confidence: 99%

Diffusion in a disk with inclusion: Evaluating Green’s functions

Stana

Lythe

2022

PLoS ONE

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We give exact Green’s functions in two space dimensions. We work in a scaled domain that is a circle of unit radius with a smaller circular “inclusion”, of radius a, removed, without restriction on the size or position of the inclusion. We consider the two cases where one of the two boundaries is absorbing and the other is reflecting. Given a particle with diffusivity D, in a circle with radius R, the mean time to reach the absorbing boundary is a function of the initial condition, given by the integral of Green’s function over the domain. We scale to a circle of unit radius, then transform to bipolar coordinates. We show the equivalence of two different series expansions, and obtain closed expressions that are not series expansions.

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“…We now use the results of the previous section to derive the T-matrix for a two sphere or sphere-plane system. We do not consider the case of one sphere inside the other (both above or below the xy plane), as in [137], but this can be treated with minor modifications. Also, bispherical coordinates cannot be used to treat touching spheres; tangent sphere coordinates may be used instead [138,139], or equivalently transformation optics (radial inversion about the contact point) [140].…”

Section: T-matrix For Two Spheresmentioning

confidence: 99%

Electrostatic images and T-matrices for simple geometries

Majic¹

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<p>This thesis is concerned with electrostatic boundary problems and how their solutions behave depending on the chosen basis of harmonic functions and the location of the fundamental singularities of the potential. The first part deals with the method of images for simple geometries where the exact nature of the image/fundamental singularity is unknown; essentially a study of analytic continuation for Laplace's equation in 3 dimensions. For the sphere, spheroid and cylinder, new deductions are made on the location of the images of point charges and their linear or surface charge densities, by using different harmonic series solutions that reveal the image. The second part looks for analytic expressions for the T-matrix for electromagnetic scattering of simple objects in the low frequency limit. In this formalism the incident and scattered fields are expanded on an orthogonal basis such as spherical harmonics, and the T-matrix is the transformation between the coefficients of these series, providing the general solution of any electromagnetic scattering problem by a given particle at a given wavelength. For the spheroid, bispherical system and torus, the natural basis of harmonic functions for the geometry of the scatterer are used to determine T-matrix expressed in that basis, which is then transformed onto a basis of canonical spherical harmonics via the linear relationships between different bases of harmonic functions.</p>

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A Green's Function for the Domain Bounded by Nonconcentric Spheres

Cited by 5 publications

References 7 publications

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

Diffusion in a disk with inclusion: Evaluating Green’s functions

Electrostatic images and T-matrices for simple geometries

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