Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

Li, Zi‐Cai; Zhang, Li Ping; Wei, Yimin; Lee, Ming‐Gong; Chiang, John Y.

doi:10.1016/j.enganabound.2015.07.001

Cited by 12 publications

(2 citation statements)

References 26 publications

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“…Moreover, the dual NFM can be applied to multiple holes, circular, elliptic and arbitrary with smooth boundary. For polygonal holes and the interior holes with corners, the dual NFM is still valid to guarantee the unique solutions, provided that the suitable singular solutions near corners are introduced into the algorithms, see [18,Section 5.4]. More exploration appears elsewhere.…”

Section: Dual Null Field Methods 415mentioning

confidence: 99%

Dual null field method for Dirichlet problems of Laplace's equation in circular domains with circular holes

Lee¹,

Zhang²,

Li³

et al. 2021

Sib. Elektron. Mat. Izv.

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The dual techniques have been widely used in many engineering papers, to deal with singularity and ill-conditioning of the boundary element method (BEM). In this paper, we consider Laplace's equation with circular domains with one circular hole. The explicit algebraic equations of the rst and second kinds of the null eld method (NFM) are provided for applications. Traditionally, the rst and the second kinds of the NFM are used for the Dirichlet and the Neumann problems, respectively. To bypass the degenerate scales of Dirichlet problems, however, the second and the rst kinds of the NFM are used for the exterior and the interior boundaries, simultaneously, called the dual NFM (DNFM) in this paper. The excellent stability and the optimal convergence rates are explored in this paper. By using the simple Gaussian elimination or the iteration methods, numerical solutions can be easily obtained. Recently, the study on degenerate scales is active, many removal techniques are proposed, where the advanced solution methods may be needed, such as the truncated singular value decomposition (TSVD) and the overdetermined systems. In contrast, the solution methods of the DNFM in this paper are much simpler, with a little risk of the algorithm singularity from degenerate scales.

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Section: Dual Null Field Methods 415mentioning

confidence: 99%

Dual null field method for Dirichlet problems of Laplace's equation in circular domains with circular holes

Lee¹,

Zhang²,

Li³

et al. 2021

Sib. Elektron. Mat. Izv.

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“…They used piecewise quadratic polynomial approximations in the boundary equation method for the solution of boundary value problems involving Laplace's equation and certain Poisson equations. The boundary element method for the Dirichlet boundary condition of Laplace's equation in elliptic domains with elliptic holes is used by Li et al [23]. In this study, they developed the null field method for the elliptic domains with elliptic holes.…”

Section: Introductionmentioning

confidence: 99%

Numerical calculations of the potential on the rectangular and ellpitic domains with various aspect ratios

Panahi¹,

Ghassemi²,

Nowruzi³

2017

J. Appl. Math. Comput. Mech.

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Abstract. In the current study, the Laplace equation is solved for rectangular and elliptical computational domains by using the boundary element method (BEM). For this accomplishment, 120 different aspect ratios in a rectangular and elliptical computational domains are designed. The Dirichlet and Neumann boundary conditions are used respectively for a rectangular and elliptical domains. Also, the Gaussian quadrature integral method is applied to solve the influence coefficient matrix in BEM. To assess a different aspect ratio on the potential solution, two different measurement positions are intended. According to our finding, with an increase of the aspect ratio, the potential value is increased for both rectangular and elliptical domains. However, a potential increment with aspect ratio enhancement is more visible in the elliptical domain.MSC 2010:76M15, 76B07

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Numerical Solution of Laplace and Poisson Equations for Regular and Irregular Domain Using Five-Point Formula

Adak¹

2022

Lecture Notes in Electrical Engineering

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Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

Cited by 12 publications

References 26 publications

Dual null field method for Dirichlet problems of Laplace's equation in circular domains with circular holes

Dual null field method for Dirichlet problems of Laplace's equation in circular domains with circular holes

Numerical calculations of the potential on the rectangular and ellpitic domains with various aspect ratios

Numerical Solution of Laplace and Poisson Equations for Regular and Irregular Domain Using Five-Point Formula

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