The method of fundamental solutions for the numerical solution of the biharmonic equation

Karageorghis, Andréas; Fairweather, Graeme

doi:10.1016/0021-9991(87)90176-8

Cited by 168 publications

(67 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Using the MFS [23], the solution (for simplicity, we use to denote each component i ) to (1) T are vectors of harmonic and biharmonic coefficients associated with these N s singularity points, i.e., the degree of the freedom to control the boundary fitting. The vanishing bilaplacian operator on is enforced by the fundamental solutions H and B, we only need to ensure the function satisfy the boundary condition.…”

Section: Solving Biharmonic Mapping Using Mfsmentioning

confidence: 99%

Biharmonic Volumetric Mapping Using Fundamental Solutions

et al. 2013

IEEE Trans. Visual. Comput. Graphics

View full text Add to dashboard Cite

Abstract-We propose a biharmonic model for cross-object volumetric mapping. This new computational model aims to facilitate the mapping of solid models with complicated geometry or heterogeneous inner structures. In order to solve cross-shape mapping between such models through divide and conquer, solid models can be decomposed into subparts upon which mappings is computed individually. The biharmonic volumetric mapping can be performed in each subregion separately. Unlike the widely used harmonic mapping which only allows C 0 continuity along the segmentation boundary interfaces, this biharmonic model can provide C 1 smoothness. We demonstrate the efficacy of our mapping framework on various geometric models with complex geometry (which are decomposed into subparts with simpler and solvable geometry) or heterogeneous interior structures (whose different material layers can be segmented and processed separately).

show abstract

Section: Solving Biharmonic Mapping Using Mfsmentioning

confidence: 99%

Biharmonic Volumetric Mapping Using Fundamental Solutions

et al. 2013

IEEE Trans. Visual. Comput. Graphics

View full text Add to dashboard Cite

show abstract

“…10,[20][21][22] Smyrlis defined appropriate discrete Sobolev norms and developed a weighted least-squares MFS algorithm. 23 The MFS was also applied to acoustic scattering problems.…”

Section: S Leementioning

confidence: 99%

Review: The Use of Equivalent Source Method in Computational Acoustics

Lee

2017

J. Comp. Acous.

View full text Add to dashboard Cite

This paper reviews the equivalent source method (ESM), an attractive alternative to the standard boundary element method (BEM). The ESM has been developed under different names: method of fundamental solutions, wave superposition method, equivalent source method, etc. However, regardless of the method name, the basic concept is very similar; that is to use auxiliary points called equivalent sources to reconstruct the acoustic pressure for radiation or scattering problems. The strength of the equivalent sources are then determined via various approaches such that the boundary conditions on the boundary surface are satisfied. This paper reviews several frequencydomain and time-domain ESMs. There are several distinct advantages in these types of methods: (1) the method is a meshless approach so that it is easy and simple to implement; (2) it does not have a numerical singularity problem that occurs in the BEM; (3) the number of equivalent sources can be fewer than the number of surface collocation points so that the matrix size is reduced and a fast computation is achieved for large problems. The main issue of the ESM is that there is no rule to find out the optimal number and position of equivalent sources. In addition, the ESM suffers from the numerical instability that is associated with the ill-conditioned matrix. Some guidelines have been suggested in terms of finding the number and position of the sources, and several numerical techniques have been developed to resolve the numerical instability. This paper reviews the common theories, numerical issues and challenges of the ESM, and it summarizes recent developments and applications of the ESM to aircraft noise.

show abstract

“…In such cases, the solution is approximated by linear combinations of fundamental solutions with singularities placed on a fictitious boundary lying outside the considered domain. MFS was successfully applied to resolve the potential flow problems by Johnston and Fairweather (1984), the Helmholtz problems by Tsai et al (2009), the biharmonic equation by Karageorghis and Fairweather (1987), the elliptic boundary value problems by Karageorghis and Fairweather (1998), the Poisson equation by Golberg (1995), the Stokes flow problems by Alves and Silvestre (2004), and the elasticity problems by Tsai (2007). A comprehensive review of MFS was presented by Golberg and Chen (1999).…”

Section: Introductionmentioning

confidence: 99%

Application of the method of fundamental solutions to the analysis of fully developed laminar flow and heat transfer

Mrozek

Mierzwiczak

2015

jtam

View full text Add to dashboard Cite

In this study, fully developed laminar flow and heat transfer in an internally longitudinally finned tube are investigated through application of the meshless method. The flow is assumed to be both hydrodynamically and thermally developed, with a uniform outside-the-wall temperature. The governing equations have been solved numerically by means of the method of fundamental solutions in combination with the method of particular solutions to obtain the velocity and temperature distributions. The advantage of the proposed approach is that it does not require mesh generation on the considered domain or its boundary, but uses only a cloud of arbitrarily located nodes. The results, comprising the friction factor as well as the Nusselt number, are presented for varied length values and fin numbers, as well as the thermal conductivity ratio between the tube and the flowing fluid. The results show that the heat transfer improves significantly if more fins are used.

show abstract

The method of fundamental solutions for the numerical solution of the biharmonic equation

Cited by 168 publications

References 28 publications

Biharmonic Volumetric Mapping Using Fundamental Solutions

Biharmonic Volumetric Mapping Using Fundamental Solutions

Review: The Use of Equivalent Source Method in Computational Acoustics

Application of the method of fundamental solutions to the analysis of fully developed laminar flow and heat transfer

Contact Info

Product

Resources

About