The method of fundamental solutions for free surface Stefan problems

Chantasiriwan, Somchart; Johansson, Björn; Lesnic, D.

doi:10.1016/j.enganabound.2008.08.010

Cited by 34 publications

(27 citation statements)

References 31 publications

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“…Recently, in [15], an MFS for the time-dependent linear heat equation in one spatial dimension was proposed and investigated. This method was extended to free surface Stefan problems in [5] and to heat conduction in one-dimensional layered materials in [16]. Encouraged by these results, in this paper we extend the approach considered in [15] to heat conduction in two-dimensional bodies.…”

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confidence: 99%

A method of fundamental solutions for two-dimensional heat conduction

Johansson

Lesnic

Reeve

2011

International Journal of Computer Mathematics

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International audienceWe investigate an application of the method of fundamental solutions (MFS) to heat conduction in two-dimensional bodies, where the thermal diffusivity is piecewise constant. We extend a recent MFS for one-dimensional heat conduction with the sources placed outside the space domain of interest (Engng. Anal. Boundary Elements 32, 697-703, 2008), to the two-dimensional setting. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be obtained efficiently with small computational cost

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confidence: 99%

A method of fundamental solutions for two-dimensional heat conduction

Johansson

Lesnic

Reeve

2011

International Journal of Computer Mathematics

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“…In (5) or (6) the function h is usually taken to be uniform, e.g. zero, such that D(t) represents a rigid inclusion for the homogeneous Dirichlet boundary condition (5) and a cavity for the homogeneous Neumann boundary condition (6). Also the Neumann boundary condition (4) may be partially limited to a portion…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…in the MFS expansion (7) are determined by collocating the initial condition (2) and either of the boundary conditions (3) or (4), and (5) or (6). In the inverse problem, the unknown coefficients (c …”

Section: Mathematical Formulationmentioning

confidence: 99%

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A meshless method for solving a two‐dimensional transient inverse geometric problem

Dawson

Borman

Hammond

et al. 2013

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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Purpose -The purpose of this paper is to apply the meshless Method of Fundamental Solutions (MFS) to the two-dimensional time-dependent heat equation in order to locate an unknown internal inclusion. Design/methodology/approach -The above problem is formulated as an inverse geometric problem, using non-invasive Dirichlet and Neumann exterior boundary data to find the internal boundary using a non-linear least-squares minimisation approach. The solver will be tested when locating a variety of internal formations. Findings -The method implemented here was proven to be both stable and reasonably accurate when data was contaminated with random noise. Research limitations/implications -Due to limited computational time, spatial resolution of internal boundaries may be lower than some similar case investigations. Practical Implications -This research will have practical implications to the modelling and monitoring of crystalline deposit formations within the nuclear industry, allowing development of future designs. Originality/value -Similar work has been completed in regards to the steady state heat equation, however to the best of the authors knowledge no previous work has been completed on a time-dependent inverse inclusion problem relating to the heat equation, using the MFS. Preliminary results presented here will have value for possible future design and monitoring within the nuclear industry.

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“…Instead, following the stationary case, linear combinations of the fundamental solution of the heat equation were used. This method has then been applied for various other direct and inverse heat problems, see, for example, [6,7].…”

Section: Introductionmentioning

confidence: 99%

Properties of a method of fundamental solutions for the parabolic heat equation

Johansson

2017

Applied Mathematics Letters

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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractWe show that a set of fundamental solutions to the parabolic heat equation, with each element in the set corresponding to a point source located on a given surface with the number of source points being dense on this surface, constitute a linearly independent and dense set with respect to the standard inner product of square integrable functions, both on lateral-and time-boundaries. This result leads naturally to a method of numerically approximating solutions to the parabolic heat equation denoted a method of fundamental solutions (MFS). A discussion around convergence of such an approximation is included.

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The method of fundamental solutions for free surface Stefan problems

Cited by 34 publications

References 31 publications

A method of fundamental solutions for two-dimensional heat conduction

A method of fundamental solutions for two-dimensional heat conduction

A meshless method for solving a two‐dimensional transient inverse geometric problem

Properties of a method of fundamental solutions for the parabolic heat equation

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