An extended method of time-dependent fundamental solutions for inhomogeneous heat conduction

Dong, C.F.

doi:10.1016/j.enganabound.2008.09.006

Cited by 6 publications

(4 citation statements)

References 16 publications

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“…where r ij is the value of r i at the point (x j , y j ) and ρ ij is the value of ρ i at the point (x j , y j ). Once the coefficients {η i } N d i=1 and {θ i } Ns i=1 are determined by solving system (22), the approximate solution φ can be obtained from Equation (13). One notices that the fictitious boundary N s and the boundary domain points N b can be chosen arbitary.…”

Section: The Fundamental Solution Methods For the Spatial Problemmentioning

confidence: 99%

“…Method of fundamental solutions has predominantly been applied to stationary problems governed by elliptic PDEs such as Laplace, Helmholtz, biharmonic, Lamé and Stokes equations [2][3][4][5][6][7][8][9]. However, in recent years some different versions of MFS used for parabolic heat-transfer problems have appeared and have been extensively treated in the literature [10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

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Combining MFS and PGD methods to solve transient heat equation

Kpogan

Tri

Sogah

et al. 2017

Numerical Methods Partial

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We propose in this article a numerical algorithm based on the combination of the method of fundamental solutions (MFS) and the proper generalized decomposition technique (PGD) to solve time-dependent heat equation. The MFS is considered as a truly meshless technique well adapted for a wide range of physical problems and the PGD approach can be considered as a reduction technique based on the separated representation of the variable functions. The proposed study relates to a separation between the spatial and temporal coordinates. To show the effectiveness of the proposed algorithm, several examples are presented and compared to the reference results.

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Section: The Fundamental Solution Methods For the Spatial Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Combining MFS and PGD methods to solve transient heat equation

Kpogan

Tri

Sogah

et al. 2017

Numerical Methods Partial

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“…Alternatively, in the MFS and BEM, it is possible to evaluate time-dependent basis functions that can directly be used for constructing the solution series. This technique was first implemented for simulating homogeneous diffusion problems [40] and later expanded to inhomogeneous heat transfer [41] and unsteady Stokes equation [42]. A similar approach is also used in the EBFs method by implementing time-dependent bases for approximating the field [43].…”

Section: Transient Green's Discrete Transformation Meshfree Methodsmentioning

confidence: 99%

A Green's discrete transformation meshfree method for simulating transient diffusion problems

Mai

Soghrati

Buchheit

2016

Numerical Meth Engineering

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SUMMARYThis manuscript presents the formulation and application of the Green's discrete transformation method (GDTM) for the meshfree simulation of transient diffusion problems, including those with moving boundaries. The GDTM implements a linear combination of time-dependent Green's basis functions defined on a set of source points to approximate the field in the form of a solution series. A discrete transformation is implemented to evaluate unknown coefficients of this series, which eliminates the need to use time integration schemes. We will study the optimal number and location of the GDTM source points that yield the highest level of accuracy, while maintaining a manageable condition number for the resulting linear system of equations. The optimal values of these parameters, which are inherently independent of the domain geometry, are determined such that the basis functions have appropriate features for approximating the field. A comprehensive convergence study is presented to show the precision and convergence rate of the GDTM for modeling various diffusion problems. We also demonstrate the application of this method for simulating three diffusion problems with complex and evolving morphologies: heat transfer in a turbine blade, thermal response of a porous material, and localized (pitting) corrosion in stainless steel.

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“…Regularization is needed to compute the source strengths as the matrix solution for source strengths is ill-posed, and the level of ill-posedness depends on the distance between the source points and the boundary. Dong [25] extended the method of fundamental solutions to the heat equation to irregularly shaped two-dimensional domains with internal sources by placing additional source points inside the domain. A weakness of the method is that the results are sensitive to the source-point locations, and the source strength calculation is ill-posed, requiring regularization for numerical stability.…”

Section: Introductionmentioning

confidence: 99%

Semi-Analytical Source Method for Reaction–Diffusion Problems

Cole

Çetin

Demi̇rel

2018

Journal of Heat Transfer

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Estimation of thermal properties, diffusion properties, or chemical–reaction rates from transient data requires that a model is available that is physically meaningful and suitably precise. The model must also produce numerical values rapidly enough to accommodate iterative regression, inverse methods, or other estimation procedures during which the model is evaluated again and again. Applications that motivate the present work include process control of microreactors, measurement of diffusion properties in microfuel cells, and measurement of reaction kinetics in biological systems. This study introduces a solution method for nonisothermal reaction–diffusion (RD) problems that provides numerical results at high precision and low computation time, especially for calculations of a repetitive nature. Here, the coupled heat and mass balance equations are solved by treating the coupling terms as source terms, so that the solution for concentration and temperature may be cast as integral equations using Green's functions (GF). This new method requires far fewer discretization elements in space and time than fully numeric methods at comparable accuracy. The method is validated by comparison with a benchmark heat transfer solution and a commercial code. Results are presented for a first-order chemical reaction that represents synthesis of vinyl chloride.

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An extended method of time-dependent fundamental solutions for inhomogeneous heat conduction

Cited by 6 publications

References 16 publications

Combining MFS and PGD methods to solve transient heat equation

Combining MFS and PGD methods to solve transient heat equation

A Green's discrete transformation meshfree method for simulating transient diffusion problems

Semi-Analytical Source Method for Reaction–Diffusion Problems

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