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Keywords: heat conduction, frequency domain, meshless, Kansa's method, radial basis functions, shape parameter.

Summary

The solution of transient heat conduction problems has been address using a large variety of methods. Direct solution of these problems in the time domain, and obtaining the solution first in a transformed domain and then performing an inverse transformation to synthesize the time response are two of the more common approaches.

More recently, a different strategy has been devised, in which a Fourier transformation is applied and then the solution is pursued in the frequency domain [1]. After solving the problem for each frequency, an inverse Fourier transformation permits synthesizing the response in the time domain. This approach has been efficiently applied together with boundary element method (BEM) and the method of fundamental solutions (MFS) formulations, but very little exists in the literature concerning its application together with other numerical methods. However, both methods require previous knowledge of fundamental solutions for the PDE, which are only known for specific situations.

Here, the authors present an implementation of Kansa's method for this type of problem, in which the solution of a given PDE is reproduced within a specific sub-domain as a linear combination of radial basis functions (RBF), and thus does not require the prior knowledge of fundamental solutions. Some of the RBFs used with Kansa's include a free (shape) parameter, whose definition is non-trivial and that greatly influences the accuracy of the computed responses. A recent work by Godinho et al. [2] addressed this question for the specific case of acoustic problems, considering a coupled approach between the BEM and Kansa's method.

In this paper, Kansa's method was used for calculation of transient heat diffusion problems, based on the solution of the problem in a transformed (frequency) domain. The implementation used an optimization scheme for the calculation of the free parameter of the RBFs, which was shown to compute different values as a function of the frequency and domain discretization. Comparison with results calculated using a BEM model revealed good responses computed by the proposed method.

Evolution of the temperature within a domain, considering an initial non-uniform distribution of temperatures, was also computed. Comparison with a standard time-marching algorithm also revealed the good accuracy of the method. One should note that, for that case, the response is first calculated in the frequency domain and then transformed to time domain by means of an inverse fast Fourier transform.

References

1: L. Godinho, A. Tadeu, N. Simões, "Study of transient heat conduction in 2.5D domains using the Boundary Element Method", Eng. Anal. Bound. Elmts., 28(6), 593-606, 2004. doi:10.1016/j.enganabound.2003.09.002
2: L. Godinho, A. Tadeu, "Acoustic analysis of heterogeneous domains coupling the BEM with Kansa's method", Eng. Anal. Bound. Elmts., 36(6), 1014-1026, 2012. doi:10.1016/j.enganabound.2011.12.017

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 100 PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping Paper 102 A Frequency Domain Formulation of Kansa's Method to Simulate Transient Heat Conduction L. Godinho¹ and F.G. Branco² ¹Research Center in Construction Sciences (CICC), ²Institute for Systems Engineering and Computers at Coimbra (INESC Coimbra), Department of Civil Engineering, University of Coimbra, Portugal doi:10.4203/ccp.100.102 purchase the full-text of this paper Full Bibliographic Reference for this paper L. Godinho, F.G. Branco, "A Frequency Domain Formulation of Kansa's Method to Simulate Transient Heat Conduction", in B.H.V. Topping, (Editor), "Proceedings of the Eighth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 102, 2012. doi:10.4203/ccp.100.102 Keywords: heat conduction, frequency domain, meshless, Kansa's method, radial basis functions, shape parameter. Summary The solution of transient heat conduction problems has been address using a large variety of methods. Direct solution of these problems in the time domain, and obtaining the solution first in a transformed domain and then performing an inverse transformation to synthesize the time response are two of the more common approaches. More recently, a different strategy has been devised, in which a Fourier transformation is applied and then the solution is pursued in the frequency domain [1]. After solving the problem for each frequency, an inverse Fourier transformation permits synthesizing the response in the time domain. This approach has been efficiently applied together with boundary element method (BEM) and the method of fundamental solutions (MFS) formulations, but very little exists in the literature concerning its application together with other numerical methods. However, both methods require previous knowledge of fundamental solutions for the PDE, which are only known for specific situations. Here, the authors present an implementation of Kansa's method for this type of problem, in which the solution of a given PDE is reproduced within a specific sub-domain as a linear combination of radial basis functions (RBF), and thus does not require the prior knowledge of fundamental solutions. Some of the RBFs used with Kansa's include a free (shape) parameter, whose definition is non-trivial and that greatly influences the accuracy of the computed responses. A recent work by Godinho et al. [2] addressed this question for the specific case of acoustic problems, considering a coupled approach between the BEM and Kansa's method. In this paper, Kansa's method was used for calculation of transient heat diffusion problems, based on the solution of the problem in a transformed (frequency) domain. The implementation used an optimization scheme for the calculation of the free parameter of the RBFs, which was shown to compute different values as a function of the frequency and domain discretization. Comparison with results calculated using a BEM model revealed good responses computed by the proposed method. Evolution of the temperature within a domain, considering an initial non-uniform distribution of temperatures, was also computed. Comparison with a standard time-marching algorithm also revealed the good accuracy of the method. One should note that, for that case, the response is first calculated in the frequency domain and then transformed to time domain by means of an inverse fast Fourier transform. References 1 L. Godinho, A. Tadeu, N. Simões, "Study of transient heat conduction in 2.5D domains using the Boundary Element Method", Eng. Anal. Bound. Elmts., 28(6), 593-606, 2004. doi:10.1016/j.enganabound.2003.09.002 2 L. Godinho, A. Tadeu, "Acoustic analysis of heterogeneous domains coupling the BEM with Kansa's method", Eng. Anal. Bound. Elmts., 36(6), 1014-1026, 2012. doi:10.1016/j.enganabound.2011.12.017 purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £50 +P&P)
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