Energetic approach to direct and inverse heat conduction problems with Trefftz functions used in FEM
Abstract
In the paper the stationary 2D inverse heat conduction problems are considered. To obtain an approximate solution of the problems three variants of the FEM with harmonic polynomials (Trefftz functions for Laplace equation) as base functions were used: the continuous FEMT, the non-continuous FEMT and the nodeless FEMT. In order to ensure physical sense of the approximate solution, one of the aforementioned physical aspects is taken into account as a penalty term in the functional, which is to be minimized in order to solve the problem. Three kinds of physical aspects that can smooth the solution were used in the work. The first is the minimization of heat flux jump between the elements, the second is the minimization of the defect of energy dissipation and third is the minimization of the intensity of numerical entropy production. The quality of the approximate solutions was verified on two test examples. The method was applied to solve inverse problem of stationary heat transfer in a rib.
Keywords
References
[1] M.J. Ciałkowski. Solution of inverse heat conduction problem with use new type of finite element base functions. In: B.T. Maruszewski, W. Muschik, A. Radowicz, eds., Proceedings of the International Symposium on Trends in Continuum Physics, 64-78. World Scientific Publishing, Singapore, New Jersey, London, Hong Kong, 1999.[2] M.J. Ciałkowski, A. Frąckowiak. Heat Functions and Their Application for Solving Heat Transfer and Mechanical Problems (in Polish). Poznan University of Technology, 2000.
[3] M.J. Ciałkowski, A. Frąckowiak, K. Grysa. Solution of a stationary inverse heat conduction problems by means of' Trefftz non-continuous method. Int. J. Heat Mass Transfer, 50: 2170- 2181, 2007.
[4] M.J. Ciałkowski, A. Frąckowiak, K. Grysa. Physical regularization for inverse problems of stationary heat conduction. J. Inv. Ill-Posed Problems, 15: 1- 18,2007.
[5] M.J. Ciałkowski, S. Futakiewicz, L. Hożejowski. Heat polynomials applied to direct and inverse heat conduction problems. In: B.T. Maruszewski, W. Muschik, A. Radowicz, eds., Proc. of the International Symposium on Trends in Continuum Physics, 79- 88. World Scientific Publ., Singapore-New Jersey- London- Hong Kong, 1999.
[6] M.J. Ciałkowski, S. Futakiewicz, L. Hożejowski. Method of heat polynomials in solving the inverse heat conduction problems. ZAMM - Z. Angew. Math. Mech., 79: 709- 710, 1999.
[7] S. Futakiewicz. Heat Functions Method for Solving Direct and Inverse Heat Conduction Problems (in Polish). PhD Thesis, Poznan University of Technology, 1999.
[8] S. Futakiewicz, K. Grysa, L. Hożejowski. On a problem of boundary temperature identification in a cylindrical layer. In:· B.T. Maruszewski, W. Muschik, A. Radowicz, eds., Proc. of the International Symposium on Trends in Continuum Physics, 119-125. World Scientific Publ., Singapore-New Jersey- London- Hong Kong, 1999.
[9] S. Futakiewicz, L. Hożejowski. Heat polynomials method in the n-dimensional direct and inverse heat conduction problems. In: A.J. Nowak, C.A. Brebbia, R. Bielecki, M. Zerroukat, eds., Advanced Computational Method in Heat Transfer V, 103-112. Computational Mechanics Publications, Southampton, UK and Boston, USA, 1998.
[10] S. Futakiewicz, L. Hozejowski. Heat polynomials in solving the direct and inverse heat conduction problems in a cylindrical system of coordinates. In: A.J. Nowak, C.A. Brebbia, R. Bielecki, M. Zerroukat, eds., Advanced Computational Method in Heat Transfer V, 71-80. Computational Mechanics Publications, Southampton, UK and Boston, USA, 1998.
[11] L. Hożejowski. Heat Polynomials and their Application for Solving Direct and Inverse Heat Conduction Problems (in Polish). PhD Thesis, Kielce University of Technology, 1999.
[12] A. Maciąg. Solution of the Three-dimensional Wave Equation by Using Wave Polynomials. PAMM - Proc. Math. Mech., 4: 706-707, 2004.
[13] A. Maciąg. Solution of the Three-dimensional Wave Polynomials. Mathematical Problems in Engineering, 5: 583-598, 2005.
[14] A. Maciąg. Wave polynomials in elasticity problems. Engrg. Trans., 55(2): 129- 153, 2007.
[15] A. Maciąg, T. Orzechowski, D. Sznajder. Analysis of the heat transfer from fin longitudinal surface. PAMM - Proc. Appl. Math. Mech., 2(1): 366- 367, 2003.
[16] A. Maciąg, J. Wauer. Solution of the two-dimensional wave equation by using wave polynomials. J. Engrg. Math., 51(4): 339- 350, 2005.
[17] A. Maciąg A., J. Wauer. Wave Polynomials for Solving Different Types of Two-Dimensional Wave Equations. Computer Assisted Mechanics and Engineering Sciences, 12: 87- 102,2005.
[18] P.C. Rosenbloom, D.V. Widder. Expansion in terms of heat polynomials and associated functions. Trans. Amer. Math. Soc., 92: 220-266, 1959.
[19] E. Trefftz. Ein Gegenstuck zum Ritzschen Verfahren. Proc. 2nd International Congress of Applied Mechanics: 131-137, Zurich, 1926.
[20] A.P. Zieliński, 1. Herrera. Trefftz method: fitting boundary conditions. Int. J. Numer. Methods Engrg., 24(5): 871-891, 1987.
Published
Jul 19, 2022
How to Cite
GRYSA, Krzysztof; LEŚNIEWSKA, Renata; MACIĄG, Artur.
Energetic approach to direct and inverse heat conduction problems with Trefftz functions used in FEM.
Computer Assisted Methods in Engineering and Science, [S.l.], v. 15, n. 3-4, p. 171-182, july 2022.
ISSN 2956-5839.
Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/726>. Date accessed: 14 nov. 2024.
Issue
Section
Articles
