2D wave polynomials as base functions in modified FEM
Abstract
The paper presents solutions of a two-dimensional wave equation by using Trefftz functions. Two ways of obtaining different forms of these functions are shown. The first one is based on a generating function for the wave equation and leads to recurrent formulas for functions and their derivatives. The second one is based on a Taylor series expansion and additionally uses the inverse Laplace operator. Obtained wave functions can be used to solve the wave equation in the whole considered domain or can be used as base functions in FEM. For solving the problem three kinds of modified FEM are used: nodeless, continuous and discontinuous FEM. In order to compare the results obtained with the use of the aforementioned methods, a problem of membrane vibrations has been considered.
Keywords
Trefftz functions, wave functions, inverse operations, FEM,References
[1] M. Ciałkowski, A. Frąckowiak. Funkcje cieplne i pokrewne w rozwiązywaniu wybranych równań mechaniki, Część I. Rozwiązywanie niektórych rownań cząstkowych za pomocą operacji odwrotnych. Współczesne Problemy Techniki. Studia i Materiały, 3: pp. 7- 69. Wydawnictwo Uniwersytetu Zielonogórskiego, Zielona Góra, 2003.[2] M. Ciałkowski, A. Frąckowiak. Heat functions and their application to solving heat conduction and mechanical problems (in Polish). Wydawnictwo Politechniki Poznańskiej, Poznań, 2000.
[3] 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.
[4] M.J. Ciałkowski, A. Frąckowiak, K. Grysa. Solution of stationary inverse heat conduction problems by means of Trefftz non-continuous method. Int. J. Heat Mass Transfer, 50: 2170- 2181, 2007.
[5] S. Futakiewicz. The Heat Functions Method for Direct and Inverse Problem of Heat Conduction (in Polish). Doctoral thesis, Poznan, 1999.
[6] L. Hożejowski. Heat polynomials and their applications in direct and inverse problem of heat conduction, Doctoral thesis (in Polish), Kielce, 1999.
[7] A. Maciąg. Two-dimensional wave polynomials as base functions for continuity and discontinuity finite elements method (in Polish) . In: J. Taler, ed., Współczesne Technologie i Urządzenia Energetyczne, pp. 371-381, Kraków, 2007.
[8] A. Maciąg, J . Wauer. Solution of the two-dimensional wave equation by using wave polynomials. J. Engrg. Math., 51(4): 339-350,2005.
[9] B. Maciejewska. Application of the modified method of finite elements for identification of temperature of a body heated with a moving heat source. J. Theor. Appl. Math., 42(4): 771-787,2004.
[10] E.B. Magrab. Vibrations of Elastic Structural Members. Sijthoff and Noordhoff, Maryland, USA, 1979.
[11] M. Sokała. Analytical and Numerical Method of Solving Heat Conduction Problems with the Use of Heat Functions and Inverse Operations (in Polish). Doctoral thesis, Poznan, 2004.
[12] M. Sokała. Solutions of two-dimensional wave equation by using some form of Trefftz functions. In: B.T. Maruszewski, W. Muschik, A. Radowicz, eds., Proceedings of the International Symposium on Trends in Continuum Physics TRECOP'07, Lviv/ Bryukhovichi, Ukraine, September 16-20, 2007, pp. 70- 71. Lviv, 2007.
[13] M. Sokała. Solutions of two-dimensional wave equation by using some form of Trefftz functions. Comput. Meth. Sci. Technol., accepted for publication.
[14] P.C. Rosenbloom, D.V. Wilder. Expansion in terms of heat polynomials and associated functions. Trans. Am. Soc., 92: 220-266, 1956.
[15] A.P. Zielinski, I. Herrera. Trefftz method: fitting boundary conditions. Int. J. Num. Meth. Engrg., 24: 871-891, 1987.
Published
Jul 20, 2022
How to Cite
MACIĄG, Artur; MACIEJEWSKA, Beata; SOKAŁA, Małgorzata.
2D wave polynomials as base functions in modified FEM.
Computer Assisted Methods in Engineering and Science, [S.l.], v. 15, n. 3-4, p. 265-278, july 2022.
ISSN 2956-5839.
Available at: <https://cames.ippt.gov.pl/index.php/cames/article/view/735>. Date accessed: 23 dec. 2024.
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