Engineering Transactions, 55, 2, pp. 129–153, 2007
10.24423/engtrans.224.2007

Wave Polynomials in Elasticity Problems

A. MACIĄG
Department of Mathematics, Faculty of Management and Computer Modelling, Kielce University of Technology, Kielce
Poland

The paper demonstrates a new technique of obtaining the approximate solution of the two- and three-dimensional elasticity problems. The system of equations of elasticity can be converted to the system of wave equation. In this case, as solving functions (Trefftz functions), the so-called wave polynomials can be used. The presented method is useful for a finite body of a certain shape. Then the obtained solutions are coupled through initial and boundary conditions. Recurrent formulas for the two- and three-dimensional wave polynomials and their derivatives are obtained. The methodology for solution of systems of partial differential equations with common initial and boundary conditions by means of solving functions is presented. The advantage of using the method of solving functions is that the solution exactly satisfies the given equation (or system of equations). Some examples are included.
Keywords: elasticity; Trefftz method; wave equation; wave polynomials
Full Text: PDF
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

References

E. Trefftz, Ein Gegenstück zum Ritzschen Verfahren, [in:], Proceedings of the 2-nd International Congress of Applied Mechanics. Zurich 131–137, 1926.

A.P. Zieliński and I.Herrera, Trefftz method: fitting boundary conditions. Int. J. Num. Meth. Eng., 24, 871–891, 1987.

P.C. Rosenbloom and D.V. Widder, Expansion in terms of heat polynomials and associated functions, Trans. Am. Math. Soc., 92, 220–266, 1956.

H. Yano, S. Fukutani and A. Kieda, A boundary residual method with heat polynomials for solving unsteady heat conduction problems, Franklin Inst, 316, 291–298, 1983.

S. Futakiewicz and L. Hożejowski, Heat polynomials method in the n-dimensional direct and inverse heat conduction problems, [in:] A.J. Nowak, C.A. Brebbia, R. Bielecki and M. Zerroukat [Eds.], Advanced Computational Method in Heat Transfer V. Southampton, UK and Boston, USA: Computational Mechanics Publications, 103–112, 1998.

L. Hożejowski, Heat polynomials and their application for solving direkt and inverse heat condutions problems (PhD-Thesis) [in Polish], Kielce University of Technology, pp. 115, 1999.

S. Futakiewicz and L. Hożejowski, 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 and M. Zerroukat [Eds.], Advanced Computational Method in Heat Transfer V. Southampton UK and Boston USA: Computational Mechanics Publications 71–80, 1998.

S. Futakiewicz, K. Grysa and L. Hożejowski, On a problem of boundary temperature identification in a cylindrical layer, [in:] B.T. Maruszewski, W. Muschik and A. Radowicz [Eds.], Proceedings of the International Symposium on Trends in Continuum Physics, World Scientific Publishing, Singapore, New Jersey, London, Hong Kong 119–125, 1999.

S. Futakiewicz, Heat functions method for solving direct and inverse heat condutions problems (PhD-Thesis) [in Polish], Poznań University of Technology 120pp, 1999.

M.J. Ciałkowski, S. Futakiewicz and L. Hożejowski, Method of heat polynomials in solving the inverse heat conduction problems, ZAMM, 79, 709–710, 1999.

M.J. Ciałkowski, S. Futakiewicz and L. Hożejowski, Heat polynomials applied to direct and inverse heat conduction problems, [in:] B.T. Maruszewski, W. Muschik and A. Radowicz [Eds.], Proceedings of the International Symposium on Trends in Continuum Physics, World Scientific Publishing, Singapore, New Jersey, London, Hong Kong 79–88, 1999.

M.J. Ciałkowski, Solution of inverse heat conduction problem with use of a new type of finite element base functions, [in:] B.T. Maruszewski, W. Muschik and A.Radowicz [Eds.], Proceedings of the International Symposium on Trends in Continuum Physics, Singapore, New Jersey, London, Hong Kong: World Scientific Publishing, 64–78, 1999.

M.J. Ciałkowski and A. Frąckowiak, Heat functions and their application for solving heat transfer and mechanical problems [in Polish], University of Technology Publishers pp. 360, Poznań 2000.

A. Maciąg and J. Wauer, Solution of the two-dimensional wave equation by using wave polynomials, J. Engrg. Math., 51, 4, 339–350, 2005.

A. Maciąg, J. Wauer, Wave polynomials for solving different types of two-dimensional wave equations, Computer Assisted Mechanics and Engineering Sciences, 12, 87–102, 2005.

A. Maciąg Solution of the three-dimensional wave polynomials, Mathematical Problems in Engineering, 5, 583–598, 2005.

A. Maciąg, Solution of the three-dimensional wave equation by using wave polynomials, PAMM – Proc. Math. Mech., 4, 706–707, 2004.

W. Nowacki, Thermoelascity, Pergamon Press, Oxford–Warsaw 1962.

M.J. Ciałkowski, M. Jarosławski, Application of symbolic calculations in generating the solution of the wave equation [in Polish], Zeszyty Naukowe Politechniki Poznańskiej nr 56, Maszyny Robocze i Transport, 2003.

M.J. Ciałkowski, A. Frąckowiak, Application of symbolic operations to the determination of Trefftz functions for a heat flow wave equation [in Polish], Zeszyty Naukowe Politechniki Poznańskiej nr 57, Maszyny Robocze i Transport, 2004.

M.J. Ciałkowski, Thermal and related functions used in solving certain problems of mechanics. Part I - Solution of certain differential equations by means of inverse operations [in Polish], Studia i materiały. Technika 3. Uniwersytet Zielonogórski, 7–70, 2003.

M.J. Ciałkowski, A. Frąckowiak, Thermal and related functions in solving certain problems of mechanics. Part II – Effective determination of inverse operations applied to harmonic functions [in Polish], Studia i materiały. Technika 3. Uniwersytet Zielonogórski, 71–98, 2003.




DOI: 10.24423/engtrans.224.2007