# Application of the Trefftz method to nonlinear potential problems

• Anita Uściłowska Poznań University of Technology

### Abstract

In this paper some types of nonlinear potential problems are discussed and some of these problems are solved by the Trefftz method. The attention is paid to Fundamental Solutions Method (FSM) supported by Radial Basis Functions (RBF) approximation. Application of FSM to nonlinear boundary problem requires certain modifications and special algorithms. In this paper two methods of treating the nonlinearity are proposed. One on them is Picard iteration. Due to some problems of application of this method the Homotopy Analysis Method (HAM) is implemented for nonlinear boundary-value problems. The results of numerical experiment are presented and discussed. The conclusion is that the method based on FSM for solving nonlinear boundary-value problem gives result with demanded accuracy.

### References

[1] M.R. Akella, G.R. Kotamraju. Trefftz indirect method applied to nonlinear potential problems. Engineering Analysis with Boundary Elements, 24: 459- 465, 2000.
[2] C.Y. Dong, S.H. Lo, Y.K. Cheung, K.Y. Lee. Anisotropic thin plate bending problems by Trefftz boundary collocation method. Engineering Analysis with Boundary Elements, 28: 1017- 1024, 2004.
[3] L. Jinmu, L. Shuyao. Geometrically nonlinear analysis of the shallow shell by the displacement-based boundary element formulation. Engineering Analysis with Boundary Elements, 18: 63- 70, 1996.
[4] J.J. Kasab, S.R. Karur, P.A. Ramachandran. Quasilinear boundary element method for nonlinear Poisson type problems. Engineering Analysis with Boundary Elements, 15: 277- 282, 1995.
[5] M. Kikuta, H. Togoh, M. Tanaka. Boundary element analysis of nonlinear transient heat conduction problems. Computer Methods in Applied Mechanics and Engineering, 62: 321- 329, 1987.
[6] S.-J. Liao. Boundary element method for general nonlinear differential operators. Engineering Analysis with Boundary Elements, 20(2): 91- 99, 1997.
[7] S.-J. Liao. General boundary element method for Poisson equation with spatially varying conductivity. Engineering Analysis with Boundary Elements, 21: 23- 38, 1998.
[8] S.-J . Liao. A direct boundary element approach for unsteady non-linear heat transfer problems. Engineering Analysis with Boundary Elements, 26: 55- 59, 2002.
[9] M. Naffa, H.J. AI-Gahtani. RBF-based meshless method for large deflection of thin plates. Engineering Analysis with Boundary Elements, 31: 311- 317,2007.
[10] M. Tanaka, T. Matsumoto, Z.-D . Zheng. Incremental analysis of finite deflection of elastic plates via boundarydomain- element method. Engineering Analysis with Boundary Elements, 17: 123- 131, 1996.
[11] M. Tanaka, T . Matsumoto, Z. Zheng. Application of the boundary-domain element method to the pre/postbuckling problem of von Karman plates. Engineering Analysis with Boundary Elements, 23: 399- 404, 1999.
[12] A. Uściłowska. Large deflection of a plate with Trefftz method. ICCES Special Symposium on Meshless Methods, 14- 16 June 2006, Dubrovnik, Croatia.
[13] W. Wang, X. Ji, M. Tanaka. A dual reciprocity boundary element approach for the problems of large deflection of thin elastic plates. Computational Mechanics, 26: 58-65, 2000.
[14] S.Q. Xu, N. Kamiya. A formulation and solution for boundary element analysis of inhomogeneous-nonlinear problem; the case involving derivatives of unknown function. Engineering Analysis with Boundary Elements, 23: 391- 397, 1999.
[15] T . Zhu, J. Zhang, S.N. Atluri. A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems. Engineering Analysis with Boundary Elements, 23: 375- 389, 1999.
Published
Jul 21, 2022
How to Cite
UŚCIŁOWSKA, Anita. Application of the Trefftz method to nonlinear potential problems. Computer Assisted Methods in Engineering and Science, [S.l.], v. 15, n. 3-4, p. 377-390, july 2022. ISSN 2956-5839. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/747>. Date accessed: 14 sep. 2024.
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