Coupling techniques of Trefftz methods

• Autorzy: Huang H. T. , Li Z. C. , Cheng A. H. D.
• Konferencja: International Conference on the Trefftz Method (5 ; 2008 ; Leuven, Belgium)
• Języki publikacji: EN

Abstrakty

EN

The Trefftz method pioneered by Trefftz in 1926 is described as follows: The particular solutions or the fundamental solutions are chosen, a linear combination of those functions is regarded as an approximate solution of partial differential equations (PDEs), and their expansion coefficients are sought by satisfying the interior and exterior boundary conditions. When the solution domain is not rectangular or sectors, the piecewise particular solutions may be chosen in different subdomains, and some coupling techniques must be employed along their interior boundary conditions. In Li et al. [1], the collocation method is used for the Trefftz method, to lead to the collocation Trefftz method (i.e., the indirect Trefftz method). In this paper, we will also discuss other four coupling techniques: (1) the simplified hybrid techniques, (2) the hybrid plus penalty techniques, (3) the Lagrange multiplier techniques for the direct Trefftz method, and (4) the hybrid Trefftz method of Jirousek [2] and Qin [3]. Error bounds are derived in detail for these four couplings, to achieve exponential convergence rates. Numerical experiments are carried out, and comparisons are also made.

Słowa kluczowe

EN

Trefftz method; computer analysis

PL

metoda Trefftza; analiza komputerowa

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Computer Assisted Mechanics and Engineering Sciences
• Rocznik: 2008
• Tom: Vol. 15, No. 3/4
• Strony: 183--213
• Opis fizyczny: Bibliogr. 75 poz., rys., tab.

Twórcy

autor

Huang H. T.

autor

Li Z. C.

autor

Cheng A. H. D.

• Departament of Applied Mechanics and Departament of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan 80424

Bibliografia

• [1] M. S. Abou-Dina. Implement at ion of Trefftz method for the solution of some elliptic boundary value problems. Math. Comp., 127: 125 147, 2002.
• [2] I. Babuška, The finite element method wit h Lagrangian multipliers. Numer. Math., 20: 179-192, 1973.
• [3] I. Babuska, A.K. Aziz. Survey lectures on the mathematical foundations of the finite element method. In: id.. Tin Mathematical foundations of the Finite Element with Applications to Partial Differential Academic Press, Inc., pp. 3-358, 1971.
• [4] I. Babuska, J.T. Oden. J.K. Lee. Mixed-hybrid finite element approximations of second-order elliptic boundaryvalue problems. Part 2 - Weak-hybrid methods. Comput. Meth. Appl. Mech. Engrg., 14: 1-22, 1978.
• [5| H.J.C. Barbosa, T.J.R. Hughes. Boundary Lagrange multipliers in finite element methods: error analysis in natural norms. Numer. Math., 62: 1-15, 1992.
• [6] J.H. Bramble. The Lagrange multiplier methods for Dirichlet's problem. Math. Cornp., 37(155): 1-12, 1981.
• [7] F. Brezzi, M. Fortin. Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
• [8] Y. K. Cheung, W. G.. Jin, O. C. Zienkiewicz. Solution of Helmholtz equation by Trefftz methods. Inter. J. Numer. 32(1): 63 78, 1991.
• [9] C.Y. Dong, S II Lo, Y.K. Cheung, K.Y. Lee. Anisotropic thin plate bending problem by Trefftz boundary collocation method. Engrg. Anal. Bound. Elem., 28: 1017-1024,2004.
• |10] J.S. Domingues. A. Portela, P.M.S.T. de Castro. Trefftz boundary element method applied to fracture mechanics. Engrg. Fract. Mech. 64: 67-85, 1999.
• [11] Hybrid finite element methods. SIAM Review, 18(3): 460-485, 1976.
• [12] J. A. T. de Freitas. Hybrid-Trefftz displacement and stress elements for elastodynamic analysis in the frequency domain. Comput. Assisted Mech. Engrg. Sci., 4: 345-368, 1997.
• [13] J.A.T. de Freitas. Formulation of elastostatic hybrid-Trefftz stress elements. Comput. Meth. Appl. Mech. Engrg., 153 : 127 - 151, 1998.
• [14] J. A. T. de Freitas. Formulation and implementation of hybrid-Trefftz stress elements for cohesive fracture. Proceedings of the Trefftz, 5th International Workshop on Trefftz methods, pp. 105-120. Leuven, March .'51 April 2,
• [15] J. A. T.de Freitas. Z..-Y. ,Ji. Hybrid-Trefftz equilibrium model for crack problems. Inter. ./. Numer. Meth. Engrg., 39
• [16] J.A.T. de Freitas, V.M.A. Leitao. A boundary integral Trefftz formulation with symmetric collocation. Cornp. .23, 2003.
• [17] I. Herrera. Boundary Method: An Algebraic Theory. Pitman, Boston, 1984.
• |18] I. Herrera. Trefftz-Herrera domain decomposition. Adv. Engrg. Softw.. 24: 1.'. 50. pi
• [19] I. Herrera. Trefftz method: A general theory. Numer. Meth. PDEs, 16(6): 561 580, 2000.
• [20] I. Herrera. M. I)i;iz. Indirecl methods of collocation: Trelli/ Herrera collocation. Numer. Meth. I'DEs. 15:709-738, 1999.
• [21] I. Herrera, J. Solano. A non-overlapping TH-domain decomposition. Adv. Engrg. Softw., 28(4): 223-229, 1997.
• [22] I. Herrera, M. Diaz. General theory of domain decomposition: Indirect methods. Numer. Meth. PDEs. 18:296-322, 2002.
• [23] J. Jirousek. Basis for development of large finite elements locally satisfying all field equal ions. Comput. Meth. Appl. Mech. Engrg., 14: 65-92, 1978.
• [24] J. Jirousek, N. Leon. A powerful finite element for plate blending. Comput. Meth. Appl. Mech. Engrg., 12: 77-96, 1977.
• [25] J. Jirousek, Q.H. Qin. Application of hybrid-Trefftz element approach to transient heat conduction analysis. Comp. Struct, 58(1): 195-201, 1996.
• [26] J. Jirousek, A. Venkatesh. A simple stress error estimator for hybrid-Trefftz p-version elements. Inter. J. Numer. Meth. Engrg., 28: 211-236, 1989.
• [27] J. Jirousek, A. Venkatesh. Hybrid Trefftz plane elasticity elements with p-method capabilities. Inter. J. Numer. Meth. Engrg., 35: 1443-1472, 1992.
• [28] J. Jirousek, A. Wroblewski. T-elements: a finite element approach with advantages of boundary solut ion methods. Adv. Engrg. Softw., 24: 71-88, 1995.
• [29] J. Jirousek, A. Wroblewski. T-elements: State of the art and future trends. Arch. Comput. Meth. Engrg. ,323-434,1996.
• [30] W.G. Jin, Y.K. Cheung. Trefftz direcl method. Adv. Engrg. Softw., 24: 65 69, 1"
• [31] Kamiya, E. Kita. Trefftz method 70 years. Adv. Engrg. Softw., 24(1 3): I 1995.
• [32] E. Kita. N. Kamiya. Trefftz method: An overview. Adv. Engrg. Softw., 24: 3 12, L995.
• [33] E. Kita, N. Kamiya, T. Iio. Application of a direct Trefftz method with domain decomposition to 2D potential problems. Engrg. Anal. Bound. Elem., 23(7): 539-548, 1999.
• [34] P. Lee. A Lagrange multiplier method for the interface equations from electromagnetic applications. SIAM J.Numer. Anal., 30(2): 478-506, 1993.
• [35] V.M.A. Leitao. Application of multi-region Trefftz-collocation to fracture mechanics. Engrg. Anal. Bound. 22(3): 251-256, 1998.
• [36] Z.C. Li. Lagrange multipliers and other coupling techniques for combined methods and other coupling techniques for combined methods for elliptic equations. Inter. J. Information, 1(2): 5-21, 1998.
• [37] Z.C. Li. Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities. Kluwer Academic Publishers, Dordrecht, Boston, 1998.
• [38] Z.C. Li. Global superconvergence of simplified hybrid combinations for elliptic equations with singularities I. Basic theory. Computing, 65: 27-44, 2000.
• |39| Z.C. Li. Error Analysis for Hybrid Trefftz Methods Coupling Neumann Conditions. Technical report. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan. 2008.
• [40| Z.C. Li, T.D. Bui. Generalized hybrid-combined methods for singularity problems of homogeneous equal Inter. J. Numer. Meth. Engrg., 26: 785-803, 1988.
• [41| Z.C. Li, T.D. Bui. Six combinations of the Ritz Galerkin and finite element methods for elliptic boundary value problems. Numer. Meth. PDEs, 4: 197-218, 1988.
• [42] Z.C. Li, T.D. Bui. The simplified hybrid-combined methods for Laplace equation with singularities. J.Appl. Math., 29: 171-193, 1990.
• [43] Z.C. Li, T.D. Bui. Coupling techniques in boundary-combined methods. Engrg. Anal. Bound. Elem., 10: 7 1992.
• [44] Z.C. Li, Y.L. Chen, G.C. Georgiou, C. Xenophontos. Special boundary approximation methods for Laplace equation problems with boundary singularities - Applications to the Motz problem. Inter. Comp. Math. Appl.,51: 115-142, 2006.
• [45] Z.C. Li, H.T. Huang. Global superconvergence of simplified hybrid combinations of the Ritz-Galerkin and for elliptic equations with singularities, II. Lagrange Elements and Adini's Elements. Appl. Numer. Math., 43: 253-273, 2002.
• [46] Z.C. Li, C.S. Huang, R.C.D. Cheng. Interior boundary conditions in the Schwarz alternating method for the Trefftz method. Engrg. Anal. Bound. Elem.. 29: 477-493, 2005.
• [47] Z.C. Li, G.P. Liang. On the simplified hybrid-combined method. Math. Comp., 41: 1-25, L983.
• [48| Z.C. Li, T.T. Lu, H.Y. Hu, A.II.-D. Cheng. Particular solutions of Laplace's equations on polygons and new models involving mild Singularities. Engrg. Anal. Bound. Elem., 29: 59-75, 2005.
• [49] Z.C. Li, T.T. Lu, H.Y. Hu, A.H.-D. Cheng. Trefftz and Collocation Methods. WITpress, Southampton, Boston] January 2008.
• [50] Z.C. Li, T.T. Lu, H.T. Huang. A.H.-D. Cheng. Trefftz, collocation and other boundary methods - A comparison. Numer. Meth. PDEs, 23: 93-144, 2007.
• [51] Z.C. I.i. It. Mat Hon. Error and stability analysis of boundary methods for elliptic problems with interfaces. Math. 54: 1 61, 1990.
• [52] Z.C Li, It. Matlion, I'. Senner, Boundary methods for solving elliptic problem with singularities and interfaces. SIAM J. Numer. Anal., 24: 487-498, 1987.
• [53] Z.C. I.i T.T. Lu. Singularities and treatments of elliptic boundary value problems. Math. Comp. Model., 31: 79-145, 2000.
• [54] G. P. Liang, P. Liang, Non-conforming domain decomposition with hybrid method. J. Comp. Math., 8(4): 363-370, 1990.
• [55] G.P. Liang, J. H. He. The non-conforming domain decomposition method for elliptic problems with Lagrangian multipliers (in Chinese). Math. Numer. Sinica, 14(2): 207-215, 1992.
• [56] T.T. Lu, H.Y. IIu, Z.C. Li. Highly accurate solutions of Motz's and the cracked beam problems. Engrg. Anal. Bounnd. Elem., 28: 1387-1403, 2004.
• [57] J. Mandel, It. Tezaur. Convergence of a substructuring method with Lagrange multipliers. Numer. Math., 73: 473-487, 1996.
• [58] I.D. Moldovan. J.A.T. de Freitas. Hybrid-Trefftz stress and displacement elements for dynamic analysis of ded and unbounded saturated porous media. Proceeding of Trefftz.08, 5th International Workshop on Trefftz methods, pp. 307 321. Leuven, March 31 April 2, 2008.
• [59] J. Pitkaranta. Boundary subspaces for the finite element method with Lagrange multipliers. Numer. Math., 33: 273-289, 1979.
• [60] J. Pitkaranta. The finite element method with Lagrange multipliers for domains with corners. Math. Comp., 37 (155): 13-30, 1981.
• [61] A. Charafi. Trefftz boundary element-multi-region formulation. Inter. J. Numer. Meth. Engrg., 45: 821-840, 1999.
• [62] Q.ll. Qin. The Trefftz Finite and Boundary Element Methods. WITpress, Southampton, Boston, 2000.
• [63] Q. H Qin, K.Y. Wang. Application of hybrid-Trefftz finite element method to frictional contact problems. Proceeding of Trefftz .08, 5th International Workshop on Trefftz methods, pp. 65-87. Leuven, March 31-April 2, 2008.
• [64] P. A. Raviart, J. M. Thomas. Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comp.,31 9138): 391-413, 1977.
• [65] S. Reutskiy. A boundary method of Trefftz type with approximate trial functions. Engrg. Anal. Bound. Elem., 26 (4): 341-353, 2002.
• [66] P. Ruge, The complete Trefftz method. Acta Mechanica, 78(3-4): 234-242, 1989.
• [67] I. Sladek, V. Sladek. H. van Keer. Clobal and local Trefftz boundary integral formulation for sound vibration. Adv. Engrg. Softw., 33: 469-476, 2002.
• [68] R. P. Shaw, S.C. Huang, C.X. Zhao. The embedding integral and the Trefftz method for potential problems with rationing. Engrg. Anal. Bound. Elem., 9(1): 83-90, 1992.
• [69] K.Y. Sze, G. H. Liu, Hybrid-Trefftz finite element models for plane Helmholtz problems. Proceeding of Trefftz.08, 5th International Workshop on Trefftz methods, pp. 401-415. Leuven, March 31-April 2, 2008.
• [70] M. Toina. I.A.T. de Freitas. Hybrid-Trefftz stress and displacement elements for transient analysis of incompress-saturated porous media. Proceeding of Trefftz.08, 5th International Workshop on Trefftz methods, pp. 401-Leuven, March 31-April 2, 2008.
• [71] E. Trefftz. Kin Gegenstuck zum Ritz'schen Verfahren. Proceeding of the 2nd Inter. Cong. Appl. Mech., pp. 131— 137. Zurich. 1926.
• [72] A. P. Zieliński. Special Trefftz elements and improvement of their conditioning. Commun. Numer. Meth. Engrg., 13(10), 765-775, 1997.
• [73] A. P. Zieliński. On trial functions applied in the generalized Trefftz method. Adv. Engrg. Softw., 24: 147-155, 1995.
• [74] A.P. Zieliński. I. Herrera. Trefftz method-fitting boundary conditions. Inter. J. Numer. Meth. Engrg., 24(5): 871-891, 1987.
• [75] O.C. Zienkiewicz, D.W. Kelley, P. Bettess. The coupling of the finite element method and boundary solution procedures. Inter. ./. Numer. Meth. Engrg., 11(2): 355-375, 1977.