Mixed finite element solution of time-dependent problems. (English) Zbl 1194.74495

Summary: A mixed formulation of the finite element method is used to establish a higher-order incremental method for the solution of second-order/hyperbolic problems. The displacement, the velocity and, optionally, the acceleration fields are approximated independently in time using hierarchical bases. The time approximation criterion preserves hyperbolicity in the sense that it replaces the solution of hyperbolic problems by the solution of uncoupled Helmholtz-type elliptic problems, which can be subsequently solved using the alternative methods currently in use for discretization of the space dimension. The development of the time integration procedure, the characterization of its performance in terms of stability, accuracy and convergence are illustrated using a polynomial time basis. In order to stress the fact that the procedure can be implemented using alternative time bases, a wavelet system is used in the solution of nonlinear, parabolic and hyperbolic problems. The method is well-suited to parallel processing and to large time stepping. The extension of its application to the solution of other than linear second-order/hyperbolic problems is discussed.


74S05 Finite element methods applied to problems in solid mechanics
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
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[1] Daubechies, I., Orthonormal bases of compactly supported wavelets, Commun. pure appl. math., 41, 909-996, (1988) · Zbl 0644.42026
[2] Freitas, J.A.T., Hybrid-Trefftz displacement and stress elements for elastodynamic analysis in the frequency domain, Cames, 4, 345-368, (1997) · Zbl 0969.74590
[3] Freitas, J.A.T., Mixed finite element formulation for the solution of parabolic problems, Comput. methods appl. mech. engrg., 191, 3425-3457, (2002) · Zbl 1101.74367
[4] Freitas, J.A.T., Time integration and the Trefftz method: part I - first-order and parabolic problems; part II - second-order and hyperbolic problems, Cames, 10, 453-477, (2003)
[5] Freitas, J.A.T.; Almeida, J.P.M.; Pereira, E.M.B.R., Non-conventional formulations for the finite element method, Comput. mech., 23, 488-501, (1999) · Zbl 0946.74068
[6] Hoff, C.; Pahl, P.J., Development of an implicit method with numerical dissipation from a generalized single-step algorithm for structural dynamics, Int. J. numer. methods engrg., 67, 367-385, (1988) · Zbl 0619.73002
[7] Jirousek, J., Basis for development of large finite elements locally satisfying all field equations, Comput. methods appl. mech. engrg., 14, 65-92, (1978) · Zbl 0384.73052
[8] Jirousek, J.; Qin, Q.H., Application of hybrid-Trefftz element approach to transient heat conduction analysis, Comput. struct., 58, 195-201, (1996) · Zbl 0900.73802
[9] Monasse, P.; Perrier, V., Orthonormal wavelet bases adapted for partial differential equations with boundary conditions, SIAM J numer. anal., 29, 1040-1065, (1998) · Zbl 0921.35036
[10] N.M. Newmark, A method of computation for structural dynamics, in: Proc. ASCE 85 EM3, 1959, pp. 67-94.
[11] Pegon, P., Alternative characterization of time integration schemes, Comput. methods appl. mech. engrg., 190, 2707-2727, (2001) · Zbl 0982.70003
[12] J.P.P. Pina, J.A.T. Freitas, L.M.S. Castro, Use of wavelets in dynamic analysis, in: Métodos Computacionais em Engenharia, APMTAC, Lisbon, 2004 (in Portuguese).
[13] Ritz, W., Gesammelte werke, (1911), Gauthier-Villars Paris
[14] Tamma, K.K.; Zhou, X.; Sha, D., The time dimension: A theory towards the evolution, classification, characterization and design of computational algorithms for transient/dynamic applications, Arch comput. methods engrg., 7, 67-290, (2000) · Zbl 0987.74004
[15] E. Trefftz, Ein Gegenstück zum Ritzschen Verfahren, in: Proc. 2nd Int. Cong. Appl. Mech., Zurich, 1926.
[16] Trefftz Net. <http://www.olemiss.edu/sciencenet/trefftz/>.
[17] Vermilyea, M.E.; Spilker, R.L., A hybrid finite element formulation of the linear biphasic equations for soft hydrated tissues, Int. J. numer. methods engrg., 33, 567-594, (1992) · Zbl 0757.73042
[18] Wood, W.L., A unified set of single-step algorithms, part 2: theory, Int. J. numer. methods engrg., 20, 2303-2309, (1984) · Zbl 0557.65042
[19] Zienkiewicz, O.C.; Wood, W.L.; Hine, N.W.; Taylor, R.L., A unified set of single-step algorithms, part 1: general formulation and applications, Int. J. numer. methods engrg., 20, 1529-1552, (1984) · Zbl 0557.65041
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