About a numeric solution of the wave equation
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The numerical solution of partial differential equations that evolve over time is a research field in constant development. In this paper on the computational solution to the wave equation, two algorithms of time integration are used: the Newmark method and the finite difference method (FDM). The Newmark method has a high precision and excellent convergence rate compared to the FDM. The FDM can be easily implemented. In an effort to compare these two methods, two typical problems using FORTRAN were implemented: first, a square membrane completely fixed at its edges with an initial velocity in the center, and second, a simply supported beam with an initial velocity at one of its ends. These test problems are discretized in the space domain through the finite element method, and in the time domain through the Newmark method and the FDM. The results show that the Newmark method allows using time steps that are greater than those of the FDM, but present a higher numerical oscillation. These results are expected to be the source of initial data for a subsequent comparison with other methods.
Wave equations, differential equations, telecommunicationEcuaciones ondulatorias, ecuaciones diferenciales, convergencia (telecomunicación)
CARRER, J. A. M. y MANSUR, W. J. Time-domain BEM analysis for the 2D scalar wave equation: initial conditions contributions to space and time derivatives. International Journal for Numerical Methods in Engineering, 1997, vol. 39, núm. 13, pp. 2188-2469.
CHAPRA, S. C. y CANALE, R. P. Numerical methods for engineers. New York: McGraw Hill, 1998.
CHUNG, J. y HULBERT, J. M. A time integration method for structural dynamics with improved numerical dissipation: the generalized-method. Journal of Applied Mechanics, 1993, vol. 30, pp. 371-375.
CHUNG, J. y LEE, J. M. A new family of explicit time integration methods for linear and non-linear structural dynamics. International Journal for Numerical Methods in Engineering, 1994, vol. 37, núm. 23, pp. 3961-3976.
DANIEL, W. J. T. The subcycled Newmark algorithm. Computational Mechanics, 1997, vol. 20, núm. 3, pp. 272-281.
DJOKO, J. K. y REDDY, B. D. An extended Hu-Washizu formulation for elasticity. Computer Methods in Applied Mechanics and Engineering, 2006, vol. 195, núms. 44-47, pp. 6330-6346.
GERSHENFELD, N. The nature of mathematical modeling. Cambridge: Cambridge University Press, 1998.
GOUDREAU, G. L. y TAYLOR, R. L. Evaluation of numerical integration methods in elastodynamics. Computer Methods in Applied Mechanics and Engineering, 1972, vol. 2, pp. 69-97.
HAHN, G. D. A modified Euler method for dynamic analysis. International Journal for Numerical Methods in Engineering, 1991, vol. 32, núm. 5, pp. 943-955.
HOFF, C. y TAYLOR, R. L. Higher derivative explicit one step methods for non-linear dynamic problems. Part I: design and theory. International Journal for Numerical Methods in Engineering, 1990, vol. 29, núm. 2, pp. 275-290.
HUGHES, T. J. R. The finite element method-linear static and dynamic finite element analysis. New York: Dover Publishers, 2000.
HULBERT, G. M. y CHUNG, J. Explicit time integration algorithms for structural dynamics with optimal numerical dissipation. Computer Methods in Applied Mechanics and Engineering, 1996, vol. 137, núm. 2, pp. 175-188.
LINIGER, W. Global accuracy and a stability of one and two steps integration formulas for stiff ordinary differential equations. Conference on the Numerical Solution of Differential Equations, Dundee University, 1969, vol. 109, pp. 188-193.
LOUREIRO, F. S. Métodos de integracão temporal baseados no cálculo numérico de funcões de Green a través do método dos elementos finitos. [MSc Thesis]. Rio de Janeiro: Universidade Federal do Rio de Janeiro-COPPE, 2007.
MANSUR, W. J. et al. Explicit time-domain approaches based on numerical Green’s functions computed by finite differences-The ExGA family. Journal of Computational Physics, 2007, vol. 227, núm. 1, pp. 851-870.
—. Numerical solution for the linear transient heat conduction equation using an Explicit Green’s Approach. Journal of Heat and Mass Transfer, 2008, vol. 52, núms. 3-4, pp. 694-701.
NEWMARK, N. M. A method of computation for structural dynamics. Journal Engineering Mechanics Division, 1959, vol. 85, pp. 67-94.
OÑATE, E. Cálculo de estructuras por el método de elementos finitos: análisis estático lineal. 2a ed. Barcelona: CIMNE, 1995.
SOUZA, L. A.; CARRER, J. A. M. y MARTINS, C. J. A fourth order finite difference method applied to elastodynamics: finite element and boundary element formulations. Structural Engineering and Mechanics, 2004, vol. 17, núm. 6, pp. 735-749.
TAMMA, K. K. y NAMBURU, R. R. A robust self-starting explicit computational methodology for structural dynamic applications: architecture and representations. International Journal for Numerical Methods in Engineering, 1990, vol. 29, núm. 7, pp. 1441-1454.
TIMOSHENKO, S. y GOODIER, J. N. Theory of elasticity. New York: McGraw-Hill, 1951.
WOOD, W. L; BOSSAK, M. y ZIENKIEWICZ, O. C. An alpha modification of Newmark’s methods. International Journal for Numerical Methods in Engineering, 1981, vol.15, núm. 10, pp. 1562-1566.
ZIENKIEWICZ, O. C. y TAYLOR, R. L. El método de los elementos finitos: las bases. Barcelona: CIMNE, 2000.
ZIENKIEWICZ, O. C. y MORGAN, K. Finite element and approximation. New York: Wiley, 1982.