Time-Discontinuous Finite Element Analysis of Two-Dimensional Elastodynamic Problems using Complex Fourier Shape Functions

Document Type: Research Paper

Authors

Civil Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran

Abstract

This paper reformulates a time-discontinuous finite element method (TD-FEM) based on a new class of shape functions, called complex Fourier hereafter, for solving two-dimensional elastodynamic problems. These shape functions, which are derived from their corresponding radial basis functions, have some advantages such as the satisfaction of exponential and trigonometric function fields in complex space as well as the polynomial ones simultaneously, that make them a better choice than classic Lagrange shape functions, which only can satisfy polynomial function field. To investigate the validity and accuracy of the proposed method, three numerical examples are provided and the results obtained from the present method (complex Fourier-based TD-FEM) and the classic Lagrange-based TD-FEM are compared with the exact analytical solutions. According to them, using complex Fourier functions in TD-FEM leads to more accurate and stable solutions rather than those obtained from the classic TD-FEM.

Keywords

Main Subjects

Agnantiaris, J.P., Polyzos, D., Beskos, D.E., Some studies on dual reciprocity BEM for elastodynamic analysis, Computational Mechanics, 17 (1996) 270–277.

Aksoy, H.G., ┼×enocak, E., Discontinuous Galerkin method based on peridynamic theory for linear elasticity, International Journal for Numerical Methods in Engineering, 88(7) (2011) 673-692.

Argyris, J.H., Scharpf, D.W., Finite elements in time and space, Nuclear Engineering and Design, 10(4) (1969) 456-464.

Bathe, K.J, Baig, M.M.I., On a composite implicit time integration procedure for nonlinear dynamics, Computers & Structures, 83(31-32) (2005) 2513–2524.

Brebbia, C.A., Nardini, D., Dynamic analysis in solid mechanics by an alternative boundary elements procedure, International Journal of Soil Dynamics and Earthquake Engineering, 2 (1983) 228–233.

Bruch, J.C., Zyvoloski, G., Transient two-dimensional heat conduction problems solved by the finite element method, International Journal for Numerical Methods in Engineering, 8(3) (1974) 481-494.

Chung, J., Hulbert, G.M., A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-a method, Journal of Applied Mechanics, 60(2) (1993) 371–375.

Chien, C.C., Wu, T.Y., An improved predictor/multi-corrector algorithm for a time-discontinuous Galerkin finite element method in structural dynamics, Computational Mechanics, 25(5) (2000) 430-437.

Dominguez, J., Boundary element in dynamics. London: Computational Mechanics Publications, Southampton, Elsevier Applied Science, 1993.

Dyniewicz, B., Efficient numerical approach to unbounded systems subjected to a moving load, Computational Mechanics, 54 (2014) 321-329.

Fried, I., Finite-element analysis of time-dependent phenomena, AIAA Journal, 7(6) (1969) 1170-1173.

Fung, T.C., Higher-order accurate least-squares methods for first-order initial value problems, International Journal for Numerical Methods in Engineering, 45(1) (1999) 77-99.

Guo, P., Wu, W.H., Wu, Z.G., A time discontinuous Galerkin finite element method for generalized thermo-elastic wave analysis, Acta Mechanica, 225(1) (2014) 299-307.

Hamzehei Javaran, S., Khaji, N., Dynamic analysis of plane elasticity with new complex Fourier radial basis functions in the dual reciprocity boundary element method, Applied Mathematical Modelling, 38(14) (2014) 3641-3651.

Hamzeh Javaran, S., Khaji, N., Noorzad, A., First kind Bessel function (J-Bessel) as radial basis function for plane dynamic analysis using dual reciprocity boundary element method, Acta Mechanica, 218 (2011) 247-258.

Hilber, H.M., Hughes, T.J., Taylor, R.L., Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Engineering & Structural Dynamics, 5(3) (1977) 283-292.

Hughes, T.J., Hulbert, G.M., Space-time finite element methods for elastodynamics: formulations and error estimates, Computer Methods in Applied Mechanics and Engineering, 66(3) (1988) 339-363.

Hulbert, G.M., Hughes, T.J., Space-time finite element methods for second-order hyperbolic equations, Computer Methods in Applied Mechanics and Engineering, 84(3) (1990) 327-348.

Hulbert, G.M., Time finite element methods for structural dynamics, International Journal for Numerical Methods in Engineering, 33(2) (1992) 307-331.

Khaji, N., Hamzehei Javaran, S., New complex Fourier shape functions for the analysis of two-dimensional potential problems using boundary element method, Engineering Analysis with Boundary Elements, 37(2) (2013) 260-272.

Lesaint, P., Raviart, P.A., On a finite element method for solving the neutron transport equation, in: C. de Boor ed., Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, NewYork, 89-123, 1974.

Lewis, D.L., Lund, J., Bowers, K.L., The space–time Sinc-Gallerkin method for parabolic problems, International Journal for Numerical Methods in Engineering, 24(9) (1987) 1629-1644.

Li, X., Yao, D., Lewis, R.W., A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non-linear solids and saturated porous media, International Journal for Numerical Methods in Engineering, 57(12) (2003) 1775-1800.

Liu, Y., Li, H., He, S., Mixed time discontinuous space-time finite element method for convection diffusion equations, Applied Mathematics and Mechanics, 29(12) (2003) 1579-1586.

Nguyen, H., Reynen, J., A space-time least-square finite element scheme for advection-diffusion equations, Computer Methods in Applied Mechanics and Engineering, 42(3) (1984) 331-342.

Oden, J.T., A general theory of finite elements. II. Applications, International Journal for Numerical Methods in Engineering, 1(3) (1969) 247-259.

Peters, D.A., Izadpanah, A.P., Hp-version finite elements for the space-time domain, Computational Mechanics, 3(2) (1988) 73-88.

Petersen, S., Farhat, C., Tezaur, R., A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain, International Journal for Numerical Methods in Engineering, 78(3) (2009) 275-295.

Reed, W.H., Hill, T.R., Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Lab, N. Mex., USA, 1973.

Saedpanah, F., A posteriori error analysis for a continuous space-time finite element method for a hyperbolic integro-differential equation, BIT Numerical Mathematics, 53 (2013) 689-716.

Shao, H.P., Cai, C.W., A three parameters algorithm for numerical integration of structural dynamic equations, Chinese Journal of Mechanical Engineering, 5(4) (1988) 76–81.

Tamma, K.K., Zhou, X., Sha, D., A theory of development and design of generalized integration operators for computational structural dynamics, International Journal for Numerical Methods in Engineering, 50 (2001) 1619–1664.

Tang, Q., Chen, C.M., Liu, L.H., Space-time finite element method for schrödinger equation and its conservation, Applied Mathematics and Mechanics, 27 (2006) 335-340.

Varoglu, E., Liam Finn, W.D., Space-time finite elements incorporating characteristics for the burgers' equation, International Journal for Numerical Methods in Engineering, 16(1) (1980) 171-184.

Wang, J.G., Liu, G.R., On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Computer Methods in Applied Mechanics and Engineering, 191(23) (2002) 2611-2630.

Wilson, E.L., Nickell, R.E., Application of the finite element method to heat conduction analysis, Nuclear Engineering and Design, 4(3) (1966) 276-286.

Zhou, X., Tamma, K.K., Design, analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamics, International Journal for Numerical Methods in Engineering, 59 (2004) 597-668.

Zienkiewicz, O.C., Parekh, C.J., Transient field problems: Two-dimensional and three-dimensional analysis by isoparametric finite elements, International Journal for Numerical Methods in Engineering, 2(1) (1970) 61-71.

Zienkiewicz, O.C., Taylor, R.L., The finite element method, London: McGraw-Hill, 1977.

Zienkiewicz, O.C., A new look at the Newmark, Houbolt and other time stepping formulas, A weighted residual approach, Earthquake Engineering & Structural Dynamics, 5(4) (1977) 413-418.