Abstract
The application of the new numerical approach for elastodynamics problems developed in our previous paper and based on the new solution strategy and the new time-integration methods is considered for 1D and 2D axisymmetric impact problems. It is not easy to solve these problems accurately because the exact solutions of the corresponding semi-discrete elastodynamics problems contain a large number of spurious high-frequency oscillations. We use the 1D impact problem for the calibration of a new analytical expression describing the minimum amount of numerical dissipation necessary for the new time-integration method used for filtering spurious oscillations. Then, we show that the new numerical approach for elastodynamics along with the new expression for numerical dissipation for the first time yield accurate and non-oscillatory solutions of the considered impact problems. The comparison of effectiveness of linear and quadratic elements as well as rectangular and triangular finite elements for elastodynamics problems is also considered.
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References
Bathe KJ (1996) Finite element procedures. Prentice-Hall, Upper Saddle River
Benson DJ (1992) Computational methods in Lagrangian and Eulerian hydrocodes. Comput Methods Appl Mech Eng 99(2–3): 235–394
Chien CC, Wu TY (2000) Improved predictor/multi-corrector algorithm for a time-discontinuous galerkin finite element method in structural dynamics. Comput Mech 25(5): 430–437
Govoni L, Mancuso M, Ubertini F (2006) Hierarchical higher-order dissipative methods for transient analysis. Int J Numer Methods Eng 67(12): 1730–1767
Hilber HM, Hughes TJR (1978) Collocation, dissipation and ‘overshoot’ for time integration schemes in structural dynamics. Earthquake Eng Struct Dyn 6(1): 99–117
Hilber HM, Hughes TJR, Taylor RL (1977) Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Eng Struct Dyn 5(3): 283–292
Houbolt JC (1950) A recurrence matrix solution for the dynamic response of elastic aircraft. J Aeronaut Sci 17: 540–550
Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice- Hall, Englewood Cliffs
Hulbert GM (1992) Discontinuity-capturing operators for elastodynamics. Comput Methods Appl Mech Eng 96(3): 409–426
Hulbert GM, Hughes TJR (1990) Space-time finite element methods for second-order hyperbolic equations. Comput Methods Appl Mech Eng 84(3): 327–348
Idesman AV (2007) A new high-order accurate continuous galerkin method for linear elastodynamics problems. Comput Mech 40: 261–279
Idesman AV (2007) Solution of linear elastodynamics problems with space-time finite elements on structured and unstructured meshes. Comput Methods Appl Mech Eng 196: 1787–1815
Idesman AV (2008) A new solution strategy and a new look at time-integration methods for elastodynamics. Comput Methods Appl Mech Eng 1–59 (submitted)
Jiang L, Rogers RJ (1990) Effects of spatial discretization on dispersion and spurious oscillations in elastic wave propagation. Int J Numer Methods Eng 29(6): 1205–1218
Krenk S (2001) Dispersion-corrected explicit integration of the wave equation. Comput Methods Appl Mech Eng 191: 975–987
Krenk S (2006) State-space time integration with energy control and fourth-order accuracy for linear dynamic systems. Int J Numer Methods Eng 65(5): 595–619
Kunthong P, Thompson LL (2005) An efficient solver for the high-order accurate time-discontinuous Galerkin (tdg) method for second-order hyperbolic systems. Finite Elements Anal Design 41(7-8): 729–762
Mancuso M, Ubertini F (2003) An efficient integration procedure for linear dynamics based on a time discontinuous Galerkin formulation. Comput Mech 32(3): 154–168
Newmark NM (1959) A method of computation for structural dynamics. J Eng Mech Div ASCE 85(EM 3): 67–94
Park KC (1975) Evaluating time integration methods for nonlinear dynamic analysis. In: Belytschko T et al (eds) Finite element analysis of transient nonlinear behavior. AMD 14, ASME, New York, pp 35–58
Vales F, Moravka S, Brepta R, Cerv J (1996) Wave propagation in a thick cylindrical bar due to longitudinal impact. JSME Int J Series A 39(1): 60–70
Wilkins ML (1980) Use of artificial viscosity in multidimensional fluid dynamic calculations. J Comput Phys 36(3): 281–303
Wilson EL, Farhoomand I, Bathe KJ (1973) Nonlinear dynamic analysis of complex structures. Earthq Eng Struct Dyn 1(3): 241–252
Zhang GM, Batra RC (2007) Wave propagation in functionally graded materials by modified smoothed particle hydrodynamics (msph) method. J Comput Phys 222: 374–390
Zienkiewicz OC, Taylor RL (2000) The finite element method. Butterworth-Heinemann, Oxford
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Idesman, A., Samajder, H., Aulisa, E. et al. Benchmark problems for wave propagation in elastic materials. Comput Mech 43, 797–814 (2009). https://doi.org/10.1007/s00466-008-0346-3
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DOI: https://doi.org/10.1007/s00466-008-0346-3