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Time discontinuous Galerkin methods with energy decaying correction for nonlinear elastodynamics. (English) Zbl 1193.74050

Summary: In this paper a new time discontinuous Galerkin (TDG) formulation for nonlinear elastodynamics is presented. The new formulation embeds an energy correction which ensures truly energy decaying, thus allowing to achieve unconditional stability that, as shown in the paper, is not guaranteed by the classical TDG formulation. The resulting method is simple and easily implementable into existing finite element codes. Moreover, it inherits the desirable higher-order accuracy and high-frequency dissipation properties of the classical formulation. Numerical results illustrate the very good performance of the proposed formulation.

MSC:

74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
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[1] Kuhl, Energy conserving and decaying algorithms in non-linear structural dynamics, International Journal for Numerical Methods in Engineering 45 pp 569– (1999) · Zbl 0946.74078
[2] Lasaint, Mathematical Aspects of Finite Elements in Partial Differential Equations pp 89– (1974) · doi:10.1016/B978-0-12-208350-1.50008-X
[3] Hulbert GM. Space-time finite element methods for second-order hyperbolic problems. Ph.D. Thesis, Stanford University, Stanford, 1989.
[4] Hulbert, Time finite element methods for structural dynamics, International Journal for Numerical Methods in Engineering 33 pp 307– (1992) · Zbl 0760.73064
[5] Borri, A general framework for interpreting time finite element formulations, Computational Mechanics 13 pp 133– (1993) · Zbl 0789.70003
[6] Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Computer Methods in Applied Mechanics and Engineering 107 pp 117– (1993) · Zbl 0787.65070
[7] Hulbert, A unified set of single step asymptotic annihilation algorithms for structural dynamics, Computer Methods in Applied Mechanics and Engineering 113 pp 1– (1994) · Zbl 0849.73067
[8] Cannarozzi, Formulation and analysis of variational methods for time integration of linear elastodynamics, Computer Methods in Applied Mechanics and Engineering 127 pp 241– (1995) · Zbl 0862.73078
[9] Fung, On the accuracy of discontinuous Galerkin methods in the time domain, Journal of Vibration and Control 2 pp 193– (1996) · Zbl 0949.65508
[10] Eriksson, Computational Differential Equations (1996) · Zbl 0946.65049
[11] Fan, A comprehensive unified set of single-step algorithms with controllable dissipation for dynamics. Part I. Formulation, Computer Methods in Applied Mechanics and Engineering 145 pp 87– (1997) · Zbl 0892.73059
[12] Fan, A comprehensive unified set of single-step algorithms with controllable dissipation for dynamics. Part II. Algorithms and analysis, Computer Methods in Applied Mechanics and Engineering 145 pp 99– (1997) · Zbl 0892.73059
[13] Li, Structural dynamic analysis by a time-discontinuous Galerkin finite element method, International Journal for Numerical Methods in Engineering 39 pp 2131– (1996) · Zbl 0885.73081
[14] Li, Implementation and adaptivity of a space-time finite element method in structural dynamics, Computer Methods in Applied Mechanics and Engineering 156 pp 211– (1998)
[15] Bottasso, Some recent developments in the theory of finite elements in time, Computer Modeling and Simulation in Engineering 4 pp 201– (1999)
[16] Chien, An improved predictor/multi-corrector algorithm for a time-discontinuous Galerkin finite element method in structural dynamics, Computational Mechanics 25 pp 430– (2000) · Zbl 0976.74063
[17] Ruge, Hybrid time-finite-elements with time-step-adaption by a discontinuity control, Computational Mechanics 17 pp 392– (1996) · Zbl 0852.70018
[18] Bar-Yoseph, Spectral element method for nonlinear temporal dynamical system, Computational Mechanics 18 pp 302– (1996) · Zbl 0884.70003
[19] Bar-Yoseph, Spectral element method for nonlinear spatio-temporal dynamics of an Euler-Bernoulli beam, Computational Mechanics 19 pp 136– (1996) · Zbl 0895.73076
[20] Wiberg, Proceedings of Complas V: Computational Plasticity, Fundamentals and Applications pp 224– (1997)
[21] Wiberg, Adaptive finite element procedures for linear and non-linear dynamics, International Journal for Numerical Methods in Engineering 46 pp 1781– (1999) · Zbl 0977.74069
[22] Bauchau, Computational schemes for non-linear elasto-dynamics, International Journal for Numerical Methods in Engineering 45 pp 693– (1999) · Zbl 0941.74078
[23] Bonelli, Explicit predictor-multicorrector time discontinuous Galerkin methods for non-linear dynamics, Journal of Sound and Vibration 256 pp 695– (2002) · Zbl 1237.65086
[24] Bursi, Analysis and performance of a predictor-multicorrector time discontinuous Galerkin method in non-linear dynamics, Earthquake Engineering and Structural Dynamics 31 pp 1793– (2002)
[25] Li, 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 pp 1775– (2003) · Zbl 1062.74621
[26] Bonelli, Iterative solutions for implicit time discontinuous Galerkin methods applied to non-linear elastodynamics, Computational Mechanics 30 pp 487– (2003) · Zbl 1038.74557
[27] Kunthong, An efficient solver for the high-order accurate time-discontinuous Galerkin (TDG) method for second-order hyperbolic systems, Finite Elements in Analysis and Design 41 pp 729– (2005)
[28] Mancuso, An efficient integration procedure for linear dynamics based on a time discontinuous Galerkin formulation, Computational Mechanics 32 pp 154– (2003) · Zbl 1038.74558
[29] Mancuso, An efficient time discontinuous Galerkin procedure for non-linear structural dynamics, Computer Methods in Applied Mechanics and Engineering 195 pp 6391– (2006) · Zbl 1122.74024
[30] Govoni, Hierarchical higher-order dissipative methods for transient analysis, International Journal for Numerical Methods in Engineering 67 pp 1730– (2006) · Zbl 1113.74025
[31] Wood, Practical Time-stepping Schemes (1990) · Zbl 0694.65043
[32] Argyris, Dynamics of Structures (1991) · Zbl 0792.73001
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