A general framework for interpreting time finite element formulations

Borri, M.; Bottasso, Carlo L.

doi:10.1007/bf00370131

Cited by 75 publications

(35 citation statements)

References 15 publications

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“…The results may not be as accurate as the result given at the end of the time step. A similar observation has been reported by Fung and Leung [22] for the higher-order accurate time discontinuous Galerkin method [13,14], and the bi-discontinuous Galerkin method [15].…”

Section: Hyperbolic Equationssupporting

confidence: 72%

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Complex-time-step methods for transient analysis

Fung

1999

Int. J. Numer. Meth. Engng.

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SUMMARYUnconditionally stable time step integration algorithms with controllable numerical dissipation of arbitrary order of accuracy are presented for transient analysis. The algorithms are based on the -method for "rst-order di!erential equations. The results at several (complex) sub-step locations are evaluated and combined linearly to give higher-order accurate results at the end of a time step. The ampli"cation factors are shown to be related to the PadeH approximations to the exponential function. The computational procedures for parabolic and hyperbolic problems are discussed. The interpolation procedure for the solutions within a time step is given. Numerical examples are used to illustrate the validity of the present complex-time-step algorithms.

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Section: Hyperbolic Equationssupporting

confidence: 72%

“…It can be seen that equation (19) is a polynomial of degree n in and is independent of from equation (15). Combining equations (18) and (19), the resultant ampli"cation factor is then…”

Section: Explicit Form Of the Amplification Factormentioning

confidence: 97%

Complex-time-step methods for transient analysis

Fung

1999

Int. J. Numer. Meth. Engng.

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“…high-frequency modes are damped out almost in one time step [3]. Nonetheless, a drawback of the TDG method is related to the size of system (8), which amounts to 4n DoF , where n DoF is the number of degrees of freedom in the spatial discretization.…”

Section: Formulationmentioning

confidence: 99%

“…The use of discontinuous displacement and momentum ÿelds leads to the class of Time Discontinuous Galerkin (TDG) methods. TDG schemes have been analysed extensively in the context of linear problems [3][4][5][6][7][8][9] and their superiority in comparison with standard ÿnite di erence methods has been demonstrated. These methods are L-stable and therefore the higher modes are quickly eliminated.…”

Section: Introductionmentioning

confidence: 99%

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Analysis and performance of a predictor‐multicorrector Time Discontinuous Galerkin method in non‐linear elastodynamics

Bursi

Mancuso

2002

Earthq Engng Struct Dyn

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SUMMARYA predictor-multicorrector implementation of a Time Discontinuous Galerkin method for non-linear dynamic analysis is described. This implementation is intended to limit the high computational expense typically required by implicit Time Discontinuous Galerkin methods, without degrading their accuracy and stability properties. The algorithm is analysed with reference to conservative Du ng oscillators for which closed-form solutions are available. Therefore, insight into the accuracy and stability properties of the predictor-multicorrector algorithm for di erent approximations of non-linear internal forces is gained, showing that the properties of the underlying scheme can be substantially retained. Finally, the results of representative numerical simulations relevant to Du ng oscillators and to a sti spring pendulum discretized with ÿnite elements illustrate the performance of the numerical scheme and conÿrm the analytical estimates.

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Complex-time-step Newmark methods with controllable numerical dissipation

Fung

1998

Int. J. Numer. Meth. Engng.

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In this paper, unconditionally stable higher-order accurate time-step integration algorithms with controllable numerical dissipation are presented. The algorithms are based on the Newmark method with complex time steps. The ultimate spectral radius ( ), the sub-step locations ( H ) and the weighting factors ( H ) are the algorithmic parameters. For an algorithm that is (2n!1)th order accurate, the sub-step locations which may be complex, are shown to be the roots of an nth degree polynomial. The polynomial is given explicitly in terms of n and . The weighting factors are then obtained by solving a system of n simultaneous equations. It is further shown that the order of accuracy is increased by one for the non-dissipative algorithms with "1. The stability properties of the present algorithms are studied. It is shown that if the ultimate spectral radius is set between !1 and 1, the eigenvalues of the numerical amplification matrix are complex with magnitude less than or equal to unity. The algorithms are therefore unconditionally C-stable. When the ultimate spectral radius is set to 0 or 1, the algorithms are found to be equivalent to the first sub-diagonal and diagonal Pade´approximations, respectively. The present algorithms are more general as the numerical dissipation is controllable and are very suitable for parallel computers. The accuracy of the excitation responses is found to be enhanced by the present complex-time-step procedure. To maintain high-order accuracy, the excitation may need some modifications.

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A general framework for interpreting time finite element formulations

Cited by 75 publications

References 15 publications

Complex-time-step methods for transient analysis

Complex-time-step methods for transient analysis

Analysis and performance of a predictor‐multicorrector Time Discontinuous Galerkin method in non‐linear elastodynamics

Complex-time-step Newmark methods with controllable numerical dissipation

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