A new algorithm for integration of dynamic systems

Gellert, M.

doi:10.1016/0045-7949(78)90126-8

Cited by 39 publications

(13 citation statements)

References 4 publications

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“…It is interesting to observe that Gellert's method is mathematically equivalent to other well-known methods for solving systems of first-order ordinary differential equations (at least when applied to equation (1) with F = 0 ) , as we now show. First, note that equation (1) can be written in the form…”

Section: The Gellert Method'''mentioning

confidence: 92%

The methods of Gellert and of Brusa and Nigro are Padé approximant methods

Thomas

Gladwell

1984

Numerical Meth Engineering

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SUMMARYWe discuss properties of methods proposed recently by Gellert' and by Brusa and Nigro' for solving systems of second-order ordinary differential equations. In both cases, we extend the derivations to obtain whole families of methods with improved stability properties (roughly equivalent to L-stability for first-order systems) and we show that obtaining improvement in stability properties results in a deterioration in accuracy and/or implementation properties. Also we bring out the relationship (and in some cases equivalence) of these methods to others proposed in the literature. In particular, we show that the methods proposed by the above authors are essentially equivalent to well-known methods of long standing, namely Pad6 approximant methods for equivalent first-order systems of ordinary differential equations.

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Section: The Gellert Method'''mentioning

confidence: 92%

The methods of Gellert and of Brusa and Nigro are Padé approximant methods

Thomas

Gladwell

1984

Numerical Meth Engineering

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“…It can be seen that if m(n, by Leibniz's theorem, the left-hand side of equation (27) would still contain (1!x) as a factor. Hence, letting x"1 would give zero value.…”

Section: Order Of Accuracy When "1mentioning

confidence: 97%

“…On the other hand, if m"n and x"1, the left-hand side of equation (27) would give (!1)L n! As a result,…”

Section: Order Of Accuracy When "1mentioning

confidence: 98%

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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“…On the basis of Hamilton's law of varying action, Riff and Baruch [2] used cubic interpolation functions to construct a time integration scheme, which was found to be fourthorder accurate and conditionally stable. Argyris et al [3] and Gellert [4] used the Hermitian shape functions and the point collocation method to derive algorithms corresponding to the Padé approximations, and the algorithms were shown to be unconditionally stable and fourth-order accurate. Also using cubic shape functions and the collocation method, Golley [5] obtained a third-order accurate and conditionally stable algorithm with the second-order Gauss quadrature points as the collocation points.…”

Section: Introductionmentioning

confidence: 99%

Precise integration methods based on Lagrange piecewise interpolation polynomials

Wang

2008

Numerical Meth Engineering

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SUMMARYThis paper introduces two new types of precise integration methods for dynamic response analysis of structures, namely, the integral formula method and the homogenized initial system method. The applied loading vectors in the two algorithms are simulated by the Lagrange piecewise interpolation polynomials based on the zeros of the first Chebyshev polynomial. Developed on the basis of the integral formula and the Lagrange piecewise interpolation polynomial and combined with the recurrence relationship of some key parameters in the integral computation suggested in this paper with the solving process of linear algebraic equations, the integral formula method has been set up. On the basis of the Lagrange piecewise interpolation polynomial, and transforming the non-homogenous initial system into the homogeneous dynamic system, the homogenized initial system method without dimensional expanding is presented; this homogenized initial system method avoids the matrix inversion operation and is a general homogenized high-precision direct integration scheme. The accuracy of the presented time integration schemes is studied and is compared with those of other commonly used schemes; the presented time integration schemes have arbitrary order of accuracy, wider application and are less time consuming. Two numerical examples are also presented to demonstrate the applicability of these new methods.

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A new algorithm for integration of dynamic systems

Cited by 39 publications

References 4 publications

The methods of Gellert and of Brusa and Nigro are Padé approximant methods

The methods of Gellert and of Brusa and Nigro are Padé approximant methods

Complex-time-step Newmark methods with controllable numerical dissipation

Precise integration methods based on Lagrange piecewise interpolation polynomials

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