Properties of time integration with first order filter damping

Krenk, Steen; Høgsberg, Jan Becker

doi:10.1002/nme.1392

Cited by 18 publications

(57 citation statements)

References 20 publications

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“…As demonstrated in [8], the leading low-frequency damping term of the generalized method is 17 16 ( ) 3 , i.e. about four times larger than that for the present algorithm.…”

Section: Asymptotic Propertiesmentioning

confidence: 64%

“…the discussion in [7,8]. In the present case, it is seen from the coefficient to the quadratic term that this implies z ∞ = −( ) −1/2 , and it is easily verified that this is a quadruple root if = .…”

Section: Spectral Analysismentioning

confidence: 82%

“…Substitution of the damping terms with the factor 1 2 leads to an interpretation of the damping parameter similar to that of the balanced dissipation algorithm of the present paper and the use in the Newmark method and its generalized alpha form [7,8]. The properties of the present algorithm are characterized by an energy balance equation and a spectral analysis to be described next.…”

Section: Extended State-space Variablesmentioning

confidence: 90%

“…This corresponds to the frequency dependence of the Newmark algorithm with alpha damping [2][3][4] and is in fact given by the same expression [8].…”

Section: Low-frequency Asymptoticsmentioning

confidence: 99%

“…In spite of the analytic solution (48), the simplest way to obtain the lowfrequency asymptotics of the free vibration response is probably by matching a direct expansion of z in terms of using the characteristic equation (42) with = . A similar analysis of the generalized alpha method was carried out in [8]. The branch containing the 'static limit' z = 0 permits an asymptotic expansion of the form …”

Section: Asymptotic Propertiesmentioning

confidence: 99%

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Extended state‐space time integration with high‐frequency energy dissipation

Krenk

2007

Numerical Meth Engineering

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SUMMARYThe introduction of algorithmic dissipation into conservative time integration algorithms is addressed in two steps. The problem is formulated in state-space, and it is demonstrated that the use of two balanced dissipation terms-one in the equation of motion and the other in the relation between the state-space variables-can produce dissipation that increases monotonically with the modal frequency. This formulation is only partially satisfactory, because it leads to dissipation also in the low-frequency range. A substantially improved form of algorithmic dissipation is then developed by representing the two balanced dissipation terms via auxiliary state-space variables related to the displacement and velocity increments by a first-order filter. The filter technique retains the high-frequency behavior of the original balanced algorithm, but reduces the low-frequency dissipation from first order in the frequency to third order. Energy balance equations are derived, and the behavior of the two algorithms is described by detailed spectral analysis with explicit asymptotic results directly in terms of the damping parameter of the algorithms. Both algorithms satisfy full symmetry of displacement and normalized velocity variables for the free algorithmically damped vibration of any single mode and, therefore, exhibit monotonic transients without 'overshoot' effects. It is also demonstrated how the algorithm can be arranged in 'single solve' format, where only the displacement update requires a matrix equation solution, while the velocity and the auxiliary state-space variables are updated subsequently in each time step by simple vector relations with scalar coefficients.

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“…As demonstrated in [8], the leading low-frequency damping term of the generalized method is 17 16 ( ) 3 , i.e. about four times larger than that for the present algorithm.…”

Section: Asymptotic Propertiesmentioning

confidence: 64%

Section: Spectral Analysismentioning

confidence: 82%

Section: Extended State-space Variablesmentioning

confidence: 90%

“…This corresponds to the frequency dependence of the Newmark algorithm with alpha damping [2][3][4] and is in fact given by the same expression [8].…”

Section: Low-frequency Asymptoticsmentioning

confidence: 99%

Section: Asymptotic Propertiesmentioning

confidence: 99%

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Extended state‐space time integration with high‐frequency energy dissipation

Krenk

2007

Numerical Meth Engineering

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Rosenbrock‐based algorithms and subcycling strategies for real‐time nonlinear substructure testing

Bursi

Cui

Vulcan

et al. 2010

Earthq Engng Struct Dyn

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SUMMARYIn this paper, Rosenbrock-based algorithms originally developed for real-time testing of linear systems with dynamic substructuring are extended for use on nonlinear systems. With this objective in mind and for minimal overhead, both two-and three-stages linearly implicit real-time compatible algorithms were endowed with the Jacobian matrices requiring only one evaluation at the beginning of each time step. Moreover, these algorithms were improved with subcycling strategies. In detail, the paper briefly introduces Rosenbrock-based L-Stable Real-Time (LSRT) algorithms together with linearly implicit and explicit structural integrators, which are now commonly used to perform real-time tests. Then, the LSRT algorithms are analysed in terms of linearized stability with reference to an emulated spring pendulum, which was chosen as a nonlinear test problem, because it is able to exhibit a large and relatively slow nonlinear circular motion coupled to an axial motion that can be set to be stiff. The accuracy analysis on this system was performed for all the algorithms described. Following this, a coupled spring-pendulum example typical of real-time testing is analysed with respect to both stability and accuracy issues. Finally, the results of representative numerical simulations and real-time substructure tests, considering nonlinearities both in the numerical and the physical substructure, are explored. These tests were used to demonstrate how the LSRT algorithms can be used for substructuring tests with strongly nonlinear components.

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State‐space time integration with energy control and fourth‐order accuracy for linear dynamic systems

Krenk

2005

Numerical Meth Engineering

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SUMMARYA fourth-order accurate time integration algorithm with exact energy conservation for linear structural dynamics is presented. It is derived by integrating the phase-space representation and evaluating the resulting displacement and velocity integrals via integration by parts, substituting the time derivatives from the original differential equations. The resulting algorithm has an exact energy equation, in which the change of energy is equal to the work of the external forces minus a quadratic form of the damping matrix. This implies unconditional stability of the algorithm, and the relative phase error is of fourth-order. An optional high-frequency algorithmic damping is constructed by optimal combination of three different damping matrices, each proportional to either the mass or the stiffness matrix. This leads to a modified form of the undamped algorithm with scalar weights on some of the matrices introducing damping of fourth-order in the frequency. Thus, the low-frequency response is virtually undamped, and the algorithm remains third-order accurate even when algorithmic damping is included. The accuracy of the algorithm is illustrated by an application to pulse propagation in an elastic medium, where the algorithmic damping is used to reduce dispersion due to the spatial discretization, leading to a smooth solution with a clearly defined wave front.

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Properties of time integration with first order filter damping

Cited by 18 publications

References 20 publications

Extended state‐space time integration with high‐frequency energy dissipation

Extended state‐space time integration with high‐frequency energy dissipation

Rosenbrock‐based algorithms and subcycling strategies for real‐time nonlinear substructure testing

State‐space time integration with energy control and fourth‐order accuracy for linear dynamic systems

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