Conservative fourth-order time integration of non-linear dynamic systems

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Abstract

An energy conserving time integration algorithm with fourth-order accuracy is developed for dynamic systems with nonlinear stiffness. The discrete formulation is derived by integrating the differential state-space equations of motion over the integration time increment, and then evaluating the resulting time integrals of the inertia and stiffness terms via integration by parts. This process introduces the time derivatives of the state space variables, and these are then substituted from the original state-space differential equations. The resulting discrete form of the state-space equations is a direct fourth-order accurate representation of the original differential equations. This fourth-order form is energy conserving for systems with force potential in the form of a quartic polynomial in the displacement components. Energy conservation for a force potential of general form is obtained by addition of a higher order secant-type correction term. The formulation leads to a consistent representation of the motion within a time increment corresponding to cubic Hermite interpolation in time. This in turn leads to excellent phase representation with only a small fourth-order error, permitting integration of oscillatory systems with only a few integration points per period. Three numerical examples demonstrate the high accuracy of the algorithm. (C) 2015 Elsevier B.V. All rights reserved.
Original languageEnglish
JournalComputer Methods in Applied Mechanics and Engineering
Volume295
Pages (from-to)39-55
ISSN0045-7825
DOIs
Publication statusPublished - 2015

Keywords

  • Nonlinear dynamics
  • Time integration
  • Energy conservation
  • Fourth-order accuracy

Cite this

@article{2ef8d60659cd40b4a2fe86b792adb413,
title = "Conservative fourth-order time integration of non-linear dynamic systems",
abstract = "An energy conserving time integration algorithm with fourth-order accuracy is developed for dynamic systems with nonlinear stiffness. The discrete formulation is derived by integrating the differential state-space equations of motion over the integration time increment, and then evaluating the resulting time integrals of the inertia and stiffness terms via integration by parts. This process introduces the time derivatives of the state space variables, and these are then substituted from the original state-space differential equations. The resulting discrete form of the state-space equations is a direct fourth-order accurate representation of the original differential equations. This fourth-order form is energy conserving for systems with force potential in the form of a quartic polynomial in the displacement components. Energy conservation for a force potential of general form is obtained by addition of a higher order secant-type correction term. The formulation leads to a consistent representation of the motion within a time increment corresponding to cubic Hermite interpolation in time. This in turn leads to excellent phase representation with only a small fourth-order error, permitting integration of oscillatory systems with only a few integration points per period. Three numerical examples demonstrate the high accuracy of the algorithm. (C) 2015 Elsevier B.V. All rights reserved.",
keywords = "Nonlinear dynamics, Time integration, Energy conservation, Fourth-order accuracy",
author = "Steen Krenk",
year = "2015",
doi = "10.1016/j.cma.2015.06.016",
language = "English",
volume = "295",
pages = "39--55",
journal = "Computer Methods in Applied Mechanics and Engineering",
issn = "0045-7825",
publisher = "Elsevier",

}

Conservative fourth-order time integration of non-linear dynamic systems. / Krenk, Steen.

In: Computer Methods in Applied Mechanics and Engineering, Vol. 295, 2015, p. 39-55.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Conservative fourth-order time integration of non-linear dynamic systems

AU - Krenk, Steen

PY - 2015

Y1 - 2015

N2 - An energy conserving time integration algorithm with fourth-order accuracy is developed for dynamic systems with nonlinear stiffness. The discrete formulation is derived by integrating the differential state-space equations of motion over the integration time increment, and then evaluating the resulting time integrals of the inertia and stiffness terms via integration by parts. This process introduces the time derivatives of the state space variables, and these are then substituted from the original state-space differential equations. The resulting discrete form of the state-space equations is a direct fourth-order accurate representation of the original differential equations. This fourth-order form is energy conserving for systems with force potential in the form of a quartic polynomial in the displacement components. Energy conservation for a force potential of general form is obtained by addition of a higher order secant-type correction term. The formulation leads to a consistent representation of the motion within a time increment corresponding to cubic Hermite interpolation in time. This in turn leads to excellent phase representation with only a small fourth-order error, permitting integration of oscillatory systems with only a few integration points per period. Three numerical examples demonstrate the high accuracy of the algorithm. (C) 2015 Elsevier B.V. All rights reserved.

AB - An energy conserving time integration algorithm with fourth-order accuracy is developed for dynamic systems with nonlinear stiffness. The discrete formulation is derived by integrating the differential state-space equations of motion over the integration time increment, and then evaluating the resulting time integrals of the inertia and stiffness terms via integration by parts. This process introduces the time derivatives of the state space variables, and these are then substituted from the original state-space differential equations. The resulting discrete form of the state-space equations is a direct fourth-order accurate representation of the original differential equations. This fourth-order form is energy conserving for systems with force potential in the form of a quartic polynomial in the displacement components. Energy conservation for a force potential of general form is obtained by addition of a higher order secant-type correction term. The formulation leads to a consistent representation of the motion within a time increment corresponding to cubic Hermite interpolation in time. This in turn leads to excellent phase representation with only a small fourth-order error, permitting integration of oscillatory systems with only a few integration points per period. Three numerical examples demonstrate the high accuracy of the algorithm. (C) 2015 Elsevier B.V. All rights reserved.

KW - Nonlinear dynamics

KW - Time integration

KW - Energy conservation

KW - Fourth-order accuracy

U2 - 10.1016/j.cma.2015.06.016

DO - 10.1016/j.cma.2015.06.016

M3 - Journal article

VL - 295

SP - 39

EP - 55

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

ER -