Global format for energy-momentum based time integration in nonlinear dynamics

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A global format is developed for momentum and energy consistent time integration of second‐order dynamic systems with general nonlinear stiffness. The algorithm is formulated by integrating the state‐space equations of motion over the time increment. The internal force is first represented in fourth‐order form consisting of the end‐point mean value plus a term containing the stiffness matrix increment. This form gives energy conservation for systems with internal energy as a quartic function of the displacement components. This representation is then extended to general energy conservation via a discrete gradient representation. The present procedure works directly with the internal force and the stiffness matrix at the time integration interval end‐points, and in contrast to previous energy‐conserving algorithms, it does not require any special form of the energy function nor use of mean value products at the element level or explicit use of a geometric stiffness matrix. An optional monotonic algorithmic damping, increasing with response frequency, is developed in terms of a single damping parameter. In the solution procedure, the velocity is eliminated and the nonlinear iterations are based on the displacement components alone. The procedure represents an energy consistent alternative to available collocation methods, with an equally simple implementation. Copyright © 2014 John Wiley & Sons, Ltd.
Original languageEnglish
JournalInternational Journal for Numerical Methods in Engineering
Issue number6
Pages (from-to)458-476
Number of pages19
Publication statusPublished - 2014


  • Time integration
  • Nonlinear dynamics
  • Energy conservation


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