Energy conservation in Newmark based time integration algorithms

Abstract

Energy balance equations are established for the Newmark time integration algorithm, and for the derived algorithms with algorithmic damping introduced via averaging, the so-called α-methods. The energy balance equations form a sequence applicable to: Newmark integration of the undamped equations of motion, an extended form including structural damping, and finally the generalized form including structural as well as algorithmic damping. In all three cases the expression for energy, appearing in the balance equation, is the mechanical energy plus some additional terms generated by the discretization by the algorithm. The magnitude and character of these terms as well as the associated damping terms are discussed in relation to energy conservation and stability of the algorithms. It is demonstrated that the additional terms in the energy lead to periodic fluctuations of the mechanical energy and are the cause of the phenomenon of response 'overshoot', previously observed empirically in the application of Newmark based algorithms to high frequency components. It is also demonstrated that the stability limit of the explicit Newmark algorithm is reached, when the stiffness term in the algorithmic energy vanishes, and that energy fluctuations take place for integration intervals close to the stability limit.In this equation M, C and K are the mass, damping and stiffness matrices of the system, and f (t) is the external load vector, conjugate to the displacements u(t). A central problem to which much effort has been devoted over the last fifty years is the numerical integration of this equation of motion. The preferred methods are of the single-step type consisting of updating the displacement, velocity and acceleration vectors u,u,ü at current time t n to the time t n+1 = t n + h, a small time interval h later. It is desirable that the algorithm has at least second order accuracy, and because the spatial discretization used in structural dynamics often leads to inclusion of high-frequency modes in the model, it is also desirable to have unconditional stability.There are two basically different approaches to the development of single-step time integration algorithms: collocation of the equation of motion at selected points in time in connection with a set of difference relations between the displacement, velocity and acceleration vectors, or the the use of an integrated form of the equations of motion whereby the acceleration is eliminated and

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

“…This behaviour does not violate the stability of the algorithm, demonstrated by the spectral analysis of Section 4.1. It has recently been established that in the Newmark algorithm and its -modified forms algorithmic damping leads to a reinterpretation of the energy, as defined by the incremental term on the left side of the energy balance equation [18]. The implication is, that although the algorithms are stable, and even dissipating, there may be local fluctuations of the mechanical energy.…”

A new family of generalized‐α time integration algorithms without overshoot for structural dynamics

“…Overshoot is obviously an undesirable feature for a numerical algorithm when it is used to solve a linear or nonlinear transient structural dynamic problem. Refer to References [14,17] for recent discussions.…”

External Coupling of Fe Codes for Structural Dynamics

“…This additional term with respect to the mechanical energy int tot kin W W W  appearing in the balance energy equation is generated by the Newmark time discretization (Krenk,[14]). The interface energy coming from the coupling algorithms corresponds to the unbalanced energy when considered only energy conserving schemes.…”