Energy conservation and high-frequency damping in numerical time integration

Momentum and energy conserving time integration procedures are receiving increased interest due to the central role of conservation properties in relation to the problems under investigation. However, most problems in structural dynamics are based on models that are first discretized in space, en this often leads to a fairly large number of high-frequency modes, that are not represented well – and occasionally directly erroneously – by the model. It is desirable to cure this problem by devising algorithms that include the possibility of introducing algorithmic energy dissipation of the high-frequency modes. The problem is well known from classic collocation based algorithms – notably various forms of the Newmark algorithm – where the equation of motion is supplemented by approximate relations between displacement, velocity and acceleration. Here adjustment of the algorithmic parameters can be used to introduce so-called α-damping, and an improved form leading only to high-frequency damping can be obtained by suitable averaging of the equilibrium equation at onsecutive time steps. Conservative time integration algorithms are obtained by use of an integral of the equation of motion and the acceleration therefore does not appear as an independent parameter of these algorithms. Typically they do not contain algorithmic parameters either. Algorithmic damping can then be introduced in two ways: either by introducing artificial damping in terms of the displacement and velocity vectors, or by introducing additional variables to represent damping. In the present paper it is demonstrated, how damping equivalent to the α-damping of the Newmark algorithm can be introduced directly via displacement and velocity dependent terms. It is furthermore shown, how this damping can be improved by introduction of a new set of variables related to the displacement and velocity vectors by a suitable first order filter with scalar coefficients. By this device an algorithmic damping can be obtained that is of third order in the low-frequency regime. It is an important feature of both algorithms that they can be arranged to require in each time step only the solution of a system of equations of the same size of the corresponding quasi-static problem, followed by one or three vector updates with scalar coefficients – the so-called ‘single step – single solve’ property.