Time Integration Algorithms |
 
|
Time Integration Algorithms |
|
For time domain solutions, the equilibrium equations of motions are integrated in time using a step-by-step procedure. So rather than solving the equations at any time t, the solution is obtained at discrete time steps, Δt apart. In order to facilitate this approach, assumptions must be made about the variation of displacement, velocity and acceleration within each time interval. Implicit integration is used exclusively by Flexcom, whereby the solution at time t+Δt is based on the conditions at time t+Δt. This means the exact equations of motion are solved for at each solution time, ensuring that precise equilibrium is achieved. This approach contrasts with that of explicit integration, where the solution at time t+Δt is based on the conditions at time t. This means the equilibrium equations are never exactly solved for at any solution time, so very fine time steps are generally required to maintain the accuracy and stability of the solution.
Flexcom provides two algorithms for the discretisation of finite element equations of motion in time, namely the Hilber et al. (1977) integration method and the Generalised-α method (Chung. and Hulbert, 1993). Both approaches are similar in many regards – the Hilber-Hughes-Taylor operator is in fact a special case of the Generalised-α method. In early versions of Flexcom, Hilber-Hughes-Taylor was the only option available, but the options have been extended to include Generalised-α as the program evolved.
The Generalised-α method achieves an optimal combination of high-frequency and low-frequency dissipation, so that for any given value of high frequency dissipation, the low frequency dissipation is minimised. This is particularly beneficial for analyses with high frequency noise, such as for example those involving intermittent contact, but should also in general allow most dynamic analyses to run with larger time steps than the Hilber-Hughes-Taylor operator. Further details are provided in Theory.
Earlier versions of Flexcom (up to and including Flexcom 8.10) used the Hilber-Hughes-Taylor operator as the default. Generalised-Alpha has since been shown to provide more effective numerical damping, particularly for sensitive models, so it is now the default method. If you are re-running some old simulations which previously used Hilber-Hughes, it is possible that you may notice some very slight differences in results.
•*TIME STEPPING is used to select the time stepping algorithm and to define associated numerical damping coefficients.