Overview

The Integrator is some middleware between Solver (in charge of solving the system) and Domain (in charge of managing the state of the system). It is mainly responsible for time integration in which the proper equation of motion can be formulated. To fulfill this task, the Integrator provides an additional layer and handles the communication between the Solver and the Domain, thus, it can also be deemed as a broker between the two. Due to this fact, a number of different (special) operations, for example, formulating the global damping model, can also be implemented via an Integrator. A number of integrators are implemented.

Newmark

But how can one determine which Integrator to use? The most widely used integrator is (probably) the Newmark integrator. Indeed, it is almost the standard practice to use it in earthquake engineering, and there is no need to justify the choice. But the Newmark method is not always the best choice.

Bathe Two--Step

If energy and momentum conservations matter, the BatheTwoStep integrator provides a very cost-efficient solution. The performance should be comparable to the Newmark integrator.

Generalized-\(\alpha\)

If one wants to customise algorithmic damping, the GeneralizedAlpha integrator can be used. By adjusting two parameters, several other methods can be recovered. Since the equation of motion is satisfied somewhere within the time step (rather than the beginning/ending), it requires roughly a factor of two more vector operations than the Newmark integrator. However, vector operations are not costly and are mostly implemented in a parallel fashion, it is not considered a severe performance issue.

GSSSS

The most general integrator is the GSSSS integrator. The optimal performance (in terms of overshoot, energy dissipation/dispersion) can be achieved by using the U0-V0 Optimal scheme.

The GSSSS integrator requires an additional iteration to synchronise the state of the system. Thus, the performance is sightly higher than that of the GeneralizedAlpha integrator.

Central Difference (Not Available)

The explicit central difference method is frequently introduced in textbooks on dynamics due to its simplicity. The major benefit(s) is that the stiffness matrix does not enter the left-hand side of the equation of motion, which means, under certain conditions, factorisation of global effective matrix would only be done once. This leads to "very efficient" solutions.

However, it is in general difficult to meet those conditions. It is not implemented, and users are discouraged from using central difference in seismic engineering.

Constant Mass and Damping Matrices

The mass matrix may be constant, but there is no guarantee that the damping matrix is constant. Thus, in the nonlinear context, it can only be assumed that the left-hand side of the equation of motion would change in each sub-step.

Convergence

To ensure convergence, the time step should be very small, and should reduce further if the mesh is refined. The maximum allowable time step is associated with the minimum period of all elements. If the mass matrix is not fully integrated ( semi-positive definite), the time step would be unnecessarily small.