Variable Time Stepping |
 
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Variable Time Stepping |
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In a variable step analysis, the choice of time step magnitude is made by the program based on a number of criteria. The time step is continuously monitored and varied as appropriate by the program within user-specified limits, to ensure a stable and convergent solution. A variable step is typically used in analyses where the structural response varies significantly during the course of the simulation. A good example might be that of a steel catenary riser with intermittent seabed contact. However, a variable time step may also be useful during the development phase for any particular model, regardless of the nature of structural response. It is useful in this context to determine if a fixed time step is potentially suitable or not, and if so, to provide an indication regarding a suitable value of time step. Some other points worth noting in relation to variable time step analyses are as follows:
•A variable time step introduces a computational overhead, due to constant monitoring of various parameters to facilitate the selection of an optimum time step (these parameters are discussed in more detail shortly).
•Changes of time-step can introduce numerical noise into the solution.
•Caution is advised against the specification of an excessively large maximum time-step value in an effort to speed up the analysis. The variable time stepping algorithm will try to achieve the maximum value at every possible opportunity, and this can actually make the overall solution less efficient, or even affect solution robustness in certain circumstances.
The relevant inputs in the context of a variable time step analysis are Suggested Time Step, Minimum Time Step, Maximum Time Step, and optionally a Step Length parameter. The first three parameters are self-explanatory. If you are using a variable time step, the program chooses an optimum time step based on two main criteria, namely (i) the number of iterations required for the last three convergent solutions, and (ii) the instantaneous current response period, which is a measure of the dominant period in the response at any particular instant.
The mathematical definition of the current period parameter 
was initially developed by Bergan et al., (1985), and applied to the analysis of impact loading on rigid beam structures. In a Flexcom time domain dynamic analysis, 
is computed at the nth solution time using the equations:
(1)
and:
(2)
where:
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is the incremental nodal displacement vector. The increment is the difference in 
for the element between the (n-1)th and nth solution times, that is, 
•
is the element stiffness matrix
•
is the element mass matrix
and the summation is performed over all the elements of the structure.
The optimum time step, when this is based on the dominant response period, is obtained by multiplying the instantaneous current period by the Step Length parameter. So for example the default Step Length value of 0.055 means the time step is approximately 1/18th of the dominant period in the dynamic response at any time. The default value is adequate in almost all cases, but may occasionally be increased (say to 0.1 or 1/10th) by experienced users who feel the resulting time step is too short.
It is shown by Bergan. & Mollestad, (1985) that 
is a weighted average of the dominant frequencies comprising the instantaneous response of the structure. So if the structure is responding in a high frequency mode, 
will be large, and consequently 
and by definition the optimum step size will be small. The opposite is true for low frequency compliant response.
The sole use of the current period parameter as a means of determining the time-step size would result in frequent and possibly large changes in the size of the step during a dynamic analysis. This in turn would introduce undesirable numerical oscillations into the solution. To counteract this, an efficient scheme has been implemented such that the size and frequency of time-step changes are moderated, and an optimum stable step is reached, if possible. This scheme is based on two main principles, namely (i) not changing the step size if the difference between old and new values is below a certain tolerance, and (ii) basing decisions on whether to change the time-step (though not decisions on the actual time-step value to use) on the variation in the current period over the last number of solution times rather than just the last one.
•*TIME is used to define time parameters for an analysis. Specifically, the STEP=VARIABLE inputs are used to specify fixed time step data.
If you would like to see an example of how this keyword is used in practice, refer to C02 - Multi-Line Flexible System.