4.8.3. Time Step Selection

This section discusses the available time stepping approaches and values for a transient simulation. For more information on how to setup this transient simulation, see Transient Analysis. See the command reference for a complete list of Aria parameters that can be specified.

4.8.3.1. Initial Time Step Selection

The analysis of a transient diffusion problem requires the selection of a suitable time step for the integration procedure. Aria provides both Adaptive Time Step Selection and Fixed Time Step Selection functionality. In either case, though, some initial estimate of an appropriate time step is required.

If the initial step is too large, a loss of temporal accuracy can occur and produce a nonphysical oscillatory response. Likewise, an inappropriately small initial time step may produce nonphysical, spatial oscillations in the early time temperature field due to the limited resolution ability (of temperature gradients) of the finite element mesh. Either of these difficulties may lead to stability problems if the boundary value problem is nonlinear. In any event, the solution during the oscillatory period is not accurate and these occurrences should be avoided.

A reasonable initial time step will typically be a function of the length scale(s) of the problem as well as its material properties. See Guidelines for advice on selecting an initial timestep value.

For both fixed and adaptive time stepping, an integration method can optionally be specified by setting the Time Integration Method. See the theory section for more details on the available integration methods.

4.8.3.2. Fixed Time Step Selection

The simplest available time stepping strategy is fixed time stepping. As the name implies, this progresses the region at a constant time step until the completion criteria is met.

# Scope: Sierra > Procedure > Solution Control Description
Begin Parameters For ...
  ...
  begin parameters for Aria region
    initial time step size = 0.15
    time step variation = fixed

    # optionally specify integration method
    time integration method = bdf2
  end
end

Note that Aria may still attempt to adjust after a failed time step A way to ensure that failing the fixed timestep is fatal is by setting this value as the minimum timestep as well

# Scope: Sierra > Procedure > Solution Control Description
Begin Parameters For ...
  ...
  begin parameters for Aria region
    # time_step = {time_step = 0.15}
    initial time step size = {time_step}
    minimum time step size = {time_step}
    time step variation = fixed
  end
end

4.8.3.3. Adaptive Time Step Selection

Unlike fixed time stepping, adaptive time stepping allows for the step size to be adjusted dynamically based on user-specified criteria. These criteria are roughly split into two parts, error-based criteria and metric-based criteria. First, the error-based (Predictor Corrector) criteria ensures that the converged Newton step solution satisfies a specified error-bound. This is accomplished by estimating the solution error as a comparison to the predicted solution obtained from the previous steps. Aside from the error-based criteria, additional Metric-Based Criteria ensure that various aspects of the solution progress as desired. Once all these metrics are defined, the procedure for Step Size Selection and Time Step Failure is discussed.

4.8.3.3.1. Predictor Corrector

Let the predictor corrector tolerance, $\epsilon$ , be specified from the input file.

$\frac{ \Delta t_{n+1} }{ \Delta t_{n} } = \left( b \; \frac{ \epsilon}{d_{n+1}} \right)^{m}$

For 1st order time integrator $m = 1/2$ , $b = 2.0$ . For 2nd order time integrator $m = 1/3$

$b = 3 \left( 1 + \frac{ \Delta t_{n-1}}{\Delta t_{n} } \right)$

Then

$d_{n+1} = \frac{1}{U_{max}} \left[ \sum_{i=1}^{N} \left( U_{i(n+1)} - U^{p}_{i(n+1)}\right)^{2} \right]^{1/2}$

so that the timestep based on predictor criteria is

$\Delta t_{n+1} \leq \Delta t_\text{pred} = \Delta t_{n} \left( b \; \frac{ \epsilon}{d_{n+1}} \right)^{m} \;\;.$

See Predictor-Corrector for more information on the prediction process.

At least one field must be included in the predictor-based time step calculations. The default is for all fields to be included, in the calculation, but alternatively fields can be explicitly included

# Scope: Sierra > Procedure > Aria Region
Begin equation system myEqSys
  predictor fields = solid_phase_voltage
  predictor fields = liquid_phase_voltage

  ...
end

or excluded

# Scope: Sierra > Procedure > Aria Region
Begin equation system myEqSys
  predictor fields = not solid_phase_voltage
  predictor fields = not liquid_phase_voltage

  ...
end

for a given equation system.

Note

The time integration method often dictates how many solution steps into a simulation adaptive time stepping can begin. In most cases one must be able to obtain at least two successive solves using a fixed initial time step before adaptive time stepping begins. In many fluid simulations one way wish to increases this number of constant steps to e.g. three or four using the Corrector Begin After Step command line.

4.8.3.3.2. Metric-Based Criteria

Max CFL

In some instances the stability of a simulation (e.g. flow, mesh displacement, etc) can be governed by a Courant-Friedrichs-Levy (CFL) number. If the maximum value of the CFL over the domain for some quantity is $\text{CFL}_\text{max,i}$ and the user specifies a limit of $\text{CFL}_\text{lim,i}$ , then the bounding time step can be computed as

$\Delta t_{n+1} \leq \Delta t_\text{max,CFL} = \min_i \left[ \frac{\text{CFL}_\text{lim,i}}{\text{CFL}_\text{max,i}} \Delta {t}_{n} \right]$

See the COURANT LIMIT commands in the command reference for possible Courant limits to impose. For example

# Scope: Sierra > Procedure > Solution Control Description
Begin Parameters For ...
  ...
  begin parameters for Aria region
    Interface Courant Limit = 0.5
    Mesh Courant Limit      = 0.5
    Courant Limit           = 0.5

    ...
  end
end

Time Step Size Ratio

When using adaptive time stepping, a time step size ratio $r$ is defined to compare the change in time step

$r = \frac{\Delta t_{n+1}}{\Delta t_{n}}$

where $\Delta t_{n+1}$ is the proposed time step provided from the adaptive time step selection process and $\Delta t_{n}$ is the timestep of the current step.

The time step size ratio is bounded in order to assume the step size does not change abruptly. The maximum value $r_\text{max}$ ensures the step does not grow too quickly

$\Delta t_{n+1} \leq \Delta t_\text{r,max} = r_\text{max} \Delta {t}_{n} \;.$

Its value will fallback to a default if not given, or can be set using the Maximum Time Step Size Ratio command. For example,

# Scope: Sierra > Procedure > Solution Control Description
Begin Parameters For ...
  ...
  begin parameters for Aria region
    maximum time step size ratio = 4.2

    ...
  end
end

Maximum Solution Increment

This criteria attempts to limit the max nodal increment of some solution DOF $b$ to be less than $(\Delta b)_\text{max}$ so that given the rate $\dot{b}$ ,

$\Delta t_{n+1} \leq \Delta t_\text{incr} = 0.95* (\Delta b)_\text{max} / \dot{b}_\text{max}$

This limit is applied for all DOFs for which a Limit Solution Increment command is specified, e.g. for temperature:

# Scope: Sierra > Procedure > Solution Control Description
Begin Parameters For ...
  ...
  begin parameters for Aria region
    #Limit change in temperature to 2 K per step
    Limit Solution Increment Temperature = 2

    ...
  end
end

Note

Although adaptive time stepping with Solution Increment requires at least one successful solve here we require at least two successful solves since control of solution error is the primary objective.

Known Time Discontinuities

In certain problems, there might be points in time where are abrupt changes to the problem. Examples of this include an abrupt change in boundary condition or applied flux. If these times are known a priori, they can declared so that Aria ensures the discontinuity is hit exactly. If the next known discontinuity will occur at $t_{disc,next}$ , then

$\Delta t_{n+1} \leq \Delta t_\text{disc} = t_{disc,next} - t_{n}$

These times are declared with the Known Time Discontinuities line command as is demonstrated in the example below

# Scope: Sierra > Procedure > Solution Control Description
Begin Parameters For ...
  ...
  begin parameters for Aria region
    # List of known discontinous times
    Known Time Discontinuities = 1.2 3.4 5.6

    ...
  end
end

4.8.3.3.3. Step Size Selection

With all conditions defined, the new time step is then determined as the largest step which satisfies all criteria, i.e. for all $\Delta t_{n+1} \leq \Delta t_\text{lim,i}$ , the new step is determined by

$\Delta t_{n+1} = \min_i \Delta t_\text{lim,i}$

This calculated step is then optionally saturated by a minimum and/or maximum time step size $\Delta t_\text{min} \leq \Delta t_{n+1} \leq \Delta t_\text{max}$ e.g.

# Scope: Sierra > Procedure > Solution Control Description
Begin Parameters For ...
  ...
  begin parameters for Aria region
    Minimum Time Step Size = 1e-6
    Maximum Time Step Size = 1e-3

    ...
  end
end

If additional output times are requested with a Timestep adjustment interval, this step might be modified slightly to exactly hit the requested time.

Note

Certain model features (e.g. Chemistry) will require that the behavior within the current time step be considered when advancing the time step. When adaptive time step selection is used, the subcycled chemistry timestep cannot be included directly in the adaptive time step selection as the chemistry time step will seldom satisfy the time step criteria. Here the minimum subcycled chemistry time step is modified to be comparable with other adaptive time step criteria. See Chemistry for more information.

The overall time stepping procedure is demonstrated in Fig. 4.28.

Fig. 4.28 Time Step Selection Schematic

4.8.3.3.4. Time Step Failure

Aria transient simulations are monitored for time step failure. For each solution time step, we expect that the specified nonlinear tolerance will be achieved in at most the permitted number of nonlinear iterations of the Newton step. If the convergence condition is not met, the time step is said to have “Failed” and solution time will not be allowed to advance.

Note

The exception to this is when the user has explicitly stated that failed time steps should be accepted with the Accept Solution After Maximum Nonlinear Iterations command line.

If the non-converged solution is not accepted, a failed time step counter is incremented, and the time step is reduced by a factor $r_\text{fail}$ set by the Failed Time Step Size Ratio command

$\Delta t_{n+1} = r_\text{fail} \Delta {t}_{n} \;.$

Rather than allow a simulation to continually fail until hitting the minimum step, the Maximum Consecutive Time Step Failures command can be set to end the simulation after a specified number of consecutive failures. See Repeated Time Step Failure for troubleshooting these timestep failures.

During phase change, the model using the energy equation for temperature employs a mushy zone formulation hence it is important that the temperature field values not bypass the range of the mushy zone so that the heat of fusion be accounted for. In this case an additional algorithm is used to detect when a nodal temperature has bypassed the mushy zone. When the mushy zone has been bypassed within a timestep, the Newton step will be marked as failed and the current time step will be scaled by a user specified PHASE CHANGE RELAXATION FACTOR in the same way that the failed time step size ratio is applied

# Scope: Sierra > Procedure > Aria Region
Begin equation system myEqSys
  PHASE CHANGE RELAXATION FACTOR = 0.75

  ...
end

In some simulations a user may wish to interrupt the Newton step (fail the step) based upon the residual behavior. For many problems an aberrant behavior is recognized early on by noting the ratio of the current residual to the first residual. In cases where this ratio continues to grow it may be appropriate to terminate the Newton step and re-try with a different time step. This ratio can be specified by the user with the Maximum Linear Residual Ratio command.

Note

Failure of a time step due to violation of time step size ratio could be due to the adaptive time stepping selection criteria. If this is frequently encountered, one should consider whether any of the selection criteria values need to be modified in the time step parameters command block(s).

Note

In problems where the adaptive time step drops dramatically at some point in the simulation, the criteria previously described may cause the time step to fail repeatedly before the simulation step finally advances. This can notably increase the overall simulation time. If these drops happen at predictable times, they can be added as Known Time Discontinuities. Otherwise, the time step size ratio criteria can be reduced to permit more abrupt changes.