Threshold violations

 

The threshold violation warnings are meant to point out potential problems with the solution. They may not actually indicate a serious problem with the solution. Alamode checks for violations by default because it is likely that all of the unknowns represent concentrations, and it is not physical to have negative concentrations. Thresholds are adjusted after a nonlinear update because negatives can cause nonlinear convergence problems.

Threshold violation warnings that are printed can usually be ignored if they do not occur on the last nonlinear update. These are just overshoot in the nonlinear update, and they do not indicate a problem with the converged solution.

Threshold violation warnings/adjustments that occur on the last nonlinear update may be an indication of a problem with the converged solution. Also, if the violations are large and they were adjusted, conservation of dose is no longer guaranteed. For this reason, -failIfHadThresholdViolation is on by default.

Suggestions:

• Check the magnitude, field, and location of the violations (turn on -printThresholdViolations). This can generate a lot of output because every nodal violation is printed.
• Change the warn threshold for offending fields (-warnThreshold). If the solution of any field can go negative, this should be set to a large negative value along with -thresholdValue for that field. Concentration fields that dissolve entirely (e.g. clusters) should have a small negative value for the warn threshold to avoid numerical noise being falsely interpreted as a failure criteria.

Threshold violations that occur with bottoming out of residual and/or update norms in the Newton solve may be due to taking too large of a timestep in either the TR or BDF2 step. Neither substep is monotonic, and a consistent application of the algorithm to an ODE system may lead to oscillating signs. The cause of these oscillations can be seen by applying TR/BDF2 to the test ODE

where . Let . The TR step is

where and . Thus, the TR sign change comes from taking a large timestep compared to the eigenvalues of the system before the modes associated with the large eigenvalues have decayed enough. The BDF2 step is

where . If the concentration has been reduced by significant fraction over the TR step , the sign on C will change.

These sign changes are a result of the consistent application of the timestepping algorithm. Adjust negative values in the nonlinear solve will not lead to convergence, because the converged values for the consistent timestep equations should be negative. Although this situation will fix itself through nonlinear failures and repeating integration over the time step interval with a smaller time step, the failures are not a productive use of computing resources.

Suggestions for dealing with threshold violations arising from the sign change conditions in TR/BDF2 include:

• Let the field's concentration go significantly negative (e.g. ). For fields representing cluster concentrations, this often does not cause problems for the nonlinear solutions (the field commands -warnThreshold and -thresholdValue are used to set threshold detection and adjustment parameters).
• Tighten the timestepping tolerance associated with field controlling the time step (-relTSTol).
• Add -avoidBDF2DecreaseViolations to the model to detect the reduction condition after the TR substep and estimate a new timestep that should not cause nonlinear failure during BDF2. The compute time spent on the TR substep is wasted, but there were usually fewer nonlinear iterations spent computing that than would be taken before giving up during the BDF2 substep. This approach is justified based on analysis of the method for the test equation and is not guaranteed to avoid all failures in BDF2.
• Use a non-zero offset field to hide large relative reductions the equivalent zero-offset field. The -offsetVariable (2.3.4) is provided to ensure that the equivalent zero-offset field's values will in computing nonlinear convergence norms and temporal local truncation error estimates that are used to control the timestep. For example, a cluster equation given by

  

  could be rewritten in terms of an offset field (O = C + o, where o is the offset which would typically be to ) as

  

  The offset value should not be too large as that will lead to problems related to floating point truncation error in the nonlinear update and temporal LTE norm calculations based on the zero-offset field.

• Use an alternative timestep algorithm. Backward Euler (-timeIntegrator backwardEuler) is a first-order method that does not appear to have a structural sign change. Since it is a first-order method, it will likely require many more time steps to achieve the same accuracy as TR/BDF2.