Under relaxation is a constraint on the change of a dependent or auxiliary variable from one solution iteration to the next. It is required to maintain the stability of the coupled, non-linear system of equations. The Relax tab in the Solver Control panel (see figure 1) allows the user to set under relaxation factors for each of the solved variables, as well as for auxiliary variables.

Figure 1.  Solver Control - Under Relaxation

The panel contains four columns: the first defines the variable, the second contains a slider bar that can be used to adjust the value, the third contains up/down buttons to adjust the order of magnitude of the value, and the fourth is a field for specifying the under relaxation value directly.

We have different methods for applying under relaxation for the solved and auxiliary variables:

Inertial under relaxation (I) is applied to variables that are directly solved for (dependent variables as determined by the active modules) during the iterative procedure, for example, velocities, pressure correction, enthalpy.

• I usually varies from 0.0 to 2.0 with a default value of 0.2.
• Increasing the value of I adds constraint. It means increasing I increases stability.
• Increasing the value of I slows convergence. It means an increase in I will result in more iterations to reach the same order of convergence.
• Values of I greater than 1.5 are allowed but not recommended.

Linear Relaxation

Linear under relaxation (L) is applied to the auxiliary variables, which are computed from the solved (dependent) variables, for example, density, pressure, temperature.

• L usually varies from 0.0 to 1.0 with default value of 1.0.
• Decreasing the value of L adds constraint. It means decreasing L increases stability.
• Decreasing the value of L slows convergence. It means a decrease in L will result in more iterations to reach the same order of convergence.
  

It all can be summarized in figure 2 and figure 3 below.

Figure 2.  Under Relaxation for Faster Convergence

Figure 3.  Under Relaxation for More Stability

Note: Relaxation can help in getting faster convergence or it may help prevent divergence. For a given problem (identical BC/VC/IC), change in relaxation values may yield more or fewer of iterations to reach convergence. However, as long as the simulation is fully converged, you will get the same result, irrespective of relaxation values.

Tips on troubleshooting your problems

The following tips are just guidelines that can help in getting a converged solution or faster convergence. The values for relaxation are problem specific, so there are no hard and fast rules as to which value should be used.

1. Simulation diverges

If you see that your problem is diverging, you can try the following:

• Make sure that you have applied correct scaling, and that all input values (BC/VC) are correct or at least within a reasonable range.
• Check the residuals and look for the variable that diverges or starts diverging first.
• Decrease the linear under relaxation (from 1.0 to say 0.7) for that variable.
• In order to make it more stable, you can also increase inertial under relaxation for the corresponding solved (dependent) variable.
• For compressible flows, decreasing linear under relaxation for density helps.
• For problem involving heat transfer, if you see enthalpy diverging, decreasing the linear relaxation of temperature from the default value of 1.0 to a smaller value like 0.7 can help in getting a converged solution.
• For Fluid/Structure Interaction problems, decreasing the linear relaxation for pressure helps moderate the pressure fluctuations seen by the stress solver, reducing the displacement fluctuations and aiding in convergence.
• If you encounter negative volumes in a Fluid/Structure Interaction problem, first try to decrease the linear relaxation for pressure to a value of 0.3 or 0.2. If the problem persists, then you can try to decrease the linear relaxation for Grid Deformation anywhere from 0.5 to 0.1. This basically restricts the grid deformation in the solid volumes to 50% (if a value of 0.5 is used) of the actual value due to sustained pressures every time you solve for stress during the time step. Upon convergence, you still get the correct grid deformation.
• For complex physics, when small changes in relaxation do not work, change the inertial relaxation values to 0.5. Also, reduce the linear factors to 0.3 and rerun. If this does not work, change the inertial factors to 0.9 and the linear ones to 0.1. These factors can be changed up to 1.5 for inertial and 0.01 for linear. Anything higher may result in a solution that is frozen to the initial field.

Another item that may help is a change to the AMG solver for pressure correction or enthalpy. If convergence problems still exist, look at the residual information, noting the location of the maximum residual. Next, examine the grid closely at this spot in CFD-VIEW and look for skewness. Sometimes, problem areas can be isolated by plotting the results every few iterations. The problem area is generally the location where the flow field first becomes unstable.

2. Slower convergence

If you see that convergence is very slow, you can try the following:

• Check the residuals and see what variable has slow convergence (might also remain flat)
• Decrease the inertial under relaxation (from 0.2 to say 0.02) for that variable.
• For conjugate heat transfer problems, decreasing the inertial relaxation of enthalpy from the default value of 0.05 to a smaller number like 1E-05 can help in faster convergence.
• When solving for the electric module, decreasing the inertial relaxation of the electric potential from 0.0001 to a smaller number like 1E-07 can help in faster convergence.

The next time you set up your simulation, consider these suggestions for setting your Solver Control parameters.

Regards,
Kartik Shah
ESI CFD Support Team