Monitor Residual Values, Solution Imbalances, and Quantities of Interest

By: Mike Kuron, M.S.M.E., Project Manager at CAE Associates.

Fluid flow problems posed by common products are highly nonlinear in nature. Only through imposing restrictive conditions can the governing Navier-Stokes equations be solved analytically. As a result, computational fluid dynamics (CFD) solutions must be calculated iteratively. This begs the question: How do I know when my solution has converged?

Since the point at which the analysis is deemed converged is defined by the judgment of the analyst, users should have a solid understanding of when the analysis has reached its final solution. Typically, when assessing the convergence of a steady state CFD analysis, at a minimum monitor the following three criteria as the analysis progresses:

- Residual Values
- Solution imbalances
- Quantities of interest

# 1. CFD Convergence using Residual Values

The residual is one of the most fundamental measures of an iterative solution’s convergence, as it directly quantifies the error in the solution of the system of equations. In a CFD analysis, the residual measures the local imbalance of a conserved variable in each control volume. Therefore, every cell in your model will have its own residual value for each of the equations being solved.

In an iterative numerical solution, the residual will never be exactly zero. However, the lower the residual value is, the more numerically accurate the solution. Each CFD code will have its own procedure for normalizing the solution residuals. It is best to check your code’s documentation for guidance on an appropriate criteria when judging convergence.

For CFD, RMS residual levels of 1E-4 are considered to be loosely converged, levels of 1E-5 are considered to be well converged, and levels of 1E-6 are considered to be tightly converged. For complicated problems, however, it’s not always possible to achieve residual levels as low as 1E-6 or even 1E-5.

For example, look at the residual behavior of various heat generating components sitting atop a printed circuit board (PCB) which is cooled by natural convection. The residual monitors in Figure 1 demonstrate monotonic convergence, indicating a well-posed problem and a tightly converged solution.

So how does the solution change as the residuals decrease? Figure 2 shows the temperature field in the components on the board at different residual levels.

With the RMS residuals at 1E-4, the qualitative behavior of the PCB can clearly be seen, but the peak temperature of the heat sink is under-predicted by as much as 8°C. As the RMS residuals decrease to 1E-5, the temperature distribution begins to resemble the more tightly converged solutions and the peak temperature is predicted to within 1°C. Finally, as the solution further converges, the difference in the temperature distribution between residual levels of 1E-5 and 1E-6 is almost negligible.

# 2. CFD Convergence using Solution Imbalances

Since our CFD analysis is solving conservation equations (conservation of mass, momentum, energy, etc.), we must ensure that the final solution does indeed conserve these quantities.

As a numerical representation of a physical system, the CFD solution imbalances will never be exactly zero. However, the imbalances should be sufficiently small before considering the solution converged. As a good practice, aim for solution imbalances of less than 1% as a starting point. Note that more sensitive applications may require tighter convergence.

As can be seen in Figure 3, after the initial startup period, the solution imbalances for the board example gradually decrease as the solution progresses. For the most part, having sufficiently small values of the solution residuals will lead to small solution imbalances.

However, for cases that have processes with widely different timescales, it is possible to have large imbalances even when the residuals are small. A common example of this occurs in Conjugate Heat Transfer (CHT) analyses, like our board example here, where the conduction timescales can be much longer than the flow timescales.

# 3. CFD Convergence using Quantities of Interest

In a steady state analysis, the solution field should not change iteration to iteration for an analysis to be deemed converged. Monitoring integrated quantities such as force, drag, or average temperature can help the user judge when his or her analysis has reached this point. In our board example, two useful monitor points might be the maximum temperature of the heat sink and the maximum temperature of one of the heat generating chips.

In Figure 4, we can see the change in the monitor point values vs. iteration number and RMS residual value. After approximately 50 iterations, the RMS residuals are reduced to 1E-4 and the chip temperature monitor point is within just a few percent of its final value.

However, the heat sink temperature is still far from its final value, so stopping the analysis here could be misleading. As the residuals decrease further, the monitor values change less and less between iterations. Once the monitor point values have “flattened out”, we can safely assume the solution is converged.

There are, of course, many other things to consider when judging the value of CFD results. Just because the solution is numerically accurate does not mean that it is a good representation of the true physical behavior. A converged solution is not very useful if it’s a byproduct of incorrect boundary conditions! Any model should be thoroughly checked, from geometry and mesh to boundary conditions and solver settings to ensure its suitability for the problem at hand.

For more simulation tips read: Debugging Complex Finite Element Analysis Using a Single Element Model.

Mike Kuron, M.S.M.E., is a Project Manager at CAE Associates, and has extensive experience performing CFD and FEA simulations in the aerospace, nuclear, defense, power generation, and electronics industries. He is currently pursuing his Ph.D. at the University of Connecticut, concentrating in the field of computational turbulent combustion.