Originally Posted by Aaron
;121326

Dave,

Here is an answer to your question.

In order to be able to conclude a grid independent solution for this problem, you will need at least three meshes. Unfortunately, the actual size of them can not be known a priori. However, it is assumed in numerical analysis that an infinitely dense mesh will yield the exact solution of the PDEs being solved, minus the truncation error. That isn't possible of course, so there may be some guidance in the literature for what size to start with and perhaps that is why you chose your sizes as you did. 200k or 800k may be reasonable sizes, but they will need to be confirmed in comparison to other grid sizes and/or in comparison to real-world validation data.

To conduct the grid independence analysis, start with the grid solution for which you have the most confidence. Then multiply that grid size by a factor of 1.3 either coarsening or refining. The value of 1.3 is known through experience to yield reasonable enough differences in grid sizes that if the calculated solutions are similar, then the actual solution is not dependent upon the grid. This is exactly what you are trying to prove.

So, for the 200,000 cell grid, refining at 1.3 gives you 260,000 cells. Now comes the question of which direction to refine. In 2D, should it be X or Y or both? For complex situations, it may be important to find out if there are directional influences. For pulsatile flow, this may indeed be the case. However, to make some initial progress, I recommend a uniform refinement in both X and Y. Solve the new grid and check to see how closely the values agree. If they are relatively close, refine the grid containing 260,000 by a factor of 1.3 giving 338,000 cells. Again solve this grid and compare values. If they are not close, conduct this analysis on your finer grid for meshes of 800K, 1040K, and 1352K. Compare the values.

In either the coarser or finer cases, you will be able to ascertain the percent relative difference ((new-old)/old) x 100 between your values of interest. How closely your values of interest are to one another are an indication of the uncertainty in your solution due to the grid. The PRD may be 10% or it may be 90%, or the deltas between values may be small or large. How acceptably large they are will be decided by some qualitative metric, such as, "Decreasing the percent relative difference by an additional 5% (from 15% to 10%) will require solving the flow problem on an additional grid of 1500K cells. Based on the previous calculations, this will necessitate an additional week of computation time, which will cost X dollars." Or you may have real-world data to compare to. In that case, again, how closely you need to get to them will determine where you stop your simulations.

As an example, lets assume you extract scalar values from three of your grids. They are, in order of increasing grid cell size, 16.23, 14.51, and 13.75. The absolute differences between these values are 1.72 and 0.76. The absolute percent relative differences are 11.85 and 5.53. From this we can see that the differences between solution values are decreasing as the total grid cell size increases. But do they continue to decrease or do they start to level off? If a fourth grid with even more cell sizes is added, and the extracted value of interest (extracted from same point in time and space as the other grids) is 13.56, the newest absolute difference is 0.19 and the newest percent relative difference is 1.4.

So, we have 1.72, 0.76, and 0.19 for absolute differences and 11.85, 5.53, and 1.4 for relative differences. From these numbers we can see that there is not a constant rate of decrease. Instead, we see that the rate of decrease is changing from, let's call it "highly sloping" to "gently sloping". What this means is that we are seeing less and less change due to the grid on the solution. To continue refining the grid at this point is likely not economic because we are not seeing much difference with new grids of increasing cell sizes. At this point, we chose the solution from the grid having the greatest number of cells and assume based on our analysis that we have a reasonable "model" of our particular problem. Some may argue that the mesh of the next highest number of cells is chosen as the "final grid" because information is known for either side of it, and that's fine, too, but if the differences are small, then it probably doesn't matter. If they're "large," continue refining. Also, if real-world data are available, then the simulation result can be compared and perhaps even "validated."

Now, recall that you have resolved the grids uniformly for X and Y, there may be a need to resolve only in one direction or the other, depending on the number of cells in the near wall region. The process described above is repeated when conducting directional grid independence analysis.

Some references to check: Patrick J. Roache: Verification and Validation in Computational Science and Engineering, 1998; Celik & Keratekin: Celik, I. and Karatekin, O. 1997. Numerical Experiments on Application of Richardson Extrapolation with Nonuniform Grids. Journal of Fluids Engineering. Vol. 19, pp. 584-590.

Final piece of advice... do not go haphazardly into the process. Instead, decide which grids you will be refining, decide your refinement factor (1.3, for example), identify your boundary conditions, and write everything down into a table. Check off each grid as it is built and solved. When all are solved, then conduct the grid convergence analysis. Don't make any changes to the grids (other than grid size) as you conduct the simulations, including the number of iterations needed to reach convergence. When you are finished you will be able to see the difference in results and make a conclusion about the grid size as a parameter influencing the results of your simulation. The above procedure can be applied to any parameter in your simulation, such as time, number of iterations, inflow velocity, etc.

The above are the basics of assessing grid independence. To determine a grid convergence that gives a useable value of uncertainty, that is, we can say that we are uncertain about only 10 or 5% of the result, requires several more steps, including demonstration of iterative independence.

Best of luck, I hope this has helped.

Aaron