I am doing a convergence analysis in on a finite element model. The converged result is the result when the element size is zero which is also when there are an infinite number of elements in the model. Therefore, there are two ways to identify the converged result:
- Linear extrapolation of the result vs. element size data.
- Exponential extrapolation of the result vs. number of elements data.
Both methods produce an estimate of the converged value with a corresponding uncertainty (i.e. the standard error of regression). My initial thought was that I should use the result with the smallest uncertainty as the actual result. However, I just realized that if I were to find a means of combining the uncertainty, then the combined uncertainty would be smaller. For example, the image below shows a red probability distribution overlapping a blue probability distribution. The combined probability distribution is shown in purple.
Now, my question is, knowing the statistical means and the standard deviations defining the red and blue curves, how do I go about calculating the statistical mean and standard deviation for the purple curve?
I know the equation for the purple line is obtained by multiplying the equations for the red and blue lines together, but that doesn't tell me what the statistical mean and standard deviation is that defines the purple curve. At the very least, it's pretty clear that multiplying the standard deviation of the two results together would give vastly different results depending on the units being used (i.e. standard deviations of +/-3 mm and +/-2 mm would give vastly different results than standard deviations of +/-0.3 cm and +/-0.2 cm).
Ultimately, I'm not extremely experienced with statistics, so I may have made mistakes in my description above, but basically, I'm trying to leverage to two separate means of extrapolation at my disposal to reduce my uncertainty.