A question about the accuracy of the gradient

[WZDDZFJ]

In the verification section, you introduced forward-mode differentiation to check the correctness of the gradients. Could you provide an example of using finite differences to directly check the correctness of the gradients? Because I followed the tutorial to analyze the two-dimensional airwing of NACA0012, but when I looked at the gradient file totalderivist.txt, I found that the FFD points in a symmetric position had different gradient values, and the gap was relatively small when it was not deformed, but the gap was large after deformation. For example, what I've framed in red is a typical set of symmetric FFD points. Usually, we use the finite difference method to verify the gradient, that is, manually give a disturbance to the FFD point, see how the objective function changes, and then obtain the gradient value. But in this case of the airfoil, do I perturb one FFD point at a time, or do I perturb a set of symmetric FFD points at the same time and then compare it to the gradient obtained by the discrete adjoint solution?

[sll426]

Hi, how to get the gradient of particular design variables value

[WZDDZFJ]

maybe in the 'totalderivist.txt'

[WZDDZFJ]

In addition to the previously asked question about the gradient of symmetric points being different, I have read your article, which contains the term FD reference, and I would like to ask how this came about. Referring to one of your previous replies:

There is no automatic way to do FD for field inversion because there are so many inputs. What you can do is to manually perturb one beta value in the runScript.py file and run the primal to get the FD derivative. Check where you prescribe the initial values for beta in the runScript.py file. You can prescribe some perturbed beta values there.

My approach was to modify the code in runScript_v2 from pts = DVGeo.getLocalIndex(iVol) indexList = pts[:, :, :].flatten() to indexList = [pts[0,0,0]] (for example) , and then use DVGeo.addLocalDV("shapey", lower=1e-5, upper=1e-5, axis="y", scale=1.0, pointSelect=PS), which perturbs a specific FFD point. After calculating the flow field and subtracting the original flow field, I divide by the perturbation value to obtain the total derivative of the objective function with respect to the design variable. However, the result I obtained does not match the value in totalderisvist.txt, but the derivative I got this way is similar to what I get when I set verify=3 in the snopt optOption. Again, my result is similar to that obtained by setting useAD": {"mode": "fd"}. The image below is the one I got using the same case but setting useAD": {"mode": "fd"}.

So to summarize the results of my experiment, this example is from the tutorials file. The final-difference result I calculated manually was similar to that of useAD": {"mode": "fd"} or verify=3 in snopt optOptions, but completely different from reverseAD. And I also tested forwordAD, and the error was also large.
The accuracy of the gradient is very important, and different results in the same case make me very confused. If my approach is incorrect, could you please let me know the correct way to obtain the "FD reference" similar to what is presented in your paper? My question is a little long, thank you Professor He for your patient reply!@friedenhe

[friedenhe]

@WZDDZFJ You can use task=checkTotal (https://github.com/DAFoam/tutorials/blob/607a4f7738d067dd0921eecd4a96750c69443811/NACA0012_Airfoil/incompressible/runScript.py#L271) in a v3 script to use FD values as references. But in the case of 2D airfoil, you can't simply perturb one FFD in FD. This is because a small perturbation of any FFD point will violate the symmetry boundary condition and return unphysical results. So the FD-based totals are not correct. What you can do is use the twist (refer to a 3D wing case on how to add a twist DV into the NACA0012 tutorial) as the design variable and then run checkTotals.