Today I tested an algorithm for this in Elmer and it seems to work nicely:
1) Create and solve the unloaded case: Ax=b
2) Create and solve a controllable load case: Ay=f
3) Compute reaction forces resulting from 1) and 2)
4) Find scaling factor c such that sum of reaction forces at selected node vanishes
5) Update x:=x+c*y, b:=b+c*f and continue iteration, if wanted
As said, almost similar algorithm has been used in Elmer for temperature control. The algo is hidded under the hood. The use of reaction forces is something new, and also the fact that this can now be used for many other Elmer modules as well.
The control is done in one sweep for linear problems. Geometrically nonlinear problems may be possible but would requite some iteration. Parallel operation is not yet supported.
I'm not a FreeCAD expert but with the free text fields of the Elmer plugin I would guess that this could in the future be done also with FreeCAD. You just need rather fresh Elmer. The feature has been in https://github.com/ElmerCSC/elmerfem
(devel branch) for 15 minutes
My test was even simpler than any on this thread. If somebody wants to try out this for real I can give some guidance.