>
> Yes, this is what I am struggling to understand, you have put it very 
> simply. 
>

Great! I'm glad that gave you a bit more insight. I only hope you didn't 
take that as a patronising comment :-)
 

> I can see how this is trivial for linear FE_Q's, and these three links 
> should make this concrete. 
>

I forgot to say that it makes sense represent the mesh-motion problem with 
linear elements, since we consider the underlying geometry to have only 
straight lines between vertices anyway.

Yes, this makes sense.  To be clear, the functionality in step 2*. A 
> function that actually moves the triangulation vertices for you based on 
> this vector*. is not strictly necessary in the present context, is that 
> right?
>

For your purposes, no I don't think that this is necessary. It would just 
be useful for testing and for demonstration in a tutorial. We can showcase 
the functionality in step-49, e.g. verifying that it undoes the distortion 
of a cartesian mesh 
<https://www.dealii.org/developer/doxygen/deal.II/step_49.html#grid_7Demonstratingdistort_random>
.

I must admit that I lost sight of the fact that you're working in codim-1. 
If you choose to go down this road (as opposed to the algorithm you've 
discussed previously) then it would be good to first check what facilities 
Mesquite has to deal with this. Maybe internally it can project the 
optimised node positions back onto the original manifold? 

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