huyanzhou,

I try to use the communicate_locally_moved_vertices, it fixes the departure
> of mesh belongs to different mesh . but there still remains some problem
>
[...]
> you can see that the mesh is consistent on the boundry, but not smooth in
> the inner mesh ,there are some shallow hole  on the surface
>
Can you explain more precisely what doesn't work as expected? Is this a
problem you only see when running with multiple MPI processes?
What is the "inner mesh" for you? Do you refer to faces with children? Can
you point out which vertices you think are set wrongly?


> my code is as follows
>
[...]
>

It seems that your idea is to first move vertices according to some
displacement function and then to "correct" hanging vertices to be placed
inside the neighboring face.
That looks OK for standard orientation but might be problematic otherwise.
You should be able to achieve the same effect when you only apply hanging
node constraints to your
displacement vector.

Best,
Daniel

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