Hi Tom,

An alternative to implemented a complex mesh update strategy is to define 
an initial mesh that, when mapped via your Euler vector, produces a higher 
quality mesh. For example, you could remove 
<https://www.dealii.org/developer/doxygen/deal.II/namespaceGridGenerator.html#ada140ece81bf38a23e738e0e57f89e97>
 the 
"top" element of the coarse mesh and substitute it 
<https://www.dealii.org/developer/doxygen/deal.II/namespaceGridGenerator.html#a843fe2c5c77374c51fcd18f05690b902>
 
with a set of elements with an o-grid topology 
<https://www.sharcnet.ca/Software/Ansys/17.0/en-us/help/icm_help/graphics/gicm_OGrid_Interpolated.jpg>.
 
Of course you would have to have some rough idea as to what your final grid 
topology will look like in order to optimise this.

By the way, in the images you've shown can you indicate where the low 
quality elements are? In Paraview there's a filter that can help with this. 
If one considers the mesh on this ellipse as something mapped from a cubic 
grid, then I presume you're referring to the elements at the cube corners.

Regards,
J-P

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