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When using the Hexpress grid generator one gets hexahedral cells, possibly with hanging nodes. Because of a smoothing step, the hanging nodes can be shifted: they are not necessarily on the straight line that connects the corners, as shown below.

cell with shifted hanging node

With FVM one would triangulate the face with 5 nodes in order to approximate its surface, face center and such, see for example the book by Ferziger and Peric, Computational Methods for Fluid Dynamics, Springer, 2001. How should one treat this situation in FEM? I'm familiar with the hexa 8 mapping to a master element using linear shape functions but that doesn't seem quite enough.

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up vote 4 down vote accepted

The best approach would be to look at adding an isoparametric element for the extra node. See, for instance,

http://www.edwilson.org/book-wilson/05-iso.pdf

Section 5.5 which has a discussion of this for the 2D bilinear element. The extension to 3D trilinear is rather straightforward.

Note that this requires some flexibility in the code in terms of local element spaces -- this element has 9 elements instead of 8. The best reference for both the math of doing this and the software design of doing this is the book:

http://www.amazon.com/Finite-Element-Method-Mechanical-Engineering/dp/0486411818

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If I understand correctly, this makes the edge with 3 nodes quadratic instead op piecewise linear, that's a major difference w.r.t FVM. –  chris Feb 21 '13 at 7:54
    
That's right. Note that if you're looking for a more direct comparison to FVM, you might be looking for a mixed form element, which would be slightly different. –  Ethan Coon Feb 21 '13 at 16:27
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