[Trilinos-Users] [EXTERNAL] Intrepid with curved elements

[Bochev, Pavel B]

Nate,

Shards defines base and extended topologies for the main finite element cell shapes. What is implemented in cellTools supports only these topologies. Basically, selection of shards topology governs the definition of the reference to physical map.
Consider for example computation of the Jacobian for the reference to physical map F on Quad cells.

There are 2 Shards topologies relevant to cellTools:

Quad<4> is the base topology with 4 vertices
Quad<9> is extended topology with 4 vertices, 4 side midpoint nodes and 1 center node


If you call setJacobian with Quad<4> then the map F is defined by the Q1 basis functions ( Basis_HGRAD_QUAD_C1_FEM )
If you call setJacobian with Quad<9> then the map F is defined by the Q2 basis functions ( Basis_HGRAD_QUAD_C2_FEM )

In the first case you get a physical quad with straight sides (restriction of bilinear F to sides is linear function)
In the second case you get physical quad with parabolic sides (restriction of biquadratic F to sides is quadratic)

Now suppose you are not happy with your physical quad having only quadratically curved sides. Say, you want your quad sides to be curved as a cubic. In this case you need to define your reference to physical map using Q3 (bicubic) elements, but shards does not have the corresponding Quad<16> topology.

The implementation strategy where cell topology serves as proxy for the order of the mapping is convenient for h-FEM where polynomial degrees are fairly low and in most cases quadratic transformation is the most you will ever need. However for p-Elements, if you want to avoid the consequences of the Strang lemma, you need to match curved domains up to the order of the finite element space. In this case a better solution is to have interface where you directly specify the degree of the transformation + some control points on the physical element to guide the “curving” of the sides and the edges.  Otherwise, you will need to create cell topologies for every transformation degree you want which does not seem very effective.

It may be worth considering this approach for inclusion in intrepid. Let us know what you think!

Pavel






On 3/8/12 1:35 PM, "Nate Roberts" <nate at nateroberts.com> wrote:

Hello all,

I've been under the impression that, by defining a custom CellTopology for my elements, I should be able to use CellTools for computing Jacobians, etc. for elements with curved boundaries.  Am I right about that?  If so, is there an example I might refer to?

In any case, I'd be glad of any advice on how to approach curved elements using Intrepid.

Thanks!
Nate

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