Serendipity elements

[Eldar Khattatov]

Dear all,

I am sorry for a somewhat silly question, but is there a reduced quadratic (2nd order serendipity) finite element implemented in Deal.II?

Thank you.

[Bruno Turcksin]

Hi,

I don't think that we have such element. You can see the list of finite element that are implemented in deal.II here https://dealii.org/8.3.0/doxygen/deal.II/classes.html#letter_F (they all start with FE_)

Best,

Bruno

[Eldar Khattatov]

Thank you for your fast reply and for the link.

Best,

Eldar

[Zhen Tao]

If you really want serendipity elements.

For example, 2D quadratic, serendipity has 8 dof, FE_Q(2) has 9 dof.

Use FE_Q(2)  and then cancel the extra degrees of freedom(x^2y^2) out by setting proper matrix constraints.

[Jonathan Russ]

Hello -

How difficult would it be to add serendipity elements to dealii? They are very useful in solid mechanics applications and can greatly reduce the computational cost. Is it very difficult to add an element that is the same is FE_Q but without the few extra basis functions when quadratic polynomials are used? i.e. 8-node FE_Q instead of 9-node FE_Q in 2D and 20-node FE_Q instead of 27-node FE_Q in 3D?

Thanks in advance for your thoughts,

Jonathan

[Wolfgang Bangerth]

On 8/3/19 1:17 PM, Jonathan Russ wrote:
>
> How difficult would it be to add serendipity elements to dealii? They are very
> useful in solid mechanics applications and can greatly reduce the
> computational cost. Is it very difficult to add an element that is the same is
> FE_Q but without the few extra basis functions when quadratic polynomials are
> used? i.e. 8-node FE_Q instead of 9-node FE_Q in 2D and 20-node FE_Q instead
> of 27-node FE_Q in 3D?

I suspect it's not going to be terribly difficult, but can you explain how the
shape functions of the serendipity element differ from the ones of FE_Q(2)?

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Jonathan Russ]

Professor Bangerth -

Thank you for your reply. The shape functions of the "quadratic" serendipity elements are very similar to the FE_Q(2) elements except they are derived without any interior nodes (i.e. in 2D the interior node is removed and the shape functions are simple polynomials with 8 undetermined coefficients instead of 9 as with a normal quadratic Lagrange element. Essentially the polynomials are not quadratic complete since they are missing the x^2 * y^2 term). It's pretty simple to express them in 2D and 3D for the "quadratic" case (I put quadratic in quotes since the polynomials are not quadratic complete) but I am more worried about how difficult it would be to get this type of element to work with all of the other functionality in dealii (e.g. the DoFHandler, Triangulation, grid refinement, etc.). Do you have a sense for whether this requires a significant amount of additional effort?

Thank you again,

Jonathan

[Wolfgang Bangerth]

On 8/4/19 5:47 PM, Jonathan Russ wrote:
>
> Thank you for your reply. The shape functions of the "quadratic" serendipity
> elements are very similar to the FE_Q(2) elements except they are derived
> without any interior nodes (i.e. in 2D the interior node is removed and the
> shape functions are simple polynomials with 8 undetermined coefficients
> instead of 9 as with a normal quadratic Lagrange element. Essentially the
> polynomials are not quadratic complete since they are missing the x^2 * y^2
> term). It's pretty simple to express them in 2D and 3D for the "quadratic"
> case (I put quadratic in quotes since the polynomials are not quadratic
> complete) but I am more worried about how difficult it would be to get this
> type of element to work with all of the other functionality in dealii (e.g.
> the DoFHandler, Triangulation, grid refinement, etc.). Do you have a sense for
> whether this requires a significant amount of additional effort?

A month at most. Maybe less.

There are many examples of elements already implemented. The easiest way to do
things is if you have a description of the polynomial space in some way. Take
a look at the FE_Poly class and how it is used in some of the other classes.
Depending on how you describe the serendipity space, this may be almost
everything you actually need -- or maybe there are more complications.

The FE interface is very self contained. You won't have to touch the
DoFHandler or any other class. All you have to describe are the shape
functions, what kind of continuity you have across faces and vertices, how
hanging nodes look like (likely the same as the FE_Q(2)), and a few small
other pieces of information that one can often ignore at first.

[Jonathan Russ]

Professor Bangerth -

Okay thank you for your advice! I'll take a look at the FE_Poly class as you suggest.

Thanks again,

Jonathan

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[Yiliang Wang]

Hi Jonathan,

I am also looking for the implementation of Hex 20 element in dealii. Did you figure it out how to do that?

Best,

Yiliang