Compute divergence of symmetric gradient of vector shape function

[Lixing Zhu]

Hello all,

I am solving a saddle point problem like a nearly incompressible elasticity. The unknown field is u (displacement) and p (pressure). For Galerkin weak form, it is very convenient in deal.ii to use FEValuesExtractors::Vector or FEValuesExtractors::Scalar and use member function gradient/symmetric_gradient/divergence to quickly compute the 1st derivative related terms in the bilinear form.

However, I am trying to implement some stabilization technique that involves 2nd order derivatives of the shape function, like the divergence of the symmetric gradient of the displacement vector (actually, the shape function). For now, I can only think of selecting the components from the Hessian of the shape function to compute the div of the symmetric gradient. Is there any way in deal.ii to directly extract such values like divergence of the symmetric gradient of shape functions?

If this is actually a quick question, a simple link to the reference or guide is also welcome.

Thanks,

Lixing

[Wolfgang Bangerth]

On 7/20/21 8:07 PM, Lixing Zhu wrote:
>
> However, I am trying to implement some stabilization technique that involves
> 2nd order derivatives of the shape function, like the divergence of the
> symmetric gradient of the displacement vector (actually, the shape function).
> For now, I can only think of selecting the components from the Hessian of the
> shape function to compute the div of the symmetric gradient. Is there any way
> in deal.ii to directly extract such values like divergence of the symmetric
> gradient of shape functions?

There is no easy way. You'll have to compute the tensor of second derivatives
and sum over the appropriate set of elements.

Best
W.

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

[Lixing Zhu]

Dear Prof. Bangerth,

Thanks for your instant hint.

I'll get the hessian first and compute the divergence of the vector gradient explicitly.

Regards,

Lixing