Hessian computation

[Shiva Rudraraju]

Hi,

I am curious where the code for the computation of Hessian of shape functions (with respect to real cell coordinates) resides. Basically, where the source code corresponding to shape_hessian() resides, which shows the implementation of the derivatives of the inverse of the Jacobian that are needed as part of this computation. 

Thanks,

Shiva

[Vivek Kumar]

You can find the source code, i.e. the corresponding include file, at the top of any class documentation. In this case it is #include <deal.II/fe/fe_values.h>. The source code can be found at https://www.dealii.org/8.4.1/doxygen/deal.II/fe_2fe__values_8h_source.html

[Vivek Kumar]

Sorry the link is for an old version. The latest version is at: https://dealii.org/developer/doxygen/deal.II/fe_2fe__values_8h_source.html

[Shiva Rudraraju]

Thanks for the pointer, Vivek. I did check the usual locations, including fe_values.h, but couldn’t locate the exact implementation of the computation for the hessian in the real cell coordinates. (Should involve the product of shape function hessians in reference coordinates, and multiplication with a whole lot of inverse Jacobean terms.)

Still looking for its exact implementation ….. as I am curious about the computational cost of this implementation.

Thanks,

Shiva

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[Wolfgang Bangerth]

On 8/13/19 5:34 PM, Shiva Rudraraju wrote:
> Thanks for the pointer, Vivek. I did check the usual locations,
> including fe_values.h, but couldn’t locate the exact implementation of
> the computation for the hessian in the real cell coordinates. (Should
> involve the product of shape function hessians in reference coordinates,
> and multiplication with a whole lot of inverse Jacobean terms.)
>
> Still looking for its exact implementation ….. as I am curious about the
> computational cost of this implementation.

The various finite element implementations provide the Hessian matrices
on the reference cell, and these then need to be mapped to the real
cell. For example, for the usual Q(k) elements, the Hessians on the
reference cell are pre-computed once at the very beginning (when the
get_data() function is called by FEValues) and the transformation
happens in fe_poly.templates.h in the function FE_Poly::fill_fe_values()
in this piece of code:

if (flags & update_hessians && cell_similarity !=
CellSimilarity::translation)
{
for (unsigned int k = 0; k < this->dofs_per_cell; ++k)
mapping.transform(make_array_view(fe_data.shape_hessians, k),
mapping_covariant_gradient,
mapping_internal,
make_array_view(output_data.shape_hessians, k));

for (unsigned int k = 0; k < this->dofs_per_cell; ++k)
for (unsigned int i = 0; i < quadrature.size(); ++i)
for (unsigned int j = 0; j < spacedim; ++j)
output_data.shape_hessians[k][i] -=
mapping_data.jacobian_pushed_forward_grads[i][j] *
output_data.shape_gradients[k][i][j];
}

This accounts for the fact that the Hessian in real space has two parts,
one that corresponds to the gradient of the Jacobian on the reference
cell, plus one that is transforming the Jacobian with the gradient of
the second derivative of the transformation implied by the mapping.

The function transform() is implemented by the individual mappings, but
in the case of the most common mappings, you'll find it in
mapping_q_generic.cc -- you just have to pick out the correct one of the
many versions of the function.

Best
Wolfgang

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

[Shiva Rudraraju]

Perfect. This is exactly what I was looking for. Thanks, Wolfgang.

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