Perfect. This is exactly what I was looking for. Thanks, Wolfgang.
> On Aug 14, 2019, at 4:58 PM, Wolfgang Bangerth <bange...@colostate.edu> wrote:
>
> 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: bange...@colostate.edu
> www: http://www.math.colostate.edu/~bangerth/
>
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