Daniel,
Thanks again. To summarize, I want to implement the gradients and the
laplacian functions for the SymmetricTensor class.
The laplacian (trace of hessian) for the fe_values block extractor is
straightforward, but the gradients (of the solution field) for the
SymmetricTensor has some issues.
e.g., what would be the type for the gradient of SymmetricTensor<2,dim> ?
how to represent this -->
[(S_1,x),(S_1,y);(S_2,x),(S_2,y);(S_3,x),(S_3,y)]... Each component of the
SymmetricTensor is a vector.
Best,
Metin
On Tuesday, October 4, 2016 at 10:15:00 PM UTC+2, Daniel Arndt wrote:
>
> Metin,
>
> I understand that the components are in space (x/y/z),
>> and (*shape_gradient_ptr++)[d] would be derivative of phi in each
>> direction; is that correct?
>>
> Yes, that is the derivative of the dth component in direction d.
>
>>
>> (compared to vectors) the implementation of divergence function for the
>> tensors is as following;
>>
>> *if* (snc != -1)
>>
>> {
>>
>> *const* *unsigned* *int* comp =
>>
>>
>> shape_function_data[shape_function].single_nonzero_component_index;
>>
>>
>> *const* dealii::Tensor < 1, *spacedim*>
>> *shape_gradient_ptr =
>>
>> &shape_gradients[snc][0];
>>
>>
>> *const* TableIndices<2> indices = dealii::Tensor<2,
>> *spacedim*>::unrolled_to_component_indices(comp);
>>
>> *const* *unsigned* *int* ii = indices[0];
>>
>> *const* *unsigned* *int* jj = indices[1];
>>
>>
>> *for* (*unsigned* *int* q_point = 0; q_point <
>> n_quadrature_points;
>>
>> ++q_point, ++shape_gradient_ptr)
>>
>> {
>>
>> divergences[q_point][jj] += value *
>> (*shape_gradient_ptr)[ii];
>>
>> }
>>
>> }
>>
> This is similar to the previous one. Note that the ith component of the
> divergence of a rank-2-tensor is defined as sum_i \partial_i T_{ij}.
> This is what is implemented for the general (non-symmetric) Tensor class.
> You have to compare this to the Vector implementation for primitive shape
> functions.
> Basically, ii corresponds to the first index of the non-zero component and
> jj corresponds to the second index of the non-zero component.
> We figure the correct component of the divergence add add the appropriate
> derivative to it.
> For a SymmetricTensor, only T_{ij} for i\leq j is stored.Therefore, we
> need to have the line
>
> if (ii != jj)
> divergences[q_point][jj] += value * (*shape_gradient_ptr)[ii];
>
> additionally to account for the contribution of T_{ji} add the same time.
>
>
>> There is only the "snc != 1" case, and it seems that the component now
>> means the tensor components and not the directions in space. Is this
>> function (for tensors) not yet implemented? or is it sth related to
>> non-primitive shape functions (I would appreciate if you could comment on
>> this as well)?
>>
> This is only implemented for primitive shape functions.
>
> Best,
> Daniel
>
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