# Confusion about bilinear form for elasticity equation in deal.ii tutorial

I'm learning how to solve vector-valued problems with deal.II library. In particular, I'm looking at the following introduction from the official website https://www.dealii.org/current/doxygen/deal.II/group__vector__valued.html#VVAlternative

Here they write the bilinear form as $$a(\mathbf{u}, \mathbf{v})=\left( \lambda \nabla\cdot {\mathbf u}, \nabla\cdot {\mathbf v} \right)_\Omega + 2 \sum_{i,j} \left( \mu \frac 12[\partial_i u_j + \partial_j u_i], \frac 12[\partial_i v_j + \partial_j v_i] \right)_\Omega$$

and then they say $$=\left( \lambda \nabla\cdot {\mathbf u}, \nabla\cdot {\mathbf v} \right)_\Omega + 2 \sum_{i,j} \left( \mu \varepsilon(\mathbf u), \varepsilon(\mathbf v) \right)_\Omega$$

but I can't understand why that double sum over indices $$i,j$$ in the last equality! I think it should not be there, also because there's no dependence on $$i,j$$.

EDIT: So the last summand is $$\sum_{i,j} (\mu \varepsilon(\mathbf{u})_{ij}, \varepsilon(\mathbf{v})_{ij})_{\Omega}$$

I'm wondering now how this double sum is translated, precisely, in the snippet

for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
for (unsigned int i=0; i<dofs_per_cell; ++i)
{
const double phi_i_div
= fe_values[displacements].divergence (i,q_point);
for (unsigned int j=0; j<dofs_per_cell; ++j)
{
const double phi_j_div
= fe_values[displacements].divergence (j,q_point);
cell_matrix(i,j)
+=  (phi_i_div * phi_j_div *
lambda_values[q_point]
+
2 *
mu_values[q_point]) *
fe_values.JxW(q_point);
}
}


Here's my attempt:

Applying quadrature formula: $$\sum_q \mu(x_q) \underbrace{\sum_{i,j} \varepsilon(\mathbf{u}(x_q))_{ij} \varepsilon(\mathbf{v}(x_q))_{ij}}_{\star}$$

Now I recognise that the $$\star$$ term is precisely $$\varepsilon(\mathbf{u}) : \varepsilon(\mathbf{v})$$

Therefore, I suspect that phi_i_symmgrad * phi_j_symmgrad is precisely the scalar product between the two symmetrized gradients, where the * operator has been properly overloaded since both terms are tensors of rank 2.

Is that correct?

• The sum symbol is a typo on that page. I submitted a pull request -- thanks for pointing it out! Jun 14, 2021 at 16:12
• @WolfgangBangerth Thanks for the check. Is my interpretation of the code (in the EDIT) right? Jun 15, 2021 at 6:35
• Yes, precisely. phi_i_symmgrad * phi_j_symmgrad has the necessary operator overload that does the double-contraction of symmetric tensors: dealii.org/developer/doxygen/deal.II/… Jun 15, 2021 at 18:16

Its not the summation that is wrong, but the lack of indices inside it. Below this expression on their site they define: $$\epsilon(\mathbf{u})=\frac{1}{2}([\nabla\mathbf{u}]+[\nabla\mathbf{u}]^{\mathrm{T}})$$
So $$\epsilon(\mathbf{u})$$ is a matrix formed from the symmetrized gradient of $$\mathbf{u}$$. But to get the sum from the earlier expression, we want to sum over the elements of this matrix. So it should really be $$\epsilon(\mathbf{u})_{ij}$$.
For the translation from formula to code, I think you have the right idea. I had initially mistaken the $$i$$ and $$j$$ for loops as the summations from the formula, but these seem to rather be contributions from different degrees of freedom of the system. I had missed that phi_i_symm was explicitly defined as a SymmetricTensor and so * must be overridden to the scalar product for this type in order for the code to make sense dimensionally.
• I don't agree with your last edit, as phi_i_symm is not a scalar, but a rank-2 tensor, as you can see from its definition right after the outermost loop. Right? Jun 14, 2021 at 8:00