Correction of symmetry plane implementation

[bigfooted]

Proposed Changes

The implementation of the symmetry plane is incomplete. We follow here the book of Blazek, Computational Fluid Dynamics: Principles and Applications . According to Blazek, (chapter 8.6) 4 conditions have to be met on a symmetry plane:

1. No flux across the boundary
2. $\overline n \cdot \nabla \phi = 0$ (gradient of scalars are zero)
3. $\overline n \cdot \nabla (\overline U \cdot \overline t) = 0$ (gradient of tangential velocity normal to boundary)
4. $\overline t \cdot \nabla (\overline U \cdot \overline n) = 0$ (gradient of normal velocity along the boundary)

Points 2-4 all deal with gradients and can be dealt with in the gradient computation, i.e. Green-Gauss or Least Squares. According to Blazek, 2 approaches can be followed. "One possibility is to construct the missing half of the control volume by mirroring the grid on the boundary. The fluxes and gradients are then evaluated like in the interior using reflected flow variables." This approach can be implemented in an easy way when computing the Green-Gauss gradients. In SU2, routines are already in place that deal with GG gradients on boundaries. Here, we just have to identify the symmetry planes and mirror the flux through the faces.
Blazek continues: "The second methodology computes the fluxes for the halved control volume (but not accross the boundary). The components of the residual normal to the symmetry plane are then zeroed out. It is also necessary to correct normal vectors of those faces of the control volume, which touch the boundary. The modification consists of removing all components of the face vector, which are normal to the symmetry plane. The gradients also have to be corrected according to Eq. (8.40) [points 2,3,4 above]"

1. Modify Green-Gauss and Least-Squares gradient computation for symmetry planes
   -> This seems to work fine
2. Fix normal components of edges connected to symmetry plane
   -> This seems to work fine
3. Fix residual components of velocity on the symmetry plane
   -> This seems to work fine
4. Multiple symmetry planes, including Euler walls
   -> This seems to work fine
5. multigrid
   -> Works fine now that we use the agglomeration as in the paper of Diskin.
6. grid movement
7. -> Works fine with the new fix to the residuals (energy update)

Related Work

#1168

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[pcarruscag]

4 loops may not be the most efficient way of doing 2 matrix multiplications.
From the way the basis/tensor is constructed there are probably some nice cancelations that make the correction simpler.

[pcarruscag]

You can probably use the gradient iterator to avoid the copy.

[bigfooted]

What do you mean with the gradient iterator?

[pcarruscag]

The 3D matrix returns an object that looks like a matrix but does not require copying the data. I can help with that sort of thing once all the tests are green.

[pcarruscag]

This is not the map transposed right? You are adding over iDim

[pcarruscag]

For example, for scalars these operations are equivalent to:

g = T' * [0 0 0; 0 1 0; 0 0 1] * T * g

(setting the first component to 0 in matrix form).
This is equivalent to

g = (I - T(1,:)' * T(1,:)) * g

(which is the same as taking out the normal component)
The tangential and orthogonal directions cancel out (as desired since they are arbitrary).

After you review the tests we can look into these simplifications.