Proposed Changes

The proposed solver is a treatment of the discrete adjoint sensitivities based on the idea of Sobolev gradient smoothing.

Sobolev smoothing of gradients computes a reinterpretation of the gradient in a different Hilbert space, see this short overview.
This is done by solving an elliptic PDE with the sensitivities, i.e., ( e1 Id - e2 Laplace) delta_x = - dF / dx, with different weights e1, e2. Inserting the sensitivities from the discrete adjoint solver for (dF / dx) leads to the gradient in a Sobolev space.
The method can be extended to incorporate the parameterization of the mesh cells x=M(p) with design values p, see here. Using the projected gradient w.r.t. the design values the systems takes the form
(dM/dp)^T (e1 Id - e2 Laplace) (dM/dp) = - dF/dp.

Such treatment can help prevent noise in the derivative computation and leads to smoother design updates. Especially, when working in a One-Shot optimization context.
Here the optimization is done with inexact function values and gradients, i.e., the flow and adjoint solvers are only partially converged between design updates. For this setting simple gradient descent with too large update steps will not work, since a few flow iterations are not enough to recover a solution from the restart file from the last optimization step.
See the graphic below for drag minimization of a NACA 0012 airfoil with constant lift, where the Sobolev solver for the gradient leads to smoother deformations and better optimization.

Application is done as a post-processing step in the SU2_DOT_AD executable.
The new solver constructs the necessary operator and solves it using the features of the finite element implementation in SU2. For this, it uses the same elements as the FEA solver but extends the definitions of the Gauss quadrature to be second order.
It can operate in two different modes:

1. The smoothing can be applied to the mesh sensitivities, either in the complete mesh or only design boundaries. See the picture below for an example of this.
   

2. It also includes the possibility to take the mesh parameterization into account, compute the derivatives (dM/dp), and smooth the sensitivities w.r.t. the design parameters. This is done by the same AD logic used in SU2_DOT.cpp::SetProjection_AD. The routines are extended, such that AD can compute arbitrary matrix-vector products with the Jacobian of the parameterization and apply the smoothing operator to the gradient.

This is a continuation of draft #1439 with a new rebased branch. It has a cleaner commit history and not so much legacy from previous versions.

The regression tests for this pull request need the files for the accompanying pull request su2code/TestCases/pull/88.

PR Checklist

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