3d Inverse design

The profiles are specified using B-Splines in a manner similar to the method already implemented in xflr5's inverse design module. This method has the advantage of simplicity and versatility.

The Cp profiles are specified at each of the wing's section, with the exception of the tip section where the evaluation of the Cp coefficients is too imprecise for use.

There is no limit to the number of sections at which the Cp profiles can be specified. Increasing the number of sections will provide more control on the target values, but may also run the risk of over-constraining the optimization. For this reason it is important to ensure that the profiles specified at all the sections are reasonably consistent.
The time to move the swarm at each iteration will not increase significantly with the number of sections, but it may cause it to wander a little more before convergence.

If a spline is not defined at a given section, the Cp profile is assumed to be free.

The bumps are described by the following set of functions with k ranging from 1 to n:

The maximum bump of function f_k is located at the fraction k/(n+1) of the chord. The width of the bump is controlled by the parameter t₂.
The shape of the functions is illustrated in the image of the right for n=5 and t₂=2.

Transpiration technique

This method allows the fast generation of an approximate solution to a panel problem. Its intent is to avoid the very lengthy generation and factorization of the influence matrix at each time step and for each particle.

A detailed description of the transpiration technique falls outside the scope of this document.
Its main idea is to add to the boundary conditions a normal blowing velocity which is adjusted to make the flow tangent to the desired modified geometry.
The method remains relatively precise as long as the modified shape does not deviate too much from the base shape, and as long as the streamwise step between two nodes remains small especially in areas of large pressure gradients.

Note: The use of the transpiration technique requires the additional storage in memory of a source influence matrix 1/6 the size of the doublet influence matrix.

The inverse design is only implemented for the linear Galerkin method applied to thick surfaces. The reason for this limitation is double:

1. it enables and simplifies the specification of Cp profiles along streamwise strips of nodes,
2. the linear order provides better accuracy at the leading stagnation point where the pressure gradient is high.

Since the goal is to achieve optimization by modification of a base configuration, i.e. by incremental changes, starting from a base configuration close to the target configuration will improve the chances of convergence. It will also reduce the error implicit in the use of the transpiration technique.

The test case is a pseudo AC75 type foil with inverse dihedral, defined by three sections. The sections are numbered starting with section 0 located at the root section. Section 2 located at the tip is excluded from the optimization process.

It is a good precaution to run a tri-linear thick surface analysis at the desired a.o.a. prior to launching the optimization module, and to check that the Cp profile at sections is smooth and is a good basis for optimization.

Activate the checkbox of a section to optimize, then click on the curve. A spline with endpoints located by default on the x-axis will be shown.

Drag the spline's control points to define the shape of the desired Cp profile.
As with all graphs in flow5, hold the 'X' or 'Y' key down while scaling to expand that axis only.

Important notes:

• The spline's endpoint must be located on the reference profile curve.
• The spline's endpoint should be located at points located on the same side of the foil.
• Specifying the profile in the areas of the LE and TE should be avoided because of the high pressure gradient and its induced high influence on the transpiration method.
• To cancel the objective at a section, use the context menu option (i.e. Right click) "Reset spline/spline #"

The last thing to do before exiting the tab is to define the maximum acceptable error for the deviation of the Cp profiles at each section.
The cumulated deviation is measured from one spline endpoint to the other.
It is possible to start with a large value and reduce it progressively without need to re-create the swarm

In the present case the maximum deviation is set to RMS = 0.025. Experience so far seems to show that the deviation should be specified as an RMS value rather than a |max.| value.

In the "Hicks-Henne" tab, select the side to which the bumps should be added. For maximum effect, the bumps should typically be added to the same surface as the one for which the target Cp was set. In this case this is the top side.
The amplitude should be set to a value sufficient to achieve the desired target profile. Too large a value however will cause imprecision in the transpiration technique and may cause the swarm to wander.
Experience so far has shown that the number of functions should preferably be kept low, e.g. less than 5, with large overlaps, i.e. t₂<2, to avoid bumpy foil shapes and to obtain smooth Cp profiles.

In the present case:

• 5 bump functions are applied to the whole extent [0%, 100%] of the foil's top surface,
• a maximum symetric amplitude equal to 2% chord length has been set. Setting the minimum amplitude to 0% will eliminate the risk of concave foil shapes, but runs the risk of reducing excessively the design space.

Important note: The range of the bumps should be similar to the range of the splines.

• If the range of the bumps is too small, the target Cp profile will not be achievable in the part of the surface which is not covered by the bumps.
• If the range of the bumps is greater than the range of the spline, the amplitude of the outer bumps will be unconstrained and will lead to wavy foil surfaces.

These parameters should preferably be left unchanged, except for the swarm size and the max. number of iterations which can be adapted to the speed of the platform and to the size of the mesh.

The detailed explanations and recommendations are given here.

At each iteration, the algorithm selects the particle in the Pareto frontier with the minimal distance to the origin, and displays this particle in all the views.

The Pareto graph has a dimension N equal to the number of sections where splines have been defined. However it can only be displayed in 2d for the first one or two sections, and in 3d for the first three sections.
In the present case, particle p6 highlighted in red is the best particle, i.e. the one closest to the origin.
Convergence is achieved when a particle enters the blue rectangle. Note that the particle's position at the sections not represented in the Pareto graph also need to enter the blue rectangle, cube or "N-hyper cube" for convergence to be achieved.

In the case illustrated in the right graph, the target error has been reached at section 0, since particle p4 for instance has Cp_0_error < 0.025, but not at section 1.
See the paragraph non convergences for common causes and fixes.

The RMS-deviations of the Cp profiles to the target profiles are displayed in the middle right graph as a function of swarm iterations.

The iterations can be interrupted and resumed at any time. It is possible to re-start iterations from the current swarm, or by first re-generating a new random one if the swarm seems to be stuck on a local minimum.

Results are generated whether the optimization process has converged or not.

In the present test case, convergence was achieved at the first run in 12 iterations.

The optimized foil shapes are displayed in the top right view.

For information, the resulting Cp colour maps and the isobars are displayed in the top right 3d view.

Since the optimization process was conducted using an approximation implied by the transpiration technique, it can be interesting to compare the results to the exact solution of the panel method.
This is done after exiting the module by defining a tri-linear analysis for the optimized plane, runnning the analysis for the specified a.o.a., plotting the results in the Cp view (F9), and overlaying manually the optimized curves using the "External curves" feature.

As it turns out, the error due to the approximation is negligible.
The slight offset of the red curves at section 1 is caused by the profiles not being evaluated at exactly the same span position, and it is not significant.