Research Fellows
Chris Kapellos
Early Stage Researcher 15 at Volkswagen
In an optimisation problem, the gradient of the objective function with respect to the design variables has to be computed. In shape optimisation, the design variables are those which parameterise the geometry to be optimised. Using the chain rule, this gradient can be written as the product of the gradient of the objective function with respect to the normal displacement of the mesh nodes, namely the flow sensitivities, and the gradient of the normal displacement with respect to the design variables, namely the shape sensitivities. In the implemented method the first gradient is computed with the continuous adjoint method, while the second one with the CAD-Finite Differences method, developed by Queen’s University of Belfast.
The method was validated at the VW S-Bend test case. The flow is laminar (Re=350) and the computational mesh is structured, comprising 710.000 cells. The optimization objective is the dissipated power minimization and the design variables are the CAD model parameters. The predicted change in the objective function for a number of perturbations of each design variable was compared with the actual change, computed with CFD analysis. This comparison concluded that the method over predicts with a factor of 1.4. Moreover, the computed gradient was compared with finite differences, when these could be computed, and with the gradient of a 3rd order fitting curve of several CFD results. The ratio of the computed gradient with the latter can be seen in the following table. For most of the design variables the error is acceptable and can be related to the accuracy of the adjoint sensitivities.
|
Design Variable |
Finite Differences |
3rd Order Polynomial |
| 1 | 0.67 | 0.67 |
| 2 | 1.55 | 1.55 |
| 3 | 1.44 | 1.44 |
| 4 | - | 4.57 |
| 5 | - | 2.48 |
| 6 | 1.75 | 1.77 |
| 7 | - | 1.03 |
| 8 | 1.19 | 1.19 |
| 9 | - | -1.12 |
The optimization loop implemented in VW workflow can be shown in fig. 3. For each optimisation cycle the CFD mesh is generated and the flow is simulated. The new objective function is computed and if the optimisation has not converged, the adjoint equations are solved. Combining the adjoint sensitivities and the CAD finite differences as presented, the gradient of the objective function with respect to the design variables is computed and used to update the CAD parameters. The new CAD geometry is then exported. The different processes are controlled through a bash shell script.
To optimise the VW S-Bend, 17 optimisation cycles were completed leading to a 3.3% reduction in the objective function. For each optimisation cycle, the time needed for remeshing, solving the primal and adjoint equations and computing the CAD gradient was around 4, 49 and 20 minutes respectively
Fig. 3: Optimisation flow chart
Fig. 4: Flow sensitivities w.r.t. power dissipation
Fig. 5: Total displacement performed during the optimisation