Gradient Approximation and Shape Optimization In Turbomachinery for 3D Incompressible Turbulent Flows
by
Bijan Mohammadi
U. Montpellier II
Coauthors: Stephane Moreau, Valeo, Mugurel Stanciu, INRIA and Valeo TM

We present our approach for the simulation and the design of industrial configurations. Optimization procedures are coupled with turbulent flow solvers in order to treat complex aerodynamics problems in turbomachinery.

Turbulent Flow Simulation

In the incompressible case, we consider the Reynolds-averaged Navier-Stokes equations coupled with a modified k-\epsilon turbulence model adapted to rotating frames through NSIKE flow solver.

To adapt to the industrial context of Valeo, the commercial flow solver TASCflow has been used. This software uses block-structured non-orthogonal grids with grid embedding and grid attaching to discretize the domain. This is also to show how to use different approaches for the state and sensitivities evaluation.

Shape Optimization and AD

We consider a shape optimization problem, where the control points are described by the mesh nodes, the state is given by the Navier-Stokes equations and the k-\epsilon turbulence model, subject to certain constraints.

We take into account the state constraints as penalty terms in the cost function and the geometrical constraints by a projection operator. A smoothing operator is defined over the shape, performing a few local Jacobi iterations wherever the shape is not smooth. Once the new shape is known, we expand these variations overall the mesh.

Using automatic differentiation (Odyssee AD tool), we compute the approximated gradient of the cost function. The dominant part in the gradient comes from sensitivities with respect to the geometrical quantities and not to the state. This avoids the calculation of an adjoint state and decreases significantly the computational cost. Different comparisons between the exact and the incomplete gradient are made for several theoretical problems and applications..

We used optimization algorithms based on the steepest descent method or the BFGS method where a quadratic approximation of the cost function is performed. For this last method, the gradient vector is deflected using an approximation of the Hessian matrix of the cost function. Further on, because for 3D problems the size of the control space is too large, we use an incomplete Hessian by keeping only the elements connected on the shape, leading to an effective improvement of the efficiency with respect to the steepest descent method.

Our platform TASCOPT adopts the unstructured CAD-free framework for shape and mesh deformations where the interface CAD is realized at the final stage of the optimization. This framework was extended to 2D and 3D multi-block structured grids and requires the development of specific interface between TASCOPT, TASCflow and CAD tools used in industrial environments.

Applications

These ingredients are illustrated through a cascade configuration blade flow simulation and shape optimization. In addition, we present an example of the application of the platform in industrial environment for the design of 3D cooling engine fan blade.

Both NSIKE and TASCflow flow solvers were used for the flow simulation for the 2D and 3D cases. Comparisons were made between the different turbulence models, as well as between the 2D-cascade and the 3D computations.

Finally, we used our platforms to optimize the 2D profiles of the blade at different sections for the cascade configuration and the entire 3D shape of the fan blade in in order to decrease the torque and keep the efficiency constant.

Our approach proved to be a powerful tool which can treat complex cases and can lead to future systematic industrial applications.

Date received: February 21, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cads-90.