Optimization methods

POD-based Surrogate Model for Aerodynamics Design

For analysing complex systems, data reduction has become a real challenge in many scientific areas. This is especially true in the context of aircraft design where the amount of data has considerably increased this last decade. At this time, POD-based Surrogate Models appear as one of the most efficient tools for achieving data reduction and data sharing for multidisciplinary optimisation. Thus, a JPOD (Jack Proper Orthogonal Decompositon) tool written in Python is currently developped and tested at CERFACS. This activity is conducted through an AIRBUS France project and in the framework of two European projects called SimSAC (FP6, 2006-2009) and ALEF (FP7, 2008-2011).

Contacts: Jean-Christophe Jouhaud, Marc Ferrier

Optimization

Cerfacs is involved in the field of optimization since 2005 and works mainly in the context of the preliminary design phase of an aircraft. Typical industrial shape optimizations encountered at this level usually consider several hundreds of design variables and all shapes must be evaluated with a high-fidelity aerodynamic solver. Because of that, it is often necessary to limit the number of cost function evaluations for computational cost issues. In this field of high-fidelity aerodynamic shape optimization, the choice of the optimization algorithm is heavily constrained by the computational cost implied by one function evaluation. Thus, gradient-based optimization algorithms are particularly appreciated for their speed of convergence especially since there exists the powerful method of adjoint that can give the gradient of the objective function with respect to the shape variables by only solving one linear system. As a comparison, the finite difference method need as many non-linear system resolutions as the number of shape variables. Numerical optimizations are performed with the OPTaliA framework of Airbus France. It includes the Onera solver elsA that handles both aerodynamic simulations and discrete adjoint state computations.

Typical aerodynamic objective functions show multiple local optima over the domain of design variables. Therefore, local gradient-based optimizer are often stopped prematurely in local areas. Moreover, these optimizers are very sensitive to numerical noise that affects the computation of the gradient. Hence, a surrogate-based optimization algorithm was developed to allow to come near the global optimum. The surrogate built with only a few hundred of samples can not accurately represent the cost function if the dimension of the design space is high, but it proves to give correct trends. Then, starting from a coarse sample database and by iteratively refining the surrogate model at promising locations it is possible to largely outperform a gradient-based optimizer.

A comparison of different sampling refinement criteria was done using aerodynamic test cases in order to finally establish an optimizer capable of proposing a population of three new shapes per process iterations. The refinement at the predicted minimum on the Kriging surrogate model proved to converge prematurely whereas the use of the Kriging standard error, which makes the algorithm to explore zones of high uncertainty, enables to ensure a more global convergence. As the accuracy of a Kriging surrogate model decreases when the dimension of the design space increases, a Cokriging model was investigated. This model interpolates both function values and gradients that are computed in our case by the discrete adjoint method.

As an example, Fig. 1 shows the results of the drag optimization of an AS28G Wing Body Pylon Nacelle configuration 48 design variables were driving the amplitude, position and width of 16 Hicks-Henne bumps distributed on the wing. A single function evaluation takes 5 hours using 10 scalar processors. Despite the fact that the Cokriging-based optimizer requires 65% more function evaluations to converge than the gradient-based reference, it finally achieves a better function improvement in less process iterations. Moreover, this optimizer proves to perform more exploration of the design space than the gradient algorithm. It converges to a shape far from the baseline configuration and more complex than the shape given by the gradient reference.

Contacts: Marc Montagnac, Julien Laurenceau