3.2 Optimization
3.2.1 Meta Models
Aerodynamic models are largely used in a lot of applications in A/C design
and interdisciplinary process (loads, MDO, Identification). In that context, CFD based metamodels
are developed at Cerfacs making use of Proper Orthogonal Decomposition (POD), Kriging and Neuronal Networks.
These models are designed to give quickly the main features of aerodynamic system and are well adapted
to interdisciplinary exchange.
Aerodynamic Metamodel using Proper Orthogonal Decomposition (POD) (J.-Ph. Boin)
The Proper Orthogonal Decomposition has been applied to a set of CFD computations in order to
extract global information and build a metamodel. According to one or several parameters (time for
unsteady flows or flight parameters for steady flows), a set of CFD computations are performed, called
snapshots. The POD decomposition is applied to these snapshots, solving the eigenvalue problem of
the auto-correlation matrix. The proper modes of this problem are orthogonal and build up the POD basis.
The POD basis could be understood as an optimal basis which contains more information than any other ones.
Each mode represents a characteristic of the set of snapshots. For unsteady turbulent flows, it represents the coherent
structures. With that basis, it is also possible to reconstruct all the snapshots and to extrapolate solutions
for a different set of parameters (metamodel).
Introduced in Fluids Mechanic by Lumley [1], the POD is mainly applied on experimental data for unsteady flow
analysis, Delville and al. [2]. More recent applications have been done for numerical steady flows
in Optimization, Bui-Thanh and al [3].
CERFACS has developed a basic functioning POD module in Python. This module has been applied to the
analysis of a turbulent flow in a combustion chamber on unstructured meshes in collaboration with
the CERFACS Combustion-CFD team and in the reconstruction of pressure distribution on a high lift
configuration in a polar computation context on multi-block structured meshes (Fig. 3.9).
The quality of the POD-based metamodel is first linked to the good choice of the snapshot distribution
(for steady problem). We aimed at putting forward systematic strategies to achieve the optimal
distribution of CFD evaluation, using sampling methods and design of experiments.
Another aspect is the use of an efficient interpolation process for the reconstruction step such as
linear and quadratic interpolation, Radial Based Function (RBF) and Neuronal Network (NN).
For the two latest topics, collaborations with the CERFACS Algo team have started.
Figure 3.9: POD: Turbulent flow analysis in a combustion chamber (left),
data reconstruction of Cp (right).
[1] Lumley J., (1967), The structure of inhomogeneous turbulent flows,
Atmospheric Turbulent and Radio Wave Propagation, 166-178.
[2] Delville J., Ukeiley L.,Cordier L., Bonnet J. and Glauser M., (1999),
Examination of large-scale structures in a turbulent plane mixing layer. part1. proper orthogonal
decomposition, Journal of Fluid Mechanics, 391, 92-122.
[3] Bui-Thanh T., Damodaran M. and Willcox K.,(2004),
Aerodynamic Data Reconstruction and Inverse Design using Proper Orthogonal Decomposition,
AIAA Journal, 42(8), 1505-1516.
Kriging surrogate-model (J.C. Jouhaud, M. Montagnac, J. Laurenceau, P. Sagaut)
The term "Kriging" denotes a family of interpolation methods where weighting coefficients are chosen to minimize
the variance of the error [1]. First applied in geological analysis, it has been extended to many fields of application,
including agriculture, human geography, epidemiology, biostatistics or archelogy. The Kriging method is a linear interpolation
method that predicts values at unknown locations (i.e. response surface construction) from data observed at known locations
(control points).
Figure 3.10: NACA (m,p,16) profile - Sample points in the parameter space with two levels of refinement (left part) and corresponding isolines of the cost function (right part).
Recently, a Kriging method has been implemented in a Kriging Computational Suite which is coupled with elsA solver.
This suite is divided in four stages :
-
Definition of the following data :
-
· Range of variation of the uncertain parameters.
- · Sampling in the selected subspace for uncertain parameters.
- CFD Computations with elsA :
-
· Realization of the simulations for each sampling point in the uncertain parameter space.
- Data Processing :
-
· Computations of the values taken at sampling points by the function to be interpolated.
- · Creation of data files for Kriging method.
- Kriging Method :
-
· Reconstruction of the Response Surface.
- · Computation of the Mean Square Error of Kriging method.
- · Visualization of the Response Surface.
- · Determination of the zone to be refined in the uncertain parameter space.
- Return to the first stage.
Figure 3.11: NACA (m,p,16) profile - Robust solution of the shape optimization (solid line) compared to symmetric shape (dashed line).
Firstly, Kriging computational suite was considered to compute the corrections needed to recover equivalent free-flight conditions
from wind-tunnel experiments. Using this approach, optimal corrected values of the free-stream Mach number and the angle of attack
from the compressible turbulent flow around the RAE2822 wing were computed [2]. It appeared that such a tool makes possible to compute optimal
corrections for wind-tunnel parameters.
Secondly, the computational suite has been applied to the case of the multidisciplinary
shape optimization of a 2D NACA airfoil [3]. The cost function is designed so that both the far-field radiated noise
and the aerodynamic forces are controlled. In order to increase the efficiency of the method, a dynamic Kriging
method has been developed, which can be interpreted as an Adaptive Mesh Refinement method in the shape optimization parameters (see Fig. 3.10 and Fig. 3.11).
[1] D. G. Krige, (1951), A Statistical Approach to some Basic Mine Valuations Problems on the Witwatersrand, Journal of Chemistry, Metal. and Mining Soc. of South Africa, 52, 119-139.
[2] J.C. Jouhaud, P. Sagaut and B. Labeyrie, (2006), A Kriking approach for CFD/Wind Tunnel Data
Comparion, Journal of Fluids Engineering, in press.
[3] J.C. Jouhaud, P. Sagaut, M. Montagnac and J. Laurenceau, (2005), A surrogate-model based
multi-disciplinary shape optimization method with application to a 2D subsonic airfoil, Submitted for
publication to Computers and Fluids.
Neural networks (F. Blanc)
Neural networks, based on a sampling, are widely used in the field of statistics to automatically
build models descibing complex relations between inputs and outputs with a low computational cost.
The basic principle of neural networks is to create an approximation of a complex function by combining
simple elementary functions --- called neurons --- through a network.
There's a lot of different structures of neural networks, each having its own advantages and drawbacks.
Among these structures, the Radial Basis Functions Network
[Blanc, 2005]
has been chosen after
comparative tests, because of its simplicity and its robustness. A Radial Basis Functions Network is
based on a layer of elementary radial functions. The output of each radial basis function depends
on the distance between an input data of the neural network and a list of elements called centers which are
defined for each elementary function. Outputs of the neural network
are computed using a weighted sum of elementary output functions.
Figure 3.12: Representation of a radial basis neural network
To create an approximation of a function, some parameters of the network (number and type of radial functions,
position of their centers, coefficients of the weighted sum) have to be computed through a process called
learning of the neural network. A completely automatic learning algorithm has been created: given a set
of samples of a function, it builds a neural network which approximates this function.
In order to reduce the cost of a genetic optimization process, neural networks have been used to predict airfoil
aerodynamic coefficients (lift and drag). The neural network inputs were 6 parameters defining the shape of the airfoil.
3.2.2 Optimization algorithms
Efficient optimization algorithms are the key features to manage computational cost since each objective evaluation called optimizer iterates
is a cfd calculation sometimes involving gradients computation. That is why optimizer must be adapted to each type of optimization problem:
DOT or CONMIN when gradients are available, trust region or pattern search for local gradient-free optimization, genetic algorithm for global
gradient-free optimization.
CONMIN/DOT (J. Laurenceau)
DOT (Design Optimization Tools from Vanderplaats R&D) or CONMIN (the free version) are the gradient-based optimizers used to drive the adjoint method of elsA and allow finding local minima in a constraigned design space. The process, using discrete adjoint state of NS equations and quasi-Newton optimization algorithm, is very efficient and precise. These optimizers are also able to compute gradients 'internally' by finite differences.
Gradient-free local optimization (J. Laurenceau)
When local minimum found by a quasi-Newton method is not satisfying or gradient can not be computed by an adjoint method, trust region (Cerfacs Algorithm Team) or pattern search (Sandia's AsynchronousParallel Pattern Search) algorithms can be more suited. Despite their higher number of evaluations to achieve convergence, these optimizers are less sensitives to local minima. Moreover, APPS is a parallel algorithm.
Genetic algorithms (F. Blanc)
During spring and summer 2005, some work on genetic algorithms has been done to evaluate their suitability
for solving optimization problems in aerodynamics. A multi objective genetic algorithm has been used as a basis for
these tests. This algorithm is an evolution of the famous genetic algorithm GADO [1] (Genetic Algorithm for Design
Optimization). It has been coupled with the software elsA to solve multi objective problems. Experiments have
been performed on a test case which consists in the optimization shape of an airfoil shape to maximize
its lift and minimize its drag.
This test has shown that the genetic algorithm coupled with elsA was able to find the Pareto's front which is
the solution of this problem, see Fig. 3.13. But the computational cost of this
optimization process was too hight, even by using all possibilities offered by parallel computation.
It was then decided to test two techniques to improve the computational efficiency of the genetic algorithm:
the use of variable fidelity and the use of surrogates
[Blanc, 2005].
Figure 3.13: Pareto front achieved by solving the multi objective problem : Optimizing an airfoil shape to minimize its drag and maximize its lift
-
Variable fidelity : With this technique, elsA is not used to evaluate the performances of all designs.
Some performances are evaluated by a low cost and low fidelity software. The technique key point is the
indicator that is needed to determine which software to use for evaluating the performance of a given design.
For the airfoil test case, a singularity method was used for the low fidelity method. The variable fidelity
technique allowed to solve this optimization problem roughly 5 times faster than with the original genetic algorithm.
- Use of a surrogate : The original genetic algorithm has been modified to solve the optimization problems
by using a neural network to perform rapid evaluations of the performances of some designs. This new version of the
genetic algorithm is able to automatically build the neural network. It can add samples to data base of the neural
network to increase its accuracy when necessary. This genetic algorithm enabled to solve the airfoil optimization
problem 2 times faster than with the original one.
[1]Khaled Mohamed Rasheed, (1998), Gado : A Genetic Algorithm For Continuous Design Optimization, PhD. thesis, New
Brunswick.
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