Evolutionary and Deterministic Methods for Design,
Optimization and Control with Applications to Industrial and Societal Problems
EUROGEN 2005
R. Schilling, W.Haase, J. Periaux, H. Baier, G. Bugeda (Eds)
c
FLM,
Munich, 2005
PROGRESS IN METAMODEL ASSISTED EVOLUTIONARY
ALGORITHMS WITH APPLICATIONS IN EXTERNAL
AERODYNAMICS AND TURBOMACHINERY
Kyriakos C. Giannakoglou†
†
Associate Professor,
National Technical University of Athens,
Lab. of Thermal Turbomachines,
P.O. Box 64069, Athens 157 10, GREECE, [email protected]
Key words: Optimization, Evolutionary Algorithms, Surrogate Evaluation Models.
Abstract. Research activities of the Design Optimization Group ( DOG) of the LTT/ NTUA
concerning the development and assessment of design optimization methods and tools are
summarized below. These are related to both evolutionary and deterministic (based on
the adjoint method) optimization methods. Particular emphasis is laid on the use of
computational intelligence to enhance the performance of evolutionary algorithms ( EAs),
by reducing their CPU cost when used along with computationally expensive evaluation
software.
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Kyriakos C. Giannakoglou
Cost Reduction in EAs
The first section presents enhanced EA–based optimization methods which may handle complex engineering problems with affordable computing cost. These are associated
with either single– (SOO) or multi–objective optimization (MOO) problems and become
advantageous whenever the evaluation candidate solutions (through CFD codes, for instance) is costly. Reduction of the optimization CPU cost is equivalent to the reduction
of the number of fitness evaluations required by the search method.
Procedures which can reduce the CPU cost of EAs are mainly based on the use of
(low–cost, approximate) surrogate evaluation models or metamodels. Through their use,
the number of calls to the costly, accurate evaluation tool (such as a CFD code) can be
kept low. Response surface models with low-order polynomials, artificial neural networks
(ANNs) and kriging are known metamodels. Existing metamodel–assisted EAs (MAEAs)
differ in the way metamodels are implemented. DOG/LTT/NTUA proposed the use of
local, on–line trained metamodels, each of which corresponds to a single population member, [1, 2, 3]. The proposed algorithm relies on the concept of the Inexact Pre-Evaluation
(IPE ) filter, [2, 3, 4, 5, 6, 7, 8]. In each generation, low-cost, local metamodels, trained
on subsets of the previously evaluated patterns, are used to screen the population and
pinpoint the top (most promising) individuals which are worth being exactly evaluated.
A separate metamodel is associated with each and every new candidate solution. By
means of the IPE, in SOO problems, the CPU cost per generation is kept even an order
of magnitude lower than by standard EAs.
Two further enhancements of MAEAs (abbreviated to EA-IPE-IF and D(EA-IPE-IF))
based on on–line trained metamodels are presented below.
1
1.1 EA-IPE-IF
An enhanced EA-IPE scheme is the so–called EA-IPE-IF one, [4, 5]. This EA-IPE
variant utilizes radial basis function networks (RBFNs) as metamodels. The EA-IPEIF concept is as follows: Modified neural networks (RBFNs) are trained to guess the
response and its gradient with respect to the inlet parameters. During the training, these
RBFNs take also into account the sensitivity of the response with respect to each one
of the design variables (i.e. their input parameters). As sensitivity measures, the partial
derivatives of the objective function with respect to the design variables are used. It is
important to make clear that the required derivatives are computed by the RBFNs and,
at the same time, used for their training. EA-IPE-IF outperforms EA-IPE which, in
turns has a much better performance than conventional EAs.
1.2 D(EA-IPE-IF)
The second promising method for lowering the cost of EAs is through their distributed variants (Distributed EAs, DEAs). DEAs rely on handling population subsets that
evolve in a semi-isolated manner, by regularly exchanging their best individuals. Distributed schemes generally outperform single-population ones. The new idea is to employ
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Kyriakos C. Giannakoglou
metamodels in DEAs, [5], especially on a cluster of interconnected processors. The new
algorithm is denoted by D(EA-IPE-IF) and incorporates all the aforementioned techniques.
1.3 A SOO Application
An indicative comparison of the previously presented algorithmic variants, which is
representative of a huge class of SOO problems, is presented. The relative positioning of
a three–element airfoil components, that yields maximum lift, is sought. The six design
variables determine the relative position of the slat and flap with respect to the main
body, [9]. In fig. 1, the optimal airfoil configuration with computed iso–Mach contours,
convergence plots for the four aforementioned algorithms as well as the computed IFs are
illustrated. The gain achieved can be interpreted in two different ways: the cost of EA–
IPE for reaching the solution quality of standard EAs is much lower or, by predetermining
the allowable CPU cost (i.e. maximum number of evaluations), a better solution can be
computed through EA–IPE. It is also clear that D(EA-IPE-IF) outperforms any other
variant.
5.3
Conventional EA
EA-IPE
EA-IPE-IF
D(EA-IPE-IF)
Lift Coefficient
5.25
5.2
5.15
5.1
5.05
0
200
400
600
800
1000
Exact Evaluations (~ CPU Cost)
Relative Importance
0.3
0.25
0.2
0.15
0.1
0.05
0
1
2
3
4
Design variables
5
6
Figure 1: Optimization of the three–element airfoil.
EA-IPE in MOO
Our experience has shown that, in MOO, the gain in CPU cost through the “standard“
(i.e. operating exactly as in SOO) IPE technique is not as noticeable as in SOO. Main
reasons are: (a) in SOO problems, the population soon becomes clustered around the
global optimum, in the vicinity of which adequate information exists to train the metamodels, which is not the case in MOO and (b) in MOO, the exact re-evaluation of just a
few population members per generation has negative impact on the number of solutions
which enter the Pareto front.
2
3
Kyriakos C. Giannakoglou
In order to improve the performance of the EA–IPE method in MOO, one should
proceed through carefully selecting the training patterns for the local metamodels (here
RBFNs), identifying and differently treating outliers, i.e. new solutions lying far from the
database (DB ) entries, selecting the RBF centers, etc. So:
(a) Selection of Training Patterns - Outliers’ Identification: In “standard” IPE the
selection of training patterns relies on distance–based criteria, with distances measured in
the design variable space. A selection scheme which is more suitable for MOO is based on
the formation of the minimum spanning tree. Through “walking” along the tree branches
while moving away from the new individual, decisions are made about which points are
to be used as training patterns, [10]. Through the same algorithm and the corresponding
criteria, outlying individuals can be identified. For the outliers, the metamodel is expected
to give rather inaccurate predictions and, thus, the exact evaluation tool should be used
instead.
(b) Selection of RBF centers and radii: It is known that RBFNs can be used as either
interpolation or approximation tools and this can be determined through appropriate
selection of their centers. So, EA-IPE for MOO should be enhanced through a new
mechanism, which automatically decides about whether the approximate fitness value
will be approximated or interpolated, [10]. Self-organizing maps (SOM ) is the heart of
this mechanism, [10].
2.1 A MOO Application
An isolated airfoil shape optimization problem is presented. The objectives are to
maximize CL and minimize CD at Re∞ = 6.2 · 106 , M∞ = 0.75 and a∞ = 2.734o. Starting
shape is the RAE2822 airfoil. At these flow conditions, the starting airfoil yields equal to
CL,ref = 0.749 and CD,ref = 0.0235. With the same CPU cost, the “standard” EA-IPE
algorithm, which employs simple RBFN fails to achieve a better front than conventional
EAs. The proposed, adaptive metamodel yields much better estimations and results to a
considerably better front than other variants, fig. 2.
3,0x10
4
2,5x10
0 ,
0
3
5
0 ,
0
3
0
0 ,
0
2
5
0 ,
0
2
0
0 ,
0
1
5
0 ,
0
1
0
Conventional EA
EA-IPE with simple RBFNs
EA-IPE with the proposed RBFNs
Reference airfoil
1500 Exact Evaluations
2,0x10
1,5x10
1,0x10
4
CD
Non-Dominated Grid Points
Conventional EA
EA-IPE with simple RBFNs
EA-IPE with the proposed RBFNs
4
4
4
1 ,
0
500
1000
2
1
,
0
0 ,
8
0 ,
6
0
,
4
1500
-CL
Evaluations
Figure 2: Two–objective airfoil optimization.
3
Development of Adjoint Methods
Continuous and discrete adjoint formulations for the inverse design of aerodynamic
shapes and the minimization of viscous losses in turbomachinery cascades, have also been
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Kyriakos C. Giannakoglou
developed. With respect to the latter, viscous losses are expressed in terms of entropy
generation caused by the boundary layer formation.
The minimization of entropy s generation is expressed as the difference of incoming
and outgoing mass averaged entropy, namely
Z
F =
Si
dṁsdS −
Z
So
dṁsdS =
Z
S
ρVn sdS
where Si,o are the inlet and outlet boundaries. Using the Gauss’ divergence theorem and
the continuity equation, this equation is transformed to a field integral as follows
F =
Z
Ω
ρui
∂s
dΩ
∂xi
and can also be written as
F =
Z
Ω
1 ∂ui
τij
dΩ
T ∂xj
(ui are the velocity components, τij are the stress components). During the formulation
of the adjoint equations, the field sensitivities are eliminated using the implicit function
theory together with the Gauss’ divergence theorem and the final gradient formula depends solely on surface terms. For the augmented objective function Faug , formed after
−
→
introducing the vector of costate variables Ψ ,
δFaug
−
→inv
−
→vis
∂fi
−
→T ∂ f i
= δF +
Ψ δ(
−
)dΩ
∂xi
∂xi
Ω
Z
the appropriate development of terms yields the objective function gradient, as follows
→T
Z −
∂U
→
−
→T −
−
→
δFaug=− Ψ f i δ(ni dS)−
Ai Tni Ψ δxk dS+
Sw
Sw∂xk
−
→vis
Z
Z
−
→T ∂ f i
δxk ni dS + Ψm qj δ(nj dS)−
Ψ
∂xk
Sw
Sw
"
!#
Z
∂ui ∂Ψj+1
∂Ψm ∂Ψi+1
∂Ψm
+uj
+
+ui
µ
δxl nj dS−
∂xi
∂xi ∂xj
∂xj
Sw∂xl
"
!#
Z
∂ui
∂Ψk+1
∂Ψm
λδij
+uk
δxl nj dS−
∂xk
∂xk
Sw ∂xl
Z
Z
∂uj
1
Rij
ni δxk dS+
Rδxi ni dS
∂xk
S
S T
Z
∂ui
where Ai are the Jacobian matrices of the inviscid fluxes, R = τij ∂x
and
j
Rij = 2(1 + δij )
∂ui
∂uj
4 ∂uk
+ 2(1 − δij )
− δij
∂xj
∂xi
3 ∂xk
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Kyriakos C. Giannakoglou
y
The proposed adjoint method is used to design a compressor cascade airfoil with minimum losses at laminar flow conditions. Geometrical constraints are taken into account
in order to avoid creating thin airfoils. Results are shown in figs. 3, 4 and 5.
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
Initial
Optimal
0.1
Initial
0.05
Optimal
0
-0.05
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
0.0007
0.000698
0.000696
0.000694
0.000692
0.00069
0.000688
0.000686
0.000684
0.000682
0.00068
0.0174
0.0172
0.017
Pt Losses
Entropy Generation
Figure 3: Initial and optimal blade contour and control points.
0.0168
0.0166
0.0164
0.0162
0.016
0.0158
0.0156
0
20 40 60 80 100 120 140 160
0
20
40
Iteration
60
80 100 120 140 160
Iteration
Figure 4: Optimization of a compressor blade: convergence plots.
0.0008
Continuous Adjoint
Finite Differences
0.0006
Gradient
0.0004
0.0002
0
-0.0002
-0.0004
-0.0006
-0.0008
0
5
10
15
Variable
20
25
Figure 5: Optimization of a compressor blade: Objective function gradient.
EAs with Gradient-Assisted Metamodels
Recently, new metamodels trained by taking into consideration available information
on both responses and their gradients have been proposed, [11, 12]. These will be referred
to as gradient–assisted multi–layer perceptron (GAMLP) and gradient–assisted radial–
basis function (GARBF) networks. These can be readily incorporated into MAEAs, in
particular those based on off-line (separately) trained metamodels. For such a MAEA
to be efficient, the additional cost for computing gradients (here, through the adjoint
method) and training the metamodel using both responses and gradients should be overbalanced by the considerable reduction in the training set size. Note that the role of the
adjoint equation solner is to compute objective function gradients to be used for training
the metamodel, rather than to guide a descent optimization algorithm. The EA undergoes
the search for the optimal solution(s).
4
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Kyriakos C. Giannakoglou
The inverse design of a turbine peripheral cascade is presented, fig 6. The target is to
capture a known pressure distribution over its surfaces (parameterized using two Bezier
surfaces with 33 control points each) at given flow conditions. The 3D Euler and the
corresponding adjoint equations are solved for each candidate blade cascade.
0.0001
IPE_RBF(15,50,10)
IPE_GARBF(15,50,10)
Steepest Descent
GARBF_OFFLINE_DB
Cost Value
1e-05
1e-06
1e-07
1e-08
0
100
200 300 400 500 600
Equivalent Flow Solutions
700
800
Figure 6: Design of a 3D turbine cascade.
Fig. 6 illustrates the convergence of the four optimization tools used. The first two
methods employ the IPE technique, in which the metamodels are (a) conventional RBF
and (b) GARBF networks. The cost for an exact evaluation using the IPE -GARBF
method is multiplied by two, to account for the cost of the adjoint equations solver.
Towards convergence, the additional information (gradient) used by the IPE –GARBF
enhances the metamodel prediction abilities and faster convergence is achieved. The
standard steepest descent method (third curve, figure 6), with gradient computed by
the adjoint method, yields a solution with very small CPU cost. The fourth method
is EAs supported by off–line trained GARBF. For the latter, an initial DB with 100
patterns is formed and objective function values and gradients are computed. Using the
so–trained GARBF, the EA computes an ”optimal” solution with almost negligible cost.
The ”optimal” solution is re–evaluated and the DB is updated. The metamodel is re–
trained and a new cycle is carried out. Eight cycles are performed at the cost of 216
equivalent flow solutions.
All EA or EA-IPE -based computations were performed using the optimization software
EASY developed and brought to market by DOG/LTT/NTUA.
Acknowledgement: This research program was co-funded by the European Social
Funds (75%) and National Resources (25%) (Program PYTHAGORAS I).
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Kyriakos C. Giannakoglou
REFERENCES
[1] Giannakoglou, K., ”Designing Turbomachinery Blades Using Evolutionary Methods”,
ASME Paper 99-GT-181, 1999.
[2] Giannakoglou, K., ”Acceleration of Genetic Algorithms Using Artificial Neural NetworksTheoretical Background, Von-Karman Institute Lecture Series (VKI-LS) 2000-07, 2000.
[3] Giannakoglou, K., Giotis, A., ”Acceleration of Genetic Algorithms Using Artificial Neural
Networks-Application of the Method”, VKI-LS 2000-07, 2000.
[4] Giannakoglou, K., Giotis, A., Karakasis, M., ”Low–Cost Genetic Optimization Based on
Inexact Pre–Evaluations and the Sensitivity Analysis of Design Parameters”, J. Inverse
Problems in Eng., 9, 389-412, 2001.
[5] Karakasis, M., Giotis, A., Giannakoglou, K., ”Inexact Information Aided, Low-Cost, Distributed Genetic Algorithms for Aerodynamic Shape Optimization”, Int. J. Num. Meth.
Fluids, 43, 1149-1166, 2003.
[6] Chiba, K., Obayashi, S., Nakahashi, K., Giotis, A., Giannakoglou, K., ”Design Optimization of the Wing Shape of the RLV Booster Stage using Evolutionary Algorithms and
Navier-Stokes Computations on Unstructured Grids”, EUROGEN 2003, Barcelona, 2003.
[7] Giannakoglou, K., ”Neural Network Assisted Evolutionary Algorithms, in Aeronautics and
Turbomachinery”, VKI-LS 2004-07, 2004.
[8] Giannakoglou, K., Papadimitriou, D., Kampolis, I., ”Coupling Evolutionary Algorithms,
Surrogate Models and Adjoint Methods in Inverse Design and Optimization Problems”,
VKI-LS 2004-07, 2004.
[9] Giannakoglou, K., ”Design of Optimal Aerodynamic Shapes using Stochastic Optimization
Methods and Computational Intelligence”, Progress in Aerospace Sciences, 38, 43-76, 2002.
[10] Karakasis, M., Giannakoglou, K., ”On the Use of Surrogate Evaluation Models in MultiObjective Evolutionary Algorithms”, ECCOMAS 2004, Jyvaskyla, 2004.
[11] Papadimitriou, D., Kampolis, I., Giannakoglou, K., ”Evolutionary Optimization using a
New Radial Basis Function Network and the Adjoint Formulation”, Inverse Problems,
Design and Optimization (IPDO) Symposium, Rio de Janeiro. 2004.
[12] Papadimitriou, D., Kampolis, I., Giannakoglou, K., ”Stochastic and Deterministic Optimization for Turbomachinery Applications based on the Adjoint Formulation”, ERCOFTAC Design Optimization Int. Conf., Athens, Greece, 2004”.
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