This document presents an adaptive model switching technique for variable fidelity optimization using population-based algorithms. The technique aims to provide reliable high-fidelity optimum designs with reasonable computational expense by leveraging multiple models of varying fidelity. It switches models by comparing the error distribution of the current model to the distribution of recent fitness function improvements over the population. The method was tested on airfoil and cantilever beam design problems, showing substantially better balance of optimum quality and efficiency than purely low- or high-fidelity optimizations.

1. Adaptive Switching of Variable-Fidelity Models in
Population-based Optimization Algorithms
Ali Mehmani*, Souma Chowdhury#, and Achille Messac#
* Syracuse University, Department of Mechanical and Aerospace Engineering
# Mississippi State University, Department of Aerospace Engineering
The AIAAAviation and Aeronautics Forum and Exposition
June 22-26, 2015
Dallas, TX

2. Making Heuristic Optimization more Viable?
2
• Population-based heuristic algorithms have been successfully
applied to diverse areas of science and engineering over the past
three decades – a core tool in designing complex systems.
• Challenge: Although proven to be effective in solving highly
non-linear problems, they often demand a high number of
function calls – computationally “too expensive” – for
complex problems.
• Need: Control the computational cost per function call, while
preserving the fidelity of the final optima.
Variable fidelity models + Model management strategies

3. Variable Fidelity Optimization (VFO)
3
• Variable fidelity models refer to models with different levels
of fidelity (low, medium, and high fidelity models)
• Model management strategies are techniques used to select
different models during the optimization process.
Medium fidelityLow fidelity High fidelity
Park model
(5 sec.)
Dynamic model
(8 mins)
CFD model
(30 hrs)
Model Management/Switching
Optimizer

4. Presentation Outline
4
• VFO: Motivation and Research Objectives
• VFO with Adaptive Model Switching (AMS)
 Methodology
 Quantifying Model Uncertainties
• Numerical Case Studies
 Aerodynamic shape optimization of 2D airfoil
 Shape Optimization of a Cantilever Composite Beam
• Concluding Remarks

5. • Trust region strategy Alexandrov et al. (1998) and Rodriguez et al.(2001)
low-fidelity model is used to define a sub-region of global optimum, then the high-
fidelity model is used to improve the low-fidelity model.
• Efficient Global Optimization Jones et. al. (1998) and Viana and Haftka (2013)
The accuracy of the low fidelity model (surrogate) is improved by running high
fidelity model in the regions of (i) global optimum and (ii) high predicted error.
• Individual-based evolution control Jin, et al. (2011)
Part of the individuals in the population are chosen and evaluated using high fidelity
model.
• Generation-based evolution control Jin, et al. (2011)
The whole population in specific iteration are evaluated using high fidelity model
Variable fidelity optimization (VFO)
5

6. Research Motivation
6
• Several of the existing VFO strategies are found to be defined
for specific types of low fidelity model (e.g., EGO works
primarily for Gaussian process-based surrogate models),
thereby limiting their applicability.
• Existing techniques generally consider the combination of only
two models of different fidelities (e.g., Trust-region methods,
and individual- and generation-based techniques).

7. Research Objectives
7
• Investigate an adaptive and model-independent strategy for managing
multiple variable-fidelity models (called Adaptive Model Switching or AMS),
with the objective to reduce the computational cost while converging to an
optimum with high fidelity function estimates.
• Implement the new adaptive variable fidelity optimization method in Particle
Swarm Optimization, and test its effectiveness through two practical case
studies.
Attributes:

8. VFO with Adaptive Model Switching (AMS)
8
• The important question is when and where to use models with different
levels of fidelity and cost.
Computationally expensive while wasting resources
Switching to higher fidelity models too early
Mislead the search process to suboptimal (or infeasible) regions
Switching to a higher fidelity model too late

9.  The AMS metric is a hypothesis testing that is defined by a comparison between
(I) the distribution of the relative fitness function improvement, and
(II) the distribution of the error associated with the model.
Rejection of the test; Don’t Switch Model Acceptance of the test; Switch Model
Fitness Func. Improvement (KDE)
Distribution of Model Error (LogN)
Adaptive Model Switching (AMS)
9

10. Adaptive Model Switching (AMS)
10
• represents a quantile function of a distribution;
• The p-quantile, for a given distribution function, , is defined by
• is an Indicator of Conservativeness (IoC).
• The IoC for the low fidelity model with an error distribution is defined as
the probability of the model error to be less than h
• The IoC is based on user preferences, and could be guided by the desired trade-
off between reliability and computational cost in the AMS-based VFO.
• Here, we defined pcr = 0.3.

11. AMS: Uncertainty Quantification
• The uncertainties associated with surrogate
models are determined using an advanced
surrogate error estimation method, called
Predictive Estimation of Model Fidelity *.
* Published in Struct. Multidiscip. Optim. (2015), Presented in AIAA SDM (2013)
• In the case of physics-based low fidelity (PLF)
models, the uncertainty in their output is
quantified through an inverse assessment, using
the same DOE as used for the surrogate model.
• Kernel Density Estimation (KDE) is adopted to model the distribution of the
relative improvement in the fitness function over consecutive t iterations.
11

12. Variable Fidelity Optimization: Airfoil Design
14 Mehmani et al.
Fig. 5 Design variables gov-
erning the geometry of the
airfoil
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
X
Y
x1
x2 x3
Table1 Design variables in airfoil optimization problem
Description Notation Lower limit Upper limit
Distance between themiddleof suction sideand horizontal axis x1 -0.01 0.01
Distance between themiddleof pressure sideand horizontal axis x2 -0.01 0.01
Distance between thetrailing edgeand horizontal axis x3 -0.01 0.01
Incidence angle x4 0◦
10◦
3.1.1 Aerodynamic modelswith different level of fidelity
To develop a high fidelity aerodynamic model for determining CL and CD (MA
HF),
the commercial Finite Volume Method package, FLUENT, is used. The Reynolds-
Shape variable angle of incidence
* The Mixed-Discrete Particle Swarm Optimization (MDPSO)[1] is used. [1] Chowdhury and Messac (2012)
Wortmann FX60.126
12
Physics-based low
fidelity model (PLF)
The fluid is steady, incompressible, and irrotational.
Computational time = 30 seconds
Surrogate model
(SM)
Kriging-Gaussian-30 sample points
Computational time = less than 0.1 second
Tuned low fidelity
model (TLF)
This model is constructed using the Multiplicative approach, as given by
onds, respectively (i.e., an order of magnitude apart). The pressure field around
the airfoil for the low and high fidelity aerodynamic models at a baseline design
(x1 = 0, x2 = 0, x3 = 0, andx4 = 5◦
) areillustrated in Fig. 7.
1.0171e+05
1.0054e+0
1.0066e+0
1.0077e+0
1.0089e+0
1.0101e+0
1.0112e+0
1.0124e+0
1.0136e+0
1.0147e+0
1.0159e+0
10.50
X
(a) High fidelity model
−0.5 0 0.5 1 1.5
−1
−0.5
0
0.5
1
x
y
1.0124
1.0126
1.0128
1.013
1.0132
1.0134
1.0136
1.0138
x 10
5
(b) Low fidelity physics-based model
Fig. 7 Pressure field around theairfoil at abaselinedesign
The third model is a surrogate model (MA
SM) constructed using a DoE of high
fidelity evaluation involving 30 sample points. The fourth model is a tuned low
fidelity model (MA
TLF). In this article, the tuned low fidelity model is constructed
using theMultiplicativeapproach, asgivenby
˜F(x,a) = f(x) × C(x) (21)
where ˜F isatunedlow fidelity model; f(x) isalow fidelity model;C(x) isanexplicit
tuning surrogateconstructedusingthehighfidelity samples, asshown below:
C(x) =
CL
CD
|HF
CL
CD
|PLF
(22)
C(x) : explicit tuning surrogate, f(x): HF model
High fidelity
model (HF)
The FLUENT is used for solving RANS eq.
Computational time = 300 seconds
Computational time ~ 30 seconds

13. Airfoil Design : Results
13
0 0.05 0.1 0.15 0.2 0.25
0
2
4
6
8
10
12
14
16
18
20
Error
PDF
0 0.05 0.1 0.15 0.2 0.25
0
5
10
15
20
25
Error
PDF
0 0.05 0.1 0.15 0.2 0.25
0
5
10
15
20
25
30
35
40
45
50
Error
PDF
Distribution of the fitness function improvements in different iterations of the airfoil optimization
0 0.05 0.1 0.15 0.2 0.25
0
5
10
15
20
25
30
35
40
Error
PDF
5th Iteration 15th Iteration
18th Iteration20th Iteration
F.F. improvement
Surrogate Error
TLF Error

14. Computational Time Function Evaluation
provides the best optimum value that is
5% better that the next best value
185% reduction in computing time
14
13th 19th
Airfoil Design : Results

15. Variable Fidelity Optimization: Cantilever beam design
Minimizing the maximum deflection of a
cantilever composite beam.
f
Weight density of epoxy resin, ρm [N/ mm3]
Thebeam optimization problemisdefined as
Minimize:
δmax
δ0
, [δ0] = 12.9
subject to
W/ W0 ≤ 1, [W0] = 2.9E4
σmax/ σ0 ≤ 1, [σ0] = 200
x4
2
1.2E6x1
≤ 1
xmin
i ≤ xi ≤ xmax
i , i = 1,2,3
In the optimization formulation, the inequality constra
arerelatedto theallowableweight, themaximumstress, a
on thebeam design (depth ≤ 10× width). Theweight an
given by
W = AρL =
12I
h2
× (12+ 5.2νf)10− 6
× L =
x1
x2
σmax =
q0L2
h
8I
=
1E6x2
8x1
Themodelsused to estimatethemaximum deflection,
3.2.1 Structural modelswith different levelsof fidelity
geometric restriction on the beam design
(depth ≤ 10 × width)
 Minimizing the maximum deflection of a cantilever composite
beam.*
 The design variables are:
(i) the second moment of area (x1),
(ii) the depth of the beam (x2), and
(iii) the fiber volume fraction (x3).
 optimization problem is
15*Zadeh and Toropov, SMO 2009

16. Cantilever beam design : Model Choices
Physics-based
low fidelity model (PLF)
Surrogate model (SM)
Tuned low fidelity
model (TLF)
High fidelity model (HF)
The models used to estimate the maximum deflection
The PLF Finite Element model is constructed using 2 beam elements in ANSYS
Computational time = 3.30 [Sec]
The HF Finite Element model is constructed using 1000 beam elements in ANSYS.
Computational time = 5.5 [Sec].
Surrogate model is constructed using Kriging with Gaussian correlation function.
Computational time less than 0.3 [Sec]
.
This model is constructed using the Multiplicative approach, as given by
Computational time less than 3.33 [Sec]
.
1.0054e+0
1.0066e+0
1.0077e+0
1.0089e+0
1.0101e+0
1.0112e+0
1.0124e+0
1.0136e+0
10.50
X
(a) High fidelity model
−0.5 0 0.5 1 1.5
−1
−0.5
0
x
y
(b) Low fidelity physics-based mod
Fig. 7 Pressure field around theairfoil at abaselinedesign
The third model is a surrogate model (MA
SM) constructed using a DoE of
fidelity evaluation involving 30 sample points. The fourth model is a tuned
fidelity model (MA
TLF). In this article, the tuned low fidelity model is constru
using theMultiplicativeapproach, asgivenby
˜F(x,a) = f(x) × C(x)
where ˜F isatunedlow fidelity model; f(x) isalow fidelity model;C(x) isanex
tuning surrogateconstructedusingthehighfidelity samples, asshown below:
C(x) =
CL
CD
|HF
CL
CD
|PLF
whereCL andCD arerespectively thelift and drag coefficients.
The surrogate model (MA
SM) and the surrogate component of the tuned lo
delity model (MA
TLF) arebothconstructedusingKrigingwithaGaussiancorrel
function.14,37 Kriging isan interpolating method that is widely used for repre
ing irregular data. Under theKriging approach, thezero-order polynomial fun
isusedasaregressionmodel. InthisarticletheOptimal LatinHypercubeisado
to determinethelocationsof thesamplepoints. ThePEMF method isthen ap
22
C(x) =
δmax|HF
δmax|PLF
The distribution of the error in the tuned low fidelity
rogate model (SM) are estimated using PEMF (Section
in Figs. 13(a) and 13(c), respectively. The distribution o
based low fidelity model (PLF) is estimated using the in
by leveragingthesame30 high fidelity samplesthat were
and SM; thePLF error distribution isshown in Fig. 13(b)
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pcr = 0.3
QP = 0.0001
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x 10
4
PDF
pcr = 0.3
QP = 0.097
0.
0
0.
0
0.
PDF
UQ: Using MC simulationsUQ: Using PEMFUQ: Using PEMF
Tuned LF Model[ Surrogate Model Low-fidelity Model
16

17. Cantilever beam design : Results
Computational Time Function Evaluation
33% lower computational
expense compared to PSO-TLF
119% lower computational expense
compared to PSO-HF
17

18. Concluding Remarks
• A novel model management technique, called Adaptive Model Switching,
was developed for variable fidelity optimization using population-based
algorithms.
• The method seeks to provide reliable high-fidelity optimum designs at a
reasonable computational expense, by leveraging multiple models.
• Model switching is guided by the comparison between:
1. Error distribution of the current model
2. Distribution of the recent fitness function improvement over the entire
population of candidate designs
• AMS was observed to substantially superior than purely low-fidelity or
purely high-fidelity optimizations, in terms of the balance between “quality
of the optimum” and “computational efficiency”.
• A more intuitive definition of the Indicator of Conservativeness (IoC) as a
function of desired trade-offs between computational expense and reliability
of optimum would further extend the practical applicability of this algorithm. 18

19. Thank you
Questions
and
Comments
19