Transcript No Slide Title

Multidisciplinary Aircraft Conceptual Design
Optimisation Using a Hierarchical Asynchronous
Parallel Evolutionary Algorithm (HAPEA)
University of Sydney
L. F. Gonzalez
E. J. Whitney
K. Srinivas
K.C Wong
Pole Scientifique - Dassault AviationJ. Périaux
Presented at the Sixth ADAPTIVE COMPUTING IN DESIGN AND MANUFACTURE(ACDM 2004)
APRIL 20th - 22nd, 2004 at ENGINEERS HOUSE, CLIFTON, BRISTOL, UK
1
Overview
PART 1
Multi-Objective Problems
PART 2
Research in Evolution
Algorithms for
Aeronautical Design
Problems (EAs)
PART 3
Test Cases and
Applications .
2
Multi-Criteria Problems
 Aeronautical design problems normally require a
simultaneous optimisation of conflicting objectives and
associated number of constraints. They occur when two
or more objectives that cannot be combined rationally.
For example:
 Drag at two different values of lift.
 Drag and thickness.
 Pitching moment and maximum lift.
3
…..Multi--Criteria Optimisation
A multi-criteria optimisation problem can be formulated as:
Minimise:
f ( x)
i
i  1,... N
g ( x)  0
j  1,.... M
Subject to constraints:
j
h ( x)  0
k
k  1,.... M
For a minimisati on problem, a vector x is said partially less than a vector x if :
1
2

f (x )  f (x )
i
i 1
i 2
* In this case the solution x dominates the solution x
1
2
Different Approaches: Traditional aggregating functions,
and Pareto and Nash.
4
Pareto Optimality
 Formally, the Pareto optimal set can be defined as the set of solutions
that are non-dominated with respect to all other points in the search
space, or that they dominate every other solution in the search space
except fellow members of the Pareto optimal set. For two solutions x
and y (in minimisation form):
.
rel  x, y   x dominates y if : f
 x   f i  y  : i 1...M
rel  x, y   x nondominated w.r.t. y if : f  x   l  f  y  :   1...M
i
i
i
rel  x, y   y dominates x if : f  x   f  y  :   1...M
i
i
i
i
For a problem in M objectives, this is called the 'relationship'
operator. In practice we compute an approximation to the continuous
set, by assembling .or each player, as can be seen in Figure 2,


whereby information is exchanged
ParetoSet  x*, x* , x* .....x*
 1

5
2
3
 
Nash Games
A Nash optimisation can be viewed as a
competitive game between two players
that each greedily optimise their own
objective at the expense of the other
player.
A Nash equilibrium is obtained when
no player can improve his own
objective at the expense of the other.
Epoch
Completed?
Player 1
Migrate and
Player 2
Exchange
6
The Problem…
Problems in aeronautical design optimisation:
 Traditional optimisation methods will fail to find the real answer in most
real engineering applications.
 Fitness functions of interest are generally multimodal with a number of
local minima. Sometimes the optimum shape/s is not obvious to the
designer. The fitness function will involve some numerical noise.
 Most aerodynamic design problems will need to be stated in multiobjective form.
 Modern aeronautical design uses CFD (Computational Fluid Dynamics)
and FEA almost exclusively.
 CFD has matured enough to use for preliminary design and optimisation.
 The internal workings of validated in-house solvers are essentially
inaccessible from a modification point of view (they are black-boxes).
7
… The Solution…. Why Evolution?
 Techniques such as Evolution Algorithms can explore
large variations in designs. They also handle errors and
deceptive sub-optimal solutions with aplomb.
 They are extremely easy to parallelise, significantly
reducing computation time.
 They can provide optimal solutions for single and multiobjective problems.
 EAs successively map multiple populations of points, allowing
solution diversity.
 They are capable of finding a number of solutions in a Pareto
set or calculating a robust Nash game.
8
What Are Evolution Algorithms?
 Based on the Darwinian theory of
evolution  Populations of individuals
evolve and reproduce by means of
mutation and crossover operators and
compete in a set environment for survival
of the fittest.
Evolution
Crossover
Mutation
Fittest
 Computers perform this evolution process as a mathematical
simplification.
 EAs move populations of solutions, rather than ‘cut-and-try’ one to
another.
 EAs applied to sciences, arts and engineering. Aerofoil and wing design,
crew scheduling, control loops,etc.
9
Why EAs? …Test Functions
Here our EA solves a two objective problem with two design
variables. There are two possible Pareto optimal fronts; one
obvious and concave, the other deceptive and convex.
10
“The Central Difficulty”
Evolutionary techniques are …
still … very … slow!
(Often involving hundreds or thousands
of separate flow computations)
Therefore, we need to think about ways of
speeding up the process…
11
Hierarchical Topology-Multiple Models
Model 1
precise model
Exploitation
Model 2
intermediate
model
Model 3
approximate
model
Exploration
Hierarchical Topology
We use a technique that
finds optimum solutions by
using many different
models, that greatly
accelerates the optimisation
process.
Interactions of the layers:
solutions go up and down
the layers.
Time-consuming solvers
only for the most promising
solutions.
Evolution Algorithm
Evaluator
Asynchronous Parallel
Computing
Parallel Computing and Asynchronous
Evaluation
12
Synchronous Evaluation
different speed
 Single population
 The whole population is passed to
the evaluator.
 All the individuals of a given
generation need to be evaluated
before proceeding to the next
generation
1 population (n individuals)
Evolution
Strategy
Synchromous
Evaluator
1 population (n individuals)
 Hierarchical populations
Sync
ES
Sync
Sync
ES
ES
ES
ES
ES
ES
Sync
Sync
Sync
Sync
 Each population has to go through
a fixed number of generations
before migration can take place
 Since migration is global, the
different populations will have to
wait for the slowest one before
exchanging individuals
Asynchronous Evaluation
different speed
 Single population
 Individuals are evaluated one by one,
and reintegrated in the population :
there is no notion of generation
 That means the ES can run on any
number of processors (whereas for a
synchronous approach, a population of
20 individuals can run on 20 processors
at the most)
1 individual
Evolution
Strategy
Asynchromous
Evaluator
1 individual
 Hierarchical populations
ES
ES
ES
Async
ES
ES
ES
ES
 Since there is no generation,
migration can take place anytime
after a minimum number of
evaluations have been performed
 There is no bottleneck
Asynchronous Evaluation…
 Fitness functions are computed asynchronously.
 Only one candidate solution is generated at a time, and only one
individual is incorporated at a time rather than an entire population at
every generation as is traditional EAs.
 Solutions can be generated and returned out of order.
No need for synchronicity  no possible wait-time bottleneck.
No need for the different processors to be of similar speed.
Processors can be added or deleted dynamically during the execution.
There is no practical upper limit on the number of processors we can
use.
 All desktop computers in an organisation are fair game.




15
….Asynchronous Evaluation…










Offspring are not sent as a complete 'block' to the parallel machines.
A candidate is generated at a time, and sent to any idle processor where it is
evaluated at its own speed.
After evaluation return to optimiser and check if accepted for insertion into the
main population or rejected.
New selector operator because offspring cannot now be compared one against
the other, or even against the main population due to the variable-time
evaluation.
Recently evaluated offspring are compared to a previously established rollingbenchmark and if successful, we replace (according to some rule) a preexisting individual in the population.
A separate evaluation buffer, which provides a statistical 'background check' on
the comparative fitness of the solution. Buffer size 2 x PopSize
We compare it with the selection buffer by assembling at random a small
subset called the tournament Q = [q1,q2,q3,…qn] and check that the individual
is not dominated by any member of Q.
Q =1/2B (Strong selective pressure), Q =1/6B (weak selection pressure).
Compare to past individuals (both accepted and rejected) -inserted or not
If accepted us strategy for replacement replace-worst-always method in this
paper.
16
Generate candidate
Send to idle
processor
If evaluation
completed send back
to optimiser
Assign fitness
Compare to a
tournament and if
successful replace
Compare to accepted
and rejected
individuals –insert
into the population
Applications-Test Functions (1)
Here our EA solves a two objective problem with two design
variables. The optimal Pareto front contains four
discontinuous regions.
17
Applications-Test Functions (2) TNK
Again, we solve a two objective problem
with two design variables and one. The
optimal Pareto front contains four
discontinuous regions and constraints
18

f1  x   x1

Minimise

f 2  x   x2



x 
C1  x   x12  x22  1  0.1cos 16 arctan 1   0


x2 
TNK :  subject to :


2
2

C2  x    x1  0.5    x2  0.5   0.5


0  x1   ,

0  x2  

Asynchronous Test: One Dimensional Nozzle
1.2
Target Mach Number
Generated Mach No.
Target Nozzle Shape
Mach Number
Generated Nozzlw Shape
1
0.8
0.6
Y-Nozzle half width
0.4
0
0.5
X-Distance
19
1
Synchronous, Single Population, Viscous model
Pop size = 20
7 processors
45mn
20
Asynchronous, Multiple Models, Viscous only
Pop size = 10
7 processors
12mn
21
CPU Times for HAPEA
Method
CPU Time
45m 27s  12m 8s
Single Pop,
Viscous,
Traditional EA
Hierarchical
Asynchronous
12m 6s  3m 58s
22
No of evaluations
2127  1137
726  112
Real –world applications
 Constrained aerofoil design for transonic transport
aircraft  3% Drag reduction

UAV aerofoil design
-Drag minimisation for high-speed transit and
loiter conditions.
-Drag minimisation for high-speed transit and
takeoff conditions.
 Exhaust nozzle design for minimum losses.
23
Real –world applications … (2)
AF/A-18 Flutter model validation.
Three element aerofoil reconstruction from
surface pressure data.
 UCAV MDO
Whole aircraft multidisciplinary design.
Gross weight minimisation and cruise efficiency
Maximisation. Coupling with NASA code FLOPS
2 % improvement in Takeoff GW and Cruise Efficiency
24
Case Studies
Multidisciplinary Aircraft Conceptual Design
Case Studies.
25
UCAV Conceptual Design.
Problem Definition:
 Find conceptual design parameters for a UCAV, to minimise two
objectives:
 Gross weight  min(WG)
 Cruise efficiency  min(1/[MCRUISE.L/DCRUISE])
 We have six unknowns:
Lower
Bound
Upper
Bound
Aspect Ratio
3.1
5.3
Wing Area (sq ft)
600
1400
Wing Thickness
0.02
0.09
Wing Taper Ratio
0.15
0.55
Wing Sweep (deg)
22.0
47.0
Engine Thrust (lbf)
30500
50000
26
Mission Definition
Cruise 40000 ft,
Mach 0.9, 400 nm
Taxi
Release Payload
1800 Lbs
Accelerate
Mach 1.5, 500 nm
Climb
Maneuvers at
Mach 0.9
Release Payload
1500 Lbs
20000 ft
Descend
Takeoff
Landing
Engine Start and warm up
Description
Range [R, Nm]
Requirement
1000
Cruise Mach Number [Mcruise]
Cruise Altitude [hcruise, ft]
1.6
40000
Ultimate Load Factor [nult]
12
Takeoff Field Length [sto, ft]
7000
27
Solver
 The FLOPS (FLight OPtimisation System) solver developed by L. A. (Arnie)
McCullers, NASA Langley Research Center was used for evaluating the aircraft
configurations.
 FLOPS is a workstation based code with capabilities for conceptual and
preliminary design of advanced concepts.
 FLOPS is multidisciplinary in nature and contains several analysis modules
including: weights, aerodynamics, engine cycle analysis, propulsion, mission
performance, takeoff and landing, noise footprint, cost analysis, and program
control.
 FLOPS has capabilities for optimisation but in this case was used only for
analysis.
 Drag is computed using Empirical Drag Estimation Technique (EDET) Different hierarchical models are being adapted for drag build up using higher
fidelity models.
28
Two Approaches
Solved via :
Nash theory
and
Pareto Optimality.
29
Implementation
Nash Approach.
Epoch
Completed?
-Two hierarchical trees, with two
Player 1
Migrate and
Player 2
Exchange
levels, population size of 40.
- Information exchanged (epoch) after 50 function evaluations.
Variables split:
-Player One: Aspect ratio, wing thickness and wing sweep; Maximises cruise
efficiency.
-Player Two: Wing area, engine thrust and wing taper; Minimises gross weight.
- Run for 600 function evaluations, but converged after 300.
30
Nash Results
31
Nash Results (2)
32
Nash Results (3)
Variables
Nash Equilibrium
Aspect Ratio
5.13
Wing Area (sq ft)
618
Wing Thickness
0.021
Wing Taper Ratio
0.17
Wing Sweep (deg)
28
Engine Thrust (lbf)
33356
Gross Weight (Lbs)
62463
Cruise Efficiency
23.9
MCRUISE.L/DCRUISE
33
Implementation
 Single population
 Pareto Optimality Approach
- Single Population.
1 individual
- Population size of 40.
Asynchromous
Evaluator
- Parallel computations, run
asynchronously.
- Run for 600 function evaluations.
34
1 individual
Pareto Optimality Results
Best for Obj 1
Nash Equilibrium
Compromised solution
Best for Obj 2
35
Comparison Results
Variables
Pareto
Member 0
Pareto
Member 3
Pareto
Member 7
Nash
Equilibrium
Aspect Ratio
4.76
5.23
5.27
5.13
Wing Area (sq ft)
629.7
743.8
919
618
Wing Thickness (t/c)
0.046
0.050
0.041
0.021
Wing Taper Ratio
0.15
0.16
0.17
0.17
Wing Sweep (deg)
28
25
27
28
Engine Thrust (lbf)
32065
32219
32259
33356
Gross Weight
(Lbs)
57540
59179
64606
62463
Nash Point
Decreasing Gross Weight
MCRUISE.L/DCRUISE
22.5
25.1
Increasing Cruise Efficiency
36
27.5
23.9
Comparison Results (2)
Upper Bound
Nash Equilibrium
Nash Design
Lower Bound
37
Subsonic Transport Design and Optimisation
Problem Definition:
 Find conceptual design parameters for a subsonic medium size
transport aircraft .
 Gross weight  min(WG)
 The aircraft has two wing-mounted engines, and the number of
passengers and crew is fixed to 200 and 8 respectively.
 The aircraft is designed to cruise at 40000 ft and Mach 0.8.
 We have six unknowns:
Lower Bound
Upper Bound
7.0
13.1
Wing Area (sq ft)
1927
2872
Wing Thickness
0.091
0.235
Wing Taper Ratio
0.15
0.55
Wing Sweep (deg)
22.0
47.0
32000
37000
Aspect Ratio
Engine Thrust (lbf)
38
Constraints and Implementation
Constraints
 Constraints in this case are minimum takeoff
distance, moment coefficient for stability and control
and range required. Violation of these constraints is
treated with an rejection criteria.
Implementation
 The solution to this problem has been implemented using a
single population and parallel asynchronous evaluation,
with the optimiser only considering a single objective.
 After an empirical study, it was found that a small
population size of 10 and buffer size of 30 produced
acceptable results.
39
Results
The algorithm was allowed to run for 1500 functions evaluations.
Broyden-Fletcher-Goldfarb-Shano (BFGS) algorithm --- > a 3.5%
improvement
 Conjugate gradient (CG) based (Polak-Ribiere) algorithm -- >
2.4% improvement
Description
EA Best
BFGS
CG_____
Aspect Ratio [ARw]
13.1
13.0
12.8
Engine Thrust [T, lbf]
34,770
38,852
39,021
Wing Area [Sw, sq ft]
1,929
2,142
2,218
Sweep
[w, deg]
27.0
28.4
27.32
Thickness [t/c]
0.091
0.112
0.096
Taper Ratio [w]
0.267
0.267
0.267
---------------------------------------------------------------------------------------------------Fuel Weight [Wf, lbs]
34,337
37,342
36,092
Gross Weight [Wg , lbs] 216,702
222,154
224,618
40
Conclusion
 The new technique with multiple models: Lower the computational expense dilemma
in an engineering environment (three times faster)
 The multi-criteria HAPEA has shown itself to be promising for direct and inverse design
optimisation problems.
 No problem specific knowledge is required  The method appears to be broadly
applicable to black-box solvers.
 As illustrated a variety of optimisation problems including Multi-disciplinary Design
Optimisation (MDO) problems can be solved.
 The process finds traditional classical aerodynamic results for standard problems, as
well as interesting compromise solutions.
 The algorithm may attempt to circumvent convergence difficulties with the solver.
 In doing all this work, no special hardware has been required – Desktop PCs networked
together have been up to the task.
41
What Are We Doing Now?
 A Hybrid EA - Deterministic optimiser.
 EA + MDO : Evolutionary Algorithms Architecture for Multidisciplinary Design
Optimisation
We intend to couple the aerodynamic optimisation with:
o Aerodynamics – Whole wing design using Euler codes.
o Electromagnetics - Investigating the tradeoff between efficient
aerodynamic design and RCS issues.
o Structures - Especially in three dimensions means we can investigate
interesting tradeoffs that may provide weight improvements.
o And others…
Wing MDO using Potential
flow and structural FEA.
42
Questions???
43
Results So Far…
 The new technique is
approximately three times faster
than other similar EA methods.
Evaluations
CPU Time
Traditional
2311  224
152m  20m
New
Technique
504  490
(-78%)
48m  24m
(-68%)
 A testbench for single and multiobjective problems has been
developed and tested
 We have successfully coupled the optimisation code to different
compressible and incompressible CFD codes and also to some aircraft
design codes
CFD
Aircraft Design
HDASS
MSES
FLO22
Nsc2ke
XFOIL
Flight Optimisation Software (FLOPS)
ADS (In house)
44
Appendix-Applications
45
Publications
ADVanced EvolutioN Team (ADVENT ) Selected Publications and Conference Papers
2003 E. Whitney, L. Gonzalez, K. Srinivas, J. Périaux: “Adaptive Evolution Design Without Problem Specific Knowledge ”,
Proceedings (to appear) of EUROGEN 2003, Barcelona, Spain.
2003 E. Whitney, “A Modern Evolutionary Technique for Design and Optimisation in Aeronautics ”, PhD Thesis, School of
Aerospace, Mechanical and Mechatronic Engineering, J07 University of Sydney, NSW, 2006 Australia
2003 E. Whitney, L. Gonzalez, J. Périaux:, and K. Srinivas, “Playing Games with Evolution: Theory and Aeronautical
Optimisation Applications”, ICIAM 2003 -- 5th International Congress on Industrial and Applied Mathematics, Sydney,
Australia, July 2003. To appear.
2002 E. Whitney, L. Gonzalez, K. Srinivas, J. Périaux: “Multi-Criteria Aerodynamic Shape Design Problems in CFD using a
Modern Evolutionary Algorithm on Distributed Computers”, Proceedings of the Second International Conference on
Computational Fluid Dynamics, Sydney, Australia.
2002 J. Périaux:, M. Sefrioui, E. Whitney, L. Gonzalez, K. Srinivas, and J. Wang “Evolutionary Algorithms, Game Theory and
Hierarchical Models in CFD”, Proceedings of the Second International Conference on Computational Fluid Dynamics,
Sydney, Australia.
2002 E. Whitney, M. Sefrioui, K. Srinivas, J. Périaux: “Advances in Hierarchical, Parallel Evolutionary Algorithms for
Aerodynamic Shape Optimisation”, JSME (Japan Society of Mechanical Engineers) International Journal, Vol. 45, No. 1.
2001 J. Périaux, M. Sefrioui, K. Srinivas, E. Whitney, J. Wang: “Recent Advances in Evolutionary Algorithms for Multicriteria
Design Optimisation in Aeronautics”, Kickoff Meeting, MACSI Working Group on Multidisciplinary Optimisation and
Inverse Problems, Vienna, Austria.
2001 M. Sefrioui, E. Whitney, J. Périaux, K. Srinivas: “Evolutionary Algorithms for Multi-Objective Design Optimisation”,
Proceedings of Coupling of Fluids, Structures and Waves in Aeronautics (CFSWA), A French / Australian workshop,
Melbourne, Australia.
2001 J. Périaux, M. Sefrioui, K. Srinivas, E. Whitney, J. Wang: “Advances in Hierarchical Parallel Genetic Algorithms and
Game Decision Strategies for Design Optimisation in Aeronautics”, Proceedings of the First French / Finnish Seminar on
Innovative Methods for Advanced Technologies, Espoo, Finland.
2000 E. Whitney, K. Srinivas: “Non-Generational Multiobjective Evolution Strategy for Aerofoil Design and Optimisation
Problems in CFD”: Proceedings of the First International Conference on Computational Fluid Dynamics, Kyoto, Japan:
46
Hierarchical Topology-Multiple Models
Model 1
precise model
Exploitation
Model 2
intermediate
model
Model 3
approximate
model
Exploration
 Interactions of the 3 layers: solutions go up and down the layers.
 The best ones keep going up until they are completely refined.
 No need for great precision during exploration.
 Time-consuming solvers are used only for the most promising solutions.
 Think of it as a kind of optimisation and population based multigrid.
47
An Example: Aerofoil Optimisation
Property
Mach
Reynolds
Lift
Flt. Cond. 1
Flt Cond.2
0.75
0.75
9 x 106
9 x 106
0.65
0.715
Constraints:
• Thickness > 12.1% x/c
(RAE 2822)
• Max thickness position
= 20% ® 55%
To solve this and other problems standard
industrial flow solvers are being used.
Aerofoil
Traditional Aerofoil
RAE2822
Conventional
Optimiser [Nadarajah
[1]]
New Technique
q
cd
[cl = 0.65 ]
cd
[cl = 0.715 ]
0.0147
0.0185
0.0098
(-33.3%)
0.0130
(-29.7%)
0.0094
(-36.1%)
0.0108
(-41.6%)
 For a typical 400,000 lb
airliner, flying 1,400
hrs/year:
 3% drag reduction
corresponds to 580,000
lbs (330,000 L) less fuel
burned.
[1] Nadarajah, S.; Jameson, A, " Studies of the Continuous and Discrete Adjoint Approaches to Viscous Automatic Aerodynamic Shape
Optimisation," AIAA 15th Computational Fluid Dynamics Conference, AIAA-2001-2530, Anaheim, CA, June 2001.
48
Aerofoil Optimisation (2)
Aerofoil Characteristics cl = 0.65
Aerofoil Characteristics cl = 0.715
Delayed drag
divergence at
high Cl
Delayed drag
divergence at
low Cl
Aerofoil
Characteristics
M = 0.75
Delayed drag
rise for
increasing lift.
49
ZDT Test Cases
50
ZDT1
 

f1 x  x1


n

9
ZDT1:  g  x   1 +
x
i
n-1

i 2

h  f1, g   1  f1/ g



51
ZDT2




ZDT 2 :  g  x  


 h f ,
1

f1  x   x1
9
1+
n-1

n
i 2
2
g   1  f1/ g

x
i

52
ZDT3
 

f1 x  x1


n

9
ZDT 3 : 
g  x  1 +
x
i
n-1

i 2

h  f1, g   1  f1/ g  f1/ g sin 10 f1 


53
ZDT4
 

f1 x  x1


n 


2
ZDT 4 :  g  x   1 + 10  n-1 
 x i  10 cos 4 x i 


i 2

h  f1, g   1  f1/ g



54


Constrained Test Cases
55
BNH



f1  x   4 x12  4 x22
 Minimise

2
2
f 2  x    x1  5    x2  5 


2
2

C1  x    x1  5    x2  5   25
BNH : 
 subject to :
2
2

C2  x    x1  8    x2  3  7.7


0  x1  5

0  x2  3



56
SRN

2
2
f1  x   2+  x1  2    x2  1

 Minimise
2

f 2  x   9x1   x2  1


SRN : 
C1  x   x12  x22  225
subject to :

C2  x   x1  3 x2  10  0


20  x1  20,


20  x2  20

57
Two Bar Truss Design

2
2
 Minimise f1  x, y   x1 16  y  x2 1  y

f 2  x   max( AC, BC )


TwoBarTruss : 
subject to : max  AC, BC  1(105)


1  y  3,


x0



A
B
x1
x2
58
Goal Programming- Test Problem P1

f1  x1, x2   2

Minimise

f 2  x1, x2   2


 subject to : S   0.1  x1  1, 0  x2  10 
P1: 

f1  10 x1

2

10   x2  5 

f2 

10 x1

59
Results. Candidate and Target Geometries
60
Results: Example of Convergence.
Mesh Adaptation : Mesh 15
61