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National Technical University of AthensSchool of Mechanical EngineeringDepartment of FluidsLab of Thermal TurbomachinesParallel CFD & Optimization Unit
Evolutionary Algorithm-based Design-Optimization
Methods in Turbomachinery
Stylianos A. Kyriacou
Supervisor: K.X. Giannakoglou, Professor NTUA
Athens, 2012
22/4709/08/12
Overview
EAs Efficient design tools for industrial scale problems:
Update EASY by:1) Knowledge Based Design (KBD)
● Use knowledge embedded in archived designs ● Automatic definition of the design variables' range ● Association of importance to the design space regions.
2) New evolution operators for ill-posed problems● Variable correlations Topological characteristics of elite set ● Use PCA to extract them● Apply Evolution operators on the principal directions (eigenvectors)
3) Efficient dimension reduction for metamodels ● Importance Eigenvalues ● Truncation of less important directions ● Suitable for high dimensional problems (>50 up to 500 dofs)● Faster & more accurate predictions
Applications / Validation ● 2D compressor cascade (1, 2 & 3)● Francis hydraulic turbine (1)● Hydromatrix® hydraulic turbine (2) ● 3D compressor cascade (3)
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EASY - (μ,λ)EA
λ offspring
Evaluation (λ CFD solutions)
Parent selection (μ parents)
Recombination /Crossover
Mutation
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EASY & MOO
Scalar cost function (SPEA II):➢
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Algorithmic basis (EASY)
Features (available prior to this PhD thesis) :● Inexact Pre-Evaluation (IPE)● Hierarchical ΕΑ
● Evaluation● Search ● Parameterization
● Distributed ΕΑ● Concurrent Evaluations (multi processor platforms)● Asynchronous ΕΑ● Memetic algorithms● Combinations of the above
Related theses by (PCOpt/NTUA)
➢ Α. Γιώτης (2003)➢ Μ. Καρακάσης (2006)➢ Γ. Καμπόλης (2009)➢ Β. Ασούτη (2009)➢ Χ. Γεωργοπούλου (2009)
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Inexact Pre-Evaluation (IPE)
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Knowledge Based Design (KBD)
Purpose:● Use knowledge embedded in archived designs
Expected gains:● Evolution speed-up (direct effect) ● Better metamodel accuracy (indirect effect)
GEOi=b1i ,b2
i , .. . ,bni
Background method● CBR method (4-R)● Automatic Revise step:
Inject archived design in 1st generation● High problem dimension (n)● User defined variable range ● Uniform importance
Proposed method:● Retrieve (m) archived designs● Reparameterize n-dimensional design vectors ● Introduce the optimization variables through
statistical analysis ● Reduced problem dimension ● Automatic definition of the
design variables' range.● Association of importance to
the design space regions
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1) Association of importance - cumulative distribution function (Φ).
2) Design variable grouping
3) Extrapolation
4) Generate candidate solutions ● Weighted sum ● w
i,k i = archived geometry , k= group
Optimization variables
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Operating conditions :➢ Μ
1= 0.54, α
1 = 44ο, Re = 4x105
Objective :➢ Minimization of ω =
Constraints :➢ Flow turning (ΔΘ) > 30ο
➢ Thickness:➢ @ 0.3/c > 0.10/c➢ @ 0.6/c > 0.08/c➢ @ 0.9/c > 0.01/c
Retrieve (4 airfoils):➢ B1: Μ
1=0.55, α
1=45.0ο, Δα=37.1ο, ω=0.01976,
➢ B2: Μ1=0.50, α
1=40.0ο, Δα=24.4ο, ω=0.01845,
➢ B3: Μ1=0.52, α
1=42.5ο, Δα=36.3ο, ω=0.02027,
➢ B4: Μ1=0.58, α
1=48.0ο, Δα=36.6ο, ω=0.01877,
Evaluation software :➢ Integral boundary layer method.
Parameterization :➢ Camber line, 3 vars➢ Width, 24 vars. (9 + 15)
KBD: Application in 2D compressor cascade
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Reuse (4 airfoils) :➢ B1 → Δα=36.3ο, ω=0.0207➢ B2 → Δα=27.8ο, ω=0.0278➢ B3 → Δα=37.5ο, ω=0.0201➢ B4 → Δα=33.1ο, ω=0.0237
4 Optimization runs :➢ EA & ΜΑΕΑ
➢ Vars : 27➢ μ=20, λ=60, λ
ε=6
➢ IPE start: 400➢ KBD-EA & -ΜΑΕΑ
➢ Vars : 13➢ μ=20, λ=60, λ
ε=6
➢ IPE start: 100
Optimal cascade :➢ Δα=30.2ο, ω=0.01834
KBD: Application in 2D compressor cascade
KBD-EAEA
KBD-MAEAMAEA
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KBD: Francis modernization/rehabilitation project
Difficulties (modernization - rehabilitation) ● Coupling with preexisting parts
● Geometric ● Hydraulic
● Small Hub & Shroud margins● No available archived designs
● Large number of design variables (336)
Optimization ● 3 Operating points
● OP 1, Best Efficiency (BE)● α, n
ed, Q
ed● OP 2, Part Load (PL)
● α, ned
, Qed
● OP 3, Full Load (FL)● α, n
ed, Q
ed● Blade number = 13
Archived design selection● Similarity regarding BE ● n
ed Q
ed
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KBD: Francis modernization/rehabilitation project
Evaluation
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Objectives 1) Draft tube coupling
Cu & C
m
2) Uniform loading .
Constraints1) Cavitation
σi
Hist < σplant
=0.2
2) Given Η ΔΗ < 1.5% Η
target (BE)
ΔΗ < 5% Ηtarget
(PL & FL)
KBD: Francis modernization/rehabilitation project
1414/4709/08/12
KBD – Francis: Reuse B1
BE operating point
ΔΗ=5.0% > 1.5% σ
i = 0.21 > 0.20 = σ
plant
ΔΗ=3.6% < 5% σ
i = 0.23 > 0.20 = σ
plant
ΔΗ=5.7% > 5% σ
i = 0.20 = σ
plant
PL operating point
FL operating point
1515/4709/08/12
KBD – Francis: Reuse B2
ΔΗ=3.8% > 1.5% σ
i = 0.22 > 0.20 = σ
plant
ΔΗ=2.6% < 5% σ
i = 0.24 > 0.20 = σ
plant
ΔΗ=4.8% < 5% σ
i = 0.22 > 0.20 = σ
plant
BE operating point PL operating point
FL operating point
1616/4709/08/12
KBD – Francis: Reuse B3
ΔΗ=6.8% > 1.5% σ
i = 0.19 < 0.20 = σ
plant
ΔΗ=6.9% > 5% σ
i = 0.21 > 0.20 = σ
plant
ΔΗ=7.1% > 5% σ
i = 0.18 < 0.20 = σ
plant
BE operating point PL operating point
FL operating point
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KBD - Francis: Results
➢ Comparison: ➢ ΕΑ (336 vars), archived designs in the first generation.➢ μ=30, λ=120 ➢ No IPE.
➢ ΜΑΕΑ-KBD (19 vars), archived designs in the first generation➢ μ=20, λ=60, λ
ε=6
➢ IPE after 150 candidate solutions in DB
A
1818/4709/08/12
KBD - Francis: Optimal Design (OD) Α
ΔΗ=1.1% < 1.5% σ
i = 0.18 < 0.20 = σ
plant
ΔΗ=1.5% < 5% σ
i = 0.193 < 0.20 = σ
plant
ΔΗ=4.1% < 5% σ
i = 0.18 < 0.20 = σ
plant
BE operating point PL operating point
FL operating point
1919/4709/08/12
KBD - Francis: OD A vs B1 & B2 (@ BE )
Α Β1 Β2
2020/4709/08/12
Ill-posed optimization problems (Variable correlations)
Ill-posed optimization problems
X Є R10
Isotropic objective function
Anisotropic objective function
Importance X2 >> X1Optimal Χ2 =! f(X1)
Problem solving difficulty (increasing vars)
Non-separable objective function
Optimal Χ2 = f(X1)
Optimal Χ1 = f(X2)
Separable F● Exponential increase of search space ● Linear increase of difficulty
Non-separable F● Exponential increase of search space ● Exponential increase of difficulty
Well-postedIll-Posted
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Ill-posed MOO problems
Welded beam example . ● Design variables (2):
● Welding length (Χ1=l).
● Cross section (Χ2=t).
● Objectives:● Minimize Cost
● Minimize displacement
● Constraints:● Maximum accepted stress (τ,σ & P
c).
● Variable correlation
X2
X1
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Estimation ● topological characteristics of elite set.● Use PCA.● Dynamic update every generation
PCA● Elite set → standardized data–set ● Empirical covariance matrix
● ● Use spectral decomposition theorem
●
● Λ eigenvalues ● U eigenvectors● U → separable directions ● Λ → importance metric
● Λ = Importance ● Λ = Importance
Separable directions estimation
ΕΑ
PC
A
2323/4709/08/12
Non-separable to Separable (New Evolution Operators)
Non-separable separable➢ Fixed Objectives (Quality metrics)➢ Change Parameterization (NO!)
➢ Lost Knowhow (based on parameterization)➢ Each problem one parameterization➢ Best parameterization unknown
➢ Update Evolution operators (YES!)➢ Retain Knowhow➢ Uniform parameterization➢ Altered offspring probability (as if separable)
Algorithm ΜΑΕΑ(PCA) ➢ Ɐ offspring ➢ If Κ
DB > K
min➢ IPE
➢ Else ➢ Evaluate all
➢ Update elite set ➢ Calculate separable direction (PCA) ➢ Align to separable direction
➢
➢ Apply Evolution operators ➢ Recombination & mutation
➢ Restore initial parameterization ➢ ,
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Non-separable to Separable (Example)
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Non-separable to Separable (Example)
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Non-separable to Separable (Example)
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Non-separable to Separable (Example)
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Non-separable to Separable (Example)
PCA-drivenRecombination
ConventionalRecombination
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Reduce metamodel (ANN) inputs
● Rotate training patterns align to eigenvectors ● Remove less important high eigenvalues
e2e1
e1
e2
PCA aided metamodels
3030/4709/08/12
➢Reduced variable range➢R13 → R27
➢Δα=30.2ο, ω=0.01834
Comparison :➢ ΜΑΕΑ.
➢ IPE if #DB > 400.➢ RBFn 20-30 t.p.. Є R27.
➢ Μ(PCA)AEA(PCA)➢ IPE if #DB >200.➢ RBFn 5-8 t.p. Є R10
Ill-posed: 2D compressor cascade
Optimal airfoil :➢ Δα=30.0ο, ω=0.01803
ΜΑΕΑμ=20, λ=60, λe=6
Χ Є R27
ΜΑΕΑ(KBD)μ=20, λ=60, λe=6
Χ Є R13
L. 1
L. 2
3131/4709/08/12
➢ Objective➢
➢ 20 design variables➢ Geometric & aerodynamic constraints
➢ Thickness (3 positions)➢ α
2 < 53ο
Ill-posed: 3D compressor cascade with tip clearance
3232/4709/08/12
Comparison :➢ ΜΑΕΑ.
➢ IPE if #DB > 220.➢ RBFn 20-30 t.p.. Є R20.
➢ Μ(PCA)AEA(PCA)➢ IPE if #DB >150.➢ RBFn 5-8 t.p.. Є R10
Optimal airfoil :➢ α
2= 52.6ο, ω=0.104
Ill-posed: 3D compressor cascade with tip clearanceResults
3333/4709/08/12
Ill-posed: HYDROMATRIX®
HYDROMATRIX®● Number of small non-regulated axial turbines ● Regulation on/off modules● Operating range (hydroelectric plant):
● Η = 5-30m, Q>60m3/s● Use pre-existing structures:
● Low installation cost ● Minimal Civil Construction work
● Minimum environmental impact● Minimal land usage
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Problem Formulation➢ 52 design variables➢ 3 Operating points
➢ BE operating point ➢ PL operating point ➢ FL operating point
➢ 2 Objectives 5 metrics➢ OP grouping weigted sum.
Quality Metrics➢ Outlet velocity profiles (Χ2)➢ Uniform loading➢ Cavitation ➢ Pumping surface (PL)
Ill-posed: HYDROMATRIX®
3535/4709/08/12
Parameterization
Regulation for BE ➢ Known (BC) Η, ω, Geometry
➢ Result Q➢ Regulate Q @ BE through α
Constraints➢ Q @ PL & FL
Ill-posed: HYDROMATRIX®
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Ill-posed: HYDROMATRIX® - Results
3737/4709/08/12
BE FL PL
Ill-posed: OD HYDROMATRIX®
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Experimental Validation HYDROMATRIX®
Problem Formulation➢ 52 design Variables➢ 3 Operating points
➢ BE operating point ➢ H = 7.47 m, Q = 9.22 m3/s, N = 375 rpm➢ nED = 0.97, qED = 0.62
➢ PL operating point ➢ H = 3.40 m, Q = 8.44 m3/s, N = 375 rpm➢ nED = 1.42, qED = 0.84
➢ FL operating point ➢ H = 8.15 m, Q = 9.35 m3/s, N = 375 rpm➢ nED = 0.92, qED = 0.6
➢ 3 objectives (5 metrics)➢ Outlet velocities.➢ Uniform loading ➢ Cavitation
➢ OP grouping through weights
Gains➢ Reduced overall design time >50%➢ Reduced cost (1/3) ➢ Suitable for Η<5m & Q< 60m3/s
➢ ~ 6,000 Gwh/year Europe Reference designDesign by proposed method
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Hydraulic parts designed by proposed method(s)
Francis Pump & Pump-Turbine
Bulb & HYDROMATRIX
Kaplan Distributor
Draft-tube
4040/4709/08/12
Single regulated machines
Problem formulation➢ 30-500 design variables➢ 1-5 OPs
➢ BE operating point ➢ OPs Cavitation danger
➢ 2-3 objectives (5 metrics)➢ OP grouping ➢ Regulated guide vanes
➢ Internal optimization (NR) ➢ Objective OP (1 dof, 1 obj)
4141/4709/08/12
Double regulated machines
Problem formulation➢ 30-500 design variables➢ 1-5 OPs
➢ BE operating point➢ OP Cavitation danger
➢ 2-3 oblectives. (5 metrics)➢ OP grouping➢ Regulated guide vanes
➢ Internal optimization ➢ Regulated rotor blades
➢ introduce β (1 / OP.) in design variables➢ Internal optimization (NR)
➢ Cu @ hub (2 dofs, 2 objs)
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Software developed during this PhD thesis
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Epilogue - Outcome
3 Innovative Methods 1) KBD
● knowledge embedded in archived design● reduced dimension● association of importance to the design space regions
2) PCA aided evolution operators● ill-posed well-posed● maintain parameterization
3) PCA aided metamodels● metamodels for high-dimensional ill-posed optimization problems
AchievedEAs Efficient design tools for industrial scale problems (ill-posed & high-dimension) ● Return high quality designs in reduced time (cost)● HYDROMATRIX : reduced time >50%
Industrial use of software developed during this PhD thesis● By Andritz Hydro (5 locations) ● Design of : Francis, Kaplan, Bulb, Pump & Pump-Turbines● Design of: Draft tubes, distributors
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Future work
● Distributed EA based on clustering of the elite set ● different PCA per deme (non linear variable correlations)
● High fatality optimization problems● different set of evolution operators during the first stages of
evolution
● Hybridization of EAs with Gradient Based Methods● adjoint regarding the quality metrics for hydraulic turbines● use HEAs (L1 EA) & (L2 GBM)
● ANN for hierarchical evaluation● ANN predict quality metric differences (from low to high fidelity)● Update targets associated with low-fidelity tool
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Publications
➢ K.C.Giannakoglou, V.G. Asouti, S.A. Kyriacou, X.S. Trompoukis: ‘Hierarchical, Metamodel–Assisted Evolutionary Algorithms, with Industrial Applications’, von Karman Institute Lectures Series on ‘Introduction to Optimization and Multidisciplinary Design in Aeronautics and Turbomachinery’, May 7-11, 2012.
➢ S.A. Kyriacou, S. Weissenberger and K.C. Giannakoglou.Design of a matrix hydraulic turbine using a metamodel-assisted evolutionary algorithm with PCA-driven evolution operators. International Journal of Mathematical Modelling and Numerical Optimization, SI: Simulation-Based Optimization Techniques for Computationally Expensive Engineering Design Problems 2011
➢ I.A. Skouteropoulou, S.A. Kyriacou, V.G Asouti, K.C. Giannakoglou, S. Weissenberger, P. Grafenberger. Design of a Hydromatrix turbine runner using an Asynchronous Algorithm on a Multi-Processor Platform. 7th GRACM International Congress on Computational Mechanics, Athens, 30 June-2 July, 2011
➢ S.A. Kyriacou, S. Weissenberger, P. Grafenberger, K.C. Giannakoglou. Optimization of hydraulic machinery by exploiting previous successful designs. 25th IAHR Symposium on Hydraulic Machinery and Systems, Timisoara, Romania, September 20-24, 2010.
➢ H.A. Georgopoulou, S.A. Kyriacou, K.C. Giannakoglou, P. Grafenberger and E. Parkinson. Constrained multi-objective design optimization of hydraulic components using a hierarchical metamodel assisted evolutionary algorithm. Part 1: Theory. 24th IAHR Symposium on Hydraulic Machinery and Systems, Foz do Iguassu, Brazil, October 27-31, 2008.
➢ P. Grafenberger, E. Parkinson, H.A. Georgopoulou, S.A. Kyriacou and K.C. Giannakoglou. Constrained multi-objective design optimization of hydraulic components using a hierarchical metamodel assisted evolutionary algorithm. Part 2: Applications. 24th IAHR Symposium on Hydraulic Machinery and Systems, Foz do Iguassu, Brazil, October 27-31, 2008.
➢ S. Erne, M. Lenarcic, S.A. Kyriacou. Shape Optimization of a Flows Around Circular Diffuser in a Turbulent Incompressible Flow. ECCOMAS 2012 Congress, Vienna, Austria, September 10-14, to appear 2012
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Acknowledgements
Development and laboratory testing of improved action and Matrix hydro turbines designed by advanced analysis and optimization tools” (HYDROACTION, Project Number 211983).
ANDRITZ HYDRO is a global supplier of electro-mechanical systems and services (“water to wire“) for hydropower plants and one of the leaders in the world market for hydraulic power generation.