Kyriacou 3melh en_pca

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

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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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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)


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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

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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

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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


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

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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


FL operating point

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KBD - Francis: OD A vs B1 & B2 (@ BE )

Α Β1 Β2

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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

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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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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

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➢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

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➢ Objective➢

➢ 20 design variables➢ Geometric & aerodynamic constraints

➢ Thickness (3 positions)➢ α

2 < 53ο

Ill-posed: 3D compressor cascade with tip clearance

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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

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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)


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Parameterization

Regulation for BE ➢ Known (BC) Η, ω, Geometry

➢ Result Q➢ Regulate Q @ BE through α

Constraints➢ Q @ PL & FL


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Ill-posed: HYDROMATRIX® - Results

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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

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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)

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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.

Kyriacou 3melh en_pca

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