Numerical shape optimization for compressible flows (Minimization of - - PowerPoint PPT Presentation

Numerical shape optimization for compressible flows

(Minimization of expensive cost functions)

• Praveen. C

praveen@math.tifrbng.res.in

Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in

Mechanical Engineering Seminar

• Dept. of Mechanical Engg., IISc, Bangalore

11 March, 2011

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 1 / 72

Collaborators

1 Regis Duvigneau

Project OPALE, INRIA, Sophia Antipolis

2 Biju Uthup

ADA, Bangalore

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 2 / 72

Outline

1 Approaches to optimization 2 Elements of shape optimization

◮ Shape parameterization ◮ Grid generation/deformation ◮ CFD solution/Adjoint solution ◮ Optimizer

3 Free-form deformation (FFD) 4 Particle swarm method 5 Surrogate models 6 Examples

• Praveen. C

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Example of optimization: RAE2822

Initial shape Solver: euler2d

• Praveen. C

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Example of optimization: RAE2822

Optimized shape Optimizer: Torczon Simplex, 20 Hicks-Henne parameters

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 5 / 72

Example of optimization: RAE2822

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 6 / 72

Approaches to optimization

Gradient-based Gradient-free Finite Difference Adjoint GA PSO

...

• Local optimum
• FD accuracy problem
• Adjoint solver required
• Issues with adjoint consistency
• Global optimum “possible”
• “Easy” to implement for engg.

problems

• Slow convergence: surrogate

models and parallelization

• Praveen. C

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Elements of shape optimization

Shape parameters Surface grid Volume grid CFD solution I Optimizer

• Praveen. C

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Shape parameterization

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 9 / 72

Shape parameterization approaches

• Aerodynamic DVs:

◮ LE radius, max camber, taper ratio

• PARSEC, Kulfan parameterization, etc.
• BSplines/NURBS
• Need to re-generate surface/volume grid whenever shape is

changed

• Or, use a free-form approach like RBF-based grid deformation
• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 10 / 72

Free Form Deformation

• Originated in computer graphics field (Sederberg and Parry)
• Embed the object inside a box and deform the box
• Independent of the representation of the object
• Deform CFD grid also, independent of grid type
• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 11 / 72

Free Form Deformation

Consistent parameterization Airplane shape DVs Compact set of DVs Smooth geometry Local control Analytical sensitivity Grid deformation Setup time Existing grids CAD connection

(Samareh)

• Praveen. C

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Free Form Deformation: Example

(R. Duvigneau, INRIA)

• Praveen. C

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Free Form Deformation

• X0(P) = coordinate of point P wrt reference shape
• Movement of point P under the deformation

X(P) = X0(P) +

ni

• i=0

nj

• j=0

nk

• k=0

YijkBni

i (ξp)Bnj j (ηp)Bnk k (ζp)

• Bernstein polynomials

Bn

m(t) = Cn mtm(1 − t)n−m,

t ∈ [0, 1], m = 0, 1, . . . , n

• Design variables

{Yijk}, 0 ≤ i ≤ ni, 0 ≤ j ≤ nj, 0 ≤ k ≤ nk

• Cannot change wing planform
• Wing twist can be added as additional variables
• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 14 / 72

Optimizer

• Praveen. C

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Particle swarm optimization

• Kennedy and Eberhart (1995)
• Modeled on behaviour of animal swarms: ants, bees, birds
• Cooperative behaviour of large number of individuals through

simple rules

• Emergence of swarm intelligence

Optimization problem

min

x∈D J(x),

D ⊂ Rd

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 16 / 72

Particle swarm optimization

• Kennedy and Eberhart (1995)
• Modeled on behaviour of animal swarms: ants, bees, birds
• Cooperative behaviour of large number of individuals through

simple rules

• Emergence of swarm intelligence

Optimization problem

min

x∈D J(x),

D ⊂ Rd

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 16 / 72

Particle swarm optimization

Particles distributed in design space xi ∈ D, i = 1, ..., Np

X2 X1

• Praveen. C

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Particle swarm optimization

Each particle has a velocity vi ∈ Rd, i = 1, ..., Np

X2 X1

• Praveen. C

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Particle swarm optimization

• Particles have memory (t = iteration number)

Local memory : pt

i = argmin 0≤s≤t

J(xs

i)

Global memory : pt = argmin

i

J(pt

i)

• Velocity update

vt+1

i

= ωvt

i + c1rt 1 ⊗ (pt i − xt i)

• Local

+ c2rt

2 ⊗ (pt − xt i)

• Global
• Position update

xt+1

i

= xt

i + vt+1 i

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 19 / 72

PSO: embarassingly parallel

xn

1

xn

2

. . . xn

Np

↓ ↓ ↓ J (xn

1)

J (xn

2)

. . . J (xn

Np)

↓ ↓ ↓ vn+1

1

vn+1

2

. . . vn+1

Np

↓ ↓ ↓ xn+1

1

xn+1

2

. . . xn+1

Np

Parallel evaluation of cost functions using MPI

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 20 / 72

Test case: Wing shape optimization

• Minimize drag under lift

constraint min Cd Cd0 s.t. Cl Cl0 ≥ 0.999

• FFD parameterization, n = 20

design variables

• Particle swarm optimization:

120 particles M∞ = 0.83, α = 2o (Piaggio Aero. Ind.) Grid: 31124 nodes

Cost function

J = Cd Cd0 + 104 max

• 0, 0.999 − Cl

Cl0

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 21 / 72

Test case: Wing shape optimization

• Minimize drag under lift

constraint min Cd Cd0 s.t. Cl Cl0 ≥ 0.999

• FFD parameterization, n = 20

design variables

• Particle swarm optimization:

120 particles M∞ = 0.83, α = 2o (Piaggio Aero. Ind.) Grid: 31124 nodes

Cost function

J = Cd Cd0 + 104 max

• 0, 0.999 − Cl

Cl0

• Praveen. C

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Wing optimization

Initial shape

• Praveen. C

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Wing optimization

Optimized shape

• Praveen. C

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PSO computational cost

• Slow convergence: O(100)-O(1000) iterations
• Require large swarm size: O(100) particles
• CFD is expensive: few minutes to hours
• Example: Transonic wing optimization (coarse CFD grid)

(10 min/CFD) (120 CFD/pso iter) (200 pso iter) = 4000 hours

• Praveen. C

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Surrogate Models

• Praveen. C

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Metamodels

• Expensive PDE-based model

Shape parameters Surface grid Volume grid CFD solution J, C

• Replace costly model with cheap model:

metamodel or surrogate model

Shape parameters Surrogate model ˜ J, ˜ C

• Approximation of cost function and constraint function(s)

◮ Response surfaces (polynomial model) ◮ Neural networks ◮ Radial basis functions ◮ Kriging/Gaussian Random Process models

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 26 / 72

Kriging I

Unknown function f : Rd → R Given the data as FN = {f1, f2, . . . , fN} ⊂ R sampled at XN = {x1, x2, . . . , xN} ⊂ Rd, infer the function value at a new point xN+1. Treat result of a computer simulation as a fictional gaussian process FN is assumed to be one sample of a multivariate Gaussian process with joint probability density p(FN) = exp

• − 1

2F ⊤ N C−1 N FN

• (2π)N det(CN)

(1) where CN is the N × N covariance matrix.

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 27 / 72

Kriging II

When adding a new point xN+1, the resulting vector of function values FN+1 is assumed to be a realization of the (N + 1)-variable Gaussian process with joint probability density p(FN+1) = exp

• − 1

2F ⊤ N+1C−1 N+1FN+1

• (2π)N+1 det(CN+1)

(2) Using Baye’s rule we can write the probability density for the unknown function value fN+1, given the data (XN, FN) as p(fN+1|FN) = p(FN+1) p(FN) = 1 Z exp

• −(fN+1 − ˆ

fN+1)2 2σ2

fN+1

• where

ˆ fN+1 = k⊤C−1

N FN

• Inference

, σ2

fN+1 = κ − k⊤C−1 N k

• Error indicator

(3)

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 28 / 72

Kriging III

Covariance matrix: Given in terms of a correlation function, CN = [Cmn], Cmn = corr(fm, fn) = c(xm, xn) c(x, y) = θ1 exp

• −1

2

d

• i=1

(xi − yi)2 ri2

• + θ2

Parameters Θ = (θ1, θ2, r1, r2, . . . , rd) determined to maximize the likelihood of known data max

Θ

log(p(FN))

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 29 / 72

Kriging: Illustration

2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 1 1.5 1.5 2 2.5 2.5 3 3.5 3.5 4 4.5

DACE predictor standard error

• f the predictor
• Praveen. C

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Exploration verses exploitation

• Need a reasonably accurate surrogate model to represent global

behaviour.

• But final goal is to find minimum of J, not to construct the most

accurate model.

• In large dimensional spaces, it is quite impossible to construct a

uniformly accurate model – Curse of dimensionality.

• Exploration refers to sampling all regions of design space.

◮ Sample in regions where σ is large

• Exploitation refers to doing greater sampling around promising

regions.

◮ Sample in regions where ˜

J is small

• First iteration: sample uniformly in design space using LHS to

create a database of design variables and objective function values

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 31 / 72

Merit functions

• Merit function based on a statistical lower bound

min

x Jκ(x) := ˜

J(x) − κσ(x) κ = = ⇒ exploitation κ = large = ⇒ exploration

• Choose a set of κ = 0, 1, 2, 3
• Minimize four merit functions

minx J0(x) = ⇒ x0 minx J1(x) = ⇒ x1 minx J2(x) = ⇒ x2 minx J3(x) = ⇒ x3

• Evaluate x0, x1, x2, x3 on exact model (CFD)
• Add J(x0), J(x1), J(x2), J(x3) to database
• Update metamodel ˆ

J

• Praveen. C

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Minimization of 2-D Branin function: Initial database

−5 5 10 5 10 15

• Praveen. C

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Minimization of 2-D Branin function: after 20 iter

−5 5 10 5 10 15

• Praveen. C

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Transonic wing optimization

• Praveen. C

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Transonic wing optimization: 8 design variables

1000 2000 3000 4000 5000 6000 7000 Number of CFD 0.5 0.6 0.7 0.8 0.9 1 Cost function PSO IPE-LB GMO-LB

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 36 / 72

Transonic wing optimization: 16 design variables

1000 2000 3000 4000 5000 6000 7000 Number of CFD 0.5 0.6 0.7 0.8 0.9 1 Cost function PSO IPE-LB GMO-LB

• Praveen. C

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Transonic wing optimization: 32 design variables

5000 10000 Number of CFD 0.4 0.5 0.6 0.7 0.8 0.9 1 Cost function PSO IPE-EI GMO-LB

• Praveen. C

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Transonic wing optimization

No of DV

• No. of CFD

Drag reduction 8, PSO 11136 0.526 8, PSO+Surrogate 181 0.523 16, PSO 9920 0.525 16, PSO+Surrogate 218 0.503 32, PSO 12736 0.483 32, PSO+Surrogate 305 0.485

• Praveen. C

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Optimization of Onera M6 wing

using FFD and twist parameters

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 40 / 72

Free Form Deformation

• Figure 2: FFD lattice around the wing: moving control points

(filled markers) and frozen control points (empty markers).

Control points moved vertically only

• Number of free control points = (ni − 1) × (nj + 1) × (nk + 1)
• We take nj = nk = 1: Linear shape change from root to tip
• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 41 / 72

Wing twist parameterization

P Symmetry plane Line of centers A C B Q Axis of rotation O D

• s = arc length of grid point P from symmetry plane

s(P) = OP/OD s(O) = 0 s(D) = 1

• Linear twist

θ(P) = (1 − s(P))θ0 + s(P)θ1

• Praveen. C

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Test Case: Inviscid, transonic Onera M6 wing

• M∞ = 0.84, α = 3.06 deg.
• Inviscid Euler model
• CFD details

◮ Structured grid FV solver ◮ Roe flux ◮ MUSCL scheme ◮ Grid size: 41 × 51 × 201 (4.2 lakh cells) ◮ Wing grid: 35 × 201

• FFD parameterization: 16 design variables

ni × nj × nk = 5 × 1 × 1

• Linear twist: 2 design variables
• FFD + Twist: 18 design variables
• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 43 / 72

Test Case: Optimization

• FFD control points restricted to [−0.05, +0.05]
• Twist variables: θ0 ∈ [−3, 0], θ1 ∈ [0, +3]

Wing root airfoil is allowed to pitch-up only Wing tip airfoil allowed to pitch down only

• Initial database generated using Latin Hypercube Sampling

Case No of DV Initial DB FFD 16 100 FFD+Twist 18 100 Twist 2 10 Database generated in parallel using 100 or 10 processors

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 44 / 72

Test Case: Objective function

• Constrained optimization problem:

min Cd Cd0 subject to Cl = Cl0 and V = V0

• Cost function using penalty approach

J = Cd Cd0 + 104 max

• 0, 1 − Cl

Cl0

• + 104 max
• 0, 1 − Vl

V0

• Unconstrained problem:

min J

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 45 / 72

Test Case: Inviscid, transonic Onera M6 wing

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 50 100 150 200 250 300 Cost function Number of CFD FFD Twist FFD+twist

Convergence of optimization iterations

• Praveen. C

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Test Case: Inviscid, transonic Onera M6 wing

(a) (b) (c) Pressure contours for (a) Onera M6 wing and (b) optimized wing using FFD, and (c) optimized wing using FFD and twist

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 47 / 72

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 20% Span Onera M6 FFD FFD+Twist

• 1
• 0.5

0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 64% Span Onera M6 FFD FFD+Twist

• 1
• 0.5

0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 80% Span Onera M6 FFD FFD+Twist

• 1
• 0.5

0.5 1 1.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 89% Span Onera M6 FFD FFD+Twist

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 48 / 72

• 0.05
• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 0.04 0.05

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 20% Span Onera M6 FFD FFD+Twist

• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 0.04 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 64% Span Onera M6 FFD FFD+Twist

• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 0.04 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 80% Span Onera M6 FFD FFD+Twist

• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 0.04 0.05 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 89% Span Onera M6 FFD FFD+Twist

(Vertical scale blown up for clarity)

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 49 / 72

Optimization of flying wing

(with Biju Uthup, ADA, Bangalore)

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 50 / 72

Application to AURA

Optimization problem

Maximize Lift/Drag subject to volume constraint

• Inviscid, compressible flow model (Euler equations)

Configuration C1A1 M∞ = 0.75, AOA = 2 deg.

• Praveen. C

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Grid for C1A1

141 × 20 × 82 On the wing: 100 × 72

• Praveen. C

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FFD Box

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 53 / 72

Optimization test

• 4 design variables
• Gaussian process metamodel
• Statistical lower bound merit function (4)
• Initial database of 100 designs using LHD

Config L 100D L/D Improve Initial 0.11529 0.47371 24.3

• Optimized

0.08523 0.28928 29.4 21%

• 13 iterations, 150 CFD solutions in total
• Intel Xeon X5482 @ 3.2 GHz
• 6 process parallel job – about 7 hours
• Praveen. C

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Initial wing

• Praveen. C

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Optimized wing

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 56 / 72

RANS computations

Studies to improve L/D for AURA : Aerodynamic Team

• Meeting

Un-optimized

• ptimized
• Praveen. C

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RANS computations

Studies to improve L/D for AURA : Aerodynamic Team

Un-optimized

• ptimized
• Praveen. C

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Transonic, turbulent airfoil

• ptimization
• Praveen. C

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TA5 test case

• Optimize RAE5243 airfoil to reduce drag under lift constraint

Mach Re Cl Flow condition 0.68 19 million 0.82 Fully turbulent

• Modify shape of upper airfoil surface by adding a bump

Xcr Xbr Xbl ∆Yh

• Praveen. C

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Reference solution: Pressure

α = 2.5 deg.

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 61 / 72

Optimization test

• 5 design variables
• Initial database of 48

using LHS

• 4 merit functions based
• n statistical lower

bound with κ = 0, 1, 2, 3

• Gaussian process

models

• Merit functions

minimized using PSO

4 8 5 2 5 6 6 6 4 6 8 7 2 7 6 8 8 4 8 8 9 2 9 6 1 1 4 1 8 1 1 2 1 1 6 1 2 1 2 4 1 2 8 1 3 2 1 3 6 1 4 1 4 4 1 4 8 1 5 2 1 5 6 1 6 1 6 4 1 6 8 5 10 15 20 25 30 Number of iterations 0.78 0.8 0.82 0.84 0.86 Objective function Annotation = Number of CFD evaluations

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 62 / 72

Shape parameters

Case Xcr Xbl Xbr ∆Yh × 10−3 Present 0.688 0.399 0.257 8.578 Qin et al. 0.597 0.313 0.206 5.900

0.2 0.4 0.6 0.8 1

• 0.05

0.05 RAE5243 Optimized

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 63 / 72

Force and Pressure coefficient

Case Cd ∆Cd Cl AOA Present 0.01266

• 22.2%

0.8204 2.19 Qin et al. 0.01326

• 18.2%

0.82

• 0.2

0.4 0.6 0.8 x/c

• 1
• 0.5

0.5 1 1.5

• Cp

RAE5243 Optimized

• Praveen. C

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Pressure contours

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 65 / 72

Optimal control parameters for cylinder flow

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 66 / 72

• Flow past 2-D cylinder at Re = 200
• Periodic vortex shedding, oscillatory forces

Ref. St Cd Bergmann et al. (2005) 0.195 1.382 Braza et al. (1986) 0.200 1.400 Henderson (1997) 0.197 1.341 Homescu et al. (2002)

• 1.440

current study 0.198 1.370

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 67 / 72

Oscillating cylinder

Oscillating cylinder: Apply oscillating velocity boundary condition to cylinder wall ω(t) = A sin(2πNt)

ω

U

Find (A, N) to minimise 1 t1 − t0 t1

t0

CD(t; A, N)dt Non-dimensional variables and bounds: A∗ = AD U∞ ∈ [0, 5], N∗ = ND U∞ ∈ [0, 1]

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 68 / 72

Optimization

Initial sample of 16 using LHS

2 4 6 8 10 12

• ptimization iterations

0.8 0.85 0.9 0.95 1

cost function

Good convergence in 3 iterations, 24 CFD solutions

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 69 / 72

Controlled case

Ref. Method A⋆ N⋆ ∆Cd Bergmann et al.(2004) POD 2.2 0.53 25% Bergmann et al.(2004) POD-ROM 4.25 0.74 30% He et al.(2002) NS 2D 3.00 0.75 30% current study NS 3D 3.20 0.80 25%

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 70 / 72

Optimization problem

Optimization

min

x∈Rd J (x, ao)

C(x, ao) ≤ 0

Robust optimization

min

x∈Rd

µJ(x) =

• Ω(A)

J (x, a) ρA(a) da σ2

J(x) =

• Ω(A)

[J (x, a) − µJ]2 ρA(a) da Prob[C(x, A) ≤ 0] ≥ p

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 71 / 72

Optimization problem

Optimization

min

x∈Rd J (x, ao)

C(x, ao) ≤ 0

Robust optimization

min

x∈Rd

µJ(x) =

• Ω(A)

J (x, a) ρA(a) da σ2

J(x) =

• Ω(A)

[J (x, a) − µJ]2 ρA(a) da Prob[C(x, A) ≤ 0] ≥ p

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 71 / 72

Summary

• Numerical optimization using CFD
• Gradient-free + Metamodels + Free-form + CFD + MPI
• Major obstacle: CAD −

→ Grid

• Other shape parameterizations + Radial basis function

deformation = free-form approach

• Develop numerical tools
• In future:

◮ 3-D RANS-based optimization ◮ Multi-point/Multi-objective optimization ◮ Optimization under uncertainties/robust optimization

• Praveen. C

(TIFR-CAM) Shape optimization IISc, 11 March 2011 72 / 72