TA5 Test Case Praveen. C 1 R. Duvigneau 2 1 Tata Institute of - - PowerPoint PPT Presentation

TA5 Test Case

• Praveen. C1
• R. Duvigneau2

1Tata Institute of Fundamental Research

Center for Applicable Mathematics Bangalore 560065

2Projet Opale, INRIA Sophia Antipolis

Integrated Multiphysics Simulation & Design Optimization Database Workshop for multiphysics optimization software validation Presentation of the Academic Test Case Results Agora, Jyv¨ asky¨ a, Finland December 3-4, 2009

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 1 / 25

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

Airfoil chord is taken to be unity

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 2 / 25

Bound constraints for bump parameters

Xcr Xbr Xbl ∆Yh

0 < Xcr < 1 0 < Xbr < Xbl 0 < Xbl < 0.4 0 < ∆Yh < 0.05

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 3 / 25

Modification of bound constraints

• If Xbr and/or Xbl − Xbr is too small,

then CFD grid will not be able to resolve the bump. Xbr > Lmin, Xbl − Xbr > Lmin

• Also, we restrict Xcr and ∆Yh

0.4 < Xcr < 0.8 Lmin < Xbr < Xbl 2Lmin < Xbl < 0.4 0 < ∆Yh < 0.01 In the computations, we use Lmin = 0.05

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 4 / 25

NUWTUN flow solver

Based on the ISAAC code of Joseph Morrison http://isaac-cfd.sourceforge.net

• Finite volume scheme
• Structured, multi-block grids
• Roe flux
• MUSCL reconstruction
• Implicit scheme, grid sequencing, multigrid
• Wilcox k − ω turbulence model

Source code of NUWTUN available online http://nuwtun.berlios.de

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 5 / 25

Grid for CFD

C-grid of size 353 × 97, 270 points on airfoil, y+ < 1.5, outer boundary at 20 chords

• 0.4
• 0.2

0.2 0.4

• 0.5

0.5 1 1.5

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 6 / 25

Validation with experiments

M = 0.68, Re = 19 million, α = 0.77 deg.

0.2 0.4 0.6 0.8 1 x/c

• 1.5
• 1
• 0.5

0.5 1

• Cp

NUWTUN Experiment

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 7 / 25

Global metamodel-based optimization

• Global models: provide global trends in objective

function

◮ Faster convergence towards global optimum

• Metamodels are approximate, inaccurate
• Not possible to construct accurate metamodel in
• ne-shot
• Difficult to construct uniformly accurate model in

high dimensions

◮ Curse of dimensionality

• Model must be accurate in regions of optima
• But need to sufficiently explore the design space
• Balance between exploration and exploitation

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 8 / 25

Gaussian process models

• Treat results of a computer code as a stochastic process !!!
• Provides an estimate of the variance in predicted value

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

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 9 / 25

Merit functions

• Statistical lower bound

fM(x) = ˜ J(x) − κ˜ s(x)

• Probability of improvement

PoI(x) = Φ

• T − ˜

J(x) ˜ s(x)

• Expected improvement

EI(x) = ˜ s(x)[uΦ(u) + φ(u)], u(x) = Jmin − ˜ J(x) ˜ s(x)

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 10 / 25

Minimization of 2-D Branin function: Initial database

−5 5 10 5 10 15

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 11 / 25

Minimization of 2-D Branin function: after 20 iter

−5 5 10 5 10 15

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 12 / 25

Grid deformation

• Interpolate displacement of

surface points to interior points using RBF ˜ f(x, y) = a0 + a1x + a2y +

N

• j=1

bj| r − rj|2 log | r − rj| where

• r = (x, y)
• Results in smooth grids

Initial grid Deformed grid

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 13 / 25

Reference solution and design variables

Result α deg. Cl Cd Cdp Cdv Present 2.5 0.8244 0.01627 0.01052 0.005757 Qin et al.

• 0.82

0.01622 0.01063 0.005586 α range for optimization: 2 < α < 3 0.4 < Xcr < 0.8 Lmin < Xbr < Xbl 2Lmin < Xbl < 0.4 0 < ∆Yh < 0.01 2 < α < 3

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 14 / 25

Reference solution: Pressure

α = 2.5 deg.

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 15 / 25

Transformed design variables

• Transformations

x1 = Xcr x2 = Xbr − Lmin Xbl − 2Lmin x3 = Xbl − 2Lmin x4 = ∆Yh x5 = α − 2.5 Xcr = x1 Xbr = x2x3 + Lmin Xbl = x3 + 2Lmin ∆Yh = x4 α = x5 + 2.5

• Bounds

0.4 < x1 < 0.8 0 < x2 < 1.0 0 < x3 < 0.4 − 2Lmin 0 < x4 < 0.01 −0.5 < x5 < 0.5

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 16 / 25

Bump function

Xcr Xbr Xbl ∆Yh

y(0)

l

, y(0)

u : lower and upper

curves of RAE5243 airfoil yu(x) = y(0)

u (x) + yb(x)

yl(x) = y(0)

l

(x) Bump function yb(x) =      x ≤ Xcr − r1 or x ≥ Xcr + r2 ∆Yh · Cubic(x) Xcr − r1 < x ≤ Xcr ∆Yh · Cubic(x) Xcr < x < Xcr + r2 r1 = Xbr, r2 = Xbl − Xbr

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 17 / 25

Optimization test

Constrained problem

min Cd subject to Cl = Cl0 We replace equality constraint with inequality constraint Cl ≥ Cl0 Constraint is enforced using penalty approach

Unconstrained problem

min Cd Cd0 + 104 max

• 0, 1 − Cl

Cl0

• At convergence, we recover Cl ≈ Cl0

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 18 / 25

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

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 19 / 25

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 P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 20 / 25

Force and Pressure coefficient

Case Cd ∆Cd Cdp Cdv Cl AOA Present 0.01266

• 22.2%

0.00680 0.00586 0.8204 2.19 Qin et al. 0.01326

• 18.2%

0.00756 0.00570 0.82

• 0.2

0.4 0.6 0.8 x/c

• 1
• 0.5

0.5 1 1.5

• Cp

RAE5243 Optimized P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 21 / 25

Pressure contours

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 22 / 25

Lift curve and drag polar

1 1.5 2 2.5 3 Angle of attack 0.6 0.65 0.7 0.75 0.8 0.85 Lift coefficient RAE5243 Optimized 0.6 0.65 0.7 0.75 0.8 0.85 Lift coefficient 0.01 0.012 0.014 0.016 0.018 0.02 Drag coefficient RAE5243 Optimized

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 23 / 25

Close-up view of grids

Initial

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 0.2 0.4 0.6 0.8 1

Optimized

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 0.2 0.4 0.6 0.8 1

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 24 / 25

Close-up view of grids

Initial

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.4 0.5 0.6 0.7 0.8 0.9

Optimized

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.4 0.5 0.6 0.7 0.8 0.9

P & R (TIFR/INRIA) TA5 Test Case 3-4 Dec, 2009 25 / 25