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Aerodynamic Design Optimization Studies at CASDE

Amitay Isaacs, D Ghate, A G Marathe, Nikhil Nigam, Vijay Mali,

K Sudhakar, P M Mujumdar

Centre for Aerospace Systems Design and EngineeringDepartment of Aerospace Engineering, IIT Bombay

http://www.casde.iitb.ac.in

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About CASDE

5 years old Master’s program in Systems Design & Engineering MDO MAV Modeling & Simulation Workshops/CEPs/Conferences

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Optimization Studies –Overview

Concurrent aerodynamic shape & structural sizing of wing FEM based aeroelastic design MDO architectures WingOpt software Propulsion system Engine sizing & cycle design Intake duct design using CFD

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Intake Design - Background Duct design practice of late 80s – based on empirical rules

Problem Revisited – using formal optimization and high fidelity analysis

Study evolved with active participation of ADA (Dr. T.G. Pai & R.K.Jolly)

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Problem Formulation

Entry Exit Location and shape (Given)

Optimum geometry of duct from Entry to Exit ?

Objective/Constraints

• Pressure Recovery• Distortion• Swirl

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Design Using CFD - IssuesSimulation Time

CFD takes huge amounts of time for real life problems Design requires repetitive runs of disciplinary analyses

Integration & Automation Parametric geometry modeling Grid generation CFD solution Objective/Constraint function evaluation Optimization

Gradient Information Finite difference – step size (??), (NDV + 1) analyses

required Exact formulations – Automatic differentiation

(ADIFOR), Adjoint method, Complex step method – All require source code

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Flow SolverDistortion & Swirl calculation requires NS solutionIn-house NS Solver

Analytical gradients possible Easy to integrate

Commercial Solvers (STAR-CD, FLUENT…) Gradients using finite difference only Difficult to integrate

FLUENT Inc. S-shaped non-diffusing duct Results validated with a NASA test case (Devaki

Ravi Kumar & Sujata Bandyopadhyay)

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StrategiesReducing Time Parameterization Variable fidelity to shrink the search space Surrogate modeling Meshing Parallel computing Continuation

Integration & Automation Wrapping executables and user interfaces Offline analysis (Surrogate models) – semi-

automatic

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Our Strategy

Variable fidelity Response Surface based design using FLUENT

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Our Methodology

Parametrization

Low fidelity Analysis

DOE in reduced space

CFD analysis at DOE points

RS for PR & DC60

OptimizationConstraints

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Parametrization

Y

X

Z

XDuct Centerline

A

X

Control / Design Variables

• Ym, Zm• AL/3, A2L/3

Cross Sectional Area

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Y

X

Z

XDuct Centerline

A

X

Control / Design Variables

• Ym, Zm• AL/3, A2L/3

Cross Sectional Area

Parametrization

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Typical 3D-Ducts

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Duct Design - Low FidelityLow Fidelity Design Rules (Constraints) Wall angle < 6° Diffusion angle < 3° 6 * Equivalent Radius

< ROC of Centerline

Objective function: pressure recovery

No low fidelity analysis for distortion or swirl

X1-MIN

X2-MIN

X2-MAX

X1-MAX

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Optimization Process – Low Fidelity

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Automation for CFD

Generation of entry and exit sections using GAMBIT

Clustering Parameters

Conversion of file format to CGNS using FLUENT

Mesh file

Generation of structured volume grid using parametrization

Duct Parameters(β1, β2, αy, αz)

Entry & Exit sections

Conversion of structured grid to unstructured format

Unstructured CGNS file

CFD Solution using FLUENTEnd-to-end (Parameters to DC60) automated CFD Cycle. Objective/Constraints evaluation

Using UDFs (FLUENT)DC60

CFD Solution

ContinuationSolution

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Automation for Design

Generation of structured volume grid using parametrizationEntry & Exit

sections

Conversion of structured grid to unstructured format

CFD Solution using FLUENT

Objective/Constraints evaluationUsing UDFs (FLUENT)

DC60

Optimization

Duct Parameters(β1, β2, αy, αz)

ContinuationSolution

Unstructured CGNS file

CFD Solution

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Results: Total Pressure Profile

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Design Space Reduction

6.19

1.42

(0.61, 0.31, 1.0, 1.0)

Optimized duct from low fidelity

24.2116.28DC60

3.532.0PLOSS

(-0.4, 1.5, 0.3, 0.6)

(0.1, 0.31, 0.2, 0.6)

P

Poor ductInfeasible duct

P – Parameters; PLOSS – Total Pressure Loss

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Optimization Post-processingDistortion Analysis

DC60 = (PA0 – P60min) /qwhere, PA0 - average total pressure at the section, P60min- minimum total pressure in a 600 sector, q - dynamic pressure at the cross section.User Defined Functions (UDF) and scheme files were used to generate this information from the FLUENT case and data file.Iterations may be stopped when the distortion values stabilize at the exit section with reasonable convergence levels.

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Huge benefits as compared to the efforts involved!!!

Methodology Store the solution in

case & data files Open the new case (new grid)

with the old data file Setup the problem Solution of (0.61 0.31 1 1) duct slapped on (0.1 0.31 0.1 0.1)

3-decade-fall 6-decade-fall

Without continuation 4996 9462With continuation 1493 6588Percentage time saving 70% 30%

Continuation Method

Generate new case file

FLUENT Solution

Duct Parameters

OldData file

Journalfile

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Simulation Time Strategies Continuation Method Parallel execution of FLUENT on a 4-

noded Linux cluster

Time for simulation has been reduced to around 20%.

0 20 40 60 80 100Time (hrs)

Time per CFD Run

Serial

Parallel

Slapping

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Sequential (Multipoint)Response Surface Approximations

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Sequential (multipoint) Response Surface Methodology

Response Surfaces generated in sub-domains around multiple pointsSurfaces used to march to optimum

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Wing aerodynamic design problem

Planform fixed2 spanwise stations4 variables for camber3 variables for geometric pre-twistMaximize cruise L/D Lift constraint

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Design Problem Statement

Maximize L/D Sub. to CL = .312

-5 r + m 5 -5 r + m + t 5

with side constraints, .05 x1 .33; .001 h1 .1

.05 x2 .33; .001 h2 .1

-2 r 5 -2 m 5 -2 t 5

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Design Tools

Lift Calculation: CL from VLMDrag Calculation: CD0 from a/c data

CDi from VLMDOE: Design Expert D-optimality CriterionResponse Surfaces: Design Expert quadratic/cubicOptimizers : FFSQP

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Overall Design Procedure

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Results - Arbitrary Starting Point 1

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Results - Arbitrary Starting Point 2

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Observations

Quadratic model found better than cubic model in subspaces. Global model inadequate.Cost of D-optimality significantSRSA seems to work well!

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GRADIENT INFORMATION BY

AUTOMATIC DIFFERENTIATION OF

CFD CODES

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User Supplied Analytical Gradients

AnalysisCode in Fortran

Manually extractsequence of mathematical

operations

Code the complex derivative evaluator

in Fortran

Manually differentiatemathematical

functions - chain rule

FORTRANsource code

that can evaluategradients

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Automatic Differentiation for Analytical Gradients

Automatically parse and extract the sequence

of mathematical operations

Use symbolic math packages to automate derivative evaluation

Automatically code the complex

derivative evaluator in Fortran

AnalysisCode in FORTARN

FORTRANsource code

that can evaluategradients

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Automatic Differentiation for Analytical Gradients

Complex AnalysisCode in FORTARN

FORTRANsource code

that can evaluategradients

Automated Differentiation

Packageeg. ADIFOR

&ADIC

Euler

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1.12 3.06 4.11

d(L/D) / d using ADIFOR  5.48 -0.38 -1.20

d(L/D) / d using Finite Difference

=0.2Value 5.09 -0.52 -1.23

% Error 7.17 38.10 2.46

=0.02Value 5.44 -0.40 -1.18

% Error 0.70 4.44 1.73

=0.002Value 5.45 -0.41 -1.18

% Error 0.61 7.08 1.56

=0.0002Value 5.56 -0.67 -1.02

% Error 1.54 77.25 15.09

Comparison of Derivative Calculation Finite Difference vs ADIFOR

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Optimization - ADIFOR vs FD

Single design variable unconstrained optimization problem Find for max. L/D for Onera M6 wing

Same starting point; FD step size 0.002

init opt L/Dopt Calls Time(min.)

ADIFOR

1.060 2.810 11.99 15 424

FD 1.060 2.810 11.99 17 111

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Thank You

Please visitwww.casde.iitb.ac.in

for details and other information

Thank Youhttp://www.casde.iitb.ac.in/mdo/3d-duct/

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Problem Statement

•Ambient conditions: 11Km altitude• Inlet Boundary Conditions

• Total Pressure: 34500 Pa• Total Temperature: 261.4o K• Hydraulic Diameter: 0.394m• Turbulence Intensity: 5%

• Outlet Boundary Conditions• Static Pressure: 31051 Pa (Calculated for the desired mass flow rate)• Hydraulic Diameter: 0.4702m• Turbulence Intensity: 5%

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Duct Parameterization

Geometry of the duct is derived from the Mean Flow Line (MFL) MFL is the line joining centroids of

cross-sections along the duct Any cross-section along length of the

duct is normal to MFL Cross-section area is varied parametrically Cross-section shape in merging area is same as the exit section

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MFL Design Variables - 1Mean flow line (MFL) is considered as a piecewise cubic curve along the length of the duct between the entry section and merging section

x

y(x), z(x)

0 LmLm/2

y(Lm/2), z(Lm/2) specified

Centry

Cmerger

y1, z1

y2, z2

Lm : x-distance between the entry and merger section

y1, y2, z1, z2 : cubic polynomials for y(x) and z(x)

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MFL Design Variables - 2• y1(x) = A0 + A1x + A2x2 + A3x3, y2(x) = B0 + B1x + B2x2 + B3x3

• z1(x) = C0 + C1x + C2x2 + C3x3, z2(x) = D0 + D1x + D2x2 + D3x3

• y1(Lm) = y2 (Lm), y1’ (Lm) = y2’ (Lm), y1” (Lm) = y2” (Lm)

• z1(Lm) = z2 (Lm), z1’ (Lm) = z2’ (Lm), z1” (Lm) = z2” (Lm)

• y1’ (Centry) = y2’ (Cmerger) = z1’ (Centry) = z2’ (Cmerger) = 0

• The shape of the MFL is controlled by 2 parameters which control the y and z coordinate of centroid at Lm/2

• y(Lm/2) = y(0) + (y(L) – y(0)) αy 0 < αy < 1

• z(Lm/2) = z(0) + (z(L) – z(0)) αz 0 < αz < 1

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Area Design Variables – 1Cross-section area at any station is interpolated from the entry and exit cross-sections

•A(x) = A(0) + (A(Lm) – A(0)) * β(x)• corresponding points on entry and exit sections are linearly interpolated to obtain the shape of the intermediate sections and scaled appropriately• Psection = Pentry + (Pexit - Pentry) * β

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Area Design Variables - 2

A0 + A1x + A2x2 + A3x3 0 β < β1

B0 + B1x + B2x2 + B3x3 β1 β β2

C0 + C1x + C2x2 + C3x3 β2< β 1β =

x

β(x)

0 LmLm/30

1

2Lm/3

β1

β2

β(Lm/3) and β(2Lm/3) is specified

β variation is given by piecewise cubic curve as function of x

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Turbulence ModelingRelevance: Time per SolutionFollowing aspects of the flow were of interest:

Boundary layer development Flow Separation (if any) Turbulence Development

Literature Survey S-shaped duct Circular cross-section Doyle Knight, Smith, Harloff, Loeffer

Baldwin-Lomax model (Algebraic model) Computationally inexpensive than more sophisticated models Known to give non-accurate results for boundary layer separation etc.

Devaki Ravi Kumar & Sujata Bandyopadhyay (FLUENT Inc.) k- realizable turbulence model

Two equation model

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Turbulence Modeling (contd.)

Standard k- model Turbulence Viscosity Ratio

exceeding 1,00,000 in 2/3 cells

Realizable k- model Shih et. al. (1994) Cμ is not assumed to be

constant A formulation suggested

for calculating values of C1 & Cμ

Computationally little more expensive than the standard k- model

Total Pressure profile at the exit section (Standard k- model)

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ResultsMass imbalance: 0.17%Energy imbalance: 0.06%Total pressure drop: 1.42%Various turbulence related quantities of interest at entry and exit sections: Entry Exit

Turbulent Kinetic Energy (m2/s2) 124.24 45.65

Turbulent Viscosity Ratio 5201.54 3288.45

y+ at the cell center of the cells adjacent to boundary throughout the domain is around 18.

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Flow Separation

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Flow Separation