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Advanced evolutionary algorithms for transonic drag reduction and high lift of 3D configuration

using unstructured FEM

21 May 2007

www.cimne.comAdvanced modelling techniques for aerospace SMEs

Gabriel Bugeda

TANK Zhili

Jordi Pons

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Index

• Mesh generation and quality aspects

• Robust design

Contributions of CIMNE to shape optimization problems in aeronautics:

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Mesh generation and quality aspects

Shape optimization problem:

f objective function

x vector of design variables

g set of restrictions

Deterministic methods

Evolutionary algorithms

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1. Total computational cost of optimizationclosely related to FE analysis cost per design.

2. Bad quality of FE analysis:

Introduce noise in the convergence

Possible bad final solution.

Evolutionary as well as deterministic methods involves the analysis (FEM) of many different designs.

Influence of mesh generation:

Mesh GenerationMesh Generation

Mesh generation and quality aspects

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Classical strategies for meshing each individual:

1. Adapt a single existing mesh to all the different geometries.

Existing strategies allow adapting an existing mesh for very big geometry modifications preventing too much distortion.

Cheapest strategy

No control of the discretization error.

2. Classical adaptive remeshing for the analysis of each design.

Good quality of results of each design

High computational cost (each design is computed more than once)

Mesh generation and quality aspects

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Adaption of a mesh to the boundary shape modifications

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Representativeof population.Representativeof population.

Generation of an adapted mesh for each design in one step using error sensitivity analysys

Mesh adaptivity based on Shape sensitivity analysis

Mesh adaptivity based on Shape sensitivity analysis

Projection parameters (sensitivity of nodal coordinates

and error indicator)

Projection parameters (sensitivity of nodal coordinates

and error indicator)

Final h-adapted mesh of representative

Final h-adapted mesh of representative

h-adaptive analysis of

representative

Classical sensitivity

analysis

Projection to individuals

h-adapted mesh for 1st individual

h-adapted mesh for 1st individual

h-adapted mesh for 2nd individual

h-adapted mesh for 2nd individual

h-adapted mesh for 3rd individual

h-adapted mesh for 3rd individual

h-adapted mesh for Pth individual

h-adapted mesh for Pth individual

in “one-step” !!

Low cost control of discretization errorLow cost control of discretization error

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Geometry:B-spline. Definition points r(i)

Geometry:B-spline. Definition points r(i)

Parameterization of the problem

Sensitivity analysis of the system of equations:

Sensitivity analysis of the B-spline expression:

Design variables:Coordinates of some definition points Design variables:Coordinates of some definition points

B-spline expression:in terms of the coordinates of “polygon definition points” ri.

B-spline expression:in terms of the coordinates of “polygon definition points” ri.

Polygon definition points vector, R:Obtained solving V=NR(V imposed conditions at r(i))

Polygon definition points vector, R:Obtained solving V=NR(V imposed conditions at r(i))

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Mesh generation and mesh sensitivity

Mesh Generator

Advancing front method

Background mesh defining the size δ at each point.

Mesh sensitivitySmoothing of nodal coordinates

Mesh Sensitivity

Boundary nodal points: obtained by the B-spline sensitivity analysis.

Internal nodal points: spring analogy (fixed number of smoothing cycles)

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The used evolutionary algorithm

Parameter vector of i-th individual

of generation t

For each individual, a new trial vector is created by setting some of the parameters up

j(t) to:

Parameters to be modified and individuals q, r, s are randomly selected

The new vector up(t) replaces xp(t) if it yields a higher fitness.

Non accomplished restrictions integrated in objective function using a penalty approach.

Evolutionary algorithm: classical Differential Evolution (Storn & Price).Evolutionary algorithm: classical Differential Evolution (Storn & Price).

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Projection to each design and definition of the adapted mesh

Representative of populationRepresentative of population pth individual of populationpth individual of population

Projection using shape sensitivity

analysis

Projection using shape sensitivity

analysis

Mesh coordinates

Error estimation

Strain energy

Generation of h-adapted mesh.

Admissible global error percentage

Mesh optimality criterion: equidistribution of error density

Target error for each element

New element size

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Pipe under internal pressure1

4

3

2

x

y P

4 design variables

Circular internal shape

P=0.9 MPa

vm 2 MPa

||ees|| < 1.0%

30 individuals/generation

Design variable

Initial Value

Data Range

Constraints

V1 20 [ 5.2 − 50.0 ]

V2 19 [ 4.0 − 50.0 ]

V3 19 [ 4.0 − 50.0 ] V3 < V1 − 0.5

V4 20 [ 5.2 − 50.0 ] V4 < V2 + 0.5

Optimal analytical solution for external surface:

• Circular shape Ropt = 10.66666

• Cross section area Aopt = 69.725903

Optimal analytical solution for external surface:

• Circular shape Ropt = 10.66666

• Cross section area Aopt = 69.725903

Minimize unfeasible designs

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Analytical Optimal shape

A = 69.725903

Optimal shape obtained

(B-spline defined by 3 points)

A = 70.049

0

Pipe under internal pressure

01 -234567891 0111 21 31 41 51 6 -1 71 81 92 0 -2 12 2 -2 32 4 -2 52 6 -2 72 8 -3 03 1 -3 53 6 -4 24 3 -5 55 65 7 -6 56 66 7 -8 18 28 3 -9 19 2 -9 59 6 -9 89 9 -1 0 31 0 4 -1 2 41 2 5 -1 2 71 2 8 -1 8 5185 generations

30 individuals/generation

only 3% individuals required additional remeshing

185 generations

30 individuals/generation

only 3% individuals required additional remeshing

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Pipe under internal pressure

0

100

200

300

400

0 50 100 150Generation

Are

a

Minimun = 69.725903

0.1

1

10

100

1000

0 50 100 150Generation

Err

or %

0.46%

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Fly-wheel

FE model of Initial design space Optimum topologyInitial design space

Initial model for further optimization (60 design variables)

8 independent design variables

60 design variables

8 independent design variables

vm 100 MPa

||ees|| < 5.0%

15 individuals/generation

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Fly-wheel

O rig . In d iv. 1-34 -67 -2 42 5-4 64 7-8 88 9 -9 29 39 4-12 71 28 -16 71 68 -17 71 78 -1 8 81 89 -20 12 02 -2 0 62 07 -2 3 42 35 -2 4 52 46 -25 92 60 -29 72 98 -3 0 0 Original

Design

OptimumDesign

1.441.451.461.471.481.491.501.511.521.53

0 50 100 150 200 250 300Generation

Wei

ght i

n kg

300 generations

15 individuals/generation

Weight reduction 1.53 1.445 kg

(0.25 0.17 in the design area)

(Deterministic: 1.53 1.45 kg)

300 generations

15 individuals/generation

Weight reduction 1.53 1.445 kg

(0.25 0.17 in the design area)

(Deterministic: 1.53 1.45 kg)

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Conclusions

A strategy for integrating h-adaptive remeshing into evolutionary optimization processes has been developed and tested

Adapted meshes for each design are obtained by projection from a reference individual using shape sensitivity analysis

Quality control of the analysis of each design is ensured

Full adaptive remeshing over each design is avoided

Low computational cost (only one analysis per design)

Numerical tests show

• The strategy does not affect the convergence of the optimization process

• Good evaluation of the objective function and the constraints for each different design is ensured

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Goal: Introducing VARIABILITY (uncertainty) of Goal: Introducing VARIABILITY (uncertainty) of parameters like Mach numbers or angle of parameters like Mach numbers or angle of attack in design optimizationattack in design optimization

Outcomes: better control of realistic product performancesOutcomes: better control of realistic product performances

Robust design

1. Performs consistently as intended (design)2. Throughout its life cycle (manufacturing)3. Under a wide range of user conditions (design)4. Under a wide range of outside influences (design)

A product is said to be Robust …

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• Taguchi methodsTaguchi methods

• Stochastic optimizationStochastic optimization

• Multi-point optimizationMulti-point optimization

• Fuzzy and probabilistic methods Fuzzy and probabilistic methods

• Bounds-based methodsBounds-based methods

• Minimax methodsMinimax methods

Two popular methodologies

Different robust design methods

Robust design

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STOCHASTIC OPTIMIZATION

• Modify the objective to directly incorporate the effects of model Modify the objective to directly incorporate the effects of model uncertainties on the design performanceuncertainties on the design performance

•Stochastic analysis of the behaviour of each designStochastic analysis of the behaviour of each design

Robust design

• Minimize the expected value of the drag over the design lifetime:Minimize the expected value of the drag over the design lifetime:

M MdDd

dMDd

dMMfMdCMdCE ,min,min

d

M

Mf

Is drag functionIs drag functionIs design vector (geometry, angle of Is design vector (geometry, angle of attack)attack)

Is uncertain parameter (Mach number)Is uncertain parameter (Mach number)

Is probability density function of Mach Is probability density function of Mach numbernumber

dC

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• Methodology:– Define probabilistic distribution of values for both geometry and

environmental parameters. All in the same analysis.

• Input variables:– Angle of attack, Mach number and Reynolds number– Knot coordinates; two points on upper profile and two points on

lower profile

• Conclusions: – Graphical representation of the [-3σ, +3σ] range and mean value– Mixed effect between geometry and environment do not define

any clear relationship.

Stochastic Optimisation

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On the X-axis the number indicates each analysis:1.- Evolution of the geometry under optimisation process.

Stochastic Optimisation

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• A disciplined engineering approach (Parameter Design) to find the best combination of design parameters (control factors) for making a system insensitive to outside influences (noise factors)

• 2 steps in the optimization procedure:

1. Reduce effect of variability on design function

2. Improve the performances

Taguchi method

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Mathematical formulation of Taguchi methods for drag reduction problem

1. Definition of design problem

2. Description of robust design problem (2 objectives)

**

1

2222

12

2

1

)1(1min

],[1min

hicknesshicknessll

K

idbidiC

K

ibb

b

idid

TTCCtoSubject

CMMCKf

MMMM

MCKCf

d

i

**

],[min

hicknesshicknessll

bbd

TTCCtoSubject

MMMatC

Taguchi method

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EXAMPLE: Robust design optimization problem

1. Find optimal airfoil geometry, which results in minimum drag Cd over a range of free flow Mach numbers while maintaining a given target lift.

2. The thickness and its position is maintained during the optimization. The NACA-2412 is the baseline profile.

3. For this example we assume that the Mach number

The Mach number can not fall outside of this interval.

4. We use a inviscid EULER solver to analyze the flow field.

]76.0,74.0[M

Taguchi method

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Pareto non dominated solutions, Nash equilibrium and Single point designed solutions

Taguchi method

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Airfoil list of non-dominated solution, single-point Airfoil list of non-dominated solution, single-point design solution and baseline profiledesign solution and baseline profile

Taguchi method

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Comparison of airfoils on Pareto front with Nash Comparison of airfoils on Pareto front with Nash equilibriumequilibrium

Taguchi method

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Drag performance of optimized airfoil

Taguchi method

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CONCLUSIONS

1. Robust design optimization is significantly more realistic for designers than the single point design optimization. This Taguchi based uncertainty methodology can identify new shapes with better performance and stability simultaneously maintained within a given range of operation.

2. Compromised solutions are captured by Pareto or Nash strategies. It is shown that a Nash equilibrium solution is also a good initial guess for capturing efficiently a Pareto non-dominated solution.

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Thank you very much