Experience in mathematical optimization

IHS-Präsentation, 2008Ruprecht

University of StuttgartInstitute of Fluid Mechanics and Hydraulic Machinery, GermanyIHS

Experience in mathematical optimization

Automatic shape optimisation

parameterized geometry

Wells-Tool



Optimisation Methods

• Directe optimisation• “Response Surface” method

– Estimation of an continous approximate function by• Neuronal net• Polynomial approach• Spline

– Search for the optimum of the approximate function



Parameter

Qu

alit

äts

fun

ktio

nberechnete Werte

Optimierung an der Response Surface

Response Surface Methode



1

2

3

assumed optimum

search direction

cost

fu

nct

ion

relaxation

• Gradient type algorithmus, with search direction• Opjective funktion is locally approximated and the minimum is

calculated along the search direction

EXTREME



Self Adaptive Evolution (SAE)

• Start with a randomly chosen population

• New population is obtained by

– Mutation

– Crossover

– Survival of the fittest– Live time of each individual is exactly 1 generation

(Comma Strategie)

Evolution methode



Parallel Optimisation

simultaneous simulation on different resources

each simulation is run in parallel

Research: Asynchronous, parallel optimisation



Parallel OptimisationGrid Compting



Applied Algorithm

randomly choseninitial parameter sets

CFD

CFD

CFD

CFD

CFD

survival of the fittest

new sets by discrete operation, e. g. mirror

new sets randomlywith weighting

CFD

CFD

CFD

CFD

gri

d p

ort

al



Example Guide vane shape



Guide vane geometry

Inlet angle, Outlet angle,

chamber line angle, Weighting factor inlet,

Weighting factor outlet,Overlapping,

Profile a, Profile b,

Trailing edge thickness

Geometry Parameterisation



Automatic block structured mesh

Automatic Grid Generation



Simulation

Results:

Flow patterns (e. g pressure distribution)

Overall quantities (e. g. efficiency, losses)

Restrictions (e. g. cavitation)

Typical computational time for one geometry: 1-4 hon a Cluster of HPC



9 free parameters:

- 45 different designs (individuals) per generation

- 8 generations- in total 360 calculations

Guide vane shape

optimized with evolution strategy



Convergence



Optimised Geometrie



Test example: Draft tube cone

Assumption:Cone length

Optimisation: Outlet diameter

L

Din

Dout



0.4

0.5

0.6

0.7

0.8

0.9

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Dra

ft tu

be e

ffici

ency

D_out/D_in


randomly chosen starting points

Cone length: 6 D_in



0.4

0.5

0.6

0.7

0.8

0.9

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Dra

ft tu

be e

ffici

ency

D_out/D_in


survivors of the first generation



0.4

0.5

0.6

0.7

0.8

0.9

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Dra

ft tu

be e

ffici

ency

D_out/D_in


survivors of the second generation



0.4

0.5

0.6

0.7

0.8

0.9

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Dra

ft tu

be e

ffici

ency

D_out/D_in


survivors of the third generation



0.4

0.5

0.6

0.7

0.8

0.9

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Dra

ft tu

be e

ffici

ency

D_out/D_in


computed pointssurvivors of the seventh generation



The draft tube contour can only be changed slightly.

Optimization of the area distribution

Draft tube area distribution

Application: Refurbishment of an existing power plant



Area

Draft tube length length

area distribution

• Area distribution represented by B-Spline curves • Inlet and outlet kept constant• other cross sections scaled up





Investigated area distribution during the optimisation

Design point




Design point

Obtained area distribution

original draft tube

maximum efficiency

minimum efficiency

draft tube efficiency increase: 8%overall efficiency increase: 0.4%



minimum efficiency

Overload

design point

part load

original draft tube


Experience in mathematical optimization

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