enhanced quadrature approach applied to a robust ... · robust compressor and turbine blade design...

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Enhanced Quadrature Approach applied to a

Robust Compressor and Turbine Blade Design

Thomas Backhaus*, Jeroen Stolwijk**, Peter Flassig***

(* TU Dresden, ** TU Berlin, ***Rolls- Royce Deutschland GmbH & Co KG)

6. Dresdner Probabilistic-Workshop, 11th October 2013

6. Dresdner Probabilistic-Workshop 2013, 10th/11th October 2013

Content

1) Motivation

2) Introduction to the URQ approach3) Performance of the URQ method4) Validation of the error5) Example: Aerodynamic blade-to-blade robust

turbine blade optimization6) Summary of the URQ method

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6. Dresdner Probabilistic-Workshop, 11th October 2013

Motivation

� Problem:− Deterministic optimization does

not feature manufacturing noise and degradation

� Solution:− probabilistic simulation describing

variations of input parameters and corresponding system responses

� Opportunity :− Monte Carlo Simulation (DS, LHC, oLHC ) − alternativ approach: Univariate Reduced Quadrature (URQ) [1]

3

compressor turbine

6. Dresdner Probabilistic-Workshop, 11th October 2013

[1] M. Padulo, M.S. Campobasso, and M.D. Guenov. Novel Uncertainty Propagation Method for Robust Aerodynamic Design. AIAA Journal, 49(3):530-543, 2011.

� very expensive

Introduction to the URQ approach 4

� Subject: Finding an alternative approach, in comparison to expensive Monte Carlo Simulations to save calculation time for estimating �� � E � � and ��� � Var � �

� Basic Idea: Choose a deterministic approach that‘s based on a Gaussian-Quadrature with 3 nodes per dimension �

�� � ���� � �� � ������ � �� � ����

�� �

�0 �� ����

Build the density function �� � of the uncertain input parameter �with 3 deterministic chosen nodes ��, ��and ��for each dimension

6. Dresdner Probabilistic-Workshop, 11th October 2013

�� � ����� � �� � ��� ������ � �� � ��� ���

���

quadrature nodes, uncorrelated casequadrature nodes, correlated case

��

Introduction to the URQ approach 5

6. Dresdner Probabilistic-Workshop, 11th October 2013

� Idea of URQ: • build 2� � 1 nodes � with corresponding weights #

• propagate each node through the response function � � of interest• build mean of system response by

Calculation cost of 2� � 1 calculations

�� � ����� � �� � ��� ��� ����� � �� � ��� ��� ��

QuestionHow to find optimal weights # and scaling parameters $ ??

�� … standard deviation of ���,� … scalingparameter

��&'()* � +�&,*� �� �- +��&,*� ��� � +��

&,*� ���./

�0�

Introduction to the URQ approach 6

6. Dresdner Probabilistic-Workshop, 11th October 2013

� Solution: comparison of coefficients between Gaussian-Quadrature and Taylor-series expansion

� �� � � ��� ��� � � �� �-1��

1������

�

2!4

�0�

� ��� � � �� �-1��1���

����

2!4

�0�

��&'()* � +�&,*� �� �- � �� +��&,* � +��

&,* �-15�1��5

+��&,* ���

5 � +��&,* ���

5

6!4

50�

./

�0�

Gaussian-Quadrature

Taylor-series expansion till 4th order

��&'()* � +�&,*� ��

�- +��&,*� ��� � +��

&,*� ���./

�0�

comparison of coefficients

Introduction to the URQ approach 7

6. Dresdner Probabilistic-Workshop, 11th October 2013

� System of eight equations, its solution leads to:� Optimal values of scaling parameter $

� Optimal values of weights w

��� �7��2 � 8�� 9

37��4

��� �7��2 9 8�� 9

37��4

+�&,* � 1 �- 1������

./

�0�

+��&,* � 1

��� ��� 9 ���+��&,* � 9 1

��� ��� 9 ���

Mean

+��&<* � 2

������ ��� 9 ����

+��&<* � ���

� 9 ������ 9 1���

� ��� 9 ����

+��< � ���

� 9 ������ 9 1���

� ��� 9 ����

Variance

7� … skewness of �8� … kurtosis of �

Introduction to the URQ approach 8

6. Dresdner Probabilistic-Workshop, 11th October 2013

� Correlation of input parameters• spectral decomposition:

• =��… input covariance matrix• > … right eigenvector of =��• ? … diagonal matrix of eigenvalues of =��

• URQ nodes with correlation:

=�� � >?>@A

quadrature nodes, uncorrelated casequadrature nodes, correlated case

�� � ����� � �� � ��������� � �� � ������

�� � ����� � �� � ��� ��� ����� � �� � ��� ��� ��

uncorrelated

correlated

� � > ?

Introduction to the URQ approach 9

This approach is called the

Univariate (no dependencies between parameters - only one dimension is altered to obtain each node → Spectral decomposition)

Reduced (as few nodes as possible, Taylor is truncate d)

Quadrature (approximation of an integral via a sum )

Method

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Performance of the URQ 10

� Compare URQ and oLHC by using CFD example� Performance:

• DoE with 19 different compressor blade profiles• each profile is defined by 14 deterministic

design parameters

• uncertainty defined with 8 parameter � � � 8

where C � C D• determine following system responses:

- mean of pressure loss- variance of pressure loss- mean of exit flow angle- variance of exit flow angle

Question: „How much additional cost is needed to ge nerate equivalent estimates?”

C � 7EFGHI�JKLM NGHI�OKLM FPQ FJQ ∈ ST

U � JKLMV

W�� �U 9 UG�.

UGHI 9 UG�.W�X �

YPQ 9 YPQG�.YPQGHI 9 YPQG�.

W�Z �YJQ 9 YJQG�.

YJQGHI 9 YJQG�.

D � W�…W�Z J ∈ S�Z

6. Dresdner Probabilistic-Workshop, 11th October 2013

chord line

camber line

Performance of the URQ 11

� Accomplishment:• reference (exact value) is oLHC with \ � 10.000• number of nodes for URQ not varying:

�'() � 2� � 1 � 17• number of nodes for oLHC varying:

�_P`a � 17 equivalentcost , 30, 50and100• each oLHC- calculation was repeated 10x to show response

variation :�10 solution, out of that

- 1 run with minimal distance to optimal solution- 1 run with maximal distance to optimal solution

• build average out of all solutions of �o_Q � 19 profiles� 1 average out of 19 profiles with minimal distance� 1 average out of 19 profiles with maximal distance

6. Dresdner Probabilistic-Workshop, 11th October 2013

Performance of the URQ 12

exit

flow

ang

levariancemean

pres

sure

loss

rela

tive

dist

ance

to e

xact

val

ue [%

]re

lativ

e di

stan

ce to

exa

ct v

alue

[%]

rela

tive

dist

ance

to e

xact

val

ue [%

]

\

rela

tive

dist

ance

to e

xact

val

ue [%

]

URQoLHC (runs with min distance to optimal solution)oLHC (runs with max distance to optimal solution)

6. Dresdner Probabilistic-Workshop, 11th October 2013

\

\

\

Performance of the URQ 13

• 10- to 20-times higher cost for oLHC- simulation to get equivalent approximations for the mean of system response

• 3- to 6-times higher cost for oLHC- simulation to get equivalent approximations for the variance of system response

• Reason: Comparison of coefficients between Gaussian- Quadrature and Taylor-series expansion doesn‘t feature cross-derivative terms of the Taylor-series.

Why is there a difference between the approximation- quality of mean and variance ??

6. Dresdner Probabilistic-Workshop, 11th October 2013

Performance of the URQ 14

� Approximations of system responses with Taylor- ser ies• Mean ��

• Variance ���

6. Dresdner Probabilistic-Workshop, 11th October 2013

Validation of the error 15

� Exact response mean and variance are calculated analytically for a test function together with a uniform probability distribution

� The input standard deviation is repeatedly halved� Result: error ratios converge to 16

→ Error of order q �IZ

6. Dresdner Probabilistic-Workshop, 11th October 2013

Aerodynamic blade -to-blade robust turbine blade optimization� Problem definition:

16

subject to

r Y� 9 Y�Ust u v w Wx�Gr y u yx�G w Wx�Gr zJQ,{{ u zx�G w Wx�Gr |GHI u |x�G w Wx�G

} � ~�,�����@~̅�,�~�,�@~�

whereW�,Q���. … isentropic total pressure at exit planeW̅�,Q … mass-averaged exit stagnation pressureW�,� … total pressure at inlet planeW� … static pressure at inlet plane

6. Dresdner Probabilistic-Workshop, 11th October 2013

Aerodynamic blade -to-blade robust turbine blade optimization

17

� Back surface diffusion d is the diffusion from Mmax to TE.

� High shape factors H indicate flow separation. HTE,SS < 2 to insure there is no separation when flow reaches next vane.

6. Dresdner Probabilistic-Workshop, 11th October 2013

Aerodynamic blade -to-blade robust turbine blade optimization� „NACA“ parameterization:

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6. Dresdner Probabilistic-Workshop, 11th October 2013

Aerodynamic blade -to-blade robust turbine blade optimization� Isight process:

19

�� ∈ S��, 2 � 1, … , 33�� , � � 1, … , 14

��, ��, �� , �� , …\&�, �*

r } u }._G , …

RR customizedCMA-ES algorithm

6. Dresdner Probabilistic-Workshop, 11th October 2013

First results of turbine optimization� First 39 CMA-ES optimization runs � Filled dots: All constraints satisfied; Green dot: Optimal solution� Wall clock time per run: approx. 30 min.

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6. Dresdner Probabilistic-Workshop, 11th October 2013

Summary of URQ method• no tuning parameters• only depending of the first four statistical moments of

design parameters• skewness and kurtosis of design parameters are covered• correlation between design parameter can be included• error of order q ��Z• computationally much cheaper than oLHC

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6. Dresdner Probabilistic-Workshop, 11th October 2013

Thank you for your attention!

6. Dresdner Probabilistic-Workshop, 11th October 2013