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© 2013 Rolls-Royce Deutschland Ltd & Co KGThe information in this document is the property of Rolls-Royce Deutschland Ltd & Co KG and may not be copied or communicated to a third party, or used for any purpose other than that for which it is supplied without the express written consent of Rolls-Royce Deutschland Ltd & Co KG.This information is given in good faith based upon the latest information available to Rolls-Royce Deutschland Ltd & Co KG, no warranty or representation is given concerning such information, which must not be taken as establishing any contractual or other commitment binding upon Rolls-Royce Deutschland Ltd & Co KG or any of its subsidiary or associated companies.
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
2
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
6. Dresdner Probabilistic-Workshop, 11th October 2013
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:
18
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.
20
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
21
6. Dresdner Probabilistic-Workshop, 11th October 2013
Thank you for your attention!
6. Dresdner Probabilistic-Workshop, 11th October 2013