1

AN EFFICIENT METHOD FOR PROBABILISTIC AND ROBUST DESIGN WITH NON-NORMAL DISTRIBUTIONS

Liping WangGeneral Electric Global Research Center

Inspection & Manufacturing TechnologiesKWD-248, One Research Circle

Niskayuna, NY 12309&

Srinivas KodiyalamSGI

HPC/CAE & MDO Applications1600 Amphitheatre Parkway, MS: 405

Mountain View, CA 94043

Abstract1

This paper details a single-level approach for probabilistic and robust design with non-normal distributions. The normal tail transformation is used to find the equivalent means and standard deviations for non-normally distributed variables. Three structural problems including a beam vibration probabilistic design, a cantilevered beam with multiple constraints and a turbine blade robust design are used to illustrate the efficiency of the proposed method. The result comparison with the commonly used two-level approaches shows that the single-level approach is much more efficient and robust.

Introduction

Over the last 10 years probabilistic and robust design has been one of the prominent applications and challenges of structural optimization. The difficulty in considering reliability constraints or variation objectives lies in the fact that modern reliability analysis methods themselves are formulated as a problem of optimization. Therefore most existing approaches model the probabilistic and robust design as two-level optimization problems. The reliability indices or failure probabilities are computed using reliability analysis methods in the first level; then, the optimum design is solved through optimization algorithms by considering them as the optimization constraints in the second level. To compute the reliability indices at each optimization run, iterations are usually required when the iterative reliability algorithm (HL-RF) [1] are applied. N+1 (N is number of random variables) function calls are needed if the AMV method [2] is used. If the efficient safety index algorithm (ESIA) [3] is applied, the

1 Copyright 2002 by the American Institute of Aeronautics and Astronautics, Inc.

number of iterations for each reliability index calculation may be reduced to 2-4 and the computational cost can be decreased.

However, if the number of active constraints is large, it will be very time-consuming to conduct reliability-based structural optimization or robust design with the two-level approaches although the efficient algorithms are used. To alleviate this problem, single-level approaches for direct reliability-based optimization have been proposed in Ref. [4-5]. These methods do not require the inner iterations for reliability index calculations during optimization runs, therefore the computational cost is significantly reduced, particularly for the problems having a large amount of constraints.

The objective of this paper is to demonstrate the efficiency of the single-level approach for both reliability-based optimization and robust designs, particularly with non-normal distributions. To directly conduct the reliability-based optimization and robust design with non-normal distributions, the normal tail transformation [6] is used to find the equivalent means and standard deviations for each variable at each optimization run. The MPP coordinates for each limit state function are determined based on these equivalent mean and standard deviation. Then, the constraints and gradients are calculated at the MPPs for the next optimization cycle. PEZ (a general purpose of the Design for Six Sigma (DFSS) tool) developed at GE Aircraft Engine) is used to drive the optimization and robust design. Three structural problems are used to demonstrate the proposed method, which include a probabilistic design of beam vibration problem with lognormal distributions, a cantilevered beam with multiple constraints and a turbine blade robust design problems.

43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con22-25 April 2002, Denver, Colorado

AIAA 2002-1754

Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

2

Traditional Two-level Approaches for Reliability-based Optimization Design

In the tradition two-level approaches, the reliability-based optimization problem of the kth stage is stated as

Minimize: f(µx(k,),p) (1a)

Subject to: Gj(µx(k), p) = βj - βj0 ≤ 0 (1b)

and µxL ≤ µx

(k) ≤ µxU (1c)

where, µx(k) is an n-dimensional vector of random

variables, which represents the mean values of random variables. p is a d-dimensional vector of design parameters. βj0 is the required safety index for the j-th constraint Gj.

The safety index βj is defined by the solution of the mathematical problem in the standard normal space (U-Space) of random variables.

Minimize: UTU (2a) Subject to: Gj(U)=0 (2b)

where U is the vector of random variables in the standard normal space.

There are many algorithms available that can solve this problem, such as mathematical optimization schemes [8] or iterative algorithms [1, 3] or the AMV method [2]. The commonly-used iteration method is the HL-RF method given in Ref. [1], which approximates the limit state surface (2b) by its tangent plane at the most probable failure point (MPP) and then an improved point is obtained by computing the shortest distance from the approximate hypersurface to the origin. For nonlinear problems, several iterations are required to converge to the final MPP because the hypersurface is approached only by its tangent plane. If the limit surface is flat, the HL-RF method converges fast. However, if the limit surface is complicated or highly nonlinear, the truncation error of the linear approximation might be large and this method may converge very slowly, or oscillate and fail to converge. The efficient safety index algorithm (ESIA) presented in Ref. [3] approximates the limit state surface using a two-point nonlinear adaptive approximation. The safety index is obtained by iteratively solving the optimum solution of the approximate safety index model with the approximate limit state surface. The numerical results presented in Ref. [3] demonstrated that this technique is significantly more efficient and stable than other iterative algorithms. Usually, it requires 2-3 iterations. The AMV method in Ref. [2] estimates a 9-point cumulative probability distribution (CDF) based on the linear approximation at

mean value points and function corrections at the selected 9 points. With a spline-fitted CDF curve, the failure probability and safety index at any given limit of the responses can be calculated based on the AMV results. This method requires N+10 performance function calls for each safety index and failure probability estimation. The number of function calls will be increased to N+1+9M if the problem has M performance functions.

The scheme of the reliability-based optimization with iterative safety index calculations or the AMV method is shown in Fig. 1. The drawback of the traditional two-level approach for reliability-based optimization is that the number of function calls will be dramatically increased if the system has a large amount of constraints. It is impractical for the problems requiring long simulation time.

Direct Reliability-based Optimization for Non-Normal Distributions

In the direct reliability optimization proposed in Ref. [4], no iterative safety index calculations are needed at each optimization iteration. The reliability index search is involved in global optimization search. The reliability based optimization problem given in Eq. (1) is stated as

Minimize: f(µx(k),p) (3a)

Subject to: Gj(X(k), p) ≤ 0 j=1,2,...,J (3b)

And: µxL ≤ µx

(k) ≤ µxU (3c)

where µx(k) are the mean values of random variables.

Like deterministic optimization problems, the constraint Gj(X

(k) is selected as the structural response, such as stress, displacement, frequency, etc. However, Gj(X

(k) is the function of the most probable failure point (MPP) at the kth optimization iteration, X(k), instead of direct design variables µx

(k). The relation between the MPP coordinates, X(k), and the design variables, µx

(k) is computed as follows.

First, at the kth optimization run, the MPP coordinates in a standard normal space (U-space) are calculated as

U* = -β α* (4)

where β is the safety index, which represents the shortest distance from the origin to the limit state surface in U-space. α* is the vector of the direction cosines of β at the MPP, which can be computed as

3

σxi ( ∂g / ∂x )*

α* = (5)

[ ∑ (σxi ( ∂g / ∂xi )* )2 ]1/2

i=1

Based on the relation between normal and standard normal random variables, the MPP coordinates in U-space can be expressed by the MPP coordinates in X-space as follows

X* - µx

U*= (6)σx

Substituting Eq. (6) into Eq. (4), Eq. (4) can be rewritten as

X* - µx

= -β α* (7)σx

From this equation, the relation between the MPP coordinates, X*, and the design variables µx can be given as

X* = µx - β α* σx (8)

In Eq. (8), X* and µx are related by β. To assure that optimization can satisfy the required reliability at the convergent point, the required safety index, β0, is used to replace the safety index in Eq. (8), that is,

X* = µx - β0 α* σx ( 9)

Since this direct reliability optimization method doesn’t require the inner iterations for the reliability analysis, itcan significantly reduce the computational cost for the problems with a large amount of variables. However, the method given in Ref. [4] can only be used for the normal distributions. In practical engineering problems, many random variables are non-normal distributions. Hence, it is important to extend the single-level reliability based optimization to the non-normal distribution problems.

Reliability-based optimization is to minimize weight of the structure subject to a given reliability and other structural performance requirements. Robust design is to minimize the variation of the performance while at the same time achieving performance requirements and meeting reliability goals. To compute the variation and

reliability of the performance, the inner iterations are required. Like reliability-based optimization, two-level approach is impractical for robust design. In this paper, the single-level reliability based optimization approach is applied for robust design with non-normal distributions.

Single-level Approach for Probabilistic and Robust Design With Non-normal Distributions

In the proposed work, the direct reliability based optimization mentioned above is applied for reliability-based optimization and robust design with non-normal distributions. Due to the non-normal distributions, the relation between the MPP coordinates, X(k), and the design variables, µx

(k) is more complicated and implicit. When the random variables are mutually independent, the non-normal variables X can be transformed to the standardized normal variables U using the following formula

U* = Φ-1 [Fx ( X *)] (10)

where Fx ( X) is the marginal cumulative distribution function. Substituting Eq. (10) into Eq. (4), Eq. (4) can be rewritten as

Φ-1 [Fx ( X *)] = -β α* (11a) or Fx ( X *) = Φ (-β α*) (11b)

A numerical method, Bisec-Raphson method, can be used to find X* of Eq. (11b). Then the equivalent means and standard deviations are computed by preserving the PDF at X*.

σµ

φσ

′Φ−=′

Φ=′

−

−

)]([

)(

)])([(

*1*

*

*1

XFX

Xf

XF

xx

x

x

(12)

Based on Eq. (12), the direct reliability based optimization problem with non-normal distributions can be given as

Minimize: f(µx(k), p) (13a)

Subject to: Gj(X(k), p) ≤ 0 j=1,2,...,J (13b)

and: µxL ≤ µx

(k) ≤ µxU (13c)

where: σαβµ ′−′= −10

kjx

kjX (13d)

The flow chart for the single-level probabilistic and robust design with non-normal distributions is shown in Figure 2. Numerical Examples

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In this paper, three examples are selected to demonstrate the efficiency of the proposed approach for probabilistic and robust design. The examples include a probabilistic design of beam vibration problem with lognormal distributions, a cantilevered beam with multiple constraints and a turbine blade robust design problems. The results are compared with the traditional approaches to demonstrate significant reduction of computational time.

Example 1: Probabilistic Design of A Tapered Vibrating Beam

The physical geometry of the tapered beam is shown in Figure 3. The linear taper from the base to the end makes the fundamental frequency a reasonably interesting nonlinear function of the geometry parameters. The random and design variables are given in Table 1. The upper and lower limits for design variables are:

8.02.00≤≤ hµ , 8.02.0 ≤≤ Lh ,

0.60.2 0 ≤≤ wµ , and 0.60.2 ≤≤Lwµ .

The first natural frequency of the beam is given as:

)(

)(4.38630

480

1

),,,,(

2

00

fac

fac

LL

KE

PEE

L

LwwhhFreq

ρπ×

=

where E is Elastic Modulus and equal to 17000000 lb/in2, ρ is Material Density and equal to 0.161 lb/in3. PEfac and KEfac represent the potential energy factor and kinetic energy factor, respectively. They can be calculated as follows.

0200

2

340

30

420

430

20

30

20

3

4340

220

20

20

2230

40

20

2340

3400

2

030

300

3

720720

48042

8020

4360

60203

802

480240240

whhwhh

whwhwhh

whwhwh

whwhhwhh

whhwhwhh

whwhwhh

whwhwhPE

LLL

LLLL

LL

LLLLL

LLLL

LLLL

LLfac

+

−++

+++

+++−+−+

−++

−+−=

πππππ

ππππππ

πππ

LLL

LLLLL

LLL

LLLfac

whwhwh

whwhwhwh

whwhwh

whwhwhKE

128128128

432

1281660

2322

000

3000

30

200

03

003

++−+−++

−−−++=

ππππππ

πππ

The weight of the beam is calculated as

)22(6

),,,,(

0000

00

LLLL

LL

hwwhwhwhL

LwwhhWeight

+++

=ρ

The beam is designed for minimum weight, subject to the constraint on the failure probability of the frequency (greater than 220 HZ) less than 0.01. With the proposed single level directly reliability-based optimization approach, the problem can be formulated as follows:

Minimize: ),,,( 00 LwLwhLhWeight µµµµSubject to: 220),,,,( **

0*0

* ≥LL hwwhLFreq

01.0≤fP

and 8.02.00≤≤ hµ

8.02.0 ≤≤ Lh

0.60.2 0 ≤≤ wµ0.60.2 ≤≤

LwµThe iteration history of reliability-based optimization design using the proposed single level approach is given in the following Table 2. The iteration history of the traditional two-level approach is given in Table 3. At each optimization run, an AMV method is used to compute the safety index, β, which takes 13 function calls. The result comparisons of the proposed single-level and the two-level approaches are given in Table 4. The single-level approach reaches the minimum weight of 0.8290 lbs with only 23 function calls, while the two-level approach takes 312 runs with the optimal of 0.8873 lb.

Example 2: Robust Design of a Cantilevered Beam With an End Load

The cantilevered beam with an end load is shown in Figure 4. The random and design variables are given in Table 5. The displacement, stress and reaction moment of the cantilevered beam are computed as follows:

5

0.44

),,,,(3

3≤=

EBH

PLHBEPLDisp

0.40006

),,,(2≤=

BH

PLHBPLStress

0.25000),( ≤= PLPLMoment

The beam is designed to minimize the variation in the displacement, stress and moment. At the same timeguarantee that the safety indices of finding the beam displacement less than 4.0 in, stress less than 4000.0 psi and moment less than 25000.0 in-lb are greater than 2.0. With the proposed single level directly reliability-based optimization approach, the problem can be formulated as follows:

Minimize:)(

3

1

),,,,( **

momentstressdisp

BH PELVariation

σσσµµ

++

=

Subject to:

0.44

),,,,(

3***

3*

****

≤=

HBE

LP

HBEPLDisp

0.40006

),,,(

2**

*

***

≤=

HB

LP

HBPLStress

0.25000),( ** ≤= LPPLMoment

0.2,0.2,0.2 ≥≥≥ MomentStressDisp βββand, 5.33.0 ≤≤ Hµ

5.44.0 ≤≤ Bµ220190 ≤≤ L

With the proposed single level approach, the optimization takes 9 runs to reach the minimum of variation, 782.2915, shown in Table 6. The iteration history of the traditional two-level approach is given in Table 7. At each optimization run, an AMV method is used to compute the safety indices, β, which takes 32 function calls for three constraints.

The result comparisons of the proposed single-level and the two-level approaches are given in Table 8. The single-level approach only takes 8 runs to find the optimal while the two-level method takes 288 runs. This example demonstrates the single-level method can dramatically reduce the number of runs for the problems with multiple constraints.

Example 3: Robust Design of A Turbine Blade

A turbine blade shown in Figure 5 has three cooling flow parameters (wa, wb, wc ) as design and random variables with lognormal distributions. Life cycles at 9 locations are computed using a set of in-house and commercial simulation codes for heat transfer and mechanical analyses. It takes about 45 minutes to complete the cooling and mechanical processes

The goal of this design is to find the cooling flow parameters that will make the life cycles at 9 locations of the turbine blade less sensitive to manufacturing or environmental variation while at the same time achieving performance requirements. In this example, only the life cycle at location 3 is selected as the design constraint and the robust design is modeled as follows

Minimize: ),,(cba wwwVariation µµµ

Subject to: requiredcba LCFwwwLCF ≥),,( ***3

001.0≥fP

and Uaw

La ww

a≤≤ µ

Ubw

Lb ww

b≤≤ µ

Ucw

Lc ww

c≤≤ µ

The result comparisons of the single-level and the traditional two-level approaches are given in the following Table 9. This example demonstrates the single-level approach can significantly reduce the computational time for the problems requiring long execution time.

Conclusions

This study investigated the use of the single-level approach for reliability-based optimization and robust design. The study shows that the single-level probabilistic and robust design is much more efficient than the traditional two-level approaches. Meanwhile, the single-level approach more likely reaches more optimum solutions than the traditional approach does by avoiding duplicated search in MPP and optimum

6

design. For some complicated engineering problems, which requires a few minutes or hours simulation, the single-level approach can dramatically reduce the computation cost over the traditional two-level approaches.

The use of the most critical failure modes to represent the system failure probability, in most situations, may provide an inaccurate estimate of system reliability, particularly when one single failure mode does not have a relatively much higher probability of occurrence than all other failure modes. Therefore, the system reliability will be considered to perform more accurate probability estimation in the future work.

Reference

1. R. Rackwitz and B. Fiessler, ``Structural Reliability Under Combined Load Sequences’’, Computers and Structures, Vol. 9, 1978, pp. 489-494.

2. Y.-T Wu, H. R. Millwater and T. A. Cruse, “An Advanced Probabilistic Structural Analysis Method for Implicit Performance Function”, AIAA J. Vol. 28, 1663-1669, 1990.

3. Liping Wang and Ramana Grandhi, “Efficient Safety Index Calculation for Structural Reliability Analysis”, Computer and Structures, Vol. 52, No. 1, 1994, pp. 103-111.

4. Xiaoguang Chen, Timothy K. Hasselman and Douglas J. Neill, ``Reliability Based Structural Design Optimization For Practical Applications’’, 38th the AIAA/ ASME/ ASCE/ AHS/ ASC Structures, Structural Dynamics, and Materials Conference, Hyatt Orlando, Kissimee, FL., AIAA-97-1403, 1997.

5. Norbert Kuschel and Rudiger Rackwitz, “Two Basic Problems in Reliability-based Structural Optimization”, Mathematical Methods of Operations Research, 46, pp. 309-333, 1997.

6. R. E. Melchers, Structural Reliability Analysis and Prediction, Ellis Horwood Limited, 1987

7. Liping Wang, Ramana V. Grandhi and Dale A. Hopkins, ``Structural Reliability Optimization Using An Efficient Safety Index Calculation Procedure’’, International Journal for Numerical Methods in Engineering, Vol. 38, 1995, pp 1721-1738.

8. Pei-Ling and A. Der Kiureghian, ``Optimization Algorithms for Structural Reliability Analysis’’, Report No. UCB/SESM 86/09, Department of Civil Engineering, University of California, Berkeley, July, 1986.

9. Liping Wang and Ramana V. Grandhi, ``Safety Index Calculation Using Intervening Variables for Structural Reliability Analysis’’, Computers and Structures, Vol. 59, No. 6, 1996, pp. 1139-1148.

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Table 1. Random and Design Variables of the Tapered Beam (Example 1)

Design Variable

Description RandomVariable

Description Mean Standard Deviation

Distribution Initial Design

-- --- L Beam Length 8.0 0.1 Extreme Value 8.0

µh0Mean of Thickness at Base

h0 Thickness at Base 0.4 0.01 Extreme Value 0.4

µw0 Mean of Width at Base

w0 Width at Base 4.0 0.2 Lognormal 4.0

µwLMean of Width at Tip

wL Width at Tip 4.0 0.2 Lognormal 4.0

hL Mean of Thickness at Tip

-- --- --- --- --- 0.4

Table 2. Iteration History of The Single-level Approach (Example 1)

RUN µµµµh0 hL µµµµw0 µµµµwL L Weight Freq1 0.400 0.400 4.000 4.000 8.000 2.061 193.9392 0.400 0.400 4.000 4.000 8.000 2.062 194.129

3 0.400 0.400 4.000 4.000 8.000 2.062 193.949

4 0.400 0.400 4.004 4.000 8.000 2.062 193.9915 0.400 0.400 4.000 4.004 8.000 2.062 193.891

6 0.468 0.200 2.000 2.000 8.000 0.861 234.513

7 0.468 0.200 2.000 2.000 8.000 0.861 234.5138 0.469 0.200 2.000 2.000 8.000 0.861 234.788

9 0.468 0.200 2.000 2.000 8.000 0.861 234.480

10 0.468 0.200 2.002 2.000 8.000 0.861 234.59111 0.468 0.200 2.000 2.002 8.000 0.861 234.453

12 0.444 0.200 2.000 2.000 8.000 0.829 220.08013 0.444 0.200 2.000 2.000 8.000 0.829 220.080

14 0.444 0.200 2.000 2.000 8.000 0.830 220.337

15 0.444 0.200 2.000 2.000 8.000 0.829 220.05116 0.444 0.200 2.002 2.000 8.000 0.829 220.152

17 0.444 0.200 2.000 2.002 8.000 0.829 220.024

18 0.443 0.200 2.000 2.000 8.000 0.829 220.00119 0.443 0.200 2.000 2.000 8.000 0.829 220.001

20 0.444 0.200 2.000 2.000 8.000 0.829 220.257

21 0.443 0.200 2.000 2.000 8.000 0.829 219.97022 0.443 0.200 2.002 2.000 8.000 0.829 220.072

23 0.443 0.200 2.000 2.002 8.000 0.829 219.944

8

Table 3. Iteration History of the traditional two-level approach (Example 1)

RUN hL µh0 µw0 µwL L Weight β Pf Freq 1 0.400000 0.400000 4.000000 4.000000 8.000000 2.105029 -0.926667 0.822950 249.124626

2 0.400000 0.400400 4.000000 4.000000 8.000000 2.106000 -0.892062 0.813820 249.345854

3 0.400400 0.400000 4.000000 4.000000 8.000000 2.105984 -0.925622 0.822679 249.120500

4 0.400000 0.400000 4.004000 4.000000 8.000000 2.106032 -0.917130 0.820463 249.188942

5 0.400000 0.400000 4.000000 4.004000 8.000000 2.106143 -0.936602 0.825518 249.053661

6 0.200000 0.439798 2.000000 2.000195 8.000000 0.877348 2.106657 0.017574 306.611865

7 0.200000 0.439798 2.000000 2.000195 8.000000 0.877348 2.106657 0.017574 306.611865

8 0.200000 0.440238 2.000000 2.000195 8.000000 0.877930 2.117673 0.017101 306.938707

9 0.200200 0.439798 2.000000 2.000195 8.000000 0.877581 2.105340 0.017631 306.564183

10 0.200000 0.439798 2.002000 2.000195 8.000000 0.877714 2.109675 0.017443 306.682632

11 0.200000 0.439798 2.000000 2.002196 8.000000 0.877792 2.104261 0.017678 306.507211

12 0.200000 0.446870 2.000000 2.000000 8.000000 0.886708 2.323303 0.010081 311.884658

13 0.200000 0.445456 2.000000 2.000039 8.000000 0.884835 2.271465 0.011559 310.828482

14 0.200000 0.445285 2.000000 2.000044 8.000000 0.884606 2.265564 0.011739 310.701706

15 0.200000 0.445730 2.000000 2.000044 8.000000 0.885201 2.281050 0.011273 311.032421

16 0.200200 0.445285 2.000000 2.000044 8.000000 0.884845 2.263770 0.011794 310.653482

17 0.200000 0.445285 2.002000 2.000044 8.000000 0.884981 2.269848 0.011608 310.772330

18 0.200000 0.445285 2.000000 2.002044 8.000000 0.885057 2.262322 0.011839 310.595072

19 0.200000 0.446972 2.000000 2.000000 8.000000 0.886847 2.327272 0.009975 311.960657

20 0.200000 0.446949 2.000000 2.000001 8.000000 0.886815 2.326368 0.009999 311.943430

21 0.200000 0.447396 2.000000 2.000001 8.000000 0.887410 2.344157 0.009535 312.277190

22 0.200200 0.446949 2.000000 2.000001 8.000000 0.887052 2.324272 0.010055 311.895735

23 0.200000 0.446949 2.002000 2.000001 8.000000 0.887187 2.331267 0.009870 312.014742

24 0.200000 0.446949 2.000000 2.002001 8.000000 0.887266 2.322646 0.010099 311.837886

Table 4. Comparison of proposed single and traditional two-level approaches (Example 1)

Method Total Number of Function Calls

Minimum Weight (lb)

Failure Probability at Optimum

Frequency at Optimum

Single-level 23 0.8290 0.010000 219.944Two-level 312 0.8873 0.010099 311.838

Table 5. Design and Random Variables Of Cantilevered Beam (Example 2)

Design Variable

Description RandomVariable

Description Mean Standard Deviation

Distribution Initial Design

µHMean of Cross-sectional Height

H Cross-sectional Height

3.0 0.3 Lognormal 3.0

µBMean of Cross-sectional Width

B Cross-sectional Width

4.0 0.4 Lognormal 4.0

L Mean of Beam Length

-- --- --- --- --- 200.0

-- --- E Young’s Modulus of Material

1.0E7 1.0E6 Lognormal 1.0E7

-- --- P Applied Force 100.0 10.0 Lognormal 100.0

9

Table 6. Optimization Iteration History of the single-level approach (Example 2)

RUN H B E P L Disp Stress Moment Variation

1 3.0000 4.0000 10000000.0000 100.0000 200.0000 6.2329 5600.4072 23990.0000 936.8390

2 3.0030 4.0000 10000000.0000 100.0000 200.0000 6.2103 5586.9173 23990.0000 936.1161

3 3.0000 4.0040 10000000.0000 100.0000 200.0000 6.2264 5594.3144 23990.0000 936.5225

4 3.0000 4.0000 10000000.0000 100.0000 200.2000 6.2516 5606.0076 24013.9900 937.7765

5 3.5000 4.5000 10000000.0000 100.0000 190.0000 2.7180 3252.6688 22790.5000 781.5097

6 3.5000 4.5000 10000000.0000 100.0000 190.0000 2.7180 3252.6688 22790.5000 781.5097

7 3.4965 4.5000 10000000.0000 100.0000 190.0000 2.7276 3260.2544 22790.5000 781.9028

8 3.5000 4.4955 10000000.0000 100.0000 190.0000 2.7208 3256.1803 22790.5000 781.6846

9 3.5000 4.5000 10000000.0000 100.0000 190.1900 2.7262 3255.9215 22813.2905 782.2915

Table 7. Optimization Iteration History of the two-level approach (Example 2)

RUN H B L E P Disp Stress Moment βdisp βstress βmoment Variation1 3.0000 4.0000 200.0000 9950370.0000 148.2950 4.5046 5017.5029 29659.0000 0.7304 0.6189 2.1242 952.3272

2 3.0030 4.0000 200.0000 9950370.0000 148.2950 4.4910 5007.3660 29659.0000 0.7404 0.6285 2.1242 951.5425

3 3.0000 4.0040 200.0000 9950370.0000 148.2950 4.5001 5012.4403 29659.0000 0.7324 0.6235 2.1242 951.9898

4 3.0000 4.0000 200.2000 9950370.0000 148.2950 4.5046 5017.5029 29659.0000 0.7304 0.6189 2.1242 952.3272

5 3.5000 4.5000 200.0000 9950370.0000 148.2950 2.5090 3264.7334 29659.0000 2.5419 2.4788 2.1242 829.5723

6 3.5000 4.5000 200.0000 9950370.0000 148.2950 2.5090 3264.7334 29659.0000 2.5419 2.4788 2.1242 829.5723

7 3.4965 4.5000 200.0000 9950370.0000 148.2950 2.5166 3271.3155 29659.0000 2.5245 2.4656 2.1242 830.0124

8 3.5000 4.4955 200.0000 9950370.0000 148.2950 2.5115 3268.0289 29659.0000 2.5366 2.4726 2.1242 829.7670

9 3.5000 4.5000 200.2000 9950370.0000 148.2950 2.5090 3264.7334 29659.0000 2.5419 2.4788 2.1242 829.5723

Table 8. Comparison of the single and two-level approaches (Example 2)

Method Total Number of Function Calls

Minimum Variation

Safety Indices at Optimum

Displacement, Stress andMoment at Optimum

Single-level 9 782.2915 2.0, 2.0, 2.0 2.7262, 3255.9215, 22813.2905

Two-level 288 829.5723 2.5419, 2.4788, 2.1242

2.5090, 3264.7334, 29659.0000

Table 9. Comparisons of Computational Costs for Example 3 (Turbine Blade)

Method Number of OptimizationRuns

Number ofFunction Calls

ComputationalTime (days)

Percentage of Variation Reduction At Optimum

Single-level 17 17 0.53 11.11%

Two-level 22 377 8.94 10.92%

10

Figure 1. Reliability–Based Optimization Using Two-level Approach

Call Optimizer

Converge?

Compute Weightat Means

Compute Sensitivities

Compute Safety Indices

Input Means, Std. Dev. & Params

Start

No

Yes

Stop

Jjj ,...,2,1, =βCompute at Meansjy

Compute Gradients at Means

MPPSearch(ESIA)

MPP Search

(HL-RF)

MPP Search(AMV)

Converge?

j < J ?

j=1

No

Noj=j+1

ii x

j

x

j

σβ

µβ

∂∂

∂∂

,

Niii xx ,...,2,1,, =σµ

YesYes

Transform

iiii xxxx ',' µµσσ →→Iterative β β β β Calculations

Call Optimizer

Converge?

Compute Weightat Means

Compute Sensitivities

Compute Safety Indices

Input Means, Std. Dev. & Params

Start

No

Yes

Stop

Jjj ,...,2,1, =βCompute at Meansjy

Compute Gradients at Means

MPPSearch(ESIA)

MPP Search

(HL-RF)

MPP Search(AMV)

Converge?

j < J ?

j=1

No

Noj=j+1

ii x

j

x

j

σβ

µβ

∂∂

∂∂

,

Niii xx ,...,2,1,, =σµ

YesYes

Transform

iiii xxxx ',' µµσσ →→Iterative β β β β Calculations

11

Figure 2. Reliability–Based Optimization and Robust Design Using Single level Approach

Call Optimizer

Converge?

Compute

Compute Direction Consine’s

Compute Gradients

Input Means and Std. Dev.

Start

No YesStop

JjGG

ii x

j

x

j ,...,2,1,, =∂∂

∂∂

σµ

Jjj ,...,2,1, =α

Niii xx ,...,2,1,, =σµ

Compute MPPs

ii xjxix σαβµ 0* −=

),...,,( **2

*1 Nj xxxy

Transform

iiii xxxx ',' µµσσ →→

Compute Weight or Variation

Call Optimizer

Converge?

Compute

Compute Direction Consine’s

Compute Gradients

Input Means and Std. Dev.

Start

No YesStop

JjGG

ii x

j

x

j ,...,2,1,, =∂∂

∂∂

σµ

Jjj ,...,2,1, =α

Niii xx ,...,2,1,, =σµ

Compute MPPs

ii xjxix σαβµ 0* −=

),...,,( **2

*1 Nj xxxy

Transform

iiii xxxx ',' µµσσ →→

Compute Weight or Variation

12

Figure 3. The Tapered Beam

Figure 4. Cantilevered Beam

w0

h0

wL hL

L

w0

h0

wL hL

L

L

P

x

BH

L

P

x

BH

13

Figure 5. Life Cycles at 9 Locations Of A Turbine Blade

2

5

3

7

1

Date: Sep 19, 1998Time: 9:12 AMFilename: /users/ae6813t/ss/runlcf/ps.f33

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