reliability based design optimization for multidisciplinary systems using response surfaces

8/7/2019 Reliability Based Design Optimization for Multidisciplinary Systems Using Response Surfaces

http://slidepdf.com/reader/full/reliability-based-design-optimization-for-multidisciplinary-systems-using-response 1/10

AIAA-2002-1755

RELIABILITY BASED DESIGN OPTIMIZATION FOR

MULTIDISCIPLINARY SYSTEMS USING RESPONSE SURFACES

Harish Agarwal ∗ John E. Renaud †

Department of Aerospace and Mechanical Engineering

University of Notre Dame Notre Dame, IN - 46556

Abstract

This paper investigates reliability based design op-

timization (RBDO) using response surface approxima-

tions (RSA) 1,2 for multidisciplinary design optimization

(MDO). In RBDO the constraints are variational (reli-

ability based) since the design variables and the system

parameters can have variation and can be subjected to un-

certainties 18. For these problems the objective is to min-

imize a cost function while satisfying reliability based

constraints. This class of problems is referred to as relia-

bility based multidisciplinary design optimization (RB-

MDO) problems 5. The reliability constraints, which

can be formulated in terms of the reliability indices or

in terms of the probability of failure, themselves repre-sent an optimization problem and can be very expensive

to evaluate for large scale multidisciplinary problems.

Response surface approximations of the constraints are

used in estimating the reliability indices or probability of

failure when solving an approximate optimization prob-

lem using FORM. In this research RSAs are integrated

within RBDO to significantly reduce the computational

cost of traditional RBDO. The proposed methodology is

compared to traditional RBDO in application to multidis-

ciplinary test problems, and the computational savings

and benefits are discussed.

∗Graduate Research Assistant, Student Member AIAA†Associate Professor, Associate Fellow AIAA

Copyright ©2001 by John E. Renaud. Published by the American In-

stitute of Aeronautics and Astronautics, Inc. with permission.

Nomenclature

x Design Variables

z Random Variables

u Independent Standard Normal Random

Variables

yd Deterministic State Variables

yr Random State Variables

p x Traditional Optimization Parameters

g R Reliability Based Constraints

p z Reliability Based Constraint Parameters

g D Deterministic Constraints

xl,xu Lower and Upper bounds on design space

f Z (z) Joint Probability Density Function of the

random variablesΦ(u) Standard Normal Cumulative Distribution

Function (CDF)

g(z) Actual Limit State Function;

Safe : g(z)> 0, Fail : g(z)< 0

βreqd Required Value of Reliability Index

˜ g(u) Approximation of the Limit State in

Standard Normal Space

Introduction

In deterministic multidisciplinary design optimiza-

tion, the designs are often driven to the limit of the designconstraints (active constraints at the optimum). These

designs may be subject to failure due to inherent uncer-

tainties that exist both in the mathematical modeling and

simulation tools and the variability in physical quanti-

ties of the actual artifact. Optimized designs determined

without due consideration of variability can be unreliable

leading to catastrophic failure. However, the existence of

physical uncertainty and model uncertainty requires a re-

1

American Institute of Aeronautics and Astronautics

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

AIAA 2002-175

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.



liability based design optimization (RBDO) to be taken

into account. The uncertainties include variational un-

certainty and simulation based uncertainty. Variational

uncertainty is mainly associated with the randomness of

physical quantities and can be easily modeled mathe-

matically by statistical means i.e., by using probabilityand cumulative density functions. Model and simulation

uncertainties are difficult to characterize and have to be

modeled using other means such as possibility theory or

fuzzy sets etc 18 because of a lack of knowledge. In this

paper only variational uncertainty is treated within the

RBDO framework. In future we plan to incorporate the

effects of other types of uncertainty in RBDO.

Literature Survey

Recently greater emphasis has been given to the de-velopment of procedures that combine multidisciplinary

design optimization techniques with probabilistic analy-

sis/design methods. Many new methods have been sug-

gested by researchers for RBDO, such as the perfor-

mance measure approach (PMA) and the reliability in-

dex approach (RIA). In the context of PMA and RIA,

several tools for probabilistic constraint evaluation have

been developed such as the advanced mean value (AMV)

method, the conjugate mean value (CMV) method, the

moving least square (MLS) method and the hybrid mean

value (HMV) method 3,4. A framework for reliability

based MDO has been suggested by Sues et. al. and

kodiyalam et. al5,6,12

. Pettit and Grandhi have im-plemented a multidisciplinary optimization approach for

the design of aerospace structures for high reliability 7.

Haftka et. al. have used response surface approxima-

tions for the reliability-based optimization of composite

laminates 8.

Background

Most engineering design problems require that de-

signers satisfy constraints imposed on the systems per-

formance. A design problem that consists of just one dis-

cipline is called a single discipline problem. When many

disciplines (structures, controls, aerodynamics, etc) in-

teract with each other, the problem becomes a multidis-

ciplinary problem. In general, a deterministic multidis-

ciplinary optimization problem can be formulated as fol-

lows.

minimize : f (x,p x,yd )

subject to : gi(x,p x,yd ) ≥ 0, i = 1..m (1)

xl ≤ x ≤ xu

In today’s competitive marketplace, it is very important

that the resulting designs are reliable. Optimized designswithout considering the variability of design variables

and parameters can be subjected to failure in service.

In RBDO, the constraints are reliability based and the

objective function is performance based. The reliability

based multidisciplinary optimization problem in terms of

RIA can be formulated as follows.

minimize : f (x,p x,yd )

subject to : g Ri = βi(z,p z,yr )−βrequired ≥ 0, i = 1..l (2)

g Di (x,p x,yd ) ≥ 0, i = l + 1,..,m

Here the constraints which are formulated in terms of

reliability indices βi are obtained as follows.

minimize : βi = u

subject to : gi(z,p z,yr ) = 0, i = 1..l (3)

The transformation of the random variables space z to

the independent standard normal random variables space

u can be obtained in general using the Rosenblatt trans-

formation 14. The probability of failure of the system

(P f )system can be estimated from the unimodal upper

bound i.e., ∑li=1Φ(−βi)), where l is the number of limit

states (reliability based constraints). For a better estimate

of the probability of failure, bimodal upper bound can be

used 10,14.

Proposed Methodology

An overview of the proposed methodology for solv-

ing RBMDO problems is summarized in Figure 1. In

traditional RBDO, the constraint values are obtained by

solving an optimization problem (Equation 3). This

sub-level optimization has to be solved many times inRBDO. This requires many system analysis calls, which

can be very expensive especially for large scale multi-

disciplinary problems. To reduce this, we approximate

the reliability based constraints by fitting a second or-

der response surface approximation for each constraint.

The second order response surface approximation is con-

structed in the standard normal space u only. Sampling

is done around the mean values of the random variables

2




Starting Point based onEngineering Knowledge

Deterministic Optimization

Use Deterministic Optimafor RBDO

Identify deterministic andrandom variables anddefine reliability basedconstraints

RBDO

Constraints

Deterministic Reliability Based

Build second order response surfaceapproximation for reliability basedconstraints about the mean valuesof the random variables

Find reliability index for eachreliability based constraint

Evaluate reliabilitybased constrants

Objective Function

Actual reliabilityconstraints satisfied

converged yesFINALDESIGN

No

Lower Leveloptimization

Do RBDO withexact constraintsusing the presentdesign point

Figure 1. Reliability Based Design Optimization

Flowchart

z in order to fit the approximation. Thus an approximateproblem is solved for each constraint evaluation as fol-

lows.

minimize : βi = u

subject to : g̃i(u) = 0, i = 1..l (4)

To solve the optimization problems (Equations 3 or 4),

is a challenge in itself. Various algorithms have been re-

ported in the literature for solving the above mentioned

problem. Kiureghian et. al has reported a list of algo-

rithms to solve this problem 21. We have used a MAT-

LAB SQP optimizer to solve this problem.

Test Problems and Results

The proposed methodology is implemented in appli-

cation to test problems. A small analytic problem and a

multidisciplinary structural design test problem are used.

Modified Barnes Problem

This is a purely analytical two-dimensional problem

and it was originally formulated by G.K. Barnes as part

of his Master’s Thesis 22. We have chosen this as a

test problem to illustrate the usefulness of the proposed

methodology. This is a highly nonlinear problem even

though it is just a two-dimensional problem. The prob-

lem is stated as follows.

Minimize :

f (x,y) = a1 + a2 x1 + a3 y4 x1 + a5 y24 + a6 x2 +

a7 y1 + a8 x1 y1 + a9 y1 y4 + a10 y2 y4 + a11 y3 +

a12 x2 y3 + a13 y23 +

a14

x2 + 1+ a15 y3 y4 + a16 y1 y4 x2 +

a17 y1 y3 y4 + a18 x1 y3 + a19 y1 y3 + a20ea21 y1

Subject to :

g Ri = βi−βreqd ≥ 0, i = 1,2

g D3 = ( y5−1)2− (

x1

500−0.11)≥ 0

xl ≤ x ≤ xu

The coefficients a in the objective function are constants

and their values is listed in Appendix. The reliability

indices to evaluate the reliability based constraints g Ri , i =

1,2 are found as follows.

minimize : βi = u, i = 1,2

Sub ject to : g R1 =

y1

z1− z2 = 0

g R2 =

x2

z3−

y4

z24

= 0

The states are calculated as follows.

CA1 : y1 = x1 x2

y3 = x22

CA2 : y2 = y1 x1

y4 = x21

y5 =x2

50

Note the mapping that the variables and parameters have

to undergo as we move from the design space to the

random variable space. In the design space the terms

zi, i = 1,..,4 are the constant parameters. They are repre-

sented as p x in the nomenclature. In the random variable

3




reliable design that is obtained from RBDO using the

RSA of constraints does not violate the actual reliabil-

ity based constraints. In the case where they are violated

due to approximation error, an RBDO using the actual

constraints, could be initiated at the reliable RSA optima

xrsa.

HPLC Structure

A high performance low cost (HPLC) structural de-

sign problem that was first introduced as a multidisci-

plinary design optimization (MDO) test problem in Wu-

jek et al15 is used for additional testing of the RBDO

method. The system is illustrated in figure 4 (Tenbar

Truss). The deterministic optimum design point is ob-

P1 P2

P3 P4

M1 M2

M3 M4

L1 L2

L3

A3

A2

A1

A4

A5

A6

A7

A8

A9

A10

Figure 4. Schematic of HPLC Structure

tained first. It is then used as the starting point for RBDO

in order to obtain the reliable design point.

The design variables in this problem come from three

different disciplines. The configuration of the structure is

varied in order to explore different topologies. Thus, the

length of the rectangular first bay ( L1) and the top andbottom lengths of the outer bay ( L2, L3) are selected as

geometric design variables. The masses placed on all of

the unconstrained nodes ( M 1− M 4) are structural design

variables representing the system payload. The areas of

the truss members ( A1− A10) make up the final category

of design variables since sizing is one of the main de-

sign considerations (see figure 4). In all seventeen design

variables are defined for this problem. A problem with

such large dimensionality is chosen to test the effective-

ness of the suggested approach, to investigate the com-

putational savings and to see whether the reliable optima

obtained using RSA is practical.

The objective is to find the size and shape of the truss

such that the weight (W tot ) of the structure is a mini-mum (low cost) and the loads (Pi) it is capable of sus-

taining and the payload ( M i) it carries are a maximum

(high performance). This multi-objective problem can

be formulated in a single objective problem by defining

a cost-performance index (CPI ) which includes each of

the objectives. The design is subject to minimum pay-

load and load requirements as well as yield stress and

first natural frequency constraints. A total of 13 inequal-

ity constraints are defined for this problem. In standard

form the deterministic system optimization problem is:

Minimize : f (x) = CPI = w1W tot +

w2

∑Pi+

w3

∑ M i(5)

Subject to :

g1 = 1.0−( M tot )min

∑ M i≥ 0.0 (6)

g2 = 1.0−(Ptot )min

∑Pi

≥ 0.0 (7)

g3 = 1.0−ω1,min

ω1≥ 0.0 (8)

g4−13 = 1.0−|σ1−10|

σ yield

≥ 0.0 (9)

x(l)i ≤ xi ≤ x

(u)i (10)

where : w1 = .003, w2 = 106, w3 = 3.5 X 106

( M tot )min = 5000 lbs, (Ptot )min = 100,000 lbs

ω1,min = 2.0 Hz, σ yield = 14,000 psi

The coefficients wi in the objective function are intro-

duced to scale the separate components so that no one

component dominates the others in driving the optimiza-

tion. The yield stress of 14,000 psi is based on the choiceof aluminum as the material for the structure. The loads

(Pi) applied to the structure are defined to be a function

of the lengths of the bays ( Li) and the payload masses

( M i) placed on the structure as shown in Equation 11.

Pi =3

∑k =1

ak i

Lk

Lre f

bk i

+4

∑ j=1

c ji

M j

M re f

d ji

(11)

5




This is similar to an aeroelastic structure in which the

loads incurred are dependent on the size and shape of the

structure. The coefficients (a,b,c and d ) in equation 11

are chosen so as to apply greater emphasis to the effect

that certain lengths or masses have on the given loads.

They are listed in the Appendix.

Deterministic Design Optimization

The deterministic optimization was run with the ini-

tial design of Li = 360 in., M i = 1500 lbs and Ai = 12

in2 (for all i). The results are tabulated in Table 3. Note

that the constraints g3,g4,g5 and g6 are active at the solu-

tion. To carry out the RBDO, the deterministic optimum

design point is chosen as the starting point.

Reliability Based Design Optimization

In reliability based design optimization, it is imprac-

tical to carry out the sub level optimization (Equation 3)

for all the constraints. This is illustrated in the two cases

that follow. We actually need to consider only the con-

straints that are active or nearly active and evaluate the

reliability index for only those constraints. In addition

significant time is saved if the deterministic optimum de-

sign is used as starting point for RBDO.

Test Case : 1

Material properties such as density, young’s modulus

and yield stress (parameters in actual design space P x)

are chosen to have randomness. The set of basic random

variables and their mean and coefficient of variation are

shown in Table 4. The coefficient of variation have been

chosen arbitrarily for this test case. All random variables

have been assumed to be lognormally distributed and are

statistically independent. The second order response sur-

face is constructed in the standard normal random vari-

able space at each step of the upper level optimization

iteration. In the reliability space the design variables x1to x17 are constant parameters p z.

It is observed in the deterministic optimization that

the constraints 3,4,5 and 6 are critical (active). So in this

test case they are treated as the failure driven constraints.

The reliability indices are calculated using Equation (3)

for the active constraints (see Table 3). The active con-

straints in this case are the first frequency constraint and

the first 3 stress constraints. Once the reliability indices

RV Description Mean Coeff. of Var.

ρ Density 0.1 0.01

E Young’s Modulus 107 0.05

σ yield Yield Stress 14000 0.05

Table 4. Random Variables (HPLC Structure)

are found, the reliability constraints are evaluated using

Equation (2). The results are listed in Table 5

MATLAB’s optimizer is used for both upper and

lower level optimization. The reliable optima is higher

than the deterministic optima. It is observed that the

value of the payloads have gone down and that of the

areas have gone up. This is expected for a more reliable

structure in which the areas of elements are larger and thepayloads are reduced so that the structure is subjected to

less stress and hence is more reliable. The reliable op-

tima obtained using RSA is better than the one obtained

using actual RBDO. This design is used to evaluate the

actual RBDO constraints. They are listed in the last col-

umn of Table 5. We observe that the actual constraints

are satisfied. Note the significant savings is computa-

tional time. The time taken in RBDO using RSA of con-

straints is one-eighth of the time required for the actual

RBDO.

Test Case : 2

The random variables are same as in Case 1. Now all

the constraints that actually depend on the random pa-

rameters are chosen as reliability based constraints. So

there are 11 reliability based constraints in this test case

i.e., the first frequency constraint and all the stress con-

straints. The results are listed in Table 6.

We observe that the reliable optima obtained using

actual RBDO and using RSA in RBDO has a higher

merit function value than that in case 1. This is expectedsince there are more reliability based constraints in case

2. The time taken in case 2 for actual RBDO is about

3.3 times more than that in case 1. We do not observe

significant difference in time for RSA in RBDO in two

cases because the system analysis gives the value for all

the constraints at any given sampling point. The reliable

design that is obtained from RBDO using RSA doesn’t

violate the actual reliability based constraints.

6




and Future”, Proceedings of the 8th ASCE Spe-

ciality Conference on Probabilistic Mechanics and

Structural Reliability, PMC2000.

15. Wujek, B. A., Renaud, J. E., Batill, S. M., “A

Concurrent Engineering Approach for Multidisci-

plinary Design in a Distributed Computing Environ-ment”,Proceedings of the ICASE / LaRC Workhop

on Multidisciplinary Design Optimization, 1995.

16. Wujek, B. A., Renaud, J. E., Batill, S. M., Brock-

man, J. B., “Concurrent Subspace Optimization us-

ing design variable sharing in a distributed com-

puting environment”, Design Engineering Techni-

cal Conference, ASME 1995.

17. Du, X., Chen, W., “A most probable point based

method for uncertainty analysis”, Design Engineer-

ing Technical Conferences, ASME 2000.

18. Oberkampf, W. L., Helton, J. C., “Mathemati-

cal Representation of Uncertainty”, Proceedings of

the 42nd AIAA/ASME/ASCE/AHS/ASC Structures,

Structural Dynamics, and Materials Conference &

Exhibit , 2001.

19. Sundaresan, S., Ishii, K., Houser, D. R., “A Robust

Optimization Procedure with Variations on Design

Variables and Constraints”, Advances in Design Au-

tomation - Volume 1, ASME 1993.

20. Haftka, R., Gurdal, Z., and Kamat, M. P., Elements

of Structural Optimization, Kluwer Academic Pub-

lishers, Dordrecht, Netherlands, Second Edition,

1990.

21. Kiureghian, A. D., Liu, P. L., “Optimization Algo-

rithms For Structural Reliability”, Journal of Struc-tural Safety, Vol. 9, pp. 161-177.

22. Barnes, G. K.,1967 M.S. Thesis, The University of

Texas, Austin, Texas.

Appendix

Load Coefficients

Coefficients for P1

a11 = 25.0 b1

1 = 4.0 c11 = 50.0 d 1

1 = 4.0a2

1 = 20.0 b21 = 3.7 c2

1 = 37.0 d 21 = 2.9

a31 = 20.0 b3

1 = 3.7 c31 = 35.0 d 3

1 = 2.9c4

1 = 37.0 d 41 = 2.9

Coefficients for P2

a12 = 17.0 b1

2 = 3.5 c12 = 25.0 d 1

2 = 2.7a2

2 = 19.0 b22 = 3.8 c2

2 = 27.0 d 22 = 2.7

a32 = 15.0 b3

2 = 3.0 c32 = 25.0 d 3

2 = 2.7c4

2 = 27.0 d 42 = 2.7

Coefficients for P3

a13 = 25.0 b1

3 = 4.0 c13 = 35.0 d 1

3 = 2.9a2

3 = 20.0 b23 = 3.7 c2

3 = 37.0 d 23 = 2.9

a33 = 20.0 b3

3 = 3.7 c33 = 50.0 d 3

3 = 4.0c4

4 = 37.0 d 44 = 2.9

Coefficients for P4

a14 = 17.0 b1

4 = 3.5 c14 = 25.0 d 1

4 = 2.7a

2

4

=15

.0 b

2

4

=3.0 c

2

4

=27

.0 d

2

4

=2.7

a34 = 19.0 b3

4 = 3.8 c34 = 25.0 d 3

4 = 2.7c4

4 = 27.0 d 44 = 2.7

a1 75.196 a2 -3.8112 a3 0.12694

a4 -2.0567e-3 a5 1.0345e-5 a6 -6.8306

a7 0.030234 a8 -1.28134e-3 a9 3.5256e-5

a10 -2.266e-7 a11 0.25645 a12 -3.4604e-3

a13 1.3514e-5 a14 -28.106 a15 -5.2375e-6

a16 -6.3e-8 a17 7.0e-10 a18 3.4054e-4

a19 -1.6638e-6 a20 -2.8673 a21 0.0005

Table 7. Coefficients for the Modified Barnes problem

10


reliability based design optimization for multidisciplinary systems using response surfaces

Documents