Sequential Reliability Analysis Framework for

Multidisciplinary Systems

Joongki Ahn∗ Jaehun Lee∗ Suwhan Kim∗

Jang Hyuk Kwon †

Korea Advanced Institute of Science and Technology, 305-701, Republic of Korea

This study presents an efficient strategy for reliability analysis under multidisciplinaryanalysis systems. Existing methods have performed the reliability analysis using the non-linear optimization techniques. This is mainly due to the fact that they directly applyMultidisciplinary Design Optimization(MDO) frameworks to the reliability analysis formu-lation. Accordingly, the reliability analysis and the Multidisciplinary Analysis(MDA) aretightly coupled in a single optimizer, which hampers utilizing the recursive and function-approximation based reliability analysis methods such as the Advanced First Order Reliabil-ity Method(AFORM). In order to utilize the efficient reliability analysis method under mul-tidisciplinary analysis systems, we propose a new strategy named Sequential Approach onReliability Analysis under Multidisciplinary analysis systems(SARAM). In this approach,the reliability analysis and the MDA are decomposed and arranged in a sequential man-ner, making a recursive loop. The efficiency of the SARAM method was verified usingtwo illustrative examples taken from the literatures. Compared with existing methods, itshowed the least number of subsystem analyses over the other methods while maintainingaccuracy.

I. Introduction

More attention has recently turned to the development of methodologies that synthesize uncertaintyanalysis and the multidisciplinary design systems.1 The techniques are optimal over the range of operatingconditions that a system may be subjected to while providing a desired level of reliability. It allows forrandom variation in the input variables and can also consider the error between model predictions andtrue system behavior. One of the earliest attempts to characterize the propagation of uncertainty for themultidisciplinary system was done by Ref. 2, it showed how to obtain the worst case linearized uncertaintyestimates for the multidisciplinary system with variation type uncertainties and model uncertainties. Gu3

developed implicit uncertainty propagation(IUP) for estimating uncertainties within bi-level optimization ofCollaborative Optimization and Simultaneous Analysis and Design framework. Since the methods are basedon worst case approximation in which the extreme value instead of uncertainty distribution are used, theygenerally give conservative results. Du4 proposed the system uncertainty analysis(SUA) and the concurrentuncertainty analysis(CSSUA) methods. The SUA and the CSSUA evaluate the mean and the variance ofsystem performance through uncertainty analysis under multidisciplinary design framework. And then themean and the variance are utilized to obtain optimal solutions based on robustness considerations. Becausethe methods evaluate only the mean and the variance of a performance distribution, they are generallyapplicable for applications where the first two moments of performance distributions are needed, such asrobustness design. Although the SUA, the CSSUA, and the IUP approach have shown good capabilities forincorporating decoupled multidisciplinary systems, they can give only the mean and the variance, or theinterval of system performance. Thus they are generally not rigorous to be used for formulating the designfeasibility under uncertainty.

To evaluate complete performance distribution incorporating probabilistic constraints, the reliabilityanalysis method is required. The term, the reliability analysis, is characterized by the use of random

∗Ph.D. student, Department of Aerospace Engineering†Professor, Department of Aerospace Engineering, Senior Member AIAA,jhkwon@kaist.ac.kr

1 of 11

American Institute of Aeronautics and Astronautics

10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference30 August - 1 September 2004, Albany, New York

AIAA 2004-4517

Copyright © 2004 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

variables which include the probabilistic distribution to describe the various sources of uncertainty.5 performReliability Based Optimization(RBO) of multidisciplinary systems. Since the reliability analysis has to beperformed for each constraint, the computational costs increases with the number of constraints. Thus, theyimplemented a constraint screening approach to reduce computational costs. Based on the literature, it canbe inferred that the reliability analysis is a critical component of MDO under uncertainty that demandsmuch more computational effort than deterministic MDO.

For this reason, many studies have been performed to integrate reliability analysis method into themultidisciplinary framework. The intuitive method for reliability analysis under multidisciplinary analysissystem is incorporating optimization algorithm with the Multidisciplinary Feasible(MDF) formulation. Inthis method, the optimizer surrogates the reliability analysis by minimizing the reliability index. By thenature of the MDF formulation, it has to perform a fixed-point iteration of subsystem analyses at each itera-tion of optimization to satisfy the interdisciplinary compatibility requirement. Therefore, a great number ofsystem analyses are involved in the process of optimization. To overcome the expensive computational bur-den, Du6 proposed the collaborative reliability analysis for multidisciplinary systems. In multidisciplinarycontext the framework corresponds to the Individual Discipline Feasible(IDF) formulation.7 They combinedthe reliability analysis with the IDF formulation in a single optimizer. The subsystem analyses interactswith the optimizer separately but there are no direct interactions among subsystems. The major advantageof the method is that the subsystem analyses can be conducted in parallel by which computational cost canbe reduced. Since the optimizer directly interacts with the individual subsystems to satisfy compatibilityrequirement, it has additional design variables and constraints. Accordingly, this may lead to more func-tions evaluation in large scale multidisciplinary systems. Ref. 5 employs the MPP-CSSO approach for thecalculation of the probabilistic constraints in RBO. The approach utilizes the concurrent subspace optimiza-tion(CSSO) framework since the reliability analysis is a kind of equality constrained optimization problems.Unlike the IDF, it does not require auxiliary variables or constraints for the compatibility.

In order to increase overall efficiency of reliability analysis under multidisciplinary systems, we proposea new strategy named the SARAM. The proposed method has an architecture in which the reliabilityanalysis and the MDA are decomposed and performed in sequential manner. The decomposed and sequentialarchitecture enables using the efficient AFORM and guarantees parallel computations of subsystem analyses.Recall that the existing methods have a nested architecture in which the reliability analysis and the MDAare tightly coupled. Therefore, they can not facilitate the recursive based reliability method. The AFORMrequires a small number of function evaluations compared to that of the conventional optimization methodin most cases.8 Furthermore, by decomposing the reliability analysis and the MDA, additional efficiencycan be achieved in both the reliability analysis and the MDA. As for the reliability analysis, the gradient oflimit-state function is obtained by the GSE that enables parallel execution subsystem sensitivity analyses.As for the MDA, since only the state variables are used to reach multidisciplinary feasibility, it is possibleto formulate parallel subsystem analyses structure. Our strategy has a similar point with that of the MPP-CSSO in that the sensitivity is obtained by the GSE. However, the SARAM performs the reliability analysisusing the AFORM by the direct calculation of subsystems. On the other hands, the MPP-CSSO utilizesthe CSSO framework in which subsystems are decomposed and approximated. Therefore, the reliabilityanalysis is conducted by the form of equality optimization problem instead of the AFORM. The efficiencyand accuracy of both strategies depend on the complexity of problems.

This paper is organized as follows. The nature of multidisciplinary system and problem formulations arereviewed in next section. And then reliability analysis methods for multidisciplinary systems are discussedwith the proposed SARAM. Two examples are used to demonstrate the effectiveness of the method.

II. Decoupling techniques in MDO

Multidisciplinary system analysis involves integrating subsystem analyses, which are coupled to oneanother through input and output. This coupling is represented by the state variables, which are variablesderived as output from one discipline but required as input to other disciplines. The state variables mustbe consistent across all disciplines to meet interdisciplinary compatibility requirement. Solving the coupledequations leads full multidisciplinary analysis usually called the MDA, in which the coupled subsystemsshould have consistency across all subsystems.

Figure 1 shows a typical two-discipline system taken from Ref. 6. Here x1 is a shared design variable xs

to subsystem 1 and subsystem 2, while x2, x3 and x4, x5 represent local design variables to subsystem 1

2 of 11

American Institute of Aeronautics and Astronautics

and subsystem 2, respectively. Variables y12 and y21 are the state variables, such that yij is an output ofsubsystem i to subsystem j. zi are output of subsystem i.

z2z1

x2, x3

y21

y12

Subsystem 2F2

Subsystem 1F1

x4, x5

x1

Figure 1. A typical two subsystems

The disciplinary input-output relations havethe following functional relationship:

zi = Fzi(xs,xi,yji)

yij = Fyij(xs,xi,yji) (1)

The coupled equations are usually solved withfixed-point iteration by assuming unknown statevariables and then the state variables are updatedby performing the subsystem analysis from whichthey are derived. This procedure is known as theMDF method. In a multidisciplinary context, theMDF optimization formulation is as follows

minx

f (x,y (x))

s.t. g (x,y (x)) ≤ 0 (2)

The function f represents the objective and g (x) is the constraint vector. In this formulation the onlyindependent variables are design variables, x, and the state variables, y(x) must be solved for every iterationin system optimization. The drawback of this formulation is the need to perform the MDA at each iterationof optimization, which can be computationally expensive.

A reduction in the overall computational time can be accomplished if subsystem analyses are done inparallel. In the IDF method, the auxiliary variables and consistency constraints are introduced to makeconcurrent subsystem analyses. The IDF formulation is given by

minx, a

f (x,a)

s.t. g (x,a) ≤ 0 (3)

aij = Fij (x,aji) for i = 1, · · · ,n, i 6= j

All state variables y(x) are replaced by the auxiliary variables a, and the additional constraints aij =Fij (x,aji) enforce interdisciplinary compatibility at the optimal solution not necessarily at each iteration.Since the subsystem interacts only the optimizer, this method enables distributed subsystem analyses, whichis a promising nature of the IDF method in applying reliability analysis.

III. Integrating Reliability analysis into multidisciplinary systems

Typical formulation of reliability optimization design has following form:

min f (d)

s.t. P [g (x) ≤ 0] ≤ pf (4)

where d is the mean of the random variable x, g (x) is the limit-state function which signifies failure tomeet the constraint, and pf is the probability of failure. A failure is defined when g (x) is less than 0. Theprobability of failure is the integration of the joint probability density function, fx(x) over the failure region:

pf =

∫

g(x)<0

· · ·∫

fx (x)dx (5)

The direct computation of pf by (5) is called full distributional approach. In general, evaluating the jointprobability density, fx(x) is almost impossible. Even if this information is available, evaluating the multipleintegral is extremely complicate. An alternative method is Monte Carlo Simulations(MCS). However, whenthe probability of failure pf is very small, the computational effort is extremely expensive. Therefore, oneapproach is to use analytical approximations of this integral that are simpler to compute. The analyticalapproximation methods contain FORM and First Order Second Moment(FOSM).

3 of 11

American Institute of Aeronautics and Astronautics

Based on the method of the FOSM,9 defined the reliability index β interpreted as the shortest distancefrom the origin to point on the limit-state surface in u-space. It reflects the point with the highest probabilityor Most Probable Point(MPP) at which the system failure occurs, that is

pf = Φ(−β) (6)

where Φ is the standard normal Cumulative Distribution Function(CDF). In this way, the problem in (5) isreduced to finding the point on the limit-state function g (x) with the smallest distance to origin, which isnothing else than a minimization problem with an equality constraint. Therefore, the optimization problemformulation yields

min β =√

uTu

s.t. g (u) = 0 (7)

where u is variables in the standard normal space which is transformed from the input random variables x.

u =x−µx

σx

(8)

where µx and σx is the mean and the standard deviation of random design variable x, respectively. Theconcept of the MPP, the limit-state function, and the joint probability density function are illustrated inFigure 2.

Figure 2. MPP for two random variables

Now we are supposed to evaluate the probability offailure of two coupled subsystems presented in Figure1. The limit-state function g (x) is given by Fz1, sothat the probability of failure is represented by

pf = P [z1 = Fz1 (x) ≤ 0]

= P [Fz1 (x,y21,y31) ≤ 0] (9)

The intuitive way of finding the MPP is substituting(7) and (9) into the form of the MDF formulation in(2). Figure 3(a) depicts the reliability analysis appliedto the MDF formulation for two coupled subsystems.In this method, the only independent optimizationvariables are standard normal variables u. To sat-isfy multidisciplinary feasibility, the state variables yij

should be solved for every iteration by the iterativemethod such as a fixed-point iteration method.

The MDF method is relatively easy to integrateexisting subsystem analyses and has a small number of design variables. However, the total number ofsubsystem analyses becomes the single set of subsystem analysis multiplied by the number of fixed-pointiterations for the MDA, then by the number of the optimization loops. And if finite difference is used toobtain gradient information, the convergence process would be repeated for each variable. As a result, agreat number of subsystem analyses are required.

To improve the efficiency of reliability analysis for multidisciplinary systems, Du6 proposed collabora-tive reliability analysis for multidisciplinary systems. In multidisciplinary context by Ref. 7 and 10, theframework corresponds to the IDF formulation. The IDF combines the MDA with the reliability analysisin a single optimizer to guarantee concurrent subsystem analyses. The schematic process is illustrated inFigure 3(b) where (7) and (9) is integrated into the form of IDF framework in (4).

The first equality constraint is the limit-state function and the others are compatibility constraints toensure that each and every state variable is equal to its corresponding auxiliary variable. The optimizereventually drives all of subsystems towards multidisciplinary feasibility by controlling auxiliary variables a.The drawback to the IDF method is that it requires a large number of optimization variables and constraints.As seen in Figure 3(b), the design variables of the optimizer contain the auxiliary variables a in addition tothe standard normal variables u. The number of auxiliary design variables is equal to the number of statevariables. The number of compatibility constraints is also equal to the number of state variables. Sinceall of the optimization process is conducted in a single optimizer, a large number of design variables andconstraints may lead difficulties in convergence.

4 of 11

American Institute of Aeronautics and Astronautics

y21

y12

u z1

Subsystem 1Analysis

Subsystem 2Analysis

( )

T

1

min

s.t. 0z

u u

u

=

=

(a) MDF

Subsystem 1 Subsystem 2

12F

y

21,u a 12,u a

( )

( )

T

1

ij ij ji

min

s.t. 0

, 0

z

F

u u

u

a u a

=

=

=

( )

( )

12 21

1 1 21

12,

,zz

yy F u a

F u a

=

=

( )21 1221,yy F u a=

21F

y

(b) IDF

Figure 3. Reliability Analysis using MDO framework

IV. Sequential Approach on Reliability Analysis Under MultidisciplinarySystems(SARAM)

The methods, both the MDF method in Figure 3(a) and the IDF method in Figure 3(b), employ acombined strategy in which the MDA is nested in the optimizer. The conventional nonlinear optimizer,such as Sequential Quadratic Programming(SQP) algorithm, performs subsystem analyses to minimize thereliability index satisfying constraints. Therefore, the optimizer surrogates the reliability analysis function inboth methods. Although applying optimization has showed good capabilities in reliability analysis,11 thereexist many efficient methods which are characterized by the recursive and function-approximation to theperformance function. It is known that the recursive and function-approximation based methods requirefewer computations at each step. Because the next point is computed using single recursive formular thatrequires only the value and the gradient of the limit-state function. The storage requirement is thereforeminimal. This method is also found to converge fast in many cases.

To utilize the recursive and function-approximation based reliability method, we decompose the reliabilityanalysis and the MDA. And then they are disposed sequentially to make recursive loop by which the efficientAFORM can be utilized. Figure 4 illustrates the generalized procedure of the proposed method, SARAM.It can be found that the MDA and the reliability analysis are globally making a recursive loop that issimilar to a reliability algorithm with a single analysis. The recursive loop makes it possible to employ theAFORM as the reliability analysis method. In this architecture, the MDA and the reliability analysis handletwo different sets of variables. The MDA updates only the state variables to meet the interdisciplinarycompatibility requirement. The random design variables given from the previous reliability analysis cycleare fixed in the MDA. While, the reliability analysis handles the random design variables with the fixed statevariables given from the previous MDA.

To distinguish between the inner loop to update state variables and the outer loop to update the MPP,we call the inner loop as iteration and the outer loop as cycle.

A. Reliability Analysis using AFORM

In reliability analysis,12 suggested an alternative Newton-Raphson type recursive algorithm usually knownas AFORM with which the safety index and the MPP are determined. The method linearizes the limit-statefunction at each cycle and uses derivatives to find next point. The AFORM update algorithm is given by

uk+1 =

[

∇ug (uk)T

uk − g (uk)]

∇ug (uk)

|∇ug (uk)|2(10)

where ∇ug (uk) is the gradient vector of limit-state function at uk, the k-th cycle point. Notice that in orderto calculate the MPP, the sensitivity information should be needed and therefore the GSE can be utilized.

5 of 11

American Institute of Aeronautics and Astronautics

Figure 4. Reliability Analysis by SARAM method

The GSE is a technique introduced by Sobieski13 fordetermining gradient information of coupled multidisci-plinary systems. The GSE approach defines the totalderivatives of output quantities in terms of local sensi-tivities. These local sensitivities are partial derivativesof each subsystem of outputs with respect to inputs. Toillustrate the mathematics of the GSE method, considerthe two-subsystem schematic presented in Figure 1. Thefunctional relationship can be expressed by expanding(1).

y12 = Fy12(x1,x2,x3,y21)

y21 = Fy21(x1,x4,x5,y12) (11)

z1 = FZ1(x1,x2,x3,y21)

Expanding (12) with first order Taylor series and express-ing in matrix form yields:

I 0 −∂y12

∂y21

0 I − ∂z1

∂y21

−∂y21

∂y120 I

dy12

dxdz1

dxdy21

dx

=

∂y12

∂x1

∂y12

∂x2

∂y12

∂x30 0

∂z1

∂x1

∂z1

∂x2

∂z1

∂x30 0

∂y21

∂x10 0 ∂y21

∂x4

∂y21

∂x5

(12)

This is called the GSE. The leftmost square matrix onthe left side of the GSE represents the isolated changesin each subsystem with respect to change in the othersubsystems. The matrix on the right hand side of theequation is the local changes in each subsystem with re-spect to design variables. Two matrices can be calculatedat each subsystem in parallel and all sensitivities are thendetermined by matrix inversion. Using the GSE, we can calculate the gradient of limit-state function withrespect to random input variables. Since we need only the value and the gradient of the limit-state functionin (10), there is no need to make the MDA in the AFORM update step.

We expect to calculate the probability of failure that z1 is less than 0. The gradient of limit-state functionwith respect to x is given by

∇xg (x) =dz1

dx(13)

where dz1/dx is calculated from the GSE. The gradient in the equivalent standard normal space, ∇ug (u)can be obtained by the chain rule of differentiation as

∇ug (u) =∂g

∂x

∂x

∂u= ∇xg (x) · σx (14)

By applying (14) to (10), we can calculate the updated values in the standard normal space uk+1. Thechanges in original space are obtained by multiplying the standard deviation of random design variables.

∆x = ∆u · σx (15)

B. Multidisciplinary Analysis

After improving design variables by the reliability analysis, the state variables should be updated by theMDA. The intuitive way is applying the fixed-point iteration method until the simultaneous equation (1) issolved. However, to guarantee concurrent subsystem analyses, we dispose subsystems in parallel and assumed

6 of 11

American Institute of Aeronautics and Astronautics

a set of state variables as initial variables. Each subsystem conducts its analysis by the input state variablesgiven from the previous cycle; then the analyses are performed in parallel. The calculated state variablesare then feed forward as the input of later analysis; this process continues until convergence is reached.At convergence, all the state variables are compatible and therefore the system satisfies interdisciplinarycompatibility requirement. One of the primal advantages of this approach is the parallel computation ofsubsystems. This approach exhibits characteristics which lie between the MDF method and the IDF method.Recall that the IDF performs subsystem analyses in parallel which reduces computational burden and theMDF method uses output values as input state variables for next iteration. In this approach, the number ofiterations in the MDA progressively decreases as the design variables approach the solution.

V. Examples

Two examples were tested to demonstrate the effectiveness of the proposed reliability analysis techniqueunder multidisciplinary systems. They include the simple two subsystems analytical problem and the foursubsystems aircraft conceptual design problem. The efficiency is measured by the number of subsystemanalyses required for convergence and the accuracy of converged solutions. The MCS were also conductedfor two test examples as the reference solution. The fixed-point iteration method was used to accomplishthe interdisciplinary compatibility in the MCS.

A. Example 1

This example, taken from Ref. 6, has two subsystems as shown in Figure 1. It has one shared variable, x1.The local variables are x2, x3 and x4, x5 for subsystem 1 and subsystem 2, respectively. Variables y12andy21 are the state variables. z1 and z2 are output of subsystem 1 and subsystem 2, respectively. As thelimit-state function is supposed to Fz1, the probability of failure is also given by expanding (9).

pf = P

[

z1 = Fz1 (x1, x2, x3,y21 (x1, x4, x5,y12))

= 5 −(

x21 + 2x2 + x3 + x2e

−y21)

≤ 0

]

(16)

All the random variables are assumed to have normal distribution. The mean values of the randomdesign variables x = {x1, x2, x3, x4, x5} are µx = {1, 1, 1, 1, 1}. The Coefficient of Variation(COV), the ratioof standard deviation over the mean, of all random variables is 0.1.

Four formulations were tested including the MCS, the MDF method in Figure 3(a), the IDF methodin Figure 3(b), and the proposed SARAM method in Figure 4. The results of reliability index β and thenumber of subsystems analyses are compared in Table 1. The converged MPP solutions are compared inTable 2. The initial values of input state variables are obtained through the MDA before actual reliabilityanalysis is started.

Table 1. Results of MPP for Example 1

method xMPP = {x1, x2, x3, x4, x5}

MCS 1.2001, 1.2299, 1.0984, 1.0735, 1.0060

MDF+SQP1 1.2348, 1.1901, 1.0951, 1.0000, 1.0000

IDF+SQP1 1.2348, 1.1901, 1.0951, 0.9991, 1.0000

SARAM 1.2348, 1.1901, 1.0951, 1.0000, 1.0000

1 SQP method is used for optimization

From Table 1, the MDF, the IDF, and the SARAM produce almost the identical results. The results ofreliability index β are also very close to those of the MCS in Table 2. For the number of subsystem analyses,the SARAM produced 148 while the one of the IDF method and of the MDF method produced 204 and 242,respectively. The SARAM has the least number of subsystem analyses among the methods. As for Example1, the proposed method appears to be computationally efficient. Since, as mentioned before, the convergencecharacteristics are not exactly the same between the nonlinear optimization algorithm based method and

7 of 11

American Institute of Aeronautics and Astronautics

Table 2. Results for Example 1

method β NSA1 Notes

MCS2 3.1633 500003

MDF+SQP 3.1671 242 Figure 3(a)

IDF+SQP 3.1671 204 Figure 3(b)

SARAM 3.1671 148 Figure 4

1 Number of subsystem analyses2 95 % Confidence interval of β is (3.0830, 3.2713)3 Number of system level analyses

the SARAM method, the number of subsystem analyses may change according to the efficiency of optimizer.To clarify the computational efficiency of proposed method, we tested more complicate example in the nextsection.

B. Example 2

Cruise Range

ESF

WTOTAL

WFUEL

Lift/Drag

WENGINE

SFC

WTOTAL ,

Drag

Lift

AerodynamicLocal DV: Cf

PropulsionLocal DV: T

Performance

StructureLocal DV: , CX

Shared DV: tc, h, M, AR, , S

Figure 5. Aircraft Design Example 2

The second example is a supersonic business jet designthat model corresponds to the problem used by NASAto present BLISS.14 It is composed of four subsystems,aerodynamic analysis, structural analysis, propulsionanalysis and performance analysis. The first 3 subsys-tems are fully coupled since they share common designvariables and exchange state variables, and the fourthsubsystem receives state variables from the others tocalculate the cruise range of the aircraft. Figure 5shows the functional relationship of the example.

This example includes 6 shared design variables,4 local design variables, and 9 state variables. Weassume that uncertainties are associated with all 10input design variables which are consisted of 6 shareddesign variables and 4 local variables. The design vari-ables are described by normal distributions and theCOV of all random variables is 0.05. The mean valueof the input design variables are as follows:

µx = {λ,Cx, Cf , T, tc, h,M,AR,Λ, S}

where

λ = 0.25, taper ratio of wing

Cx = 0.98, wing box cross section

Cf = 0.985, skin friction coefficient

T = 0.47, throttle setting

tc = 0.05, thickness to chord ratio

h = 5.7×104, cruising altitude (ft)

M = 1.22, Mach number

AR = 5.27, aspect ratio of wing

Λ = 60, sweep angle of wing (deg)

S = 1000, wing area(

ft2)

The deterministic value of the cruise range calculated at the mean input design values is approximately1500nmi.

8 of 11

American Institute of Aeronautics and Astronautics

In this example, 1000nmi of cruise range is chosen for the value of limit-state. Therefore, the probabilityof failure that can not meet the limit-state of cruise range, 1000nmi is given by

pf = P [CruiseRange − 1000nmi ≤ 0] (17)

Table 3 shows the results of the reliability index and the number of subsystem analyses required to obtainthe MPP.

Table 3. Results for Example 2

method β NSA Notes

MCS1 -2.3632 500002

MDF+SQP -2.3359 2728 Figure 3(a)

IDF+SQP -5.9745 2168 Figure 3(b)

SARAM -2.3429 540 Figure 4

1 95% confidence interval of β is (2.3305, 2.3986)2 Number of system level analyses

The values of the reliability index from the proposed SARAM method are very close to the one fromthe MDF method and from MCS. While, the value from the IDF method is greater than the others. TheIDF method seems to fall into the local minimum. Although many different variations including changesof the bounds, reformulations of constraints, and normalization of design variables were tested on the IDFmethod, none led to better results. It seems that the increased number of design variables and additionalconstraints in the IDF method hampers to reach the minimum solution. In this example, the IDF methodhas 19 design variables and additional 9 constraints while the MDF method and the SARAM method hasonly 10 design variables. It is known that the method has practical limitations for complex problems dueto the interdisciplinary coupling formulations.10 A definite cause of the local convergence calls for furtherinvestigation.

To reconfirm the accuracy of the proposed method, we calculated the higher reliability region of Example2. The results of the reliability index with respect to the limit-state values are displayed in Figure 6(a). Thelimit-state values in (17) are changing from 500nmi to 1000nmi. It can be seen that the SARAM methodproduced nearly the identical results with those of the MDF method and the MCS. On the other hand, theresults of the IDF method are far below than the others, which means that it fell into the local minimum.The results in Figure 6(a) and Table 3 clearly show that the proposed SARAM method can be reliable fora complex multidisciplinary problem and a high reliability region.

As for the cost of computation, the total number of subsystem analyses for the SARAM method isonly 540, while the one for the MDF method is 2728 and the IDF method is 2168. The SARAM methodproduced more than five-folded reduction and three-folded reduction over that of the MDF method and theIDF method, respectively. The reduction of the number of subsystem analyses is mainly caused by the useof AFORM which requires only the value and gradient of limit-state function at each cycle. However, theconvergence characteristics between the SARAM method and the existing methods are different, we illustratethe history of convergence of three methods in Figure 6(b).

It shows the history of reliability index β according to the number of subsystem analyses. In spite ofthe different nature of three methods, we can confirm that the SARAM method has the fastest convergencenature. The results indicate that the substantial efficiency exists in the proposed SARAM method. Fur-thermore, since all the subsystem analyses are disposed in parallel, we have an opportunity to establish aparallel computing environment.

VI. Conclusion

The purpose of this research is to develop an efficient strategy for reliability analysis under multidisci-plinary analysis systems. The measure taken by the proposed strategy is separating the reliability analysisand the MDA. The decomposed architecture enables to employ the efficient AFORM instead of nonlinear

9 of 11

American Institute of Aeronautics and Astronautics

(a) Higher reliability region (b) Convergence History

Figure 6. Result of Example 2

minimization algorithm. Furthermore, the use of the GSE to get the gradient of the limit-state function en-ables parallel execution of subsystem sensitivity analyses. The subsystems for the MDA are formulated alsoin parallel to guarantee concurrent subsystem analyses in which the state variables are recursively updateduntil the interdisciplinary compatibility requirements are satisfied. The design variables are updated by thegradient information in the AFORM; and then the updated design variables are used to the MDA to updatethe state variables. By doing this procedure, the design variables are progressively converging to the MPPmaintaining concurrent analysis of subsystems.

The proposed SARAM method has several advantages over the existing reliability analysis method.

• The method can utilize the AFORM which enables a small number of subsystem analyses at each stepand fast convergence.

• The method performs concurrent subsystem sensitivity analyses by using the GSE.

• The subsystem analyses in the MDA are also conducted concurrently.

The tests showed that the proposed SARAM method conducts the least number of subsystem analysesover the existing reliability method under the multidisciplinary system while accuracy is maintained.

It should be noted that this research is concerned with the reliability analysis under the multidisciplinarysystem not with MDO under uncertainty. The proposed method can be used to evaluate probabilisticobjectives and probabilistic constraints in MDO under uncertainty. Further research may be explored forincorporating the proposed method with MDO architecture efficiently.

References

1Zang, T.A., e., “Needs and Opportunities for Uncertainty-Based Multidisciplinary Design Methods for Aerospace Vehi-cles,” 2002.

2Gu, X. e., “Investigation of Multidisciplinary Design Subject to Uncertainty,” Aiaa paper 98–4747, 1998.3Gu, X. and Renaud, J., “Implementation Study of Implicit Uncertainty Propagation(IUP) in Decomposition-Based

Optimization,” Aiaa paper 2002–5416, 2002.4Du, X. and Chen, W., “Efficient Uncertainty Analysis Methods for Multidisciplinary Robust Design,” Vol. 40, 2002, pp.

545–552.5Padmanabhan, D. and Batill, S. M., “Decomposition strategies for Relaibility Based Optimization in Multidisciplinary

System Design,” Aiaa paper 2002–5471, 2002.6Du, X. and Chen, W., “Collaborative Reliability Analysis for Multidisciplinary Systems Design,” Aiaa paper 2002–5474,

2002.7Alexandrov and Lewis, “Algorithm Perspectives on Problem Formulation in MDO,” Aiaa paper 2000–4719, 2000.

10 of 11

American Institute of Aeronautics and Astronautics

8Halder, A. and Mahadevan, S., “Reliability Assessment Using Finite Element Analysis,” John Wiley& Sons, New York,2000, pp. 71–86.

9Hasofer, A. and Lind, N., “Exact and Invariant Second-moment Code Format,” Vol. 100, 1974, pp. 111–121.10Mahadevan, S. and Smith, N., “System Risk Assessment and Allocation in Conceptual Design Analysis,” 2003.11Liu, P. and Der Kiureghian, A., “Optimization Algorithm for Structural Reliability,” Vol. 9, 1991, pp. 161–177.12Rackwitz, R. and Fiessler, B., “Structural Reliability under combined Random Load Sequences,” Vol. 9, 1978, pp.

489–494.13Sobieszczanski-Sobieski, “Sensitivity Analysis and Multidisciplinary Optimization for Aircraft Design: Recent Advances

and Results,” Journal of Aircraft , Vol. 27, No. 122, 1990, pp. 993–1001.14Sobieszczanski-Sobieski, J., “Bi-Level Integrated System Synthesis(BLISS),” 1998.

11 of 11

American Institute of Aeronautics and Astronautics