MDO formulations: an evaluation study

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MDO formulations: an evaluation study

Renaud, G.; Shi, G.

MAIN Se..-TL501 R428 LTR-SMPL-2001-0264 National Research Council Canada Conseil national de recherches Canada

Institute for lnstitut de

Aerospace Research recherche aerospatiale

taC·CtaC

Unclassified Unlimited

MDO Formulations: An Evaluation Study

LTR-SMPL-2001-0264

G. Renaud and G. Shi

INSTITUTE FOR AEROSPACE RESEARCH Pages: 26 Tables: 15 Figures: 7 For: Pour: Reference: Reference:

REPORT

RAPPORT

STRUCTURES, MATERIALS AND PROPULSION LABORATORY

LABORATOIRE DES

STRUCTURES, DES MA TERIAUX ET DE PROPULSION INSTITUT DE RECHERCHE AEROSP A TIALE Report No.: L TR-SMPL-200 1-0264 Date: Dec. 2001 Classification: Unclassified Distribution: Unlimited

L TR-SMPL-200 1-0264

MDO Formulations:

An

Evaluation Study

Submitted by:

Presente par: D.L. Simpson

Director/Directeur

Approved by:

Approuve par: W . Wallace

Director General/Directeur general

THIS REPORT MAY NOT BE PUBLISHED WHOLLY OR fN PART WITHOUT THE WRITTEN CONSENT OF THE INSTITUTE FOR AEROSPACE RESEARCH.

Author(s): Auteur(s):

G. Renaud G. Shi

CE RAPPORT NE DOlT PAS ETRE REPRODUIT, NI EN

ENTlER NI EN PARTIE, SANS UNE AUTORISATION ECRITE DE L 'fNSTITUT DE RECHERCHE AEROSPA TIALE.

MDO Formulations: An Evaluation Study

Guillaume Renaud and Guoqin Shi Structures, Materials and Propulsion Laboratory

Institute for Aerospace Research National Research Council Canada

Ottawa, On KIA OR6

ABSTRACT

Several MDO formulations have been proposed in the literature to overcome the difficulties of optimizing coupled engineering systems. The objective of this study is to compare some ofthese formulations to determine which ones should be envisaged for establishing a practical MDO capability. The selected methods are the Multidisciplinary Feasible method (MDF), the Individually Feasible method (IDF), the Collaborative Optimization (CO), the Concurrent SubSpace Optimization with Response Surfaces (CSSO/RS), and the Bi-Level Integrated System Synthesis with Response Surfaces (BLISS/RS). The methods are tested with two demonstrative examples. The results show important differences in the behavior of each method, especially concerning reliability and efficiency. The MDF, IDF, and BLISS/RS formulations appear to be viable options. For the considered implementation, the CO and CSSO/RS formulations were unable to solve the problems in several cases. It appears, for theoretical and practical reasons, that the formulation that should be retained is the BLISS/RS. Modifications are proposed to improve the models when approximated global constraints are used. Further investigations concerning neural networks are recommended.

1. Introduction

Multidisciplinary design optimization (MDO) is a methodology developed to address the computational and organizational complexity of complicated engineering systems. The objective of a MDO process is to determine the design that optimizes a certain system performance measure while considering the effects of several mutually interactive disciplines. Conceptually, many approaches can be taken to address this type of problem.

Several MDO formulations have been proposed in the recent years. Each proposes its own way of decomposing the problem into various subtasks, resulting in specific characteristics concerning its particular ease of implementation and use, complexity, robustness, and computational efficiency. Moreover, some of these formulations permit the use of autonomous concurrent disciplinary optimizations, allowing parallel computational work for each of the disciplines. With these decomposition approaches, each team of disciplinary experts can be in charge of its own aspects of the project and totally free to choose the set of methods that are best suited for its particular subsystem [1].

In this study, five existing MDO methods are described, evaluated and compared. They are the MultiDisciplinary Feasible (MDF) method [2] , the Individual Discipline Feasible (IDF) method [3], the Concurrent SubSpace Optimization with Response Surfaces (CSSO/RS) [4], the Collaborative Optimization (CO) [5] , and the Bi-Level Integrated System Synthesis with Response Surfaces (BLISS/RS) [6]. Furthermore, several variations of these methods are tested. The objective of this work is to determine which methods are most promising for establishing an aerospace MDO capability at the Institute for Aerospace Research (IAR) of the National Research Council (NRC) of Canada. The selected and developed MDO strategies will be applied to an aircraft wing design optimization with structures and aerodynamics interaction. It

is intended, as an initial step, to implement such a capability in the MATLAB programming environment [7], coupled to various disciplinary analysis codes. For this reason, the evaluation presented in this paper has been entirely performed using MATLAB and its optimization toolbox [8].

2. Multidisciplinary design analysis and optimization

In general, three types of variables can be defined in a multidisciplinary problem. The first type corresponds to the global or shared input variables z, required by more than one discipline or by system-level calculations. Conversely, disciplinary or local input variables x; are used for calculations concerning the i1h discipline only. Finally, state or behavior variables y;,

corresponding to the disciplinary responses, are subsystem output values that can be used as input parameters to other disciplinary calculations as, for aN-discipline problem,

Yi = y;(X;,yj,z), j = 1,. 0 .,N, j "# i.

The analysis of a multidisciplinary system consists in determining the disciplinary state parameters y that correspond to a specific set of local and shared input parameters x and z. The conventional solution approach results in a sequence of iterations between the various disciplinary analyses. At convergence, all state variables are compatible and the system is in a so-called state of multidisciplinary feasibility.

The optimization of such a system is the process of determining the values of x and z, within certain limits, that minimize or maximize an objective function j(z,y(x,y,z)) representing the system performance, potentially subjected to some global constraints. Furthermore, each subsystem is also subjected to local constraints g;. To be acceptable, the final design must be multidisciplinary feasible, must satisfy all constraints, and must utilize the same values for the shared variables z in all subsystems. A typical multidisciplinary analysis is sketched in Figure 1 for a three-discipline system.

! Initialize y

I

J ' I x,y, z I L j Update y

f l

I

I

SYSTEM ANALYSIS ._], -.[, ..;,

I

Subsystem 1 Subsystem 2

I

Subsystem 3

analysis analysis analysis

I

yl (xl,z,y2,y3) yix2,z,y 1,y3)

I

y,(x3,z,y,,y2)

I I I

ｃｯ ｮｶ ･ｲ ｧ･､ｾ＠

No

Yes

Figure 1. Multidisciplinary analysis.

As mentioned in the introduction, there exist several approaches to formulate multidisciplinary system optimization. Each is based on a different concept, resulting in a certain level of complexity and efficiency. The formulations consist in sequences of subtasks including disciplinary analyses, disciplinary and system sensitivity analyses, optimization and approximations at the subsystem (discipline) and system (coordination) levels. The next section briefly describes some of the most cited techniques in aerospace MDO.

3. MDO Formulations

It was stated in the introduction that several approaches exist for optimizing coupled problems. This section briefly describes the five MDO formulations evaluated in this study. For more details, the interested reader should consult the references included within each description.

The MultiDisciplinary Feasible (MDF) or All-in-One (A-i-0) [2] method is the simplest way of performing MDO. The multidisciplinary problem is fully solved by conventional iterative methods at each optimization step, resulting in a system that is always multidisciplinary feasible.

It is not a decomposition method as such and does not exploit the modularity of the problem. Therefore, all variables and constraints are treated at the global level. The main drawback of MDF is its expense, especially if system derivatives are calculated with finite differences, where a full multidisciplinary analysis has to be performed for each design variable perturbation.

The MDF optimization can be stated as

mm1m1ze subject to

j(z,y(x,y,z)) g(z,y(x,y,z)) ::; 0

where

f

is the objective function and g represents all system and/or disciplinary constraints. Figure 2 schematizes this method.

SYSTEM ANALYSIS (Fig. 1)

Figure 2. Multidisciplinary feasible method.

The Individual Discipline Feasible (IDF) [2][3] method uncouples the disciplinary analyses but keeps the optimization at the system level to form a single-level decomposed optimization problem. The subsystems are individually analyzed and the optimization is performed for the system as a whole with constraints imposing multidisciplinary feasibility, using extra coupling variables that are introduced in the formulation. The disciplines are always feasible individually but the complete system may not be feasible until the optimization process converges.

The IDF optimization statement is

m1mm1ze subject to

j(z,y(x,y' ,z))

g(z,y(x,y' ,z))::; 0

y' - y(x,y' ,z) = 0

where y' are auxiliary disciplinary input variables corresponding to the various state variables. The second constraint ensures multidisciplinary feasibility when y is equal to y'. The IDF procedure is illustrated in Figure 3.

w

Subsystem 1 analysis I

w

Subsystem 2 analysis I

"'

I ConvergedO No J Yes 'V -.lt Subsystem 3 analysis I I

Figure 3. Individually feasible method

The Collaborative Optimization (CO) [2],[5] ,[9],[ 10],[11 ], which decomposes and reformulates the problem as a bi-level optimization, is one of the most studied and documented MDO formulations . The system is optimized at the coordination level by determining target values for subsystem responses and shared design variables, with compatibility constraints ensuring multidisciplinary feasibility. The optimization objectives of the subsystems are to match as closely as possible these target values while satisfying local disciplinary constraints. As in the IDF method, multidisciplinary feasibility is achieved at the end of the process. If the target

values corresponding to the shared and state v ariables are z andy', respectively, the system level

optimization problem can be written as

m1mm1ze subject to

j{z,y')

c(z,/ ,y' ,y(x/ ,y' ,z/)) = 0

where c represents the compatibility constraints, one for each subsystem, of the form

( ·)2 (y' ( • ' .))2

C; = Z - Z;

+

-

y; X; ,y ,Z;

where the asterisks indicates optimal subsystem values.

Similarly, at the disciplinary level, the i1h subsystem optimization can be stated as

m1mm1ze subject to

c;(z,z;,y' ,y(x;,y' ,z;))

g(x,z,y(x,y,z)) ｾ＠ 0

where the objective c;is ofthe same form as the constraints at the global level. The CO procedure can be illustrated as in Figure 4 .

Subsystem 2 analysis Subsystem 3 analysis

ｾ＠

l

OPTIMIZER

I I

OPTIMIZER

＾Ｍ ＭＭＭＭＧ

Ａｃｯｮｶ･ｲｧ･､ｾ＠

Converged? >-No_ ----' Yes I System evaluation

I

OPTIMIZER

I

Yes ｃｯｮｶ･ｲｧ･､＿ ｾ＠ No ＾ＭＭＭＭ ＭＭＭ ＭＭＭＭＭＭＭＭＭＭＭｾ＠ Yes -.v

Figure 4. Collaborative optimization.

The Concurrent SubSpace Optimization with Response Surfaces CCSSO/RS) [ 4] is also a true decomposition strategy. The system and subsystems are optimized sequentially, using their specific objective functions, constraints, and variables. A specific system performance is approximated in each subsystem optimization, using Response Surface (RS) models for simulating other disciplines state variables. Similarly, the system is optimized at the coordination level using RS models replacing the required disciplinary analyses. As the analyses and optimizations are performed, more information is known about the actual system, permitting to update the various models. The i1h discipline optimization can be stated as

m1mm1ze subject to

j{ z,y; x ;,yj ( app ) app) ,z; ,yj

g,{X;,Z,Y,{X;,y/PP ,Z;),y/PP) ｾ＠ 0

where y/ PP=y/PP(z,xj) represents the other disciplines approximated state responses. A complete multidisciplinary analysis is performed for each subsystem optimal design to generate a set of multidisciplinary feasible designs that will improve the quality of the approximation models. A system optimization is then performed as

m1mm1ze subject to

f{z,yapp) g(z,yapp) ｾ＠ 0.

Another multidisciplinary analysis is performed with the system optimal design to improve further the models, and the whole process is repeated until convergence, as schematized in Figure 5.

t

SYSTEM ' ANALYSIS (Fig. 1) SYSTEM ANALYSIS K- -- - ----, (Fig. 1) MODEL UPDATE

t

System approx. 2 ｾ＠ OPTIMIZER

I

Yes I SYSTEM ANALYSIS (Fig. 1) System

I

approximation ｾ＠

l

I

OPTIMIZER

I

Converged?

I

No _>-:-=---- ---' Yes No Yes

y

System approx. 3 SYSTEM ANALYSIS ! (Fig. 1) ｾ Ｌ＠

Figure 5. Concurrent subspace optimization with response surfaces.

Finally, in the Bi-Level Integrated System Svnthesis with Response Surfaces CBLISS/RS) [6] each subsystem is optimized with respect to local variables, holding the shared variables constant, to minimize the system objective under local constraints. The shared variables are utilized by the system level optimization only. Total derivatives (ref) are used to predict the

effects of each set of variables on the objective function. The optimization of the

/h discipline

takes the form

mm1m1ze subject to

D(f ,x,) T !:J. X;

g,{x;) ｾ＠ 0

where D(f ,x;) T is the total derivative of the objective function with respect to the local variables

of the disciplines. It includes the indirect effects of these variables on other subsystems. The term D(f,x, / Ax; corresponds to the first order predicted objective function change due to a change in x ;. The optimization at the system level takes the same form with the shared variables, using RS approximation models for system performance estimation. A simplified flowchart of the BLISS/RS is presented in Figure 6.

SYSTEM ANALYSIS r - - - ' ) 1 (Fig. 1) Total derivatives evaluation Yes y

MODEL UPDATE

System approximation

I

OPTIMIZER

I

ｃｯｮｶ･ｲｧ･ ｾ Ｌ［ＮＮ［ＮＮｎｯ ＧＭＭＭ Ｍ ＭＭ 1 Yes Discipline 2 evaluation Discipline 3 evaluation

"j

I

I

ｏｐｔｾｅｒ＠

Ill

＾Ｍ Ｍ｟｟［

ｃｯｮｶ･ｲｧ

ｾ＠

I

I Yes I

I

I

ｾＭＭ ＭＭＭＭ ｾ ｖａｒｉａｂｌｅｓ＠ ｕｐｄａｔｅ ｾＭＭＭＭＭＭｾ＠

Comparison of features

All the approaches listed above possess different characteristics with respect to their applicability and practicability. Next is a series of key points that can be considered when selecting a formulation among the proposed ones.

One important feature that the chosen formulation can offer is the possibility of decomposing the problem in lines of individual disciplines to allow potentially dispersed experts to perform their tasks concurrently. Ideally, the formulation should provide discipline autonomy with respect to subsystem analyses and optimizations.

Moreover, it is desirable that the chosen formulation ensures constant multidisciplinary feasibility. This permits the termination of the optimization work at anytime for latest design acceptation or for human intervention in the process.

Finally, it is a good feature to have relatively few design variables at the system level. This permits the effective use of an approximation technique, such as Response Surface Methodology (RSM), at the coordination level.

The formulations are compared in Table 1 with respects to the requirements listed above. It should be noted that the effectiveness or the ease of implementation is not considered in this comparison. Such evaluations, requiring the testing of the methods on real MDO problems, will be presented in the next section.

REQUIREMENT MDF IDF

co

CSSO/RS BLISS/RS

Discipline autonomy: analysis X X X X

Discipline autonomy: optimization X X X

Constant multidisciplinary feasibility X X X

Few design variables at the system level X

Table 1. Comparison of MDO formulations.

It should be noted that the original versions of the CSSO [12] and BLISS [13],[14],[15 ] do not include Response Surface methodologies. This addition is however expected to be particularly efficient. Indeed, such approximations usually improve the convergence characteristics if the number of parameters used to build the approximation stays limited. BLISS/RS is particularly promising because the number of system variables is kept low, resulting in an effective use of such an approximation method.

4. Evaluation

In order to make direct comparisons, all formulations were implemented in a MATLAB environment, with the same general characteristics. For instance, the same procedure was invoked each time a multidisciplinary analysis was required. Also, if not otherwise stated, the same bounds and initial values were taken for the various variables.

Two examples were implemented to test the five formulations. The efficiency of each method is evaluated by the number of subsystem evaluations required to obtain a solution. It is assumed that the number of disciplinary analyses is the most determining factor concerning computational efficiency.

Example 1: Simple two-discipline problem

The first optimization problem corresponds to the test case that was used to demonstrate the capabilities of CSSO/RS [ 4] . It consists of two simple subsystems, coupled through shared and state variables. The problem can be stated as

m1mm1ze

subject to the constraints

and to the side constraints

ＭＱ Ｐｾｸ _Ｑ＠ ｾ ＱＰ＠ Ｐｾ＠ x2 ｾ＠ 10

ｏ ｾｸＳｾＱＰ＠

and where the state variables y1 and y2, evaluated in subsystems 1 and 2, are defined as

y1

=

ｸｾ＠ + x2 + x3 - 0.2y2 1/ 2

Y2 = Y1 + X1 + X3 •

The y1a and Y2a parameters were not given in the aforementioned reference. The following

values were used, as they result in the same solution as in the paper.

Implementation

Y1a = 8

Yza

=

24.

Several decisions were taken concerning the implementation of the methods for solving this problem. The following points were adopted for all methods:

1 All three input variables were treated as global variables. Indeed, they are either used in both disciplines (x1, x3), or used in the global objective function (x2, x3) . Consequently,

2 The termination criterion for multidisciplinary analysis (Fig. 1) is that the change of each Yi between two consecutive iterations must be lower or equal to 0.0001.

3 The global objective function was evaluated outside the disciplines, at the coordination level.

4 The upper and lower bounds for the state variables (IDF, CO), were set to positive and negative infinity, respectively. This avoids any beforehand estimation of the final values of these variables.

5 No scaling was made. The variables were used as specified.

6 The initial values of y were obtained, when required, through a full multidisciplinary analysis. The number of needed disciplinary analyses was included in the total number of subsystem evaluations.

MDF

MDF optimization was performed on the problem as stated above.

IDF

Two formulations were used to test IDF optimization. The differences between the formulations lie in the way they' additional state variables are forced to converge to the actual state variables. Table 2 describes the constraint formulations.

Method Constraint formulation IDFl yl- yl ' = 0

y2- y2 ' = 0 IDF2 _{(yl- yl ' /} ｾ＠ 0

(y2- y2' )2 ｾ＠ 0 Table 2. IDF formulations.

co

Four formulations were used to test CO. As with IDF, equality and inequality constraints were tested, as well as analytical gradients (ref) instead of finite differences for system constraints. Table 3 describes the constraint formulations.

Method Constraint formulation Constraint gradient calculations

COl c:::;o finite differences

C02 c:s;O analytical

C03 c=O finite differences

C04 c=O analytical

Table 3. CO formulations.

CSSOIRS

It is possible to use several techniques to build the model approximations required in the CSSO. The chosen method corresponds to the response surface method proposed in reference [ 6] and consists in three steps:

1) Build a linear approximation using the initial design and a perturbation of the initial design for each design variable. This step necessitates as many multidisciplinary analyses (Fig. 1) as there are x and z variables, plus one.

2) Construct a quadratic approximation with the additional information coming from the various local and global optimizations. The diagonal terms of the quadratic polynomial are calculated first.

3) Work with a quadratic best fit when the number of evaluated designs exceeds the required number for establishing a quadratic approximation.

Furthermore, decisions have to be made on the trust region on the various approximation models . Two formulations, described in Table 4, were retained for comparison purposes. For both formulations, the termination criterion was set as the change off between two consecutive iterations must be lower or equal to 0.0001.

Method Characteristics

CSSOl - Perturbation of +50% of each variable to build linear approximation.

- Move limit of± 50% of each variable for system and subsystem optimizations. CSS02 - Perturbation of +50% of each variable to build linear approximation.

- Move limited by the maximum design space defined by the maximum and minimum variable values used so far.

Table 4.- CSSO/RS formulations.

BLISSIRS

Results

The results from the implementations defined above are summarized in Table 5. The number of required disciplinary analyses is written in the last column as "Nda".

Method XI X2 XJ YI Y2 I 2 Nda MDF 8.0029 3.0284 0.0000 0.0000 8.0000 5.8569 0.0000 -0.7560 290 IDFl 8.0029 3.0284 0.0000 0.0000 8.0000 5.8569 0.0000 -0.7560

88

IDF2 8.0029 3.0284 0.0000 0.0000 8.0000 5.8570 0.0000 -0.7560 170 COl C02 8.0027 3.0283 0.0000 0.0000 7.9999 5.8568 0.0000 -0.7560 2075 C03 10.0519 3.3293 0.1833 0.0270 9.9899 6.5172 -0.2487 -0.7285 6270 C04 8.0029 3.0282 0.0015 0.0000 8.0000 5.8567 0.0000 -0.7560 2432 CSSOl 8.0029 3.0284 0.0001 0.0001 8.0000 5.8569 0.0000 -0.7560 1205 CSS02 8.0029 3.0284 0.0001 0.0001 8.0000 5.8569 0.0000 -0.7560 936

Table 5. Example 1 results.

It can be seen that, although most of them result in the same solution, the formulations show substantial differences in terms of their numerical efficiencies. It appears from that study that the most efficient method is IDF1 , for which the number of disciplinary analyses was only 88.

Furthermore, it is seen that the Collaborative Optimization (CO) based on finite differences failed to converge in one case, and converged to a wrong solution in the other. It is also seen that even when it worked, CO was the slowest formulation to converge, with at least 2075 analyses.

Example 2: Supersonic business jet preliminary design (four disciplines)

The second optimization problem corresponds to the problem used by NASA to present BLISS [13]. It consists in a maximization of the range of a supersonic business jet. Four coupled subsystems representing "structures", "aerodynamics", "propulsion", and "range" by semi-empirical relations are used to determine the multidisciplinary state of the aircraft. The first three disciplines are fully coupled, as they share common variables and exchange computed values, and the forth discipline receives information from the others to evaluate the performance of the design.

The various disciplines include 6 shared variables, 4 local variables, and 9 disciplinary state variables that act as input variables to other disciplines. The performance value is treated as a 1Oth state variable, evaluated by the "range" discipline. Table 6 summarizes the characteristics of these variables.

Variable Name Definition Lower bound Initial value U erbound

z(1) tic thickness to chord ratio 0.01 0.05 0.09

z(2) h altitude 30000 45000 60000

z(4) AR aspect ratio 2.5 5.5 8.5

z(5) A wing sweep 40 55 70

I

z(6) Sref wing surface area 500 1000 1500

I

XI (I) /... wing taper ratio .10 .25 .40

_I

XI(2) X wing box x-section area 0.75 1.00 1.25

I

xz(1) Cr coefficient of friction 0.75 1.00 1.25

I

X3(1) T throttle setting 0.1 0.5 1.0

YI(1) WT total aircraft weight 5.1417e+4

YI(2) WF fuel weight 7.3062e+3

YI(3)

e

wing twist 1.0000

yz( 1) L lift 5.1417e+4

I

Yz(2) D drag 1.3479e+4

I

yz(3) LID lift to drag ratio 3.8146

y3(1) SFC specific fuel consumption 1.1075

Y3(2) WE engine weight 7.0592e+3

Y3(3) ESF engine scale factor 0.5558

Y4(1) R range 484.0863

Table 6. Variables in Example 2.

It should be noted that there are a number of discrepancies between the equations or references [6],[13][14],[15] and the program presented at the end of reference [13]. Furthermore, an error was reported [16] in the program itself. Table 7 lists the relations used in this study for the various disciplinary calculations. They correspond to the program of reference [13], with the above-mentioned correction. STRUCTURES b / 2

=

ｾｓｲ･ｦ

ａｒ＠

12; 0 = pf(x,b i 2,R,L); Fo = pf(x) R

=

1+2A-3(1 +A-) Ww = 0.0051(W7N2 ) 0557

Sre/649 AR0'5(t I c ) -oA (1 + A-)0'1 (O.l875Sref )0'1 Fo

cos( A) WFW

=

(5Sref I 18)(2! / 3)42.5; WF

=

WFW + WFO W₇ = W₀ + Ww + WF +WE; 0"1 ｾ＠ a5 = pf(t I c,L,x,b i 2,R) Constraints 0"1 ｾ＠ O"s ｾ＠ 1.09 096 ｾ＠

e

ｾ＠ L04

AERODYNAMICS if h < 36089ft: V

=

Mlll6.39.Jl- (6.87e- 06)h p

=

(2.3 77 e - 03)(1- ( 6.87 5e- 06h)) 4·2561 if h > 36089ft: V

=

M968.1 p = (_{2.3 77} e _ ₀₃)e - <h -36089)120806.7 Fol

=

pf(ESC,C₁) CDmin

=

CDmin,M<IFol + 3.05(t I c) 513 cos(A)312 k

=

1 I (n-0.8AR); Fo2

=

pf(0) CD = (CDmin + kC/)Fo2; L

=

WT D=CD0.5pV2Sref ; dp l dx=pf(t l c) Constraint dp l dx ｾ＠ 1.04 PROPULSION T

=

Tl6168.6; Temp = pf(M,h,T); ESF=

ｄｾ

Ｓ＠ T

SFC

=

1.1324 + 1.5344M - (3.2956e- 05)h - (1.6379e- 04)T- 0.31623M2 _{+ (8.2 138e - 06) Mh}

- - - 2

- (10.496e - OS)TM - (8.574e - 1l)h2 + (3.8042e- 09)Th + (1.0600e - 08)T

WE = 3W₈EESFI.05; T uA

=

11484 + 10856M -0.50802h + 3200.2M2 -0.29326Mh + (6.8572e - 06)h2 Constraints 0.5 ｾ＠ ESF ｾ＠ 1.5 ｔ ｾｔ ｵ ａ＠ ｔ ･ｭｰｾ＠ 1.02 RANGE if h < 36089ft: 8

=

1- 6.875e - 06h if h > 36089ft: 8

=

0.7519 R = M(L I D)661J9 In Wr SFC WT - WF

Ｍ Ｍ ｾ＠

A number of constants are used in the equations of Table 7. The values of these constants are

WFo = 2000

Wo

=

25000 Nz =6

WBE = 4360

C nmin.M<J = 0.01375.

Furthermore, a number of parameters are calculated using polynomial functions "pf", which are given in the program of reference [13). The independent variables of these polynomials, given in the parentheses, are normalized by their initial values. However, the initial values that correspond to disciplinary state variables were not mentioned in any paper. Based on the fact that the system responses of the initial design must be equal to the initial values of y, a simple iterative sequence of multidisciplinary analyses was performed to determine the initial values of these variables, as listed in Table 6.

Implementation

Due to the nature of the problem, the methods were not implemented exactly as for the first demonstrative example. The following considerations concern all methods:

1 The termination criterion for multidisciplinary analysis (Fig. 1) is that the change of each y; between two consecutive iterations must be lower or equal to 0.0001 .

2 The global objective function was evaluated inside one of the disciplines. 3 All variables were scaled using their initial values for all calculations.

4 When needed, the initial values of y were obtained through a full multidisciplinary analysis before the actual optimization was started. The required disciplinary analyses were included in the total number of subsystem evaluations.

The remainder of this section gives the results obtained by the various methods. Specifications of the methods are also given when needed.

MDF

MDF optimization was performed on the problem as stated above. The results are presented in Table 8.

XJ : 0.4000 0.7500

X2 : 0.7500

X 3 : 0.1562

4.6616e+004 4.6616e+004 0.9239 3.2355e+003 1.9351e+004 0.9889 6.7110e+003 6.9463 1.1512e+004 0.8855 gl: 0.9684 0.9725 0.9770 0.9804 0.9830 0.9889 g 2: 1.0400 g 3: 0.8855 0.8367 0

Number of disciplinary analyses: 1072

Table 8. MDF results for problem 2.

IDF

The two variations of IDF were tested. To simulate absolute ignorance of the expected state variables, the upper and lower bounds for they design variables were set to positive and negative infinity, respectively. The IDF2 method failed when the variable representing the total weight was set to be negative. However, IDF1 performed very well, as shown in Table 9.

X1: 0.4000 0.7500 X2: 0.7500 X3 : 0.1562 z: 0.0600 60000 1.4000 2.5000 70 1500 4 .66 16e+004 4.6616e+004 0.9239 3.2355e+003 1.9351 e+004 0.9890 6.7110e+003 6.9463 1.1512e+004 0.8855 g l : 0.9684 0.9725 0.9770 0.9804 0.9830 0.9890 gz: 1.0400 g3: 0.8855 0.8367 0

Number of disciplinary analyses: 744

Table 9. IDF1 results for problem 2.

co

The four variations of CO were tested. Unlike IDF, this method was unable to behave well when the bounds on they variables were set to large numbers. And even for a restricted design space, this method either failed or was unable to fmd the true solution. For example, the total weight becomes less than the fuel weight when using CO 1, and C03 was not able to converge to a solution within the default maximum number of iterations allowed in MATLAB. Many different

vanat10ns were tested, including changes on the bounds, addition of constraints, and reformulation of existing constraints, but did not lead to better results.

Table 10 and Table 11 present the results for the two variations that converged to a solution. For these particular tests, the upper and lower bounds for the y variables were set to 5 and 0.5 times their initial values, respectively, except for the upper bound for y4 , which was set to 10 times its

initial value. X t : 0.3430 0.7500 X2: 0.7500 X3 : 0.2519 z: 0.0557 5.5599e+004 1.5934 3.4996 69.5729 1.1475e+003 3.5193e+004 3.3755e+004 1.1075 1.4496e+003 1.0682e+004 1.0300 7.0692e+003 4.7750 7.0592e+003 0.5558 g l: 0.9227 0.9521 0.9654 0.9730 0.9779 1.0300 g2: 1.0225 g3: 0.5597 0.8761 0 Number of ､ｩｳ｣ｩｰｬｩｮｾ＠ ana!yses: 124341

Table 10. C02 results for problem 2.

Xt: 0.19490.7500 Xt: 0.7500

X t : 0.2397

z: 0.0610 5.6215e+004 1.4900 2.5110 68.3650 1.4714e+003

Yt: 4.4208e+004 1.9924e+004 1.0300 y2: 4.5323e+004 7.32 15e+003 6.1904 y3: 1.1075 7.0592e+003 0.5558

y4: 2.7917e+003

gl: 0.9024 0.9211 0.9357 0.9461 0.9537 1.0300 g2: 1.0400

g3 : 0.5626 0.8700 0

Number of disciplinary analyses: 96635

Table 11 . C04 results for problem 2.

It should be noted that the presented optimal shared variables are the ones used as targets at the global level. Conversely, the presented state variables are the true disciplinary outputs. The additional variables used for the various constraint formulations are not listed.

CSSO/RS

The CSSO/RS method did not appear to be well suited for this particular problem. For each subsystem optimization, the state variables and the constraints from other disciplines were approximated through models. The response surface approximation used for problem 1 appeared to be effective but not reliable, as most attempts caused premature mathematical problems as encountered with IDF2 and COl. The initial sample, to define the linear approximation model, was taken as random values over the design space. Table 12 lists the results from one of the runs that converged to a solution.

Xi: 0.3039 0.8168 X2: 1.2500

X3: 0.1814

z: 0.0900 59766 1.5400 2.5000 70 1500

yi : 5.7381e+004 2.8124e+004 0.0001e+004

y2: 5.7381e+004 0.8725e+004 0.0007e+004 y3: 0.0001e+004 1.4484e+004 0.0001e+004 y4: 3.1755e+003

gi: 0.9675 0.9751 0.9802 0.9836 0.9860 0.9700

g2 : 1.0500

g ) : 1.10200.84310

Number of disci linary analyses: 2248

Table 12. CSSO/RS results for problem 2.

BLISS/RS

The version of BLISS with response surfaces was tested because it was shown [ 6] to be more reliable and effective than the initial version of BLISS. The approximation models were as described in the CSSO/RS section of problem 1. The perturbation for the initial model sampling was set to 15% of the design range of each variable. Similarly, the allowable design range during system optimization was set to ±15% of each variable. Also, the g2 constraint was

calculated at the system level, as it depends solely on global variables. The results are reported in Table 13.

X i : 0.4000 0.7500 X2: 0.7500

X3: 0.1562

z: 0.0600 60000 1.4000 2.5000 70 1500

Yi: 4.6616e+004 1.9351 e+004 0.9889

y3: 0.9239 1.1512e+004 0.8855

y4: 3.2355e+003

0.9684 0.9725 0.9770 1.0400

0.8855 0.8367 0

Number of disciplinary analyses: 425

0.9804 0.9830 0.9889

Table 13. BLISS/RS results for problem 2.

Again, MDF and IDFl appear to be computationally efficient for solving MDO problems. However, as pointed out in the next section, they are probably not the best choice for establishing a practical MDO capability because of their lack of modularity.

Two methods, CO and CSSO/RS, failed to obtain the exact solution. It should be said that most of the times the problems came from mathematical difficulties inherent in the equations. In

particular, equations in the "structures" and "range" require good behavior of the various weight state variables.

Finally, BLISS/RS was able to get the right solution in the smallest number (425) of iterations. Furthermore, it did appear to be reasonably insensitive to the approximation models. Indeed, several tests using different models and initial points all converged in less than 1000 disciplinary analyses.

5. Discussion

The two examples showed that the methods differ greatly with respect to thei r efficiency and reliability. Figure 7 summarizes the number of disciplinary analyses that were required by each method. The blue bars correspond to the successful methods, the red bars correspond to the unsuccessful methods, and the purple bar corresponds to an unreliable method .

.,

g 2500 + -,..

_..

" ｾ＠ 2000 + -QJ c 0 ﾷｾ＠ 1500 + -c .a 0 .,

.2

2 5 0 0 +

-..

" ｾ＠ 2000+---c .g _u 1500 1 -c .a 0 1000 ; ..0

5

500 z 0

Figure 7. Symmary of method performances.

ｾ＠ Ul en :J Cll ｾ＠ Ul en :J Cll

Several remarks and recommendations concerning the tested methods can be made, both from the numerical tests and from the experience gained in their implementation and use. Obviously, the demonstrative problems included in this study might not reflect the characteristics of large-scale problems. This is why caution should be applied when extrapolating these recommendations to real problems.

First, it is important to mention that all the effort invested in developing these methods aim at: 1) Increase computational efficiency in comparison to the "all-in-one" approach (MDF).

2) Distribute the computational effort over several computers or processors to exploit modularity of the system and perform calculations concurrently.

Next is a list of comments concerning each method.

Contrary to what is usually stated in literature, the MDF appears to be one of the most efficient approaches, while being, by far, the simplest one to implement and use. It is a method worth considering when decomposition and multiprocessing are not desired or impossible.

However, one of the most efficient methods was undoubtedly IDFl , for both tested problems. It

was shown to be very stable and reliable since no predetermined bounds were necessary to be set for the state variables. It is also very simple to implement and use, although extra variables are needed to model the state variables. The main problem with this method is that convergence of the process is needed to ensure multidisciplinary compatibility.

Although these two methods were shown to work well with these two simple problems, care should be taken concerning their application in large-scale MDO. For instance, a common optimization method has to be defined for all disciplines. Furthermore, all variables are treated simultaneously, which could lead to less efficient processes as the number of variables increases. Similarly, although worth considering, IDF could be much less efficient when the discipline outputs is in form of fields represented by a large number of behavior variables

The other three methods offer the type of decomposition that allows using specific methods for each discipline.

The CO approach was shown to be ineffective and unreliable. This behavior is not surprising since problems in CO have been reported in the literature. For example, CO has been qualified as "inherently difficult to solve by means of software intended for conventional, single-level, nonlinear programming problems" [2]. It is reported in the same reference that "much fine-tuning would be required to implement the method for a specific problem and that convergence behavior of conventional optimization methods applied to the CO formulation might be erratic". A set of properties of the CO formulation that explains its "erratic behavior" has also been identified [ 1 0] . They report in particular that SQP optimization methods, such as the one employed in the current study, have difficulty solving the CO problem at the coordination level.

The CSSO appeared to be a better option than CO for decomposing the problem. It was however not efficient in solving problem 2, due to some mathematical difficulties included in the equations. This method should be further tested with other approximation models. Another problem with CSSO is that all variables are used at the system level. This strategy could lead to very expensive approximation models in a real-problem situation.

Finally, BLISS/RS has demonstrated the best abilities to solve problem 2 in a fully decomposed manner. It also possesses the desirable characteristics of generating a multidisciplinary compatible design at each iteration cycle, and of dealing with x and z design variables only. It is for these reasons that BLISS/RS appears to be the most promising formulation for implementing true MDO capability. Care should however be taken concerning the formulation of the approximation models, as it could affect the overall behavior of the method to a great extent, especially when solving more realistic problems. Next section discusses one aspect of BLISS/RS that should be considered when implemented in a real-problem situation.

6. BLISS/RS with improved approximation model

The good behavior of BLISS/RS with the aforementioned implementation can be partly explained by two facts . First, no discipline calculations had to be performed in the aerodynamics discipline. Indeed, the aerodynamics constraint depends only on z. Second, it was assumed that it is possible to calculate this constraint outside the aerodynamics discipline, at the system level. This assumption may not be valid in a real-problem situation.

A more realistic way of defining the problem is to use a response surface approximation to model the aerodynamics constraint at the system level. The actual constraint values that are used to build this model are calculated at the disciplinary level during the various System Analyses (SA). Three ways of building the response surface were tested:

1) Initial linear model. Linear best-fit approximations when more points are added.

2) Initial linear model. Linear best-fit approximations until enough points ·are known to build a quadratic model. Quadratic best-fit approximations when more points are added.

3) Initial linear model. Quadratic term addition (starting with the diagonal ones) with each new point until the full quadratic model is built. Quadratic best-fit approximations when more points are added. This method is the same as described earlier for CSSO/RS and BLISS/RS. Table 14 presents the behavior of BLISS/RS for the various ways of calculating the aerodynamics constraint at the system level. Note that the tolerance criteria for stopping the BLISS/RS process is that the difference between two consecutive objective values must be less than one unit.

Constraint Results evaluation

Exact ｃｯｮｶ･ｾｧ｟･､＠ to exact solution in 425 disc!plinary analyses Approx. 1 Failure due to mathematical difficulties in the equations Approx. 2 Failure due to mathematical difficulties in the equations Approx. 3 Does not converge to the right solution

Table 14. BLISS/RS results for different global constraint evaluation methods.

It is seen that the BLISS/RS method was unable to converge to a solution with any response surface constraint approximation. This can be due to the fact that the system objective and constraints are approximated in the z space only while they are, in general, functions of both z

and x. For example, the design at two distinct iterations can have quite different objective and constraint values even though they have the same or similar z vectors. This can cause excessive distortion in the approximation models, where the approximated functions must vary significantly over a short distance. Furthermore, the used objective and constraint values that were used to build the initial linear model may not be representative at subsequent iterations, where x is updated, even if the z values are similar.

An extra step was added in the response surface building procedure to improve the overall behavior of BLISS/RS when faced with approximated global constraints. The idea is to make sure that none of the already stored points are in the neighborhood (in the z space) of the new point that is to be added to the approximation modeL If so, the old point is removed and replaced with the new point. This procedure ensures that the approximation model uses the most current information for a certain design space region. Table 15 presents the results of this method. The chosen criterion for replacing a point by the newest point is that the difference between the z components must be less than 10% of the range of the corresponding variable.

Constraint Results

evaluation

Exact Converged to exact solution in 425 disciplinary analyses A_22rox. 1 Converged with an error of 3.5% in 971 disciplinary analyses A_Qprox. 2 Converged with an error of3.5% in 971 disciplinary analyses Approx. 3 Converged to exact solution in 1279 disciplinary analyses

Table 15 . Modified BLISSIRS results for different global constraint evaluation methods.

It can be seen that with the proposed modifications the BLISS/RS is more reliable when approximated global constraints are used. Furthermore, the quadratic approximation leads to the best results, reported in

Figure 7 as "BLISS/RS*". Actually, the number of iterations was not high enough for the first two approximations to reach the point where they differ.

7. Recommendations

Based on this study, it appears that only MDF, IDF, and BLISS/RS can be considered as reliable methods to solve MDO problem in a MATLAB environment. The following guidelines can therefore be established:

1) BLISS/RS should be envisaged if full system decomposition is desired. It is, however, not as easy to implement as the other two methods.

2) IDF could be envisaged if it is possible to define the system responses by a limited set of behavior variables.

3) MDF should be envisaged otherwise.

Two other recommendations can be stated concerning future work:

• It is suspected that the bad performances of CO were mainly due to the optimization

algorithm (SQP) that was used. The methods should be therefore tested with another optimization method.

• Improved approximation methods should be tested for CSSO/RS and BLISS/RS, as they appear to be sensitive to the particular method employed. This sensitivity could become a bigger concern as the size of the problem scales up. Tests using Neural Networks have already been performed and suggest that these methods can take advantage of better approximation methods.

8. Conclusion

Five MDO formulations were implemented and tested with two demonstrative problems in a MA TLAB environment. Results indicate that substantial efficiency and reliability differences exist among the evaluated methods.

The tests performed in this study were able to show that three methods are worth considering when establishing a true MDO capability. The methods are the MultiDisciplinary Feasible method (MDF), which correspond to the traditional "all in one" approach, the Individually Feasible method (IDF), and the Bi-Level Integrated System Synthesis with Response Surfaces (BLISS/RS). More work should be done before determining if the two other methods could be retained as good candidates for a real MDO implementation.

After comparisons, it appeared for theoretical and practical reasons that the best formulation for true MDO capability is the BLISS/RS. Modifications were proposed to improve the models when approximated global constraints are used. Extra work should however be done to improve the approximation features in order to make this method more reliable and possibly less dependent on preliminary model decisions. For example, the use of soft computing techniques

for approximations, such as neural networks, should be experimented with to strengthen the optimization models.

9. References

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th

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[3] Cramer, E. J., Dennis, J. E., Jr., Frank, P. D. , Lewis, R. M. and Shubin, G. R., Problem Formulation for Multidisciplinary Design Optimizations. SIAM Journal on Optimization, Vol. 4, No. 4, Nov. 1994, pp. 754-776.

[4] Sellar, R. S., Batill, S.M. and Renaud, J. E. , Response Surface Based, Concurrent Subspace Optimization for Multidisciplinary System Design. 34th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, Jan. 15-18 1996.

[5] Braun, R. D. , and Kroo, I. M., Development and Application of the Collaborative Optimization Architecture in a Multidisciplinary Design Environment. In Multidisciplinary Design Optimization: State of the Art, N. M. Alexand.rov and M . Y, Hussaini, eds. , SIAM,

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[7] MATLAB the Language of Technical Computing, Using MATLAB Version 6, The Math Works Inc., 2000.

[8] Optimization Toolbox for use with MATLAB, User's Guide, The Math Works Inc., 1999. [9] Alexand.rov, N. M. and Lewis, R. M., Comparative Properties of Collaborative

Optimization and Other Approaches to MDO, First ASMO UKIISSMO Conference on Engineering Design Optimization, July 8-9, 1999.

[10) DeMiguel, A.-V. and Murray, W., An Analysis of Collaborative Optimization methods.

gh

AIAAIUSAFINASAIISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA, 6-8 Sept. 2000.

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[12] Sobieszczanski-Sobieski, J., Optimization by Decomposition: A Step from Hierarchic to Non-Hierarchic Systems. Second NASA/Air Force Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, Hampton, Virginia, Sept. 1988.

[13] Sobieszczanski-Sobieski, J. , Agte, J. and Sandusky, R., Jr., Bi-Level Integrated System Synthesis (BLISS). NASA Report TM-1998-208715, 1998.

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ih

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[15] Sobieszczanski-Sobieski, J., Emiley, M. S., Agte, J. and Sandusky, R. , Jr., Advancement of Bi-Level Integrated System Synthesis (BLISS). AIAA paper 2000-0421

[ 16] Chan, L., Evaluation of Two Concurrent Design Approaches in Multidisciplinary Design Optimization. IAR-LM-A-077, National Research Council Canada, 2001.