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J;p/

./ NASA

Contractor

ICASE

Report

Report

195015

No. 94-96

S AN ALGORITHM PROBLEM

FOR SOLVING

IN MULTILEVEL

THE SYSTEM-LEVEL

OPTIMIZATION

R. J. Bailing J. Sobieszczanski-Sobieski N95-18108

(NASA-CR-195015) AN ALGORITHM FOR SOLVING THE SYSTEM-LEVEL PROBLEM IN MULTILEVEL OPTIMIZATION Final Report

(ICASE}

26

Unc|as

p

G3/64

Contract December

NAS 1-19480 1994

Institute

for Computer

NASA Hampton,

Applications

Langley

Research

VA

23681-0®_

Operatedby

in Science

and Engineering

Center

Universities

Space

Research

Association

0034988

AN

ALGORITHM PROBLEM

FOR SOLVING IN MULTILEVEL

THE SYSTEM-LEVEL OPTIMIZATION*

R. J. Balling Brigham Young University Provo, UT 84602 J. Sobieszezanski-Sobieski NASA LangleyResearchCenter Hampton, VA 23681-0001

ABSTRACT A multilevel optimization approachwhich is applicableto nonhierarchiccoupledsystems is presented. The approachincludesa generaltreatment of design(or behavior) constraints and coupling constraints at the discipline level through the use of norms. Three different types of norms are examined-the max norm, the Kreisselmeier-Steinhauser(KS) norm, and the lp norm. The max norm is recommended.The approachis demonstratedon a classof hub frame structures which simulate multidisciplinary systems.The max norm is shown to produce system-levelconstraint functions which are non-smooth. A cutting-plane algorithm is presentedwhich adequately deals with the resulting cornersin the constraint functions. The algorithm is tested on hub frameswith increasingnumber of members(which simulate disciplines), and the results are summarized.

*Thisresearch wassupported by theNationalAeronautics andSpace Administration underNASAContract No. NASl-19480 while the first author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001.

I.

Introduction

This paper coupled

is concerned

modules,

to engineering "discipline"

with the optimization

each transforming

disciplines

optimization

or physical

to this problem approaches

optimization

problem

components,

can be divided

optimization,

problems

equality

and optimizations

applications problems

(Thareja

(Schmit

approaches

presented

identified

discrepancy

suggested

"system"

optimization equations

approach

of a norm

Such a formulation

Steinhauser

Three

level

The

into

In that method,

the

at the system

1.

The

constraints

systems,

The need to satisfy arose

in

optimization

the design (or behavior)

optimization

multilevel systems.

associated

(see Figure

problem

and discrepancy

systems,

with the coordination

may be coupled

than

approach

In nonhierarchic

of the

to every other

system by treating 1).

of the design constraints

optimization

approach

optimization

as a nonhierarchic

as the other disciplines

violation

while satisfying

that occasionally

multilevel

each discipline

the treatment

discipline-level

the discipline-level

The

and optimization

The

the

multilevel

and coupling

is formulated

in the coupling

always exists for the discipline-level

norms will be examined---the

1983), and the lp norm.

The focus of

1993).

multidisciplinary

solution

a three-level

problem

were performed

equations while satisfying

discipline-level

that a feasible

It may be

the discipline-

optimization

in Figure

formulations,

may be viewed

of both design constraint

guarantees different

system

level.

three levels.

1982).

difficulties

1993.

here also generalizes

at the discipline-level.

minimization

problem.

presented

as shown

a more general

to nonhierarchic

on the same

large

and analyses

of numerical

in Sobieszczanski-Sobieski

are on the same level, and analysis

as a discipline

a single

1973; Sobieszczanski-Sobieski

The traditional hierarchic

approach,

is used for comparison.

in the design (or behavior)

in the coupling

system is implied (see Figure 1). In nonhierarchic discipline.

optimization

were coupled to the system but not to each other.

In alternative

of this paper is to present

here has been extended

all disciplines

1986).

are solved

passed from the system to the discipline.

as a source

and Ramanathan

The first objective the

variables

and Haftka

seek to minimize

constraints

The disciplines

problems

It this case,

beyond

(Sobieszczanski-Sobieski

sought to minimize violation

was

The term

only a single

occur at the system-level,

optimization

decomposing

to as a "hierarchic system"

on the coupling

constraints

of

and multilevel

In the former,

at the discipline

extendable

single-level

was suggested

into disciplines,

was referred

equality constraints

1994).

optimizations

although

level and within each of the disciplines.

optimization

modules.

approaches

of subdisciplines.

is readily

for linearly

problems

system was decomposed

wherein

This scheme

ago, a method

optimization

Such a system

optimization

Thus, in a two-level

as a system itself composed

may be employed

this paper is on multilevel

multilevel

into single-level

and Sobieszczanski-Sobieski

level, and the subdiscipline-level.

is an assembly

usually corresponding

within these

at the system level, and there are optimizations

approach

a decade

and optimizations,

may be executed

as well as for the system as a whole.

to view a discipline

Over

model

to mean such a module.

(Balling

there is an optimization

optimization

Analyses

mathematical

is solved for the entire system, while in the latter, optimization

within the disciplines

possible

input to output.

will be used throughout

Approaches

of systems whose

as the

equations. optimization

max norm, the KS norm (Kreisselmeier

and

Thesecondobjectiveof

this paper is to present

optimization

problem.

It will be demonstrated

optimization

problems,

the system-level

non-smooth. adequately

embedded

treats non-smooth

functions.

Results

begins by presenting systems.

and optimization

problem.

The cutting-plane

will be presented,

Single

Consider

algorithm

and numerical

and Multilevel

the three-discipline

associated

analysis

variables.

A three-discipline

keep the discussion The system

program

single-level

of sensitivities

j.

which

analyses.

By associating

variables

system shown computes

because

output

coupling

single-level between

or multilevel

each coupling

The vectors analyses. vectors

against

unacceptable

that each constraint than zero.

such as the maximization

a selected

for the discipline-level

The the data

is a good disciplines.

optimization

optimization

problem

complicate

coupling

of the

it is small enough

to

to every

other

discipline,

and no

i which

are needed

a corresponding

analyses

of the disciplinary vector

may be executed

of coupling

in parallel.

Each

as output.

One of the tasks of the

is to satisfy coupling

constraints

which

coupling

enforce

sets of design variables

variables

needed by more than one discipline,

Only inequality

needed by Disciplines functions.

These

constraints

the design

of benefits and the minimization

represent

as input to the while

the constraints

These represent

It is assumed

1

|!

which

when less objectives

that each objective

and the value of zero is associated

target value.

the

iaere, and it is assumed

value, and it is satisfied

functions.

of costs.

needed

1, 2, and 3, respectively.

are considered

objective

minimization,

equality

function.

exclusive

through

as input to

functions

the design constraint

such that it is improved

from input values

the order of execution

functions

design variables

fl, f2, and f3 contain

has an

because

in Discipline

the disciplinary

approach

system design

in this system

pattern.

is coupled

computed

and its corresponding

behavior.

of the functions

has been formulated such that zero is its allowable

The vectors

been formulated

values

of coupling

x, x 1, x2, and x3 are mutually

g_, g2, and g3 contain

for

The vectors Y12,Y13,Y21,Y_, Y31,and Y32are the coupling

which

Y32"),

xi, x 2, and x3 contain disciplinary

The vectors pard

functions

optimization

Note that x contains

because

of coupled

for the system-level

to see a general

as input and computes

variable

approaches

or finite elements

in Figure 2. Each discipline

each discipline

with each vector

variables

strategy

size.

will be discussed.

composed

as a basis for discussion

those functions

(Y_2*, Y13", Y21*,Yz3*,Y31*, and

receives

the three norms

system was chosen

It is these coupling

of substructures

which

of increasing

optimization

are

Approaches

simple but large enough

Discipline

for test problems

of a system

which

results will be discussed.

coupled

Note that Yijcontains

functions

strategy will be presented

and multilevel

composed

and move-limit

discipline is viewed as being "above" the others. functions.

the system-level

of a hub frame which was selected

to examine

Optimization

is nonhierarchic

constraint

for the approaches

and optimization

will then use this example

possesses

within a move-limit

of a structure

solving

the max norm is used in the discipline-level

problem

on an example

of the data flow in the analysis

The paper

II.

The calculation

for efficiently

will be presented

the general

will then be demonstrated

flow in the analysis analog

optimization

algorithm

nonhierarchic approaches

that when

A cutting-plane

The paper

an algorithm

has with

The optimization

Single-Level

problem

Optimization

may be solved on a single level as follows:

Problem

Find:

f,X, X1,X2,X3,YI2

Minimize:

f

Satisfy:

g_ < 0,

g2

fl