Effective Three-Dimensional Finite Element Analysis and Optimum Design of Composite Patch for Repairing Structural Panels(Composite 4)

Effective

Three-Dimensional

Finite

Element

Analy

Optimum

Design

of

Composite

Patch

for

Repairing

Structural

Panels

sis

and

Hideki

SEKINE

and

Daisuke

AOKI

Depatrtment

of

Aeronautics

an

Aoba-yama

Old

Spa

£e

Engineering,

[[bhoku

University,

,

Sendai

980-8579,

Japan

Composite

patch

repair

of

aircraft

structural

_panels

has

recently

received

wide

attention.

When

this

repair

technique

is

applied

to

the

aircraft

structural

_panels,

it

is

imperative

to

design

composite

_patches

which are reliable

enough

to

operate

in

severe

environmental

conditions.

In

designing

composite

_patches,

the

accurate

stress

fields

in

the

repaired

structural

_panels

and

the

bonded

composite

_patches

should

be

analyzed

since

highly

complicated

three-dimensional

stress

fields

are

developed.

Tb

obtain

_the

accurate

stress

fields

in

repaired structural

_panels

and

bonded

composite

patches,

usually

the

conventional

threedimensional

finite

element

method

is

used.

However,

the

conventional

three-dimensional

finite

element

method

involves

with

a

long

computational

time

_for

_iterative

stress

analysis.

In

this

_paper,

we

develop

an

effective

threGdimensional

finite

element

method

based

on

the

concept of

domain

decomposition

with

independent

interfaces.

Using

_this

method,

we

carry

out several

paraJnetric

studies

to

examine

the

efiect

of

_patch

shape and size on

the

stress

fields

in

a repaired structural

_panel

and

bonded

composite

_patches.

The

method

is

also used

to

determine

the

optimum

_patch

shape

and

size

so

as

to

endure

maximum

tensile

applied stress

by

a

mathematical

_programming,

Kley

illordS:

3D-FEM

analysis,

Domain

craft

structural

_panel,

Optimum

_{designdecomposition,}

Composite

_patch

repair,

1

Introduction

Recently,

composite

_patch

repair

of

aircraft

struc-tural

_panels

has

received

wide attention,

The

repair

using composite

_patches

causes

less

damage

to

the

re

paired

structural

_panels

than

that

of repair

using

me

chanical

joints,

In

addition,

the

composite

_patches

are

comparatively

lightweight

due

_to

_their

higher

specific stiffhess and

strength.

In

the

damaged

structural

_panels,

a

circular

hole

is

made

by

cutting

the

damaged

portion,

Se

far,

a

few

studies of composite

_patch

repair

of structural

_panels

with

a

circular

hole

have

been

carried

out

by

some

investigators(i)-(5).

Chue

and

Wang(2),

and

Chue

et

al,(3)

determined

the

optimum

ply

orientation of

the

composite

_patch

by

_parametric

calculations using

fi-nite element analysis.

Further,

Soutis

et al.(4)

deter-mined

_the

optimum

overlap

lengt]h

for

adhesion

by

a shear

lag

stress

analysis.

However,

the

study of

op-timum

_patch

shape

and size

has

still

remained

to

be

made.

In

designing

composite

_patches,

the

accurate stress

fields

in

_the

repaired

structural

_panels

and

the

bonded

composite

_patches

should

be

analyzed

since

a

highly

complicated

three-dimensional

stress

field

is

developed,

The

conventional

threedimensional

finite

element

method

to

calculate

these

stress

fields

involves

with

a

long

computational

_time

for

iterative

stress

analysis.

In

this

paper,

we

develop

an

effective

three

dimensional

finite

element

method

based

on

the

con-cept of

domain

decomposition

with

independent

in-terfaces.

Then,

the

finite

element

analysis

fbr

subdo-mains cam

be

carried

out

separately.

This

is

particu-larly

very usefu1

in

the

optimum

design

of

composite

patches

where

only

the

shape

of

the

_patch

is

needed

_to

be

changed

in

optimization

_process

keeping

the

other

pordons

ofarepaired

structural

_panel

unchanged.

car-NII-Electronic Library Service

ried

out

to

examine

the

effect

of

_patch

shape and size on

the

stress

fields

in

a

repaired

structural

_panel

and

bonded

composite

_patches,

and

_the

optimum

design

of a composite

_patch

by

a mathematical

_programming

is

fbllowed.

2

EffectiveThree-DimensionalFiniteElement

Method

Based

on

the

Dornain

Decomposition

So

far,

several

finite

element

methods(6)-(iO)

based

on

the

concept

of

domain

decomposition

have

been

proposed.

Among

these

methods,

the

FETI

method(iO)

is

advantageous as

it

can analyze

the

prob-lems

with

hundreds

of

thousand

degrees

of

freedorn.

In

this

method,

the

continuity

of

displacement

among

the

subdomains

is

satisfied

by

making

use

of

Lagrange

multipliers.

However,

using

of

Lagrange

multipliers,

the

formulation

is

complicated

to

solve

the

floating

problems.

Thus,

it

is

not easy

to

develop

the

com-puter

prograin

of

this

method

based

on

the

conven-tional

finite

element

method.

This

diMculty

urges

to

introduce

a

method

whose

computer

program

can

be

easily

developed

based

on

the

conventional

finite

ele-ment

method.

In

this

paper,

we

develop

an effective

three-dimensional

finite

element method

based

on

the

con-cept of

domain

decomposition

with

independent

in-terfaces,

The

_penalty

function

method(ii)

is

used

to

account

for

the

continuity

of

displacement

aJnong

the

subdomains.

As

to

the

independent

interfa

£

es,

two-dimensional

ones

are

introduced

to

analyze a

three-dimensional

problem,

Furthermore,

to

enable

the

ef

fective

analysis

of

a

decomposed

body,

a

special

_pr}

cedure

is

introduced

to

analyze

the

subdomains

sepa-rately.

A

schematic view of

dornain

decomposition

is

shown

in

Fig.1

which exhibits an

independent

inter-face

£ i and only

two

subdomains

9i

and

st2

in

IVa

subdomains.

The

subdomains

are

connected

through

the

independent

interface

in

which

the

constraints of

the

continuity

of

displacement

among

the

subdomains

are

imposed

by

the

_penalty

function

method.

The

po-tential

energy

ll

of

the

decomposed

body

including

the

independent

interface

can

be

_given

by

"-

£

.,

Hgt

+Sic

fllll,

(v

-

u,)2

dA

(1)

llt V U2

Fig,

1

Decomposed

body

where

k

is

the

_penalty

_parameter,

ui

is

the

disp)ace-ment

vector

of

the

subdomain

sti,

and

v

is

the

dis-placement

vector

of

the

independent

interface.

In

order

to

develop

a

threedimensional

finite

el-ement method

based

on

the

domain

decomposition,

it

is

first

necessary

to

discritize

Eq.(1),

which can

be

performed

by

substituting

the

fo11owing

relations

into

Eq.(1).

ui=Niqi,

(2)

v=Tqi.

(3)

In

Eqs,(2)

and

(3),

_qi

and

_qi

are

the

nodal

displace-ment vectors of

the

subdomain

9i

and

the

indepen-dent

interface

Zi,

respectively;

Ni

and

T

are

the

shape

function

matrices.

Further,

we

haye

aiso

the

foilowing

relation:

6filq,,q,

==O

(i=1i'''ilVa)'

(4)

Now,

the

equilibrium

equations can

be

obtained

by

substituting

the

discretized

form

of

Eq.(1)

into

Eq.(4)

as

follows:Kl

O.,,

O

-PII

O

K2

...

O

-PI2

tt

t

t

ttt

tt-

t

o

O

O

KN,

-PIN,

-pT,-PF,...-PF..

_Ki

matrix

Ki

is

always

symmetric

qlop:'dqNdql

fifp

fyo

(5)

where

Ki

matrix

as-sociated

dependent

inter-face,

coupling

term

between

dependent

inter-fhce

andG

In

Eq,(5),

the

stiffuess

and

_positive

definite.

obtained

bya

simple

direct

solver even which

has

no

boundary

floating

problerns

are encountered

penalty

function

method.

Algebraic

manipulation

of

Eq.(5)

yields

(Ki

r

IS.

i,

PF,K,-

iPit

)

qi

=

£

.

,

P7,K,-

i

fl

₍₆₎

qi

=K,i iPiiqi

+K,,

i&

(i

==

1,...,Nh).

(7)

When

the

inverse

matrices

of

each

subdomain

are

calculated and substituted

into

Eq,(6),

we

obtain

the

nodal

displacement

vector

qi

of

the

independent

inter-face,

Next,

the

nodal

displacement

vector

_qi

is

sub-stituted

into

Eq,(7)

to

obtain

the

nodal

displacement

vector

_qi

of each subdomain.

Here,

it

is

noted

that

the

analysis

of

the

whole

decomposed

body

can

be

carried

out

by

analyzing

the

subdomains

separately,

Thus,

for

the

model

in

which

the

mechanical

characteris-tics

and

_geometry

in

some subdomains

are

changed

in

-

₆₉₁

[[hble

1Dimension

of

the

model

H(mm)PV(mm)d(mm)A(mm)B(mm)tp(mm)tA(MM)tR(MM)

240

240

10

16t--7016･v70

3

O.1

O.2t-.

[[bble

2Mechanical

_properties

El(GPa)

fa'pEa9

%

£

S3

C23(GPa)

?s

7075-T6

Vl?)V13

71.02

71,02

26.90

26,90

O.32

O.32

boronlepoxy

208

25.44

7,24

4.94

O.035

O,17

AF163-2K

L17

1.17

O.44

O.44

O,34

O.34

Table

3

Strength

of

aluminium

alloy7075-T6

and

boronlepoxy

composite

cr+o x=aCr570 cra zz cr-a x=aCrr cr-azz

7075-T6

(MPa)

aCr.

cra ==a:r

570

570570570570330330330

boron/epoxy

_(MPa)1,26060

_{602,500202202676767}

twc{;fey

x H

1I/I$gg:/fx

AW

Fig.

2

Model

of

bonded

composite

_patches

and

repaired

structural

_panel

each

iteration,

it

is

necessary

to

calculate

the

inverse

matrices

of

all

the

subdomains

at

the

first

iteration.

After

that,

the

inverse

matrices

of only

the

subdo-mains

whose

mechanical

characterictics

and

_geometry

are

changed

are necessary

to

calculate

in

the

next

it-erations,

This

feature

of

the

_present

method makes

it

more effective

_than

_the

conventional

finite

element

method

in

the

optimization

_process.

3

NumericalExamples

3.1

Model

of

bonded

composite

_patches

and

paired

structural

_panels

A

model of aircraft structural

_panels

with

a

circu-lar

hole

repaired with

two

elliptic

composite

_patches

bonded

to

each side of

the

panel

is

shown

in

Fig.2,

The

major and

the

minor

axes

of

the

elliptic

compos-ite

_patches

are

A

and

B,

respectively.

The

thickness

_of

the

_panel,

the

_patch

and

the

adhesive

are

tp,

tR

and

tA,

respectively,

The

diameter

_of

_the

_circular

_hole

of

the

_panel

is

d.

The

_panel

_is

_subjected

_to

_a

uni-form

applied

tensile

stress ayeey.

The

deflection

of

the

loading

edges

is

constrained

to

be

zero.

Due

to

sym-metry,

only

oneeigth of

_the

model can

be

considered

S)3'

94-

Y"<E.ix

nl

snl

Fig.

3

Mesh

configuration

for

the

finite

element

analysis

which

is

decomposed

into

four

subdomains

as

shown

in

Fig.3,

This

figure

also

shows

the

mesh

configuration

used

in

the

_present

analysis,

The

_subdomain

9i

_represents

_the

_portion

of

the

_panel

which

includes

oneforth

of

the

_patch.

In

the

finite

element

analysis,

20-noded

isoparametric

threedimensional

brick

elements

are

used

in

the

_panel

and

the

_patch.

The

adhesive

is

assumed

to

consist

of

linear

spring

elements.

The

mesh configuration

used

in

the

conventional

finite

element

method

is

also

the

same

except

that

the

indepenedent

interface

has

no

existence,

i,e,

the

_subdomains

_are

directly

_connected.

The

various

dimensions

of

the

model

are

shown

in

[Ibble

1.

The

_panel

and

adhesive

materials

are,

re-spectively,

aluminium

alloy

7075-T6

and

AF163-2K.

The

_patch

is

made of a

boron/epoxy

composite

in

which

fibers

are

_parallel

to

the

_y-axis.

The

mechan-ical

_properties

of

_these

materials

are shown

in

[fable

2,

Also,

the

strength

of

an

aluminium

_alloy

_7075-T6

and

a

boron/epoxy

composite

is

shown

in

Table

3.

3.2

Vlerificationofefficiencyandaccuracy

A

comparison

of

the

number of nodes

and

el-ements, _and

computational

time

required

by

the

present

method and

_the

conventional

fininte

eiement

method

is

shown

in

Table

4.

The

number of

nodes

in

each subdomain

for

the

_present

_method

_is

_smaller

than

half

of

that

for

the

conventional

finite

element

method.

The

computational

_time

_is

also

_shorter

_for

NII-Electronic Library Service

[lable

4Comparison

of

the

number

of

nodes

and

elements,

and

computational

time

Method

Domain

decomposition

91st2st3st4

£I[[btal

Present

method

Applied

Node3,0308018011,7811056,518

Element592154154336681,304

Time39,95.04.625,61.276.3

Conventional

FEM

Not

applied

Node

6,126

Element

1,100

Time

100

1.02

7,eifgsg

O.97"oEE

O･9

log,e(kIE)

Fig.

4

Relationship

between

the

normalized

penalty

parameter

and

the

normalized

stress

ference

can

be

observed

from

the

table,

Compared

to

the

conventional

finite

element method,

the

computa=

tional

time

for

the

_present

method

is

about

80%

when

al1

the

subdomains are considered and about

40%

when

only

the

subdomain

sti

and

the

independent

in-terface

Zi

which are calculated

iteratively

in

the

op-timum

design

of a composite

_patch

are considered.

Thus,

it

is

realized

that

the

eMciency

of

the

_present

method

is

much

better

than

that

of

the

conventional

finite

element

method.

It

should

be

mentioned

that

this

difference

occurs when

the

modified

Cholesky

de

composition combined with

the

band

method

is

used

for

calculation.

The

effect

of

the

normalized

_penalty

parameter(icIE)

on

the

normalized

stress(ayylagy)

is

shown

in

Fig.4.

The

_penalty

_parameter

k

is

normal-ized

by

the

Ybung's

modulus

E

of

aluminum

alloy

7075-T6.

The

stress

at

the

edge

of

the

circular

hole

ayy(x=d12,

_y=e,

x=O)

is

normalized

by

the

stress

crgy(x=:d12,

_y=O,

z=O) obtained

by

the

conventional

finite

element method,

In

this

figure,

the

open circles

represent

the

results of

the

_present

method

and

the

solid

line

represents

the

results of

the

conventional

finite

element method.

In

the

numerical

examples

considered

from

here

onward,

the

calculations

are

carried

out

by

using

the

penalty

parameter

corre-spending

to

the

value

5

on

the

horizontal

axis of

1.4

t's

_1.2ifs"1gs

O.8xk-g

o.6z

O.4

O

10

20

30

40

SO

60

x

(rnrn)

Fig.

5

Distribution

of normalized stress

(ayylayOOy)

along

the

x-axis

(y=O,z=O)

Fig.4.

Shown

in

Fig.5

is

the

normalized stress

distribu-tion

along

the

x-axis,

In

this

figure,

the

open and close circles, respectively, represent

the

normalized stress

distributions

in

the

subdomains

sti

and

st2.

The

solid

line

shows

the

normalized stress

distribution

obtained

by

the

conventional

finite

element methed.

It

is

noted

that

the

difference

between

the

results obtained

by

the

present

and

the

conventional

finite

element method

re-mains within

1%.

This

means

that

the

accuracy

of

the

present

method

is

the

same

as

that

of

the

conventional

finite

element

method.

3.3

Preliminaryanalysis

As

a

_preliminary

analysis

for

the

optimum

design

of

a

composite

_patch,

several

parametric

studies

are

carried

out

to

examine

the

effect

of

the

major axis

A,

the

minor

axis

B

arid

the

thickness

tR

of

the

elliptic composite

_patch

on

the

stress

fields.

The

relationship

between

the

major axis

A

arid

the

maximum normalized stress(ayMyaXlaCr)

is

shown

in

Fig,6(a),

where aCr

is

the

strength of

the

aluminium alloy

7075-T6.

The

figure

shows

that

the

ma)cimum

normalised

stress

_gradually

decreases

as

the

major

axis

A

inc;eases.

The

relationship

between

the

minor

axis

B

and

the

maximum

normalized

stress(ayMya'`laC')

is

shown

in

Fig.6(b).

In

this

figure,

the

maximum

normalized

stress

is

a concave

function

of

B.

This

is

because

the

shear stress

developed

in

the

elliptic

com-posite

patch

rises

with

the

minor

axis

B

which

in

turn

-

₆₉₃

.Ag

_on

k)g

o,6stheaB o.6.NaF2 o,sssE'Res o.sE 20 30 40

SO

60

A(mm)

(a)

Majer

axis

A{tR=O.4mm)

Fig.

6

Effect

of

shape and size

a,OO,==245(MPa) 1.4

ig

l.3oe 12a>: IJ

ST,

ts-

OJbv o.6s$euB

o.6.NreE2

o.ssi.E:

O.5E

gF

g･

go=E=ES5

O.95O.9O.8SO.BOJSO.7O.6SO,6o.ss

20

30 40 SO 60

B(rnm)

(b)

Minor

aJcis

B(tR=O.4mm)

of composite

_patches

on

the

maximum normal

k ℃

sgi.gtsE

O.9 20 30 40 SO 60

A

(mm)

(a)

Major

axis

A<tR=O.4mm)

Fig.7

Relationship

of

fracture

con

a,co,

=245(MPa)

1.41.31.21.1

O.9D.8O.7

20 30 40

SO

60

B(mm)

(b)

Minor

axis

B(tR=O,4mm)

dition

of

composite

_patches

with

produces

a

larger

stress

in

the

_panel,

The

relationship

between

the

thickness

tR

and

the

maximum

normal-ized

stress(oyMyaX!aCr)

is

shown

in

Fig.6(c).

It

is

seen

from

this

figure

that

the

dependency

of

the

thickness

tR

on

the

maximum

normalized

stress

is

significant.

The

fracture

condition

for

a

composite

according

to

Hofiman

criterion

can

be

written

as

F

=

Ci

(ayy

-

a.x)2

+

C2

(a..

-

axx)2

+

Cb

(a..

-

ayy)2

+

(]4a.x

+

Csayy+

Cba..

+

C7a,2.

+

Cba.2.

+

Cb

ag,

(s)

where

the

coeficients

Ci

to

Cb

are

obtained

from

the

strength of

the

cemposite.

A

composite

will

fracture

when

the

_value

of

F

exceeds

unity.

The

relationship

between

the

major

axis

A

and

the

maximum value of

F

in

the

elliptic

composite

_patch

is

shown

in

Fig,7(a).

This

figure

shows

that

the

maximum value of

F

grad-ually

de

¢reases

as

the

major axis

A

increases.

The

relationship

between

_the

minor

axis

B

and

the

max-imum

value

of

F

in

the

elliptic composite

_patch

is

shown

in

Fig.7(b).

In

this

figure,

the

maximum value

of

F

is

a

concave

function

of

B.

This

is

because

the

shear

and

peel

stresses

become

larger

at

the

edge

of

the

_elliptic

composite

_patch

due

to

the

bending

effect as

the

minor axis

B

becomes

larger,

The

relationship

].B l.64ts

1.4s->1.2a=El.-ajE

o,se,6

O O.2 e.4

O,6

O.8 1 ]2

tR

(mm)

(c)

Thickness

tR

ized

stress

(ayrnyaxlaCr)

fbr

O O,2 O.4 O.6

O,8

1 1,2

tR

_(mm)

(c)

Thickness

tR

the

_parameters

_A,

_B

and

tR

for

between

the

thickness

tR

and

the

maximum

value of

F

in

the

elliptic

composite

patch

is

shown

in

Fig.7(c),

This

figure

shows

_that

_the

ma)[imum value

of

F

grad-ually

increases

as

the

thickness

tR

increases.

F)rom

the

above results,

it

is

realized

that

the

shape

and

size

of

the

elliptic composite

_patch

have

great

effects

on

the

stress

fields

in

the

repaired

struc-tural

_panel

and

the

bonded

composite

_patches.

3.4

Shapeoptimizationofacompositepatch

3.4.1

Formulation

In

this

article, we outline

the

design

_procedure

for

obtaining

the

optimum

_patch

shape and size

so

that

it

can

endure maximum

ten-sile

applied stress.

The

_parameter

Ik

controling

the

size

of

the

elliptic composite

_patch

is

introduced

as required

in

formulating

the

optimization

_problem

as

fo11ows:

Maximize:

aco yy

Subjectto:

F<1

(9)

ayMyaxlaCr

<

1

ABtR

=

Iii{

where

the

design

variables

are

A,

B,

tR

and

ayOOy,

In

this

shape

optimization,

the

side constraints of

the

design

variables

A,

B

and

tR

as

16(mm)

_-<

A,

B

S

70(mm)

and

tR

)

O.2(mm).

The

first

constraint

of

the

NII-Electronic Library Service

340

330A

320esS

310L-/

3008,M'

290g,:

280S

27o

260

250

240

O

100

200

300

Vit

(mm3)

Fig.

8

Ma)cimum

applied

tensile

against

the

_parameter

VR

400500

Stress

ayOoy

fracture.

The

second constraint restricts

the

_panel

from

failure

by

overloading,

the

third

constraint

con-trols

the

size of

the

elliptic composite

_patch.

The

op-timization

_problem

is

solved

by

using

an

optimization

program

ADS

in

which

the

method

of

feasible

direc-tions

is

used

as

an

optimizer

while

the

polynomial

interpolation

is

used as a one-dimensional search.

3.4.2

Results

and

discussion

The

relationship

between

the

maximum applied

tensile

stress

ay'Oy

and

the

_parameter

Vk

is

shown

_graphically

in

Fig.8.

This

figure

shows

that

the

_gradient

of

maximum

ayOOy

against

iik

decreases

as

the

parameter

ltk

increases.

This

indicates

that

too

large

patch

is

not

effective

to

repair

the

panel

with

a

circular

hole.

Figure

9

shows

the

relationship of

the

_parameters

A,

B

and

tR

with

the

ma)cimum

applied

tensile

stress

ay"Cy,

Flrom

this

figure,

it

is

seen

that

the

major

axis

A

increases

while

the

minor axis

B

almost remains

the

same as

the

maximum applied

tensile

stress ayOey

increases.

This

is

because

the

maximum stress

devel-oped

in

the

_panel

and

the

maximum value

of

F

in

the

composite

_patch

are

1arge

when

the

minor axis

B

is

too

large

or

too

small,

The

thickness

tR

remains

con-stant up

to

a certain value of

the

ma)[imum applied

tensile

stress after which

it

increases

as

the

ma)cimum applied

tensile

stress

increases,

It

is

observed

that

the

optimum

_patch

shape

is

circular

when

the

maximum

applied

tensile

stress

is

smaller and elliptic with

the

minor axis along

the

load-ing

direction

when

the

maximum

applied

tensile

stress

is

higher.

4

Conclusions

In

this

paper,

we

have

developed

an effective

three-dimensional

finite

element method

based

on

the

concept of

domain

decomposition

with

independent

interfaces,

The

method

is

applied

to

carry out astress

807060AE

sovco

302010

O.35O,3025

o.2

go.is

lny

O･1O.05

o

o

250

275

300

325

Maxlmum

oyco..

(MPa)

Fig.

9

Relationship

of

the

parameters

A,

B

and

tR

with

the

maximum

applied

sile

stress

aO

yy

analysis

of

a

repaired

structural

panel

and

bonded

composite

_patches

from

which

it

i$

observed

that

the

eMciency of

the

method

is

much

better

whereas

the

accuracy

is

the

same

in

comparison with

that

of

the

conventional

finite

element

method,

Several

paramet-ric

studies

are

also

carried

out

to

exarnine

the

effect of

patch

shape and size on

the

stress

fields

in

a repaired structural

_panel

and

bonded

composite

_patches.

It

is

found

that

the

stress

fields

are

_greatly

influenced

by

the

_patch

shape

and

size.

Finally,

the

method

is

used

to

determine

the

optirnum

_patch

shape and size

so as

to

endure maximum applied

tensile

stress

by

a mathematical

_programming.

References

(1)

Paul,

J.,

Bartholomeusz,

R.

A.,

Jones,

R.

and

Ekstrom,

M.,

Engng.

Ftact.

Mech.,

Vbl.48,

No.3(1994),p.455-461.

(2)

Chue,

C.

-H.

and

Wang,

S.

-C.,

Engng.

F\acture

Mech.,

Vbl,48,

No.4(1994),p.515-522.

(3)

Chue,

C.

-H.,

_Lin,

_L.

-A.

and

Wang,

S.

-C.,

Engng.

Flracture

Mech.,

VdL48,

No.1(1994),p.91-101.

(4)

Soutis,

C.,

Duan,

D.

-M,

and

Goutas,

P.,

Compos.

Strut.,

VbL45(1999),p.289-301.

(5)

Hu,

F.

Z.

and

Soutis,

C.,

Comp,

Sci.

Technol.,

Vbl.60(2000),p.1103-1114.

(6)

Farhat,

C,

and

Wilson,

E.,

Int.

J.

Numer.

Methods

Eng.,

VoL24(1987),p.1771-1792.

(7)

Ransom,

J,

B,

and

Knight

Jr.,

N.

F.,

Comput.

Struct,,

Vbl.37,

No.4(1990),p.375-395.

(8)

Yagawa,

G.,

Soneda,

N.

and

Y6shimura,

S.,

Comput.

Struct.,

VoL38,

No.5(1991),p.615-625.

(9)

Aminpour,

M.

A.,

Krishnamurthy,

T.

and

Fadale,

T.

D.,

AIAA-98.2060"CP,

(1998),p.3014-3024.

(10)

Farhat,

C.

and

Roux,

F.

-X.,

Int.

J,

Numer.

Methods

Eng.,

Vbl,32(1991),p.1205-1227.

(11)

Cho,

M.

and

Kim,

W.

B.,

AIAA-98-2061-CP,

(1998),p.3025-3032.