Computational Solid Mechanics I' (Finite
Elements & Boundary Elements): Present
Status and Future Directions '
S, N, ATLURI '*
-.
"..
In Ih.:: firs! p:H1 ,J ) this plP'::L .;ert:1I1l developments Jnd trel1d~ In .:ompuIJ·
1l!JIlJ1 m<.'thl)J~ themselvt's Jre t.hscussed. These mclude IOPI":S In fum . . ~Iemem
Jnu 'luunuJry eio.'menl le~-hllologles (or solid mechanl":s: Ii) LBB ..:onditlon<; lor
, ','ec-..... '
~l'Il('rJI IlIl n.::
dl'rncl1I method s: Iii)
ka ~l·order.
s!lblc. 1I1\'HllrH. l>uparJmetrl":.
,lr~~,-t>J\<."d. h~bT1lJ
IllLx.::d <!i<.'1l1ems : (I ii) use u! symholh: m:lI11pulJIIOIl (1\1
JJJr'Il\':> 'l1('sil re l lrll!menl. (VI tr:lIlSlent dynamic respnn~: Jnd In) bound..lr:-·
<,kr~l<''11 :ll<,!htld~ lor lineJr dJ~Il<.:il\. JS well J5 tor linn.:: -ITJ1Il ?whl<'m) 11
1J1(',J,t:, :n.U':T1Jh. In th,: ~,:~o no part u r the papn. ~ertal:; tOPt~S m Ihl! rnl!~hJn:
'(' "t' ,,)liJs. wi1<.!relll (U!llputJl1u nal methods h;.lve pbyed. Jno jr..: exp~.::eJ
'1!!lllfi(J1l! rok" Jre JI~..:usr.ed. These mdudl! (n cUmtllUll\e :TI!)Jl!i·
Inc ,\ IneIJ)II'; mJtertJI I)du\"lor. (Ii ) mechanICS ~) I lIleiJ)tl( Jnd JynJrr.lc
t"fJdUrl!. tlli) n\)niln<'Jr contmuum me~hanl~sJnd )hellthem~. Jnd It\)strtLdUral
,ulltr"l. tll\'olvll1g ~olllr()1 ,lI JynJlmc response ot' brge 'PJ(~ Jnd ':Jrlh )tTU~'
IUf,"
:0 "IJ\
~t l ~ H1J ~ ( ~~H ~ 7[,& i
lJLJ? ,&
W~ 7[ I
**
•
~~ Ii jJ rtiJ
- ..
""
,
- _ ,',
lit _,!:.! LBP
,',ttL':'
~ 6.:£ f. "Ii
'
i
;-~ ..:~'f'
'-
,
!! 1;1, '.
"
•
,'IIIJI1I<1I1, lur -;cn<'rJlll1l11,:.:k:1\"1I1 "h'[h,),h,IU!
,[:.1:"111'\
"TRODL G IO\
','J',
'J,.:<1
'i..:"'I'!lI"
"
!Ill
I.n
'"
"\ I'
:' _I P ..':
1:'\
:":"':
",' ~ \
11 ' "
,III.
,
."::,.
~J
"
'
.,
'':
''1h
>t"c'
,"
,
·<.'1\'
~,',
',\1':
.1
'1':[1
(
"LIJ
<'~ ..'.I:
~.Ii!1e-J
lh',h.ITH,·,.
-ilJI1'Il.lII,'Tl ,\IIi!
,Lllk,I":'1
:h<, (\111..::
,II
,m:'II:.:lh'l1
IS(\p.IIJmc'!r;~
til)
~1<'111<,1l!
JJJro11\1!
:1\<,,:1
;nJ 1\11
r",1
~!·Tl·
"'''Ln\.!. \.
JI':.'
j
,"
;H::nJr!j~
III
\ ,11',111,,,"1\,'1'1
~,\<'
hi
JI',.:u"
,'enJIll IOPI,', [11 th,' 'lh',:I.m:.
'I ." I,h .. ~llI~h 11:1\<.' h.:,·n. Jnd Jr ',' 11\..(:\ 1(1 :,,' ,'Ill"I' • ..!
:":1:;,111,\
llruugh the J<.'\'.:ltJplll<'l:t
II dlmpIIIJ;. tlLl1
"kll <1,
Th.::>1.' mduo<,
(I) ~OIh,:tull\l' Ill"J,: :!,~ I]
h',' 11:\1 'I\hkl~h ilL.!
','r tlh'
hJst'u
IllJtlIPUIJll'lIl
II J 'UII~lent Jynal11k fl"r'U\'-l'
,
"c',';·>l1
"'11\,'
_ 1I1l:".Il.lli'1IIJ.
"II,.::
\111"nll(
:,>:~,0111 :ll<.'tlhHh 1m 11Il11<' ~trJm In,':J'!k pr,'hknh
"
' \ I 'III
,Ill
,I
_lfl,,1l
[ II~
!ht"i.k.-mcllt
hllJI
. " I) ':",t ", '11.
"
'\Jol.:, lIl \'afl.l lll. h~omJ 1!1!'(<'u hOt"JfJP·,'t::,
.)1) J~,llm,'d 'Ir.:\'i~'~, j , Jilar~~ 1-'
'h,'
'1,1,"
IIJI<'nJI
Ih,"J_;:,
IIiLI \\tlJlllk
_:
\ 1",:\'111
n.:hJ\[o r.
I rJ.:tu re
(11)
1Tl<',:IJIlI" ,)1
liii) nUnlineJf
~')Tlltllllllm
:1;~[J"i,
'1\,'.1:.1111.,
.1 ,..:
. I ,)
"tl",I>.:l.l,',
II
,11
h
nIl.'
t"ll'.I.
.,lIlt,',\
h,,·
tI
'lUll ~ ,1<'
In ""
0.1"\"]"[,111':111
-·d'Il1I"I'.j:.IlJJ
.!l
::'IJIIIo.1.',
c m lPUT \TIO NAL \lE THOIl S
1I11~.\ \IJL'
1
"
.h,:h, .en.lIll
,'IlH"llIJII"llJl
"un:""
:1'~.IIJ"
"i"ll.Ii '<',',,·II':,.n [it,:
.lIt"U'I'I'"
'Ul,th"<J
h..:
rJN"
lie.. ·.• ,\,' Ji::..:u~~ rt'ct'1l1 Jd\311~~'i JnJ 'r,'II.j, n '::,:1<,
nn undJ r\ <,I<,mclll t<,dlll(\iul!l<'~,
If.'
·..:nu, ,Ill,:
'hcf1\'<'!\'"
',::"
'i<'~~,":1 .11l,j
II I
Fitlll(' - Elemenl T!!chnology
'J :, 'nJi Con!,"
JI
~k~
,11 fI'~
:-:::,J:
..
;Ih. \ [':Jllt:..; \1c ,:~,J!I;,
LlIlJIl. I{
• • 1,1<'
\1c: Jl"
'''-<''''
,n 'he rnii"v.mc
'Ii
,I ... i \ '.1
,
I,
',;111,'
"
'J .. ,'.
"11" .. .
'J,,..:
!
'P\'(j]~,
' :'1,
L.JJ"/hl'I"",I\J BJh'''''J· lk,·1!1
\ ' It..,
,\
methods: Oi) element development based on non·standard
(hybrid/mixed) formulations. to avoid klfiematic deforma·
tlon modes. etc..
(iii) use of symbolic mampulation;
(iv) adaptive mesh refinement: and (v) transient dynamic
analysis.
1. 1.
lB B Co nditi o ns Consider. for Instance. a
linear elastic solid undergomg mfimtesima! deformation.
Let the wlid.
be discretlZed. for purposes of generatIng
an :lpprmumaling solullon. mto a finlle number of elements
Urn 1m '"
[U" ~ n J. and nm be the boundary
n.
L.;·n.
a
m en
of nm In general. an m = Pm+Sum + Stm. where Pm is
+
J
S urn a i j n j (ii i - ui) ds
(' )
Case
1:
The Irial functions ui,
arbitrary.
ai j . and E i j are 111
We m~roduce Lagr.ange multiplier s
f
IP at
Pm. such that (Tip)+ + (Tip)-=Oat,vm' apnoti.
The variallonal prinCiple governing (I.a to I.h) may be
5tatedasOL(u.E.a.T p ) =O.where:
the inter element boundary. and 5tlll and ~lIn are those
v. hlch :Ire oommon With the external
segments 01
boundary segments 5t (where tractions are prescTlbed) and
Su (where displacements are prescTlbed). respectively. As
stated m [I J. the govermng equations :lnd boundary conditions fo r the limte·element am'mbl~ are as follows:
a nm
FUIU
+
( I .a)
(I.b)
(I.e)
Case
3:
n J .:
a IJ .
J
.j.
II
I ,
,,\ I
• 0
a, ) • a ) '
",•
•
(a
I)
",
nm
m
m
"
n J) ~ +W
( I.J)
Stm
JI
(I J)
1)-
= 0
JI
P m
Let the lTlal fun~llon u l obey (I.g, a prion.
The variatio na l prtnclple governtng the remalllder of lhe
equallons may be ~llted JS'
CJ:.C
LiUE.a) =
Ui Tip ds
(3)
Here also Ihe tnal funcllons ui. ai j. and €jJ
displacement mmpallbility condillons, ( u+i = Uip ) and
) at Pm. through arbllTary Lagrange
pliers lip and Tip on either side Pm'
prtnciple in this case becomes:
multi'
The V'JT1a1!OIIal
liUU,E.O.U p , Tp)=O, where
( l.h)
where Ihe 1I0men.:iJlure IS JS tollo\\~ a iJ (stress tensor);
Elj htrJIIl L u l (dlspiJ~emellll. (+) Jnd (-) denote.
ar tlllTartl~. the two ~ I ue~ vI Pm Eq . (l.g) IS the tntere!·
emem dlsplacemenl wmp:lllbdtty ~ondlt1on_ while (l.h)
IS lhe Intere1emelll .. tracllon-reClproCllY" condl1ton_
A~ ~h()\... n III [I. :: \. Eqs. (I .J to I.h) may be treated
as the Euler - Lagrange equations and natural boundary
(.:onuJllons o f geneT:lI \arlatlonal prtnClples in four different
wa~ S H summarIZed below
li[Llu.E,al]" 0,
JP m
(I.,)
Pm
II
ti ui ds -
(I.e)
11m
I)
tm
IJ n--IJ (ii I - u)ds
I
0"
are all arbitrary. We Introduce :In Independent. ullique.
displacement field uip at Pm and enforce the lnterelement
(u l = U ip
o iJ
um
- f5
E. u)ET x E x V .. uch that
(a.
Js
(4)
Case
4'
Here again the trial functions uL OJ rand €i J
Jre arbitrary. In this case. which is a SimplificatiOn of Case
3 above. Inasmuch as TiP and Tfp are ar bitrary on either
Side of
Pm.
we set
liP =(oi j nj)+ and lip
= (oi Jnj)-Jt
Pm' The vanationai lprinciple in this case becomes:
li L (u. E. 0. U p) = O. where
where
~~J [1-:E1JkIiEklEIJ
11m
+0 IJ 1U l l .J)
E IJ )
IIUll u\
( 5)
Note that in Eqs. (3), (4), and (5), the integrals
over the e!ement ~ volume, n m' are identical to that in
Eq. (2) and hence are not repeated.
The variational problems involved in (2) to (5)
are multiple S.lddle ~ point problems and represent the
most general bases for formulating a wide variety of
finite element methods such as the mIXed, hybfld. and
mIXed~hybrid as well as the compatible displacement
methods. The question naturally arises, if the multifield
variational pnnclples as in (2 to 5 ) are used, are there
anv crite·T1a that one must use 10 Jpproximale each 01
where
(6J)
( Ker(C))'
=!v
(K"(P))'=
€
V
,~Jpm Vi tip ds =0, V tp€ Ker(D)j
IrE T, ~ lIn
m
(6,g)
m
r IJ '('l.J') d,
th~
several vaTlJbles (ui' €ij, Oij. Uip,TiP,Tip"et c. )
In each element. such that the finite element solution
eXists and is stable? Even though a wide variety of hybnd
and mIXed elements have been developed over the last
20 years [see Rei. 3 for J comprehensive summary],
several of these were based on t rial and error and were
plagued at tLmes by lack of stability and appe:nance of
zero~energy (kmematic) modes (other than the flgldbody modes) al the elemem - leveL due 10 the lack of
mathematical answers 10 the quemon posed above . In
this context. BrezZl [41 prescmed what appears 10 be a
ploneermg study on the existence. uniqueness. and stability of SOlUll0ns to a variaIJonal problem with a single
constraint. i.e. a vartational problem with two vaTLable
fields. Later, Ying Jnd Atlun [5J extended Brezzi·s work
to slUdy the problem of J hybnd - finne - element solullon
of Stoke·s flow vf In incompresSible flow - a problem
which Involves three vanable fields and two Lagrange
multipliers. Recently. the eXIstence and stability condi·
tlons for discrete solutIons based on the most-generJi
multiple saddle-pomt problems 01 solid mechanics. based
on Eqs. (2 10 5), have been systematically explored [6 to
81. Here we present these wnditions for finne element
soJullons based on the variatll.mal principle m Eq. (4)
which. m J way. IS the more general of Ihe four cases
presented Jbove. Thus. finne dement >alUllons based
on Eq. (4) eXlSt. Jre stable . lnd converge. provided vnly
the follOWing ..:o nditlonsare met
[Ker(L))'
=j€ €E,
~ f €ij Tij =O. V T € (Ker(p))lf
(6.i)
When the constant 11 in the above relations is positive
and independent of the finite element mesh used. conver·
gence of the solution is achieved. The above conditions
are somel1rnes referred to as the Ladyzhenskaya [9J.
Babuska [9), and Brezzi [4) conditions. The reduction
of the above global condit ions 10 an element level is discuss·
ed In [5.6.7,8j.Tt IS important to note that while Eqs. (6.a
- 6.i) Jnd theu local coun terparts. specify the conditions
to be S.ll1sfied by the respective fields approximated in
each nnlle element, they do no!. unfortunately. give any
clues as 10 how one should go about choosing these fields
to sal1sfy the stated cTlteria. ThiS question has been a subJect of much recent study [10-14], and some frulllul
results are summar IZed below.
1.2, Elemem Development Based on Hybrid-\lixed
Formulations to Satisfy Element Stabilit y Conditions ::md Avoidance of Kinematic Modes We mall
consider two specific examples: (i) a mIXed finite element
method. wheretn the fields Qjj and Uj. satisfying the strainstress relallon. and the Eqs. (I.e. l.f, l nd \.g). J prion are
assumed tn each element: and (ij) a hybrid fintte element
mel hod w herein an equ Ilibrated 0i j that satisfies Eqs.(Le::l nd
\ .0.
pnorl. is assumed tn eJch n m . and an tnterelemem·
that satisfies (I ...: Jnd
..:ompa\lble displacement field
J
iiw
Sup
V v € \'
rn
JPm vi tIP;;' a
a
It ·1
Vlp€Ker(D)(t'-l b )
P T(pi
\' ~ V
l.g,. a priori. is assumed at nm only. TIle vaT1ational
baSIS of the above mIXed method can be deduced from
Eq. (2) \0 be:
( J )
j3 II v il V V v € (ker (C) )'
Sup
Ve f
(0.":)
Likewise. t he vaTlallonal basis of the hybTld method
mentiOned above can be deduced from Eq. (5) to be:
(
"
)
wherem UIP is assumed such Ihat Utp : ui at Sum' and
~
el )
oil okQ ==
W I.;
~f n
nl
m
0 1J vlJdn
0., ~
TVo€kenGI
?n~111\'e
Jre
vDVVv€V,,;o
14,:1 I
Jn d\\'~,IOIJ,olJ';,a a
""h el<'
;.,3
'
T
'0 1)
-
- -
r.J
m 11
(Jnu mdependent
01
mesh I, Jnd
O"V('
m
(0) IS the .;omplemenlary energy
density
According to [he geneTJI theory for eXistence and
stability [6,- ,8,]l ndicJted in Eqs, (6 ), the finite element
soluILons based on (7) eXIst and are stable If:
Vo€T
vl B o:
I)
(1 ~ ., )
·)dv
I,)
°
mould be 'n'. Since vi"
at Sum for veY..: o , It can be
seen that Vi appearing in (II) do not mc1ude any global
rigid body modes, assummg that the original bou ndJr y
conditions at Su are such t hat they preclude any rtgld
mOllon o f the solid as a whole, However, e\'en while global
rigid mOllon may be precluded, such rigid motion may
be ~o,nsidered at the element ,level. T~u.s. v0.Jl '"
for 'r ngld modes (r '" 6 for .>- D and _, lor IWO dun<!ns'
10r:~ ) for each element displacement. v m , Furthermore,
a! J IS arbitrary and mdependent for each element. <,x~ept
that wllhlll e'J ch element, am e To. Thus. (II) m3~ be
wrmen as:
°
Sup
¥ a € To
~J
III
nm un
I)
VI'!', d
.»0
" j
'"
V €
Y
\ I :.bl
'0
...
where ~d denote non'T1gld modes III each element.
suffiC ient cllnd Lllo n for (I ~,b) to hold is:
A
(9.d\
( 13 )
":OTTe,p\l!lJl!l~
bJ~('d
~(\ndIPons
for 1111l1(, eicmc rll
;,olull()n~
"n 1 'II JT('
Lei )1 ~ be the number of stress modes 3ssumed III
<'3lh element n m , :md leI Nqbe the numot'r oJ! dispb . :.
n
md
ement modes III e3ch Hm. Then Ihe dtm<!nslOn 01 Vi
'vo€
IS
(:\q-r). Thu s.
( IO,J I
i lO,bl
"h er('
( 1.1 )
\- , t \
.<'
Thu s. lor (13) [0 hold. ) 1{3 :;.. (;\q_rl JIlJ ,he 'Jnk
01 !.J~ m :)/lOuld be (Nq -r ).
I
t IIh l
Rem3rk 1 If We (Q", q) III e:a~h element. \-'
ed lS Wc Iq, ql . ..:an be wrmen as:
T,"'jr 1J c'H rr!m 1 r iJ,J""J
W IJ
- 7'1·
I))
Hl Q m V m
I
IIO.d\
m
W,
"
\'~() lSJ~J,'IIIll'J
JnJ
.)J
J
then 11 .::a n he ~hown
t..m ":3 n be wrLllen as:
_
_ \
k_ m : Bm
H
!J 111
Oq\(L.)\>OV\ E V..: o
am
i 15}lh31
Bmt
the demelll
>tl11ne~)
I'!lJt!IX
( Ib I
( II I
le! 'lie JU!\o.'mI\JI\ ,11 01 T" .11 1111 he HI Jnu Ih:l1 1\
'"Ie':1. fil<'!l 1III :!lll"'l1es !lUI ill n JIlJ thJ! tho.' fJn"
'I\e ':lJtr;\ [3 III
'0
Hm
( 1:' )
Soo
\
0 1 '" a mt
T\).denol.
tnl'IJI
In :he :\\":IJ 'nethoJ ~J~L'd 0111 \"1\. 11 11 is pos~lhle
to ':'I<' :~<.l ~:p 'cHm Pm UllIqU<'I\ ,nlll rl m . 11 Ii rm111edl'
J[e!\ ,e,'n 'h:H 'h .... .:ondllilln ,10JI t-e(omes ~~nonYIl1\JU~
"lIh r<l J 1 B,'ln ..:nnJalon:; lmpi\ thJI
V r} E T(nrT,,)~
(0,
OE
"here !.J m
IS
defined 'lhrou2h
~
,- I
Here. ~m mcludes both rigid and non-rigid mOOes.
and Jim (-::m):; Nq . Note lhat the rank o f ~ IS the same
as thai 01 B*1n
Smce Wo,;m (0. a) IS poSlilve delinue.
II follows IhJI Ihe rank of Ihe element suffness malnx
IS (Nqr I. provided Ihat 01 ~m i~ (Nq- r).
Remark ~ '\IOle that both 0IJ Jnd vi In \ I-n are compo·
nent s In the '::lTleSlln system '<I T he momentum olilnce
o.:ondlllon Involves Jllferenuauon 01
0lj W.r.! X1' while
the ~lTlHIS v \LJ)abo Involve dllt'ere nlla\lon W.f.t '<J' In
thc usual lsoparametm: dement formuiltlon. the geomel·
flCl1
transformation
between
the (no ndimenSlonal) ..
p;Hent" ekment Jnd that In th..:
ph~slcal
Jomatn IS '<J
"'Xl t E k) ..... here EJ... usually laken 10 be - I";;; Ek ..;; I. are
o:uf\llinear
..:oordtnate~.
In J . hsplacement iormulatlon.
one usuall~ aSsun1<.'S u\ = u l (~l. Jnd In an isoparam etTlc
-r eprt"se11lat Ion t h ... repre>entatton ror Xl J S well as u tCO nlal1l
all eq ua l number L11 bhlS fun..:tl\l!lS m ~k
T he stillness
matflX .)1 t he d ... ment In Ihe Jlsp!Jcem ... nt !ormu!Juon.
whl,'h Jepends un W( th l aXk l
..:a n be IDo ..... n 10 be ob·
are 384 c hoices for a stress field wllh M(J =54 ; all of which
lead to stable and objective elements. The 'best' selection
among all these chOIce s may depend upon : (j) Ihe lowest
e lgenv:alue o f the matrIX ( B·m) ( B*ml) ( B*m betng
dcflllcd tn Eq. ( 14) 1 and (ii) the capabililY of Ihe ca ndldale
stress field to represelll the cardinal states o f stress of
pure tensIOn. shear. bending. and lorsion in each eleme nt.
A ..:omprehensive study at suc h tests is gIven in (I -:. 131·
Remark -l . Consider a mIXed hybrid element 01' a general
curv Ll mear shape and tntroduce a geometric mappmg ot'
the type "i = xi( ~k). wah - I ~ ~k
~m = O. then. II has been sho .... n [!~ 1.J1 that requlIe·
menlS 01 in\lanance may be mel b} represenllng the str~~s
tensor 11l t he alternall\'c !orms:
a I) e
O=O Ij(Xk)~i~j'
=OlJt~k)~i~)
.
a Ij
€ T
I n ISUp;ITJlIletTl': dlspb.: ... ment tormubuon In th ... XI .:oor·
ditute ~y~t ... rn. then liS r ... pre~ntJIl\ll1 In any uther o:afte~lan
= olj ,~k) ~ 1 ~j ,
a' J
,T
,T
=al]\~k)~ 1-::]
2
01j
e
lllthll ~onal
To malntJIn the Obj"'CIlVlI} .)1 the demen t ~lLlfness
1l1alrLX In J mlxed·hybrld formulallon. 11 has been ~h l)wn
[II-loll that the ,ITeh tensor. q. Jlou!d he Jssum ... d 111
In ... lem ... nt·I\)~al .:o\)rdlnate ~ystem Jnd .!!2.!. III J glob:.1i
.:oordln:at ... ~~~tem
el ... m... ms "t :;quJte Jnd ~unlc (or re..:t JngulJr Jnd
r"'dJng uiJr pth1l1t ,hlp"'s. re~p"'~ll vei\
Here. the Ih~'H\
,It ,ymrnetrt,' ~fllUpS ha., h... en demonstrJled [11 .1 ..'1
it) 0 ... J u')<!tu! tooi trI _hll')'il1l~ i.:J)t-order ~trc))
!1 ... ld,
( \l iJ = " 4 /I that I"'Ju 10 Ih ... ·lllalrLX ~",m l see eq IJ)
tlr ran k ("4
rI. Jnd a ,ulfness malH" ~m. Whl..:h IS nb·
.
a)
( I~,b)
a'J
9 IS
( I">
To
t~k ) glg).
~_
~'m = q~'1l ~T ..... here
~k (~m)
en
=0 IJ
h}
Lei
and I!~ (~m) be the covana nt and ..:ont ravanant base vectors
. respecllvely. of the ..:urvtltnear ..:oorduut es
Let ~k
represent the ,;;ovana nt baseveclOrs at the centrOid. I.....
£k = .! k t ~m = 0), and lei ~k be a ..:arlCSlln system Jt
Ject l\t' o r ()b~ef\er HWar lJnt
- -ThJ I meJIIS. I t k m IS the dement stlffn ... ss malflX In
s~~I ... m ~. = q~ IS given
"I.
( 1~.c)
( 1:-'.J I
T
Ol her possible re presentatio ns are discussed 111 [ loll.
In l ib) To IS the spa ce 01
v.hue T
IS that II! differentable
,tresses.
It IS )<!en Ihal (18. J and 18.b ) can eaSily represent stat~~
')1 .;onsta1ll ~'Hess tn the ..:art eslJn ,:oofdlnate S}Slem Jnd
h ... ,ll· ... _In pass the so ·..:alled .. ..:onsI3111 stress" pah'h
Ilrlll...
leC\l\'e and JI\\\ <I t Jnk ".:j
r l In th\) ..:aSt'. J ..:anC,lan
":llll1dlllJt~ ,ntem lo.;ated Jt the ~entrtlld 1)1 Ih ... c'l ... m ... 1l1
Jnd Jlllng Ihl' J x ... ~.1\ ,ymmt"l1\ III t he eh~m ... nt. I~ u.,ed In
[11 1-lI.bolhOIJJnd\I.J[ ( 1.]lmplytngiH I i,,] IJre
J"'Cllmpo,ed 1IlII) l1lvarlant Irr ... du":lblc ,pJ..:e, uSing .;rllup
IhellT\
In lerm'!! III IheS<.' Irr ... du":lbk repre:.entJIHln.,.
the maITI'. '\.3\ I B-*) ' . .:orrespo ndln~ 10 Bl a. \" ) IlU e:a..:h
element. b ... .:nme'i "4UJ,1 ·JIJl!.onar: nm s. group Ihellr~
... nJol",., ,HI'" to pld, 0 1] 11l ~J..:h ... Il.'men!. for J gl\en \\
, u..:h thJt I h ... rcs ultmg elcmem lormulJuul1 IS tn\l,lrtJnt
JIlO'IJbk. It ha,h ...en;huwnll: 1-l]Ih.:tt (I)IOIJIOUI nllu ... J :>quar ........ I\h :-':q - r '" ~. Ihere Jr ... 1..... \1 po)slbl ...
.:h\)\~~, ttll J IlW-paIJmet ... ! c'qulhbrat~d ,tr ... ss lle1d
(HI :m In ~u!.ht '1Od ... J "<lUJI'" v.uh "'q _ f '" l.t. there
Jre ':1 ..:hOl":"', ;,H J 1; -OJrJlllel"'l ,tress li~ld.lll1 ) I.l! In
<!l\!hl lII.d ... d
..:tlh· th~!" Jr ... eu::ill ..:hol ~ il)T J "r ... "
h ... ld With \1 3 = I . . Jnt.! 11\1 t'li J 20 l!uu ... J .uhe· dl'::lc'
(ulhIJ ... rmg
~1 ~J
. v.here
J
(I)
~1ale III ..:onstalll stress. :>.I).
Q = (II
He o.:01lStJ1II S. it IS se ... !! \hal re preSt"!II·
Ilio n ( I~.:) .:In pass the pal(h lest If 0 1) (~ I tndude~
tun..:t IOns suo.:h thai
tJ9.a I
ap,
til an lsoparam ... utc 10rmuiallOn. ,a'm
)Implc pol}nomlJl til
11 is pOSSible. In genenL
thai J polynorlllal re pr ese ntat IOn ~X15tS for ail l~kl tn
( 18 . ..: ) whICh passes the patc h tesl. Ho wever. th ... mess
ticid ..... 111 nOI be. In general. nt the "least-order"
SI!1( ....
1~
J
r.
Ot! the u lher hand. t Ih.d) ..:an pJSS the pal..:h lest
a I) I ~ k) tndudes fune! IOns such I mt
a 11 1 ~k
,=Cmn(~:n.~I) I ~n
~J t
:i
(~O.J
I
a t'- a ti
",e rnn -
aXm aXn
For the usu:lI isoparame tric formul:l1ion.
It
is secn
that ali I ~k) of C~O.b) 3re no longer smlple polynormals.
Hence. represemat lOn (IS.dl wilh po[ynot1llJi funCtions
a ij i~") \\lll nol. In gener:il . pass the patch test. However.
(lb,el will pa~s Ihe pat~h lesl. ~mce . in this..:ase.
\
whi.'r~
limn Jr.:
~lllhtJIHS .
pol~normJI
order)
Jnd hct1l:C
re pr ~~en[Juon
J
~
I )
sunple / .. ,>,en least-
\m duJing
":OIlSlao\
terrns l \\!llsut"ii..:e (or O IJ I~kl.
Re m:.H" :)
To formulate In lsoparametric curvilinear
n1l.xed.hybnd demem. o ne rna) use JltcrnatJve represent-
ations for
)lr~S'
JS
III
(l8.a
(~k)
18.e) Jnd JSSUTne "i
to be III ihl! ,,;.Ime (orm 3.S '(i(~k). NotethJI "iJrecartesian ,"omponcms or displacement.
For the J!tern:Hlve
repro:s~lltJll0 lb ,)! ~trc~S::lS In lIb), the bilineJr form B(
o. \' 1. r'or .!Jch element . take s () n the respe(\!ve re present31 10n
:J nmOmnl~k.hLk. Jk/J~:J~/(det J)d~l d~d~3
C.:!.:!·c)
:
J n omnr ..I:!-.· \\lnJ
.
nn Idet
~~11l
v.lwe
I
I.m Jo!note~ 0 ( )
(~~.d)
a ~ . J mJ = (a '\ m a ~J)
Jnd
:1 m): JmJI~k.=O I.
In R<.'mJrk.3 ,:olh:ermn!! >qua res Jnd ~ubes. J group IheorellC:li method which .!nables a choice 01 a Ij (Xk). for
gl\"('n vi / xk). tll:l! gives the ra nk tN q - r) to B"m was
descnbed For such squares and cubes. the bilinear (orm
IS ~ompuled uSing (14). Compa rlllg (l ..Q Jnd 1.:!2l. It can
be seen lhal there exists no sunple way of choosmg the
stress 3S m (18 ) lor .:urvlline3r elements -;uch that the
rank ot B"m is delermtned :l pnori. However. it has been
de monstr:lted 1Il[1.3.14]that If 0i) (~k).LIr olj O~k) of
(I i'.>.b-I b.e ) IS chosen to be 01 the SJme polynomial form
(i.e. by "eplaclllg "(k b) ~):iS that 01 0ij (xk) which IS
Jd!WU ~y uSllIg ~roup theory lor squ:Hes and cubes.
th<'n the fanl-..: .)f B*1l1 IS mallllal!lo!d 10 he (Nq- r )
even ',n .en ,evereiy Jistorted eie!1le!lh. Funher. 1\
11J~ -ee~l .. i<'ari\
JeTll(llhlT:lteJ tlut :h~ leJ~t .)rder.
J
invar ia nt . isoparamet ric, curvilinear mixed-hybrid elements
are less d istortion- sensitive and lead to more accurate
results compared to the standa rd d isplacement elements III
a variety of examples. Methods for suppreSSIon of zero
energy modes m hybrid-st ress elements. based on heunst\c
reasoning, have ;tlsa been mdependently suggested recently
in{161·
1.3. Use of Symbolic Manipulatio n In recent YellS.
there has bee n a surge in research activity. likely 10 burgeon
tn the (uture. in the area of co mpuler symbolic manipu.
lanon: (i) in the evaluation of element stiffness coel"fiCients
for HOlte elements to elimillJ t e errors introduced by
nume rical quadrature and to improve the el"ficiency 01
the
generation of relevant element propertleso. (ij) to
compare the performance of different elements and to
synthesize desirable elements. and (iii) to capitalize on
the symmetry and /o r other properties of a panicu!ar
elemem or group of elements 10 generate thelT chara.:teristic arrays in an efficient way.
The computer
symbolic
manipulation
systems
MACSYM A [17]. INTER. and FO RM AC have been used
in several studies. Ref. {181 used MACSYMA 10 gener:lte
the stiffness coefficients of finite elements as funcl ions
of the specified material and geometric parameters while
earlier studies (19] employed simple algebraic polynomial
mampulators. The manipulator INTE R was used in [.:!O]
while FORM AC was used in [:!l]. While most of these
studies pertam 10 linear problems. the use of MACS't'MA
in nonlinear finite element analysis. wherein the geometnc
nonline:H1ties lead 10 cubic and higher-order Sllffness
terms. has been discussed III Ref. [2:!] which also presents
a sununary of studies employing computerized symbolic
manipulation up to 1980. More rece nt studies emploY lllg
symbolic manipulatlon have been presented by Park and
colleagues [23 - 25] who used 5lJch methods for J Founer
analYSIS of spurious mecha nisms and locking III the (inlle
element method as well as to construct a rank-5lJITicient.
one-pomt integrated. four-noded plate- bending element
based upon a discrete Founer analySIS techmque by which
the uncoupled discrete o perator governing the transverse
displacement is directly wmpared wnh Ihe corresponding
contllluum operator.
1.4. Adaptive Mesh Refinement The economics
01 tinae element comput;ttions dictate that a given acc uracy be achieved with the mimmum man-hour as weI!
as .:omputer costs. Convergence of fimte element resu lts
is usually sought in several ways: (i) h-convergence.
i.e. retimng t he spatial mem while keeptng lhe order 01
interpolation in each element the same.:md (ii) p -.:onver·
gence. i.e. increasmg the order o f function-mle rpolauon
in each element while keeplllg the spatial mesh the same.
or (iii) a co mbinatlon o f both. In any event. the IOta I
number o f degrees of freedom is progressively increased.
Early studies [26] focused on the determlllatlon
o( o ptunal finite element meshes (h-verslon) which mml'
mize the error of the fimte elemem solulion ior a given
numbe r of deg rees of freedom. The cost 01 this optlmlZ'
ation itself is often prohibitive. :md pracllcal "gUidelines"
(:!7] are often used to generate efficient meshes. An
lVenue of research 01 greal promIse. that has been recently
pursued and one tml is likely to rece tve much Jltenllon.
C("Inc.:,,~ ad~p\lh'
letl;;ement of fimte element meshes [
::!lS ~91
fhe mLsh ,efmement (h or p. or a combuutlon
:hdfo\, IS .,d:lptive In ,he ~nse that each step depends
UII :t.e 'nformatlOh con ... erlllng the error mdicators and
"( torrl){ "stulI.tlors prov 'dt'd by the previous ones, The
Nu b:.:,[,
hI! srlf adapllve (:!S .:!91 In the sellse thai
ilL Usel
n,cr .. ctloll l~ ne..:,;ssary to trigger the adaptive
mesh re!:n<:ment process.
1.5.
Tra nSien t
Uyna mic
Analysis The standard
approach 10 recent years for transient dynamic response
Lit solids. and transient analYSIS of solid-tlUid IOleraCtlon
problems. has been 10 first discrellze. in space. the govern·
109 partl:ll different III equatIons. In space and tllne ..:oor·
duutes. to obtalll a system o! ..:oupled (nonlinear) ordinary
diffcrenllal equations 10 lime. There have been Significant
stndes made In Integrating these time- differential eqw·
tlolIS uSlIIg impliclt. expliCit. and mIXed forms. ImpoTiant
results have also JPpcared recently concerning operatorsplitting and panllionmg methods. For an excellent sum·
mary o! these Jnd related tOpiCS. the reader is referred
to J recent monograph [30]
2. Boundary-Eleme nt Techn o logy
Since about
the early 1970·s. the applic:Lllon of weighted reSidual
methods. wherelll only boundar)' resldua.is enter IIItO
..;onSlderallon. ha.\e re":~lved Ihelr overdue allenlLon \
31 33]'
Contrary 10 ..:bllilS often made III ..:ommercI:LI
sort ware - vendor crrcies. the boundary element method
is nell her "'different" nor "better" than the tinne element
method -II IS. III J way. a (inl1e element method If the
modern :Jc:Jdemlc vIew th:lt JII discrete methods :Jre somt
(orm of weighted resldu:J1 methods IS taken. ThiS IS.
III fact. Ihe VIew that led to the lII\ToduclJon of gradwu
courses IIIled "Finlte Elements. Boundary Elements. and
Other Computational \lei hods III ~teclunLcs" IIItO the
currH.:ulum at G<!orgla [nslllute of Technoloogy in 1<)-There Jre "everal ways HI which J so-..:alled bound:lTY
<!lemerll method rna) be deVised \3:]. Two ot thes<! .HC
ba~ed on
(i) the use u t J "SlllEula.r" solution to wher
th<! ~nltJe differential <!qu3tlon \)j- the problem or to the
dL1fcrenual operator of Ihe problem th:J! ..:ol\taUlS the
tughe~1 order deTlvatlves. as test !UIICILOns m the weIghted
resldu31 approach. Jnd luI the use ot asymplOllc solullons
to Ihe differentul cqualLon ot' the problem lor )Impi<!
domams such as seml-mllllLle domaills. elc .. as \TuJ tunc·
lions III Ihe weighted reSidual approa..:h. These are hrL<!tl~
dls..:ussed belo .....
Consider the prohlem ,II linear 'sotroPIC elastl(Ll}.
Fllr an appmxlm:Jte )OlutLon. consider a global ITJallUllc·
11011 Uk (Le. valid over the entITe domaml Jnd J glohal
te'St fUn(tloll vk' Let the ..:ompallbllity condition and
Slress .. stram rei:Jtlon be S.lllslied a pTlon. i.e ..
n
0IJ(u\.,)=Eljkr u(k.\:)
wht:r1;'111 a II I Uk)
't ~
11 E .. I
Implle~
'"
0ij(Uk).j
a 'J
U
,
~
, fi
~
oJ
"
"
Uj
'" 0
'" n
s,
"
(:!S.a)
(25.b)
Su
(:5.c)
The .... elghled reSIdual forms of (:5a- ~ Sc) are
I ~6.a.)
I ~6,b)
and
J
SU
(ui - uJtj{vk)ds'" 0
(';b.c)
where Ii (vk) '" OlJ (vk)nj' USing the divergence theorem
III C6.J) . one Cln "add" Eqs. (::!6.a-~6d to ob taLlI a
summed weighted residual equJtion:
SIII>:I! Ihe mJter]JI is linear. we ha\'e:
0IJIU\.,JVII.J) '" EljUU(k,~)v(i.j)
'" a U. (VI'
U
(k.~)
(~8)
()SLIIg t ~8) III (:!7) Jnd uSing the divergence theorem
on the 'Iolume IIItegral. II IS eas)' to obtain:
-f
(l9)
Su
'" frz
"d
°IJ (\\.,):EIJk\\(1.; \\
::;: E k 2 i j, etc" the stresses in (23) and (~4) are symmet ric
tensors and hence 53l1Sfy the angular momentum balance
conditions of the noo- polar contmuum. The only other
condl1lons to be met by the trial function Uk are:
!Oij (Vk).jUi + fiVild!2+JantiiUJ..,)v\JS
n
(30)
I he .tress a q JefLved from U\.;..
~:l' th~ "U;II ?r"!,en\ El\.;.I =
E";"I
"'rlen' 'n Jetimtloo.
an =
lhat the values II (uk) and u l 01 the trial solulion may be
prescTlbec + at approprllte ~gments of
The key step IS now to make a specifi..: o.::holce for
the ~ iunLtton vk We WIll o.::hoose vk to be Ihe )mgular
solution tor a po lOt load m an mfimte spa..:e. In the
present 3-dimenslonal o.::Jse. let the po lOt load he UI the ~~
direction Jt the 10CatlOll '<m := ~rn' thus. 11 IS seen thai
vk SJllsfies the equatlon
an.
rO!J\Vk ll] ~ Oh' m
~mH5ll el = 0
lor tl"'I.: ..~1
(31)
.... here 0 '''m - ~ml IS the DLTJ": (un":I]!Jn. 6 r l I) Ihe
Kwne..:ke r deltJ. Jnd H denoles simp ly Ihe direction
Ot Ih ... load. U)1I11! 1.'11111 tJOI Jnd reo.:Jlling the property
01 DlrJo.: iUTl..:tloTls. 1I I:) )eeTl dut
wheretn the superscript b denotes a ··boundlry-value··.
The so-called ''boundary-e lement method" seeks to
satisfy Eq. (34) in a weighted residual sense. ~ote rust
that while l volume integral does appelr 111 (34). 11 does
not involve t~ function Uk. Eq. (34) may be satlsfied
in Jny number of ways In a discrete sense l3~]. includ ing
c01l0callon and Galerkln and "variallonal- techmques.
When a Galerkin scheme IS used. on each boundary
element. arbitrary -order Interpolallons may be used for
u~ (e
b)and ti (x! ). In
terms of I heu respecllve nodal·
values It the element nodes. TIllS results In an equation
01 the form:
(35)
~9*=~9*
where ~( is a vector 01 boundlry nodal Ji5pll~ements (
is the vector ot" boun·
some 01 which lre speCIfied). and
dary nodaltraCltons (some of which Jre speCified)
[t 15 Important to note that a singular solullon [0 Eq .
(31) IS tmposslble to obtain when Eijk( is stro ngly am·
sOtr0PIC lS In modern compoSile mateflals. Theretn lies
the essenual limttatlon 01 the boundary element melhod.
Ano[her simple 3nd more direct "boundary -element"'
method may be developed when the trial iun..:tlons Uk
<;r
The .;,oluIIlln {or a pomt ILIad In 1Illtnile :ipa~·e. when
the mJ[ er III ~ lSO[ fOPlo.::. I. e wh ... n the [ensor Ei j kQ, In \ ~3
:ltId :·H IS ISlHroPh: Jnd !TI\"\11\"esonl~ 1.... 0 ..:onstants E Jnd
v WJ, ~.I\"en b~ Lord Kelvlll more Ihan J ..:entury Jgo (3~].
In KdvUl') ,OIUlllll1, the dl)pIJ..:ernent m the lth dlTeo.::llon
al 'm due to J um! I"Jd III the I.th dlTe..:tton JI ~n 15 given
b} VII. t'<Ill' ~m). Jnd the tra..:ilon ill the lIh JlreCllon
011 Jnlflemeu >Urlao.:~ 'Wllh normal ,;osme) 11k at '<m.
du ... to J unll IOJU 1fI the lth dtreo.:llon at ~m ~s given b}
tCit'<1l1.~rnl T hu~.irom ( 30 )JndI 3:) \.\ehJ\e
themselves satlsiy the Navier differentlal equallon. but
!!2.! Ihe boundary condit iOn, i.e ..
(36)
lsymptotic solU1ions that satisfy (36) may be Jemed for
seml-tnltmte strIpS. near holes and voids. ~rlC\... etc.
[34.35]. When (36) IS used tn (~7). one.)btalfls the ..
bound3ry - weighted -residual" equat ion.
In 1;.;1. ~ml~Jn~ poml.\.\hd'::'mis Ihe
n
duw-fr.\ \anar-ie C'uher .n
"r ,)11 )!1 h .nuI,,:a led . ' ote
thJ! lie Ke:\\n 'lliu[IIIIl, \11. Jnd III Jre 'lfI~ulJr JS Em
- '\~II
\\ h.:::: ~ III b IJken !IIlhe lirml to JPPfl1J..:h J bound·
Jf\ ppml ,lne mt!'>1 ~nmld~r (Ju..:h~ prlll..:Jpal value~
ot the .nt~~rlb in (33) Jnd !hu~ obl all1 l31. 3:1 Ihe '0..).
.;:.olled hounOJf\ IIllegralequJIl()Il.
The methodology Ifl1phed III Eqs. P6) l!1d (37)
has been labeled "the edge-function method" m[34. 35].
Similar ~OllceplS may. of o.:ourse. be e,<t.!nded to
either geometrtc311y or m~tertally nonlmeJr problems.
Hcre we illustrale Ihe case Llf finne deiormalton mel3s,
IIcilY. using the rate t'ormulallon [36.37]wherem [he rates
Jre rele rred 10 the currently deiormed conrlguranon.
Usmg a cartesian system for slfT1pliclly.let Vi be the velOCity
oi a matertal panicle as a iuncl10n of ~urrent wordtoates
'<k With Lij = Vi.] being the veloclt)' gradient. Dij '" II:.
(L I] .,. LJj ) the rl[e 01 deformation lensor. and WI]::
(LIJ -
II1J\' )1 ~uur~<! I"h' lerllle>.l <!J~tl\. thJt e\'en II one had
~IJ rteJ ,Jut .\ Ilh tTlll lUIl..:Jlom Uk th;1t ;,a\Jsty the JIspIJ'
~"'J:l"'llI '111UllL1Jl\ dJIlUJIlon 1 ~~ ... l J rflorl. llle >1111 end,
111 ,nill:OI
.,.
l;i ) bemg
W ij)'
... I[
the Spin tensor. such that Lij "'( Dij
Let Si] be the rlte o r Second Plaia - Kirchhoff
stress tensor and TiJ be the CJu~hy stress In Ihe (ur rent
..:onllgur3 tion. The tield equallOIlS to Ihe rlle larm Jrc
I·; -.." J
o
(L:-'IB):(Sij+ T1 kVj.k)j+ f)= O.
(46)
Oil ::: f(Dkf2. Ilk£ . oH )
(1.)"'3()3x)
(30)
"d
(30)
Ilij ::: g(DU, O:kf2. 0ke)
(40 )
wherein both f and g are isotrOpic tensor functions . It
has been de monstrated [39 . .11\ that. as long as the te nsor
functions f and g o f (46) and (47) are general enough.
one may use anyone of the mfinnely many stress-rates.
such .l S (43t on the left hand side of (46) and (47).
The general rebtlons (46) and (47) produce physically plaUSIble stresses in homogeneous deformallon problems.
L e. nono~lliJtory stresses In Simple filllte-shell.
In a iinne-deformalion melastic problem. one m:ty
decompose Ihe melastlc SIT3ms as:
o
(A]\IB).
5 lJ :: 5) i
(CompatibIlity): DlJ = '-:tv iJ + vpl
(Boundar~ ("011(1Itlon51 IldSI)'" Tlk Vj.kJ
::, J at
51
In linLle delmmatton problems. one needs to ..:onSlder
the obJ~..:tlvlt} Jnd ffiJt;:n:!I Ir3memdirler;,~nce 01 the
constitutive lJy, To IhlS end. one may ..:onslder an oblee·
live fal e of the Klrchhorf stress 0 1) 1= J TI J' where J IS
Ihe
JawhlJn \) 1' deforffiJIlOn grJdlem In the ~urre!lt
conllguratl\ln) Lei the ub)~ctlve rate be 0i J. y, hlch ..:an be
anyone ,)1 Jn tnlLntte!y many po:,slblhlles. For l!lstance
13Q J'
(~3.a )
,
a'i •
Lf..IOkJ ·Olklkl
(47)
d.,1
(48)
wherem )uperscnpn e and 1 denote ebsll<.: and melastL~.
respectl\'el~
Henceiorth. we Will consider. wlIhoUI loss
01 gener3Iit~·. an lsotroplc-hardenmg eustoplasllc Tebllon
101 metals twnh small elastIC stlJlns) ]37. 3<)1 as.
tCu!ter-RI\Jinl
(43.b)
,
,
,
°'1
Llf..of..J· 0lf.. LkJ
(.n.e l
aIJ ... Lf..1 0Jf..
OlJ
(mLxed)
-
a if.. lJk
\\ d, Of..] -O'k\\ Jf..
imlxed)
( 43.d)
dJumJnn- 'all
Z:HemDa)
(.1',~
1
0lf.. nJf.. IGreen \I ~Glilnhl
where 0::: 1 If the matef\.31 undergoes pbsllc strain. p.
and A 3rc Lame constants. OISlhe yield-mess m uniaXIal
tenSIon. k IS the slope 0 1 the true stress versus logarithmIC
plasllc )trall1 curve m unuxlJI tension. and O"k ~ IS the
de\lator of Kirchhoff stress. When elaStiC 'MJmS are I'en
'i1ll311. JS \I.e shall assume. oil m (49) maybeldentlfud
JS !he )Jumann late 01 Stress \3 Q ], wuhout In} lI1.:on·
~ISh'rk\ "or slmplicll},.( 49 1 rna} bewrttten.1S:
Q,
a I I I) :he m:II~:lJj f.ltC
nil:: R. I ).. R,I" y,here RJk b
y,h~r~
,
I -ok~o lJ) Dkl:
t.1 ',n
!
1501
'Jl Klr~hhllif mes:.. Jnd
the tool ~otat10n:n 'he
t 5 1)
~urr~nt ~Omll!.UrJIi\ln. h
Jelermmed Irom J pnl:ll ·de':lJmnl the ~urrent d~lormaIK)n gladlem
l. n ,)blC(I1\<, ~()rmllull\e rCIJIHHl ,)1 lh~ tlte IOfm I~
pO~lulaled. 1II\"1f..1rl\! th~ pr1rlelpl~ ll[ 'lbJe..:tI\'lI~ 1401 JS.
The deltnll ion of BIlk( IS app3rem by
po~lIlon
p4,
~O
~ompaTlng
Jnd 51).
""',lte that E1)k( m (50) and (51) IS the Iilstalltaneous
tensor ,II IsotrOPIC eI35I1cII),. when the m31Cfl31 is ~onSLder·
eu 10 he tnlll3l1y IsotrOPIC.
TIle relation between Slj and the laurnann rate OIl IS
In IsotropIC temor lun~l!on j3q . .JO] .. c. under
ob'>~r\'er Iran~l orm.1tLon 01 (l1gld rotalloni O mn.
where
I h
t.1q
II ~IIJH1 mdu.:ed .ll1lSOIHlP~ 1~ ~o he
~onskicred.
,me rna\ ,ntr<luu~e ,11l~rnal \"artahle~ illch .lS the had;
,Hess u~\ :11 .lfll;"l1rO(ll( h;:LTJ~nlt1!! ?I.1'Il":lI~ Jnd 1\Tlle
Ihe ;O!l~tlIUI1\'" ;.1\\, .1,
\3'. 3' I
( 52,a I
t 5':.b)
(52.d)
In wr!llOg (52.b), (51) has been used. The meamng
of FiJk~ In t52.d) is apparent. Note that 5iJ In (52)
is symmetric. by defilHllon.
~o.".. let 'Ii the test functlon for veloclIY. Without
loss of generality, we :lS5um e thai the thesl funclLons
saliS!} the geometric boundary co nditi£'O' (42) . . Let
d tv) be the veloell} strains der tveo Trom vi and 51) (v)
1J
be the second P- K mess-rate denved from v. USing (52.d)
Thu~. the onl~ tield ~quallons to be satisfied 3TC:
[SIJ {\"'+T tk"J,kl.l'" f) "' 0
In
n
( 5j)
Now. we may choose the test funCllon w I such that
It is a solullon of the KelVin problem:
[E ij kQD J I(W m)],Q +8(x m - Em} o..:.p e p = 0
(p • 1,:.3)
(60)
"d
(5.)
l':l \', I be the lest fun":llon. whICh. In general. may
not valH~h Jl Su The summed weighted resldlLJl forms
ot \ 5.; IJndt:'-ll Jlt.'
f n jIS1Jt \)+T 1 k\ J. kl.l'"
:\fw J
( 56 )
0;
f Ii \\
Q
'
,-,lte Ihlt E IJ k \. IS Ihe IsotrOPIC tensor ot hne:lT elas·
11(1\\
Th u~,
.1. \ k J
n
(58)
It IS then immediately apparent that an Integul relation for vk tn Sl and hence an tntegral equation ror vk
on
results from (59) and (60) in just the same way as
in Eqs. (31 to 34). In (59) it is understood that tj (v) would
be spectfied at ~ and vk would be specified at Su. whue
the test functX)n wk is non-zero at Su, and [Elj H Dj i (w)
J are non-zero at~.
Now several conclusions may be drawn. In ISOtrOPIC
Eq. 33). the unknown trial funClion
uQ. appears only on the boundary.
Hence any discrete method to SOItisfy (33), such as the
Galerkin method. involves only boundary- tnlerpolallon
for uQ and. hence. the name "boundary - element method"
On the other hand. when lar e deformations or tnel·
asticlly. are present (see Eq. 59 , the unknown trial [unclln
(for partIcle velocity from the current configuration) vk
appears not only on the boundary. bu t also 10 the inteIlOr.
Hence. in a direct deveJ0Jlment of a discrete method . the
trial functnn vk must oe discrellzed not only on the
boundary, but also In the mterior of the domam. Hence.
\I IS no longer 3. "boundary·element'· method. but rather
a finite element method of 3. specIal kind. If. on the Oter
hand. an lIeratlve solullon IS used and the boundary·
mlegral equatIon resultmg from (59) is VIewed JS l rei·
ation for Ith Iterative value for vk al a n on the lelt-Iund
side. tn terms of the known (i- I) Ih Iterallve \'alue lor
Dk.£ (v) and vk.2 in non the right-hand Side. then o ne
m3.Y consIder. in a way, a "boundary-element" method
analogous to that of hnear elastlcay wllh a rather complex
system 01 "body forces" due the effects of nnlle delor·
mat Ions a nd melasllcuy (corresponding to the (i-I ) th
mlerallve solullon).
In thiS case. the methodOlogy IS
similar to the finite element method wherem only t he
linear -elastiC st iffness matrIX IS used throughOUt. and
the solution for fume deformations and tnelasllclty IS
obtained through a slow process of "modified" Ne Wlon·
Raphson Iter:lIio ns. On the other hand. when finite ele·
ment Interpolations are used In the mtenor of the domain.
it is common practice to obtatn "tangent -stiffness" matrices to account for Hnlle deformations and inelasllclIY,
10 speed up convergence. All saId and done. Lt LS perhaps
approprl:lle 10 label the solullon methodology based o n a
duect apphcanon of (59) and (60) as a "mIXed fmlteelement {boundary element method".
OccaSIOns. wherein a pure boundary element method
(with bounda ry interoolatlOn only for tTial tun~l!ons)
an.
,
and a pure finite element method (with intenor interpol·
ations for trial funldons ) may be combmed JudicIOusly.
are discussed in [23. 4::!J.
COMPUTATIONAL MECHANICS
Here we discuss certam topics in the mechanics of
solids. wherein computJlIonal 10015 have played In the
recent years. and are expected to play In the ooming
years. a slgmficant. but secondary. role.
1. Constitutive Modeling While. until a fe w years
back. simple constnutlye relations such as lSOUOPIC harden·
ing or linear kmemallc hardening plasticity were the mam·
stay of computer programs. currently there is a widespread
interest m the constllullve modeling of expenmentally
observed behaVior of materla !s Im'olving plastic and creep
detormallons under monolOnic and cyclic loading.
The general theory of inter nal va riab les. as skctched
Endochrumc Theory:
tr (
in (46) and (47) fo r instance, has played a key role in the
development of more and more realistic constitutive
models to characterize inelastic behavior. Such internal
variables that are being widely employed include. for
instance: 0) the te nsor locating t he center of the yield
surface in the stress space ("the back-stress"), Oi) the
parameters that characterize the expansion of the yield
su rface. (iii) the parameters that characleTIZe the "bound109-su rface" in multi- yield-surface theones {43-45J
oi plasllcity. and (iv) the back-stress and drag- stress
used to characterize creep rurface. etc.
The multitude of conslltutive relations for melasticity.
proposed in literature. appear on the surface to be unTelat·
ed to each other and to be based on totally diverse concepts
II has been found recently {46,47Jthat such is not the
case and that the "internal-time" (endochronic) theory
{48.49J. the multi-yield-surface theories {43 10 451. and
the internal variable theories {50. 51 J are essenllally the
same. wllh only mmor variations. It is also >hown that
lhe difierentla l forms of the suess-strain relations for
{( . ) denotes a derivative of ( ) with respect to
Newtonian-time or a Newtonian·time-like extem:tl
parameter such as cxtcrnalloadJ
g) = t~'"b + jAn) tr (V· where ~o. Ao are
f(O = (I +
Lame contants
J3r> (linear) or an = {a + t I - a)cxp ( - TnJ(expon)
, S~ldfd\)
c=l
r = ~ r(D . h* = ~
..
I
-
~
KmemallC
H ardenm~
i l l) = ~Jl PI
-
- a
1
ep -
1~
Q.
r(d
.
-'--~ (eP· ~p) :-:( no ~um on 1) for i = l.~ ......
f
-
i= ~, ~(i)= ~J.loPl (o)~P _!,
IsotropIC Hardenmg:
o
Sy = SyP (!p ~p) l-: (linear i)
= TIs: - S; {a "-il-a)cxp ( Tn!}
Jable I SUmm3l\ rJ! th.:
Proe~!
(~p ~P)~(-;JtuT31ed t)
Imclnal·Timc Theory 'll PbSllClty
plastlclty lnd the differenllal forms of evolution equations
for mternal variables given m [46, 47] mclude, as special
cases, the multi-yield-surface theories of Mro z (43],
Krieg [44], 3nd Defalias and Popov [45]: the nonlinear
kinelllallc hardemng theories of Chaboche et al. [50,51];
and Ihe classiC31 Pr:lger - Melan hardenmg rule.
\\ e present here 1 synopsIs of the Iheory developed In
[-46.-P I. We restm:t ourselves to small strams and deform·
atlons. Let ~ be tile small -str:lln tensor Jnd g liS rate. Let
the dev\JlOr of ~ be ~. Let Q be the stress and Q Its rate.
and the deviator of 0 is denmed as S. The rate fo rm 01
the ~tress--str:lm relatIOns tor .:omblned lsotr0plc-kmem·
au.: hardening plaso":IIY denved In [-46. -47] are summaTlz·
ed in the Jbove labk.
The re iJllon s given III Table 1 have been shown [-46} to
be no more Jlffi..:ult to implement In a computatlonal
algorithm than the .:l:mical plasticllY theory. However.
In ..:onlTlst to the d:lsslcal theory. the relalions m Table
I predici ..:ydic h:udnlllg. cross-hardemng. etc. as lccur:lIely .IS desued.
A umfied creep-pbsllclly-theory.
arull)~OUS to Ihat In Table I. has also been presemed [-46.
J-\.
-\ nmher IOpl": In ..:ons\ltutlve modeling that h:ls engendered mu..:h Jis,,:usslOn In the past three years concerns
linue )train pIJSlI":lIY. The "controversy" surrounded
th(' results tor ,he:1T mess. which were found to be osctll·
Jlor~ 1!I !IIne[~:]. In fimte sunple she:lr. when Ihe Jaumann
- \ \111 ::lte wa) u~ed for the emlUllon equallon tor backSlre~) 2. III J Simple generJliZJl!on of Prager-~lebn linear
klllenlJtl": hard(,lllnl! rulc to finite deformallons. This
"Jn omal~" has prompted Lee el al. {53] to derive modified humann rales Jnd \)(hers to "Ihrow Ihe bbme" on the
JaumJnn rale Jnd IIlstead to use the Green-~kGinms
rale
[I has hetn dernonstr:lted In [3 q l thaI none o f Ihe
strt~, rates Jre 10 b~ "blamed". and in fa,t no o ne stressrale ~s preferable 10 others. prOVIded lhat the isolToplC
len",-IT funclions. sUlh as r :lnd g lS III 1-46) Jnd P7). :lTe
e'\panded JPproprlllel~ un the TIght ·hand sides 01 (-46)
Jnd ,,,-) lrrespe.:I1\·e ,)1 .... hlCh obJectlw rale IS used on
:he kit hand -;lde \)1 Ph 1 Jnd (-n).
I 'll\g tluly geneTJI ~'\pans lon s fOI t' and g as In P6)
Jllo.i (J-) Jnll i..I~l!lg. lor e'\lmple. the humlnn rales In
the .elt hanJ ~Ide~ "1 1-46) Jnd tJ""l. modeling .)\ test
JalJ !II l finne deformallon dastlc-plasl!": tenSlOn!\IT'l\lll 'e5t has been pertormed successfully lor the lirst
11J1le ,n luerJture. Jnd Ihe ,;oc31led S.... \1t dfe<:t ha, been
J~''''uTltely modeled
Slullle~ Jlong Ihc lbo\'e lines [ll. l6 . .17] Jre expect ·
cd "1 1'13\ J "ilglllllo.:Jm role III the ne3! tuture III underslJ.nll!n~ materul beh:l.\lor m brge ~lTaIllS.
"'l
COlllputJtio nal FrJ CllI rc MecilJn ics The
rliedlJrlh,;S \)1 lrJOUle (especially ot lIlelasli..: dynamic
lnu 1hret'-Junenslonal 1rJ..:1 ure 1 IS one area where compUl·
Jtl~lIlJI method,; ha\'e played an unquestionably tmportant
10k Jnd Jrt' likdy to contmue 10 do:;o. A ..:omprehenslve
,urnmary ,)t .:ompu t:llloIlJI mcthods In the met.:hamcs of
tradure hasre.:entl~ he~n prepared {5-41
Here >\e "T1ell~ touch upon 1.... 0 topiCS: modeling
,)1 tlvnaml': ..:ra.:),; propagallon Jnd three-dimenSIOnal
~ronkms 01 ~mnedJed ,IT 'illTTa..:e Tbv.s.
In rhe phI 'LX yeJrs ,lr )\). "l1IovlIlg ,mguIJr dement"
procedures of analyzing dynamiC crack propagation and
arrest have been ext ensively developed {55 -59 ]. In this
procedure, the eigenfunctions for displacement. velocity,
and acceleration near the tip of a dynamically propagating
crack, as derived in (60], lre used 3S basis functlons of the
"slngular element" which surrounds the crack-tip. The
Singular element may move by an arbitrary amount of
crack length 1Ilcrement .6.30 in each time increment ..It of
the numerical time-integration procedure. The movmg
Singular element. within which the crackttp always has a
fixed locatIOn. retains its shape 3t all times: but the mesh of
regular (isoparametTlc) finite elemenl5. surrounding the
moving singular element. deforms accordingly. To Slffiulate
large amountS of crack-propagallon. the mesh pattern
of regular elements is readjus ted perIodically.
The analYSIS of dynamic crack-propagation has.
however. been greally simplified due to the recent develop·
ments [60-641 concermng far-field path-mdependem
Integrals governtng the fields neal the proplgltmg cracktip. 3nalogous 10 the well- known J of elasto-statics.
For instance. for elastodYllJnuc crack propagJllon. the
integrll which IS eqUivalent to the ra le of energy-release
IS given by
where \II IS Ihe stram energy densay, T the klnellc energy
denSIty, ti the tractlons. ui the displacements. (.) denOies
IS a near-field path. and
the 131a nme-denvallve.
field palh. The above far -field Inlegr:lis eruble In a~.:urJte
:lnd eificlent analYSIS of dynamiC crJck propagaTion.
uSing ordinary (non- smgulal) isoparametrl': finne ele·
menlS [65. 66].
Another interesting and Impurtlnt development In
.:ompulatlOnal fracture mechamcs has been the enhancement of Ihe Schwarz-Newmann alterllJtlng method tor
solvmg three-dImenSional problems of embedded 01
surface flaws In structural components.
[n the alternatlng method [671. the analytlc3l solution
for an embedded elliptical ..:rack m an injimtc elamc
medIUm. which IS a basic soluuon reqUired 111 the alternat·
Ing techmque. has been limited to 1 .:ublt.: Varll110n 01
normal pressure on the crack surface [68]. This limitation
IS thought to be one of the major reasons for the rebuve
maccuracy of Ihe aiternallng mel hod as t.:ompared to
hybrid finite element procedures or Ihe boundarY-integral
equallon procedures.
Recently a general solullon procedure has been deTIVed
in [69\ for the problem of an mlinlle daSllc medIUm with
an embedded ellipllcal crack. the faces of which are subleCt
to arbItrary 'larllIlOnS of normal lS well as shear tractions.
Later a more detailed solullon. lS .... ell as a general procedure for the evaluallon of the requued eJhplI": Integrals.
wJsobtalned m (""0\.
r€
r
...
I~-
-
Since 1971 no work has appeared in literature to
generalize the solution in [68] to an arbitrary pressure
variation on the crack surface due to the seemingly insurmountable mathematical and algebraic difficulties. While
the analytical solution [69, 70J can be reduced to a closedform solution for a relatively simple loading such as constant or linear variation of the tractions, for a high-order
polynomial variation of the tractions_ the solution procedure requires a digital computer. To obtain the stress
components at a given pOint by uSing a computer. a general
evaluallon procedure [70J for obtaining the partial denva·
tives 01 the potential functions used In the formuhtion
is also one of the key algebraic steps in the successful
applicalion of the present analytical solulion.
Recently a major improvement of the alternating
method has been made in [70. 71 J. In the new alternating
method [70. 71 J, the complete . general analyl1cal solution
[69. 70] for an elliptical crack explained e:ulier was imp].;~mented in conjunction with the finite element method.
The major steps requITed in the nOite element alterna t·
ing method are gIVen In the foHowmg: (i) Solve the uncracked body under the given external load by using the
finlle element method. To save computation time in
solvmg the nnlle element equation for multlple right·
hand SIdes. J spet'ial solution techmque was implemented
PO]. (ij) Using the finite element :.olution. compute the
residual ~tresses at the locallon uf the origmal crack 10
the uncracked solid: (iii) Compare the reSIdual messes
calculated m Step (Ii) With J permissible stress magn itude:
(iv) To SJtislY the suess houndJry condition on the crack
surface. reverse the reSidual stresses. Then determine the
analytical :;olutlon for the crack subjected 10 these revers·
ed reslduJI stresses: (v) Evaluate the slress lOtenslty factors
in the JnalYllcal solution for the current iteration: (VI)
Calculate the residual stresses 011 external surt"aces of lhe
body due to the applied stresses on the crack surfJce In
Step (iv). To sallsfy the stress boundary condlllOn o n
the externJl surfaces 01 the body. reverse the reSidual
stre)ses Jnd cakubte the equIvalent nodal forces: and (Vii)
Cunslder these nodal furces JS external applied loads acting
on the uflcrJcked body. Repeat all steps 10 the Iterallon
pro..:edure unlll the reSidual stresses on the .:rack surtJce
become tleghglble (Step iii). Tu obtam the rinal solUllon.
Jdd the s\T('ss lOtensity lactors of alillerallons.
In the Jbove. ,everal novel wmputJtJOnal techniques
were al:;o Implemented to save the computJtton ume Jnd
to Impruve the convergency Jnd accuracy ur the present
linlle demem .lltemaling method [70- "' 3]. Sma.l very
W;Jr~e mc~h ~ an be used to analyze the uncracked body.
the JiteTtiallng method becomes :.I very lnexpenSI\e pro·
cedure for rounne evaluation ot Jccurate ,;tress mtenslty
factors tor Ibws In structures. It was fo und [70] IhJt
thiS new Jlterm.lIng method IS .It least an order 01 mag·
tlnude lOexpeoslve compared 10 the eJrlier hybTld element
prllceduf~ ]-"'] .
Several )tudies have .1150 been completed !-5. 7h]
m USltlg the alternating method for muillple sem l-ellipncal
cr Jck s 10 pressure vessels Jnd for thermJI "hock analysis
01 sunace tlaws In pressure ve~sel~.
lna~n uch .:!..:' no tlllmertcal modelin!! .)1' crac'k from
Sll1gUiar!lll"S IS pellvrmed. the present tmne-eiement
JlternJtln~ .t1ethuti hJ~ hec n IOU110 to he h\ f:H the ka~t·
expenSive .IS ,~ell I~ th ..' ,nll~t .iL..:urJte m'~thlld J~ ~(lm·
pared to th ree-dimensional hybrid-crack element procedu res [77J or three-dimensional boundary element
approaches [31. 781.
3. Nonlinear Continuum Mechanics and SheUTheory
The works of contemporary mechanicians such as T ruesdeU,
Noll , Hill. and othe rs have done much in plaCing the
theories of nonlinear behavior of continua on a fi rm.
rational basis. The advent of nonlinear computational
mechanics in the past decade or so has not only
brought these aesthellcaUy appealing theoretical develop·
ments to the pract ical level of a technologist but also has
enriched. and has the enormous potential for enrichmg.
these theones through the search for aJternallve formula·
tions that may lead to computationally more-efficient
algorithms.
On a broad philosophical level. Gne may
observe that most theories of modern conunuum mech·
anics IOvolve differenllal or integral operators. whereas
computatIOnal mechanics relies on discrete operators.
Thus. it is fruitful to think of formulallng theones of
mechanics from a discrete viewpoint. A case 10 pOint
is the question of objectivity of stress-rates used in continuum mechanics. whereas computational mechamc s
involves the employment of stress-increments whICh
remam objectlve over finite time-steps [79, SO].
A recent monograph [Sll exemplifies the diversity
of work In nonlinear compulallonal mechanics.
The
future work will undoubtedly involve development of
techniques for large-strain large·rotatlon analySIS of :.hells
ustng alternate theories [S2J and the development of
Jlternate finite element methodologies to treat largestram behaVIOr of metallic mat enals.
TIle use of Bubnov-Galerkin techniques. using glObal·
local approXimatIOns. for studying the global nonlmeJr
response of structures [S3J and development of :.olullon
::ligOTlthms. such as [84], are topics that deserve a Wider
attenuon.
.t. Stru c tu ral Cont rol
The subject 01 "aeroelasllcllY
"- a ,;tudy of the interaction of fleXible elastic bodies
with the surrounding Iluid medium-has blossomed 10
the iJte 1940's and early 1950'5 and played an unponant
role In the design 01 modern aITcraf!. In aeroebS\lCllY
the Jerodynamic forces actmg on the elastic body have
an eqUIValent intluence of "negallve dampIng". I.e .. the
surrounding flUId medium acts as an energy supplier to the
Vibrating elastlc body. A problem of the opposite vanety
arises in the deSign of large space structures( LSS) Intended
for a vanety of operJ\lons 10 outerspace. These structures
are enVisaged 10 be as large as Manhattan Island. for m·
stance. and 10 be very tlexlble. Other examples of LSS
lre the large space antennae. The ce ntral problems 1Il
the design of these LSS are vibratlon suppression and shape
control. when the LSS are subject 10 disturbances such
as due to unbalanced rotating machinery on board. thruster
finngs, sleWing/pointing maneuvers. thermal transIents.
etc. Vibrallon suppressIon and shape control 01 LSS are
sought through either active or passive (or a combination ot"
the two) types of control mechan!sms. Thus. in parallel
With the ,;ohJect of aeroelasllcny. we have the emergmg
5uhJect o! ser\,oelaS1lcny - thai of (ontrol ,)1' dynamiC
mOllon ' It detormabk '!Tuctures. In Jt.lditlOn to LSS.
slll1!hr problems aT!se in the deSign ,)1 tall hudd1Ol!S nn
earth, wherein It is required to co ntrol t he dynamic motion,
say under se IsmIc loads. to amplitudes within the bounds of
human comfort levels.
The tOpiCS germane to the Issue of LSS contro llability
are lS follows: (i) Wlule for the usual free or forced vlbra·
tlon res ponse analysIs of structures. efficient algorithms
based on many thousands o f degrees of freedom eXIst.
the algornhms for optimal control are current ly limIted
to bUl a few degrees of freedom. In the traditional finite
element modelling of a Slructure. several hundreds or
thousands of element-nodal degrees of freedom may
have to be used to obtam even the (irst few (sat. 10 or ~O)
fundamental frequencies and global mode shapes. Thus.
there IS J need tor Jiternallve :lpproaches for reduced·
order modelling ut the suucture-1.e .. its stiffness and
mertll. (11) Design of :algortthms for irnplememallon of
optunJI -:ont rol of systems of moderate dimension of.
sa~
50-100 (I.e .. 50-100 global modes); (iii) The ml)fC
prevalent concepts (85. 861 have been to either Ignore
dampmg or conSider it bemg proportional to mass or
sllffness Jnd obtain the "normal modes" of the (linear)
struo.:ture. Based on the orthogonality of these normal
mode s. Ihe system of (linear) ordinary differenCtal equa·
lions are com pletely decoupled. and conlIol of response
of ~ach decoupled equation IS attempted mdividuaJly.
Tlu s -:oncep! is labelled the so-o.:alled " Independent Modal
Splo.:e Control'"
Tlus approach. whtle mathematically
sImple. depends un the use of as many ,ontrol-iorce
actuators as the numb..:r of decoupled modes bemg COntIOl!·
ed On the other hand. If damping exists due to deliberate
deSIgn oJf pass\\'e dampers or due to deliber:Jle deSIgn 01
JOIntS. 1\ may be ot "nonproporllonal'" type; and thus.
the (linear ordinary differential) equations of motIon
cannot be decoupled. Thus. one needs 10 implement
control based on the system ot coupled equatiOns of
mOllon Jnd wl\h an JrbllTary number of active control·
force actuators that IS perlups much smaller than the Older
of the ,ystem bemg o.:ontrolled: (iv) The effect of no·
n!ine:mlles In the system. such 11u! the discrellZed equa·
110m 0\ mOllon are nonlinear ordinary differential equa·
oons m Hme
D"pendlng un the spatlJl and temporal vari:l!10ns 01
the dlSlurbJncrs. Ihe LSS may be modeled as a thlre·
dlmenslonJI nctwork 01 beams and bars [87. 881. or.
aiternatlvel) . an eqUivalent continuum model such as
a piatt' ma\ be used. Recently [891 a methodology was
presented wherein
(i) an eqUIvalent contmuum mod~l
III the form 0\ J flee-tree plate is used. (ii) the linear trans·
lent JYLUmh': response 01 the plate is modeled by a boun·
dan -dement technique based on the singular soiullon
:)1 J blharmOnLC operator. (iii) a nodal-control. I.e .. control
01 nodal response based on the completely coupled nodal
system eli equallons IS used. (Jv) non proportlonal dampIng
is consldrred. Jnd (v ~ the tina! lime III the control algoTtthm
IS set to be miLnuy so that a neady-state Rlccati eqUlllon
IS solved.
In another recent study [901, a singular-solution
approach was used to drnve un" discrete coupled ordinary
dlfferenllal equauons governmg {he tranSient dynamiC
response 01 3n IIllllally stressed !lat plate that is assumed
to be l continuum model of:1 large space Structure. ~on­
propoIllonal dampmg IS assumed. A reducedorder model
ot \I (" n ) coupled urdinary Ji(fercntlal equations lre
derived in terms of t he amplitudes of psuedo- modes
of the nominally unda mped system. Optimal co ntrol is
implemented using m « N ) cont ro l-force actuators. in
addilion to a possible number p « m) paSSive VISCOUS
dampers.
Algorithms fo r efficient solution of Riccati
equatio ns are implemented.
The problem of control
spillover is discussed. Several example problems involving
sup pression of Vibratio n of "free-free" and "simply·
supported" pla les were presented and discussed.
Structura l control. Lnyolving Ihe active or paSSive
control of no nlinear dynamic response of deformable
bodies. is likely to be a subject of intense research actiVity
of the next decade.
ACKNOWLEDGEMENTS
The studies which are bmfly discussed herein, were
made possible through past and present research support
from the Office of Nayal Research. the Air force Office
o f Scientific Research. the National Science Foundation.
and NASA. The encouragement of Drs. Y. Rajapakse.
A. Ku shner, A. K. Amos, C. C. Charms, l. Be rke. and C. J.
Astill, oyer the past several years. is thankfully acknowledg.
ed. II is a pleasure to thank Ms. 1. Webb for her assIstance
in the preparallon of this paper.
REFERE NCES
I.
,
3.
4.
5·
6.
7.
8.
Atlufl. S.N .. On hybrid fiOlte element models III
solid mechanics, III Vichnevetsky, R. (ed.), Advances
III Compute r Methods for Partia l DIfferential Equallons
, AIC A. Rutgers Umverslt y, NJ , 346·355 (1975).
Atlu fl . S.N .. Tong, P.• and Murakawa. H" Recent
studies of hyb rid and mixed fi nite element methods
in mechanics. in Alluri. S.N. el al. (eds.). Hybrid
and Mixed FillLte Element Methods (John Wiley &
Sons. 1983) 51-72.
At lufi. S.N .. Gallagher. R.H. .. a nd Zienkiewicz. O.C.
(eds.). Hybnd and Mixed Fillite Element Melhods
(John Wiley & So ns. 1983).
Brezz!. F., On the existence. uniqueness. and approx·
ImJlIon of saddle-point problems ariSlIlg from Lagr'
ange multipliers. R.A .I.R.O. 8- R:!. 129- 151 (1974).
Ying, L-A. and Atluri. S.N .. A hybfld fi lllte element
method for Stokes flow : pan lI-slability and com'erg'
ence studies. Camp. Meth. App\. Mech. & Engg ..
36.39-60 (1983).
Xue. W- M" Ka rlovitz, L. A., and Atlufl. S.N .. On
t he exist ence and stability condit io ns for mix,ed·hybnd
fi nlle element solulio ns ba3ed on Relssner's valla·
tional prinCIple. 1m!. Jrn!. of Solids & Structures (
Spec13 l Issue in honor 01 professor Eric Relssner) (
in press).
Xue. W- M.. ExIstence. Uniqueness, and StabilllY
Co nd itions for General Finite Element Methods in
U nca r ElaSllcity, Ph.D. Thesis. Center for the Advan·
cement of Computational Mechanics. GeorglJ Tech
( 1984).
Xue. W- M. and Atlun. S.N .. Existence. uniqueness,
and stabIlity condllions for fimte element methOds
based on the Hu- Washizu prmciple (1984)(1Il prepara·
tion).
ladyzhenskaya. O.A .. The Mathematical Theory of
Viscous IncompressIble Flow (Gordon and Breach.
NY. 1969).
10. Babuska. I.. The fimte element method with lagrange
multIpliers. Numer. Math .. 20. \79-19"2 (1973).
II. Rubinstein. R .. Punch. E. F ., and Atluri, S.N., An
analysis of. and remedies for. kinematic modes in
hybrid - stress finite elemems: selection of stable.
mvanant stress field s. Compo Meth. App!. ~1ech. &
Engg .. 38. 63-9"2 (i98~).
I~. Punch. E.F .. Stable. Invariam. Least - Order lsoparame·
trie Mlxed - Hybnd Stress Elements: Linear Elastic
Conunua. and Fimtely Deformed Plates and Sheils.
Ph.D. ThesIS. Cemer for the Advancement or Com·
putalional Mechanics. Georgia Tech (1983).
13. Punch. E.F. and Atlun. S.N., Development and testing
of stable. invariant. isoparametric curvilinear 2- and
3- D hybTld stress elements. Compo Meh. App!.
Mech. & Engg. (1984) (in press).
14. Punch. E.F. and Atluri, S.N .. Applications of isopar·
ametric three-dimensional hybrid - stress finite elem·
ents with least - order stress fields. Compu ters and
Structures (1984) (in press).
15. Pian. T.H.H . and Tong, P., Ba sis of finite element
methods for solid continuua. Inll. l rn!. for Num.
Meth. In Engg .. I. 3 - ~8 (1969).
16, Pian, T.H.H. and Chen. D.P .. On the suppression of
zero energy delormallon modes, IntI. lrn!. for ~um.
Meth. in Engg .. 19. 12. 1741 - 1752 (1983).
17 . MASYMA Reference Manual. The Mathlab Croup,
lab for Computer SCience. M.LT., Version 9 (1977),
18. KorncofL A.R. and Fenves. S. J .. Symbolic generatlon
of finlle element stlffness matrices. Computers &
Structures. 10. 119-1 24 (1979).
19. Luft. W .. Roesset. 1.M.. and Connor. J.J .. Au tomatic
generatIOn of iinne element matrices. 1m\. StrUCI.
Div .. ASCE. 349 - 36~ (1971),
~O. Cecchi. M,~ . and Lam!. C .. Automatic
generatIon
of mffness matrices for fimte element analYSIS. Int!'
lrnl. ~um. Meth. Engg .. 11,396- -100 (1977).
Chepurny. N., Evaluation of high-order polynomIal
triangular fimte elements using FORM.Ac' Proc. Sec.
Symp. on Symbolic Algebraic Mampul.atlon, 365·371
(1971).
"I"
NoOT. A.K. and Anderson. c'M .. ComputerIZed sym·
bolic manipulatIon In nonlinear finite element analYSIS,
Computers & Structures. 13,379,403 (1981).
Park. K.c' and Flaggs. D.L.. An operational procedure
for the symbolic analYSIS of the finite element method.
Camp. Meth. Appl. ~e ch. & Engg .. 42. 37-l.6 (1984),
~4. Park. K.c' and Flaggs, D.L.. A Founer analySIS of
SPUTIOUS mechanisms and locking in the finite element
method. Compo Meth. App!. ~ ech. & Engg. (in press).
25. Park, K,C. and Flaggs. D.L.. A symbolic Fourier
synthesls of a one-poml integrated quadrtlateral
pla l e element, Compo Meth. Appl. Mech. & Engg.
(to appear).
~6. Felippa. C.A .. OpttrmzatJOn of fimte elemem grids
by direct energy search, Applied ~ ath. \10delling,
1 (19 781.
,- ~telosh, RJ Jnd \lar cal. PV .. An energy bam for
me sa reimemenl >1 structural .;ontlnua. Int i 1mJ.
'\UI11 \k th .n E;'I\.' ~
\ 1.1 083, 109 1 (19 r ).
9.
28 . Kelly, D.W. , de S.R. Cago, J.P" Zienkiewicz, D.C ..
and Babuska, I., A posteriori error analysis and adaptive processes in t he finite element method: part [·error
analysis. IntI. hnl. Num. Meth, in Engg .. 19, 16~1'
1656 (1983).
29. de S.R. Gago. J.P ., Kelly, OW., Zienkiewicz, O.c.. and
Babuska. L , A posteriori error analySIS and adapHve
processes in the finite element method: part lI ·adaptive
mesh refinement, IntI. Irl. Num. Meth. Eng. 19.
1657·1669 (1983).
30. Belytschko, T. and Hughes. T.1 .R. (eds.). Computa,
tional Methods for Transient Analysis (North· Holland.
1983).
31 . Cruse. T.A .. Mathematical foundations of the boun·
dary integral equation method in solid mechanic s,
AFOSR TR 77·IOO"2. U.S. Air Force (1977).
32. Atluri. S.N. and Crannell. 1.1.. Boundary element
methods (BEM) and combination of BEM~FHL
Report No. GIT- ESM - SA- 78-16. Center for the
Adv. of Compo Mech., Georgia Tech. Atlanta. GA
(1 978).
33. Brebbia. C.A" The Boundary Element ~ethod ror
Engineers (pentech Press 1978).
34. QUillian, PM .. Fitzgerald. 1,E., and Atlurt, S.N .. TIle
edge, function method, Proc. IntI. Symp. on Innova·
tive Num. Methods in App!. Engg, Science, Versailles,
France (1977).
35. Qumlan, P,M .. Grannell. 1.1 .. Atluri. S.N .. and Fltz,
gerald. 1.E" The edgefunction method for three'
dimensional stress analysis. including embedded
elliptical cracks and surface flaws, in Brebbia. C.A.
(ed.), Boundary Element Methods in Engineenng
(Springer-Verlag 1982) 457·471.
36. Atlurl, S.N. and Murakawa. H.. On hybrid finite
element methods in nonlinear solid mechamcs. in
Bergan, P.C. et al. (eds.), Fimte Elements in Nonlinear
Mechanics. I (Tapir Press. Trondheim. Norway 1977)
3 '44.
37. Atlur!. S.N., On some new general and complementary
energy theorems for the rate problems in !lnlle stralll.
claSSIcal elastoplasticity , JmJ. Structur:1l Mechantcs,
8.1 . 6 1·92 (1980).
38. Atluri. S.N .. On rate principles for finite stra in analy sis
of elastic and melastic nonlinear solids. Recent Re·
search on Mechanical Beh..viour of Solids (professor
H. Miyamoto's 60th Anniversary Volume). (Umv. of
Tokyo Press 1979) 79·107.
39. At luri. S.N., On constitutive relations 3t finne stram:
hypo,elasticity and elasto·plasticity with isotropic or
kinematic hardemng, Comp, Me th. App!. Mech. &
Engg ...B, 137·171 (1984),
40. Truesdell. C, and Noll. W.. The nonlinear field theoTle s
of mechanics. in Flugge, S, (ed,), Encyclopedia Df
PhYSICS (Springer. Berlin 1965).
41. Reed, KW . and Atluri. S.N., Constitutive modelmg
and computational implementation in [imte strain
plasticllY. Inti. lrnl. of Plasticity (Special issue m
honor of Professor I .F. Bell's 70th Anmversary)
(m press).
J"2 Atlun. S.N. and Nishioka, T., Hybrid metho ds of
Jnaly sls. m Kardestuncer. H. (ed.). Umficatio n o t
Finll e Element Methods (j .H. Argyris' 70th An·
nlversa ry
Volume) (North·llolbnd 19H-l) 05,06.
.l _~ \1r ,ll . l. . An Jltempl to J escnbe the f.o",hJ\lOur 11
metals under cycl ic loads usmg a more general workhardemng model. Acta ~te chamca. 7.199-212 (1969).
~4. KrIeg, R .O .. A practical two surface plasllclty theory,
hnl. of Appl. Mech .. 4::::.64 1-646 (1975).
~5. Oafalias. Y.F and Popov. EP .. PI:lSIic internal variables
formalism of cyclic plastlcLty. J rn!. of App!. Mech .. 43,
645-651 (1976).
-Ul. Watanabe . D. and At iuTL. S.N. constit utIVe modeling
of .::yclic p[aslI'::LlY and .::reep. usmg an mternal tLme
..:oncept. Inil. Jrnl. 01' Solids and Structures (1984)
1m revLew).
4- Walana be. O. Jnd AtluTl. SS .. InternaltLfTle. general
mternal variabl~. Jnd mull1·yield·surface theones of
plaslL":LlY and .::reep
J umfication of concepts.
GeorgLa Tech. C~nw lor the Ad,·. of Compo ~ I ech.
R~poTl (Augu~t 1(84)
48. V;ilJnts. K. C. Fundamental ..:onsequence of a new
mtnnSI':: (Jme measure plamcuy as J ILmlt of the
~ndo..:hrontc theory. Ar .::h. of Mech. J2. 2.171-191 (
19bO)
.,lq Wu. H.C Jnd YIp . .\l..: .. Endochronic desCflptlon of
.::ychc hardenmg behavIour of metallic materials.
Jrn!. 01 En~g. ~hterlJl s & Technology. T rans. ASME.
103. J.~I::::-~17 (1 98J).
50. Chaboche. J.L. and Rou.seher. G .. On the plastic and
vLscopbstic .::onStltU!LVe equations. part I: rules devel·
oped WIth mlernal \'Jrtable concept. part II : applica·
\lon 0 1 mternal varuble .::oncepts to 316 stamless
steel. m B:.Iyla..:. G. (cd .), InelastIC Analysis and Life
Pr edIctIon III Elevated Temp Design. AS~I E - PVP­
Vol. 59. 33·55lI9S: t
51 Chaboche. J.L.. VLscoplaSllc con' t!1Ul!ve equations
for the deserlpt Lon of cy..:lic and alllsotrOplc behaviour
of metals. Bulletm de L' Academle PolonaIse des
Sclences. Senes de SCIences Techniques. XXV, I . JJ.
4:::: I 19,-).
~:::: 'Jagtegaa l. JC and de Jong. J.E.. Some aspects o i
n01HsotroPlc workhardenmg til finlle stram plaslLcLty.
III Lee. E Hand \h llc::(
R.L. (eds.). PlastlcLty of
\\e(als J( F11Iue Stram' Theon. Expenment. and
(ompUIJILOll (O\'n. 01 .-\ppl. \ I<,th .. Stanford. lnd
Dc::pt \1 <'..:h. Engg .. RPL IQ8:) o5·10:!.
53 L<'<'. E.I\.. ~IJ llett. R .L.. and W<'Tthelmer. T.B .. Stress
JnalY~Ls lor JllLsotr0PLC hardenmg 111 finne deformatton
plaslI..:Lty. lrnl. of App!. '\1<'ch .. 50. 554-560 0983).
::;4 Atlur1. S.:'\' (ed.). (omputalLonal .\1ethods III the
\1<,..:hJIl1":s 01 Fracture (~orth. H olbnd) (to appe:.lr).
~, At lurL SS . 'Jr shlOkJ. T.. Jnd :"Jakagakl . .\1 .. Numencal
\todehnl! 01 dynamLc and nonhnear cr:ick propagatron
m tinae bodLes. by movll1g stngular elements. 111
Perrone. 'J . :.Ind Atlun. S 'J.led s.\. AS1'-tE- AMO- Vol.
.UI:\.S~IE [9 -9) 37·06
50. 'JrmlOkJ. T . .lnd AtiuTl. S.;-": .. Numerrcal modehng o i
Jynarm..: ~ra.::k propagatlon 111 t'\nne bodies by movmg
~1I1gular dements. part [ formulation: part 11: results.
1rn!. 01 App!. .\i ech .. 47 i 1980) 570·576 and 577-582
. respe..:trvely
5- .... LshLoka. T .. Stonesrler. R.B .. Jnd AtluTl. S.N .. An
<,va]uJllon 01 :,everJI movmg s1I1gu]arny [lilLIe element
procedures tor Jnalysl~ 01 last Ira..:ture. Engg. Fr:.lcture
\I e.::h" IS. ~05·~1 b t Iqhl)'
511. \1"I)LokJ. T. Jnd .-\tlun. S.:"\ -";umert..:al JnalY~ls 01
J\n:.lllll.:: crJ..:k PlOp:.lCJ\ml1 ~ell<,r31l0n Jnd predLctron
,1:1...ile· Em:; F'J~'Jr'.' \k~:\Jnt..:,. Itl ..:O.~·_~_~:::: r [lhe).
59. Nishioka, T . and Atluri, S.N., Finite element sLfTlula·
tion of fast fracture in steel OCB specImen. Engg.
F racture Mechanics. 16. 157·17 5 (1982).
60. Nishioka, T. a nd Atluri, S.N., Path· iade?endem in·
tegrals. energy release rates, and general solutions of
near·tip fields in mixed mode dynamic fracture mech·
all1cs. Engg. Fracture M echanic~. 18, !. 1·12 (1983).
61. Atlun. S.N" "P ath·independent integrals 111 finite
elasticity and inei.:tsticity with body forces and ar·
bltrary crack-fa ce conditions, Engg. Fracture Mechan·
ics. 16. J. 34 1·364 (1982).
62. Atlurl. S.N .. Nishioka. T .. and Nakagaki . .\1.. Recent
studies m energy Integrals and therr applicallons.
Plenary Paper .6th IntI. Congress on Fracture (Perg.
amon Press. New Delhi. 1984) (10 appear).
63. Atlun. S.N. and Nishioka. 1.. On path'lndependent
Lntegrals in the mechanics of elamc. melastLc. and
dynamiC fracture. Jrnl. of the AeronaulLcal Society
of India (Special issue in honor of Professor G.R.
lrwLl,) (l984)(in press).
64. Atlurr. S.N .. Nishioka. T., and Nakagaki. ~1.. Incremental path·independent integrals In melastlc and
dynamic fracture mechanics. Engg. Fracture ~I e{.:h·
antcs (in press).
65. Nishioka, T. and Atluri. S.N .. DynamIC crack propaga·
tlon USLng a new path·independent lnlegral and moving
isoparametric elements. A IAA hnl .. 2~. 3. -l()9415
(1984).
66. NLshloka. T. a nd Atiuri. S.N., A numerJcal study of
the use of path-independent integrals In elaslO·dynamL{.:
crack propagation, Engg. Fracture Me..:hantcs. 18. I.
~3·33 (1983).
67. Shah. R.C. and Kobayashi. A.S., On the surface Oaw
problem. 111 Swedlow. J.L. (ed.). The Suriace Cra{.:k.
PhYSical Problems and Computational Studies (AS~\E
1972) 79·124.
68. Shah. R. C. and KobayashI. A.S .. Stress intensLty fa.:tor
for elliptical crack under arbitrary normal loadmg,
Engg. Fracture Mechanics. 3. 71-96 (1971).
69. Vijayakumar. K. and AtluTL. S.N .. An embedded
ellip(ical tlaw in an infinite solid. subJe~t 10 arbiITar~
crack'Iace tracttons, Jrnl. of Applied .\\echanl":s . .l8.
88·06(1981).
70. Nishioka. T. and AtiurL. S.N., Analyu..:al ,;oIUILon
fOI embedded elliptical cracks. and finue element
aiternatLng method for elliptical surface t13 ..... s. ,ubJe..:t·
ed to arbnrary loadings. Engg. Fra cture Mechanr..:s. I"
~47·268(1983).
., I. :'-lishtoka. T. and AtiuTl. S.N .. A maJO! de\'elopmelll
towards a cost·effectlve alternating method for Ira..:ture
analySIS of st eel reactor pressure vessels. Paper G·I ~.
Trans. 6th SM IRT .Pans(1981) .
n. NLshlOka. T. and AtlurL. S.N .. Ana.lysls ot SUf(J(e t1m.
111 pressure vessels by a new 3·dimenSlonal 3lternJILng
method. J rn!. of Plessure Vessel Technology. 104.
299- 307 (1982).
73. NishLoka. T .and Atluri. S.N .. AnrnexpenslVe 3·D 111lne
element alternallng method for the analym of SUr!·J..:e
nawed 3Ifcralt structural components. AlA:\. lrni..
~l. 5,749.758(1983).
7 ..t At luTL. S.N. lnd Kathiresan. K. 3·0 anal\\LS o{ mrlJ.::e
I1lws III thICk·wailed rel.::tor pf('ssure' ves..els u~mg
drspllcement·hybnd filllte element method. :,\udear
Em.!!! & D':slgn. 51.~. 163')-01 Iq-qf
75. O'Do noghue , P.E., Nishioka . T., and Atluri. S.N. ,
Multlple coplanar embedded elliptical cracks in an
mfinlte solid subject 10 arbitrary crack face tractions,
Imi. Jm!. Num . Meth. in Engg. (1984) (in press).
76. O'Donoghue. P.E.. Nishioka. T., and At luri. S.N ..
~I ultlple surface cracks In pressure vessels, Engg.
Fr:u;ture Mech. (1984) (in press).
77. Allun, S.j'J lnd Kathiresan. K .. 3·D analyses of surface
!laws tn thlck·walled reactor pressure vessels UStng a
dlsplacement·hybrld finne element method. Xuclear
Engg. & DeSign. 51. t 63·176 (1979).
-So HelioL J Labbcns. R.C. and Pellissler-Tallon. A..
SemL-elhptL..:a.1 ..:ra.cks tn a cylinder subjected to stress
gradients, Fracture ~lecharlLCS ASTM 5TP 677, 34 1·
364 (l97Q I.
-Q RubtnSletn. R. and AIlur\. S.N., ObJectlvllY of mcremental ~onSIL tutlve relalio ns over finlle time steps tn
.:omputltlonal nnlle deformalLon analySIS. Compo
\I elh. tn Appl. ~I ech. & Engg .. 36,:!77·:90 t1983)'
SO. Reed. K.W. lnd AlluT!, S.N., AnalYSIS of large quasi·
~1atlc de(ormatLons of inelastic bodies by a new hybridstress (jnne element algomhm. Compo Meth. 1O Appl.
Mech. & Engg .. 39. ~45·295 (1 983), and "applications
·'.-10.171·198(1983).
81 AtluT!. S.N. and Perrone. N._ {eds.}. Computer ~I elhods
tor SlonlineJr Sohd and Structural Mechanics. ASr..IE·
.\MD. 54 IAS\lE 1'183) 5~4 pp.
~~
-\tluTl. SOS. Al lerrutc stress measures. conjugate
)H:lln meJ~urrs. :Ind mL'(ed vaTLalLonal 10rmulaILons
tnvolvlOg rigid rotatIons. (or .:omput:nion:l! analyses vI
IlOnely delOrmed sohds. wllh applicalLon to plates
Jnd ~ell~·part L theory. Computers & Structures,
IS.1.93·llbC]Qb4).
S3 \;ovr. .-\ K. Re..:ent Jdvances III reduction methods
ror nonltne:1T problems. Comput ers & Structures. 13,
31-44 (1981) ,
84. Hughes, TJ .R., Winget. J .. levit . I.. and Tezduyar.
T.E .. New allernalLng direClio n procedures III finlle
element analysis based o n EBE approximate factorizations. III Atlun, S.N. and Perro ne . N, (eds.). Computer
Methods fo r Nonlinear Solids and Suuclural Mechanics
,ASME- AMD , 54 (1983),
85. Melrovltch. L. . Baruh, H., and Oz. H., A comparison of
control techntques for large fleXIble systems. Jm!.
GUidance. Control. and DynamiCs, 6. 4, 30::!·3 l 0 tl983
),
86. \Ie\!ovllch. L.. Baruh . H., Montgomery. R.L and
Williams. 1 p " Nonlinea.r natu ral cont rol of an experlm·
ental beam. Jrn!. GUidance. Control. and Dynamics.
7.4. 43-'--+4 3 (1984).
87. Kondoh , K. and Atlur\' S.N., Influence of local buckling on global instability : simplified fi n1le deformation.
post-buck ling analyses of trusses. Computers & Stru·
ctures (in press).
88. Ko ndon. K. and Alluri. S.N ., A simplified finite element method for large defOlmatlon post·buckling
ana lysis of large frame structures. uSillg explicitly
denved tangent stiffness matrices. In tI. 1m!. Num.
Meth. in Eogg. (in Te'o'lew).
89. O'Donoghue , P.E. and Atlu rt. S.N .. Connol of tranSIent dynamic response of structures. ~ UMET A85.
Swansea. UK (1985) (in press) .
QO. O'Donoghue. P.E . and AliuT!. S.N .. Servo-elastlc
osc~la t1ons: control of tranSient dynam1c mOllon
of a plate. Advances in Aerospace StructureS-bolo
ASME WAM (December I 984)(tn press).
Manus.:npt recel\'ed December
publicalLo n January ~4. 1985.
~l.
1984. accepted for