ZII1:N(:, (', rJI, '.1".'11 "f lid rt" ,Iu,,/ 'f ,bar-arl. fI5:tf·~ for Qmltll1tl'Jtl oj ,to/ut,. !Ta!",.port HI lhn" dWdlUI'HU. (:'01111.1 \V.lkr.H. (1!~II]), PI> ,Jf,t; '1G:i

YOHIS'h, V-C. ASII A,S ("Oli" ..... ·111 l1.un!ytlcit! _\0"1.':1011 }or hllflll Tl.'al,.r!lf)(')li ,r.dlli.1U'I tilr ,flrd,\ i.J ((111l1!a:":~ 11t"f.Hlitf, s... I'dl' I·,n~ J" 2:i, (1~)R.l). pp, 115124,

\,VARMING H F ANC' U,J Ih'ETT. Tnr. mvdl.t~I'd r'qliatn111 apprtl;uh to th,. Jta6ilit,nlld Ilrr'Tlrnry Ilna/y ...!... oj /onlf dlfft1'~I""r. mtOw·f.~, .1, COfllp Phys_.l.1, (19i4).i'P, I.ll:) t 7~}

\VEITEIUNK. J ,J, ANI) D, SUPA, ,-",Tt,'::,:,lolt hl.ll/lH anin' Pdr'~f'-Glllfrknt m~tAoitfor flu' ,H'/llt:Or: of fhr fran."" III r(,I.'Il'l"(hIHl d'1711,QOn uillatHlfl, Inl.. J NUlUer.l\'lctllOd~ I~llf:.., 28, (19S,9). pp. 1077 1101

\\·IIELI.CH, [\'1., (I'd.), N/j".rJUG/ Sln.l,iI.'~!I)11 1110:1 l'O'Ol'("1'), Spring(T.V,'dilg, Bnlin,N(,,, Y'III., NY. I(ISH

J TINSLEY ODEN'. \\'FIHAN Wll" ANI' 1\11\111"·\IN~\\()ln III

THREE-STEP Ii·}' ADAPTIVE STIlATEGY FOil THEIJ'!COMPRESSIBLE NAVIEil-STOKES EQUATIONS

J·I'

A.b~t ..:'l't. Hf"t"'lItly, ,\ ,di.tl,II' a po,drnorl '"no! ..~t im,l(l' W.l:"t d,'w·lt.",..t. 1lI,lin!"~ _llllh.- .,'kllwnL 't'~~dll.-,1 mdlwd. (ot' ;~cl.,-~s o( ~tt'~Ht)' :'iLII." 1I11'olllpn'~:'o",I.· N.wl:"c;tokes cqU.ltl .. ll~ In tills I,.ifw_l, uSing tlt.lt> nidi (':.11111.1(", ,1 ~)Il(T-~;lf'P 1.-,. ".1.11'1.1\'.'

~j\tl'&y i~ dr\',~lor>c,1 10 slllvf· IJlr.~mpr(,sslbk o.)W J\I'~,blt'll\S, I h," ~("d .of Clf"\'~'JI:P:llP.~ h-p nd:,ptl\'(' !".~r<llq;y I~; In,,}~t:un <lCCllrntl~ apprtlxlIll,d"o s'llull'ifl'" \,1111.· 11.'11111111:1.1111;

CGII'PIlIl\tlOn~ I~o:-.b. TIlt' b'U;H~ ld~a.. of llw thn·.I·-:'il<·P !,' ..p_.-ul ..tplJVe };tl·~tes)· J:. ~IO ~O)Vf'

I thf: to:ystcln on thl' th,,'c' COntit'Clll1\',~ IIlcsllt's, 1.1:.lUI 1I1Itl:') nwsh. an Inlt'l"lm"rlulll' II -;,'apLi'W mesh, llud a final hop a.~a.pti ...,(· m(,~h. Eadl III"W ad.~ptiv(' Ill(:sh is ohl'lincd by

UJT1~ling lh~ .'rrnr on lh(, prn'wus IIlt'sh and ('xecutillR ,l sl1lgl,' Ii - "I' P - rdllWllwllttSroc~dure (111111l~p,,~\'inll~ 111":,>11,wconlil1S to the l'(·:,.ults tlf tt ... -ndnplivf' :;..r,alrsy. Nu-~('ric.a.l rf":>ull:;. indic:at.· (bllL t 1\1' Pl"p')~'f'd ll~n'I"sc"P I1d.i1pLI\'I~sl u1I "P:;Y1'10.]11.'( ....:'·~nlrHII'

solutioll:> \"laih- k""J1ing till' tl,I:1I <'omlllll;Ltu,ll.1.I {',.sl,:,> \111<1"1 ("('Iltn')

]. Introduction. Adaptl\'l' methods 1Il romput.al iOllal lIuid <I)'ualll-ICS are dpvclo!,l>d with an aim of produc,iug good IW'sl",,, t hal can ddivcrsolut.ions wit-h high accuracy wit.h fewf~r di'gl"'I'S (If frr-f'dOIH tlt,,") COIIVel1-

tional techlllqu"s, If h I' fintl" c1ell1ent.s arc used. II", 1I",,,h SlZ" h aud I h,'dem"nl sl,,·ct.ral (>nl<-.. l' arc adapt",.1 t.o cont rol ,'rror Willdl IS ""lIllIillo'd us-iog an a PO,dtTl0n error f'stlluat.or H'·{,(,lIt.ly, an n po-"l( non ~'rrc)r ('sl iJllat..'for tlw :-:It>ady incoIlIpr"~;slhlt' Na.vifl'r-St.okt·~ ('qU;ttlOIl:-i wlt.h :-;ll1all data \\';L"-;

devd(lJ>I~d. If t.11t' dl~tral'lItt!Hl of ('rrOr II) a rlH'~..dl i~kllOWlI, j I.,· IS~IIt> of 110\\tlwerror ran lit' sysl.t'lllatically r,'du"I'd to :lclII1'Y" a Ilri':"1't 11·\-'I·ln:tlur;t1I):m~('_'i. III tlai~ not.,. \\~' dt'~i'flhj' :t t.hn'!'-~t,·p " 11 ;1,lap!.I\'(· :-;t.r;I1,~'J.~Ythatlt~ads to good Ii-II Ilw~lli':-; for t Ill' two dllll1'IISlullal sf(';l.dy "Ictlillprl':->~dll,'

~avi('r·Stc)k(·:; !·iluat.i(Hl:-;

Follo\\'lllg tills illtr1ldlh'tlflll, S(JIIH' 1I{)t.;dll}n~; :1.11..1d"fillltl\\lI:-> II:-:cd ill thIs

paper are pr,'st'lI1.t'd III S(·.'! ion"2 III SpdlUII :~, it :-italldard Iini1t' ('It'IIWIII

approxililatiofl of Nav",r-Sv,kes ('C}lJ:tl lOtlS IS (b':-irriLi,d III S('c1 inll ii, t.Jw-orcms 011a pOStlTIOTl error ,·sti,uat.(·s for st.ea.dy statl' Navwr-Slokl~S (>,pJa-lions ar" given Th.' h-1' adapli\'!' st.rategy for the Navwr Sto\; .." "'1l1aliou"1$ described 1[1 Sed,loll G. Some nurnerical r'-'slllt~ an'). di.sclls~fl'(J in Si'ctioli6

2. The Navicr-Stok('" ('quation". The st..·ady sl,aLe Navi." Stnh·sequatinus 011 a bounded Lipschitz domain n E 111". u "" '2 or :1, for inCOlIl-

- T('.\:ll-.. III:-.t.itllk fOI ('Olllplll,~tioll:d '(U1d Applif'd ~'1athl'1lI:\tiq;, Tit" lIltl\·,·I· ..iLy .If

Tl:Xas:l.l Au:-.till, A\I~till. TX 78712 U_S_A,I D"IJ.;u·luu'lll ,.r fl..1'-lllw111.ltie" lJI1IVfTliity of l,f·i'~I·,:,t'·r.lJ.I\:

.......-MANFIlEIl "oell 's.:RIISShl" I.F AND ~I.F \VIU'_hl.m~. Flrllt, ~!(lll'tlt 'lndfanit: diDnrnrr rn-:t4o , .....

c(lnltnlI0l.i$ flow.~ 11_1 porQ~:\ mr:~la, In: 1 h(" TII.lt)H'lTudIIS !)f ,I-'Sl'rVOiTSitlltU::.1tp

Soclt~ty fUI Illdllstrll:11.U1d Apph,..d ~bthf'lllatlCs. Plu!;\d .....lpluil.. PA. EWin IGIa,(,.<1). (l;lS1). 1'1' 1S Ill'> g.Il..£.,

SI-:\\L.:l.L, G .• Tilt HtJmulc4/.~o/l.llltljl (>j orJIIlO'1"!/ IHld l'orltrd dlDnf'1ltld Clll.:oljA"HI.'BU' Pf'/'s~, 's.'\li DIe'hu, l'lSS 0....

SLI'T.o\'lI, A, A~I~ l'( AZlz. Tru:lmlnt (·f f10rdWriH' t{l'm.~ 111 O,r Tt'ln/H:tua.l '0/"oj ;1a,T'tlrJt .bDr-n:r.tl('/ rqlidtIOr/:\ fur Hld"l,},a.1r /llllil I'll IJl)n.'ll,~ lflr-d •..: In ta ••!\1ulllph;\s(' Flnw, I, (1~)7!.). pp.l'l.17 ~qL ' t. J.

S~UI·II. H,M ANIl A.(i_ IhITrON. 'lk, llliHun"lll tlf(llmn/! oj a,'h,rt10TL .4jnrm01"Cf C''''Hl11''11.~(lU ,,} cunnlt md/H,J,\, Ntlllll'r, 1I"at T'-Hl1~" ;', (1982) Pf?-34~l ,1'.1 .PIJ.,

S<.lD. G _A., Nun. (T"'it! mdll~d, .. i,: flllld. dYllwmlu, Initi."t.l'llld illltiaJ LOUl1dar -valprobl(~IllS, Cambridge Ulllv~ ..~it)' 1-'1l'S:>;, Camhridge, MA, 1985, y-

TJo:ZDUYAH, T.E. AND D.I'_ GANJI). P,tp-,;'·(."nl,rkln J'lllll'U/a"tlf)n.~ with tIIt;i,'ti~f1t1ldHI1J$ dqll'tl4flll 'j~flll :\rfftlil/ ~lfIC{ tnn.'IOt'L"l1 dluutJ;;atirln: .4pp/1catioJl •• :irafl.,Hod COil-'H'dHltl rllDu.1WrI pT-/lMr-m.~, (Oll1pllt M,·tlwd~ AIJpl. M"dl. ~c,''. (1"8G),I'I' ~1971 .,

VAN CENlc-'IITI':N. f\.-1.Tu,. ·1 do, ..,;l }III',')I ''1utl!lOli JM prr'dlrtrug tll( Itydrc.u/H: eo._4uctn'lty oj UJLHrtU1'f.'!tJ ~{lfi:f, Soil Sn_ S(h' Am ,I. •. 1'1. (l980). pp. 892-89&.

:11(;

.~hi' .J'~~.·-:l:':1JI&:l,' ,

ltrJ.'~i;It:tl:pi:

t1f:'',~ f"

f\i;i"

'I"S.I.'! !•1'1 ,;:!·.I~;}; ,

i~i,HI:!i,,,', •to' •.~~. ,

jt. > '~.;l..'~t!·J.;J:i .,,:1 I•~;

r:-

pll'~Slhk f1qW pr(/hh'lll~ call h,' \\'fllt"IJ .IS [1,110\\""

('2 :1)'1'1,.. rOrllJS "(. ).b(. ), and f() ow' 'OlltIIlIlO"S alld b(~_·) satistie'S theI11f"1I' (Olld,ltOIl [10.11J TI ... trilinear fOrlll r( , . ) IS cOlltinuous and wed1'1I0l1'

(II V)II - V /TIll. 1') f11111

I:.! 11 V II (I

\\ 'II.ul"') u' till ct{!

:\.1\1

o for 1\ =J l.

,,!/;

Slip 1ft 1')1I'f /I/o 1"1,

IIfl,'

IIfll'

blq "' ~ II

dll ",")-<11".") - b(I' ,,) ~ f(")

fl-::(/I '(11))"

VIIHI (u.d fcc \' x (>511.1t tltat for

,'wry (".'1) C \- > (,.

lIlHEE.S I EI' 1f~1' AIMI' liVE S IliA'! H;Y

N(P)

n = U ilK wIt It ilK nnl1\"- I

3. Finit .. dpltlent approximation, 1..1 P I,,' a hilito' ..J'·m'nt par-tltllHI of dOJllal1l 12 such that

till' slliut lOll I'" ullique

UOI)

\\Ith !lOrnl d. linf·d ;L'

tilldf'f II", "ho\"~ dpfillltiol1:-> 1i11d/"nlldltl(HIS, proldl'llI (:2 1) has at Jt'a ..c.;t <llH

solullnll(".}')\ \' (J,alldlf

wh'·r,· NCP) '" 1.11<' tolal 11111111... , of ..J"IlI'-lIb In II'" partill"l1 l' "lid I'adl

(~II'111I'lIt11,\ ( (If sirllply writh'rl ;t..'\ 1\" ) IS ;ll~')IIVI'X HlIbduIllain,

(111

']'Ill' I"HI} 101<" f IS d..tillt',j "lIclt 11'''1 Illt'rr' "XlSts a fUllftloll;d.

1'11f'1I II ... wrak !'H1J1Ulal'''1I of N,t\wl' SI,.k,·s "'111:11'''115 ('2.1) IS

~ ...Il .h"! ".!'-, I.illf'ar, hililH'ar• JI,J [ 'I, "',

I~(u. ", w) IIllit 1"11 11/1 It

SlipU,V lL'( IIc\'

-,

\. \' /Ji, "(11..") I 2" D(u) D("l dr!l

lJ \ iii, b(,/. 1'1 I 'IV "dxn

\' , \' \ . - II r( It, ". II') I" v" lD dx

\ . - R, f(") I f-,·dxn

I I INSI,I·.Y O[lhN WEIlI·\N WI'. "Nil M'\H1, i\':>IS\\'Olfili

h

"

f

\',11111 II l' .\rt tlH' \1'lo(J1~ itlld prt'~"'lIr,·; "(fI.l') is rlw (".,uchy stress,1,1111'd .IS ·It' D( II) - 1'1 wil I' I I.. klll<'lII.11 Ie \ IS( oSlly I' ;. 0, 1.1'" stram ra~t"IIS01 D(u) --(Vu + VitI )/2. ,,"d th,' 1111,1 1"lIsor I, ;11111 f is lhp bOdyfOIlI' '( U slIllpllfy Ilw a.nalysls, IlOlnl)~~lJwOllS I )Irlrld.·t IU)Ulhlary conditionsiH" d:-':->IIIJ1,·d I III the an

"1/- IIOW illt.r.,dtJcf· t lv~ followlIlg SjHI(,('S .llId .·C)rrl~p.)ndillg norms forr. '1'111111;d 11',1( t.h .. w..ak rOrtll of "qu,,' iOlls (2 I);

:1Iii

I- ( /{,\(ll))"H - {I'1-:: I' V /' III

Q = {qC U(ll) !'ldJ =- OJII

I ~, ~l I f VI-i",; =- V".I.r

I .' - :; q' d.r~q :I~..

\\'h,·f" d:J' t!;J']dJ'/ dr'l :ill.) VII v",tlld Irlllll".lr fllnllS ;\r4' d,'fiJlf,d .\.....

The fintl.l' ,·I'·lIl"lIt "pproxllnaltoll of (24) call I,,· e1f'~frilwd 3.<fullows

.);,)

c' - l' - l'

fI( t', ,,) n (lU,', ,.) + r(", ", ,,) - r( 1/., " .• ')

-b(q. ,'1

f' -:- fl ~ It

q 'I Ill' "rror (c. E) IS dl'fllll'eI ..,

TIlHEh S I hI' If-I' AIJAI'TIVE 5THATI'X;Y

{·1(cp I')III, q)

(11! " i,', 1 J I. Icp'!'. + II~";,

wllh

(1 ~i) IIcpll,; A(lf'.lf')lI'ill;, 8(,'.';')

{ ".. ", .I 2/1D(u) D(,·) ,h

(41) II

1311'.'/) J pq dxn

(n)

4. Au 11/'O .. t( 110' t ("('rOl' ('stilllatf> for :;(eady st ate Navi.·r·Stokt·sl'<Juatinus. 'I (I t(lll~t rlJct flli' II l/O .. I( rzon "rror /'slllnal,' for HI"'Illy sf ill (

Navier Slnk.':-\ j'qlJallOI1S, \\'t' IIltroduce (,wo hdJlIl'ar fOI'lIlS,

and t.It,· I'd,r (cp. ~,,) EO \. )<, (,11 .. J". I IIf' $Oll1t,lon, "I'

(-1.6) (IlJ' 0'(1/,1'») ~ "li [( I - "J{L)(") O',,(tI,I') I oJO.(·') O'J (1/,1')]

for e"l'r) (,. q I -= I

(42)

\\"e nott Ih,tl III ord«'(' to ohl.lIl1 <,oIlVt'rgcnl approxunat!.: finIte t')f'IIll'II(

saillt 1011'. Ih,' p"l~II' '1I1I,-.I~of eI"gr,'" kli for v"!ortty allellK for I'r"s""r" ..r.'chos"11 "~II It lha: til'" d"cr<,t, 'I/Is,,1' condItion IS <"tl,Ii.·d. , .... iI2]

wlwrl' O'J, I" the Cauchy st.r'·,,,' '" nJ\ at boundary .• (c I J,'1., tTJ_ IS t.11<' "'1[/'-

spolldlli/\ sl ress III neighboring ,·I,·ment ~)L at. til<' "aniI' hOllndary I'''Sltloli.', and II" " th .. Uillt outward 1I0rrllili ,",,('tor "11 llnli The paralll<'t.rl('functions" on l'K L, dl'no"''' "Ii I , are construct,'d such that the "lo-n1<'111alld boulld"ry res,dual are b•.I.lIl,.'d 111 th .. Sl'ns.' of [1:1]- TIIPII (-1.6) r,,!""-sent, a IIII<'"r ",,,"l>llIalll>1I I>fd"IIll'IIt. nux Oil tI... boundary rliL at.' 111particular. If all cornpcII1I'ntsllf 'tl\l. arc~, thell (I) 6) IS ~i'"ply all a.Vl'ragl•

boundary lIux {)J) 1'1\ l. •

Freull tilt' pr('cf'din~ d.-finll iOI1:-;, \\0'1' then h:1V('" till' follnwing tlworclIls

wh"[f' (11,1') \rl'lh,' ,'xacl ",1111'011' of (2 4) an,{ (11.1') ar .. Ihl' a""roxi 11... 1,50111111>11'1>1(:; cl Th,' ,tar ""rl" "f (I> c') I~ ",'Iil",d .L'

"'f' <list, .1'1111' lit., "i\.ra,g" 'l,pr«IXlltla.tf' nux Ilfl tlu bl)llndar~ I hi II

dOli' 1.1!11 •. 1'

q 1,11i 1'':; <II .I.'nc,l,all.1 (I ro'~1 rl, 1.·eI to OK

,"Qh =- II Q~

"-I

J :'I'DIU) D(,,) ,I> Vu••• E VI.

/,

,.,.

II I'J~'K-l

/\-=1

bl,(q, "I -, J q"il,,, dr "''1 E QI .. t. E \'1.

I,

/1.(11) - J f '''/1 '0'.'1 \'1,

'·I,(U.", wi - J " "il •• ,,/olt/' ,IT V.t. ",'0 E \'1.

I.

h;;;;l

(I" (11, p)

N

L /'hk ,,)

= L f,,(1I)

N

~'h

"

= L aI-Itt t'l1,_1

L CldU,t',W)I{-:;:J

I'mel ("h, ph) E I'h > Q" 'llch t hat for 1'\'O'r~

(,,I', ,/h) E I'" A (i'd"h,,,h .• ,") f "(1/",.,1') -b("h,ll/.) - f(',1')

b(.l", "I.) ~ (l

J I IN51.I';Y ODEN, WEIIIAN Wlf. AND MAHK AINSWORTII

b(l/. ,,)

11Ill,tlj

TI ...II till' J ..liOitIOIiS (23) (.111 I... "rit t"l1 ,IS

1(")

c(u, n, ",)

(2)

(3 1)

:1:>0

\dl"r,· I" - I In,,) ,Uld (h (q':; l~I!!,,1tlw Cl.rn"'p11Ildll1g local "'p;\('( S flf fUllctlons In \.r!~Spt 'ct 1\,,1)

F()lln\\'lH~ 1h,' ~talldard {lIl1h- "'1'111"111 pro("f·dllr'·l w·' d"filll" twu liI1l1;'C]<'IlH'lIt appf( IXllllal il)[1 span's \ J~' ilnd Q~ \\ II II pll'('f'WI:l(' polYIU~lnials Iffdc>grJ'" {'f, :tltd 11\. r"spl'cLlv('ly, spilllllf·d 1)\"'1(·I('IIII'lIt /\' "lJc'l! Ihal \'t L \'"

<tlld ()~ I ()" '1111' gl(Jkll flllit. I I. Illl'lIt apl'fc,xllla.,t«' "'p.tt fO, \ .,. ;lIId Q~,tl\l'll ;·Ul'

is;:

+

-",

;:

"'"'IIE

:'" .-

"W~'.1

I-

:..

~

- ...;;;.;

~~~

;:

-"

'.

'-'::~- ....= -=-=--

,~

,~

'-r.

..:

- 7:~ c

7 ~1: ~- -=

:: ""'- ":;; ""C r.

"".;:>~ ...... '""-2~I ~

... -~;:: ;:;, c

-<:: ....;:,

=

II

VI

1-

.... ;::~~.:.: -.. -=~ ~=::~

-'

z...

;J'15

.'\'11"" 'Jl

2::"/- '\'"h 1\

1\::...1 1\

01.: = 1\ ('PI.:' <ii,' lLK) 11.,1{

11(", c·)I1·'1 ,- IR~~0if

!I{ " J,'I:I: .

!\

L {II (<pl.:. dit· "I.:) Ii? I.: + II (1f',;1.: d", iLl.:) 11:./,-}, = I

(J

of 2.-1/\ ('I'I.: .If',,/, )}

""RhE-S'1 EI' /1./' ADAPTIVE STIlATEC;Y

."1(1'1

., "" 0"o· '" L.J i"K~l

III It. I')11:'

whidl is cslimate,1 by "'HlII'utlllg rl ".S d.·lil1"d ill (1,25)To dev"lop th,' adilptiv,' stral<-gy. Wt· ,hall III "k" thl' following "'''''"ll1p'

lions:

(5.3)

with

The Nror ind"x '/ d,'fin"d in (<lIS) i,

(52)

(5 I)

\\lwn' "h alld lJh ill" t hI' 1I1,'sh paralJll'tI'ls <llld rl'pr.'sl"11 till' 1Ilt'~1t ~IZI'

ano ~l'(dr.lll'r(i\ rIll .\11 ·1'11Il"I11 illlll'J ;-tJHII' dr" par,lIlld"rs dqIPlldlllgon thl' rr~ulanty I'.f 'I .Jut lOll'" ;lIld approXIIHltf' "'IH'd ral Ordt'f1 S{'(> {K,9J forddads. AI" "a 1.,..,,1 <0,,,1;1111 W .. "Isll d,.f,w' "gloloal .. rror IIldlcator 0and a loc"l ,'rror Indicator (h as

(41ti)

". Th ... ·.. ·~te\l 1/- /' '\llal'tiv.· ~trat"l~Y. Ilt-r.·. IV" ,·XI."lId tl ... thret-step h,/, ad .. pl.lV<' str ...tegy lIltrodw'ed III [9] alld us"d hy [8,14! TIll' goalof til<' three st.-I' adaptl\<' ,I.r"tegy is to reach a Pl'<'51't target I'rror ofthe probl~1Il by <>nly soh illg Ih,. syst"lll on t.hp'" e'H1S(·cut.ve hop adapl,""m""hes, lIamely 11ll' illitial me.,h Pl. tlH' liltermed,al" h adapt IV" Illl'sh1'1. alld ,he fill,,1 h-T' "dapl.ivf' JI1esh P:! at II ... II"Lsl p'lssihl,' cost

Th" JI1ltlalllll'sh PI. ",llIell IS SIJI1ply a rOllgh (hscr,'1 "''''lOll of I.hl' prohlem. I~ g,-It'c(c<d 10 I'rod 11('(' .. r .. latl\·,·ly IIWX!"-IISI\'" approxlIllatloll of th,.$vstcm. but It dOl'S provid(' SCHill' rflficallnf'l)flliatloll for later adapti\'l' n~-r;nenlcnts rlH' Internl,~dl(ll(' IIwsh P'! IS the ~1'coJ)d 1I11~h produo'd withonly h - ,,,\;.pli\,· r.>lin'>III'·lIls Oil lit,' 1Il1llalnl<>sll 'I h.' IIn,,1 nlt'sh p" "allII-p adolp' "',' 1Il,>sh (,Iolall ... d loy ,IPply,"g I' ,ulapt"" .. nnChlll('nls 011 II ... 111-

Imn,·dial<' nll'slt i\ graph,ca' ,J,·scrlpl .. ,n orlltr'-'> ""I' adapli,,1' "I~orrlhlllcan 1)(' :;"cU III rlgUJI' :> I

To d('\'f.:'l,)p tlw s( IH'IIH', \\'(' COIl~Hl"r "global a IJflOn " ,; ('nor ('Stllllah'

in trrm~ nf :·dar 1I0flll

o

:11'P,.'P.) A('P",<p"I+·lA\'I',;.If',,1A{<;, •• 'P.) - .\('f'",'P" I + '!l H'f', ;,::•• 1 + I(lf'u,CPuJ)

.\(;,::,.If'.) + Illf'".If'"H Llllf' .. lf'ul

{AI'i'"I')H(~'" (/)

'\lNSI.EY onl·;N. WEill AN WI'. AN!) MAHh ~lN~WOII (II

(·1 :':\1

Frill1\ d'l' U 0, I'qnatloll (·t l1) Iw,·otu(·s

11(1I./,)I.!: '" '\('p" 'P,.) I /1(1:,,, .• 1,.1'\(If',.'P,) 1 .\('1'" CP,.) -+ ;'.1(", •. ",,,)

\!If', II:, ' \I1f'" 11'\ ' .} \1 'P. 'Po. I

ll,,"!; II ... .1"'""11"1" (I 1) alI<I (1 1:\). w,' h\\," 1 11\)

.I( If'". 'P" I(1:m

1'100f "r"lIl II ... ""'"1'ul"I .•./ual IH,.ut pI' \I"W, {I II ,'1Il1 (1]6/ are!'fllUjlUI1.d h~ ~,cjhITt~~till l()rr(:-;l'flllli\lq.~ \t).-,\I pr..\.l. 1!1'" n\' r th.· tnl~h ll~'

\lSI II!!, (-I II), ( 1 1'1) (I 17) ow,1.~'

il(lt.p)ll; L: (.1J, (If'". '1'" H lh ('i' .. , ./ .. )}

(4 '2-1) /\f,ilL. \II (If"d, ,dll' U/\ ) 1\:,/\ 1/\ ~l

Wf' d(~hn(~ till' ~\pproxttnat(" effOf ll1(h'x a.....

t.' }l(l'. ~~L {1I(v".-.d"'u,.)II~,,,l .,---,:_ _ _ 1,~ -- ------ ---- ;,:~'(:

I:{,,,,,.,,,,,.,."'.'."'.",..'""'.",,,'",.•.•",}.:'.,.;h-l I"

and (,JwlI

(-t 21)

(-l.10)

fnr "\"'r~ (t', q \ ~ \' x Q Iku"-

A('i',.'i',) -- \('i'" 'i',) AI'i'i •• <P·)A('i' •. 'i',,) =, :11'i'".<P,,)' A('i'"'<P,,)A('i', .. <p,,) A(<p",'i',;) - A(<p",<P,,)

ItwlI, by luallg l")'lIllllf'trac prop/'rty or lnllllt':lr fOrlll At . ). Wo' h:\\"4"

A(If',.'i',) = :1('i'" ,'1',,) + A(cp",If',,) - 1:1(1f'".1f',,)

(11\1)

TIl~:ORI.M .1 -\ F,,"" tI" "b"1< d'JlIllllOll' tI" ,II'''' ",da '/1>

Il(e, f:)I\. ( \\'1',1\'\ ~ I\~',I\~, )l/~'I = - --- - = --., -- > ~- - -

\\(u , 1')\\. 1I'F,lh + 1I'i',;\I'\ of. I('i'.''i'..\

1/1n 'i'. (JIld ~'. "I( "1II1'/Y'i' tllld v f""" U .!,I"l/"f From till' d"lil\lllOn, (,1'1), (1 If,). (IIi). Wt' h;\".'

A('i'". "I - .1\'i'".I')fJ( 'i'" , 'I) .. H( ~"',. 'I)

(.1 IS)

:151;r;·~I I;n I

'I'rl -1

iH;.[1

:1.'i7

[

'CPI (II' ):.'" .!..h...\/,L- ])1'

1\ = 1 1\

for lit,· lillalll"'slt 1'.1

1I1I11 •.! 1111, trll· "l.lll. ,tlJd fll1;1111I""'!1

'ql,1I IIIIUdq'r nt' "j.'IIIi'lIts for .'ad;111":0;11

glllL.d ,\ "j~ ".1. h IIlI':";!1

;lpprUXlIlldk µ;Iofl.tl'·rrol' J•• , I':lel!IIH'SJI

th,," tOllJputt',j rf'pr.~el)(atl\" !"'rrOT

and apprOXIIllal,' ",Iutlons for each1l1C'01i

appr(IXllllatp SOIII' ion}; for ."a.chmesh

"p,.rOXIIII:<',· ex""t, ,·"luUon III s'ar1l0rnl for I'aeh cHI'sh

,,~0" = -A/\

I'v/\

N(P)

L Oi,A"= I

\, ,\, \,01,1/:>.(1 .•

l' p) 1"'1

'\1 \", :\'1

II/Iii, F;,S'J EI' 11,1' AIJM' /'IVE S I HA11~{;Y

0.'

'/7 = W ". I )11.Ii( II i;jll

01'

(it, ,I'J ), (II~.1'!), (II" .I',d

and th,' 1I111'rnll·dl.lll' "rrnr IlId.·x 1/1 rur IJIi' 11Ifi'lJllf'dJ.'111' IIll'sh t',.Dl"i.Il' 1111'"()"(l\\III~ lIol.lllt>llS'

"'t:l·tp(~. tpt'.'l'~lll'tp,~. I..pli,

• (;;""11 tit., III,tiallll,·,1t '1',. ,,,Iv,· lit,· 'y,t,·", alid obl,llI' tlte ,'pprox,;llIat ...solullon, (Ill.pd

ConJput/· tilt' error .·stlrrl~1t!..S OJh and OJ

\Ve now d"ScfJh" lilt' "hrl'!' :-ill'p ;ldapt,l\t' :d,c;orit 11111;t.~;ft,II!.)\\,:-;

• Th,- a.'~IllI,I<lI'C prror ""llll"I,' (:, I) IS Ir",,!<'d 'L< '"I "quality alldIS .. ,.,,/i,·d '" 1/,,· "/"111<'111 1,'\0'/

• '1'1", .Icillal /"".11 "rr,.! 11<". E)/i. /\ ,'<tll Ill' ,'pproXIIII;lI"d by Ih,.f~1 J JlJat('J ('rror IJHhra Ior III (~) 2)

'Ilwrefor.·, by IJ~IIIA I h(·~1' .\.-.;stJlllP' HillS, W(' Iia\t· t Iii' fojfl>\\'/lIg 10('.IIJ ('rr(lrr:itllllafp,

(5 1)

Aft..r COIIII""I1'g FJ:» ... ,,1 (.j ,I) w" obl.uII

(aJ)

SkI' 0: A',"~II II", "ffP" lokr...",.

\\'1' IH'~1l1 h.:- as.....I~llIlJ~ ri'l 'ar~f'( \'f1tlr Ill,i,'x

(Hi)

lI(ul,pdl/•. 11<1I~. h)II., III U.l.IJ.l)lI.

(5,7)

Step 1: Illit,allllcsh 1',

...,"

.oJ."..

,'!' ,:~:1·'''':1~::

mtermed"le h'ldined mesh

I I 1:-..; p

,-'I-I~EB' .ITU'__,'b;i

tL]

L J Jp

J TINSLEY OJ>I>N. WEIlIAN \\'1;, 'Nil ~1\Il" AIJ'l:SWOIlJ II

mlli31 mesh

-'--"--II ,

fiul3l hop adapuve mesh

Fit; ') 1 j\fo/tl!'!t from lb. ... thlr.~-~rri' h " cdcphn a/90,.;chm

CJ=r=1 p3 4 5

:I:;()

.\,

Stl'p 2: In\I rll1<,,\Iatc' )fi"sh P:• (;ivI'1I the ,nl"rllw.hak ll1('sh p", ,.,h.> th,' s)sl"II' and ohtallllb~

apprOXllnatc sollll.lOIl~'elL:" 1'~)('OI1\put(\ tlw f'rror !"stlluatl's O}f- and 0'1

• ('ompllt'. 'I' for "al'lI ,.11'111<'111 :llllllllll?"2>1j. loy

'IINSI,I;;\' om,N. WEIlIAN \\1'. AND MAI\K AINSWORIH

• COll1plllc' AI' for ..,,/ h ")1'111<'111all<l!i( "1./'1 )1\. by

359

)

'/1

I H/,lIP"h''''"''')}

f--

fIlHEI~~ rLP II,/' ADAPHVE STHATE(;)

1.1" U/µ-'"'I,

( ,",L {h;" "'Ii ('P"'h .·100 u") 1\:,/,I c:...1

0,IIG:;"}IJ.

PH•. hI g"nn,tr,' oj flo I!, (IV"'" rllHtty 11'10.. Rr ~ .1£10

'f,

11/.

!I(UJ.f·.llll·

• If ( 'Ia > 'n )lh"n r"l"'i1t 111<'h /' .•dapll\'· pro,c8S al1d dll'ck th,'stoppiuJ!;. crllt'r1oI1 aft;'r 1';lC'h 111"'0 Illf's11

• "lop

~

ThiS ;l('t,ual CIlSt..,f" or JI ,L(Lq)fl\'lly I~ Ili·e;,hµ;lhl.· \\'IWll f'otnparcd tothe comput,allcHI;ll (1).:-1.>f "'lhlll~ Ih.· systl'lll TI\l' two parall\f'h'r~ II ,nHiµ ill (5 I) wJlIch alff'II till' dl·g.'l'" I)r II or I' rdiu( 111'"11\ t~an Iw «'stnnit.t.t·dthrough t,wo IlllI!;" r"~lllar ""·s",,s Th.' d,·L.ds can I", fOllndln [9.R]

6. NunH'rical r(·:,;;u!I.:-.. \\(0 fOIl~Jd!'r ;l Jlf.d}JI'11i (Jf 11+)\\' I'Vt'l' cavity

lb, R..ynolds '"I1I1I"'r I' "'I 10 I.., ·100 \\'.11. ..haractt'r"t,c v.'I ..coly alldlength ha.,ed Oil rill' ,I\"'ritg.'d IlIfl,,\\' v..roc,ty alld IIlliow ch""I ...1 widt hA paraholtc IlInow alld 1I111f.. rrll ,,"t.liow I,olllld",.y condItion are u,,'d. Adetailed geolJlelrlc dcscrrpt.loll ,~shown In Flgllrl' (j I Th,' 1l1ltiallJlP.$h IS ''''-

lected with s<>nll' lo('al h .1IIdl' - rdin,·m,·nt arolllld tl ... corrll'r sllIglllarityThe fiow pro hiI'm IS ~olv"d I,y lISllIg a stalldard NI'wton-Raphson it ..rativcIOlverwith contlllllou, "docil les and pressurl's '1'1... vel')l'Ily Iprf'Ssur .. COlJl-

binations on a f<'f,·rl·nc,· ,","IlI"nl 1\ arc Qk(l\)/Q,(I\) wlwr(' QdJ\) IS til<'Ip&CCof tcnS<Jr produI'" of illlcgratl'd Leg"ndr,· polynotnmals of degr('" kForthis cxampll" I I\<'followillg velocity Ipr,,,,sllr,' spacl's ,,('re used. Q2/QI.Q3/Q2, Q4jQ:3 a,nd Qp/Q(p.:!) for p ;:. 5 A c"ntlllllatioll method 1" us"dlo reach a steady ~tal(' solulJon at R,' = 400 for tIl<' millal 1Il1'sh. howI'ver.forthc int"rtIlc(iiat,' ,.nd filial Ill,>,h. t h,' prn'IOIlS approxllllal.I' solul")I) IS

lI8ed a., all illitial gll,'S~ for the NI'wtoll,llal'hsoli solwr Each Newton-Raphson it ..ration is s,,\,,'d b) :0 direct frolltal solwr 1'h,· t.wo p"rallwl.crs

(512)

~ --- 14 ---I

1E'"1L."'/_'L'LL//LLIL/LILLLLLLLli=,,~JllIr.lI>,J" ,nn.,.

9 I/1lll7l777l/3vmvmh /~-

)'/'!,I:,('P••• ''P'.,)}

)

'/1

1. ,dil' u.'l\ 11I:,h • ! "'(~"': .....,..,••",,)}

(0'[~,OI'~) ,!+t

(~~ )\/ ../'1> "- -O,.:CI,/;,.\I'

I",

III fj':'h,/.

(

N,2, Ioi" ~ II' " ..

!:1.llli~\ I.-

('"L: {<Ii I> + \'i('Pu, ,.,.In UI, \ ',1:" ~1\--1

Ah

• ('OIIlPIlt. •. 1,1... r.·,plII<,<1 /'/'. for th,' til1.'\ 1111.,,10 h)

('mllPIlI" Ill> ,,".1 Nf h~

\

Ill'

N,L"I, .., N2h:: 1

wll<'''' ,,/,IS I.h.. r"<Ill1T<·,1I, rdilwd ..1.. "wllllluml ...r in p~ for Ib~I-\.-II1('lItK III 1',. ,J = '2 for I D . I for ·2n. ,\1\'\ 'l(l for 3D, see [8,9]{or dl'r1vatloll, alld 0/ = !JIl\(lt1,1'1)\\'

." r,.fin" th .. ll11'sh 1', aCfo/(hn~ '0 11/, and ubta," p~

"h('r" Or = 'n \I( It? ,I'"lIl,• p _ rdin(' Ih.' lII..sh ac,ordll1g 1.0 1'/, al1'\ ohl ain 1'3

St.ep 3: FlIlal (lIc"h P:\• (;I\"'n thl> tlnal \lle~h P3. so\w II", system an.) ohl"in the appr<i·'

illla'c ~Ol\ltlons (tL31 PJ)('OIl1pll\.'> th .. ,>nor ",,\Imal'''' 03, and 03

('olllpnL<> AI' for "ach ckmcnl and \\(113,1'.1)\\. hy

I,I,IU,: 1I.'}I!.

(5 II)

,'-,1111

F,9)

~I

lI(u,.p, )11,

I:, ~)

:1;'8

[)elf u~

'1II/lEI.:,STEi' 1/,/' ADAPTIVE Sl nA II'CY

,H'.,~hJIH tlufI",'trl' a/g"nt!1I1l Inr ,aOl/' O/ffT ,·a.t!1tg (11, = 400)

361

1"11 1"II11

J,d.,.LI /"1,,,: '

(a) 1/111,,,,, III"'" PI

1_ .1.",","""1

. :~~.:iil:if1l::ia:I-=j.i' j- t- t-~ -i""1-=1:- T ". ."1 I

f h) 1I11.rlll>',IJ.II.· 11I,;.;h p,

I'r-I L/'rl~_111J Lil 111':~~~lrjl"!1'; f, : "",t

FIG.6 . .?

A .. knowl,·,t"nH·nt: I'h, "1['I",rl ..f 1111' " .. rk I,~ 1),\111'.\ IJnd~r tht( ... lIlr .... ( #IlABlld " ... 11111:1" 1;111.11,11> ... kll.nd,·dr.,,1

J. •J~

~:}."

" and v art' SI't to I 'I he local r"sldual prohl"1II IS calrulat.·d hy USingIh,' ,·qUlhbr.lllonleclllllqu.· d.·v,'lop,'d in [1:1). ,,".1 a "pt·rtral order of I'i I.'Iclliellt is IIsed for ~olvlIIg I h.-. loc ..! rl'sldual prohlt'lII wh.'n the IH.I, ord~rI'!c'nwnt for wloeit,y .,pproxunatloll IS us<'d for solvlllg II." ~Iobal systl'ln

The UlIllal lIIesh 1'1 h ..s &:17 .t 0 f s p"r cOlllpon"lIt B} "''It Ing tltellll<'rme,halt' '11 and lill,'\ '1/ to I... 10 pl'r<'I'lll~ .tIlJ :1 pt·rc.'nls, "t' ob-lain II", IlIkrnl<'dlal<' 1I ... "h p~ and filial nH'sh p,\, as shown in Figur(' 62wlLh d<,!~r""s of fr""dolll 10 h,· !:,Xil 'lIld H~:l:" r,·slwct.IvI'ly Ahh'}IJgh tl'e(I IIO~dt1'1(111 error f·~t 1Illat,-. hold:-; (lilly Budl'f til(' a.....SIJ1I1plioIlS or hOinoge-.""OUS boulldar} conditions w,lh slllall enough uu,,;h SlZ", till' results of the.'SIIl1,al"d "rror from t.he prohll'nl. shown III FIgure 6 3, g.ve a go<,d er.ror ,'stullat., eV<'n thollgh the apprOXllllatl' Solllllon frolll III<' Initial meshIS 11Ighly os"l1ator~', "" shown in Figure 6 '1. ,.nd nonhomogeneous inflowlogether With an outflow boundary rondltlon .He u""d Aft .. r a first h -r<'linemcn(, In Skp I. the "scallall,m ohsl'rwd from 11\1' pr''$sure is undercotll.rol. Tlw ~!'cond], r~·fil1el1l\·nl, IS appli"d 1(1Ol.t:\l1I th«' approximatet-i(l!ut.ioll Within tlw larg,·t,·d f;rrOr \V«' Ilnf,f' that it i:, nol 1H'1't,:,:-:arylo com.p\l(;~ til<' (~t lI11:tu'd errlir for (~:t('h t'1t'IlH'llt for «·ad. st('1> In pract.ice, on~can avoid r"calculallll!; Ih~ error for delllelll, wllh wry slllall ""tillla~,'rrar aft"r II", fir~1 skI' alld ;\.""111" I hat I h,' "rror al t.1", 1"lf'r steps IS lh.sanwas t Iw first step II hI' final \l,lof'lty \("'dllJ "idUff' I~slul\\'n as FigureIi:) The filial preSSllr", as shown ('lgllr<' Ii 1. I'n" 10'''' a ;:1>,,<1 approXlIIll-tlon wlH'11 corllpared \\'Ith a :o::eparatcapprnxllIl;ltf· ~nlllllflll:o. obtain{'d from

SI,!VllIg !'IJI' prohicill 011 a Illll('h fillf'r IIw:-,;h

3()0 J TINSI.EY ODEN, WEIIIAN Wti. ANI> MAIlI, AINSWORTHI:i't I

.II

F'. "1' ,

: :

"'f',, t

'-

..,Q...~".~1~~ e~-......~"'''', .....

~~~-. ...E ';:

~ §, ..., ,

'" ;!:i ~... ""

..... ::;:.

I!

bI •.... '

•;,",

u

"

_I

I~1

'1:'

'.';;;:Co.<c«

~z<

z(,:

co

:::

IINSLI';Y ODEN. WPIIIAN \\'lI. ANIl M,\III\ AINS\\'OIlI'lI36:)

MIN .. , ......l\.O4

MA\;_4'l\1~".

'I~I L " - - J ..

......., 1"0)'\'.~"_u ., .....

(c) finallll"sh p"

THREE-STEP /I-I' ADAPT/VI'; STHAJ'EGY

FIG 6') FHlal:df.)Clty fir/d Jorfl(IU dIn rOllt) I R~ = .{OO)

FIG, f, ... Appl'Orllllldr lHtHur.>: 111:/·Joltl fUll/flit/lit w /v, th,n .~tr-J' algonthl)j Jllr jl'J<L'Ot'rr C4t'Jlr ( u, = 400 ) (cont'd)

o.l'!'i .... ~

..t"' .•~I~l1

... ,"'_ OJ'1l"'11

"'<\):-.~"~:A

lr_ ft~ rh""-slc;; clg"r:ttmJ,rf..-.. dUTl"·l_,,"n'l:

(,,) illll.ial'II,·sh 1',

...'"j~

(1)) IllkrlIl,·,liatl' lIl<'sh P2

rr. II "

\ -'I ..~~ ....\ ~...'if:

'~'.).'~",if",

11(; b i4pP".-;;rU1ldtO' J'n'H'\r" Inntr'r (at'dl' t Rt = 400 )

:W4

"-,r ~.

~-' .

(I) M AII'''\\'OI<'" ANI> J rOO' N. <\ PO,""'''' Em" E'"'''6In''./''' SInk., P",ble",f1l'OM IeI'M'. Tlt9J,Ol. Auslm 1993 .

121 It L BANl< ".0 110 WE~""'t1. ~ po,''''"'' "rr .. rr""""lor, (nc Sh,keHq"·'",1UIt. compaIl50II, ('Olll(l Mf,tll Appl Mt'dl 1:l\g 8: (l~JO). pp 32.1-'\40.

['11 It [0; BANK AND B.1l WI~I<I·I:"T. A po."<TIon e.-rore.ol,m",,,,,(,,,Sloke>eq".'IIO<,,

SIAM J. "" Nnmrr .. ,,1 AI,""Y." 28 (l!*H). PI' !>91 f.1.l '(.IJ It \'En,.'''t1 II .. 4 po.,""o" error ,,1,"Io,IOn< (0' Iile Sloke< «,".,,,..,,,. Num<ritcht

l\b,hem,'lIk !>:, (1989). PI' 309-'125(~j It '<if.1\pijlt1l1. II l",l,,,on error es'm .."or' ("r tIl< SI ..k,·" '·'1""'; ....' /I "0"",,,

rmnllugdisa("(izlltion'>, NllllH~I'i!'>d\t'M.'\lh"ItlJ\tik GO (1~l92). pp. 235-249.[61 W ltACI10WICZ. J T. 0o,,1'1 ANI) I.. J)EMKOWI<Z. f<n"" ,I 10 ..... ,.,."".11,.1' bd,'ph",

rr"jtc d('U1L'f1t strAtt'g)'; part J. A :otud,\' of tile doi&fI of '~.p meshe:., Comp,Mdh Appl M.:d" Eng. 77 (No 2) (1~l89). 1'1' 181-'111

[7] l.T 01)\ N. W. WI' AND M AI"SWOltn ••. <\" • po,I",or' ,'''01' e,Ii'''4'' ("r 60;"elc:OlC'ut approxiuliItJoll!o of ti,e NJl\-ier-Stnke .. ('quill;(III~. COlllp. !\.kth. Appl

Meeh Ens 111 (N..s. 1-2) (199.1). pp Jl!~-202.[HI W WII. 1/_1' ArI"pt;",· Mdl,,,rls (or '10""""'1'",,,1>/, \ "ru".' F' ... • Pn,bkm> (Ph.D.

DI'>:>;I'rl:lt IOu) 'IiII' t1lliV("I~llY ur rc:\,\~ dl Am.tlll. A\I~hn. 'ft"x,,"' 19__3[~,] J I, OO\.N. A. PATHA ANI> Y S. F.:"". An 1/,(' .,,1,,1'''''' st'dlcl>.'" in Ad'ph .. ,

Millt JIc-\'d JlIld Hll ..r ..,rdlical (\unputlllHlfidl S'tmlf"SW" (A .I{ - NOf1R .. rd.) AMD.

\tu\ 157, ASME l'uhliolli'-\l\S l~N~. pp 2J ·F'[10) It '1', ,,,M, N''''lCr,S'okc., E'I,""i ...." TI........ ,,,,,/!,,,,,,,"".' '\1041,<'sl> (.«and

printl1lf!.) NOItll-11uJlawl. Alll"t"ldaI1l19Sr

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lrr 1(1\2

APPLICATIONS OF AUTOMATIC MESII Gt~NERATIONAND ADAPTIVE METHODS IN COMPUTATIONAL

MEDICINE

Introductiou, ('ornplltatlonal Mi·dlnlW LSillll'lllPrg,lIlJ; 'Ut·a r)f"('Wl1

tifie eompu' Jn~" Ilich otf"rs HI>' IIIV,'S('1''11or opport IInlll"S f,.r dls'o\'<'r~ and.pplieatlOn III bnl.h llIed,CIIW alld In 1lIlIlI>'rI"al allal~slS ('(Jlllputal1onalM~hcll1e CIICPIlIfHI'S'<' a hro;ul rall~(, of tnpi/s IlIcllldlllg h .. ,,,I.'clnc li,·ldsimulatIOn (1I10dl'l'II" of .·I,·elrlcal and l1Iaglwllr [i"lrls fr'HIl th, bra,n "lidbrart. aod Jllo.I,·IIIl~ of i'xc:ifahlt:, tisslWS), hll)t\lf'(-h.\l\\(';llln\)lkll1l,~ (bIlIJlI!"

challlcs. t'rgollollll(~. and ht'fn.)dyuanllrs). Innlf'nllaT Illultl;!.)' , .Hld 1J1I;lgIIJ,l..',

leclmol"gy [l.:!,:ll.~I! It i, n .. t th .. purl""'" ()f II", 1'''1'''1 I.. ,JIS\II',' I",'flltirr fif'ld fir cn(llput.tboll~d llH"dlrtB~, hut 10 1n\",·~tIgalt' ,1("(Urac}' l:-,slH'S

10blo('lcdrir ~l1l1l1l(\tt()ns IS:-;'lJ('~ that 111lp:td Tum.1 oth;'r fi,ld p/ohl/'fIlS lit

computatlonalllJ"di,"""Computatl'lu.d I i,f l1o]Jhy.'>Iolot}l} call b" hl(l:-;I'!} dl'hlwd :L" Ilw ~llIlllla-

don and ulod<,'l1ng of nlarrn~C()plC alld JlIICr')s("(Jprc h'I)I')I'dri( ti/·ld" 1"IH'ekctrical acllvit} 111 the 11I11lian body" ,ndu('cd by 1.10,' flow of chargedms across membranes of mdivldual cells. Th,· collectlw ml.eract iOIl of all\he current~ from groups of cells I(IVCS ris" 1.0 ,·I,·cl.nr I'okntials wlllch ar ..measured from. for o>xampic, th,< ,;ku II (E EG) or til>' t '>rs,) slIrfae" (E(,C)The registration of normal alld abnormall"o,",ectncal Signals has he"" IIsed• a diagnostic 1.001 for many years lind <'xl"nsiv,' catalogs Imkmg Signals10pathologlt·s haw evolved More r<,·C'·ntly. invpsligators have be!;l1l1 \.0~Iop mathematical d''ScnptHlns of the Mi",in of th,> electric signals theyIleaBuw, Th" hope "XlSts that thes,' lllathcmalleal d ..scnptions Will pro-

J.A. :;C1iJ\IJD". ('II JOIINSON'. J C EASON' AND Il S MA('[.E()l):

Abstr:u·t. 11l1pClrthll( prol,l<,m~ 111 (\,IIIJHII ..afIOllid ~kdltltll" ('xlsl tll.al / lUI h~l1t'hl

from th~ ImplrUH'ull\llOn \}f ",dnpllvc mesh f(,filH'IIWlll 1('I"llIqll~ BI"JOglt'll '>ystl'!ll'l al"50 iuher<':llth' t~OlHpl,'x tha.t .ml)" ('lIictf"nl modrb. 'UIlOIl1'; Oil stl:1l~ ur lilt" .'11'1 Il.'Ird\\ilP'

C&JIbl,gill ((I ..i'lllll.ltl' ('(,/llit\' Tu ltlckk- ll1 ... <ompl(');. gromdri"1\ il~~O"'lllll:d with IllNijc/l1applkntj("I~ We pi r"~'I'.·nlA g(·,WU.J I~"rpo~w nwsh g{,lIeHl.tion~('h,.mf' Ua.!>("dupon th(" Drlall~n,"y tCPlltclIa!lOn aJgonthm ilIl~ lin II_eraljvf' puWl g(~fJt"ratur. In llddill'H\. automatiC, t.Wo

and thre('~dlln~n"'>Ionill ndaptlv(" nH"sh tdin('lllcllt 11U"1tlOd~ aTe pl'e~('Jltt:d lh.ll Me dellv.·dfrom local and global ~tim ..\t~ of th,. fluitt' cklJlelH en·o •. !\l('!"oh g('IH:,.-ll,on mlt! a,lnl'-live refin('nH~lIt tedmiql!("$, are Iltili'l.~cI to ohtall1 ."l.t-nlf-.'ll r~;Ipptn~ 1Il11lthl"~ of bi(wl''':l'l,.,,-

ftdds within lHllltOllllt'a1I" f'(lfTl"'d I1lncit..'l .. of th,' he;)_lt ..,nd hUl1l,lIl I hOl ,LX, Sp,'e.findly..1: ('xl'lop' till' ~ltll"l"lc i'HI (If canJuLC d,·fihnll.ltioll alld tlw gt9'\t'rA! ft..rwdrd And 1II\'('r'>('probkltl$ 111 de( tr'H.:.'l,.dlllgrllpfl)' (ECG). C'UHI1lUISIJ1lo, ''''!wt'r''l\ UUI{.Il'IU .md ''\({~~1''1\'

refi,n('m("nt It-dlJllqH~ o'lrl'" rWld(' to IHgfdlghl th,! ,-.)1nl'tll'1I1011IU ('f/iewl\! Y ;Uld i.!c('tU,U",,,

01ada,pti,r u\dhuttl'o 11\ lh(': ~nllltlOlI qr fi(·ld plof}ll'lIl~ Jll I <JIlIPlllililollillm,·,Ji, lilt'

REFf.Rf.NCl:S

llN:,LEY OOEN. Wl~IHAN WU. A;';O M"Rt< AIN!>WOIITII:166