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Tow ard D y n a m ic S p e c tr a l /h p R efinem ent: A lg o r ith m s and A p p lica t io n s to F lo w -S tru ctu r e In teraction s

byR o b e rt M . K ir b y I I

Se.M .. C o m p u te r Science. B ro w n U n iv e rs ity 2001 S c.M .. A p p lie d M a th e m a tic s . B ro w n U n iv e rs ity 1999

B .S .. M a th em a tics and C o m p u te r and In fo rm a tio n Sciences. T he F lo r id a S ta te U n iv e rs ity 1997

Thesis

S u b m itte d in p a r t ia l fu lf i l lm e n t o f th e requ irem en ts fo r the degree o f D o c to r o f P h ilo so ph y

in th e D iv is io n o f A p p lie d M a th e m a tic s a t B ro w n U n iv e rs ity

M a y 2003


UMI Number: 3087288

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© C o p y r ig h t

by

R o b e rt M . K ir b y I I

2003


T h is d is s e rta tio n by R ob e rt M . K ir b y I I is accepted in its present fo rm by th e D iv is io n o f A p p lie d M a th e m a tic s as s a tis fy in g the

d is s e rta tio n re q u ire m e n t fo r th e degree o f D o c to r o f P h ilo so p h y

D a t e I 1 k ( "LO ^ 2^G eorge Em K a rn ia d a k is . D ire c to r

R ecom m ended to tin* G ra d u a te C o u n c il

D a te 0 8 / t ___________________D a v id G o tt l ie b . Reader

D ate 0 ? / l b / V-CC ^ - v ___^— Jan S. H t's thaven . Reader

A p p ro ved by th e G ra d u a te C o u n c il

n„„ 1Peder J . E s tru p /

Dean o f th e G ra d u a te School and Research


T h e V ita o f R o b e rt M . K irb y I I

C itizenship:

U n ite d S tates o f A m e rica

Education:

• B ro w n U n iv e rs ity . M as te r o f Science degree in C o m p u te r Science. M ay 2001.

P ro je c t T it le : "V is u a liz in g F lu id F lo w D a ta : F rom the C anv;is to the C’ave"

A d v is o r: P rofessor A n d rie s van D am

• B ro w n U n ive rs ity . M as te r o f Science degree in A p p lie d M a th e m a tic s . M ay 1999.

A d v is o r: P rofessor G eorge E m K a rn ia d a k is

• T h e F lo r id a S ta te U n ive rs ity . B ache lo r o f Science degree: M a jo rs : A p p lie d M a th e

m a tics and C o m p u te r and In fo rm a tio n Sciences: G ra d u a te d Sum m it C um Lauda.

P ub lica tions

Book

1. George E m K a rn ia d a k is and R o b e rt M . K irb y . Para l le l S c ie n t i f ic Com p a t in if in C-r +

and M P I . C a m b rid g e U n iv e rs ity Press. J u ly 2002.

Book C hapters:

1. R .M . K irb y . G .E . K a rn ia d a k is . O . M ik u lc h e n k o and K . M a y a ra m . " In te g ra te d S im u

la t io n fo r M E M S : C o u p lin g F lo w -S tru c tu re -T h e rm a l-E le c tr ic a l D o m a in s ". The C R C

Handbook o f M E M S . C R C Press. B oca R a ton . F L . 2001. E d ito r : M . G a d -e l-H ak

2. R o b e rt M . K ir b y and George E m K a rn ia d a k is . "U n d e r-R e s o lu tio n and D iagnostics

in S p e c tra l S im u la tio n s o f C o m p le x -G e o m e try F lo w s". T urb u len t F low C om pu ta t ion .

K lu w e r A cadem ic P ub lishers . T h e N e the rla nd s . 2001. E d ito rs : D. D rik a k is and B.

G e u rts

iv


P eer-R ev iew ed Journal P ub lications:

1. I. L o m te v . R . \ I . K irb y , and G .E . K a rn ia d a k is . "A D is co n tin u o u s G a lo rk in A L E

M e th o d fo r C om press ib le V iscous F low s in M o v in g D o m a in s ''. J o u rn a l o f C o m p u ta

t io n a l Physics. Vo l. 155. 128-159. 1999.

2. R .M . K irb y . T .C . W a rb u rto n , I. L o m te v . and G .E . K a rn ia d a k is . "A D is c o n tin u

ous G a lo rk in S p o c tra l/h p M e th o d on H y b r id G r id s " . J o u rn a l o f A p p lie d N u m e rica l

M a th e m a tic s . .‘13:393-405. 1999.

3. R .M . K irb y . G .E . K a rn ia d a k is . O . M ik u lc h e n k o . and K . M a ya ra m . "A n In te g ra te d

S im u la to r fo r C ou p le d D o m a in P rob lem s in M E M S ” . J o u rn a l o f M ic ro e le c tro m e

chan ica l System s. Vol. 10. 3:379-399. 2001.

P eer-R eview ed C onference P ublications:

1. R .M . K irb y . H. M a rm a n is and D .H . L a id la w . "V is u a liz in g M u lt iv a lu e d D a ta from

2D Incom pre ss ib le F low s U s ing C oncep ts fro m P a in t in g " . P roceed ings o f IE E E V i

su a liz a tio n 1999. San F rancisco. C A . O c to b e r 1999.

2. G-S K a ram anos . C. E vange linos. R .C . Boes. R .M . K irb y a nd G .E . K a rn ia d a k is .

"D ire c t N u m e ric a l S im u la tio n o f T u rb u le n c e w ith a P C /L in u x C lu s te r: Fact o r

F ic t io n ? " . P roceedings o f S u p c rC o m p u tin g 1999. P o rtla n d . O R . N ovem ber 1999.

3. A . Forsberg. R .M . K irb y . D .H . L a id la w . G .E . K a rn ia d a k is . A. van D am . and .1. E lio n .

"Im m e rs iv e V ir tu a l R e a lity fo r V is u a liz in g F low T h ro u g h an A r te r y " . P roceedings

o f IE E E V is u a liz a t io n 2000. S a lt Lake C ity . U T . O c to b e r 2000.

4. D .H . L a id la w . R .M . K irb y . J.S . D av idson . T .S . M ille r . M . d a S ilva . W .H . W arren ,

and M . T a r r . "Q u a n t ita t iv e C o m p a ra tiv e E v a lu a tio n o f 2D V e c to r F ie ld V is u a liz a t ion

M e th o d s ". P roceedings o f IE E E V is u a liz a t io n 2001. San D iego. C A . O c to b e r 2001.

O ther C onference P u b lica tion s (E x ten d ed A bstracts R efereed):

1. T .C . W a rb u rto n . I. L o m te v . R .M . K ir b y a nd G .E . K a rn ia d a k is . " A D iscon tinu ou s

G a le rk in M e th o d fo r th e C om press ib le N av ie r-S tokes E q u a tio n s o n H y b r id G r id s " .

v


P roceedings o f th e T e n th In te rn a tio n a l C onference on F in ite E lem ents in F lu id s .

Ja n u a ry 1997.

2. I. Lom tev . R .M . K irb y , and G .E . K a rn ia d a k is . "A D isco n tin u o u s G a le rk in M e th o d in

M o v in g D o m a in s ". In D iscon t inu ou s G a le rk in Methods: Theory, C om pu ta t ion and

A p p l ica t ions . Eds. B. C o c k b u rn . G .E . K a rn ia d a k is . and C.-VV. Shu. S p rin ge r-V e rla g .

N Y . 1999.

3. R .M . K irb y . I. L o m te v . C. Evange linos. G-S K a ram an os and G .E . K a rn ia d a k is .

"P a ra lle l D N S o f F lo w -S tru c tu re In te ra c tio n s " . P roceed ings o f th e D oD H P C M P

Users G ro u p C onference. June 7-10. 1999.

4. I. Lom tev . R .M . K irb v . and G .E . K a rn ia d a k is . "A D isco n tin u o u s G a le rk in A r b it ra ry

L a g ra n g ia n E u le r ia n F o rm u la tio n fo r V iscous C om pre ss ib le F lo w s ". P roceedings o f

th e 2nd In te rn a tio n a l C onference on D N S /L E S . June 7-9. 1999.

5. R .M . K irb y . T .C . W a rb u rto n . S.J. S h e rw in . A . Beskok and G .E . K a rn ia d a k is . "T h e

N e k ta r C ode: D y n a m ic S im u la tio n s w ith o u t R em esh ing ". P roceed ings o f th e 2nd In

te rn a t io n a l C onference on C o m p u ta tio n a l Techno log ies fo r F ln id /T h e rm a l/C h e m ic a l

System s w ith In d u s tr ia l A p p lic a tio n s . A u g u s t 1-5. 1999.

5. R .M . K irb y . Y . D u . D . Luco r. X . M a. G-S K a ram anos a nd G .E . K a rn ia d a k is . "P a r

a lle l D N S a nd LES o f T u rb u le n ce and F lo w -S tru c tu re In te ra c t io n s " . P roceedings o f

th e D oD H P C M P Users G ro u p C onference. June 5-8. 2000.

7. .J. Boyan. A . G reenw a ld . R .M . K ir b y and .1. R e ite r. "B id d in g A lg o r ith m s fo r S im n l-

taneous A u c tio n s " . P roceedings o f I . IC A I W o rksho p on E co no m ic Agents. M ode ls ,

and M echan ism s, pages 1-11. 2001.

8. I. P iv k in . R .M . K irb y , and G .E . K a rn ia d a k is . "H ig h -o rd e r D iscon tinu ou s G a le rk in

M e th o d : S im u la tio n o f C o il F lo w s ". D N S /L E S : Progress and Challenges. T h ir d

A F O S R In te rn a t io n a l C onference. A r l in g to n . T X . A u g u s t 5-9 2001.

9. R o b e rt M . K irb y and George E m K a rn ia d a k is . "A D y n a m ic S p e c tra l V a n ish in g

V is co s ity M e th o d fo r L E S " . A IA A 40f/l Aerospace Sciences M e e tin g and E x h ib it .

v i


Reno. Nevada. Ja n u a ry 14-17. 2002.

C onference A bstracts:

1. R .M . K irb y . T .C . W a rb u rto n . I. L o m te v . and G .E . K a rn ia d a k is , " A D is c o n tin u

ous G a le rk in S p e c tra l/h p M e th o d on H y b r id G r id s -’ . P resented a t the In te rn a tio n a l

C onference on S p ec tra l and H igh O rd e r M e th od s . June 1998.

2. R .M . K irb y . A . Beskok. T . W a rb u r to n and G .E . K a rn ia d a k is . "F lo w P ast a C y l in

der w ith a F le x ib le S p lit te r P la te ” . Presented a t the 51st A n n u a l M e e tin g o f the

A m e rican P hys ica l S oc ie ty 's (A P S ) D iv is io n o f F lu id D yna m ics . N ovem ber 1998.

8. I. L om tev . R .M . K irb v . and G .E . K a rn ia d a k is . "A D iscon tinu ou s G a le rk in F o rm u

la t io n for C om press ib le F low s Past a .'ID F le x ib le W in g ” . Presented a t the F if th

O ff ic ia l Congress o f th e U.S. A sso c ia tio n fo r C o m p u ta tio n a l M echan ics (U S A C M ).

A u gu s t 4-6. 1999.

4. J. A lle n . G .E . K a rn ia d a k is . R .M . K ir b y and A . S m its . "P ie zo -e le c tr ic Eels as Power

G enera to rs ." Presented a t th e 52nd A n n u a l M e e tin g o f the A m e ric a n P h ys ica l So

c ie ty 's (A P S ) D iv is io n o f F lu id D yna m ics . N ovem ber 21-23. 1999.

5. G. K a ram anos. R .M . K irb y , and G .E . K a rn ia d a k is . "A S p e c tra l V a n ish in g V isco s ity

M e th o d fo r L E S -F E M ". Presented a t F in ite E lem ents in F lo w P ro b lem s . A u s tin .

T X . A p r i l 30 - M ay 4. '2000.

6. D. X iu . R. M . K irb y , and G . E. K a rn ia d a k is . “ A S e m i-La g ra ng ian S p e c tra l/h p E le

m ent M e th o d fo r A d v e c t io n -D iffu s io n " . Presented a t F in ite E lem ents in F lo w P ro b

lems, A u s tin . T X . A p r i l 30 - M ay 4. 2000.

7. A . Forsberg. A . van D am . R .M . K ir b y and G .E . K a rn ia d a k is . "S im u la tio n S tee ring

in C F D ." P resented a t th e U ndersea W eapon S im u la tio n Based D esign W orkshop .

June 7-9. 2000.

8. R. M . K irb y and G. E. K a rn ia d a k is . "F lo w Past a B lu f f B o d y w ith a R ig id and a

F le x ib le S p lit te r P la te " . Presented a t IU T A M S ym p o s iu m on B lu f f B o d y Wakes and

V o rte x -In d u ce d V ib ra tio n s . June 13 - 16. 2000.

v ii


A cknow ledgm ents

A cknow ledgm en ts are those th in g s w h ich are o fte n w r it te n la s t b u t are p resented firs t

in th e docum en t. I am t r u ly blessed to have had the o p p o r tu n ity to in te ra c t w ith m any

people in m y quest fo r advanced degrees in A p p lie d M a th e m a tic s and C o m p u te r Science,

a nd fo r th is I am t r u ly g ra te fu l.

F irs t o f a ll. 1 m ust express m y g ra t itu d e to Professor G eorge E m K a rn ia d a k is . m y

P h .D . a d v iso r in A p p lie d M a th e m a tic s . T h ro u g h o u t m y five years a t B ro w n , he has

been a fa ith fu l a d v iso r and m e n to r, re fin in g m y s k ills as a s im u la t io n s c ie n tis t. I am

a lso g ra te fu l to h im fo r a ffo rd in g me the o p p o r tu n ity to w o rk on an advanced degree in

C o m p u te r Science w h ile also w o rk in g on m y P h .D . in A p p lie d M a th e m a tic s . He has b o th

encouraged and su p p o rte d m y c o lla b o ra tio n w ith sc ien tis ts in C o m p u te r Science and in

o th e r fie lds. I look fo rw a rd to c o n tin u in g o u r p ro fess iona l re la t io n s h ip as c o lla b o ra to rs

and scholars, and c o n tin u in g o u r persona l re la tio n s h ip as fr iends .

Secondly. I w ou ld like to th a n k Professor A n d rie s van D am . m y a d v is o r in C o m p u te r

Science. F rom the ve ry b e g in n in g . A n d y m ade me feel "p a r t o f th e te a m ": I was no t

considered an A p p lie d M a th e m a tic ia n t r y in g to do C o m p u te r Science, b u t ra th e r I w;is

respected and encouraged as a s im u la tio n sc ien tis t w ho co u ld co m b in e s k ills in b o th areas.

A n d y c o n tin u a lly encouraged me to seek o u t those th in g s w h ich I do w e ll, and do them

b e tte r : to o b je c t iv e ly see those areas w h ich need im p ro ve m e n t, a nd to re fine th em . H is

m e n to r in g is ve ry m uch a pp re c ia te d .

T h ird ly . I w ou ld like to th a n k P rofessor D a v id L a id la w o f C o m p u te r Science. I have

had the o p p o r tu n ity to in te ra c t w ith D a v id on m any d iffe re n t levels as a s tu d e n t in the

c lassroom , as co -a u th o r on several papers, as c o n s u lta n t on a v a rie ty o f p ro je c ts , and m ost

im p o r ta n t ly , tis a fr ie n d . H is s c ie n tif ic in s ig h t, his hum or, h is ca n d o r, and m ost o f a ll h is

fr ie n d s h ip are g re a tly a pp re c ia te d .

A m o ng the A p p lie d M a th e m a tic s fa cu lty . I w ou ld s p e c ific a lly like to th a n k several

professors w ho have in flu en ced me as a sc ien tis t. F irs t. I w ou ld lik e to th a n k Professor

C h i-W a n g Shu fo r p ro v id in g me w ith b o th gu idance and m e n to r in g as a s tu d e n t. I w ou ld

lik e to th a n k Professor Jan H esthaven fo r be ing a fa ith fu l so u n d in g b oa rd fo r m y ideas

v i i i


and p ro v id in g co rrec tio ns to m y w ayw a rd th o u g h ts as needed. Las tly . I w ou ld like to

th a n k P ro fessor D a v id G o tt l ie b fo r a llo w in g me to q ues tion h im co nce rn ing the d e ta ils o f

m a th e m a tic a l ideas, to w h ich he a lw ays p ro v id e d ins ig h t.

G ra d u a te s tu d e n t experiences are crea ted no t o n ly d u r in g research and in the class

roo m . Q u ite o fte n a s ig n ific a n t p a r t o f the g ra du a te school experience consists o f th e

people w ho m you meet a long th e way. I am th a n k fu l to have been p a r t o f the a m az ing

C R U N C H g roup , d ire c te d by G eorge K a rn ia d a k is . F irs t. I w ou ld like to acknow ledge th e

a lu m n i m em bers o f the g ro u p fro m w ho m I have gained m uch adv ice and s u p p o rt: D r.

A l i Beskok. D r. Spencer S h e rw in . and D r. Igo r L om tev . I th a n k D r. T im W a rb u rto n

fo r b e in g m y g ra d u a te s tu d e n t m e n to r d u r in g m y firs t years in the C R U N C H g ro u p and

fo r p re p a rin g me to co n tin u e th e deve lopm ent o f A/"e n ' T a r : I th a n k D r. H a ra lam bos

M a rm a n is fo r o u r w o n d e rfu l d iscussions a b o u t a m u lt itu d e o f th in g s b o th s c ie n tific and

o th e rw ise : I th a n k D r. C 'onstan tinos Evange linos fo r a lw ays b e ing w ill in g to p rov ide com

p u t in g assistance: and I th a n k D r. M a X ia fo r a lways be ing w ill in g to lend a h e lp in g

hand . I w ou ld also like to acknow ledge sp e c ifica lly those in the lab w ith w hom I have

in te ra c te d d a ily - D id ie r L u co r. D o n g b in X iu . and Ig o r P iv k in - each person c o n tr ib u t in g

to m a k in g la b -life s u rv iva b le . L a s tly . I acknow ledge a ll those g ro up m em bers w ith w ho m I

have in te ra c te d - some o n ly b r ie f ly and some fo r p ro longed am oun ts o f t im e - each person

has in flu en ced m y experience here a t B row n .

I w o u ld like to acknow ledge and th a n k the fo llo w in g fo r th e ir he lp in va rious capac ities

as I a tte m p te d to fin ish m y degrees: A n d y Forsberg. Sam Fu lcom er. George Lorio t.. L o r in g

H o lden . M e lih B it im . and Joe L a V io la .

I w an t to acknow ledge and th a n k those fu n d in g agencies w h ich have s u p p o rte d the

research presented in th is thesis:

• A F O S R A ase rt: F 49620 -96-1-0267

• A F O S R : F49620-98-L-0315

• A F O S R : F 49620-01-1-0035

• D A R P A / IL L : 98-240

ix


L a s tly , b u t c e r ta in ly n o t least, I w o u ld like to th a n k m y lo v in g w ife A lis o n . She has

s u p p o rte d me c o n s ta n tly and w a ite d p a tie n t ly (w e ll, m ost o f th e tim e ) w h ile I s low ly

m oved to w a rd m v goa l o f a P h .D . I am th a n k fu l to her for her pers is tence th ro u g h th is

p a r t o f o u r jo u rn e y to ge th e r, a nd loo k fo rw a rd to the m any new b eg in n in gs th a t aw a it us.

x


C ontents

A cknow ledgm ents viii

1 In troduction 1

1.1 O b je c t iv e s ..................................................................................................................................... 3

1.2 O u tlin e ........................................................................................................................................ 4

2 D iscontinuous G alerkin M eth od S tu d ies 6

2.1 E x a m in a tio n o f the S te n c i l ................................................................................................... (J

2.2 E x a m in a tio n o f the E ig c n s p e c t ru m ................................................................................. 10

2.3 Convergence S t u d ie s ............................................................................................................... 15

2.3.1 A S tu d y o f h -C o n v e rg e n c e ...................................................................................... 15

2.3.2 A S tu d y o f p -C o n v e rg c n c e ...................................................................................... 10

2.-1 E x a m in a tio n o f S ta b il iz a t io n F a c t o r s ............................................................................. 19

2.-1.1 Test Ouse: V iscous B u rge rs E q u a tio n w ith u = l ( ) “ ‘ / 7 r ................................ 20

2.4.2 Test Case: E le m e n ta l In te rfaces w ith V a ria b le p - O r d e r ..................................21

3 A rbitrary L agrangian-E ulerian Form ulation for C om pressib le F low s 25

3.1 D iscon tinu ou s G a le rk in A L E fo r A d v c c tio n ....................................................................26

3.2 G r id V e lo c ity A lg o r i t h m ............................................................................................................ 28

3.3 F lo w S im u la t io n s ........................................................................................................................... 30

3.3.1 2D F lo w A ro u n d a P itc h in g A i r f o i l .........................................................................30

3.3.2 3D F lo w A ro u n d A W in g w ith E n d p la t e s .......................................................... 33

x i


4 F lu id -S tru ctu re In teraction for C om pressib le F low s 36

4.1 T h e S tru c tu ra l S o lve r - S tressC heck .................................................................................. 37

4.2 F lu id -S tru c tu re C o u p lin g ........................................................................................................38

4.2.1 O ne-w ay C o u p l i n g ..........................................................................................................38

4.2.2 T w o -w ay C o u p l in g ..........................................................................................................42

4.3 O pen Is s u e s ..................................................................................................................................... 43

5 P olynom ial D e-a liasing 44

5.1 A ccuracy. S ta b il i ty a n d O v e r - In te g ra t io n .......................................................................... 45

5.2 O v e r- in te g ra tio n S tu d y u s ing B u rge rs E q u a t io n ........................................................... 47

5.2.1 C o n tin u o u s G a le rk in M e t h o d ....................................................................................48

5.2.2 D iscon tinu ou s G a le rk in M e th o d ................................................................................ 51

5.3 P o ly n o m ia l D e -A li; is in g in F lo w S im u la t io n s ...................................................................52

5.3.1 T ra n s it io n a nd T u rb u le n c e in a T r ia n g u la r D u c t ........................................... 52

5.3.2 2D F low A ro u n d a P itc h in g A i r f o i l .........................................................................54

6 S p ectra l V anishing V isco sity 58

6.1 S V V fo r the C o n tin u o u s G a le rk in M e t h o d .......................................................................60

6.1.1 New S V V F o rm u la t io n fo r C G M ........................................................................... 61

6.1.2 N u m e rica l E x a m p le D e m o n s tra tin g C G M - S V V ............................................... 63

6.2 SVV ' fo r the D isco n tin u o u s G a le rk in M e th o d ................................................................64

6.2.1 S V V F o rm u la t io n fo r D G M ........................................................................................64

6.2.2 N u m e rica l E x a m p le D e m o n s tra tin g D G M - S V V ............................................... 65

6.3 D yn a m ic S V V ...................................................................................................................................68

6.3.1 B u rge rs E q u a t io n .............................................................................................................. 68

6.3.2 A lte rn a t iv e LE S Im p le m e n ta t io n ............................................................................. 71

6.4 Use o f S V V in F lo w S im u la tio n s ..................................................................................... 72

6.4.1 S V V -L E S C oarse R e so lu tio n S im u la tio n s ...........................................................72

6.4.2 2D F low A ro u n d a P itc h in g A i r f o i l ......................................................................... 75

6.4.3 D yn a m ic S V V Used fo r 2D F low A ro u n d a N A C A 0012 A ir fo i l . . . 77

x i i


7 N on-C onform ing D iscretiza tion s 80

7.1 N o n -C o n fo rm in g D is c re tiz a tio n s W h ic h M a in ta in

p-C onvergence ........................................................................................................................... 82

7.2 N o n -C o n fo rm in g D is c re tiz a tio n s U s ing F in ite Vo lum e h -R e f in e m e n t........................85

7.2.1 N u m e rica l E x p e rim e n ts U s ing F in ite Vo lum e h-R efinem ent. .......................88

7.2.2 F lo w S im u la t io n s ............................................................................................................ 95

8 Sum m ary 100

A D o cu m en ta tion o f th e C om pressib le A f e k ’T olt C ode 103

A . l In t r o d u c t io n .....................................................................................................................................108

A .2 C o m p ila tio n G u id e ...................................................................................................................... 104

A .2.1 V e c l i b .................................................................................................................................105

A .2.2 H l i b .................................................................................................................................... 105

A .2.8 Source D i r e c t o r ie s .......................................................................................................105

A .8 E xe cu tio n G u i d e ..........................................................................................................................106

A .8.1 T h e rea F i le ..................................................................................................................... 106

A .8.2 R u n n in g th e C o d e .......................................................................................................109

A .4 P ro g ra m m in g G u id e ' .................................................................................................................. 109

A.4.1 M isce llaneous P ro g ra m m in g N o t e s .......................................................................110

x i i i


List o f Figures

2.1 L D G and B a um a nn -O de n s tenc ils ( le ft) and B assi-R ebay s te n c il ( r ig h t) .

T h e e lem ent c o n ta in in g the b lack d o t denotes the e lem ent on w h ich the

s o lu tio n is be ing co m p u te d , and shaded areas denote th e e lem ents from

w h ich in fo rm a tio n is req u ired fo r c o m p le tin g th a t c o m p u ta t io n .............................. 9

2.2 E igenvalues o f the o p e ra to r .4 in e q u a tio n (2 .9 ) w hen us ing Bassi-R ebay

fluxes to fo rm u la te .4. Ten e lem ents were used in a ll cases: each p lo t denotes

a different, value' o f the num be r o f m odes (A f) . T h e o rd in a te is the com p lex

im a g in a ry axis, and the abscissa is tin * co m p le x rea l a x is ........................................... 11

2.4 E igenvalues o f the o p e ra to r .4 in e q u a tio n (2.9) w hen us ing L D G fluxes

to fo rm u la te .4. Ten e lem ents were used in a ll cases: each p lo t denotes a

d iffe re n t va lue o f the n um be r o f m odes ( 4 / ) . T h e o rd in a te is the com p lex

im a g in a ry axis , and the abscissa is th e co m p le x rea l a x is ........................................... 12

2.4 E igenvalues o f the o p e ra to r .4 in e q u a tio n (2.9) w hen us ing B a um ann -O den

fluxes to fo rm u la te .4. T en e lem ents were used in a ll cases: each p lo t denotes

a d iffe re n t value o f the n um be r o f m odes (A /) . T h e o rd in a te is the com p lex

im a g in a ry axis, and the abscissa is th e co m p le x rea l a x is ........................................... 13

2.5 M o d u lu s o f the m a x im u m eigenvalues versus the n u m b e r o f m odes per e l

em ent fo r Bassi-R ebay ( le ft) . L D G (ce n te r), and B a um a nn -O de n ( r ig h t) .

T h e sym bo ls denote the a c tu a l m o d u lu s o f the e igenvalue, and th e so lid

line denotes a least-squares A / 1 f it w here A/ is the n u m b e r o f m odes used

per e le m e n t........................................................................................................................................ 14

x iv


2.6 C o m p a riso n o f th e ra te o f convergence fo r Bassi-R ebay ( le f t ) . L D G (cen te r),

and B a u m a n n -O d e n ( r ig h t) fluxes fo r tw o modes (M = 2 ) a nd th re e modes

( \ I = 3 ) p e r e lem ent. T h e Lo e rro r is g iven on the o rd in a te , a nd th e num be r

o f e lem ents is g iven on the abscissa, k denotes the m a g n itu d e o f th e slope. . 16

2.7 C o m p a riso n o f the ra te o f convergence fo r Bassi-R ebay ( le f t ) . L D G (cen te r),

and B a u m a n n -O d e n ( r ig h t) fluxes fo r viscous B u rge rs e q u a tio n w ith v =

10~‘ / t t . S ix u n ifo rm a lly spaced e lem ents were used. T h e L > e rro r is g iven

on the o rd in a te , and the num be r o f m odes per e lem ent is g iven on th e abscissa. 18

2.8 E xact s o lu tio n u (s . t j ) = s in (2 T rx )s in (2 n y ) o f d if fe r in g p -o rd e r tos t p ro b le m

( le ft) : O n e -d im e n s io n a l s lice o f the exact s o lu tio n sh ow ing th e e lem en ta l

d eco m p o s itio n in the x -d ire c tio n ( r ig h t ) ................................................................................21

2.9 T he f irs t co lu m n denotes e lem en t-w ise values o b ta in e d fo r (dem ent tw o.

and th e second co lu m n denotes e lem en t-w ise values o b ta in e d fo r ('lenient,

th ree. T h e e rro r ( to p row ) a nd L> e rro r (b o tto m row ) are g iven on

the o rd in a te , and the va lue o f the r/t, p a ram e te r in the s ta b il iz a t io n fa c to r

is g iven on the abscissa (and is deno ted as the re sca lin g ). T h e sym bo ls are

e xp la in e d in the te x t .................................................................................................................. 23

3.1 N o ta tio n fo r a tr ia n g u la r e le m e n t.............................................................................................27

3.2 G ra p h sh ow ing ve rtices w ith associated ve loc ities and edges w ith associated

w eights ..............................................................................................................................................29

3.3 F u ll d o m a in ( le ft) and loca l te sse lla tio n ( r ig h t) fo r th e s im u la t io n a round

the N A C A 0015 p itc h in g a ir fo il . A l l d im ens ions are in u n its o f chord le n g th . 31

3.4 C o m p a riso n o f the l i f t coe ffic ien t versus ang le o f a tta c k in degrees fo r the

9(/l o rd e r case (so lid line ) versus th e c o m p u ta tio n a l a nd e x p e rim e n ta l resu lts

presented in [71] (sym b o ls ). T h e chord R eyno lds n u m b e r is Re = 45 .000.

and th e M ach n um be r is M a = 0 .2 ..........................................................................................32

3.5 D e n s ity co n to u rs fo r angles o f a tta c k a - 10.4 ( le ft) , n — 26.1 (ce n te r), and

a = 52.2 ( r ig h t ) ................................................................................................................................. 33

xv


3.6 Ske leton mesh fo r How past a th re e -d im e n s io n a l N A C A 0012 a ir fo i l w ith

endp la tes ( to p and m id d le ). Iso -co n tou rs fo r x -c o in p o n e n t o f m o m e n tu m

fo r M — 0.5 How past a th re e -d im e n s io n a l N A C A 0012 a ir fo il w ith e ndp la tes

(b o t to m ) ................................................................................................................................................34

3.7 In i t ia l c o n fig u ra t io n ( le ft) and de flec ted ( r ig h t) N A C A 0012 w in g section

between tw o e nd -p la tes . Iso -con tou rs o f s tream w ise m o m e n tu m are show n. . 35

4.1 D iag ra m sh ow ing th e one-w ay c o u p lin g o f A /*etc7_Q r w ith S tressC heck. . . 39

4.2 Screen snap -sho t sh ow ing th e m od e l crea ted in S tressC heck ( le ft) ; Screen

snap-shot, sh ow ing th e d e fo rm a tio n o f the s tru c tu re based u p o n u n it n o rm a l

load ing on one sub -pane l o f the s tru c tu re ( r ig h t) .........................................................40

4.3 T im e h is to ry o f the d isp lacem en t o f a p o in t on the a ir fo il . D isp lacem en t

and tim e are b o th in n o n -d im e n s io n a l u n its ........................................................................41

4.4 T im e h is to ry o f th e coeffic ien t o f l i f t fo r a s ta tio n a ry a ir fo i l c o n fig u ra tio n

(so lid ) versus the H u id -s tru c tu re m ode l (c lashed)............................................................. 41

4.5 D iag ram sh ow ing th e tw o -w ay c o u p lin g o f A /*£ K 7 *Q r w ith S tressC heck. . . 42

5.1 C om pa riso n o f th e d iffe rence in m o d a l coe ffic ien ts w hen d iffe re n t num bers

o f q u a d ra tu re p o in ts are used. Q u a d ra tic n o n lin e a r ity is show n on th e le ft

and cub ic n o n lin e a r ity is show n on th e r ig h t ...................................................................... 48

5.2 S o lu tio n o f th e v iscous B urgers e q u a tio n w ith u - 10“ ° e va lua te d a t T -

0.5. In .4. Q = 24 q u a d ra tu re p o in ts are used fo r in te g ra t in g b o th the

advection and d iffu s io n te rm s: and in D . Q — 24 q u a d ra tu re p o in ts are

used fo r th e a dve c tion te rm , and o n ly Q — 17 p o in ts are used fo r the

d iffu s io n te rm s ................................................................................................................................... 49

5.3 S o lu tio n o f th e v iscous B u rge rs ecp ia tion w ith u = 10_ 2/ 7r eva lua ted at

T = 0.5. In .4. Q = 17 q u a d ra tu re p o in ts are used fo r in te g ra t in g b o th the

advec tion and d iffu s io n te rm s: in B . Q = 24 q u a d ra tu re p o in ts are used

fo r in te g ra tin g b o th the a dve c tion and d if fu s io n te rm s: and in C . Q = 24

q u a d ra tu re p o in ts are used fo r th e a d ve c tio n te rm , and o n ly Q = 17 p o in ts

are used fo r the d iffu s io n te rm s ................................................................................................. 50

x v i


5.4 S o lu tio n o f th e in v is c id B u rge rs e q u a tio n eva lua ted a t T = 0.5. F ive equal

spaced e lem ents were used w ith 16 modes in each e lem en t. O n th e o rd in a te

we p lo t the L > n o rm o f th e s o lu tio n , and on th e abscissa we p lo t the n um be r

o f q u a d ra tu re p o in ts used fo r n um erica l in te g ra tio n . U n s ta b le so lu tio n s are

deno ted by *. O bserve th a t a fte r Q — 24 po in ts , th e L > n o rm o f th e s o lu tio n

does no t change.................................................................................................................................52

5.5 M o d a l coe ffic ien ts in th e m id d le e lem ent o f the in v is c id B u rge rs s o lu tio n a t

tim e T = 0.35. O v e r- in te g ra tio n us ing Q = 3 A //2 q u a d ra tu re p o in ts leads

to a s tab le s o lu tio n u n lik e th e Q = A / + 1 case................................................................. 53

5.6 D u c t flow d om a in : T h e cross-section is an e q u a la te ra l tr ia n g le and the

s tream w ise len g th is th re e tim es the tr ia n g le edge. O n th e le ft we show

a fram e o f the e n tire d o m a in w ith flood c o n to u r cu t-p la n e s o f the f lu id

v e lo c ity in t in ' s tream w ise (w ) d ire c tio n . In the cen te r and on the r ig h t we

present flood c o n to u r c u t-p la n e s o f the flu id v e lo c ity in the s tream w ise (w )

d ire c t io n w ith a rrow s d e n o tin g the ve lo c ity in th e crossflow (u .v ) d ire c tio n s

a t : = 1 and : — 2 re s p e c tiv e ly ............................................................................................. 54

5.7 W a ll shear forces as a fu n c t io n o f t im e fo r (a) (Q = M + 1): (b ) {Q = 2 A /) :

and (c) {Q = 3 A / /2 ) ........................................................................................................................55

5.8 F u ll d o m a in ( le ft) and loca l tr ia n g u liz a t io n ( r ig h t) fo r th e s im u la tio n a ro un d

th e N A C A 0015 p itc h in g a ir fo i l . A l l d im ensions are in u n its o f chord le n g th . 56

5.9 C o m p a riso n o f the l i f t co e ffic ie n t C /. versus ang le o f a tta c k in degrees fo r

Cases A -C ( le ft) and Case A com pared w ith the c o m p u ta tio n a l and e xp e r

im e n ta l resu lts p resented in [71] ( r ig h t ) ................................................................................ 57

6.1 S o lu tio n o f in v isc id B u rge rs e q u a tio n a t t im e T = 0.5 u s in g co n tin uo us

G a le rk in w ith o u t ( to p ) and w ith (b o tto m ) S V V . F ive e q u a lly spaced ele

m ents sp an n ing [—1. 1] were used, each o f w h ich co n ta ine d 16 m odes................. 63

6.2 C o m p a riso n o f e ig e n sp e c tru in o f the tw o ope ra to rs 5V 'V 'i and S V V ' i . D e ta ils

o f th is case are presented in the te x t ......................................................................................65

6.3 S o lu tio n o f in v is c id B u rge rs e q u a tio n a t tim e T = 0.5 u s in g the d is c o n tin

uous G a le rk in m e th o d w ith o u t ( to p ) and w ith (b o tto m ) S V V ..................................66

x v i i


6.4 M o d a l coe ffic ien ts in the m id d le e lem ent o f the in v is c id B u rge rs s o lu tio n a t

tim e T — 0.35. B o th o v e r- in te g ra tio n and S V V lead to a s ta b le s o lu tio n

u n lik e th e c o llo c a tio n a p p ro a ch ............................................................................................

6.5 M o d a l co e llic ie n ts in the m id d le e lem ent o f the in v is c id B u rge rs s o lu tio n a t

t im e T — 0.35. R esu lts fo r S l 'V 'i and S V 'V j are p resen ted .....................................

6.6 P lo t o f th e dynam ic: coe ffic ien t c ( x . t ) a t the fin a l t im e T = 0.5. T h e th ree

eases are e xp la in ed in the te x t ..............................................................................................

6.7 Segm ent o f the u n s tru c tu re d mesh fo r flow past an a ir fo il a t 10 degrees

ang le o f a tta c k and Re = 10 .000 ..........................................................................................

6.8 A m p litu d e ' a t one' t im e ins tance o f s p e c tra l va n ish in g v iscos ity in How p ;is t

an a ir fo il at 10 degree's ang le o f a tta c k and Re — 10 .000 ........................................

6.9 M esh in the- crossflow p lane fo r tu rb u le n t, channel flow a t R eT : 180 . . . .

6.10 M e a n -ve lo e ity p ro file fo r th e tu rb u le n t channe l flow . T h e sym bo ls co rre

spond to the b enchm ark s o lu tio n s o f K im . M o in & M oser [46]. T h e s o lid -lin e

co rresponds to the unde r-reso lved D N S. the d o tte d - lin e to ( M s v v — f —

1 /8 ) . the d o t-d a sh e d -lin e to (A /.v t'V = 5.e = 5 /8 ) . and the dashed -line to

( A / . s r r = 5 . f = 9 /8 ) ..................................................................................................................

6.11 T u rb u le n ce in te n s itie s fo r th e tu rb u le n t channel flow . T he sym bo ls co rre

spond to the b enchm ark s o lu tio n s o f K im . M o in &: M oser [46]. T h e s o lid - lin e

co rresponds to the under-reso lved D N S. the d o tte d - lin e to (A / .v v f = 2 . f =

1 /8 ) . the d o t-d a sh e d -lin e to ( M s \ \ ' = 5.e — 5 /8 ) . and th e dashed -line to

( A / s i t = 5. e = 9 /8 ) ..................................................................................................................

6.12 F u ll d o m a in ( le ft) and loca l te sse lla tio n ( r ig h t) fo r th e s im u la tio n a ro un d

th e N A C A 0015 p itc h in g a ir fo il . A l l d im ensions are in u n its o f chord le n g th .

6.13 C o m p a riso n o f the l i f t coe ffic ien t C /. versus ang le o f a tta c k in degrees fo r

S V V (s o lid ) , o ve r-in teg ra te d (dashed), and th e c o m p u ta tio n a l (c irc le ) and

e x p e rim e n ta l (as te risk ) resu lts presented in [71]..........................................................

6.14 C o n to u rs o f ins tan taneous s tream w ise m o m e n tu m fo r d y n a m ic S V V s im u

la t io n a t tim e T = 131.5...........................................................................................................

x v i i i


6.15 C o n to u rs o f in s tan taneous s tream w ise m o m e n tu m fo r s im u la tio n w ith over-

in te g ra tio n a t t im e T = 131.5.....................................................................................................78

6.16 T im e h is to ry o f l i f t coe ffic ien t ( C l ): T h e so lid line co rresponds to over

in te g ra tio n a n ti th e d o tte d lin e to d y n a m ic S V V ............................................................. 78

6.17 T im e h is to ry o f d ra g coeffic ien t ( C / j ): T h e so lid line co rresponds to over

in te g ra tio n and the d o tte d lin e to d y n a m ic S V V ..............................................................79

7.1 D ia g ra m sh o w in g how n o n -c o n fo rm in g d is c re tiz a tio n s are "m o rta re d " to

gether. N o ta t io n used in th e d ia g ra m is e xp la ined in the te x t .................................. 83

7.2 T h re e d if fe re n t d is c re tiz a tio n s used fo r d e m o n s tra tin g the e ffectiveness o f

'h ' n o n -c o n fo rm in g and *p ’ n o n -c o n fo rm in g d is c re tiz a tio n s . T h e th re e cases

A -C ( le ft.c e n te r, and r ig h t resp ec tive ly ) are discussed in the te x t ........................... 84

7.3 E r ro r in th e b roken H \ n o rm versus the n um be r o f modes p e r element, fo r

d is c re tiz a tio n s 7.2. T h e sym bo ls fro m ease A and B o v e rla p ......................................84

7.1 D ia g ra m sh o w in g th e s p li t t in g scheme fro m a tr ia n g le i l h a v in g a P \ o rde r

p o ly n o m ia l to a co lle c tio n o f .U 2 f in ite vo lum es D j. FT1 and IT2 represen t the

fo rw a rd a nd b ackw ard tra n s fo rm a tio n s resp ec tive ly fo r th is s p l i t t in g scheme. 86

7.5 D ia g ra m sh o w in g th e s p li t t in g scheme fro m a q u a d r ila te ra l H h a v in g a P \

o rd e r p o ly n o m ia l to a c o lle c tio n o f A /2 f in ite vo lum es f l j . n l a n d n 2 rep re

sent the fo rw a rd and backw ard tra n s fo rm a tio n s resp ec tive ly fo r th is s p li t

t in g schem e......................................................................................................................................... 86

7.6 O r ig in a l fo u r ('lenient, mesh w ith 7tfl o rd e r p o ly n o m ia ls pe r e lem en t ( le ft) :

o r ig in a l mesh p a r t it io n e d in to e ig h t f in ite vo lum es per e lem en t (cen te r):

o r ig in a l m esh p a r t it io n e d in to 32 f in ite vo lum es per e lem ent ( r ig h t ) ......................89

7.7 L~c e rro r versus th e n um be r o f f in ite vo lum e e lem ents per s ide ( M ) in to

w h ich each m acro -e lem ent was p a r t it io n e d ......................................................................... 90

7.8 M acro-e lem ent, d is c re t iz a tio n co n s is tin g o f tw o tr ian g le s , each w ith 7tho rd e r

p o ly n o m ia l expansions ( le f t ) . F in ite vo lum e p a r t it io n in g c o n s is tin g o f 49

f in ite vo lum es p e r m acro-e lem ent (cen te r). R econstruc ted s o lu tio n using

n 2 o p e ra to r ( r ig h t ) ..........................................................................................................................91

x ix


7.9 M acro -e lem en t d is c re t iz a tio n co n s is tin g o f fo u r q u a d r ila te ra ls , each w ith 7th

p o ly n o m ia l expansions ( le ft) . F in ite vo lum e p a r t it io n in g co n s is tin g o f 04

f in ite volume's per m acro -e lem ent (cen te r). R econs truc ted s o lu tio n using

n J o p e ra to r ( r ig h t ) ..........................................................................................................................91

7.10 M acro -e lem en t d is c re t iz a tio n co ns is ting o f fo u r q u a d r ila te ra ls , each w ith 7tk

p o ly n o m ia l expansions ( le ft) . F in ite vo lum e p a r t it io n in g c o n s is tin g o f 49

f in ite vo lum es per m acro -e lem ent (cen ter). R econs truc ted s o lu tio n using

n J o p e ra to r ( r ig h t ) ..........................................................................................................................92

7.11 D ia g ra m show ing w h ich d is c re t iz a tio n m ethods were used in the s o lu tio n o f

th e p ro b le m discussed above ........................................................................................................ 93

7.12 C o m p a riso n o f the e xact s o lu tio n to the th ree cases d iscussed in the te x t

eva lua ted a t T = 0 .3 .......................................................................................................................94

7.13 F u ll d o m a in ( le ft) and loca l te sse lla tio n ( r ig h t) fo r the supe rson ic s im u la tio n

a ro u n d the N A C A 0015 a ir fo il. A l l d im ensions arc; in u n its o f cho rd leng th . 95

7.14 F u ll d o m a in ( le ft) and loca l tesse lla tion ( r ig h t) fo r the supe rson ic s im u la

t io n a ro u n d the N A C A 0015 a ir fo il a fte r n o n -co n fo rm in g re fin em en t h;us

been accom plished. N o n -c o n fo rm in g re finem en t fo r case one is show n. A ll

d im ens ions are in u n its o f chord le n g th ................................................................................ 96

7.15 D ia g ra m show ing w h ich d is c re t iz a tio n m ethods were used in th e s o lu tio n o f

th e p ro b le m discussed be low . A l l d im ensions are in u n its o f cho rd len g th . . 96

7.16 D e n s ity con tours fo r supe rson ic How past a N A C A 0015. T h e f in ite vo l

um e s o lu tio n us ing 12.000 q u a d r ila te ra ls ( le ft) and the case tw o m ixed

n o n -c o n fo rm in g f in ite v o lu m e /h ig h -o rd e r e lem ent c o m p u ta tio n ( r ig h t) are

p resen ted ..............................................................................................................................................97

7.17 M ach n um be r co n to u rs fo r supe rson ic flow past a N A C A 0015. T h e fin ite

vo lum e s o lu tio n us ing 12.000 q u a d rila te ra ls ( le ft) and th e case tw o m ixed

n o n -co n fo rm in g f in ite v o lu m e /h ig h -o rd e r e lem ent c o m p u ta tio n ( r ig h t) are

p resen ted ..............................................................................................................................................98

7.18 L in e slices o f the d e n s ity ( le ft) and M ach num be r ( r ig h t) taken a t y = 0

(w here th e a ir fo il ce n te rlin e is a t y = 0. A l l fo u r c o m p u ta tio n s are presented. 99

x x


7.19 D en s ity co n to u rs fo r th re e d im e n s io n a l superson ic flow past a N A C A 0015.

O n th e le ft we present lin e co n to u rs o f th e d en s ity a t tw o p o s itio n s on the

w in g (a t 20% and 80% o f th e le n g th ). T h is is done so th a t tin? ftd l le n g th

o f the w in g can be seen. O n th e r ig h t, we present a Hood c o n to u r c u t-p la n e

o f the d en s ity taken a t th e m id d le o f th e w in g ..........................................................

A . I T h e c u rre n t fu n c t io n a lity tree fo r the com pressib le A f e n . 'T a r code. . . .

x x i


List o f Tables

2.1 Proposed D G m e thodo log ies fo r e l l ip t ic p rob lem s and the f lu x choice's th e y

rep resen t........................................................................................................................................... 8

2.2 L> e rro r and va lue o f | '^ j(0 ) | fo r th e Bassi-R ebay. L D G . and O d e n -B a u m a n n

fluxes. T h e test p ro b le m is e x p la in e d in the te x t ......................................................... 18

2.3 P aram ete rs used fo r the resu lts presented in tab le 2 .4 .................................................. 20

2.4 R esu lts fo r B tiss i-R ebay f lu x w ith s ta b il iz a tio n and o r s ta b il iz a t io n lis t'd . 20

2.5 L-y and terrors fo r d if fe r in g values o f p\ and p>. T h e d e fin it io n s o f p\

and p-y are d iscussed in the te x t ................................................................................................ 22

3.1 S im u la tio n pa ram ete rs fo r com press ib le flow past a N A C A 0012 a ir fo il w ith

e nd p la tes ...............................................................................................................................................33

5. 1 N u m b e r o f q u a d ra tu re p o in ts necessary fo r G a uss -Lo b a tto -Le ge nd re q u a d ra

tu re to be exact fo r g iven p o ly n o m ia l orders o f the in te g ra n d ................................... 45

5.2 N um be r o f q u a d ra tu re p o in ts necessary fo r G a uss -Lo b a tto -Le ge nd re q u a d ra

tu re to be exact, in te rm s o f th e n um be r o f modes for th e o r ig in a l expans ion

fo r g iven p o ly n o m ia l o rders o f th e in te g ra n d ......................................................................40

5.3 Param ete rs used fo r the re su lts presented in figure 5 .2 ................................................. 49

5.4 E rro r in the d isc re te n o rm fo r th e th ree cases presented above ........................51

5.5 M ean shear forces on each w a ll versus the q ua d ra tu re ' o rd e r em p loyed. . . . 54

5.6 P o ly n o m ia l o rd e r p e r tenso r p ro d u c t d ire c t io n and n u m b e r o f q u a d ra tu re

p o in ts per d ire c t io n used fo r cases A -C .................................................................................56

x x ii


6.1 C o m p a riso n o f L± and L ^ e rro rs fo r th e in v is c id B u rge rs e q u a tio n us ing

d y n a m ic S V V (S V 'V 'i) w ith M s \ \ ■ = 8 and e = 1 /1 6 ................................................. 70

6.2 C o m p a riso n o f L > and L zc e rro rs fo r the in v is c id B u rge rs e q u a tio n using

d y n a m ic S V V ( S l ’ lV ) w ith M s \ \ ' = 8 and e = 1 /1 6 ................................................. 70

x x i i i


C h ap ter 1

Introduction

M o st o f th e num erica l ana lys is w o rk fo r e ng in ee rin g a p p lic a tio n s over th e last. 41) years

have' been geared tow ards the developm ent, o f m ethods w h ich are tier urate., f lex ib le , and

robust. M ost m ethods have ;is th e ir p r im a ry focus one o f these* c r ite r io n . a nd fo r some

m e th od s even tw o o f tlie'se c r ite r ia are m et. However, th e c o m p u ta tio n a l m e th o d w h ich

encapsu la tes a ll thre*e c r ite r ia is q u ite e lus ive, fo r s a tis fy in g a ll th ree goals q u ite o ften

requ ires com prom ise . We w ill now present a b r ie f (and c e rta in ly incomplete?) ove rv iew

o f some? o f the a tte m p ts th a t have been made! to meet the! th ree goals m en tion ed above',

anel w i l l show how. in th is thesis, we have a tte m p te d tej design a m e th o d o lo g y fo r the

s o lu tio n o f the llu ie l-s tru c tu re in te ra c tio n p rob lem s w h ich con ta ins th e c u lm in a t io n o f a ll

th ree ideas.

In th e se?arch fo r accura te m e thods , the g lo b a l sp e c tra l m ethods o f G o tt l ie b and O rszag

[34] are th e ieie'al m e th od fo r s o lv in g sm oo th p rob lem s in regu la r geom etries. G ive n su f

ficient. re g u la r ity o f the s o lu tio n , increas ing th e num ber o f degrees o f freedom in the ap

p ro x im a tio n leads to e xp o n e n tia l convergence to the exact, so lu tio n . In a d d it io n , it. has

been a rgued by Kre iss e t a l. [49] th a t fo r th e lo n g -tim e in te g ra tio n o f t im e dependent

s o lu tio n s , h ig h -o rd e r m ethods are th e m ost cost-e ffective in te rm s o f a m o u n t o f w o rk p e r

fo rm ed to a tta in a g iven level o f e rro r . In te rm s o f f lu id s im u la tio n s , s p e c tra l m ethods

have been successfu lly used in th e d ire c t n u m e ric a l s im u la tio n s o f tu rb u le n c e [34. 16]. A l

th o u g h m ee ting one o f the goals above, g lo b a l sp ec tra l m ethods are ra th e r re s tr ic t iv e in

th a t o n ly w ith g reat e ffo rt, i f a t a ll. can th e y be a pp lie d to com p lex geom etries. Secondly.

1


the convergence p ro p e rtie s o f those m ethods req u ire h ig h re g u la r ity o f th e s o lu tio n . W hen

a re d u c tio n in th e re g u la r ity o f th e s o lu tio n occurs, such as w ith shocks in a com press ib le

How. w igg les m ay deve lop w h ich ren de r the m e th o d uns tab le .

In th e search fo r f le x ib le m e thods , f in ite e lem ent and f in ite vo lum e m ethods p ro v id e a

trem endous a m o u n t o f f le x ib i l i ty in h a n d lin g co m p le x g eo m e try dom a in s . T ra d it io n a lly ,

b o th o f these; m ethods have used lo w -o rd e r p o ly n o m ia l expansions fo r c o n s tru c tio n o f th e ir

loca l bases. lin e a r modes in the case o f f in ite e lem ents and constan ts in th e case o f f in ite

vo lum es. In a d d it io n , the use o f th e low -o rde r bases tends to make th e m ethods som ew ha t

ro b u s t. In te rm s o f f lu id s im u la tio n s , these types o f d is c re tiz a tio n s have been successfu lly

a p p lie d to a e ro d yn a m ic p ro b le m s [41]. A lth o u g h fle x ib le , s ta n d a rd f in ite e lem ents and

f in ite vo lum es e x h ib it ra th e r s low convergence as th e n u m b e r o f (dem ents increases com

pared to the p re v io u s ly m en tioned sp e c tra l m ethods. T h is re s tr ic t io n m an ifests its e lf in

p rob lem s like' e lec trom agne tics in w h ich fo r lo n g -tim e in te g ra tio n o f h ig h -fre q u e n cy wave;

s o lu tio n s a p ro h ib it iv e ly large n u m b e r o f e lem ents is needed to reduce the d is s ip a tio n and

d isp e rs io n effects o f the lo w -o rd e r schemes.

T h e confluence o f those tw o concepts, the h p ve rs ion o f the fin ite ’ e lem ent m e th od , was

p ioneered by B .A . Szabo. Seeking to m erge the s tre n g th s o f th e sp e c tra l and f in ite (dement,

m ethods. Szabo fo rm u la te d a m e th o d o lo g y w h ich a llow ed th e f le x ib i l i ty o f a s ta n d a rd f in ite

e lem ent m e th o d to be com b ined w ith the convergence p ro p e rt ie s o f a sp ec tra l m e th od .

Ins tead o f re s tr ic t in g onese lf to a lin e a r basis as in the s ta n d a rd f in ite element, m e th od .

Szabo fo rm u la te d a new hp f in ite e lem ent m e th o d w h ic h accom m oda tes h ighe r o rd e r p o ly

n o m ia l expansions on each e lem ent. T h is new hp m e th o d o lo g y a llow ed fo r a dua l p a th to

convergence - a llo w in g an increase in p o ly n o m ia l o rd e r (p re s o lu tio n ) w hen the re g u la r ity

o f th e s o lu tio n is h ig h w h ile s t i l l m a in ta in in g th e f le x ib i l i t y o f an /(-d is c re tiz a tio n b o th in

m o d e lin g co m p le x geom etries and in s o lu tio n reg ions o f low re g u la rity . Several versions o f

th is a pp roach have been successfu lly a pp lie d to b o th s o lid m echanics a nd f lu id d yna m ics

[5. 4. 61. 44],

T h e d isco n tin u o u s G a le rk in m e thods p rov ides a h ig h -o rd e r e x tens ion o f the f in ite vo l

um e m e th o d in m uch the same w ay as Szabo's w o rk ex ten de d s ta n d a rd f in ite e lem ents. In

th e ir o ve rv ie w o f the deve lopm en t o f the d isco n tin u o u s G a le rk in m e th o d (D G M ). C ock-


3

b u rn et a l. [19] trace th e deve lopm ents o f D G M and p ro v id e a su cc in c t d iscuss ion o f th e

m e rits o f th is e x tens ion to f in ite vo lum es. T h e D G m e th o d o lo g y also a llow s fo r the d u a l

p a th to convergence, m a k in g i t h ig h ly d es ira b le fo r c o m p u ta tio n a l a e rod yna m ics p rob lem s.

I t has been proven th a t D G M satisfies a ce ll e n tro p y in e q u a lity (w h ic h is s tro n g e r th a n L~

s ta b il i ty ) fo r genera l sca lar n o n lin e a r conse rva tions law s in m u lt ip le space d im en s ion s [42].

and D G M satis fies L 1 s ta b i l i ty fo r general sca la r c o n v e c tio n -d iffu s io n e qua tions in m u lt i

d im ens ions [20]. T h o u g h b o th accura te and fle x ib le , th is m e th o d is no t a lw ays ro b u s t.

T he m e th o d o lo g y is conse rva tive , b u t n o t a lw ays m o n o to n ic ity -p re s e rv in g .

In th is w o rk , we focus on th e use o f the d isco n tin u o u s G a le rk in m e th o d fo r s o lv in g

the com press ib le N av ie r-S tokes equa tions fo r m o v in g geom etries. F o llo w in g the w o rk o f

L o m te v e t a l. [52]. we e m p lo y th e a rb it ra ry L a g ra n g ia n -E u le r ia n m e thod fo r s o lv in g flow

p rob lem s in m o v in g geom etries. We fu r th e r ex ten d L o m te v 's w o rk by a c c o m p lis h in g the

f iu id -s t ru c tu re c o u p lin g w ith th e / ip -F E M code S tressC heck deve loped under the d ire c t io n

o f B .A . Szabo. In a d d it io n to th is e x tens ion , we a t te m p t to address some o f th e ro b u s t

ness issues o f the d isco n tin uo u s G a le rk in m e th o d in fo u r ways: th ro u g h an e x a m in a tio n

o f th e fluxes used in the d isco n tin uo u s G a le rk in fo rm u la t io n , th ro u g h an e x a m in a tio n o f

o v e r- in te g ra tio n as a means o f a lle v ia t in g th e effects o f p o ly n o m ia l a lia s in g , th ro u g h an

e x a m in a tio n o f the use o f s p e c tra l va n ish in g v isco s ity as a means o f m a in ta in in g m o n o to n ic

ity . and th ro u g h an e x a m in a tio n o f the use o f n o n -c o n fo rm in g f in ite vo lum e d is c re tiz a tio n s

fo r s o lv in g p rob lem s w h ich c o n ta in shock d is c o n tin u it ie s . In the next sec tion , we present

the e x p lic it goals o f th is thesis.

1.1 O b je c t iv es

T h e goals o f th is thesis are:

• To exam ine th e ra m ific a tio n s o f d iffe re n t f lu x choices w hen s o lv in g e ll ip t ic and

p a ra b o lic p rob lem s us ing the d isco n tin u o u s G a le rk in m e th od .

• To e x ten d th e a rb it ra ry L a g ra n g ia n -E u lc r ie n (A L E ) w o rk o f L om tev et a l. [52] to

encom pass a ll e lem ent types by fo rm u la t in g a genera lized g raph th e o ry a lg o r ith m

fo r mesh m ovem ent.


4

• T o coup le the s p e c tra l/h p e lem ent f lu id code {J\[ekT cit ) w ith the h p -F E M s tru c tu ra l code S tressC heck.

• T o u n d e rs ta n d th e ro le o f p o ly n o m ia l a lia s in g in the s im u la tio n o f h ig h R eyno lds

n um be r f lu id flow co m p u ta tio n s .

• T o p ro v id e a fo rm u la tio n o f sp ec tra l v a n ish in g v iscos ity (S V V ) fo r the s o lu tio n o f

th e inco m p ress ib le N avie r-S tokos equ a tio ns in w h ich the f i lte r in g a llow s fo r scale-

sepa ra tion .

• T o fo rm u la te d S V V fo r the d isco n tin uo u s G a le rk in m e th od , and d e m o n s tra te its use

in the s o lu tio n o f the com pressib le N av ie r-S tokes equations.

• T o im plem ent, n o n -co n fo rm in g d is c re tiz a tio n s w ith in th e d isco n tin uo u s G a le rk in fra m e

w o rk fo r so lv in g the com pressib le N av ie r-S tokes equa tions unde r supe rson ic c o n d i

tions .

1.2 O u tlin e

T h is w ork is o rgan ized as fo llow s. In ch ap te r tw o we present a co lle c tio n o f s tud ies o f d i f

fe ren t fluxes fo r s o lv in g e ll ip t ic and p a ra b o lic p ro b le m s us ing the d isco n tin uo u s G a le rk in

m e th o d . In ch ap te r th ree we present th e a rb it r a ry L a g ra n g ia n -E u le ria n fo rm u la tio n fo r

th e d isco n tin uo u s G a le rk in fo rm u la tio n o f th e com press ib le N avie r-S tokes e qua tions , and

present a genera lized g ra ph th e o ry a lg o r ith m fo r c o m p u tin g the mesh m ovem ent. In chap

te r fo u r we present th e co u p lin g o f the f lu id so lve r A f £ K l ~ a r us ing the A L E a lg o r ith m

w ith th e h p -F E M s tru c tu ra l so lve r S tressC heck. In ch ap te r five we exam ine p o ly n o m ia l

a lia s in g and its effect on b o th incom press ib le and com press ib le flow s im u la tio n s . In ch ap te r

s ix we present a sp e c tra l va n ish in g v iscos ity fo rm u la t io n fo r b o th the co n tin uo us G a le rk in

a nd d isco n tin uo u s G a le rk in fo rm u la tio n s , a nd p ro v id e b o th incom press ib le and com press

ib le How exam ples resp ec tive ly w h ich d e m o n s tra te S V V 's effectiveness. In ch a p te r seven

we present a n o n -co n fo rm in g im p le m e n ta tio n o f th e d isco n tin uo u s G a le rk in fo rm u la tio n

fo r the com press ib le N av ie r-S tokes. and p ro v id e c o m p u ta tio n a l exam ples d e m o n s tra tin g

its use fo r th e s im u la tio n o f superson ic flow s a ro u n d a ir fo ils . We conclude in ch a p te r seven


b y s u m m a r iz in g w h a t was accom p lished in th is w o rk , p o in t in g o u t th e re levan t in d iv id u a l

c o n tr ib u t io n s th a t were made.


C h ap ter 2

D iscontinuous Galerkin M ethod

Studies

A lth o u g h the o r ig in a l th ru s t o f m ost d isco n tin uo u s G a le rk in research was in so lv in g h y

p e rb o lic p rob lem s, the genera l p ro life ra t io n o f the D G m e th o d o lo g y has also spread to th e

s tu d y o f p a ra b o lic and e ll ip t ic p rob lem s. For exam p le , w orks such as [7]. in w h ich th e

viscous com press ib le N av ie r-S tokes e qua tions were so lved, re q u ire d th a t a d isco n tin uo u s

G a le rk in fo rm u la tio n be ex ten de d beyond the h y p e rb o lic adve e tio n te rm s to the viscous

te rm s o f th e N avie r-S tokes e qua tions . C o n c u rre n tly , b o th in [20] and [10] o th e r d is c o n tin

uous G a le rk in fo rm u la tio n s fo r p a ra b o lic and e ll ip t ic p ro b le m s were proposed. In an e ffo rt

to c la ss ify a ll th e e ffo rts m ade to w a rd th e use o f D G m e thods fo r e ll ip t ic p rob lem s. A rn o ld

e t a l.. f irs t in [2] a n il then m ore fu l ly in [3j. p u b lish e d a u n ifie d ana lys is o f d isco n tin u o u s

G a le rk in m ethods fo r e ll ip t ic p rob lem s.

In [3] a m a th e m a tica l fra m e w o rk is p ro v id e d fo r s tu d y in g a v a r ie ty o f the d iffe re n t

d isco n tin u o u s G a le rk in approaches fo r e ll ip t ic p rob lem s. In an a tte m p t to ascerta in w h ich

fo rm u la t io n was a p p ro p r ia te fo r us to use in the s o lu tio n o f th e viscous com press ib le N av ie r-

Stokes equa tions , we set o u t to s tu d y several o f th e d iffe re n t fo rm u la tio n s presented in [3].

T o accom p lish th is goal, we f irs t recognize from [3] th a t th e p ro b le m o f s o lv in g

6


7

— A u = f i n H ( 2 . 1)

u = 0 on Oi1

can be fo rn n ila te t l in t l ie d iscre te case as fo llow s.

Assum e we are g iven a tesse lla tion 7j, = { A '} o f the d o m a in i l . W e define the fo llo w in g

tw o spaces

w here P( K ) = P f,(A ') is th e space o f p o ly n o m ia l fu n c tio n s o f degree a t m ost p > 1 on K

and (C(A’) = [P ;,( K) ]~ . F o llow in g [3] we now define the d isc re te s o lu tio n o f e q u a tio n (2.1)

as th e p ro b le m o f f in d in g iq, G \ ' n and

8

M e th o d UK ° KBassi-R ebav [7] { “ /»} { a h )

B rezz i et al. [15] { “ /.} {(7/ J - a r ([[(q ,]])L D G [20] { u h } - 3 • [[iq,]] W h } + f3- [[

9

In th is ch a p te r, we w il l exam ine a v a r ie ty o f fac to rs such as th e s te n c il w id th , e igenspec-

t ru m . h -convergence p ro pe rtie s , and p-convcrgence p ro p e rtie s o f the d iffe re n t n u m e rica l

fluxes to exp lo re th e d ifferences betw een th e d iffe ren ce choices presented.

2.1 E x a m in a tio n o f the S ten c il

T h e f irs t o b se rva tio n th a t can be m ade im m e d ia te ly upion e x a m in a tio n o f e q u a tio n (2.7)

is th a t b o th L D G and B aum ann -O den have s h o rte r s tenc ils th a n Bassi-R ebay. L D G and

B a n m a n n -O d e n req u ire in fo rm a tio n o n ly fro m nearest n e ig h b o rin g e lem ents, hence p ro

d u c in g a th re e e lem ent s te nc il, w h ile B ass i-R ebav requ ires w h a t equates to a five e lem en t

s te nc il. In the tw o -d im e n s io n a l case, as show n in figu re 2.1. L D G and B a u m a n n -O d e n

re q u ire o n ly lo ca l e lem en ta l c o m m u n ic a tio n , w h ich fo r tr ia n g u la r meshes re q u ire o n ly a

fo u r e lem ent s te n c il. For Bassi-Rebay. how ever, in fo rm a tio n fro m as m any as ten e lem ents

m ay be req u ired fo r the c o m p u ta tio n o f th e s o lu tio n on a s ing le e lem ent.

F ig u re 2.1: L D G and B a um a nn -O de n s te n c ils ( le f t ) a n d B assi-R ebay s te n c il ( r ig h t) . T he e lem ent c o n ta in in g the b lack d o t denotes the c le m e n t on w h ich th e s o lu tio n is be ing c o m p u te d , and shaded areas denote th e e lem en ts fro m w h ich in fo rm a tio n is re q u ire d fo r c o m p le tin g th a t c o m p u ta tio n .

T h is fac t conce rn ing the d iffe re n t m e th o d s is im p o r ta n t w hen co ns ide rin g p a ra lle l

c o m m u n ic a tio n costs: d epend ing on th e w ay in w h ic h the c o m m u n ica tio n s are im p le


10

m ented. the Bassi-R ebay m e th od req u ires tw ic e th e c o m m u n ica tio n its b o th th e L D G and

B a um a nn -O de n m ethods.

2.2 E x a m in a tio n o f th e E ig e n sp e c tr u m

To u nd e rs ta nd th e ra m ific a tio n s o f choos ing each typ e o f flu x , we beg in by s tu d y in g the

one -d im ens iona l p a ra b o lic e qu a tio n

On 0~u „- — v — r = n u e n . u > o ( 2 .8 )Ot Ox-

on [0 . 1] w ith p e r io d ic b o u n d a ry c o n d it io n s . We w r ite o u r n u m e ric a l a p p ro x im a tio n o f

e qu a tio n (2 .8 ) in th e fo llo w in g fo rm :

d i i

d t" - A u „ (2.9)

where ii,; denotes t in ; co nca ten a tion o f m o d a l co e ffic ien ts o f each e lem ent (hence i f g iven

N e lem ents, each h av ing M m oda l co e ffic ie n ts , th e size o f i i g is N x .V /). and .4 is a

s ize (u 9) x s ize (iiy ) square m a tr ix . We now e xa m in e the eigenvalues o f the o p e ra to r .4 fo r

th e th ree d iffe re n t fluxes fo r the case in w h ic h v = 1. and ten e q u a lly spaced e lem ents

are used. E ig en spe c tra o f .4 fo r one to n in e m odes per e lem ent are p resented fo r the

Bassi-R ebay (B R ). L D G (L D G ). and B a u m a n n -O d e n (B O ) fluxes in figures 2.2. 2.3. and

2.4 respective ly .

E x a m in a tio n o f th e e igenspec trum leads to th e fo llo w in g obse rva tions :

• B o th the B assi-R ebay and L D G fluxes are p u re ly d iffus ive (a ll e igenvalues lie on the

negative rea l ax is ) up to m ach ine p re c is io n . T h is is cons is ten t w ith th e fa c t th a t fo r

b o th B ass i-R ebay and L D G the m a tr ix .4 is b o th real a nd s y m m e tr ic . T h e use o f

B a um a nn -O de n fluxes fo rm s a rea l, n o n -s y m m e tr ic o p e ra to r .4. w h ich is e v id e n t by

eigenvalues w h ich have non-zero im a g in a ry com ponents.

• W hen using o n ly one m ode ( f in ite vo lu m e s). B a um a nn -O de n reduces to an in co n

s is ten t scheme, w h ich is deno ted b y a ll th e eigenvalues ly in g a t (0 .0 ) . For m odes

g rea te r th a n o r equal to tw o. B a u m a n n -O d e n fluxes p ro v id e a cons is ten t scheme.


11

M = 1 M = 2 M = 3

o coo on j

-15 -10

'p

M = 4200

100

E

-100

- 200 '— -2000 -1000 -4000

M = 7

1

0.5

0

-0.5

M = 5400

200

-200

-400-2000Re(X)

M = 8

-1000-10000

*300 00308)

-400

M = 6

-5000

M = 92000 4000 5000

20001000

-1000 -2000

-4000 -5000-5000 -2

-2 0 0 0 -----10000

CEO

Re(/i) x 10 x 10

F ig u re 2.2: E igenvalues o f th e o p e ra to r A in e q u a tio n (2.9) w hen u s ing Bassi-R ebay fluxes to fo rm u la te A. Ten e lem ents were used in a ll cases: each p lo t denotes a d iffe re n t va lue o f the n u m b e r o f m odes (A /) . T h e o rd in a te is the co m p le x im a g in a ry ax is , a nd th e abscissa is th e co m p le x rea l axis.


12

-15

E

M = 1 M = 2 M = 31 1 1

0.5 0.5 0.5._„

l 0o o 3 o o r 3 0 GDO O CO 0DCQ t 0 CEGD G E d )

-0.5 -0.5 -0 .5

-1 -1 -1-10 -100

M = 4

-50 Re(A.)

M = 5

-400

M = 7

Re(A)

M = 8

-200

M = 6

M = 9

200 400 1000

100 200 500

-100 -200 -500

-10 00 -2000 -5000-400 '—

-4000-1 0 0 0 -----

-10000

2000 4000 5000

1000 2000

0 oa>

-1000 -2000

- 2000 '— -10000

-4000 -5000-5000 0 -2

4 4x 10 x 10'

F ig u re 2.3: E igenvalues o f th e o p e ra to r .-1 in e q u a tio n (2 .9 ) w hen us ing L D G fluxes to fo rm u la te .4. Ten e lem ents were user! in a ll cases: each p lo t denotes a d iffe re n t va lue o f the n um be r o f modes (A / ) . T h e o rd in a te is the com p lex im a g in a ry ax is , and the abscissa is th e com p lex rea l axis.


13

M = 1 M = 2

r <

E

-100

-10000

M = 4200

100

Eo n i

-100

- 200 '— -2000 -1000 0

M = 7

-400 -4000

GE

-200

M = 6

Q X D I I CHOC I

-500

-5000

M = 92000 4000 5000

pi1000 2000 %

a.©

'X 0 OCCD « t 0 OSD C ► 0 m o c ic

-1000 -V -2000 f 9a y ■*y'

-2000 -4000 -5000-5000 -1

Re(>i)-3 -2

x 10

-1 0

x 104

F ig u re 2.4: E igenvalues o f the o p e ra to r .4 in ec jua tion (2.9) w hen us ing B a u m a n n -O d e n fluxes to fo rm u la te .4. Ten elem ents were used in a ll cases: each p lo t denotes a d iffe re n t va lue o f th e n um be r o f modes (A /) . T h e o rd in a te is the co m p le x im a g in a ry ax is , and the abscissa is the co m p le x real axis.


14

• T h e B a u m a n n -O d e n Hux. a lth o u g h d ispers ive , does n o t show s ig n ific a n t d ispers ion

fo r low n um be r o f modes (M odes = 2 .3). I t is conce ivab le th a t th is fac t can be ex

p lo ite d to crea te a s y m m e tr ic p re c o n d itio n e r w h ich w o u ld acce le ra te th e convergence

o f im p lic i t m e th od s w h ich use B a um a nn -O de n fluxes.

• C om pa riso n o f the e ig ne spe c trum o f th e o pe ra to r fo rm ed u s ing L D G fluxes and the

o p e ra to r fo rm ed u s ing B assi-R ebay fluxes shows th a t L D G requ ires a m ore s tr in g e n t

tim e step i f an e x p lic it t im e s te p p in g m e thod is used fo r the advancem ent o f the

O D E system g iven by e q u a tio n (2.9). In one -d im ens ion , th is fa c t can be ra tio n a lize d

by e xa m in in g th e w id th o f the s te n c il th a t is crea ted (as is done in [65]) bv the tw o

m ethods cons ide red . As [ jo in te d o u t ea rlie r, fo r L D G th e w id th o f the s tenc il is

three*. where;is fo r B assi-R ebay the w id th o f the s te n c il is five. Hence the effective*

A x fo r the L D G o p e ra to r is s m a lle r th a n th a t o f the B ass i-R ebay o p e ra to r. T h is fact

re*quires th a t whe*n us ing an e x p lic it t im e s te pp ing scheme, th e tim e step o f the L D G

m ethod w il l be s m a lle r th a n th a t e>f the; Bassi-R ebay m e th o d so th a t the* d iffu s io n

num be r l im it can be* m a in ta in e d .

• In figure 2.5 we* p lo t the m o d u lu s o f th e m a x im u m e igenva lue versus the* num ber

o f modes pe*r e lem ent fo r Bassi-R ebay ( le ft) . L D G (cen te r), and B a um ann -O den

( r ig h t) .

—. f; • t

F ig u re 2.5: M o d u lu s o f the m a x im u m eigenvalues versus th e n u m b e r o f m odes per e lem ent fo r Bassi-R ebay ( le ft) . L D G (ce n te r), a nd B a u m a n n -O d e n ( r ig h t) . T h e sym bo ls denote th e a c tu a l m od u lu s o f the e igenvalue, and th e so lid lin e denotes a Ieast-squares A/ 1 fit w he re M is the n um be r o f m odes used per e lem ent.

A l l th ree Hux choices show a A/ 1 sca lin g where M is th e n u m b e r o f m odes used per

e lem ent, how ever th e co e ffic ie n t is d iffe re n t between th e fluxes. For b o th Bassi-R ebay


15

and B a um a nn -O de n . th e lea d in g coe ffic ien t in the least-squares a p p ro x im a tio n is 1.35

whereas in the case o f L D G , th e lea d in g coeffic ien t is a ro u n d 4.1.

2.3 C on vergen ce S tu d ies

T o con tin ue o u r s tu d y , we now exam ine th e convergence p ro p e rtie s b o th fo r h-convergence

and p-convergence.

2 .3 .1 A S tu d y o f h -C o n v e r g e n c e

T o s tu d y the h-convergence o f the Bassi-R ebay. L D G . O d e n -B a m n a n n fluxes, we cons ider

th e one -d im ens iona l p a ra b o lic e qu a tio n

i)'n d 211 ~—------- v — 7 = 0 v € TZ. v > 0Ot O x1

on [0. 1] w ith p e r io d ic b o u n d a ry co n d it io n s . A n in i t ia l c o n d it io n o f u (x ) = sin(2Trx)

was used, and th e e rro r was e xam ined a t T = 0.1. T o d e te rm in e th e convergence ra te

fo r each flu x choice, we h o ld the n u m b e r o f modes per e lem ent fixed (b o th M = 2 and

. \ [ = 3 m odes) w h ile successive ly d o u b lin g the num be r o f e lem ents used in the sp a tia l

d is c re tiz a tio n . A p lo t o f th e L < e rro r versus the num be r o f e lem ents is presented in figure;

2.6 .

O bserve th a t b o th B ass i-R ebay and O d en -B au m an n have d if fe re n t convergence rates

d e p en d in g on w h e th e r th e n u m b e r o f m odes is o dd o r even w h ile L D G m a in ta in s a cons tan t

convergence ra te . T h is b e h a v io r is cons is ten t w ith the resu lts show n in [65]. Bassi-R ebay

achieves M tlx o rd e r convergence w hen A / is oven and (A / — I)* * o rd e r convergence when

M is odd . B a u m a n n -O d e n achieves (.V/ — I ) ' 7* o rd e r convergence w hen \ [ is even and

M lfl o rd e r convergence w hen M is o dd . L D G achieves A [ lfl o rd e r convergence fo r b o th

o d d and even A /. In th is exam p le , fo r an o d d num be r o f m odes. A / = 3. B ass i-R ebay and

L D G have the same convergence ra te a nd have s im ila r a bso lu te e rro rs . T h e L D G Hux.

in genera l, shows s u p e rio r p e rfo rm an ce to b o th the B assi-R ebay and th e B a um a nn -O de n

fluxes in th a t i t has a co ns is te n t convergence ra te n o t d ependen t o n th e p a r ity o f the

p o ly n o m ia l o rd e r em p loyed.


16

B o s M -R c b jv Baumann-Oden

jk=|slopc|

M * 2 M » 3

F ig u ro 2.6: C o m p a riso n o f tin* ra te o f convergence fo r B ass i-R ebay ( le ft) . L D G (cen ter), and B a nm a nn -O de n ( r ig h t) fluxes fo r tw o m odes ( \ I = 2 ) a nd th re e m odes ( \ I = 3 ) per e lem ent. T h e L-> e rro r is g iven on th e o rd in a te , and th e n u m b e r o f e lem ents is g iven on th e abscissa, k denotes th e m a g n itu d e o f the slope.

2 .3 .2 A S tu d y o f p -C o n v e r g e n c e

T o fu r th e r o u r e x a m in a tio n o f d iffe re n t flu x te rm s used fo r d if fu s io n in the d iscon tinuous

G a le rk in m e th od , we e xam in e d the s o lu tio n o f viscous B u rge rs e q u a tio n :

as a test p ro b le m in [6 ] fo r e v a lu a tin g the effectiveness o f d if fe re n t s p a tia l d isc re tiza tio ns .

centered a t th e o r ig in a ro u n d th e tim e t, = 1/ t t (an a p p ro x im a tio n w h ich comes from the

in v is c id th e o ry fo r B u rge rs e q u a tio n ), and the m a x im u m g ra d ie n t o f th e s o lu tio n occurs a t

a p p ro x im a te ly 0.5. As in [6 ]. th e C ole tra n s fo rm a tio n is used to o b ta in th e exact s o lu tio n

to e qu a tio n (2 . 10):

On 1 ch i* d 2u

d t 2 Ox Ox2( 2 . 10 )

For th is , . l ent . th e v iscos ity is taken to be u = 10 2/ 7t. T h is p ro b le m was used

As is p o in ted in th a t pape r, fo r u = 10 2/ ~ the s o lu tio n deve lops in to a sa w too th wave

u (x . t ) = —J-Ztc s in (7 r(x - q ) ) f ( x - t } ) e xp ( — T]2 / - j u t ) dr]

J T x f ( x ~ n) c x p ( - r ] '2/ 4 u t ) drt(2 . 11)


88

INFORMATION TO USERS - Brown University€¦ · Methods". Proceedings of IEEE Visualization 2001. San Diego. CA. October 2001. Other Conference Publications (Extended Abstracts Refereed):

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