Differential Quadrature Review Paper With Examples

8/18/2019 Differential Quadrature Review Paper With Examples

1/28

D i f f e r e n t ia l qua dr a t ur e m e t hod

in c o m p u t a t io n a l m e c h a n i c s A r e v ie w

C h a r l e s W e r t a n d M o i n u d d i n M a l ik

School o f Aerospace and M echanical Engineering

University of Oklahoma Norman O K 73019-0601

T h e d i f f e r e n t i a l q u a d r a t u r e m e t h o d i s a n u m e r i c a l s o l u t i o n te c h n i q u e f o r i n i ti a l a n d / o r b o u n d a r y

p r o b l e m s . I t w a s d e v e l o p e d b y t h e l a te R i c h a r d B e l l m a n a n d h i s a s s o c i a t e s in t h e e a r l y 7 0 s a n d ,

s i n c e t h e n , t h e t e c h n i q u e h a s b e e n s u c c e s s f u l ly e m p l o y e d i n a v a r i e t y o f p r o b l e m s i n e n g i n e e r i n g

a n d p h y s i c a l s c i e n c e s . T h e m e t h o d h a s b e e n p r o j e c t e d b y i t s p r o p o n e n t s a s a p o t e n t i a l a l t e r n a t i v e

t o t h e c o n v e n t i o n a l n u m e r i c a l s o l u t i o n t e c h n i q u e s s u c h a s t h e f i n i t e d i f f e r e n c e a n d f i n i t e e l e m e n t

m e t h o d s . T h i s p a p e r p r e s e n t s a s t a te - o f - t h e - a r t r e v i e w o f t h e d i f f e r e n t i a l q u a d r a t u r e m e t h o d ,

w h i c h s h o u l d b e o f g e n e ra l i n t e r es t t o t h e c o m p u t a t i o n a l m e c h a n i c s c o m m u n i t y .

C O N T E N T S

1 INTRODUCTION........................................................................... 1

2 THE DIFFERENTIALQUADRATUREMETHOD ...................... 2

2. I The qu adra ture rules ............................................................... 2

2.2 Weig hting coefficients and sam pling points .......................... 4

2.3 Exam ples of d i fferential quadrature solutions ........................ 5

Exam ple 1 He at trans fer in a trian gular fin ............................ 5

Examp le 2: Torsion of a rectangular-cross-sectionshaft ........ 7

Examp le 3: A freely vibrating cantilever beam .................... 10

Example 4: Steady-state heat conduction n a slab with

temp erature-d epen dent con duc tivity .............................. 1I

Exa mp le 5: An integro-d ifferen tial equa tion ........................ 13

Example 6: Cooling/heating by combined convection and

radiation .......................................................................... 15

Exam ple 7: One-dim ensional, time-dependent heat diffusion

in a sphere ...................................................................... 16

3 CHRONOLOGICALDEVELOPMENTOF THE DIFFERENTIAL

QU AD RA TU RE ME TH OD ....................................................... 17

4 GEN ER AL RE M AR KS ................................................................ 22

5 CL OS UR E .................................................................................... 25

ACKNOWLEDGMENTSAN D DED ICATIO N ............................. 25

REF ERE NC ES ................................................................................. 25

1 I N T R O D U C T I O N

A l o n g w i t h t h e e v e r g r o w i n g a d v a n c e m e n t o f f as t er c o m p u t -

i n g m a c h i n e s , t h e r e s e a rc h i n t o th e d e v e l o p m e n t o f n e w

m e t h o d s f o r n u m e r i c a l s o l u t i o n o f p r o b l e m s i n e n g i n e e r in g

a n d p h y s i c a l s c i e n c e s a l s o i s a n o n g o i n g p a r a l l e l a c t iv i t y .

S u c h r e s e a r c h i n te r e st s , o f c o u r s e , r e m a i n m o t i v a t e d b y

n e e d s o f m o d e r n t e c h n o l o g y . A s a n e x a m p l e , s i m u l at i on o f

m a n y d y n a m i c s y s t e m s o f t e n r e q u i r e s v e r y f a s t n u m e r i c a l

s o l u t i o n o f th e e q u a t i o n s o f th e s y s t e m m a t h e m a t i c a l m o d e l s .

A n o t h e r e x a m p l e i s t h e c o m p u t e r a i d e d d e s ig n p r o c e s s i n

w h i c h t h e d a t a b a s e o f t e n r e q u i re s l a r g e c o m p u t e r s t o r a g e a n d

t h e i n t e r p o l a ti v e m a n i p u l a t i o n s f o r t h e o p e r a t i n g d e s i g n p a -

r a m e t e r s m a y b e l e s s a c c u r a t e a s w e l l a s q u i t e t i m e c o n s u m -

i n g . I n s u c h c a s e s f a s t n u m e r i c a l s o l u t io n o f th e s y s t e m

e q u a t i o n s o f f e r s t h e p o s s i b il i t y o f m o r e a c c u r a t e a n d e f f i -

Transmitted by A ssociate Editor Isaac Elishakoff

ASME Reprint No AMR182 22

Appl M ech Rev vo149, no 1, January 199 6

c i e n t r e a l- t im e a n a l y s i s a n d d e s i g n , b y p a s s i n g f u l l y o r p a r -

t i a l ly t h e n e e d o f a d a t a b a s e . T h i s p a p e r f o c u s e s o n t h e d i f -

f e re n t ia l q u a d r a t u r e m e t h o d B e l l m a n , 1 9 7 3 ; B e l l m a n a n d

A d o m i a n , 1 9 8 5 ; B e l l m a n a n d R o t h , 1 9 8 6 ) w h i c h h a s a re l a -

t i v e l y r e c e n t o r i g i n a n d i s g r a d u a l l y e m e r g i n g a s a d i s t i n c t

n u m e r i c a l s o l u t i o n t e c h n i q u e f o r t he i n i t ia l - a n d / o r b o u n d -

a r y - v a l u e p r o b l e m s o f p h y s i c a l a n d e n g i n e e r i n g s c i e n c e s .

T h e p r o b l e m a r e a s i n w h i c h t h e a p p l i c a t i o n s o f th e d i f f e r e n -

t ia l q u a d r a t u r e m e t h o d r e f e r r e d t o h e r e a f t e r , f o r b r e v i t y , a s

t h e qu a d r at u r e m e t h o d o r s i m p l y a s th e D Q M ) m a y b e f o u n d

i n t h e a v a i l a b l e li t e r a tu r e i n c l u d e b i o s c i e n c e s , t r a n s p o r t p r o c -

e s s e s , f l u i d m e c h a n i c s , s t a t i c a n d d y n a m i c s t r u c t u r a l m e -

c h a n i c s , s t a t i c a e r o e l a s t i c i t y , a n d l u b r i c a t i o n m e c h a n i c s . I t

h a s b e e n c l a i m e d t h a t t h e D Q M h a s t h e c a p a b i l it y o f p r o d u c -

i n g h i g h l y ac c u r a t e s o l u ti o n s w i t h m i n i m a l c o m p u t a t i o n a l e f -

f o r t . T h e m e t h o d h a s s e e m i n g l y a h i g h p o t e n t i a l a s a n a l t e r -

n a t iv e t o t h e c o n v e n t i o n a l n u m e r i c a l s o l u t i o n t e c h n i q u e s

s u c h a s t h e f i n i t e d i f f e r e n c e a n d f i n i t e e l e m e n t m e t h o d s .

T h i s p a p e r p r e s e n t s a s t a t e - o f - t h e - a r t r e v i e w o f t h e d i f f e r -

e n t i al q u a d r a t u r e m e t h o d . I n t h e f o l l o w i n g , f i r s t t h e b a s i c

m a t h e m a t i c a l c o n c e p t s u n d e r l y i n g t h e D Q M a r e p r e s e n t e d .

T h e i m p l e m e n t a t i o n o f th e m e t h o d f o r th e s o l u t i o n o f a c t u a l

p r o b l e m s i s e l a b o r a t e d t h r o u g h s o m e e x a m p l e s . D u e t o i t s

r a t h er r e c e n t o r i g in , t h e D Q M i s p o s s i b l y n o t w e l l k n o w n t o

t h e c o m p u t a t i o n a l m e c h a n i c s c o m m u n i t y . F o r t h is r e a s o n ,

t h e p a p e r a l s o a i m s t o f a m i l i a r i z e t h e r e a d e r s w i t h t h e D Q M

a n d , t h e r e f o r e , t h i s s e c t i o n i s w r i t t e n i n a p e d a g o g i c a l m a n -

n e r. I n t h e n e x t s e c ti o n , a r e v i e w o f th e c h r o n o l o g i c a l d e v e l -

o p m e n t o f th e m e t h o d i s p r e s e n te d . T h e p a p e r i s c o n c l u d e d

w i t h r e m a r k s o n t h e i s s u e s t h a t c o n c e r n t h e a p p l i c a t i o n a n d

f u rt h e r d e v e l o p m e n t o f t h e D Q M .

T h e m e a n i n g s o f t h e s y m b o l s u s e d i n t h e p a p e r a r e d e -

f i n e d w i t h i n t h e t e x t .

© 1996 Amenca n Society of Mechanical Engineers

wnloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 04/25/2014 Terms of Use: http://asme.org/terms


2/28

2 Bert and Mal ik : Dif ferent ial quad rature metho d in computat ional mechanics

Appl Mech Rev vol 49 no 1 Janu ary 1996

2 T H E D I F F E R E N T I A L Q U A D R A T U R E M E T H O D

2 . 1 T h e q u a d r a t u r e r u l e s

T h e n u m e r i c a l m e t h o d s f o r t he s o l u ti o n o f i n i ti a l- a n d / o r

b o u n d a r y - v a l u e p r o b l e m s , i n g e n e r a l , s e e k t o t r a n s f o r m ,

e i t h e r t h ro u g h a d i f f e r en t i a l o r an i n t eg ra l fo rm u la t i o n , t h e

g o v ern in g d i f f e r en t i a l an d /o r i n t eg ro -d i f f e r en t i a i eq u a t io n s

in to an an a lo g o u s se t o f f i r s t -o rd e r o r a lg eb ra i c eq u a t io n s i n

t e r m s o f t h e d i s c r e t e v a l u e s o f t h e f i e l d v a r i a b le ( t h e f u n c -

t i o n ) a t s o m e p r e s p e c i f i e d d i s c r e t e p o i n t s o f t h e s o l u ti o n d o -

m a in . In t h e d i f f e r en t i a l q u ad ra tu re m e th o d , t h i s i s acco m -

p l i s h e d b y e x p r e s s i n g a t e a c h g r i d p o i n t , t h e c a l c u l us o p e r a -

to r v a lu e o f a fu n c t io n w i th r e sp e c t t o a co o rd in a t e d i r ec t i o n

a t a n y d i s c r e t e p o i n t a s t h e w e i g h t e d l i n e a r s u m o f t he v a l u e s

o f t h e fu n c t io n a t a ll t h e d i sc r e t e p o in t s ch o s en in t h a t d i r ec -

t ion .

I n o r d e r t o g o i n t o t h e m a t h e m a t i c a l b a s i s o f t h e D Q M ,

c o n s i d e r a f u n c t i o n W = q ( x , y ) h av in g i t s f i e ld o n a r ec t an -

gu lar do m ai n 0 _< x _< a, 0 _< y _< b. L et, in the giv en do m ain ,

t h e f u n c t i on v a l u e s b e k n o w n o r d e s i re d o n a g r i d o f s a m -

p i in g p o in t s , w h ere , a s sh o w n in F ig 1 , t h e g r id i s o b t a in ed

b y t a k i n g N x a n d N y p o in t s i n t h e x an d y d i r ec t i o n s , r e sp ec -

t i v e ly . T h en , a r t h -o rd e r x -p a r t ia l d e r iv a t i v e o f t h e fu n c t io n

~P(x,y)

a t a p o in t x = x i a lo n g a n y l i n e y =

y j

paral le l to the x -

a x i s m a y b e w r i t t e n a s

x

° v

0x r Ix=n,

k = l

i = 1 , 2 . .. .. N x

an d a s th -o rd e r y -p a r t i a l d e r iv a t i v e a t a d i sc r e t e p o in t y = y j

a lo n g an y l i n e x = x i p a ra l l e l t o t h e y -ax i s m a y b e w r i t ten a s

GAs i. / Nv

.O s)~..l/

= ~ ~ j ¢ x i ( ~ , y

=y, j = l , 2 . . . . .

P=I

w h e re A ¢ [ ) an d B ( '0 a r e t h e r e sp ec t iv e w e ig h t in g co e f f i -

je

cien ts . A lso , W . ,. /= q~(xi,yi) .

E q u a t io n s (1 ) an d (2 ) ex p res s t h e q u a d r a t u r e r u l e s fo r t h e

d e r iv a t i v es o f a fu n c t io n a t d i sc r e t e p o in t s o f th e fu n c t io n a l

d o m a i n a n d a r e t h e v e r y b a s i s o f t h e d i f fe r e n t ia l q u a d r a t u r e

m e t h o d . I n o r d e r t o i m p l e m e n t t h e D Q M , o n e n e e d s t o k n o w

th e w e ig h t in g co e f f i c i en t s f i p r io r i . T h i s can b e d o n e b y t h e

fu n c t io n a l ap p ro x im a t io n s i n t h e r e sp ec t iv e co o rd in a t e d i -

r ec t i o n s . T h e ap p ro x im a t in g fu n c t io n s a r e k n o w n as t h e t e s t

(o r t r i a l ) f u n c t io n s , an d th e p r im ary r eq u i r em en t fo r t h e

c h o i c e o f t h e t e st f u n c t io n s i s c o m p l e t e n e s s i n t he s a m e s e n s e

as o n e n eed s fo r t h e i n t e rp o l a t i o n fu n c t io n s i n f i n i t e e l em en t

a n a l y s i s . F o l l o w i n g H u e b n e r ( 1 9 7 5 ) , t h e c o m p l e t e n e s s r e -

q u i r em en t i n t h e p resen t co n t ex t m ay b e s t a t ed a s fo l l o w s .

C o m p l e t e n e s s r e q u i r e m e n t :

T h e t e s t f u n c t io n s

sh o u ld r ep resen t t h e p o ss ib l e u n i fo rm s t a t e s o f f i e ld

v a r i ab l e s an d h av e d i f f e r en t i ab i l i t y u p t o th e h ig h es t

o r d e r d e r i v a t i v e a p p e a r i n g i n t h e g o v e r n i n g d i f f e r -

en t ia l equat ion .

A l t h o u g h t h e r e c a n b e m a n y c h o i c e s o f t h e t e s t f u n c ti o n s ,

t h e m o s t c o n v e n i e n t c h o i c e i s w h a t a r e c o m m o n l y r e f e r r e d t o

as t h e p o ly n o m ia l t e s t f u n c t io n s . T h u s , t h e fu n c t io n q ( x , y )

m a y b e e x p r e s s e d a s

V ( x , y ) = F ( x ) G ( y ) (3 )

w h e r e F ( x ) an d G ( (v ) a r e t h e t e s t f u n c t io n s i n t h e x an d y d i -

r ec t i o n s , r e sp ec t iv e ly , su ch th a t

F ( x ) = x U - I ; u = 1 , 2 , . . . , N x (4 )

an d

G ( y ) = y ~ - I ; g = 1 , 2 , . . . , N y (5 )

I t m ay b e seen t h a t t h e t e s t f u n c t io n s i n o n e i n d ep en d en t

( 1 ) v a r i a b l e a r e as m a n y a s t h e n u m b e r o f s a m p l i n g p o in t s i n

th a t d i r ec t i o n an d th a t t h ese fu n c t io n s a r e ac tu a l l y t h e m o -

n o m i a l s o f a p o l y n o m i a l o f o r d e r o n e l e s s th a n t h e n u m b e r o f

s a m p l i n g p o in t s . O b v i o u s l y , f or c o m p l e t e n e s s , t h e m i n i m u m

n u m b e r o f s a m p l i n g p o i n t s i n a c o o r d i n a t e d i r e c t io n s h o u l d

b e eq u a l t o o n e p lu s t h e h ig h es t o rd e r d e r iv a t i v e w i th r e sp ec t

t o t h e c o r r e s p o n d i n g i n d e p e n d e n t v a r i a b l e i n t h e g o v e r n i n g

2 )

d i f f e r en t i a l e q u a t io n .

S u b s t i t u t i n g E q s (3 ) t h ro u g h (5 ) i n E q s ( I ) an d (2 ) , o n e

o b t a i ns t h e f o l lo w i n g s y s t e m s o f V a n d e r m o n d e e q u a t i o n s

~ ( x U - I ] A (r) O r ( x U - I l l • i , u = l , 2 . .. .. N x

k ] i k = O X ' ~X I l x = x i '

k = l

(6 )

a n d

N y

2

j=l

i=1

xi.Yj

[ , _

2 . . N× --x

Fig 1. Quadrature grid for a rectangular region

v

t= l OYS y ; j g

= 1 2 . . . N y

(7 )

w h i c h m a y b e s o l v e d f o r t h e w e i g h t i n g c o e f f i c i e n t s . Q u i t e

o b v i o u s l y , f o r t h e a s s u m e d f o r m o f t h e t e s t f u n c ti o n s , E q s

(4 ) an d (5 ) , t h e w e ig h t in g co e f f i c i en t s a r e ze ro fo r an y d e -

r i v a ti v e o f o rd e r e q u a l t o o r g r e a t e r th a n t h e n u m b e r o f

sam p l in g p o in t s .

I t m a y b e s e e n f r o m t h e a b o v e d e s c r i p t i o n t h a t t h e x - d e -

r i v a ti v e w e i g h t i n g c o e f f i c i e n t s A ~ ) d e p e n d o n l y o n t h e

s a m p l i n g p o i n t s xi; i = 1,2 ..... N x t ak en in 0 < x < a , a n d th e



3/28

App l Mech Rev vo149 no 1 Janua ry 1996 Ber t and Ma l ik : D i ff e ren t ia l quadra tu re method in com puta t iona l mec han ics 3

same A)~ )t. coe f f i c i en t s may ho l d f o r t he co r r e spond i ng

po i n t s on t he y = cons t an t l i nes p r ov i de d t ha t t he x - bou nda-

r i e s o f the d om ai n a r e t he s ame and a r e pa r a l l e l to t he y ax i s .

S i mi l a r l y , t he y - de r i va t i ve w e i gh t i ng coe f f i c i en t s B (s) de -

~

p e n d o n l y o n t h e yj; = 1,2 .... Ny tak en in 0 _< y _< b, an d the

same B ( ) coe f f i c i en t s m ay h o l d f o r t he cor r e spon d i ng po i n t s

jc

on t he x = cons t an t l i nes p r ov i ded t ha t t he y - bou ndar i e s o f

t he do mai n a r e t he s am e and a r e pa r a l le l t o t he x ax i s . I t i s

t hus i m pl i c i t in t he qua dr a t u r e r u l e s , E qs ( 1 ) and ( 2 ) , t ha t the

bou ndar i e s o f t he f i e l d va r iab l e dom ai n a r e a l i gned wi t h the

x and y co or d i na t e axes . H owe ver , a s such t he r e is no r e -

s t r i c t i on t ha t t he r e f e r ence axes be Car t e s i an ; one can a l so

have ob l i que and cur v i l i nea r r e f e r ence axes . T hus , f o r ex -

ampl e , o ne can ob t a i n t he we i gh t i ng coe f f i c i en t s f o r t he de -

r i va t i ves w i t h r e spec t t o t he ob l i que coor d i na t e s f o r use i n a

pa r a l l e l ogr am dom ai n , F i g 2 , and f o r de r i va t i ves w i t h r e spec t

t o po l a r coor d i na t e s f o r use i n a s ec t o r ia l domai n , F i g 3 . F or

f u t u r e r e f e r e n c e , th e d o m a i n s h a v i n g t h e b o u n d a r ie s a l i g n e d

wi t h t he r e f e r e nce axes sha l l be r e f e r r ed t o a s regular d o -

mai ns o r r eg i ons .

I t f o l l ows f r om E q ( 1 ) t ha t t he quadr a t u r e r u l e s may be

wr i t t en co l l ec t i ve l y i n ma t r i x f o r m as

d

w h e r e { W } j a n d { W i t ) ) , a r e t h e c o l u m n v e c t o r s o f t h e Nx

J

va l ues each o f t he f unc t i on an d i t s r th - o r de r x - pa r t ia l de r i va -

t i ve , r e spec t i ve l y , a t t he s am pl i ng po i n t s on a l i ne y =

yj.

Als o, [A(r)] is the

Nx x Nx

m a t r i x o f w e i g h t i n g c o e f f i c ie n t s o f

t he r t h - o r de r de r i va t i ves . F ur t he r , no t i ng t he de f i n i t i on o f the

d i f f e r en t i a l ope r a t o r s

O x r

~r~j~, Nx Nx Nx N~

O X r x : x i : Z 4 > 2 ) Z A r - l , t . l J

, km

- - m j = Z A ~ ;- I> Z A I ) q I

z a k m - - m j

k = l m = l k = l m = l

(9)

N o w f o l l o w i n g E q s ( 8 ) a n d ( 9 ) , o n e m a y e a s i l y o b t a i n t h e

f o l l owi ng r ecur r ence r e l a t i onsh i ps f o r t he we i gh t i ng coe f f i -

c i en t s

[ A , , > = [ A , , , ] [ A ' r ] = [ A < r } [ A ' } 0 >

I t m ay be s een t ha t hav i ng t he ma t r i x [ AO)] o f f i r s t - o r de r

de r i va t i ve we i gh t i ng coe f f i c i en t s , one can ob t a i n t he

w e i g h t in g c o e f f ic i e n t s o f t h e h i g h e r - o r d e r d e r i v a t iv e s b y

succe ss ive mu l t ipl icat ions o f the [ ,4(1) ] ma t r ix by i t se l f . Eqs

( 8) t h r ough ( 10) a r e g i ven f o r t he x - pa r t i a l de r i va t i ves ; t he

equa t i ons f o r t he y - pa r t i a l de r i va t i ves f o l l ow i n an i den t i ca l

manner .

I n i t s usua l s ense , t he t e r m quadr a t u r e r e f e r s t o t he ap-

p r o x i m a t i o n o f a n i n t eg r a l o f a f u n c t i o n b y a l i n e a r w e i g h t e d

s u m o f th e f u n c t i o n v a lu e s a t s o m e s a m p l i n g p o i n t s t a k e n

be t we en t he l i mi t s o f i n t egr a ti on . I t i s i n t e r e s ti ng t o m ent i on

t ha t the quad r a t u r e r u l e f o r f unc t i on de r i va t i ves was a c t ua l l y

f o r mul a t ed a s an ana l ogous ex t ens i on o f i n t egr a l quadr a t u r e

by B e l l man and Cas t i (1971) . T h e x - i n t egr a l o f t he f unc t i on

W(x,y) on any l i ne y = y j i s

N

fxa__oW(x ,y j )dx=ZCkqJk j

k = l

and t hey - i n t egr a l on any l i ne x = x i i s

( 11)

~y~ W (xi y)d y= Z DeWi

( 12)

= 0 f = l

E qs ( 11) and ( 12) a r e t he r u l e s o f i n t egr a l quadr a t u r e . Her e ,

C k and D e a r e t he we i gh t i ng co e f f i c i en t s f o r i n t egr a l s i n t he

x and y d i r ec t ions , r e spec t i ve l y . Us i ng E qs ( 3 ) t h r o ugh ( 5 ) i n

E qs ( 11) and ( 12) , one ob t a i ns

one m ay a l so exp r es s t he quadr a t u r e r u l e f r om E q ( 1 ) a s

N

j = It2

Fig 2. Quadrature grid for a parallelogram region

O

l

Fig 3. Quadrature grid fo r a concentric, circular, sectorial region



4/28

4 Bert and Mal ik: Di f ferent ia l quadrature method in computat ional mechanics

Appl Mech Rev vol 49, no 1, January 1996

a n d

~ ( x O - i ) a °

=T;

k = l

o = 1 , 2 . .. .. N x

N r

• )~-I 1,2 ,. . . ,N y

- - ; g =

C = l I t

The quadrature rules as given by Eqs (1) , (2) , (13) , and

( 14) m a y a l so be used f o r l inea r com bina t ions o f the f unc t ion

der iva t ives and in tegr a l s, bu t wi th r e spec t to one independent

var iable on ly. That is to say, a quadrature rule is not given in

the f o r m of Eqs ( 1 ) and ( 2 ) , f o r exam ple , in the case o f a

m ixe d pa r t ia l de r iva t ive o f the type

~ r + s ~ t } l/ O x r ~ y ~ .

However ,

f o l lowin g the de f in i t ion o f ca lcu lus ope r a to r s , the d i f fe r en t ia l

quadr a tur e ana log o f a m ixed de r iva t ive m ay be ob ta ined a s

o ( r + s ) ~ I / x , , y i

o r

( O s k I ~

[

Nx Nv

=

- ~ ~( r) ~-~ n('~)~u

(15)

OxrOyS Oxr t ~yS )1 x i 'Y' - ~ ' ~ik ~ u j£ - -kg

k=l

t = l

and , s im i la r ly , f o r m ixed in tegr a tion a s

( 1973) . The we igh t ing coe f f ic ien t s f o r the de r iva t ives m a y

be ob ta ined d i r ec t ly , and m os t accur a te ly , i r r e spec t ive o f the

( 13) num ber and pos i t ions o f the s am pl ing po in ts , f r om the ex-

p l ic it f o r m ulae ( Quan and C hang , 1989a ; S hu and R icha r ds ,

1992a-b) . These extremely useful formulae, taken f rom Shu

and R ichards (1992a-b) , are given here for the interes t of the

readers . These are given with respect to the x-coordinate

on ly ; the f o r m ulae wi th r e spec t to the y - coor d ina te would

(14) fol low in an identical mann er .

The of f - d iagona l t e r m s of the we igh t ing coe f f ic ien t m a-

tr ix of the f i r s t-order der ivat ive are given by

I-I(xi)

f o r

i , k = l , 2 , . . . , N x

a n d k i ( 17 )

A 7 =

wher e

r l ( x i ) = H ( x i - x o ) ' n ( x k ) = H ( x * -x o )

t ) = l . u ~ i

u=l u~k

18)

The o f f - d iagona l t e rm s o f a we igh t ing coe f f ic ien t m a t r ix

of the s econd- and h ighe r - or de r de riva t ives m a y be ob ta ined

thr ough the f o l lowing r ecur rence r e la t ionsh ip

gx g v

I~ ' f~ q J ( x 'y ) d x d y = Z C k Z D e ~ k '

(16)

= 0 = 0 k = l f = l

Using the quadr a tur e r u le s f o r the va r ious o r de r de r iva -

t ives , one m ay wr i te the quadr a tur e ana log o f a g iven d i f f e r-

en t ia l equa t ion a t each gr id po in t o f i ts so lu t ion dom ain and ,

consequently, obtain a se t of f i r s t-order or a lgebraic equa-

t ions in t e r m s of the g r id - po in t func t ion va lues. One m ay

a lso f o r m the quadr a tur e ana log equa t ions o f the bound ar y

condi t ions . The f i r s t - o r de r equa t ions m ay be in tegr a ted in

t im e or the a lgebr a ic equa t ions o f the d i f fe r en t ial equa t ions

and the i r boundar y condi t ions be so lved s im ul taneous ly to

ob ta in the unknown gr id - po in t f unc t ion va lues . Obvious ly ,

the same procedure a lso applies to integral or integro-dif fer -

ent ia l equat ions .

2 .2 W e ig ht ing c o e f f i c i e n t s a nd sa mpl ing po in t s

Tw o ex tens ive ly dec i s ive f ac to rs in the accur acy o f the d i f -

f e r en t ia l quadr a tur e so lu t ions a re : one , the accur acy of the

we igh t ing coe f f ic ien t s and two, the cho ice o f s am pl ing

points . 1 In order to obtain th e w eightin g coef f ic ients , one

m a y so lve the Vander m onde sys tem of equa t ions, such a s

Eqs (6) , (7) , (13) , and (14) , us ing the usual l inear equat ion

so lve r s . However , Vander m onde m a t r ices a r e known to be

inhe r en t ly i l l -condi t ioned ( P r es s

e t

l 1988

and, in fact, it is

expe r ienced tha t the w e igh t ing coe f f ic ien t s ob ta ined by a d i -

r ec t so lu t ion of the V ander m o nde eq ua t ions becom e inc r eas-

ing ly inaccur a te wi th an inc r eas ing num ber o f s am pl ing

poin t s . Be t te r accur acy in the w e igh t ing coe f f ic ien t s m a y be

obta ined us ing the ana ly t i ca l so lu t ion m e thod of Ham m ing

IThese two issues will be discussed further n the n ext wo sections.

A )[ ) = r[ A f f=OA )~ ) x ,A ~ [-O

f or i , k = l , 2 . .. .. N x a n d k i

(19)

wher e

2 _


5/28

a n d

Bert and Malik: Differentialquadrature method in computat ional mechanics 5

r ou nd i n g- o f f e r r o r s , t han t ha t by t he r ecur r ence r e l a t i onsh i p

o f E q 1 0 ) .

A na t u r a l , and o f t en conven i en t , cho i ce f o r t he s ampl i ng

po i n t s i s t ha t o f the eq ua l l y spaced po i n t s ; t hese a r e g i ven by

a n d

i - 1

x i = - a ; i = 1 , 2 . .. . N x 2 1 )

N x - 1

i - I

Y i = ~ b ; i = 1 ,2 . .. . N y 22)

N y - 1

i n t he x and y d i r ec t i ons , r e spec t i ve l y . Qui t e f r equen t l y , the

d i f f e r en t i a l quadr a t u r e so l u t i ons de l i ve r mor e accur a t e r e -

su l t s w i t h unequa l l y spaced sampl i ng po i n t s . A l t hough such

po i n t s may be s e l ec t ed by t r i a l , a r a t i ona l bas i s f o r t he s am-

p l i n g p o i n t s i s p r o v i d e d b y t h e z e r o s o f t h e o r t h o g o n a l p o l y -

n o m i a l s . F o r e x a m p l e , o n e m a y o b t a i n t h e s a m p l i n g p o i n t s

b y t h e e q u a t i o n

a n d

l - c ° s [ ( i - 1 ) x / ( N x - l ) ] a ; i 1 , 2 , . . . ,N x 23)

X i =

2

l - c ° s [ ( i - l ) x / ( N y - l ) ] b ; i = l , 2 , . . . , N y 24)

Y ; = 2

i n the x an dy d i r ec t ions , r e spec t i ve l y .

I t shou l d be ment i oned t ha t i n t he quadr a t u r e so l u t i ons ,

t he s ampl i ng po i n t s i n t he va r i ous coor d i na t e d i r ec t i ons may

be d i f f e r en t i n number a s we l l a s i n t he i r t ype . I n f ac t , one

ma y ev en use d i f f e r en t t ypes o f t e s t f unc t i ons i n the va r i ous

coor d i na t e d i r ec t i ons .

2 3 Exam ples of Differential Quad rature Solutions

I n o r de r t o i l l us t ra t e t he app l i ca t i on o f t he DQM , sev en ex-

a m p l e s o f d i v e rs e t y p e s o f p r o b l e m s a r e n o w p r e s e n te d . I n

t h e f i r st t w o e x a m p l e s , o n e o f t he h e a t t ra n s f e r p ro b l e m o f a

f i n and t he o t he r o f the t o r s i on p r ob l em of a r ec t angu l a r -

c r os s - sec t i on sha f t , t he gove r n i ng d i f f e r en t i a l equa t i ons a r e

o f s e c o n d o r d e r a n d t h e i m p l e m e n t a t i o n o f t h e b o u n d a r y

cond i t i ons i s r a t he r s t r a i gh t f o r war d . T he t h i r d exampl e i s o f

t h e e i g e n v a l u e p r o b l e m o f a f r e e l y v i b ra t in g c a n t i l e v e r b e a m .

T h e g o v e r n i n g e q u a t i o n o f t h e p r o b l e m i s a f o u r t h -o r d e r d i f -

f e r en t i a l equa t i on and , i n t h i s case , t he i mp l ement a t i on o f the

b o u n d a r y c o n d i t i o n s n e e d s s o m e c a r e fu l c o n s i d e ra t io n . T h e

f our t h p r ob l em dem ons t r a t e s t he app l i ca ti on o f t he quadr a -

t u r e me t h od t o t he so l u t i on o f a non l i nea r o r d i na r y d i f f e r en-

t ia l equa t i on f o r hea t cond uc t i on i n a s l ab . T he f i f t h p r ob l em

pr esen t s t he so l u t i on o f an i n t egr o- d i f f e r en t i a l equa t i on .

T h e s e f i rs t f i v e e x a m p l e s a r e a c t u a ll y b o u n d a r y - v a l u e p r o b -

l ems . T he s i x t h exampl e i s an i n i t i a l - va l ue p r ob l em. L as t l y ,

t he s even t h exa mp l e i s an i n i t i a l- bounda r y- va l ue p r ob l em.

O f these exam pl es , t he f ir s t fi ve p r ob l ems and t he s even t h

pr ob l em h ave ana l y t i ca l so l u t ions . F or t he s ix t h p r ob l em, ac -

cur a t e numer i ca l so l u t i ons a r e ava i l ab l e . T hus , a l l o f t hese

exam pl es s e r ve we l l t o a s ses s t he numer i ca l acc ur ac y o f the

d i f f e r en t i a l quadr a t u r e so l u t i ons . F r om t he exac t va l ue a t a

g i ven sampl i ng) po i n t , t he pe r cen t e r r o r i n a quad r a t u r e so -

l u t ion may be ob t a i ned a s

~ F ] e x a c t k l I D Q M X | 2 5 )

8 =

[ exact

w h i c h i s u s e d i n t h e f o l l o w i n g e x a m p l e s f o r t h e e r r o r a n d

conv er genc e ana lys i s o f the DQ so l u t ions .

E x a m p l e 1. H e a t t r a n s f e r in a t r i a n g u l a r f i n

Cons i de r a one - d i mens i ona l t h i n t r i angu l a r f i n , shown i n F i g

4 , i n whi ch hea t i s t r ansmi t t ed a l ong i t s l eng t h by con duc t i on

and d i s s i pa t ed f r om i t s l a t e r a l su r f aces t o t he su r r ound i ngs

b y c o n v e c t i o n . T h e e q u a t i o n g o v e r n i n g t h e t e m p e r a t u r e i n

t h e f in m a y b e o b t a i n e d b y a n e n e r g y b a l a n c e a n d w r i t te n i n

a d i mens i on l es s f o r m as L i enha r d , 1987)

•d ®

O

d - - ~ - + - ~ = m 2 ® , 0 _< ,~ _< 1 2 6 )

wher e ® i s t he nond i mens i ona l t emper a t u r e and ~ i s t he

nondi mens i ona l ax i a l coor d i na t e . A l so , m i s a d i mens i on l es s

p a r a m e t e r g i v e n b y

m 2 = h L 2

k 8

w h e r e t h e L a n d 8 a r e t h e g e o m e t r i c p a r a m e t e r s o f th e f i n

F i g 4 ) . A l so , k and h a r e t he t he r ma l conduc t i v i t y and f i n -

t o - ambi en t hea t t r ans f e r coe f f i c i en t , r e spec t i ve l y .

T h e b o u n d a r y c o n d i ti o n s f o r E q 2 6 ) a re

Fig 4. A triangular fin

d__OO= 0 a t ~ = 0 27)

d ~

® = l a t ~ = 1 2 8 )

T h e s o l u t io n o f E q 2 6 ) s u b j e c t t o th e b o u n d a r y c o n d i -

t i ons 27) and 28) is g i ven i n t he f o l l owi ng f o r m of t he

B e s s e i f u n ct i o n s o f f i rs t k i n d w i t h c o m p l e x a r g u m e n t s

L i enha r d , 1987)

Appl Mech Rev vo149 no 1 January 1996



6/28

6 Ber t and Mal ik : Di f ferent ial quadrature method in compu tat ional mecha nics

Appl Mech Rev vo l 49 no 1 Janua ry 1996

® j0 (2 m ~..L_]. (2 9)

F or the d i f fe ren t i a l quadra tu re so lu t ion o f the sys tem

equat ions , Eqs (26) through (28), f i rs t the requis i te quadra-

ture rules for the f i rs t - and second-order derivat ives are

wr i t t en f rom Eq (1) a s

d

~ ~ Z

A ( I ) o j ' d 2 0

Z

A ~ 2 ) O J ;

d~ 2

~ ~ i

j = l

j l

(a,b)

i = 1 , 2 . . . .. N

where N i s the nu m be r o f s am pl ing po in t s in the dom ain 0 _<

~_


7/28

A p p l M e c h R e v v o l 4 9 n o 1 J a n u a r y 1 9 9 6 B e r t a n d M a l i k : D i ff e re n t ia l q u a d r a t u r e m e t h o d i n c o m p u t a t i o n a l m e c h a n i c s

T A B L E I C o n v e r g e n c e a n d e r r o r a n a l y s i s o f th e D Q s o l u t i o n fo r t e m p e r a t u r e d i s t r i b u t i o n i n a t r ia n g u l a r f i n , m = 1 .0

a ) S o l u t i o n w i t h e q u a l l y s p a c e d T y p e I p o i n t s

P e r c e n t e r r o r c i n q u a d r a t u r e s o l u t i o n

E x a c t N u m b e r o f s a m p l i n g p o i n t s, N

0 l l 2 0

3 1 4 0 5 1 5 5

0 . 0 0 . 4 3 8 6 7 6 - 9 . 5 4 3 - 5 . 1 6 5 - 3 . 3 1 9 - 2 . 5 8 1 - 0 . 9 1 1 7 4 . 3 0

0 .1 0 . 4 8 3 6 5 3 - 3 .8 6 3 - 1 .6 9 3 - 0 . 9 5 1 - 0 . 6 9 4 - 0 . 2 3 3 2 7 . 7 1

0 . 2 0 . 5 3 0 8 9 7 - 2 . 1 9 9 - 0 . 9 7 6 - 0 . 5 4 6 - 0 . 3 9 9 - 0 . 1 7 8 1 9 . 0 9

0 . 3 0 . 5 8 0 4 8 5 -1 . 4 0 1 - 0 . 6 2 9 - 0 . 3 5 0 - 0 . 2 5 7 - 0 . 1 5 1 1 4 .9 1

0 . 4 0 . 6 3 2 4 9 4 -0 . 9 2 6 - 0 .4 2 2 - 0 . 2 3 3 - 0 . 1 7 2 - 0 . 1 3 5 1 2 . 4 2

0 . 5 0 . 6 8 7 0 0 3 - 0 . 6 1 5 - 0 . 2 8 7 - 0 . 1 5 6 - 0 . 1 1 6 - 0 . 1 2 5 1 0 . 7 9

0 . 6 0 . 7 4 4 0 9 6 - 0 . 4 0 0 - 0 . 1 9 3 - 0 . 1 0 3 - 0 . 0 7 8 - 0 . 1 1 8 9 . 6 6 3

0 . 7 0 . 8 0 3 8 5 5 - 0 . 2 4 4 - 0 .0 5 0 - 0 . 0 6 5 - 0 . 0 5 0 - 0 . 1 1 3 8 . 8 4 7

0 . 8 0 . 8 6 6 3 6 7 - 0 . 12 8 - 0 .0 7 4 - 0 . 0 3 6 - 0 . 0 2 9 - 0 . 1 0 9 8 . 2 4 0

0 . 9 0 . 9 3 1 7 1 8 -0 . 0 3 9 - 0 .0 3 6 - 0 . 0 1 5 - 0 . 0 1 3 - 0 . 1 0 6 7 . 7 7 8

b ) S o l u ti o n w i t h u n e q u a l l y s p a ce d T y p e l i p o i n t s s o lu t i o n

P e r c e n t e r r o r ~ i n q u a d r a t u r e s o l u t i o n

E x a c t N u m b e r o f s a m p l i n g p o in t s, N

O 1 1 2 0 3 1 4 0 I 5 1 1 0 0

0 . 0 0 . 4 3 8 6 7 6 - 3 . 0 2 3 - 1 . 0 0 2 - 0 . 4 5 1 - 0 . 2 8 4 I - 0 . 1 8 3 - 0 . 0 5 3

I

0 . 1 0 . 4 8 3 6 5 3 - 0 . 6 3 5 - 0 . 1 6 4 - 0 . 0 7 1 - 0 . 0 4 1 - 0 . 0 2 5 - 0 . 0 0 6

0 . 2 0 . 5 3 0 8 9 7 -0 . 3 8 8 - 0 . 0 8 8 - 0 . 0 4 1 - 0 . 0 2 2 - 0 . 0 1 5 - 0 . 0 0 4

0 . 3 0 . 5 8 0 4 8 5 - 0 . 1 8 4 - 0 . 0 6 4 - 0 . 0 2 5 - 0 . 0 1 6 - 0 . 0 1 0 - 0 . 0 0 2

0 . 4 0 . 6 3 2 4 9 4 - 0 . 1 9 4 - 0 . 0 4 7 - 0 . 0 1 7 - 0 . 0 1 0 - 0 . 0 0 6 - 0 . 0 0 2

0 . 5 0 . 6 8 7 0 0 3 - 0 . 1 1 0 - 0 . 0 2 4 - 0 . 0 1 2 - 0 . 0 0 6 - 0 . 0 0 4 - 0 . 0 0 1

0 . 6 0 . 7 4 4 0 9 6 - 0 . 0 4 8 - 0 . 0 2 2 - 0 . 0 0 8 - 0 . 0 0 4 - 0 . 0 0 3 - 0 . 0 0 1

0 . 7 0 . 8 0 3 8 5 5 - 0 . 0 6 0 - 0 . 0 1 2 - 0 . 0 0 5 - 0 . 0 0 3 - 0 . 0 0 2 - 0 . 0 0 0

0 . 8 0 . 8 6 6 3 6 7 - 0 . 0 2 6 - 0 .0 0 5 - 0 . 0 0 3 - 0 . 0 0 1 - 0 . 0 0 1 - 0 . 0 0 0

0 . 9 0 . 9 3 1 7 1 8 - 0 .0 1 0 -0 . 0 0 3 - 0 .0 0 1 - 0 .0 0 1 - 0 . 0 0 0 - 0 . 0 0 0

o l ) = i

r e spec t i ve l y . I n T ab l e 1 , a ) and b ) , t he re su l t s a r e g i ven a t

= 0 .1 i n t e r va ls . F or t he cases o f T y pe I po i n t s i n whi ch t he

n u m b e r o f in t e r v a ls N - I ) a re m u l ti p le s o f 1 0 , t h e s e v a l u e s

wer e ob t a i nab l e d i r ec t l y f r om t he quadr a t u r e so l u t i ons .

H o w e v e r , f o r t h e c a s e s o f T y p e I p o i n ts i n w h i c h t h e n u m b e r

of i n t e r va l s N - 1 ) wer e no t mul t i p l e s o f 10 and f o r the case

of T y pe 11 po i n t s , t he qua dr a t u r e so l u t i on r e su l ts a t N po i n t s

wer e u t i l i zed t o ob t a i n t he t emper a t u r e va l ues a t ~ = 0 .1 i n -

terva ls ov er the leng th 0 _< ~ _< i v ia the L agra nge interpo la-

t i o n s c h e m e .

T he r e su l t s o f t he p r esen t p r ob l em, g i ven i n T ab l e 1 , a )

and b ) , con t a i n t he exac t t emper a t u r e va l ues and t he e r r o r s

i n t he quadr a t u r e so l u t i ons ob t a i ned wi t h t he T y pe I equa l l y

spaced) and T ype 11 unequa l l y spaced) s ampl i ng po i n t s . I t

ma y be s een t ha t quadr a t u r e so l u t i on r e su l t s a re o f good ac -

c u r a c y . T h e m a x i m u m e r r o r i n th e q u a d r a t u r e s o l u ti o n i s a t

= O, ie ,

a t t he t i p o f t he t r i angu l a r f i n . T he va r i a t i ons o f t he

m a x i m u m p e r c e n t e r r o r w it h N f o r t he t w o t y p e s o f s a m p l i n g

po i n t s a r e sho wn i n F i g 5 . I t is i n t e r e s t ing t ha t t he quadr a t u r e

s o l u t io n y i e l d s r e s u lt s o f h i g h e r a c c u r a c y , o f o n e o r d e r o f

m a g n i t u d e o r m o r e , w i t h u n e q u a l l y s p a c e d s a m p l i n g p o i n t s

as compar ed t o t ha t w i t h equa l l y spaced sampl i ng po i n t s . I t

ma y be s een f r om F i g 5 t ha t w i th T y pe 1 po i n t s , t he so l u t ion

e x h i b i ts u n i f o r m c o n v e r g e n c e u p t o N = 4 1 o n l y ; m o s t a c c u -

r a t e r e su lt s a r e ob t a i ned wi t h N = 51 and f o r s ampl i ng po i n t s

N > 51 , t he so l u t i on s t a r t s de t e r i o r a t i ng r ap i d l y . How ever , on

t he con t r a r y , t he so l u t i on w i t h unequa l l y spaced sampl i ng

p o i n t s sh o w s a m o n o t o n i c c o n v e r g e n c e w i t h i n c r e a s i n g

num ber o f po i n ts ; no t e t ha t T ab l e I b ) i nc l udes r e su l t s f o r

t he T ype 1 I s ampl i ng po i n t s o f N = 100 i n number .

E x a m p l e 2. T o r s i o n o f a r e c t a n g u l a r - c r o s s - s e c t i o n s h a f t

Cons i de r a p r isma t i c i so t r op i c sha f t o f r ec t angu l a r c r os s s ec -

t i on o f si des 2a and 2b , F i g 6 , whi ch i s sub j ec t ed t o un i f o r m

t wi s t ing o ve r i t s en t ir e l eng t h . T he s t a t e o f s t r e s s i n t he sha f t

0 . 0 ~ , - -

T y p E s 1. U n e q ua lly p a tte d / ~ /

-2 .0 / s a m p l i n g V

~ , - 4 . 0

~ T y p e : Equatlyspaced

.~ ~

s a m p l t n g oints

6 .0

~ , 8 .0

a ,

r I t

- 1 0 . 0 1 0 2 0 3 0 4 0 5 0 6 0

N u m b e r of sampling p o i n t s , N

Fig 5. Convergence of the DQ solution for the tr iangular f in

problem



8/28

8 Bert and Ma lik: Differential quadrature method in compu tational mechan ics Appl Mech Rev vo149 no 1 January 1996

i s desc r i bed a s ( Chou an d P agano , 1967)

T h e t w i s ti n g m o m e n t i s g iv e n b y

O~ = £ 0dp (40)

xYz = - 0--~-' %x Oq

w h i c h a r e t h e o n l y n o n z e r o s t r e s s c o m p o n e n t s w i t h r e s p e c t

t o t he

x , y , z

refe ren ce f ram e. H ere , ~b = ~(~ , r l ) i s the P iand t l

s t r e s s f unc t i on whi ch i s gove r ne d by P o i s son ' s equa t i on

O~ 2 (~lq2 = - 2 , - 1_


9/28

Ap p l M e ch Re v vo l 49 n o 1 Jan u ary 199 6 Ber t an d M a l i k : D i f fe ren t ia l q u ad ra t u re met h o d i n co mp u t a t io n a l me ch an i cs

T A B L E 2 C on ve r ge n c e an d ac c u r ac y o f th e D Q s o l u t ion for th e tors i on p r ob l e m o f a r e c tan gu l ar

s h a f t w i th od d n u m b e r o f s am p l i n g p o in t s ~ = 1 .0

0.2

0.4

0.6

0.8

1.0

T

x~ b y d i f f e re n t i a l q u ad r a tu r e s o lu t i on w i th s a m p l i n g p o i n t s N o f

T yp e i

T y p e II

e xac t 7 1 15 19 7 11 15 19

0.20292 0.20291 0.20292 0.20292 0 .2 02 92 0. 20 29 8 0.20292 0.20292 0.20292

0.4 233 2 0.42321 0 . 4 2 3 3 2 0 . 4 2 3 3 2 0 . 4 2 33 2 0 . 4 2 3 6 6 0 . 4 2 3 3 2 0 . 4 2 3 3 2 0 . 4 2 3 3 2

0 .6 78 36 0 . 6 7 7 8 9 0 . 6 7 8 3 6 0 . 6 7 8 3 6 0 . 6 7 8 3 6 0 . 6 7 8 8 4 0 . 6 7 8 3 6 0 . 6 7 8 3 6 0 .6 78 36

0 .9 8 36 4 0 .9 8 31 4 0 .9 8 36 5 0 . 9 8 3 6 4 0 . 9 8 3 6 4 0 . 9 8 3 4 6 0 . 9 8 3 6 6 0 . 9 8 3 6 4 0 .9 8 36 4

1.35062 1.35399 1.35052 1.35063 1.35063 1.34973 1.35052 1.35061 1.35062

2. 24 92 3 2 .2 4 78 0 2.24918 2.24923 2.24923 2.24941 2.24923 2.24923 2.24923

~ ~ , 0 ) = 0

I t m a y b e n o t e d t h a t t h e a n a l o g s o f t h e b o u n d a r y c o n d i -

t i o n s , E q ( 4 2 ) , is, E q s ( 4 8 ) a n d ( 4 9 ) , a r e u t i l i z e d i n E q s ( 5 1 )

a n d ( 5 2 ) .

T h e r e s u lt s o f t h e p r e s e n t e x a m p l e f o r a s q u a r e s h a f t ( ~ =

i . 0 ) o b t a i n e d w i t h e x a c t a n d q u a d r a t u r e s o l u t i o n s a r e p r e -

s e n t e d i n F i g s 7 a n d 8 , a n d T a b l e 2 . I n a l l t h e c a s e s , t h e s a m e

t y p e o f s a m p l i n g p o i n t s a r e u s e d i n b o t h t h e x a n d y d i r e c -

t i o n s a n d a l s o a r e t a k e n t o b e t h e s a m e i n n u m b e r i n t h e t w o

d i r e c t i o n s ,

is , Nx = N y = N.

T h e r e s u l t s a r e o b t a i n e d w i t h b o t h

T y p e I ( e q u a l l y s p a c e d ) a n d T y p e I I ( u n e q u a l l y s p a c e d )

3.0

2 . 0

E

•~ t o

E

.~.

b

®

0.0

-t .0

5

x

~ Even ointsnvelope

i i

10 15 20

N u m b e r o f s a mp l in g p o in ts N

Fig 7 . Convergence o f the DQ so lu tion wi th Type I sampl ing

points for the torsion problem o f a square shaft

6 . 0 i

4 . 0

x

. / Evenpointsenvelope

2 °

~ 0 . 0

i ~ Oddpolnlsnvelope

i

-2 .0 1 0 15 20

N u m b e r o f s a m p l i n g p o i n t s

N

Fig 8 . Convergence o f the DQ so lu tion wi th T ype I I sampl ing

points for the torsion problem of a square shaft

s a m p l i n g p o i n t s ; f o r th e p r e s e n t p r o b l e m t h e s e p o i n t s a r e

g i v e n b y

Type I: Equally space d sampling poin ts

i - I

~ i , q i = - l + 2 N _ l ; i = i , 2 , . . . , N

a n d

Type 11. Unequally spa ced sampling p oints

~ i 1 ] i

=

--COS ; i = 1,2 . . . . . N.

I n F i g s 7 a n d 8 , t h e p e r c e n t e r r o r i n q u a d r a t u r e s o l u t i o n

v a l u e s o f t h e m a x i m u m s h e a r s t r e s s 2 is p l o t t e d a g a i n s t t h e

n u m b e r o f sa m p l i n g p o i n t s o f T y p e s I a n d I I , r e sp e c t i v e l y .

T a b l e 2 g i v e s t h e e x a c t a n d q u a d r a t u r e s o l u t i o n v a l u e s o f t h e

s h e a r s t r e s s 3 Xzx a n d t h e t w i s t i n g m o m e n t T i n w h i c h t h e

q u a d r a t u r e s o l u t i o n v a l u e s a r e t h e o n e s o b t a i n e d f o r o d d

n u m b e r o f b o t h T y p e s I a n d I I s a m p l i n g p o i n t s . N o t e t h a t t h e

s h e a r s t r e s s v a l u e s a r e g i v e n a l o n g t h e 1 1 ( o r t h e y ) a x i s a t r I =

0 . 2 i n t e r v a l s a n d w h e r e v e r n e e d e d , t h e s e v a l u e s a r e o b t a i n e d

b y t h e L a g r a n g e i n t e r p o l a ti o n f r o m t h e q u a d r a t u r e s o l u t i o n

resu l t s .

I t m a y b e s e e n f r o m T a b l e 2 t h a t t h e q u a d r a t u r e m e t h o d

y i e l d s r es u l ts o f g o o d a c c u r a c y w i t h s a m p l i n g p o i n t s a s s m a l l

a s N = 7 . A l s o , f o r s m a l l e r v a l u e s o f N , t h e q u a d r a t u r e s o l u -

t i o n s w i t h e q u a l l y s p a c e d ( T y p e I ) p o i n t s a r e b e t t e r t h a n

t h o s e w i t h u n e q u a l l y s p a c e d ( T y p e I I ) p o i n t s . T h i s f e a t u r e

m a y b e s e en m o r e c l e a r l y b y c o m p a r i s o n o f F i g s 7 a n d 8 .

T h e c o n v e r g i n g t r e n d o f th e q u a d r a t u r e s o l u t i o n s w i t h i n -

c r e a s i n g n u m b e r o f s a m p l i n g p o i n t s i s o b v i o u s i n T a b l e 2

a n d F i g s 7 a n d 8 . I t m a y a l s o b e o b s e r v e d f r o m F i g s 7 a n d 8

t h a t b e tt e r a c c u r a c y a n d f a s te r c o n v e r g e n c e i n th e q u a d r a t u r e

s o l u t io n o f th e p r e s e n t p r o b l e m a r e o b t a i n e d w i t h a n o d d

n u m b e r o f s a m p l i n g p o i n t s p o i n t s th a n w i t h a n e v e n n u m b e r

o f p o in t s . T h i s i s i n c o n t ra s t t o t h e f in p r o b l e m o f E x a m p l e 1

i n w h i c h , a s s ee n f r o m T a b l e 1 a n d F i g 5 , t h e c o n v e r g e n c e

t r e n d , p a r t i c u l a r l y w i t h T y p e I I p o i n t s , i s f o u n d t o b e u n i -

f o r m w i t h t h e i n c r e a si n g n u m b e r o f s a m p l i n g p o i n t s . I n t h e

2The maximum shear stress occurs at the m idpoint of the longer side o f the

rectangular cross section.

For a square sha ft, I~yzl=

ITzxl

at the corresponding points on the ~ and r1

axes, respectively.



10/28

10

presen t case , l a rge r e r ro r wi th even num b er o f po in t s re su l ts

pos s ib ly due to the exc lus ion o f the po in t ~ = r l = 0 a t which

the f im c t ion ~ has a m ax im u m va lue .

Exam ple 3: A f ree ly v ibrat ing cant i lever beam

The l inea r f ree v ib ra t ion p rob lem of a th in p r i sm a t ic

Bernoul l i -Eule r beam i s desc r ibed by the fo l lowing e igen-

va lue d i f fe ren t i a l equa t ion

Bert and M alik: Differential quadrature method n comp utational mechan ics

d4w - ~'22W

(53)

d ~ 4

where w = w(~ ) i s the d im ens ion les s m ode func t ion of the

la tera l de f lec t ion , ~ i s the d im ens ion les s coord ina te a long the

ax i s o f the beam , and ~ i s the d im ens ion les s f requency of

the beam v ibra t ions .

F or a can t i l eve r beam , F ig 9 , the boun dary condi t ions a t

the two ends a re

d w

w = = 0 at %= 0 (54)

d~

due to the de f lec t ion and ro ta t ion bo th be ing ze ro a t the

c l a m p e d e n d , a n d

d 2 w d 3 w

= 0 a t ~= 1 (55)

d~ 2 d~ 3

due to the bending m om ent and shea r fo rce bo th van i sh ing a t

the f ree end .

The analyt ical solut ion of Eq (53) subject to Eqs (54) and

(55) y ie lds the f requency equa t ion

cosh13co s13 + i = 0, 13

= ~-~4

(56)

which m a y be fou nd in m os t v ib ra t ion t ex tbooks ; s ee fo r ex-

ample , Meirovi tch (1986)•

In the quadra tu re fo rm ula t ion o f the p resen t p rob lem , i t

shou ld be no ted tha t the govern ing d i f fe ren ti a l equa t ion , Eq

(53) , be ing a four th -orde r equa t ion , invo lves two condi t ions

(on the f ie ld variable) , as give n by Eqs (54) an d (55), a t each

b o u n d a r y p o i n t,

ie,

a to ta l o f four boun dary condi t ions • Thus ,

o f the needed N quadra tu re ana log equa t ions , four equa t ions

ough t to be ob ta ined f rom Eqs (54) and (55) , and the rem a in-

ing (N-4) equat ions from Eq (53)• For this purpose, le t the

the quadra tu re ana log of Eq (53) be wr i t t en a s

N

Z A ~ 4 w .= ~ 2 2 w i ;

=3 4 . . . . .

( N - 2 )

¢j J

J =

(57)

L ~1

Fig 9. A cantilever beam

x

~=x / L

/

/

f

/

/

/

f

Appl Mech Rev vol 49 no 1 January 1996

l eav ing two sam pl ing po in t s a t each end and , the reby , y ie ld -

ing (N - 4) l inear equations•

The quadra tu re ana logs o f o f the boundary condi t ions a t

= 0, Eq (54), are written as

N

w

= 0 ,

Z A ~ ) w j = O ,

i = i . ( 5 8 )

j = l

The q uadra tu re ana logs o f o f the boundary con di t ions a t ~ =

1, Eq (55), are written as

N N

A~3)w

~2)w.=O,

Z Y j = 0 , i = N

j = l j = l

(59)

The a s sem bly of Eqs (57) th rough (59) y ie lds the fo l low-

ing s e t o f l inea r equa t ions

1 0 0 0 0 0

A l l , A l l 2 A l ; v , ) A I d A l l ; . l )

. . . I(N-2)

A(2) ,(2) .( 2 ) .(2) A(2) ~(2)

N I ~ N 2 ~ I N ( N - I )

A N N * N 3 . . . .

N ( N - 2)

A(3) ,(3) (3) A(3) ,(3) (3)

N I q N 2 A N ( N - I ) N N q N 4 ' A N ( N - 2 )

A(4) .(4) A(4) . i(4) A ~ ) .(4)

31 A32 ~ 3( N -1 ) /13N ' A3(N -2)

A~41> A ~ ) A(4> A(4> AI~ ) .,4(4)

~4( N - I ) - - 4N

. . . .

4(N-a)

( 4 ) , ( 4 ) . i( 4 ) A ( 4 ) A ( 4 )

A ~ ) = 2 ) N _ 2 )

A(N-2)I ~(n-2)2 -'(N-2)(N-0 ( N- 2)N (N-2)3

'

W

W2

W N-0

W N

%

w4

W N-2)

. = ~ 2

0

0

0

0

a,,

W/N-2)

(60)

where i t m ay be s een tha t the four q uadra tu re ana log equa -

t ions, Eqs (58) and (59), o f the boun dary condi t ions ac tua l ly

rep lace the quadra tu re ana log equa t ions o f the govern ing d i f -

ferent ia l equat ion a t the boundary points , i=1 and N, and

the i r im m edia te

adjacent

poin t s , i=2 and (N-I ) . A ques t ion-

ab le m a t te r whe the r the quadra tu re ana log equa t ions o f the

boundary condi t ions can rep lace a rb i t ra r i ly chosen quadra -

tu re ana log equa t ions o f the govern ing d i f fe ren t ia l equa t ion ,



11/28

Appl Mec h Rev vo149, no 1, January 1996

Bert and Ma lik: Differential quadra ture method in comp utational mech anics 11

wil l be d i s cus sed shor t ly . However , i t m us t be borne in

mind , as i t is appare nt from Eqs 58) and 59), that in the

quadra tu re a na log equa t ions o f the bou ndary condi t ions , the

we igh t ing coe f f i c ien t s m us t cor re spond to the boundary

point i tse lf .

Equa t ion 60) m ay be wr i t t en a s

F ig 10. When the 5 -po int s a re in t roduced on a g r id o f

equal ly spaced points ,

ie,

in to the Type 1 po in t s o f Eq 38) ,

then the coord ina te s o f the s am pl ing po in t s a re g iven by

Type I lL Equal ly spaced sampl ing points wi th adjacen t (5-

points

-[Sbb][Shw]]f{ }~-f

0} ~L

6 1 )

I n w }

where the subsc r ip t s b and d ind ica te the g r id po in t s used fo r

wr i t ing the quadra tu re ana log o f the boun dary condi t ions and

the govern ing d i f fe ren t i a l equa t ion , re spec t ive ly . By e l im i -

na t ing the 4 x 1 ) co lum n vec tor

{wb},

Eq 61) is reduced to

the fo l low ing s tanda rd e igenva lue equa t ion

[ S ] { w a } - - ~2 [ l ] {Wd

}= {0} 62)

where [S ] i s o f o rde r N - 4 ) x N - 4 ) . The e igenva lues ,

which a re the f requency squa red va lues , and the e igenvec tor

{wa},

which desc r ibes the m ode shapes o f the f ree ly v ib ra t-

ing beam , m ay bo th be ob ta ined s im ul taneou s ly f rom the [S ]

m a t r ix . S tanda rd t echn iques fo r th i s purpose , such a s inve rse

i tera t ion with shif t ing, are avai lable in the l i tera ture; see for

exam ple , Ba the 1982) .

I t m a y seen tha t the e igenvec tor

{wa}

in Eq 62) does not

con ta in the d e f lec t ion a t the s am pl ing po in t s which a re u t i l-

i zed fo r invoking the boundary condi t ions in the DQ form u-

la t ion o f the p rob lem . Cons ide r a pos s ib le s i tua t ion in wh ich

for a pa r t i cu la r m ode of v ib ra t ion , the exc luded po in t s m ay

be loca ted in a ha l f -wave and the ha l f -wave i s e l im ina ted

fu l ly o r pa r t i a l ly f rom the e igenvec tor . Consequen t ly , in tha t

case , the i t e ra tion p roces s on the e igenva lue m a t r ix m ay no t

converge a t a l l. I t i s a l so pos s ib le tha t the convergence m a y

occur to a low er m od e or to som e phys ica l ly unrea l i st i c

m ode . I t i s appa ren t tha t th i s m a t te r would be m ore p rob-

lem a t ic in s cann ing h ighe r m odes . F ur the r , th i s ind ica te s tha t

the rep lacem ent o f the quadra tu re ana log equa t ions o f the

govern ing d i f fe ren t i a l equa t ion by the quadra tu re ana log

equa t ions o f the bounda ry condi t ions can no t be a rb i t ra ry .

The m os t appropr ia te cho ice o f the po in t s fo r invoking

the boundary condi t ions would be o f the ones o f ze ro d i s -

p lacem ents ,

ie,

the noda l po in t s inc lud ing the suppor ted

ends /po in t s ) o f the v ib ra t ing beam . How ever , even i f the

nodes a re suf f i c ien t in num ber to accom m oda te the bounda ry

condi t ions , the loca t ions o f such nodes , s ave a l im i ted num -

ber o f cases , would no t be known h p r io r i . A prac t i ca l ap-

proach would be to have add i t iona l

adjacent

points as c lose

as pos s ib le to a bo unda ry po in t such tha t a l l the po in t s m ake

a to ta l o f the num ber o f bound ary condi t ions ex i s t ing a t tha t

boundary po in t . The c losenes s be tween the ad jacen t po in t s

cou ld of the order 5 ~ 10 5 on a norm alized spat ia l variable);

then these po in t s would v i r tua l ly cor re spond to a s ing le

point ,

ie,

the boundary po in t i t s e l f . F or convenience , the

close ad jacen t points are here inafter referred to as

&points.

For the present problem, on ly two 5-points , one each with

the two end po in t s , a re needed . These po in t s a re shown in

~l =-0 ,

~2

=-(5 ~N-I =1- - (5 , ~N - - - - l

i - 1 63)

~ i - N _ 3 ; i = 3 , 4 , . . . , ( N - 2 )

With the 5 -po in t s in t roduced on a g r id o f unequa l ly spaced

points ,

ie,

in to the Type I I po in t s o f Eq 39), the coord ina te s

of the s am pl ing po in t s a re g iven by

Type IV. Uneq ually spa ced sampling poin ts with adjacent (5-

points

~ , l = 0 , 2 = 5 , N - I = 1 - 5 , ~ o = l

l - c ° s [ ( i - l ) r c / ( N - 3 ) ] i

3,4 . . . . . N 2

64)

~ ; =

;

=

_

2

The re su l t s o f quadra tu re so lu t ions wi th four typ es o f

samp ling points , Eqs 38), 39), 63), and 64), for the f i rs t

s ix m ode f requenc ies o f the can t i l eve r beam a re g iven in

Table 3 . Also inc luded he re a re the exac t f requenc ies ob-

ta ined f rom Eq 56). B y com par ing the frequenc ies o f the

Types I and I I po in t s wi th those o f the Types I I I and IV

poin t s, re spec t ive ly , i t m ay be s een tha t fo r a g iven num ber

of s am pl ing po in t s, the DQ so lu t ion wi th the inc lus ion of 5-

po in t s p roduces m ore accura te f requenc ies

(ie,

closer to the

exact values) . In comparison to the Type I and III points ,

ie,

equa l ly spaced po in t s wi thou t and wi th 5 -po in ts , even the

unequa l ly spaced po in t s o f the Type I I l ead to be t t e r accu-

racy of the DQ so lu t ion . Need les s to s ay m o s t accura te quad-

rature solut ion resul ts are obtained with Type IV points .

S im i la r to the p rev ious exam ples , the conv ergence o f the

quadra tu re so lu t ion fo r the p resen t e igenva lue p rob lem m ay

a l so be obse rved f rom Table 3 . I t m ay a l so be s een tha t fo r

the h ighe r m odes , equa l ly spaced po in t s o f bo th Type I and

Type l l I becom e inc reas ing ly l e s s e f fec t ive . In fac t , the i t -

e ra t ions on the e igenva lue m a t r ix wi th the Type I po in t s

s h o w a c o n v e r g e n c e p r o b l e m e v e n f o r t h e s e c o n d m o d e a n d

do no t converge a t a l l fo r the th i rd and h ighe r m odes .

Example 4: Steady-state heat conduction in a slab

with temperature-dependent conductivity

Cons ide r one -d im ens iona l s t eady-s ta te hea t conduc t ion in a

s lab in which the the rm a l conduc t iv i ty depends l inea r ly on

the t em pera tu re . The equa t ion govern ing the t em pera tu re ®

= ® ~: ,) in the s lab is cons idered in the fol lo win g nond ime n-

s ional form Finlayso n, 1980)

O

i = 1 2 3 J N

( N - l )

Fig 10. A one-dimen sional quadrature grid with adjacent 5-points



12/28

12 Ber t and Mal ik : Di f ferent ial quadrature method in compu tat ional mecha nics Appl Mech Rev vo149 no 1 Jan uary 1996

T A B L E 3 C o n v e r g e n c e a n d a c c u r a c y o f t h e D Q s o l u ti o n i n f r ee v i b r a ti o n a n a l y s i s o f a c a n ti l e v er b e a m

Mode I:

E x a c t f r e q u e n c y , ~ 1 = 3 . 5 1 6 0 1 5 3

Q u a d r a t u r e s o l u t i o n s :

S a m p l i n g p o l n t s t y p e

I I I I 1 IV

3 . 4 7 3 6 5 7 3 . 4 8 6 2 6 0 3 . 4 8 6 2 6 0 3 . 5 0 8 7 5 2

3 . 5 2 2 3 6 6 3 . 5 1 9 9 7 0 3 . 5 1 9 5 9 9 3 . 5 1 6 7 0 8

3 . 5 1 7 2 4 1 3 . 5 1 6 6 9 8 3 . 5 1 6 5 5 1 3 . 5 1 6 0 8 8

3 . 5 1 6 0 7 4 3 . 5 1 6 0 4 5 3 . 5 1 6 0 3 5 3 . 5 1 6 0 1 7

3 . 5 1 6 0 0 3 3 . 5 1 6 0 1 0 3 . 5 1 6 0 1 2 3 . 5 1 6 0 1 5

3 . 5 1 6 0 1 6 3 . 5 1 6 0 1 6 3 . 5 1 6 0 1 5 3 . 5 1 6 0 1 5

N

7

8

9

10

I1

12

Mode 4:

E x a c t f r e q u e n c y , ~ 24 = 1 2 0 . 9 0 1 9 1 6

Q u a d r a t u r e s o l u t io n s :

S a m p l i n g p o i n t s t y p e

N I I I I I IV

1 2 1 3 1 . 5 3 6 4 1 1 2 5 . 4 6 2 0 6 1 2 1 . 3 1 9 2 3

3 . .. . 1 2 0 . 0 3 6 8 4 1 2 0 . 8 3 4 2 2

14 . . .. 120 .3 9889 120 .878 21

1 5 1 2 1 . 2 6 4 8 1 1 2 0 . 9 7 8 7 2 1 2 0 . 9 0 5 1 9

16 12 I . 14675 120 .9 4106 120 .90331

1 7 1 2 0 . 8 7 2 8 0 1 2 0 . 8 9 8 2 5 1 2 0 . 9 0 1 8 0

Mode 5:

E x a c t f r e q u e n c y , f~ 5 = 1 9 9 . 8 5 9 5 3 0


S a m p l i n g p o i n ts t y p e

N I I I I I IV

4

. .. . 195 .942 63 199 .5 1674

5

. .. . 1 9 7 . 4 5 8 4 2 1 9 9 . 7 7 5 0 8

1 6 2 0 4 . 2 4 0 2 4 2 0 0 . 4 1 8 1 0 1 9 9 . 8 7 8 1 9

1 7 2 0 2 . 3 0 0 0 5 2 0 0 . 1 3 5 5 5 1 9 9 . 8 6 7 9 5

1 8 1 9 9 . 4 3 9 9 2 1 9 9 . 8 1 9 4 9 1 9 9 . 8 5 8 4 2

1 9 1 9 9 . 5 5 1 8 3 1 9 9 . 8 3 8 6 9 1 9 9 . 8 5 9 0 4

Mode :

E x a c t f r eq u e n c y , ~ 6 = 2 9 8 . 5 5 5 5 3 1



N 11 IV

5

. . . . 2 9 7 . 2 8 2 0 5

1 6 2 9 0 . 4 1 9 6 9 2 9 8 . 3 8 3 2 9

1 7 3 0 1 . 4 1 9 5 0 2 9 8 . 6 2 2 1 4

1 8 2 9 9 . 8 2 9 5 3 2 9 8 . 5 8 9 3 2

1 9 2 9 8 . 3 0 1 3 5 2 9 8 . 5 4 9 2 4

2 0 2 9 8 . 4 2 6 3 3 I 2 9 8 . 5 5 2 8 8

M ode 2:

Exact frequency, ~2 = 22.0344916

Quad rature solutions:


I I l l I I I V

. . . . . . . . 2 1 .2 5 8 7 2 9 2 1 .9 2 2 7 0 7

. .. . 2 2 . 2 3 5 7 8 8 2 2 . 1 9 4 7 2 2 2 2 . 0 5 8 8 3 0

. .. . 2 2 . 1 3 0 2 9 8 2 2 . 0 9 8 3 0 8 2 2 . 0 4 1 8 0 2

2 2 . 0 3 0 2 1 2 2 2 . 0 3 2 3 1 4 2 2 . 0 3 3 2 9 2 2 2 . 0 3 4 3 7 9

2 2 . 0 2 5 3 0 1 2 2 . 0 3 0 3 9 3 2 2 . 0 3 2 6 7 2 2 2 . 0 3 4 3 6 4

2 2 . 0 3 5 2 4 0 2 2 . 0 3 4 8 0 4 2 2 . 0 3 4 6 0 4 2 2 . 0 3 4 4 9 8

N

8

9

10

I1

12

13

Mode :

E x a c t f re q u e n c y , ~ 3 = 6 1 . 6 9 7 2 1 4 4

Q u a d r a t u r e s o l u t i o n s :


I I I I I IV

. . . . . . . . 6 1 .7 3 2 9 7 5

6 5 . 0 5 7 3 5 8 6 3 . 6 9 5 3 8 8 6 1 . 8 3 4 6 2 9

6 3 . 3 1 7 5 3 2 6 2 . 5 6 2 0 5 1 6 1 . 7 8 0 0 9 1

6 1 . 4 9 9 0 9 8 6 1 . 6 0 5 9 0 0 6 1 . 6 8 9 8 1 5

6 1 . 5 3 3 8 2 9 6 1 . 6 3 9 3 8 1 6 1 . 6 9 3 8 3 6

6 1 . 7 1 6 8 1 1 6 1 . 7 0 2 7 8 0 6 1 . 6 9 7 5 0 2

N

9

10

II

12

13

14

~ d 2 ®

d ® ) 2

w i t h t h e b o u n d a r y c o n d i t i o n s

® = 0 at ~ = 0

a n d

(65)

(66)

O = 1 a t ~ = 1 . 6 7 )

The exac t so lu t ion to the above pr ob lem i s g iven a s

(Finlayson, 1980)

® = - 1 + 1 + ~ ~ (68)

Equation (65) is a nonlinear dif ferent ia l equat ion and,

the r e f or e , i ts num er ica l so lu t ion should f o l low som e i t e r a tive

pr ocedur e . F or th i s pur pose , Newton s appr oach i s adopted

wher e in , beg inn ing wi th an a s sum ed tem p er a tur e f i e ld ® ( ~)

cons i s ten t wi th the bo undar y condi t ions , Eqs ( 66) and ( 67) ,

one obtains success ively ref ined solut ions through the fol-

lowing i t e ra t ion schem e

®(n+0 = ®(.) + 0(.) (69)

where 0 = 0(~) is the temperature ref inement and n is the i t -

era t ion count .

The tem per a tur e r e f inem ent i s de te r m ined by the so lu t ion

of the f o l lowing equa t ion wr i t t en in the op e r a to r f o r m as

0L ( ® ) + L( ® ) = 0 ( 70)

where L(® )is the lef t side o f Eq (65) ,

ie,

2 2

L o ) = 1 + o ) + d ° )

d~ ~ ka Y

(71)



13/28

App l Mec h Rev vo149 no 1 Janua ry 1996 Ber t and Ma l ik : D i ff e ren t ia l quadra tu re method in comp uta t iona l mec han ics

T A B L E 4 C on ve r ge n c e an d e r r or an a l ys i s o f th e D Q s o l u t i on for t e m p e r a tu r e d is t r i b u ti on i n a s lab w i th n on l i n e ar h e a t c on d u c t i on

13

0.1

0.3

0.5

0.7

0.9

0.0

0.2

0.4

0.5

0.6

0.8

1.0

E xac t

solution

0.140175

0.378405

0.581139

0.760682

0.923538

1.5

Pe r c e n t e r r or e i n q u ad r a tu r e s o l u t ion w i th s am p l i n g p o i n t s

o f

T y p e I I Type I I

5 7 l l 5 7 I I

T e m p e r a tu r e

@

1.227

0.223

0.044

-0.025

-0.066

0.184

0.031

0.007

-0.003

-0.009

0.005

0.001

0.000

0.000

0.000

-0.012

-0.303

-0.034

0.077

0.002

-0.039

0.006

0.000

-0.003

0.002

H e at f l u x Q

2.614

-0.340

-0.274

-0.157

-0.201

-0.394

1.613

0.567

-0.057

-0.030

-0.039

-0.034

-0.046

0.327

0.027

-0.001

-0.001

-0.001

-0.001

-0.001

0.015

0.944

-0.680

0.379

0.681

0.580

-0.354

0.568

0.085

0.042

-0.003

-0.055

-0.031

0.046

0.048

0.000

0.000

0.000

0.000

0.000

0.001

-0.001

0.000

-0.001

0.000

0.000

0.000

o ( o ) = o , o ( o ) = l

a n d L ' ( 0 ) i s t h e F r e c h e t d e r i v a t i v e d e f i n e d a s ( F i n l a y s o n ,

1 9 7 2 )

, . q

0 L ' ( O )

= 0 ~ L ( O + ~ 0 )l ~ 0 . ( 7 2 )

I n o r d e r t o e v a l u a t e t h e F r e c h e t d e r i v a t i v e , ® i s r e p l a c e d

b y ® + c 0 i n E q ( 7 1 ) t o o b t a i n

2

L ( O + E 0 ) = ( i + O + c 0 ) d ~ 2 Jr

w h i c h i s d i f f e r e n t i a t e d p a r t i a l l y w i t h r e s p e c t t o ~ a n d t h e n ,

i n t h e r e s u l t i n g d e r i v a t i v e , ~ i s s e t e q u a l t o z e r o . S u b s t i t u t i n g

t h e F r e c h e t d e r i v a t i v e s o d e t e r m i n e d a n d E q ( 7 1 ) i n E q ( 7 0 ) ,

o n e o b t a i n s t h e f o l l o w i n g e q u a t i o n

• d 2 0 d O dO d 2 0 ^

(1 + ® ) d ~ + 2 d ~ a t + - ~ - u

( 7 3 )

. d 2 0

( d O ] 2

w h i c h i s a l i n e a r e q u a t i o n i n t h e t e m p e r a t u r e r e f i n e m e n t 0 .

I n a s m u c h a s 0 i s a c o r r e ct i o n v a r i a b le , i t s b o u n d a r y c o n d i -

t i o n s a r e s i m p l y

0 = 0 a t ~ = 0 a n d 1 . ( 7 4 )

T h e q u a d r a t u r e a n a l o g o f E q ( 7 3 ) i s n o w w r i t te n a t a p o i n t

= ~ i a s

( 2 ) [ ( + O i ) A ~ .2 ) +

20 iA~ ) ]Oj+O ~Oi

j=2

= - 1 2

w h e r e

( 7 5 )

N d 2 ® N

k=2 k=2

( 7 6 )

a n d i = 2 , 3 . .. .. ( N - I ) . A l s o O N = I .

I t m a y n o t e d t h a t in E q ( 7 5 ) t h e b o u n d a r y c o n d i t i o n s o f

t h e 0 - v a r i a b l e , E q ( 7 4 ) , a r e i n c l u d e d . S i m i l a r l y , t h e b o u n d a r y

c o n d i t i o n o n ® a t ~ = 0 , E q ( 6 6 ) , i s i n c l u d e d i n E q ( 7 6 ) .

T h e c o m p u t e d r e s u l t s a r e g i v e n i n T a b l e 4 . T h e t a b u l a t e d

v a l u e s s h o w t h e p e r c e n t e r r o r s e i n t h e q u a d r a t u r e s o l u t i o n

w i t h r e s p e c t t o th e e x a c t v a l u e s f o r t h e t e m p e r a t u r e d i s t r ib u -

t i o n i n t h e s l a b . T h e s e r e s u l t s i l l u s t ra t e t h e c a p a b i l i t y o f t h e

D Q M i n t h e so l u t i o n o f a n o n l i n e a r p r o b l e m . I t m a y a l s o b e

s e e n th a t T y p e I I s a m p l i n g p o i n t s y i e l d b e t t e r a c c u r a c y t h a n

t h e T y p e I p o i n t s .

A s a f u r th e r c h e c k o n t h e a c c u r a c y o f t h e D Q s o l u t io n ,

T a b l e 4 a l s o i n c lu d e s p e r c e n t e r r o r s in t h e q u a d r a t u r e v a l u e s

o f t h e h e a t f lu x d i s t r i b u t i o n w i t h r e s p e c t t o t h e e x a c t s o l u t i o n

v a l u e o f 1 .5 . T h e ( d i m e n s i o n l e s s ) h e a t f l u x is g i v e n b y

d O ( 7 7 )

Q = ( 1 + ® ) - - ~ -

I n th e c a l c u l a ti o n o f q u a d r a t u r e r e s u l ts o f T a b l e 4 , t h e

f o l l o w i n g c o n v e r g e n c e c r i t e r io n w a s u s e d f o r t h e i te r a ti o n

s c h e m e , E q ( 7 0 ) , o f N e w t o n ' s m e t h o d .

N-102

i =2 i < 1 0 _ 6 ( 7 8 )

•

N -102

i=2 i

W i t h a n i n i ti a l g u e s s ® i = ~ i, a c o n v e r g e d s o l u t i o n c o u l d a l -

w a y s b e o b t a i n e d i n a m a x i m u m o f t h r e e i te r a ti o n s .

Exam ple 5: An integro-dif ferent ial equat ion

T h e i n t e g r o - d i f f e r e n t i a l e q u a t i o n b e i n g c o n s i d e r e d h e r e i s

o n e t y p e o f B o l t z m a n n e q u a t i o n e n c o u n t e r e d i n th e k i n e t i c



14/28

1 4

T A B L E 5 C o n v e r g e n c e a n d e r r o r a n a l y s i s o f t h e D Q s o l u t io n f o r t h e

B o l t z m a n n e q u a t i o n

( a ) E x a c t s o l u t i o n

.f

f

Ber t and Ma l ik : D i f f e ren tia l quadra tu re method in computa t iona l mechan ics

0.1 0.2 0.3 0.4

0 . 5

1.12293188 1.26820896 1.43698648 1.63045897 1.84987730

0.6 0.7 0.8 0.9 1.0

2.09656797 2.37195479 2,67758437 3.01515270 3.38653610

App l Mech Rev vo l 49 no t Janua ry 1996

N

I ~ e - n f ( q ) d q = Z C j f j ( 8 2 )

j=l

w h e r e f j = f l r l j ) . U s i n g t h e p o l y n o m i a l t e s t f u n c t i o n s , o f t h e

f o r m o f E q ( 4 ) , i n E q ( 8 2 ) , o n e o b t a i n s

( b ) P e r c e n t e r r o r e i n D Q s o l u t i o n w i t h e q u a l l y s p a c e d T y p e I p o i n t s

5

0.1 0. 60 x 10 1

0.2 0.75 x 10 l

0.3 0.73 x I0 1

0.4 0.68 × 10 I

0.5 0.65 x 10 1

0.6 0.6 4× 104

0.7 0.63 × 10 1

0.8 0.61 x 104

0.9 0.5 8x 104

1.0 0.5 6 x 10 l

N u m b e r o f s a m p l i n g p o i n t s, N

6

-0.4 9 x 10 2

-0.55 x 10 2

-0.52 × 10 2

-0.49 × 10 2

-0.48 x 10 2

-0.47 × 10 2

-0.45 x 10 2

-0.4 4 x 10 -2

-0.4 4 x 10 2

-0.43 x 10 2

7

0,14 x 10 3

0,15 × 10 3

8

-0.50 x 10 5

-0.4 9 x 10 5

0,14 x 10 3

0.14 × 10 3

0.13 x 10 3

0,13 x I0 3

0.12 x 10 3

0.12 x 10 3

0.12 x 10 3

0.1 2 x 10 -3

-0. 47 x I 0 5

-0.4 6 x 10 5

-0. 45 x 10 5

-0. 43 x I 0 5

-0.42 x 10 5 ]

-0.41 x 10 5

-0.40 x 10 ~

-0.40 x 104

9

0.50 x 10 6

0.48 × 10 6

0.4 6 x 10 -6

0.45 × 10 6

0.4 4 x 10 -6

0.43 x 10 -6

0.42 × 10 6

0.41 × 10 6

0.40 × 10 6

,0.39 × 10 6

( c ) P e r c e n t e r r o r ~ i n D Q s o l u t i o n w i t h u n e q u a l l y s p a c e d T y p e 1 p o i n t s

N u m b e r o f s a m p l i n g p o i n t s , N

5 6 7 8 ] 9

0.32 × 10 -0.16 x 10 -2 0.26 x 10 4 -0.41 x 10 6 0.15 x 10 7

0.31 x10 1 -0.98x10-31 0,82 xl0 -5 -0.1 1xl0 -6 0.15 x10 -7

0.21 x 104 -0.4 9 x 10 -3 0,83 x 10 5 -0.36 x 10 6 0.32 x 10 7

0.14 × l0 1 -0.5 2× 10 3 0.1 6x 10 .4 -0.4 6x 10 6 0.22 × 10 7

0.1 2x I 0 1 -0.7 7× 10 3 0.1 9x 10-4 -0.3 2x 10 6 0.14 × 10 7

0.14 × 10 1 -0.93 × 10 -3 0.16 x 10 4 -0.23 x 10 6 0.19 × 10 7

0. 16× 10 l -0.91 x 10 3 0,1 2× 10 4 -0. 27× 10 6 0.2 2× 10 7

0.1 8x 10 1 -0. 80x 10 3 0.1 2x 10 4 -0.32 × 10 6 0.1 8x 10 7

0 . 17x10 1 -0 . 7 3x l0 3 0 . 13x10-4 -0 . 29×10 .6 0 . 18x10 7

0.1 7× 10 1 -0. 74 × 10 3 0.1 3× 10-4 -0.28 × 10 6 0.18 x 10 7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

. r iO =

t h e o r y o f g a s e s . In a d i m e n s i o n l e s s f o r m , t h e e q u a t i o n i s

g i v e n a s

f

I o e ~ - n f ( r l ) d r l = s i n ~ , 0 _ < ~ _ < ( 7 9 )

a t

w h e r e r I i s a d u m m y v a r i a b l e t o ~ .

T h e e x a c t s o l u t i o n o f E q ( 7 9 ) s u b j e c t t o t h e b o u n d a r y

c o n d i t i o n

f = l a t ~ = 0 ( 8 0 )

i s ( C i v a n a n d S l i e p c e v i c h , 1 9 8 6 )

f ( { ) = 3 e - 4 + c o s l - s i n 1

e a - r i e - l ) + 2 - c o s

81)

w h e r e i n E q s ( 7 9 ) a n d ( 8 1 ) , e i s t h e b a s e o f t h e n a t u r a l

l o g a r i t h m s .

F o r t h e q u a d r a t u r e f o r m u l a t i o n o f E q ( 7 9 ) , th e q u a d r a t u r e

r u l e f o r t h e i n t e g r a l t e r m i s w r i t t e n a s

N

j=[

f r o m w h i c h t h e w e i g h t i n g c o e f f i c i e n t s C j m a y b e d e t e r -

m i n e d . I t s h o u l d b e n o t e d t h a t i n t h i s e q u a t i o n , t h e i n t e g r a l

t e r m

I u = I 2 e n l l U l d r l

i s a d e f i n i t e i n te g r a l a n d i s , t h e r e f o r e , a c o n s t a n t t e r m . I t m a y

b e e v a l u a t e d a s

l l = l - l - , l u = - l - + ( u - 1 ) l ~ , _ l ; u = 2 , 3 . . .. , N

e e

U s i n g t h e q u a d r a t u r e r u l e f o r t h e f i r s t - o r d e r d e r i v a t i v e

a n d f o r t h e i n t e g r a l f r o m E q ( 8 2 ) , o n e o b t a i n s t h e q u a d r a t u r e

a n a l o g o f E q ( 7 9 ) a s

, I , c , } , ,

e ~ i = s i n ~ ,i i = 1 , 2 . . . . . N . ( 8 4 )

Z.at /J

j = l

W r i t i n g t h e b o u n d a r y c o n d i t io n , E q ( 8 0 ) , a s

f j = 1 , j = l ( 8 5 )

a n d u s i n g i n E q ( 8 4 ) , o n e c a n e x p r e s s t h e q u a d r a t u re a n a l o g

o f E q ( 7 9 ) a s

N

Z { A ~ ') - e ~ ' C j } f: = s i n a i - A }~ + e L ' C , ; i = 2 , 3 . .. . . N ( 8 6 )

j=2

i n w h i c h , n o w , t h e b o u n d a r y c o n d i t i o n , E q ( 8 0 ) , h a s b e e n i n -

c o r p o r a t e d .

I t i s s e e n th a t E q ( 8 6 ) m a y b e w r i t t e n a s

N

Z L , jf j = sin ~,i - L a ; i = 2 , 3 . . . .. N ( 8 7 )

j=2

w h e r e

L/j = d~. ) e ~ i C j ( 8 8 )

so tha t L u a r e t h e w e i g h t i n g c o e f f i c i e n t s o f t h e i n t e g r o - d i f -

f e r e n t ia l o p e r a t o r

d f I ie ~ _ n f ( r l) d q

~

w h i c h m a y b e o b t a i n e d f r o m t h e e q u a t i o n s

N

• I _ -I

j=l

u = l , 2 . .. .. N .

( 8 9 )



15/28

Appl Mech Rev vo149, no 1, Janua ry 1996 Bert and Malik: Differential quadrature method in computational mechanics 15

T he r e su l t s i n T ab l e 5 sh ow t he exa c t va l ues o f the f unc t i on

f i t ) a t t = 0.1 interva ls in 0.1 _< t -< 1.0 and the p erce nt er rors

i n t he numer i ca l so l u t i on va l ues wher e , a s be f or e , t he nu-

mer i ca l va l ues a t t he g i ven i n t e r va l s wer e ob t a i ned by t he

L agr ange i n t e r po l a t i on f r om t he quadr a t u r e so l u t i on va l ues .

I t may seen t ha t t he e r r o r s i n t he quadr a t u r e so l u t i on r e su l t s

a r e v e r y s m a l l e v e n w i t h t h e n u m b e r o f s a m p l i n g p o i n t s a s

sma l l a s N = 5 . A l so , t he e r r o r s w i t h T ype I I s ampl i ng po i n t s

so l u t i ons a r e abou t an o r de r o f magni t ude l e s s t han t ha t w i t h

T y pe 1 s amp l i ng p o i n t s so l u t i ons . I n f ac t f o r N _> 9 , T y pe I I

DQ so l u t i ons ma t ch t o e i gh t o r mor e dec i ma l p l aces w i t h t he

exac t so l u t i on .

I t shou l d be n o t ed t ha t t he d i f f e r en t i a l quadr a t u r e so l u t i on

o f t h e f o r e g o i n g e x a m p l e w a s f ir s t c o n s i d e r e d b y C i v a n a n d

S i i epcev i ch ( 1986) . However , t he p r esen t so l u t i on i s modi -

f i ed i n t he quadr a t u r e f o r mu l a t i on o f t he i n tegr a l t e r m of t he

gover n i ng equa t i on . Consequen t l y , t he r e su l t s p r e sen t ed i n

T ab l e 5 a r e mor e acc ur a t e t han t hose i n t he c i t ed wor k .

Example 6: Co®ling/heating by

combined convection and radiation

C o n s i d e r a b o d y s u b j e c t e d t o h e a t t ra n s f e r b y c o m b i n e d c o n -

v e c t i o n a n d r a d i at i o n . A s s u m i n g a l u m p e d c a p a c i t y m o d e l o f

t h e b o d y , t h e t i m e r a t e o f c h a n g e i n t e m p e r a t u r e o f t h e b o d y

i s g i v e n b y ( L ie n h a r d , 9 8 7 )

d O = _ O _ O a ) __ _ ~ O 4 _ O 4 )

dE ~ (90)

whe r e O = O( T ) is t he abso l u t e t emp er a t u r e o f t he body , x i s

a nond i mens i ona l t i me , and Oo and Os a r e t he abso l u t e t em-

pe r a t u r es o f , r e spec t i ve l y , t he ambi en t gas and d i s t an t su r -

r ound i ngs w i t h whi ch r ad i a t i on exchange occur s . A l so , h i s

t he con vec t i o n hea t t r ans f e r coe f f i c i en t , F i s the v i ew f ac t o r ,

and ~ i s t he S t e f an- Bol t zm ann cons t an t.

I n t he so l u t i on o f E q ( 90) , one woul d be i n t e r e s t ed i n

k n o w i n g t h e t e m p e r a t u r e v e r s u s ti m e h i s t o r y o f th e b o d y f o r

some g i ven va l ue o f i t s i n i t i a l t emper a t u r e . A l ong- t e r m i n -

t egr a t i on o f E q ( 90) shou l d i nd i ca t e a s t eady- s t a t e o r f i na l

t e m p e r a t u r e ® f w h i c h t h e b o d y w o u l d e v e n t u a l l y re a c h .

H o w e v e r , t h e f in a l t e m p e r a t u r e O r o f t h e b o d y m a y b e o b -

t a i ned d i r ec t l y by app l y i ng t he phys i ca l cond i t i on

d O

O - - - ~ Of , d~ - - - ~ 0as ~ - - ~ oo ( 91)

t o E q ( 90) g i v i ng t he qua r t i c equa t i on

0 4

0 4 _ h _ 0 = 0 9 2 )

' f + T ~ f - O s - Fc~ a

w h i c h m a y b e s o l v e d a n a l y t i c al l y f o r O h I t m a y b e n o t e d

f r om E q ( 92) t ha t the f i na l temp er a t u r e i s i ndepen den t o f any

i n i ti a l body t emp er a t u r e , wha t soeve r .

F or the D Q so l u t i on , E q ( 90) m ay be t r ea t ed a s a boun d-

ary- valu e pro blem taking the t im e d om ain as 0 _< x _< ~f

wher e "~r s t he t i me up t o whi ch t empe r a t u r e ve r sus t i me r e -

cor d i s needed . I f th i s t i me i s su f f i c i en t l y l a rge , t hen t he

t emp er a t u r e va l ue a t T = '~ ; ma y p r ac t i ca l l y r each t he s t eady-

s t a te va l ue

Of

N o w , d e f i n in g a n o r m a l i z e d t im e a s

T

= - - 93 )

T . f

and substituting in Eq 90), one obtains the gove rning equa-

tion in the no rmalized tim e d om ain 0 _< ~ _< I as

dO

4 . x f o + F o o 4 1 _ x . f o a + F o 0 4

O .

( 94)

d t • h T j

L et the bo dy hav e an i n it ia l t empe r a t u r e Oo so t ha t t he

boun dar y cond i t i on f o r E q ( 94) i s

O 0 ) = O 0 95 )

Equation 94) is nonlinear and the its DQ solution m ay be

obtained in a manner identical to that of the nonlinear prob-

lem of Example 4. Thus, the iterative solution of the equa-

tion fol low s the scheme given by Eq 69), that is,

O( n+O = 0 ( ' ) +0 ( " )

w h e r e , f o l lo w i n g t h e p r o c e d u r e o f E x a m p l e 4 , t h e e q u a t i o n

g o v e r n i n g t h e te m p e r a t u r e r e f i n e m e n t 0 = O ( t ) m a y b e o b -

ta ined as

( 96)

4 X f

1 + 4 0 = -

d r h a t

l - ~ f ( O a -F hFOO4~s

w i t h t h e b o u n d a r y c o n d i t io n

0( 0) = 0 ( 97)

T h e q u a d r a t u r e a n a l o g o f E q ( 9 6 ) m a y b e w r i t te n a t a p o i n t

t i a s

o i

A ~ . t ) 0 j + x . f 1 + 4 h

j= 2

= - O ' i - ~ f ( O i 4 ~ O I D - t - ' ~ f I O a ' l - h s )

wh ere i = 2 ,3 . .. .. N, and

(98)

N

d O

k=l

i n whi ch O ; = Oo f o r i = l .A l so , i t i s no t ed t ha t the bo und ar y

cond i t i on , E q ( 97) , i s i nc l uded i n E q ( 98) .

F o r th e p u r p o s e o f n u m e r i c a l s o lu t io n o f t h e a b o v e p r o b -

l em, a numer i ca l exampl e i s adop t ed f r om L i enha r d ' s t ex t -

book ( L i enha r d , 1987 ; page 217) . Cons i de r a b l ack sphe r e i n

an enc l osur e w i t h amb i en t gas t emp er a t u r e a t O® = 2 93° K

and t he wa l l o f t he enc l osur e ma i n t a i ne d a t Os = 1000° K.

T he co nvec t i ve hea t t r ans f e r coe f f i c i en t i s h = 3 00 W / m2- ° K.

As t he r ad i a t i on exchange i s be t ween t he sphe r e and i t s en -

c l osur e on l y , F = 1 . A l so , t he S t e f an- B ol t zma nn cons t an t i s

= 5 .669 7 x 10-S W / m2- °K4. F or t hese da t a , t he so l u t i on o f



16/28

16

Bert and Ma l ik : D i f ferent ial quadrature method in computat ional mechan ics Appl Mech Rev vol 49, no 1, January 1996

T A B L E 6

Convergence of DQ solution forconvection-radiation

prob lem and comparison with two other solutions

N

6

11

16

21

Dimensionless t i m e , x

1 . 0 2 . 0 3 . 0 14 0 x7

® o = 293°K

O x ) b y D Q solution in°K

3 9 2 . 2 7 9 4 4 6 . 3 4 8 4 7 1 . 0 3 0 4 7 8 . 3 3 7 4 7 2 . 1 2 7

4 1 0 . 1 3 5 4 5 1 . 1 4 3 4 6 5 . 2 1 4 4 7 0 . 1 0 6 4 7 2 . 5 6 4

4 1 0 . 1 1 2 4 5 1 . 1 5 6 4 6 5 . 2 6 9 4 7 0 . 0 8 4 4 7 2 . 5 6 5

4 1 0 . 1 1 2 4 5 1 . 1 5 6 4 6 5 . 2 6 9 4 7 0 . 0 8 4 4 7 2 . 5 6 5

Comparison data, Lienhard 1 9 8 7 )

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