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P U F E M i n O n e - D i m e n s i o n

N . S u k u m a r

A p r i l 5 , 1 9 9 6

1 P r o b l e m S t a t e m e n t

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

L u +

2

u = q i n ;

u ( 0 ) = 0 ;

u ( L ) = 0 ;

( 1 )

w h e r e L d

2

= d x

2

i s t h e L a p l a c i a n o p e r a t o r i n 1 D , a n d = f x x 2 ( 0 ; L ) g

2 D i s c r e t e S y s t e m

L e t t h e t r i a l f u n c t i o n u

h

( x ) b e g i v e n b y :

u

h

( x ) =

X

I

I

a

I

; ( 2 )

w h e r e

I

=

I

f 1 x : : : g i s t h e s h a p e f u n c t i o n v e c t o r a t n o d e I , a n d a

I

a r e t h e

c o r r e s p o n d i n g v e c t o r o f u n k n o w n c o e c i e n t s a t n o d e I . W e c h o o s e

I

= w

I

=

P

K

w

K

t o b e t h e p a r t i t i o n o f u n i t y f o r p a t c h

I

. B y s u b s t i t u t i n g t h e t r i a l a n d t e s t f u n c t i o n s

i n t h e w e a k f o r m , t h e f o l l o w i n g d i s c r e t e s y s t e m o f e q u a t i o n s i s o b t a i n e d :

K a = f ( 3 )

w h e r e

K

I J

=

Z

( B

T

I

B

J

+

2

T

I

J

) d

f

I

=

Z

T

I

q d

( 4 )

1

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3 D i s c r e t i z a t i o n a n d N u m e r i c a l R e s u l t s

T h e d o m a i n i s d i s c r e t i z e d b y n n o d e s ( F i g u r e 1 ) . L e t t h e s p a c i n g b e t w e e n a d j a c e n t

n o d e s h =

L

n 1

. T h e n o d a l c o o r d i n a t e s a r e x

j

= ( j 1 ) h ; j = 1 ; 2 ; : : : ; n . A l s o

d e n e x

0

= h a n d x

n + 1

= 1 + h . C o n s i d e r p a t c h e s

j

= f x x 2 ( x

j 1

; x

j + 1

) g . L e t

t h e l o c a l s p a c e o n e a c h p a t c h b e d e n o t e d b y V

j

T h e p a r t i t i o n o f u n i t y

I

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

f u n c t i o n w ( x ) . F i v e - p o i n t G a u s s q u a d r a t u r e w a s u s e d i n t h e n u m e r i c a l i n t e g r a t i o n .

I n E x a m p l e s 1 , 2 , a n d 3 , t h e d o m a i n w a s d i s c r e t i z e d b y 1 1 n o d e s ( n = 1 1 ) , w h i l e

1 0 1 n o d e s w e r e u s e d i n E x a m p l e 4 . A c o n j u g a t e g r a d i e n t l i n e a r e q u a t i o n s o l v e r w i t h

b l o c k J a c o b i p r e c o n d i t i o n e r ( P E T S c p a c k a g e ) w a s u s e d i n t h e c o m p u t a t i o n s .

3 . 1 E x a m p l e 1

C o n s i d e r t h e B V P ( 1 ) f o r

= 0 ; q ( x ) = 2 ; a n d L = 1 ( 5 )

T h e e x a c t s o l u t i o n i s :

u ( x ) = x x

2

;

d u ( x )

d x

= 1 2 x

( 6 )

T h e l o c a l s p a c e s V

i

a r e c h o s e n a s :

V

1

= s p a n f x ; x

2

g o n

1

\

V

j

= s p a n f 1 ; x x

j

; ( x x

j

)

2

g o n \ ; j = 2 ; 3 ; : : : ; n 1

V

1 1

= s p a n f x ; x

2

g o n

1 1

\

( 7 )

T h e d i s p l a c e m e n t a n d s t r a i n r e s u l t s a r e s h o w n i n F i g u r e s 2 a n d 3 , r e s p e c t i v e l y . C o n -

v e r g e n c e w a s a t t a i n e d i n o n e i t e r a t i o n : R e s i d u a l n o r m K a f

L

2

= 2 . 1 E - 1 5 .

3 . 2 E x a m p l e 2


= 0 ; q ( x ) =

3

4

p

x

; a n d L = 1 ( 8 )

T h e e x a c t s o l u t i o n i s :

u ( x ) = x x

3 = 2

;

d u ( x )

d x

= 1

3

2

p

x

( 9 )

2

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i


V

1

= s p a n f x ; x

3 = 2

g o n

1

\

V

j

= s p a n f 1 ; x x

j

; x

3 = 2

g o n \ ; j = 2 ; 3 ; : : : ; n 1

V

1 1

= s p a n f x ; x

3 = 2

g o n

1 1

\

( 1 0 )



L

2

= 9 . 2 E - 1 5 .

3 . 3 E x a m p l e 3


= 1 ; q ( x ) = 2 ; a n d L = 1 ( 1 1 )

T h e e x a c t s o l u t i o n f o r q = 2 i s :

u ( x ) = 2 ( 1 c o s h x )

2 ( 1 c o s h L )

s i n h L

s i n h x ;

d u ( x )

d x

= 2 s i n h x

2 ( 1 c o s h L )

s i n h L

c o s h x

( 1 2 )


i


V

1

= s p a n f s i n h x ; 1 c o s h x g o n

0

\

V

j

= s p a n f 1 ; s i n h ( x x

j

) ; c o s h ( x x

j

) g o n \ ; j = 2 ; 3 ; : : : ; n 1

V

1 1

= s p a n f 2 ( 1 c o s h x ) ;

2 ( 1 c o s h L )

s i n h L

s i n h x g o n

1 1

\

( 1 3 )

T h e d i s p l a c e m e n t a n d s t r a i n r e s u l t s a r e p r e s e n t e d i n F i g u r e s 6 a n d 7 , r e s p e c t i v e l y .

C o n v e r g e n c e w a s a t t a i n e d i n o n e i t e r a t i o n : R e s i d u a l n o r m K a f

L

2

= 1 . 4 E - 1 3 .

3 . 4 E x a m p l e 4


= 1 ; q ( x ) = 2 ; a n d L = 1 0 ( 1 4 )

T h e e x a c t s o l u t i o n i s g i v e n i n e q . ( ? ? ) . T h e l o c a l s p a c e s V

i


V

1

= s p a n f 2 ( 1 c o s h x ) ;

2 ( 1 c o s h L )

s i n h L

s i n h x g o n

1

\

V

j

= s p a n f 1 ; s i n h ( x x

j

) ; c o s h ( x x

j

) g o n \ ; j = 2 ; 3 ; : : : ; n 1

V

1 0 1

= s p a n f 2 ( 1 c o s h x ) ;

2 ( 1 c o s h L )

s i n h L

s i n h x g o n

1 0 1

\

( 1 5 )

3

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L

2

= 1 . 1 E - 5 .

n

xh

x = 0

1

x = L

2 3 4

x j

n-2 n-1

F i g u r e 1 : N o d a l D i s c r e t i z a t i o n

4

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0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

Displacement

x

EXACT SOLN.PUFEM (5 PT. QUAD.)

F i g u r e 2 : D i s p l a c e m e n t a l o n g t h e 1 D b a r f o r e x a m p l e 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Strain

x


F i g u r e 3 : S t r a i n a l o n g t h e 1 D b a r f o r e x a m p l e 1

5

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0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.2 0.4 0.6 0.8 1

Displacement

x



-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Strain

x



6

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0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

Displacement

x



-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Strain

x



7

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0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 2 4 6 8 10

Displacement

x



-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10

Strain

x



8

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