Some Thoughts on t h e Convergence o f t h e C l a s s i c a l R a y l e i g h - R i t z Method and t h e F i n i t e Element Method*

Leonard M e i r o v i t c h t and Moon K. Kwak t t Department o f E n g i n e e r i n g Science and Mechanics

V i r g i n i a P o l y t e c h n i c I n s t i t u t e and S t a t e U n i v e r s i t y B lacksburg , VA 24061

A b s t r a c t

The R a y l e i g h - R i t z method i s a t e c h n i q u e f o r app rox ima t i ng t h e e i g e n s o l u t i o n a s s o c i a t e d w i t h a d i s t r i b u t e d s t r u c t u r e . The method amounts t o ap- p r o x i m a t i ng t h e s o l u t i o n o f a d i f f e r e n t i a l e i gen - va lue p rob lem t h a t has no known c l o s e d - f o r m s o l u t i o n by a f i n i t e s e r i e s o f a d m i s s i b l e f u n c t i o n , t h u s r e p l a c i n g t h e d i f f e r e n t i a l e i g e n v a l ue prob lem by an a1 g e b r a i c e igenva l ue problem. The f i n i t e element method can be r e - garded as a R a y l e i g h - R i t z method, a t l e a s t f o r s t r u c t u r e s . The main d i f f e r e n c e between t h e f i n i t e element method and t h e c l a s s i c a l Ray le igh - R i t z method l i e s i n t h e n a t u r e o f t h e a d m i s s i b l e f u n c t i o n s . An i m p o r t a n t q u e s t i o n i n bo th t h e c l a s s i c a l R a y l e i g h - R i t z method and t h e f i n i t e element method i s t h e speed o f convergence. It i s dernonst r a t e d i n t h i s paper t h a t convergence o f t h e c l a s s i c a l R a y l e i g h - R i t z method can be v a s t l y im- proved by i n t r o d u c i n g a new c l a s s of admiss i b l e f u n c t i o n s , c a l l e d quas i -compar ison f u n c t i o n s . F a c t o r s a f f e c t i n g t h e convergence of t h e f i n i t e e lement method a r e a1 so d iscussed.

1. I n t r o d u c t i o n

The e i g e n v a l u e prob lem f o r a d i s t r i b u t e d s t r u c t u r e i s d e f i n e d by a d i f f e r e n t i a l e q u a t i o n and c e r t a i n boundary c o n d i t i o n s , and i s known as t h e d i f f e r e n t i a l e i genva lue proh lem (Ref. 1 ) . More o f t e n t h a n not , p a r t i c u l a r l y f o r nonun i f o rm s t r u c t u r e s c h a r a c t e r i z e d by c o e f f i c i e n t s depend ing on t h e s p a t i a l v a r i a b l e ( s ) , a c l o s e d - f o r m s o l u t i o n o f t h e d i f f e r e n t i a l e i genva lue p rob lem may no t be f e a s i b l e . The R a y l e i g h - R i t z method i s a t e c h n i q u e f o r o b t a i n i n g an approx imate s o l u t i o n o f t h e e i genval ue problem. The method c o n s i s t s o f app rox ima t i ng t h e s o l u t i o n o f a d i f f e r e n t i a l e i genva lue prob lem t h a t has no known c l o s e d - f o r m s o l u t i o n by a f i n i t e s e r i e s o f t r i a l f u n c t i o n s m u l t i p l i e d by undetermined c o e f f i c i e n t s . There a r e two genera l approaches t o t h e d e t e r m i n a t i o n o f t h e approx imate s o l u t i o n . One approach i s t o work w i t h t h e d i f f e r e n t i a l e q u a t i o n and t o s e l e c t t h e t r i a l f u n c t i o n s from t h e space o f comparison f u n c t i o n s , namely, f u n c t i o n s t h a t a r e 2p t i m e s d i f f e r e n t i a b l e and s a t i s f y a1 1 t h e boundary c o n d i t i o n s , where 2p i s t h e o r d e r o f t h e d i f - f e r e n t i a l e q u a t i o n (Ref. 1 ) . The o t h e r approach c o n s i s t s o f d e f i n i n g and r e n d e r i n g a R a y l e i g h q u o t i e n t f o r t h e system s t a t i o n a r y . I f t h e R a y l e i gh q u o t i e n t i s g i v e n i n te rms o f exp ress ions r e l a t e d t o t h e p o t e n t i a l and k i n e t i c energy, t h e n t h e t r i a l f u n c t i o n s can be s e l e c t e d f rom t h e space

o f a d m i s s i b l e f u n c t i o n s , namely, f u n c t i o n s t h a t a r e p t i m e s d i f f e r e n t i a b l e and s a t i s f y t h e geomet r i c boundary c o n d i t i o n s a1 one (Ref. 1). Because a d m i s s i b l e f u n c t i o n s a re c o n s i d e r a b l y more abundant t h a n comparison f u n c t i o n s and t h e y a r e g e n e r a l l y e a s i e r t o produce, we choose t h e second approach.

The f i n i t e element method can be regarded as a R a y l e i g h - R i t z method, a t l e a s t f o r s t r u c t u r e s . The main d i f f e r e n c e between t h e f i n i t e element method and t h e c l a s s i c a l R a y l e i g h - R i t z method 1 i e s i n t h e n a t u r e o f t h e a d m i s s i b l e f u n c t i o n s (Ref. ? ) . Indeed, i n t h e c l a s s i c a l R a y l e i g h - R i t z method t h e a d m i s s i b l e f u n c t i o n s a r e g l o b a l f u n c t i o n s , i.e., t h e y a r e d e f i n e d ove r t h e e n t i r e domain o f t h e s t r u c t u r e . D i f f e r e n t i a b i l i t y i s seldom an issue, i f ever . On t h e o t h e r hand, i n t h e f i n i t e element method t h e a d m i s s i b l e f u n c t i o n s a re l o c a l f u n c t i o n s , i.e., t h e y a r e de f i ned over s m a l l e r subdomains, such as two ad jacen t f i n i t e elements. The admiss i h l e f u n c t i o n s a re low-degree p o l y - nomia ls , s a t i s f y i n g t h e minimum d i f f e r e n t i a b i l i t y requ i rements . They a r e o f t e n r e f e r r e d t o as i n t e r p o l a t i o n f u n c t i o n s .

The f i n i t e element method has proved enorm- o u s l y success fu l i n s t r u c t u r a l dynamics. One o f t h e reasons f o r i t s success i s t h a t i t can s o l v e problems i nvo l v i ng pronounced parameter non- u n i f o r m i t i e s and/or comp l i ca ted geometr ies , where o t h e r methods cannot cope. Another reason i s t h a t i t s use has become mechanized, w i t h t h e cho i ce o f i n t e r p o l a t i o n f u n c t i o n s f o r a g i v e n prob lem b e i n g ve ry narrow. I n f a c t , t h i s i s t h e reason t h a t has i n h i h i t e d t h e more f requen t use o f t h e c l a s s i c a l R a y l e i g h - R i t z method. Indeed, i n t h e c l a s s i c a l R a y l e i g h - R i t z method t h e r e i s a wide cho i ce o f a d m i s s i b l e f u n c t i o n s f o r any g i ven prob lem and t h e r e a re no good c r i t e r i a t o h e l p w i t h t h e s e l e c t i o n . Yet, i n problems i n which t h e parameter n o n u n i f o r m i t i e s a r e n o t very pronounced, t h e c l a s s i c a l R a y l e i g h - R i t z method i s capab le o f y i e l d i n g s u p e r i o r accuracy f o r t h e same number o f degrees o f freedom, o r t h e same accuracy w i t h fewer degrees o f freedom.

One of t h e ques t i ons i n b o t h t h e c l a s s i c a l R a y l e i g h - R i t z method and t h e f i n i t e element method i s t h e speed of convergence. I n t h e c l a s s i c a l R a y l e i g h - R i t z method, and perhaps more i n d i r e c t l y i n t h e f i n i t e element method, t h e q u e s t i o n i s r e l a t e d t o t h e completeness o f t h e s e t o f admis- s i b l e f u n c t i o n s . A l t hough t h e i n d i v i d u a l admis- s i b l e f u n c t i o n s need n o t s a t i s f y t h e n a t u r a l

*Sponsored i n p a r t by t h e USAF/ASD & AFOSR Research Grant F33615-86-C-3233 mon i to red by Drs. V. B. Venkayya and 4. K. Amos, whose suppo r t i s g r e a t l y app rec ia ted .

t u n i v e r s i t y D i s t i n g u i s h e d P ro fesso r . F e l l ow AIAA.

t t G r a d u a t e Research A s s i s t a n t . S tudent Member AIAA.

Copyright O American Institute of Aeronautics and Astronautics, Inc., 1981. All rights reserved.

boundary c o n d i t i o n s , convergence can be v a s t l y improved if t h e p o s s i b i l i t y e x i s t s t h a t a l i n e a r comb ina t i on o f them can s a t i s f y them. T h i s paper i s concerned w i t h t h e q u e s t i o n o f convergence and i n p a r t i c u l a r how completeness a f f e c t s con- vergence.

The concept o f completeness i s more q u a l i t a t - i v e than q u a n t i t a t i v e i n na tu re . Moreover, whether a s e t i s comple te o r not can o n l y be answered i n connec t i on w i t h a g i v e n prohlem. Even then, t h e r e seems t o e x i s t a degree o f comple te- ness, as a s e t t h a t i s comple te f o r a c e r t a i n prob lem may no t be as comple te f o r a somewhat d i f f e r e n t prohlem. T h i s q u e s t i o n i s i m p o r t a n t because i n many cases i t i s p o s s i b l e t o use as a d m i s s i b l e f u n c t i o n s f o r a g i v e n system t h e e i g e n f u n c t i o n s o f a re1 a t e d s i m p l e r system.

T h i s paper i s concerned w i t h t h e convergence o f t h e R a y l e i g h - R i t z method. I t i s demonstrated t h a t convergence o f t h e c l a s s i c a l R a y l e i g h - R i t z method can he v a s t l y improved by i n t r o d u c i n g a new c l a s s o f a d m i s s i b l e f u n c t i o n s , c a l l ed quas i - comparison f u n c t i o n s . The quasi-cornpar i son f u n c t i o n s a re a d m i s s i b l e f u n c t i o n s such t h a t , a l t h o u g h i n d i v i d u a l l y t h e y do not s a t i s f y t h e n a t u r a l boundary c o n d i t i o n s , f i n i t e l i n e a r com- b i n a t i o n s t h e r e o f are capab le o f s a t i s f y i n g them t o any d e s i r e d degree o f accuracy. Convergence o f t h e f i n i t e element method seelns t o be a f f e c t e d more by t h e a b i l i t y o f s a t i s f y i n g t h e d i f f e r e n t i a l e q ~ ~ a t i o n s than t h e boundary c o n d i t i o n s .

A numer ica l example c o n s i s t i n g o f a rod i n a x i a l v i b r a t i o n f i x e d a t one end and w i t h a s p r i n g a t t ached a t t h e o t h e r end i s cons ide red . The e i g e n v a l u e p r o b l e ~ n i s so l ved by b o t h t h e c l a s s i c a l R a y l e i g h - R i t z method and t h e f i n i t e element method. R e s u l t s demonst ra te t h e advantages o f u s i n g quas i -compar ison f u n c t i o n s .

2. The D i f f e r e n t i a l E igenva l ue Problem

We a r e concerned w i t h t h e d i f f e r e n t i a l e i genval ue prob lem

where i s a 1 i n e a r homogeneous d i f f e r e n t i a l o p e r a t o r o f o r d e r 2p, i n wh ich p i s an i n t e g e r , W i s t h e d isp lacement o f a g i v e n p o i n t x o f t h e e l a s t i c s t r u c t u r e , x i s a parameter , m(x) i s t h e mass d e n s i t y a t p o i n t x and L i s t h e l e n g t h o f t h e s t r u c t u r e . The d isp lacement i s s u b j e c t t o t h e boundary c o n d i t i o n s

where Bi a re 1 i near homogeneous d i f f e r e n t i a1 o p e r a t o r s o f maximuin o r d e r 2p - 1. The s o l u t i o n o f t h e e igenva lue probleln, Eqs. ( 1 ) and (2), c o n s i s t s o f a denumerably i n f i n i t e se t o f e i gen - va lues A and a s s o c i a t e d e i g e n f u n c t i o n s W,(x) ( r = I,?,...):

To p e r m i t an expanded d i s c u s s i o n o f t h e e igenva lue prob lem and t h e v a r i o u s p r o p e r t i e s o f i t s s o l u t i o n , we w ish t o rev iew b r i e f l y c e r t a i n p e r t i n e n t d e f i n i t i o n s . To t h i s end, l e t us c o n s i d e r two f u n c t i o n s f and g t h a t a r e p iecew ise con t i nuous i n t h e domain 0 c x < L and d e f i n e t h e

i n n e r p roduc t

I f t h e i n n e r p roduc t van ishes, t h e f u n c t i o n s f and g a re s a i d t o be o r thogona l ove r t h e domain 0 < x < L. Next, l e t us c o n s i d e r a s e t o f l i n e a r l y independent f l ~ n c t i o n s @ 2 , . . . and d e f i n e

where cr ( r = 1,2, ... n) a r e cons tan t co- e f f i c i e n t s . Then, i f by choos ing n l a r g e enough t h e mean square e r r o r

can be made l e s s t h a n any a r b i t r a r i l y sma l l p o s i t i v e number E , t h e se t o f f u n c t i o n s

$2 , . . . i s s a i d t o be complete.

There a r e t h r e e c lasses o f f u n c t i o n s o f s p e c i a l i n t e r e s t : 1 ) e i g e n f u n c t i o n s , 2 ) compar ison f u n c t i o n s and 3) a d m i s s i b l e f u n c t i o n s . As i n d i c a t e d e a r l i e r , t h e e i g e n f u n c t i o n s must s a t i s f y b o t h t h e d i f f e r e n t i a l equa t i on , Eq. ( I ) , and t h e boundary c o n d i t i o n s , Eqs. ( 2 ) . T h i s i s a r e l a t i v e l y smal l c l a s s o f f u n c t i o n s , and more o f t e n t h a n not i t i s no t p o s s i b l e t o s o l v e t h e e i genval ue prob lem e x a c t l y and o b t a i n t h e e i g e n f u n c t i o n s . I n such cases, we must be c o n t e n t w i t h an approx imate s o l u t i o n , wh ich can be ob ta ined i n t h e fo rm o f a l i n e a r comb ina t i on o f members f rom t h e two o t h e r c l a s s e s o f f u n c t i o n s . The c l a s s o f c o m ~ a r i s o n f u n c t i o n s comor ises f u n c t i o n s t h a t ake 2p t i m e s d i f f e r e n t i a b l e and s a t i s f y a l l t h e boundary c o n d i t i o n s , b u t no t n e c e s s a r i l y t h e d i f f e r e n t i a l equa t i on . C l e a r l y , t h e c l a s s bf comparison f u n c t i d n s i s cons ide rab l y l a r g e r t han t h e c l a s s o f e i g e n f u n c t i o n s . To i n t r o d u c e t h e t h i r d c l a s s o f f u n c t i o n s , i t i s necessary t o exami ne t h e boundary c o n d i t i o n s c l o s e r . There a r e two t y p e s o f boundary c o n d i t i o n s , name1 y , geomet r i c boundary c o n d i t i o n s and n a t u r a l boundary c o n d i t i o n s . As t h e name i n d i c a t e s . t h e f i r s t t y ~ e i n v o l v e s t h e s a t i s - f a c t i o n of some geomet r i c c o n d i t i o n a t t h e boundary, such as ze ro d i sp lacemen t o r z e r o s lope. I n t h e case o f a geomet r i c boundary c o n d i t i o n , t h e o p e r a t o r Bi i s o f maximum o r d e r p - 1. On t h e o t h e r hand, t h e second t y p e r e f l e c t s f o r c e o r bend ing moment ba lance a t t h e boundary. I n t h e case o f a n a t u r a l boundary c o n d i t i o n , t h e o p e r a t o r Bi i s o f maximum o r d e r 2p - 1. The c l a s s o f a d m i s s i b l e f u n c t i o n s comprises f u n c t i o n s t h a t a r e p t i m e s d i f - f e r e n t i a b l e and s a t i s f y t h e geomet r i c boundary c o n d i t i o n s on l y . It i s obv ious t h a t t h e c l a s s o f a d m i s s i b l e f u n c t i o n s i s f a r l a r g e r t h a n t h e c l a s s o f comparison f u n c t i o n s .

Next, l e t us c o n s i d e r two comparison f u n c t i o n s u and v and d e f i n e t h e i n n e r p r o d u c t

( u , L v ) = / u L v dx 0

( 6

Then, t h e d i f f e r e n t i a l o p e r a t o r i s s a i d t o be

s e l f - a d j o i n t i f As a c o r o l l a r y , t h e e i g e n f u n c t i o n s W,, W ,,... a r e I L

r e a l . M o r e o v e r , i f t h e s y s t e ~ n i s s e l f - a d j o i n t , t h e e i g e n f u n c t i o n s a r e o r t h o g o n a l (Re f . 1 ) . I f t h e e i g e n f u n c t i o n s a r e n o r m a l i z e d so t h a t t h e y

L s a t i s f y J0 mu2 d x = 1 ( r = 1,2, ...), t h e n t h e y r become o r t h o n o r m a l , where t h e o r t h o n o r m a l i t y p r o p e r t y i s e x p r e s s e d by

W h e t h e r Eq. ( 7 ) h o l d s o r n o t can be v e r i f i e d t h r o u g h i n t e g r a t i o n s by p a r t s , w i t h due con- s i d e r a t i o n o f t h e b o u n d a r y c o n d i t i o n s . 4 mere f u n c t i o n , such as t h e mass d e n s i t y m, i s s e l f - a d j o i n t by d e f i n i t i o n . Hence, i f i s s e l f - a d j o i n t , t h e n t h e s y s t e m i s s e l f - a d j o i n t . I n t h i s case , i n t e g r a t i o n s by p a r t s y i e l d

where 6rS i s t h e K r o n e c k e r d e l t a . I n v i e w o f Eq.

( I ) , i t f o l l o w s t h a t

L 1. / wrf.,ws d x = j !,I L. W d x = ArSrS, o o S

where a k ( k = O,l, ..., p ) and b,(e = O , l , ..., p - 1 )

a r e i n g e n e r a l f u n c t i o n s o f x. O b s e r v i n g t h a t t h e r i g h t s i d e o f Eq. ( 8 ) i s s y m m e t r i c i n u and v, we c o n c l l ~ d e t h a t s e l f - a d j o i n t n e s s imp1 i e s symmetry of t h e e i g e n v a l u e p r o b l e m i n t h e sense i n d i c a t e d b y Eq. (8 ) .

Because t h e e i g e n f u n c t i o n s a r e o r t h o g o n a l o v e r 0 < x < L , t h e y c o m p r i s e a c o m p l e t e s e t . T h i s f a c t ~ e r m i t s t h e f o r m u l a t i o n o f t h e f o l l o w i n a e x p a n s i o n t h e o r e m E v e r f u n c t i o n W w i t h c o n t i n u o u s m d + g t h e h o u n d a r y c o n d i t i o n s of t h e s y s t e m can he expanded i n a b s o l u t e l y and u n i f o r m l y c o n v e r g e n t s e r i e s

t h e I t w i l l p r o v e c o n v e n i e n t t o i n t r o d u c e t h e e n e r g y i n n e r p r o d u c t d e f i n e d h y

L c = / mWWr d x , r = 1,2, ...

0 and we o b s e r v e t h a t t h e e n e r g y i n n e r p r o d u c t i s d e f i n e d even when u and v a r e a d m i s s i b l e f u n c t i o n s r a t h e r t h a n c o m p a r i s o n f u n c t i o n s . The t e r m e n e r g y i n n e r p r o d ~ ~ c t can be e x p l a i n e d by t h e f a c t t h a t t h e p r o d u c t o f u w i t h i t s e l f , name ly , 3. The R a y l e i g h - R i t z Method

L e t us c o n s i d e r once a g a i n t h e e i g e n v a l u e p r o b l e m d e s c r i b e d by Eqs. ( 1 ) and ( 2 ) , i n w h i c h t h e o p e r a t o r L. i s s e l f - a d j o i n t , and d e f i n e t h e R a y l e i g h q u o t i e n t

r w , MI K (11) = ---------

( J ~ w , m) c a n be i n t e r p r e t e d as t w i c e t h e maximum p r o t e n t i a l e n e r g y o f t h e s y s t e m whose e i g e n v a l u e p r o b l e m i s d e s c r i b e d by Eqs. (1) and ( 2 ) . Now, l e t u s c o n s i d e r t h e s e t o f f u n c t i o n s , $ ?,... and

where W i s a t r i a l f u n c t i o n , [U, \ I ] i s t h e e n e r g y i n n e r p r o d ~ r c t and (JmW, J m l l ) i s t h e i n n e r p r o d u c t o f JGW w i t h i t s e l f (Re f . 1 ) . I J s i n g t h e e x p a n s i o n t h e o r e m , E u . ( 1 4 ) . i t i s p o s s i b l e t o show t h a t

d e f i n e t h e a p p r o x i m a t i o n un t o 11 ~ a ~ l e i g h ' s has s t a t i o n a r y v a l u e s a t t h e s y s t e m e i g e n f u n c t i o n s and t h a t t h e s t a t i o n a r v a l u e s a r e p r e c i s e l y t h e s y r t e m ~ R e f . 1 ) . Hence, as an a l t e r n a t i v e t o s o l v i n g t h e d i f f e r e n t i a l e i g e n v a l u e p r o b l e m , Eqs. ( 1 ) and ( 2 ) , one can c o n s i d e r r e n d e r i n q R a v l e i q h ' s a u o t i e n t

where d r ( r = 1,2, ..., Q) a r e c o n s t a n t c o e f f i c -

i e n t s . Then, i f by c h o o s i n g a s u f f i c i e n t l y l a r g e

- . - s t a t i o n a r y . T h i s v a r i a t i o n a l a p p r o a c h i s p a r t i c u l a r l y a t t r a c t i v e when t h e o b j e c t i s t o o b t a i n an a p p r o x i m a t e s o l u t i o n t o t h e d i f f e r e n t i a l e i g e n v a l ue p r o b l e m , because t h e s o l u t i o n u s i n g R a y l e i g h ' s q u o t i e n t c a n be c o n s t r u c t e d f r o m t h e space o f a d m i s s i b l e f u n c t i o n s i n s t e a d o f t h e space o f c o n p a r i son f u n c t i o n s .

n t h e e n e r g y i n n e r p r o d u c t , [ u - un, u - un] can be made l e s s t h a n any a r b i t r a r i l y s m a l l p o s i t i v e number E , t h e s e t $ ?,... i s s a i d t o be c o m p l e t e i n energy .

As a consequence o f t h e s e l f - a d j o i n t n e s s of t h e o p e r a t o r L- , and hence t h e s e l f - a d j o i n t n e s s o f t h e sys tem, t h e e i g e n v a l u e s A,, A?, . . . a r e r e a l .

Y e x t , l e t u s c o n s i d e r a t r i a l f u n c t i o n W f r o m t h e space o f a d m i s s i b l e f u n c t i o n s . F o r p r a c t i c a l

reasons , we c a n n o t c o n s i d e r t h e e n t i r e space b u t o n l y a f i n i t e - d i m e n s i o n a l subspace, where t h e subspace i s spanned by t h e f u n c t i o n s $2 ,. . .$n,

. . . . The f u n c t i o n s a r e such t h a t : ( i ) any n e l e m e n t s $ $? ,. .. ,$,, a r e l i n e a r l y i n d e p e n d e n t

and ( i i ) t h e seqluence $?, ...,$,,,... i s

c o m p l e t e i n energy .

The R a y l e i g h - R i t z method i s a p r o c e d u r e f o r d e r i v i n g an a p p r o x i m a t e s o l u t i o n o f t h e e i g e n v a l u e p r o b l e ? . F o r t h e r e a s o n s s t a t e d above, we a r e i n t e r e s t e d i n s o l v i n g t h e e i g e n v a l u e p r o b l e m b y r e n d e r i n g R a y l e i g h ' s q u o t i e n t s t a t i o n a r y . To t h i s end, we c o n s i d e r a t r i a l f u n c t i o n i n t h e f o r m o f t h e f i n i t e s e r i e s

where ui a r e c o e f f i c i e n t s t o be d e t e r m i n e d and 4 . 1

a r e known a d ~ n i s s i b l e f u n c t i o n s ( i = 1,2, ..., n ) ; t h e mean ing o f t h e s u p e r s c r i p t n i s o b v i o u s . I n s e r t i n g Eq. ( 1 7 ) i n t o Eq. ( 1 6 ) , we o b t a i n

i n w h i c h

m . . = m . . = (dm$ 1 J 1 J

L = iommiaj dx

a r e s y m m e t r i c s t i f f n e s s and mass c o e f f i c i e n t s , r e s p e c t i v e l y , where t h e symlnet r y i s a d i r e c t con- sequence o f t h e systern b e i n g s e l f - a d j o i n t . T h e c o n d i t i o n s f o r t h e s t a t i o n a r i t y o f R a y l e i g h ' s q u o t i e n t a r e

E q u a t i o n s ( 2 0 ) l e a d t o n h o m o g ~ n e o u s a l g ~ b r a i c e q u a t i o n s (Re f . I ) , w h i c h can be i d e n t i f i e d as r e p r e s e n t i n g t h e a l g e b r a i c e i g e n v a l u e p r o b l e m and e x p r e s s e d i n t h e m a t r i x f o r m

where K = [ k . . ] and M = [ m . ] a r e n x n r e a l 1 J 1 J

s y m m e t r i c s t i f f n e s s and mass m a t r i c e s ,

r e s p e c t i v e l y , ; = [ u I, 9, ... , u n l T i s t h e n-

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

and A ~ S a p a r a m e t e r . Hence, s p a t i a l d i s - c r e t i z a t i o n has t h e e f f e c t o f r e p l a c i n g a d i f f e r e n t i a l e i g e n v a l u e p r o b l e m by an a l g e b r a i c one. The s o l u t i o n o f t h e a l g e b r a i c e i g e n v a l u e

n n p r o b l e m y i e l d s t h e v a l u e s hl, A?, ..., A: o f t h e

p a r a m e t e r ; t h e y a r e known as t h e e s t i m a t e d e i g e n v a l ues, o r computed e i g e n v a l ues, as opposed

t o t h e a c t u a l e i g e n v a l u e s hl, h2,. . . ,A, o f t h e

d i s t r i b u t e d system. C l e a r l y , we c a n o n l y e s t i m a t e t h e n l o w e s t e i g e n v a l u e s , and t h e r e a r e no

e s t i m a t e s f o r x n+l, A n+2,. . . . TO t h e e i gen-

n n n v a l u e s hl, A?, . . . ,hn b e l o n g t h e e i g e n v e c t o r s

3, ,... ,U r e s p e c t i v e l y . They c a n be used t o - n Y p r o d u c e e s t i m a t e s o f t h e e i g e n f u n c t i o n s o r computed e i g e n f u n c t i o n s . Indeed , i n t r o d u c i n g t h e

- r n - v e c t o r 9 = [$1 $2 ... m n ] o f a d m i s s i b l e

f u n c t i o n s and r e c a l l i n g Eq. ( 1 7 ) , we c a n w r i t e t h e e s t i m a t e d e i g e n f u n c t i o n s , o r computed e i g e n - f u n c t i o n s

The q u e s t i o n r e ~ n a i n s as t o how t h e e s t i m a t e d e i g e n v a l u e s and e i g e n f u n c t i o n s compare w i t h t h e a c t u a l ones. It can be shown ( R e f . 1) t h a t

o r t h e e s t i m a t e d e i g e n v a l u e s p r o v i d e u p p e r bounds f o r t h e a c t u a l e i g e n v a l u e s . U n f o r t u n a t e l y , t h e e s t i m a t e d e i g e n f u n c t i o n s do n o t l e n d t h e m s e l ves t o such m e a n i n g f u l c o m p a r i s o n . It c a n be s t a t e d , however , t h a t i n g e n e r a l t h e e s t i m a t e d e i g e n v a l ues a r e b e t t e r a p p r o x i m a t i o n s t o t h e a c t u a l e i g e n v a l u e s t h a n t h e e s t i m a t e d e i g e n f u n c t i o n s a r e t o t h e a c t u a l e i g e n f u n c t i o n s . The a c c u r a c y can be i m p r o v e d by i n c r e a s i n g t h e number n o f t e r ~ n s i n Eq. ( 1 7 ) . I n d e e d , because t h e a d m i s s i b l e f u n c t i o n s $2 ,.. . ,$n,.. . a r e f r o m a c o m p l e t e

s e t , we can w r i t e

n l i m x r = x r , r = 1,2 ,..., n n i -

q u a r a n t e e d i f t h e a d m i s s i b l e f u n c t i o n s (I 2,... a r e f r o m a c o m p l e t e s e t , b u t

o f c o n v e r g e n c e depends on t h e c h o i c e o f a d m i s s i b l e f u n c t i o n s .

A l t h o u g h n o t s t a t e d above, one can t b e a d m i s s i b l e f u n c t i o n s m r ( r = I,?,..

ues - S

t h e r a t e

t h e

i n f e r t h a t ,q ) a r e

g l o b a l f l n c t i o n s , i n t h e sense t h a t t h e y e x t e n d o v e r t h e e n t i r e l e n g t h L o f t h e s t r u c t u r e . M o r e o v e r , t h e c o e f f i c i e n t s u r ( r = 1,2, ..., Q) do

n o t r e p r e s e n t a c t u a l d i s p l acernents. I n d e e d , t h e y a r e a b s t r a c t q u a n t i t i e s p r o v i d i n g o n l y a measure o f t h e c o n t r i b u t i o n of t h e i n d i v i d u a l a d m i s s i b l e f u n c t i o n s t o t h e d i s p l a c e m e n t . We r e f e r t o t h i s v e r s i o n as t h e c l a s s i c a l R a y l e i g h - R i t z method.

The f i n i t e element method can a l s o be regarded as a R a y l e i g h - R i t z method. I n c o n t r a s t , however, i n t h e f i n i t e element method t h e a d m i s s i b l e f u n c t i o n s a r e l o c a l f u n c t i o n s , i n t h e sense t h a t t h e y ex tend ove r g i ven segments o f t h e s t r u c t u r e o n l y , where t h e segments a r e known as f i n i t e elements. I n a d d i t i o n , t h e a d m i s s i b l e f u n c t i o n s t e n d t o be ve ry s imple . I n f a c t , t h e y t e n d t o be low-degree po l ynomia l s , s a t i s f y i n g t h e minimum d i f f e r e n t i - a b i l i t y c o n d i t i o n s . F i n a l l y , t h e c o e f f i c i e n t s ui

r e p r e s e n t a c t u a l d isp lacements a t p o i n t s shared by two ad jacen t elements, where t h e p o i n t s a r e known as nodes. Whereas i n b o t h t h e c l a s s i c a l Ray le igh - R i t z method and t h e f i n i t e element method accuracy i s improved hy i n c r e a s i n g t h e number o f a d m i s s i b l e f u n c t i o n s , i n t h e f i n i t e element method t h i s i s ach ieved by i n c r e a s i n g t h e number o f e lements, wh ich amounts t o u s i n g a f i n e r f i n i t e element mesh.

To examine how accuracy improves as t h e number o f a d m i s s i b l e f u n c t i o n s i nc reases , we propose t o s o l v e t h e e igenva lue prob lem f o r t h e system o f F ig . 1 by b o t h t h e c l a s s i c a l R a y l e i g h - R i t z method and t h e f i n i t e element method. T h i s example w i l l p r o v i d e t h e m o t i v a t i o n f o r t h e r e a l o b j e c t o f t h i s i n v e s t i g a t i o n , namely, t o deve lop t h e framework f o r an approach capab le of a c h i e v i n g improved convergence c h a r a c t e r i s t i c s .

The system o f F i g . 1 rep resen ts a nonun i f o rm rod i n a x i a l v i b r a t i o n w i t h one end f i x e d and t h e o t h e r end r e s t r a i n e d by a s p r i n g . The parameters a r e as f o l l o w s :

where EA(x) i s t h e a x i a l s t i f f n e s s , i n wh ich E i s t h e modulus o f e l a s t i c i t y and A (x ) i s t h e c r o s s - s e c t i o n a l area, m(x) i s t h e mass p e r u n i t l e n g t h and k i s t h e s p r i n g cons tan t . The energy i n n e r p r o d u c t , appea r i ng a t t h e numerator o f R a y l e i g h ' s q u o t i e n t , has t h e fo rm (Ref. 1 )

so t h a t p = 1. Yoreover, t h e we ighted i n n e r p roduc t , appea r i ng a t t h e denominator o f R a y l e i g h ' s q u o t i e n t , i s

For f u t u r e re fe rence , t h e boundary c o n d i t i o n s a r e (Ref. 1)

where (28a) i s recogn i zed as a geometr ic boundary c o n d i t i o n and (2%) as a n a t u r a l boundary c o n d i t i o n .

I n t h e c l a s s i c a l R a y l e i g h - R i t z method, we assume an approx imate s o l u t i o n i n t h e f o rm

(26

whe

where 2 i s an n - v e c t o r o f c o e f f i c i e n t s and $ ( x ) an n - v e c t o r o f a d m i s s i b l e f u n c t i o n s , and we n o t e t h a t t h e s u p e r s c r i p t n was o m i t t e d f rom \I f o r sim- ~ l i c i t y o f n o t a t i o n . I n s e r t i n g Eq. (29) i n t o Eq.

), we o b t a i n t h e energy i n n e r p r o d u c t

i s recogn i zed as t h e s t i f f n e s s m a t r i x . Moreover, i n t r o d u c i n g Eq. (29) i n t o Eq. ( 2 7 ) , we o b t a i n t h e we igh ted i n n e r p roduc t

T ( J d 4 , J a ) = _ a T ( J $ , ~ ~ T ) _ a = _ a M _ a (32 )

where

i s t h e mass m a t r i x .

The a d m i s s i b l e f u n c t i o n s must s a t i s f y o n l y boundary c o n d i t i o n (28a) . We choose t h e admis- s i b l e f u n c t i o n s

$ i ( ~ ) = s i n ( 2 i - 1 ) n x 2 L , i = lY2,...,n(34)

wh ich a re recogn ized as t h e e i g e n f u n c t i o n s o f a r e l a t e d system, namely, a u n i f o r m rod i n a x i a l v i b r a t i o n w i t h t h e l e f t and f i x e d and t h e r i g h t end f r e e . The s t i f f n e s s and mass m a t r i c e s a r e o b t a i n e d by i n t r o d u c i n g Eqs. (34 ) i n t o Eqs. (31 ) and (33 ) , r e s p e c t i v e l y.

I n t h e case o f t h e f i n i t e element method, we c o n s i d e r t h e approx imate s o l u t i o n

T W(x) = bJjk(x), ( j - l ) h < x < j h ,

where h = L /n i s t h e w i d t h o f t h e f i n i t e element and

i n wh ich W j W l and W j a r e t h e va lues o f t h e

d i sp lacemen t W(x) e v a l u a t e d a t x = ( j - l ) h and x = j h , r e s p e c t i v e l y , and

a r e a d m i s s i b l e f u n c t i o n s hav ing t h e same shape f o r every element, namely, s t r a i g h t l i n e s (Ref. 1). They a re commonly known as l i n e a r i n t e r p o l a t i o n f u n c t i o n s . F o l l o w i n g t h e same p a t t e r n as above, t h e energy i n n e r p roduc t can be w r i t t e n i n t h e fo rm

n T T

[ W , H] = 1 E . K . W . = W_ K[ j = 1 J J-J

where K j a r e 2 x 2 element s t i f f n e s s

h a v i n g t h e t y p i c a l exp ress ion

(38 )

m a t r i c e s

T . Moreover, W = [Y1 Y2 . .. W n ] 1s an n - v e c t o r o f

nodal d i sp lacemen ts and K i s t h e o v e r a l l s t i f f n e s s m a t r i x , ob ta ined f rom t h e element s t i f f n e s s m a t r i c e s t h rough an assembl ing process (Ref. 1 ) . S i rn i l a r l y , t h e we igh ted energy i n n e r p roduc t can he w r i t t e n as

where !I. a r e t h e 2 x 2 element mass m a t r i c e s J

and M i s t h e o v e r a l l mass m a t r i x , o b t a i n e d f rom t h e element mass m a t r i c e s t h r o u g h t h e same assembl i ng process ment ioned above.

The e i g e n v a l u e prob lem was so l ved by b o t h t h e c l a s s i c a l R a y l e i g h - R i t z method and t h e f i n i t e element method. The l o w e s t 3 n a t l l r a l f r e q u e n c i e s a r e l i s t e d i n Tab le I f o r n = I , ? , ..., 30 i n columns denoted by CRR and FEM, r e s p e c t i v e l y . We observe t h a t t h e FEM y i e l d s s l i g h t l y b e t t e r con- vergence than t h e CRR f o r t h e f i r s t f requency, bu t poo re r convergence f o r t h e second and t h i r d . A c t u a l l y , t h e CRR g i v e s b e t t e r convergence f o r a1 1 f r equenc ies h i g h e r t h a n t h e f i r s t . Note t h a t , because t h e computed n a t u r a l f r e q u e n c i e s p r o v i d e upper bounds f o r t h e a c t u a l n a t u r a l f r e q u e n c i e s , l ower es t ima tes imp ly more a c c u r a t e es t ima tes . 4s i t t u r n s ou t , convergence i s r e l a t i v e l y poo r b o t h f o r t h e CRR method u s i n g t h e a d m i s s i b l e f u n c t i o n s g i ven by Eqs. (34) and f o r t h e FEM u s i n g 1 i n e a r i n t e r p o l a t i o n f u n c t i o n s . I n t h e nex t s e c t i o n , we l o o k i n t o t h e reasons f o r t h i s poor convergence c h a r a c t e r i s t i c s and propose ways f o r a c c e l e r a t i n g t h e convergence.

4. Convergence C o n s i d e r a t i o n s

As can be v e r i f i e d f rom Tab le I , convergence o f t h e c l a s s i c a l R a y l e i g h - R i t z method i n con- j u n c t i o n w i t h t h e a d m i s s i b l e f u n c t i o n s r e p r e - s e n t i n g t h e e i g e n f u n c t i o n s o f a u n i f o r m f i x e d - f r e e r o d i s r e l a t i v e l y slow. Convergence o f t h e f i n i t e element method u s i n g 1 i near i n t e r p o l a t i o n f u n c t - i o n s i s s l i g h t l y b e t t e r f o r t h e f i r s t n a t u r a l f requency, b u t i t worsens p r o g r e s s i v e l y f o r h i g h e r n a t u r a l f r equenc ies . I n t h i s s e c t i o n , we seek an e x p l a n a t i o n f o r t hese poo r r e s u l t s and propose ways f o r improv ing them.

l n seek ing t o e x p l a i n t h e r e s u l t s o b t a i n e d i n

S e c t i o n 3, we w ish f i r s t t o l ook i n t o t h e n a t u r e o f t h e approx imate s o l u t i o n . I n t h e Ray le igh -R i t z method, bo th t h e c l a s s i c a l and t h e f i n i t e element method, we c o n s t r u c t an approx imate s o l u t i o n as a 1 i near colnbi n a t i on o f a d m i s s i b l e f u n c t i o n s , i .e., f u n c t i o n s t h a t a r e comple te i n energy. Such f u n c t i o n s s a t i s f y t h e geomet r i c boundary c o n d i t - i o n s by d e f i n i t i o n , b u t t h e q u e s t i o n remains how we1 1 t h e y approx imate t h e d i f f e r e n t i a l equa t i on and t h e n a t u r a l boundary c o n d i t i o n s . I n t h i s rega rd , t h e concept o f completeness o f f e r s no he lp . Indeed, completeness i s a mathemat ica l concept t h a t i s q u a l i t a t i v e i n na tu re . It s t a t e s what happens t o t h e e r r o r caused by t h e approx- i m a t i o n o n l y as n becomes l a r g e . From a com- p u t a t i o n a l p o i n t o f view, t h e concept o f completeness i s o f dub ious va lue, as t h e whole o b j e c t o f an approx imate t e c h n i q u e i s t o produce s u f f i c i e n t accuracy w h i l e t h e number o f terms i n t h e app rox ima t i on i s s t i l l r e l a t i v e l y s m a l l . To e b r i n g t h i s p o i n t i n t o sha rpe r focus, we c o n s i d e r Eqs. (17 ) and (34 ) and w r i t e

and ohserve t h a t eve ry t e rm i n t h e s e r i e s i s zero a t x = I., except when i approaches i n f i n i t y . The i m p l i c a t i o n i s t h a t t h e s l o p e a t x = L i s ze ro f o r f i n i t e n, so t h a t t h e n a t u r a l boundary c o n d i t i o n , Eq. (28b) , cannot be s a t i s f i e d e x a c t l y f o r f i n i t e n.

As f a r as t h e approx imate s o l u t i o n o b t a i n e d i n S e c t i o n 3 by means o f t h e f i n i t e element method i n c o n j u n c t i o n w i t h 1 i n e a r i n t e r p o l a t i o n f u n c t i o n s , we conc lude t h a t t h e d i sp lacemen t i s approx imated by a cu rve c o n s i s t i n g o f s t r a i g h t l i n e s , and hence d i s p l a y i n g a c o r n e r a t eve ry nodal p o i n t . The a c t u a l d i sp lacemen t curve, however, must be smooth. Hence, t h e d i f f e r e n t i a l e q u a t i o n cannot be s a t i s f i e d e x a c t l y f o r f i n i t e n.

The above d i s c u s s i o n p0int .s t oward some ways o f imp rov ing t h e app rox ima t i ons . I n t h e c l a s s i c a l R a y l e i g h - R i t z method, t h e need i s f o r a b e t t e r approx imate s a t i s f a c t i o n o f t h e n a t ~ l r a l boundary c o n d i t i o n , Eq. (28b ) . To t h i s end, we c o n s i d e r f i r s t s a t i s f y i n g t h e n a t u r a l boundary c o n d i t i o n e x a c t l y by u s i n g t h e comparison f u n c t i o n s

4 . = s i n B . X , i = l , ? ,..., n 1 1

(43

i n s t e a d of a d m i s s i b l e f u n c t i o n s , where 8 . a r e such t h a t Eq. (281)) i s s a t i s f i e d , o r 1

The f i r s t t h r e e n a t u r a l f r e q u e n c i e s computed by t h e c l a s s i c a l R a y l e i g h - R i t z method u s i n g t h e comparison f u n c t i o n s d e f i n e d by Eqs. (43 ) and ( 4 4 ) a r e d i s p l a y e d i n Tab le I 1 f o r v a r i o u s va lues o f n. As c o u l d be a n t i c i p a t e d , t h e convergence c h a r a c t e r i s t i c s a r e marked ly s u p e r i o r t o those based on t h e a d m i s s i b l e f u n c t i o n s s i n ( 2 i - l ) n x / Z L .

Improved f i n i t e element app rox ima t i ons can be o b t a i n e d by u s i n g h ighe r -deg ree i n t e r p o l a t i o n f u n c t i o n s (Ref. 1 ) . T h i s r e q u i r e s t h e i n t r o d u c t -

i o n o f i n t e r n a l nodes. Q u a d r a t i c i n t e r p o l a t i o n f u n c t i o n s r e q u i r e one i n t e r n a l node p e r f i n i t e element, so t h a t t h e nodal c o o r d i n a t e v e c t o r and t h e v e c t o r o f i n t e r p o l a t i o n f u n c t i o n s have t h e fo rm

F -7 F 7

where t h e i n t e r p o l a t i o n f u n c t i o n s have t h e exp ress ion

X = ( j - ;)[2(j - K) -

X = 4 ( j - ;)[I - ( j - 5

X $13 = 1 - 3 ( j - F ) + 2 ( j

S i m i l a r l y , c u b i c i n t e r p o l a t i o n two i n t e r n a l nodes, so t h a t

f u n c t i o n s r e q u i r e

where

The f i r s t t h r e e n a t u r a l f r e q u e n c i e s cornpllted by means o f t h e f i n i t e element neth hod u s i n g q u a d r a t i c and c u b i c i n t e r p o l a t i o n f u n c t i o n s a r e a1 so shown i n Tab le 11.

From Tab le 11, we conc lude t h a t t h e c l a s s i c a l R a y l e i g h - R i t z method u s i n g comparison f u n c t i o n s g i v e s somewhat b e t t e r r e s u l t s t h a n those ob ta ined by means of t h e f i n i t e element method u s i n g q u a d r a t i c i n t e r p o l a t i o n f u n c t i o n s and comparable r e s u l t s t h a n those ob ta ined by u s i n g c u b i c i n t e r p o l a t i o n f u n c t i o n s . I t i s c l e a r t h a t t h e accuracy d i s p l a y e d i n Tab le I 1 was never ach ieved i n Tab le I.

5. A New Class o f Admiss ib le F u n c t i o n s

Acco rd ing t o t h e R a y l e i g h - R i t z t h e o r y , i n f o r m u l a t i n g t h e a l g e b r a i c e igenva lue p rob lem by r e n d e r i n g t h e R a y l e i g h q u o t i e n t s t a t i o n a r y , where t h e q u o t i e n t i s i n te rms o f t h e energy i n n e r p r o d u c t [W, W ] i n s t e a d o f t h e i n n e r p r o d u c t ( W , W) , t h e a p p r o x i m a t i on can be c o n s t r u c t e d f r o m t h e space o f a d m i s s i b l e f u n c t i o n s , r a t h e r t h a n comparison f u n c t i o n s . The main d i f f e r e n c e between

a d m i s s i b l e f u n c t i o n s and comparison f u n c t i o n s l i e s i n t h e f a c t t h a t a d m i s s i b l e f u n c t i o n s need s a t i s f y o n l y t h e geometr ic boundary c o n d i t i o n s and t h e comparison f u n c t i o n s must s a t i s f y a l l t h e boundary c o n d i t i o n s . O f course, t h e r e i s a l s o t h e q u e s t i o n o f d i f f e r e n t i a b i l i t y , bu t i n t h e c l a s s i c a l R a y l e i g h - R i t z method t h i s q u e s t i o n seldom a r i s e s , as t h e f u n c t i o n s used t e n d t o have s u f f i c i e n t smoothness t o ensure t h e e x i s t e n c e o f d e r i v a t i v e s o f h i g h order .

I n u s i n g a d m i s s i b l e f u n c t i o n s i n t h e case o f t h e system o f F ig . 1, t h e convergence was p a i n f u l l y slow. C o n s i s t e n t w i t h t h i s , s a t i s f a c t i o n o f t h e n a t u r a l boundary c o n d i t i o n a t x = L was no t p o s s i b l e f o r a f i n i t e number o f terms i n s e r i e s (17 ) . On t h e o t h e r hand, t h e use of comparison f u n c t i o n s y i e l d e d r e l a t i v e l y f a s t convergence. However, t h e f a c t t h a t each o f t h e comparison f u n c t i o n s must s a t i s f y a1 1 t h e boundary c o n d i t i o n s can be q u i t e a burden. A l though i n t h e case o f t h e system o f F ig . 1 t h e burden was n o t p a r t i c u l a r l y heavy, i t was s t i l l necessary t o s o l v e a t r anscenden ta l equa t i on , Eq. (44) . I n o t h e r cases, such as i n two- and th ree -d imens iona l s t r u c t u r e s , t h e s a t i s f a c t i o n o f n a t u r a l boundary c o n d i t i o n s can p resen t a s e r i o u s problem. Hence, t h e q u e s t i o n i s whether a way o f c i r c u m v e n t i n g t h i s prob lem e x i s t s .

To improve convergence, we must a t t emp t t o produce a s o l u t i o n t h a t s a t i s f i e s t h e n a t u r a l boundary c o n d i t i o n s as c l o s e l y as p o s s i b l e s h o r t o f u s i n g comparison f u n c t i o n s . To t h i s end, we i n t r o d u c e a new c l a s s of a d m i s s i b l e f u n c t i o n s . T h i s i s t h e c l a s s o f a u a s i - c o m ~ a r i s o n f u n c t i o n s d e f i n e d as a d m i s s i b l e f u n c t i o n s o f such a n a t u r e t h a t f i n i t e l i n e a r comb ina t i ons t h e r e o f a r e capab le o f s a t i s f y i n g t h e n a t u r a l boundary c o n d i t i o n s as c l o s e l y as des i red . I n essence, quas i -compar i son f u n c t i o n s can be regarded as b e i n g comple te i n boundary c o n d i t i o n s , i n a d d i t i o n t o b e i n g comple te i n energy. C l e a r l y , s i n ( 2 i - l ) n x / 2 L do no t qua1 i f y as quas i -compar ison f u n c t i o n s . as no f i n i t e l i n k a r comb ina t i on o f t hese f u n c t i o n s can produce a nonzero s lope a t x = L.

As a s e t o f quas i -compar ison f u n c t i o n s f o r t h e system o f F ig . 1, we c o n s i d e r

i a x $i ( x ) = s i n - , i = 1 , . . n (49 2L

We n o t e t h a t none o f t h e above f u n c t i o n s s a t i s f t h e n a t u r a l boundary c o n d i t i o n o f x = L i n d i v i d u a l l y . However, t h e l i n e a r comb ina t i on

f o r example, does s a t i s f y t h e n a t u r a l boundary c o n d i t i o n , Eq. (28b) . The f i r s t t h r e e n a t u r a l f r e q u e n c i e s computed by t h e c l a s s i c a l R a y l e i gh- R i t z method u s i n g t h e quas i - compar i son f u n c t i o n s g i v e n by Eqs. (49 ) a r e shown i n Tab le 111. As can be conc luded f rom Tab le 111, convergence i s v e r y r a p i d .

The i n t e r e s t i n g t h i n g about t h e r e s u l t s o b t a i n e d by means o f t h e quas i -compar ison f u n c t - i o n s i s t h a t , except f o r v e r y l o w n, t hey a re somewhat b e t t e r t h a n even t h o s e o b t a i n e d by means o f comparison f u n c t i o n s , as can be concluded by comparing r e s u l t s i n Tab les I 1 and 111. The e x p l a n a t i o n i s t h a t t h e quas i -compar ison f u n c t i o n s

a r e capab le o f app rox ima t i ng t h e s o l u t i o n t h rough - o u t t h e domain 0 < x < L a l i t t l e b e t t e r t h a n t h e comparison f u n c t i o n s . F i g u r e 2 shows a p l o t o f t h e l o g a r i t h m o f t h e mean-square e r r o r o f t h e f i r s t computed e i g e n f u n c t i o n versus t h e number o f t e rms f o r bo th quas i -compar ison f u n c t i o n s and comparison f u n c t i o n s . The p l o t shows s m a l l e r mean-square e r r o r f o r quas i -compar ison f u n c t i o n s .

I n examining Eqs. (34) and (49 ) , we observe t h a t t h e s e t o f quas i -compar ison f u n c t i o n s s i n i n x / 2 L c o n t a i n s t h e s e t o f f u n c t i o n s s i n i nx /L , i n a d d i t i o n t o t h e a d m i s s i b l e f u n c t i o n s s i n ( 2 i - l ) n x / 2 L . The f u n c t i o n s s i n i s x / L a r e zero a t x = L, bu t t h e i r s l o p e i s not . Hence, t h e a d d i t i o n o f t h e s e t o f f u n c t i o n s s i n i n x / L p r o v i d e s t h e c a p a b i l i t y o f s a t i s f y i n g t h e n a t u r a l boundary c o n d i t i o n a t x = L. I n v iew o f t h i s , one can rega rd t h e se t o f quas i -compar ison f u n c t i o n s s i n i ~ x / ? L and b e i n g a "more comple te s e t " o f a d m i s s i b l e f u n c t i o n s t h a n t h e s e t of a d m i s s i b l e f u n c t i o n s s i n ( 2 i - l ) ~ x / 2 L .

6. Summary and Conc lus ion

I n p roduc ing an approx imate s o l u t i o n o f a d i f f e r e n t i a l e i genva lue prob lem by t h e R a y l e i g h - R i t z method, i t i s advantageoils t o use a v a r i a t i o n a l approach as i t p e r m i t s t h e use o f a d m i s s i b l e f u n c t i o n s i n s t e a d o f comparison f u n c t i o n s . The a d m i s s i b l e f u n c t i o n s need s a t i s f y o n l y t h e geomet r i c boundary c o n d i t i o n s and must be f rom a comple te s e t . I n t h e c l a s s i c a l R a y l e i g h - R i t z method t h e r e i s a wide c h o i c e o f a d m i s s i b l e

f u n c t i o n s . A l t hough completeness o f t h e s e t o f a d m i s s i b l e f u n c t i o n s guarantees convergence, t h e convergence r a t e can be slow. I n some cases, t h e poor convergence can be t r a c e d t o t h e i n a b i l i t y o f s a t i s f y i n g t h e n a t u r a l boundary c o n d i t i o n s w i t h a f i n i t e number of a d m i s s i b l e f u n c t i o n s .

To a c c e l e r a t e convergence i n t h e c l a s s i c a l R a y l e i g h - R i t z method, a new c l a s s of a d m i s s i b l e f u n c t i o n s i s i n t r o d u c e d i n t h i s paper. T h i s i s t h e c l a s s of quas i -compar ison f u n c t i o n s de f i ned as a d m i s s i b l e f u n c t i o n s such t h a t , a l t h o u g h i n - d i v i d u a l l y do no t s a t i s f y t h e n a t u r a l boundary c o n d i t i o n s , l i n e a r comb ina t i on t h e r e o f a r e capab le o f s a t i s f y i n g them t o any d e s i r e d degree o f accuracy. Convergence of t h e f i n i t e element method seems t o be a f f e c t e d more by t h e a b i l i t y of s a t i s f y i n g t h e d i f f e r e n t i a l e q u a t i o n t han t h e boundary c o n d i t i o n s .

A numer ica l example demonst ra tes t h a t t h e c l a s s i c a l R a y l e i g h - R i t z method i n c o n j u n c t i o n w i t h quas i -compar ison f u n c t i o n i s capab le o f p roduc ing s u p e r i o r r e s u l t s .

7. References

1. M e i r o v i t c h , L., Computa t iona l Methods i n S t r u c t u r a l Dynamics, S i j t h o f f & Noordhof f , The Nether1 ands, 1980.

2. S t rang, G. and F i x , G. J., An A n a l y s i s o f t h e F i n i t e Element Method, P r e n t i c e - H a l l , Inc. , Engl ewood C l i f f s , NJ, 1973.

Figure 1 - Nonuniform Rod i n Ax ia l V ibra t ion

1 2 3 4 5 6 7 8 9 10

Number o f Terms

Figure 2 - Logarithm o f the Mean-Square Er ror o f F i r s t Computed Eigenfunct ion

Tab le I - F i r s t . Second and T h i r d N a t u r a l Freauencies Comouted bv t h e c l a s s i c a l R a y l e i g h - R i t z Method (CRR) and t h e ~ i n i t e ~ l e m e n t Method (FEM).

CRR --

2.32965 2.27291 2.25352 2.24369 2.23781 2.23391 2.23115 2.22910 2.22751 2.22625 2.22523 2.22439 2.22367 2.22307 2.22254 2.22209 2.22169 2.22133 2.22101 2.22073 2.22047 2.22024 2.22003 2.21983 2.21966 2.21949 2.21934 2.21920 2.21907 2.21895

FEM CRR FEM

6.27163 5.68345 5.43128 5.31181 5.24676 5.20758 5.18218 5.16479 5.15237 5.14318 5.13620 5.13076 5.12646 5.12298 5.12014 5.11778 5.11581 5.11414 5.11271 5. Ill49 5.11042 5.10950 5.10868 5.10796 5.10733 5.10676 5.10625 5.10580 5.10539

CRR FEM

Tdble I 1 - F i r s t Three N a t u r a l Frequencies Computed by t h e C l a s s i c a l R a y l e i g h - R i t r Method Using Comparison Func t ions (CRRC) and t h e F i n i t e Element Method Using Q u d d r a t i c (FEMQ) and Cubic

(FEMC) I n t e r p o l a t i o n Func t ions

n W1

-

F EMQ

n W 2

FEMQ -- -

6.27984

5.18817

5.12170

5.10695

5.10265

5.10105

5.10036

5.10002

5.09983

5.09973

5.09966

5 .OW62

5 .OW60

5.09958

5.09956

n W3

F EMU -- -

CKRC -

2.22297 2.21647 2.71573 2.21559 2.21555 2.21553 2.21553 2.21553 2.21553 2.21553 2.21552

FEMC FEMC CCRC --

5.10630 5.10070 5.09984 5.09964 5.09957 5.09955 5.09954 5.09953 5.09953 5.09953 5.09953 5.09953 5.09953 5.09953 5.09953 5.09953 5.09952

CRRC FEMC

Table 111 - First Three Natural Frequencies Computed by t h e Classical Rayleigh-Ritz Method Using Quasi-Comparison Functions