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Finite element method - a Galerkin approach Hutton, Stanley George 1971

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F I N I T E ELEMENT METHOD - A G A L E R K I N APPROACH by  STANLEY GEORGE HUTTON B . S c . , U n i v e r s i t y o f N o t t i n g h a m , 1963 M.Sc., U n i v e r s i t y o f C a l g a r y , 1966  A T H E S I S SUBMITTED IN P A R T I A L F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY  in  the  Department of  CIVIL ENGINEERING  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o the r e q u i r e d standard  THE U N I V E R S I T Y OF B R I T I S H COLUMBIA AUGUST 1971  In p r e s e n t i n g  this thesis in partial  fulfilment  o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e  University  o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t freely available for reference that permission  for extensive  s c h o l a r l y p u r p o s e s may  and s t u d y . copying  be g r a n t e d  of t h i s t h e s i s  by t h e Head o f  D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . that copying  I further  w i t h o u t my w r i t t e n  S. G.  Department of C i v i l  Engineering  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a August,  1971.  Columbia  for  my  It is understood  or p u b l i c a t i o n of t h i s t h e s i s f o r  g a i n s h a l l n o t be a l l o w e d  agree  financial permission.  Hutton  F I N I T E ELEMENT METHOD - A G A L E R K I N APPROACH  ABSTRACT This s t u d y i s c o n c e r n e d w i t h d e f i n i n g the mathem a t i c a l framework i n which the f i n i t e element procedure can  m o s t a d v a n t a g e o u s l y be c o n s i d e r e d .  It is established  t h a t t h e f i n i t e e l e m e n t m e t h o d g e n e r a t e s an a p p r o x i m a t e s o l u t i o n to a g i v e n e q u a t i o n which i s d e f i n e d i n terms of a s s u m e d c o - o r d i n a t e f u n c t i o n s and unknown p a r a m e t e r s .  The  a d v a n t a g e s o f d e t e r m i n i n g t h e p a r a m e t e r s by G a l e r k i n ' s m e t h o d a r e d i s c u s s e d and t h e c o n v e r g e n c e  characteristics  of t h i s method a r e r e v i e w e d u s i n g f u n c t i o n a l a n a l y s i s principles.  *  C o m p a r i s o n s a r e made b e t w e e n t h e G a l e r k i n and  R a y l e i g h - R i t z p r o c e d u r e s and t h e c o n n e c t i o n b e t w e e n v i r t u a l w o r k and G a l e r k i n ' s m e t h o d i s i l l u s t r a t e d .  The  convergence  r e s u l t s p r e s e n t e d f o r the G a l e r k i n procedure are used to provide s u f f i c i e n t c o n d i t i o n s that ensure the convergence of a f i n i t e element s o l u t i o n o f a g e n e r a l system o f time independent l i n e a r d i f f e r e n t i a l equations.  Application  of the p r i n c i p l e s d e v e l o p e d i s i l l u s t r a t e d w i t h a c o n v e r g e n c e proof f o r a f i n i t e element s o l u t i o n of a non-symmetric e i g e n v a l u e p r o b l e m and by d e v e l o p i n g a c o m p u t e r p r o g r a m for the f i n i t e element a n a l y s i s of the two-dimensional s t e a d y s t a t e f l o w o f an i n c o m p r e s s i b l e v i s c o u s f l u i d . ii  T A B L E OF CONTENTS Page ABSTRACT.  i i  T A B L E OF CONTENTS  i i i  L I S T OF T A B L E S  v  L I S T OF F I G U R E S  '.  NOTATION  vi v i i  ACKNOWLEDGEMENTS  v i 1 i  CHAPTER 1.  2.  3.  INTRODUCTION  1  1 .1  Background  1  1.2  Purpose and Scope  2  1.3  Limitations .....  3  MATHEMATICAL P R E L I M I N A R I E S  4  2.1  B a s i c C o n c e p t s and D e f i n i t i o n s  4  2.2  V a r i a t i o n a l Formulation  of the Problem. ...  11  APPROXIMATE SOLUTION T E C H N I Q U E S  24  3.1  R a y l e i g h - R i t z Method  25  3.2  G a l e r k i n ' s Method  30  iii  CHAPTER 4.  Page THE F I N I T E ELEMENT PROCEDURE 4.1  Generation of a Finite  54 Element  Approximation  5.  ,  55  4.2  G e n e r a l Remarks  60  4.3  Convergence C r i t e r i a  63  A P P L I C A T I O N OF THE G A L E R K I N PROCEDURE  TO  PROBLEMS WITH MIXED AND N0NH0M0GENE0US  6.  7.  BOUNDARY CONDITIONS  73  5.1  Homogeneous Mixed Boundary C o n d i t i o n s . . . .  73  5.2  Nonhomogeneous Boundary C o n d i t i o n s  79  BOUNDARY R E S I D U A L CONCEPT AND V I R T U A L WORK . . . .  86  6.1  Boundary R e s i d u a l Concept  86  6.2  V i r t u a l Work  90  F I N I T E ELEMENT SOLUTION OF A NONSYMMETRIC PROBLEM  8.  93  A F I N I T E ELEMENT SOLUTION OF THE L I N E A R VISCOUS FLOW PROBLEM 8.1  106  Generation of the Equations Governing a F i n i t e Element S o l u t i o n  9.  8.2  Development  o f a F i n i t e E l e m e n t Model  8.3  C o m p a r i s o n w i t h Known S o l u t i o n s  106 ....  SUMMARY  114 121 133  BIBLIOGRAPHY  136  APPENDIX:  138  COMPUTER PROGRAM FOR L I N E A R VISCOUS FLOW . . . . iv  L I S T OF T A B L E S Table 7.1  Page Determinant Value Versus Aerodynamic Parameter  102  7.2  E i g e n v a l u e R e s u l t s when B = 0  7.3  Eigenvalue Coalescence Results  v  i . . . 105 105  L I S T OF FIGURES Figure  Page  8.1  D e f i n i t i o n of Area Co-ordinates  8.2  Assumed Domain S u b d i v i s i o n s  8.3  2 Element P a r a l l e l Flow S o l u t i o n .  127  8.4  8 Element P a r a l l e l Flow S o l u t i o n  128  8.5  32 E l e m e n t P a r a l l e l F l o w S o l u t i o n  129  8.6  V e l o c i t y D i s t r i b u t i o n i n Channel f o r Parallel  8.7  Flow  ....  126  130  V e l o c i t y D i s t r i b u t i o n i n Channel f o r C o u e t t e Flow  8.8  120  131  P r e s s u r e D i s t r i b u t i o n i n Channel  vi  132  NOTATION The s p e c i f i c u s u a g e and m e a n i n g o f s y m b o l s  is  d e f i n e d i n the t e x t where they are i n t r o d u c e d . The s u m m a t i o n c o n v e n t i o n h o l d s f o r s u b s c r i p t e d v a r i a b l e s w i t h r e p e a t e d lower case i n d i c e s ; i t does a p p l y t o r e p e a t e d u p p e r c a s e i n d i c e s . The r a n g e summation i s i n d i c a t e d where the v a r i a b l e s are i ntroduced.  vi i  of first  not  ACKNOWLEDGEMENTS The a u t h o r w i s h e s t o e x p r e s s  his gratitude to  h i s a d v i s o r D r . D.L. A n d e r s o n f o r h i s i n v a l u a b l e and g u i d a n c e thesis.  advice  d u r i n g the r e s e a r c h and p r e p a r a t i o n o f t h i s  T h a n k s a r e a l s o d u e t o D r . N.D. N a t h a n f o r s t i m u -  l a t i n g the author's  interest i n the f i e l d of f i n i t e  elements  and f o r h i s a s s i s t a n c e d u r i n g t h e p r e p a r a t i o n o f t h i s work. The  f i n a n c i a l support  of the National  C o u n c i l o f Canada i n the form o f a Research is g r a t e f u l l y  acknowledged.  A u g u s t 1971 Vancouver, B r i t i s h  Columbia  viii  Research  Assistantship  CHAPTER  1  INTRODUCTION' 1 .1  Background The e q u a t i o n s t h a t a r e e n c o u n t e r e d i n e n g i n e e r i n g  p r a c t i c e a r e , i n g e n e r a l , o f s u c h a n a t u r e t h a t no c l o s e d form s o l u t i o n i s a v a i l a b l e .  In o r d e r t o o b t a i n a n s w e r s  t o s u c h p r o b l e m s r e c o u r s e must be made t o a p p r o x i m a t e techniques.  solution  T h e f i n i t e e l e m e n t m e t h o d was d e v e l o p e d , on  t h e b a s i s o f p h y s i c a l i n t u i t i o n , as s u c h a t o o l f o r a p p l i c a t i o n i n the a n a l y s i s of complex s t r u c t u r a l systems. d e v e l o p m e n t o f the method  i s well documented  The  and Z i e n k i e w c z  (20) p r o v i d e s a c o m p r e h e n s i v e l i s t o f r e f e r e n c e s i n h i s review of the  method.  In r e c e n t y e a r s t h e w o r k s o f M e l o s h (8) and  Keys  (6) i n p a r t i c u l a r h a v e s e r v e d t o a s s o c i a t e t h e m e t h o d  with  the R a y l e i g h - R i t z p r o c e d u r e .  S u c h an a s s o c i a t i o n  enabled r i g o r o u s mathematical arguments  has  t o be u s e d t o  j u s t i f y t h e u s e o f f i n i t e e l e m e n t s and t o p r o v i d e s u f f i c i e n t c o n d i t i o n s that ensure convergence of the approximate 1  2 i  s o l u t i o n to the c o r r e c t one. is notable 1.2  The w o r k o f O l i v e i r a  in this respect.  P u r p o s e and  Scope  The p u r p o s e o f t h i s t h e s i s i s t o e x t e n d of previous  element procedure  framework i n which the  can most a d v a n t a g e o u s l y  be  generates  t h a t i s d e f i n e d i n terms o f assumed  co-ordinate  This approximation  may  i n c o n j u n c t i o n w i t h a number o f m e t h o d s f o r t h e  determination  o f t h e unknown p a r a m e t e r s .  The r e l a t i v e  a d v a n t a g e s o f t h e G a l e r k i n , R a y l e i g h - R i t z , and work p r o c e d u r e s judged  element  an a p p r o x i m a t e s o l u t i o n f o r a g i v e n  f u n c t i o n s and unknown p a r a m e t e r s . be u s e d  finite  considered.  I t w i l l be e s t a b l i s h e d t h a t t h e f i n i t e equation  the work  i n v e s t i g a t o r s i n t h i s f i e l d and t o d e f i n e  p r e c i s e l y the mathematical  procedure  (13)  are presented  preferable.  virtual  and t h e G a l e r k i n m e t h o d i s  Convergence r e s u l t s e s t a b l i s h e d  Mi k h1 i n ( 9 ) f o r t h e G a l e r k i n p r o c e d u r e  are reviewed  by using  f u n c t i o n a l a n a l y s i s p r i n c i p l e s , and i t i s d e m o n s t r a t e d t h a t t h e y c a n be a p p l i e d when a f i n i t e e l e m e n t m e t h o d i s used to g e n e r a t e  the a p p r o x i m a t e s o l u t i o n .  Particular  a t t e n t i o n i s p a i d t o t h e b o u n d a r y c o n d i t i o n s t h a t m u s t be s a t i s f i e d by t h e a s s u m e d c o - o r d i n a t e shown t h a t by s u i t a b l y f o r m u l a t i n g  functions. the G a l e r k i n  It is procedure  i t i s o f t e n s u f f i c i e n t to s a t i s f y only the p r i n c i p a l ones.  3 T h e c o n v e r g e n c e r e s u l t s p r e s e n t e d by M i k h l i n a r e extended to i n c l u d e problems w i t h non-homogeneous  boundary  c o n d i t i o n s a n d t h e e q u a t i o n s , and c o r r e s p o n d i n g c o n v e r g e n c e c r i t e r i a , t h a t govern a f i n i t e e l e m e n t s o l u t i o n o f a g e n e r a l system of l i n e a r  differential  equations i s presented.  This system o f e q u a t i o n s i n c l u d e s problems to which the R a y l e i g h - R i t z procedure i s not a p p l i c a b l e . A p p l i c a t i o n of the p r i n c i p l e s developed i s illustrated  with a convergence proof f o r a f i n i t e  s o l u t i o n o f a non-symmetric  element  e i g e n v a l u e p r o b l e m a n d by  d e v e l o p i n g a computer program f o r the f i n i t e  element  analysis of the two-dimensional steady state flow of an i n c o m p r e s s i b l e v i s c o u s f l u i d . 1.3  L i m i t a t i ons A t t e n t i o n w i l l be c o n f i n e d t o t h e c o n s i d e r a t i o n  o f p r o b l e m s t h a t a r e c h a r a c t e r i z e d by t i m e i n d e p e n d e n t linear  differential  equations.  4  CHAPTER  2  MATHEMATICAL P R E L I M I N A R I E S The s u b s e q u e n t d e f i n i t i o n s a n d l a t e r p r o o f s a r e b a s e d u p o n t h o s e p r e s e n t e d by Mi kh1 i n ( 9 , 1 0 ) a n d f u r t h e r m e n t i o n o f t h e s e r e f e r e n c e s w i l l be o m i t t e d . 2.1  B a s i c C o n c e p t s and D e f i n i t i o n s In a t t e m p t i n g t o p r e d i c t t h e b e h a v i o u r o f a  p h y s i c a l s y s t e m by means o f m a t h e m a t i c a l a n a l y s i s i t i s n e c e s s a r y t o i d e a l i z e t h e system i n a manner t h a t r e n d e r s the a n a l y s i s t r a c t a b l e .  T h i s i d e a l i z a t i o n i s known as t h e  m a t h e m a t i c a l m o d e l o f t h e s y s t e m a n d i n many c a s e s i t i s a differential equation.  The f i n i t e element method w i l l  be p r e s e n t e d as a means o f g e n e r a t i n g an a p p r o x i m a t e  solu-  t i o n f o r m t o s u c h an e q u a t i o n a n d t h u s t o t h e p h y s i c a l system.  In o r d e r t h a t t h e f i n i t e e l e m e n t p r o c e d u r e may  be u t i l i z e d t o i t s f u l l p o t e n t i a l a c o m p l e t e u n d e r s t a n d i n g of i t smathematical b a s i s i s d e s i r a b l e .  With t h i s i n mind  a number o f r e l e v a n t m a t h e m a t i c a l c o n c e p t s a n d d e f i n i t i o n s w i l l be i n t r o d u c e d .  5 The p r o b l e m a t h a n d i s t h e d e t e r m i n a t i o n  of  some f u n c t i o n t h a t s a t i s f i e s a g i v e n d i f f e r e n t i a l  equation,  w i t h i n some r e g i o n , a n d c e r t a i n c o n d i t i o n s on t h e b o u n d a r y of that r e g i o n .  T h e r e g i o n w i l l be a s u r f a c e i f t h e  f u n c t i o n s o u g h t d e p e n d s upon two i n d e p e n d e n t o r a v o l u m e i f t h e number o f i n d e p e n d e n t  variables  variables i s three.  I f t h e f u n c t i o n d e p e n d s upon f o u r o r more v a r i a b l e s the domain i n which the f u n c t i o n i s d e f i n e d i s a  then  hyperspace.  The c o n c e p t • o f r e g i o n , o r d o m a i n , c a n be f o r m a l i z e d by t h e f o l l o w i n g two p r o p e r t i e s : (i)  I f some p o i n t P b e l o n g s  to the domain^  then a l l p o i n t s s u f f i c i e n t l y c l o s e to P belong  to the  d o m a i n; (ii)  A n y two p o i n t s i n t h e d o m a i n c a n be j o i n e d  by a l i n e l y i n g e n t i r e l y w i t h i n t h e d o m a i n . The f i r s t p r o p e r t y i s e q u i v a l e n t t o s a y i n g t h e d o m a i n i s open, o r c o n s i s t s o f o n l y i n t e r i o r p o i n t s ; and t h e second property specifies  t h e d o m a i n be c o n n e c t e d .  The boundary  o f t h e d o m a i n i s d e f i n e d as t h a t s e t o f p o i n t s i n a n y neighbourhood o f which there a r e both p o i n t s belonging and n o t b e l o n g i n g fined  to the domain.  to  A t t e n t i o n w i l l be c o n -  t o those problems where the curve o r s u r f a c e  forming  the b o u n d a r y i s e i t h e r smooth i . e . i t has a c o n t i n u o u s l y turning tangent  or tangent  plane, or i s piecewise  smooth,  6 i . e . i t c o n s i s t s o f a f i n i t e number o f s m o o t h p i e c e s . a n d t h e b o u n d a r y by 5  d o m a i n w i l l be d e n o t e d by Si  t h a t t h e domain does not i n c l u d e t h e boundary. of points that are obtained  .  .  Note  The s e t  by c o m b i n i n g t h e d o m a i n a n d  i t s b o u n d a r y i s known as t h e c l o s e d d o m a i n a n d w i l l d e n o t e d by SI  The  O n l y f i n i t e d o m a i n s w i l l be  i . e . d o m a i n s t h a t c a n be i n c l u d e d  be  considered,  in a sufficiently  large  sphere. The  s o l u t i o n of the given equation  is  accomplished  by f i n d i n g t h a t f u n c t i o n w h i c h when a c t e d upon by a g i v e n o p e r a t o r y i e l d s a known f u n c t i o n . r e s t r i c t e d to thoses  Attention will  be  f u n c t i o n s t h a t a r e s q u a r e summable  over the domain i . e . to those  functions  w h e r e k" i s a f i n i t e c o n s t a n t  and Lebesgue i n t e g r a t i o n  i s e m p l o y e d , and t h e r e p e a t e d  lower case  summed.  The f u n c t i o n s c o n s i d e r e d  vector valued  and U  with components  u,, U , 4  u  c  L  such  that  indices are  w i l l , i n g e n e r a l , be  thus represents  L  Lt  a column .  vector  Thus  The c l a s s o f s q u a r e s u m m a b l e f u n c t i o n s o v e r SI , w h i c h w i l l be d e n o t e d by  L^si)  , c o n s t i t u t e s a vector space  over  7 the f i e l d o f r e a l numbers. Thus i f of  L- (si)  and  2  C  ( V  +•  L  c  UJ  i s a r e a l number, )  L  =  c  v  .  +  and t h e v e c t o r w i t h c o m p o n e n t s L  member o f  a n d u;^ a r e members  c  c ( +  (52.) . T h e n o t a t i o n  2  be e m p l o y e d t o mean  then ^  u>i ) u  L  i s also a  € L  s  (si)  i s a member o f t h e s e t L  u-  will (si)  e  .  When d i s c u s s i n g a p p r o x i m a t e s o l u t i o n o f a g i v e n equation  i t becomes n e c e s s a r y  mate a n s w e r s .  t o compare d i f f e r e n t a p p r o x i -  I t i s thus necessary  the " d i s t a n c e " between f u n c t i o n s . can be o b t a i n e d L  2  (si)  t o be a b l e t o m e a s u r e The s t r u c t u r e f o r t h i s  by i n t r o d u c i n g t h e c o n c e p t  • This i s accomplished  inner product  into the space.  functions  , ]/  is  L  by f i r s t i n t r o d u c i n g an  An i n n e r p r o d u c t  function  a  o f a norm i n t o  (Ui  ,  )  o f two defined  such that (i)  (Ui',  (ii)  ( Ui  (iii)  V  ,  [U-i U-i  L  )  bv£  , Ui) =  =  +  (V  c u)  )  L  ^  O  ,  L  =  U) L  b ( u c , V i )  L,  C  where the e q u a l i t y only holds i f  O  For f u n c t i o n s i n  +C(U. ,VL)  f  define  8  (2.1)  52  The norm o f a f u n c t i o n i s any f u n c t i o n  || Ui f|  satisfying  the axioms (i)  >/ O  , where the equal i t y o n l y holds i f  ( i i ) '/[ Ui + Vi [\ £ ( i i i ) jj c Ui jl  -  +  Ui  = O  fli/c/l  \c\\\uif\ , w h e r e  c is a constant  I t c a n r e a d i l y be s e e n t h a t d e f i n i n g (2.2) s a t i s f i e s these axioms. may  The d i s t a n c e b e t w e e n two f u n c t i o n s  t h e n be c h a r a c t e r i z e d by t h e norm o f t h e i r d i f f e r e n c e ,  i.e.  || Ui ~ Vi If  La  (SI)  .  A complete  v e c t o r space, such  as  , w i t h an a s s o c i a t e d i n n e r p r o d u c t i s known as  a H i l b e r t space.  It i s p o s s i b l e to d e f i n e a l t e r n a t i v e i n n e r  p r o d u c t s and c o r r e s p o n d i n g norms t o t h e o n e s p r e s e n t e d , as w i l l be d i s c u s s e d i n s u b s e q u e n t  chapters.  C o n s i d e r now more s p e c i f i c a l l y t h e t y p e o f e q u a t i o n s t h a t w i l l be c o n s i d e r e d .  A t t e n t i o n w i l l be p r i m a r i l y  c o n f i n e d t o e q u a t i o n s t h a t c a n be e x p r e s s e d i n t h e £  SI  form (2.3)  )  w h e r e to  a r e members o f L (si) a n d l  a n d ..  L  L  A-  known.  fi t  2  is  i s a l i n e a r d i f f e r e n t i a l operator o r matrix  L  J  T h e l i n e a r i t y o f A^-  of such o p e r a t o r s .  implies (2.4)  (ii) < A , A ,--- A ><c, ... c u >=, c.Ai.a^.... +c A a u  l2  where t h e  C  l(  52  t  t  c  are constants.  Together within  U|r  with reducing  E q . 2.3 t o a n i d e n t i t y  , o-> w i l l a l s o b e r e q u i r e d t o s a t i s f y c e r t a i n  boundary c o n d i t i o n s .  Thec l a s s o f functions that 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 the problem and possess t h e r e q u i r e d c o n t i n u i t y p r o p e r t i e s t o make t h e e v a l u a t i o n o f A ij Uj definition of if  A-  p o s s i b l e i s known a s t h e f i e l d o f A^-  and i s denoted by  were a d i f f e r e n t i a l o p e r a t o r o f t h e f o u r t h  order then f u n c t i o n s i n D  ft  m u s t be c o n t i n u o u s  o r d e r d e r i v a t i v e s a t e v e r y p o i n t i n 52 A •• J  . For example,  w i l l be c o n s i d e r e d  . In general,  d e f i n e d f o r a d e n s e s e t o f some  H i l b e r t space H . A s e t M i f every element i n H  fourth  i s s a i d t o be d e n s e i n  can be o b t a i n e d  a sequence o f f u n c t i o n s from  M  H  as the l i m i t o f  u  e  10 The f o l l o w i n g t y p e s o f o p e r a t o r s w i l l p l a y an important r o l e i n the subsequent  discussions.  An o p e r a t o r  i f ( A i j U j , \/t) = (Acj V j , u-,) f o r a l l u  (i)  symmetric,  (ii)  p o s i t i v e d e f i n i t e , i f ( A g Uj , U; ) ^ 0 f o r a l l Uj, £ D  £ 1  ^fiD ft  w h e r e t h e e q u a l i t y s i g n h o l d s o n l y i f u. = 0 (iii)  p o s i t i v e bounded below, i f ( for a l l u  c  D  The c l a s s o f pletely continuous  y  where  A  , u- ) ^ y  2  L  ( Uj (ij ) 7  i s a positive constant.  o p e r a t o r s t h a t a r e known a s com-  o p e r a t o r s w i l l a l s o be o f i m p o r t a n c e i n  the f o l l o w i n g p r e s e n t a t i o n , and t h e r e f o r e a d e f i n i t i o n o f such  o p e r a t o r s w i l l be g i v e n .  i n some H i l b e r t s p a c e  An o p e r a t o r T' , d e f i n e d  H , i s s a i d t o be d e g e n e r a t e  if it  c a n be r e p r e s e n t e d i n t h e f o r m  where N i  "T j t  i s f i n i t e and both  ^  , cj)^ £ H  i s then completely continuous  .  An o p e r a t o r  i f f o r any e y o  i t can  be r e p r e s e n t e d i n t h e f o r m  where  i s degenerate  (/IT": Uj II < * /|Ui /I) S  .  a n d t h e norm o f T t j i s l e s s t h a n  6  A  T o g e t h e r w i t h Eq. 2.3 t h e p r o b l e m o f d e t e r m i n i n g t h e e i g e n v a l u e s and e i g e n v e c t o r s o f t h e o p e r a t o r A;.- w i l l J  a l s o be c o n s i d e r e d .  That i s , the s o l u t i o n of the  equation  (2  A j W j - w< w i l l be d i s c u s s e d . parameter  In t h i s e q u a t i o n  )\  i s a numerical  and t h e s o l u t i o n o f t h e e q u a t i o n e n t a i l s  d e t e r m i n a t i o n of those values of  A  say  )\  the  f o r which  t h e r e e x i s t c o r r e s p o n d i n g non t r i v i a l s o l u t i o n s f o r say  .  Such  )\ u[  corresponding  a r e c a l l e d t h e e i g e n v a l u e s and  t h e e i g e n v e c t o r s o f t h e o p e r a t o r A--  In t h e f o l l o w i n g d e v e l o p m e n t s  t h e use o f  s c r i p t e d v a r i a b l e s w i l l be a b a n d o n e d , e x c e p t w h e r e sary for  the .  subneces-  c l a r i f i c a t i o n , and a l l t h e f u n c t i o n s e m p l o y e d  a s s u m e d t o be v e c t o r v a l u e d .  T h u s f o r e x a m p l e Eq. 2.3  will  be w r i t t e n  A 2.2  w  = f  V a r i a t i o n a l F o r m u l a t i o n of the The p u r p o s e  Problem  of t h i s s e c t i o n i s to i n t r o d u c e  concepts t h a t are n e c e s s a r y f o r the l a t e r development of the G a l e r k i n procedure  and a t t h e same t i m e t o d i s c u s s t h e  range of a p p l i c a b i l i t y of the R a y ! e i g h - R i t z procedure.  In  12  o r d e r t o do t h i s t h e c o n d i t i o n s u n d e r w h i c h t h e s o l u t i o n of a given d i f f e r e n t i a l equation c o i n c i d e s with the f u n c t i o n that minimizes  a known f u n c t i o n a l w i l l be d i s c u s s e d .  It.  w i l l f u r t h e r be shown t h a t i t i s p o s s i b l e t o o b t a i n a s o l u t i o n v i a t h e v a r i a t i o n a l f o r m u l a t i o n e v e n when no solution of the o r i g i n a l equation exists i n D  A  .  Consider the equation  A  0 —  UJ  w h e r e u> a n d | tu  (2.6)  SI  €  a r e members o f some H i l b e r t s p a c e  H  , and  i s p r e s c r i b e d t o s a t i s f y c e r t a i n homogeneous boundary  condi t i ons. Let A  Theorem 2.1.  be a s y m m e t r i c p o s i t i v e b o u n d e d b e l o w  o p e r a t o r d e f i n e d f o r some d e n s e l i n e a r s e t D  ft  of H  I f E q . 2.6 h a s a s o l u t i o n ( i n D ) t h e n t h i s s o l u t i o n m i n i ft  mizes the f u n c t i o n a l F(u) Conversely,  =  ( Au,u)  -  <?(u,| )  i f there exists i n D  A  a £ D  a f u n c t i o n which mini-  m i z e s F(u) t h e n t h i s f u n c t i o n s a t i s f i e s E q . 2.6. proof:-  R  (2.7)  13 Assume t h a t  u , g e D  fl  and s e t  u - U J =•  Thus  u = w + rj  w h e r e uj i s t h e s o l u t i o n o f E q . 2.6.  = Ffu,) But  A U J - ^ - O  F( J  =  U  Now  A  4 2 ( A * - / , 5) + ( A } . ) ) by h y p o t h e s i s , h e n c e  j)  (Aj,  +  F H  i s p o s i t i v e bounded below  F(u)  Thus  ^  F(u>)  Hence t h e f u n c t i o n  ,  whence  with the e q u a l i t y only v a l i d a t t a i n s i t s minimum  ifj=o.  v a l u e when u. »  w  C o n v e r s e l y , assume t h e f u n c t i o n a l a t t a i n s i t s minimum where  v a l u e when u, - cu . A  is a constant.  F ( w + > rj)  %  L e t rj e. "D  ft  T h e n by F  and t h u s uu+-^rj d D  hypothesis  (w)  whi ch r e d u c e s t o  T h e l e f t - h a n d s i d e i s a non n e g a t i v e q u a d r a t i c for the real parameter  A  .  Thus  function  ft  14 But  fj  i s an a r b i t r a r y f u n c t i o n f r o m t h e d e n s e s e t  and t h e o n l y f u n c t i o n o r t h o g o n a l the zero f u n c t i o n .  A *  to a l l such functions i s  Therefore  =  T h a t UJ  |  i s the only function that s a t i s f i e s  Eq. 2.6 c a n be s e e n by a s s u m i n g t h a t u j  A ( cu -  T h i s impl i e s The  ) = O whence  oJ  0  p o s i t i v e bounded below nature  Thus  w = w B  i s also a solution.  0  ( A (  of  UJ-I*J ), 0  w  w  0  = o  situation consider  ^ 6 H  there  i n the f i e l d of d e f i n i t i o n  t h a t w i l l s a t i s f y E q . 2.6.  of a uniform  )= o  .  does n o t e x i s t a f u n c t i o n A  UJ-UJ*  then r e q u i r e s  A  I t c a n o c c u r t h a t f o r some f u n c t i o n s of  ,  As an e x a m p l e o f t h i s  the problem of p r e d i c t i n g the d e f l e c t i o n  c a n t i l e v e r e d beam u n d e r t h e a c t i o n o f d i s -  t r i b u t e d l o a d <j(x)  .  The e q u a t i o n s  characterizing the  d e f l e c t i o n o f t h e beam a r e  £1  =  U  m  Lflo) =  Lf (0) l  ^X)  -  The d i f f e r e n t i a l e q u a t i o n  U»(L)  X  6  [O.L]  = ^'"(L) =  ( 2  i s d e r i v e d by c o n s i d e r i n g t h e  e q u i l i b r i u m o f a n . i n f i n i t e s i m a l l e n g t h o f t h e beam u n d e r s e c t i on.  )  (2.8b)  0  the assumption t h a t t h e l o a d i n g i s continuous  8 a  across  such a  15 In t h i s c a s e t h e o p e r a t o r  =  A  and  EI  j*  (2-9)  i t s f i e l d of d e f i n i t i o n  functions  is  D  i s the t o t a l i t y of  ft  d e f i n e d o v e r [ o, L ]  with continuous  v a t i v e s t h a t s a t i s f y the boundary c o n d i t i o n s T h u s i f c^(x) i s c o n t i n u o u s D  b u t i f <^(x)  A  found in D  .  A  is discontinuous  of the  manner t h a t a g e n e r a l i z e d J u s t as a d i s c o n t i n u o u s  deriproblem.  no s o l u t i o n c a n  be  T h i s d i f f i c u l t y c a n be o v e r c o m e by  p o s s i b l e to formulate  that l i e in  .and  the f u n c t i o n a l  F(u)  s o l u t i o n o f Eq. 2.6 l o a d may  l i m i t of a sequence of continuous discontinuous  fourth  then there e x i s t s a s o l u t i o n i n  s i d e r i n g l i m i t s of f u n c t i o n s then  those  con-  i t is in such a  is  be c o n s i d e r e d  obtained.  as  the  l o a d s , so f u n c t i o n s  f o u r t h d e r i v a t i v e s are introduced  that  are  the l i m i t s o f s e q u e n c e s o f f u n c t i o n s w i t h c o n t i n u o u s derivatives. new  I t c a n t h e n be a s s e r t e d  s e t of f u n c t i o n s  s o l u t i o n i f i t i s not i n D For the example c o n s i d e r e d s q u a r e summable o v e r these  ideas  ) o f Eq. 2.6  fl  H  [ o,L]  i s taken .  The  fourth  t h a t amongst  l i e s the s o l u t i o n (or  with  the  generalized  f o r any j £ H 1  as t h e s e t o f  formal  functions  development of  follows.  A new energy product,  inner (or s c a l a r ) product,  called  is introduced  D  f o r the s e t  ft  .  the Recalling  16  that the operator A  i s symmetric  i t i spossible, using  i n t e g r a t i o n by p a r t s , t o w r i t e ( A u , v  B  in which  =  )  f (Bu)(  i sa differential  iei ,  bv)  operator.  u , / fe D  A  The energy  p r o d u c t , w h i c h w i l l be d e n o t e d b y s q u a r e b r a c k e t s , i s t h e n d e f i n e d as [u,v] The  A  =  J  (Bu)(Bv)dsi  u,/feD  ?  (2.10)  A  si  e n e r g y norm, w h i c h i s d e n o t e d by b o l d v e r t i c a l  lines,  is d e f i n e d as  I ui - n*> \  -  a  <2  A  n)  The e n e r g y p r o d u c t a n d e n e r g y norm s a t i s f y t h e a x i o m s d e f i n i n g an i n n e r p r o d u c t a n d a norm p r e s e n t e d i n t h e p r e vious  section. I t may b e t h a t t h e s p a c e  i s incomplete with  r e s p e c t t o t h e e n e r g y norm i . e . n o t a l l C a u c h y in then  converge to a f u n c t i o n i n D Dft  i s completed by d e f i n i n g  A  sequences  . I fthis i s so  u t o b e a member o f  the s p a c e i f I  - U I  A  —>  O  as  n ->  a>  (2.12)  17 where  u  /u„}  i s a t y p i c a l member o f a s e q u e n c e  f t  member o f w h i c h i s i n D  .  A  each  The c o m p l e t e d s p a c e so  o b t a i n e d i s a H i l b e r t s p a c e and i s d e n o t e d by H  to  A  emphasize  i t s dependence  A  upon  .  The e n e r g y p r o d u c t i n  Eq. 2.10 i s o n l y d e f i n e d f o r f u n c t i o n s i n D  A  b u t may i n  ah o b v i o u s f a s h i o n be d e f i n e d f o r a l l f u n c t i o n s i n H  :  fl  ( B tin) (BVn) fi  —>  ,  A*-  U ,V n  n  fe D  (2-13)  ft  00  SI T h u s t h e e n e r g y p r o d u c t a n d e n e r g y norm h a v e m e a n i n g f o r any f u n c t i o n i n H  •  A  Their d e f i n i t i o n ensures that they  s a t i s f y a l l t h e r e q u i r e d p r o p e r t i e s o f i n n e r p r o d u c t s and norms. The f i e l d o f d e f i n i t i o n o f t h e f u n c t i o n a l F ( u ) o f Eq. 2.7 c a n now be e x t e n d e d f r o m 2.1  D  A  to  H  and Theorem  A  becomes:  Theorem  2.1A.  If A  i s a symmetric  p o s i t i v e bounded  operator, then o f a l l of the functions i n  H  A  below  t h e one t h a t  minimizes the f u n c t i o n a l . F M  =  K  "3  A  i s t h e s o l u t i o n o f E q . 2.6.  -  ,  «• £ H  A  (2-14)  18 proof: T h e o r e m 2.1 d e m o n s t r a t e s t h a t i f E q . 2.6 h a s a solution F(u)  tu  in  D  this solution uniquely  ft  i n the class of functions c o n s t i t u t i n g the f i e l d of  definition of A F(u)  .  I t w i l l be shown t h a t t h e m i n i m u m o f  i n the wider class  function  UJ  and by ~d  H  i s n o t a l t e r e d and t h a t t h e  A  o n l y g i v e s t h e minimum  D e n o t e by cl  minimizes  value.  t h e minimum v a l u e o f f ( " )  d  t h e minimum i n H  .  A  T h e n as H  in  includes  A  D D  A  A  d  4  Assume d fc H  fl  [ u, B u t as  cl < d  such that  F (u) <  d  , i.e.  (u,  Cl € H  ft  { n\ u  i t - f o l l o w s that there e x i s t s a sequence such t h a t  6  which a l s o implies bounded below.  Then there e x i s t s a f u n c t i o n  2 f) = U l ' - 2fu, /) < &  a] -  of functions  .  II U - Cl II  o as A  R  |U I  Thus  | il I  ft  |u  Therefore f o r s u f f i c i e n t l y large  n  and  a  - a I -» o is positive ( U , -j?)  d i c t i o n shows  as  U  n  ( u , f)  a  F ( u ) and F ( u ) n  by an a r b i t r a r y s m a l l a m o u n t a n d i t f o l l o w s t h a t This however i s impossible  ,  e D  A  .  differ  F(u ) ^ n  ^  The c o n t r a -  cl = d  To show t h a t t h e m i n i m u m o f t h e f u n c t i o n a l i s unique assume t h a t  tu fc. H  ft  also minimizes the f u n c t i o n a l .  19 (^UJ  From t h e p r o o f o f T h e o r e m 2.1 function  rj e. D  ( f i ?  =  ij = UJ  In p a r t i c u l a r s e t t i n g [ w, w ]  fjn} C  L  y  £  D  =  (  A  such t h a t  iQn.J=  (-^i  Qrv)  ( ^, a;)  | j _ j) J  fj = CD, UJ  )  functions and  a  t o t h e l i m i t g i v e s Eq.  ( / ,  *  H  2.15  . Thus  A  )  (  with  2  '  1  7  )  -F-(u-)  t h e i d e n t i t y C uJ, yl = ( j n) i s  a s i n Eq. 2.14  o b t a i n e d where  5  ft  By r e p e a t i n g t h e p r o o f o f T h e o r e m 2.1 expressed  1  H . In f a c t  which i s the v a l i d f o r a r b i t r a r y f u n c t i o n s i n =  >  (2.16)  o , II rj - g I I o  • Pi^eceeding  [>,*>]  2  gives  then there e x i s t s a sequence of  A  any  )  i s a l s o v a l i d f o r any f u n c t i o n i n  /j £ H  if  for  . T h i s r e l a t i o n may b e w r i t t e n  A  [ > , n ]  Eq. 2.15  - |, i j ] = o  7  rj i s a n a r b i t r a r y f u n c t i o n i n H  A  .  Putting  gives [ CD , CD ]  =  ( -j , &)  (2.18)  1  (2.19) S u b t r a c t i n g Eq. 2.18 2.16  gives  from  Eq. 2.17  and Eq. 2.19  f r o m Eq  20  [  UJ -  UJ  ,  UJ  ] =  F i n a l l y s u b t r a c t i n g these whence  O  ;  [ w - u>,  two e q u a t i o n s  UJ ]  gives  =  0  L"UJ-U3,  U-W]  - o  u) = u; I f t h e minimum o f t h e f u n c t i o n a l e x p r e s s e d i n  Eq. 2.14 i s g i v e n b y a f u n c t i o n t h a t i s n o t i n t h i s f u n c t i o n i s known a s a g e n e r a l i z e d  s o l u t i o n o f Eq. 2.6.  As a n i l l u s t r a t i o n o f t h e s e c o n c e p t s again  , then consider  t h e p r o b l e m o f t h e b e n d i n g o f a beam d e f i n e d b y  Eqs. 2.8.  (Aui.v)  =  v £1 u  tut w  At  o  ~  +  (vElu'-v'EW'J  f f l u V A  Thus u" The o p e r a t o r  £1 /"  (2.20)  ix  i s thus symmetric and i s i n f a c t a l s o p o s i t i v e  bounded below.  Thus t h e f u n c t i o n a l  F(u) i s U  q Jt  (2.21)  Jo  which i s twice t h e p o t e n t i a l energy o f the system. case f u n c t i o n s , U , i n H  ft  a r e d e f i n e d such  that  In t h i s  21 f  )  (CL"-  £ D  in which the  C(X  2  as  O  -5>  M  —>  co  and t h e r e f o r e have c o n t i n u o u s  A  fourth  d e r i v a t i v e s 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 o f t h e problem. HL  S u c h a d e f i n i t i o n means t h a t t h e f u n c t i o n s i n  have g e n e r a l i z e d  second d e r i v a t i v e s which i n this  case  Pi  i m p l i e s t h a t they have c o n t i n u o u s  first derivatives.  f u n c t i o n s m u s t t h e r e f o r e s a t i s f y t h e same b o u n d a r y  These condi-  tions involving the f i r s t derivative of the function or the f u n c t i o n i t s e l f as t h e f u n c t i o n s does not imply  . The d e f i n i t i o n  t h a t t h e f u n c t i o n s must s a t i s f y  boundary conditions  those  i n v o l v i n g second or t h i r d d e r i v a t i v e s .  T h e o r e m 2.1A s t a t e s t h a t t h e f u n c t i o n t h a t mizes the f u n c t i o n a l given  i n E q . 2.14 i s t h e s o l u t i o n o f  Eq. 2.6 a n d t h a t t h i s f u n c t i o n c a n be f o u n d u n i q u e l y the elements to i n t r o d u c e in  H  A  of  H  .  ft  In t h i s c o n t e x t  the concept  i t is  amonst  convenient  of a complete s e t of functions  . £ 4k1  A s e t of functions s a i d t o be c o m p l e t e i n norm) i f f o r e v e r y M  mini-  and c o n s t a n t s  H  =  1S  (with respect to the energy  ft  V 6 H  ft  a,,  ^  £ yo  and a  M  t h e r e i s an i n t e g e r  such that  22 In o t h e r w o r d s , any f u n c t i o n i n  H  c a n be  A  approximated  a r b i t r a r i l y c l o s e l y , i n e n e r g y n o r m , by a l i n e a r t i o n o f members o f a c o m p l e t e s e t i n H Consider operator product  the g e n e r a l  .  A  case where the  i s o f o r d e r 2. tn .  A  Expressing  i n i t s symmetric form would then  t i v e s o f maximum o r d e r m  and f u n c t i o n s  combina-  symmetric  the  energy  involve in  H  derivaw o u l d be  A  l i m i t i n e n e r g y o f f u n c t i o n s w i t h <?m  derivatives  s a t i s f y a l l the boundary c o n d i t i o n s .  Such f u n c t i o n s  m ih  generalized  that  t h a t do n o t i n v o l v e d e r i v a t i v e s  o r d e r and h i g h e r .  The b o u n d a r y c o n d i t i o n s  i n v o l v e d e r i v a t i v e s of order g r e a t e r than or equal known as n a t u r a l  possess  o r d e r d e r i v a t i v e s and m u s t s a t i s f y a l l  those boundary c o n d i t i o n s the m  the  for  A  .  The r e m a i n i n g  boundary  that t o m.  are  conditions  are c a l l e d the f o r c e d or p r i n c i p a l boundary c o n d i t i o n s . known as p r i n c i p a l d e r i v a t i v e s .  Thus f u n c t i o n s  s a r i l y s a t i s f y the f o r c e d boundary c o n d i t i o n s s a t i s f y the n a t u r a l . techniques  In  t h o s e d e r i v a t i v e s o f o r d e r l e s s t h a n m.  an e q u i v a l e n t way  when c h o o s i n g  of  T h i s i s an i m p o r t a n t  in  are  neces-  but need  not  consideration  t r i a l s o l u t i o n s f o r the approximate s o l u t i o n  t h a t w i l l be d i s c u s s e d The  eigenvalue  i n the next  chapter.  problem that is represented  by  the  equation /\  uj  =  )\ uJ  (. 2  23 c a n a l s o be e x p r e s s e d i n a v a r i a t i o n a l m a n n e r i f A i s s y m m e t r i c and p o s i t i v e bounded below.  The d e v e l o p m e n t s  a r e p r e s e n t e d by M i k h l i n ( 9 ) a n d w i l l n o t be r e p e a t e d h e r e as t h o s e c o n c e p t s n e c e s s a r y f o r t h e f u r t h e r d e v e l o p ment o f t h i s t h e s i s h a v e a l r e a d y b e e n i n t r o d u c e d i n t h e preceeding discussion.  CHAPTER  3  APPROXIMATE SOLUTION  TECHNIQUES  The p u r p o s e o f t h i s c h a p t e r i s t o p r e s e n t m e t h o d s t h a t c a n be u s e d 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 f o r E q . A g a i n i t i s c o n v e n i e n t t o assume t h a t A  2.6. ential are  operator of order 2m  homogeneous.  is a differ-  and t h a t t h e boundary c o n d i t i o n s  The d e v e l o p m e n t s  presented herein follow  t h o s e g i v e n by M i k h l i n ( 9 ) . Many a p p r o x i m a t e m e t h o d s a r e b a s e d upon t h e c o n cept o f assuming a s o l u t i o n i n the form ten  in which the  a r e unknown p a r a m e t e r s a n d t h e c j ^ a r e  known c o - o r d i n a t e f u n c t i o n s . T h i s f o r m i s v a l i d i f u;  is  a single function.  u>  H o w e v e r , as i n d i c a t e d i n C h a p t e r 2  may be c o n s i d e r e d a s a v e c t o r q u a n t i t y w i t h more t h a n o n e component.  In t h i s c a s e an a p p r o x i m a t i o n o f t h e f o r m o f  E q . 3.1 m u s t be a s s u m e d f o r e a c h c o m p o n e n t o f i n g e n e r a l , i f u; by  oJ  L  LP  .  Thus,  i s a v e c t o r q u a n t i t y i t w i l l be d e n o t e d  where  _ UJ •  =-  <  ai, ,  ,  OJ  C  T  >  and w h e r e e a c h c o m p o n e n t i s t h e n a p p r o x i m a t e d by  24  25  Once a g a i n  i n the i n t e r e s t s of a l g e b r a i c s i m p l i c i t y  f o l l o w i n g d e v e l o p m e n t s w i l l be i n t e r m s o f Eq.  the  3.1.  T r e a t m e n t o f such a p p r o x i m a t e methods can found i n the works of C r a n d a l l Scriven  (4).  Attention  Galerkin.  3.1  Rayleigh-Ritz The  the e q u a t i o n  Finylason  i n t h i s t h e s i s w i l l be  t o t h e d i s c u s s i o n o f two and  (2) and  r e l a t e d methods :  and confined  Rayleigh-Ritz  Method  R a y l e i gh-^Ri t z m e t h o d i s a p p l i c a b l e o n l y i f t o be s o l v e d has a s o l u t i o n t h a t c o r r e s p o n d s  t h e s t a t i o n a r y v a l u e o f some k n o w n • f u n c t i o n a l . then  c a l c u l a t e s t h e unknown  a p p r o x i m a t e s o l u t i o n UJ s t a t i o n a r y i n the M co-ordinate  functions  renders tj^  The  the g i v e n  states that i f A  F(u)  The  H  is symmetric  which m i n i m i z e s the  A  =  Rayleigh-Ritz  a p p r o x i m a t e s o l u t i o n uj F(u5)  £ u,  u]  A  £  -  functional  (a,|)  i n t o t h e f u n c t i o n a l and <Xu  and  is  p r o c e d u r e i s to s u b s t i t u t e  with r e s p e c t to the  the  .  p o s i t i v e b o u n d e d b e l o w t h e n t h e s o l u t i o n o f Eq. 2.6 that function in  the  functional  s u b s p a c e s p a n n e d by  fc=l,....M  to  method  i n such a manner t h a t  dimensional  T h e o r e m 2.1A  minimize  be  .  Hence  then  the  26  Now,  i f the  ^  are l i n e a r l y  independent  F(u>)  is stationary  when  =  111*) 3a  -  0  I,....  "  k  Thus M  J"  M  ^  Using the symmetry o f the energy p r o d u c t g i v e s  w h i c h i s a s y s t e m o f l i n e a r e q u a t i o n s f o r t h e QJ, w h i c h a u n i q u e s o l u t i o n as t h e ^ linearly  have been assumed  t o be  independent. If the s e t of c o - o r d i n a t e f u n c t i o n s  the  d?£  has  are l i n e a r l y independent, is complete in H  > where A  the  a p p r o x i m a t e s o l u t i o n o b t a i n e d by t h e R a y l e i g h - R i t z m e t h o d c a n be made a r b i t r a r i l y c l o s e i n e n e r g y norm t o t h e e x a c t s o l u t i o n by i n c r e a s i n g N  sufficiently.  proof: The m i n i m u m v a l u e o f F ( u ) U =  OJ  , the s o l u t i o n of  /\u)=-^  .  i s o b t a i n e d when  27  F(w) - [w,^ ] - 2 L e t fil = - | IU |  2  AUJ)  which i s the exact . Hence i f f  Flu)  functional  (UJ,  =  M  2  lower bound o f the  i s an A r b i t r a r y  p o s i t i v e number, then t h e r e e x i s t s i n H  ft  such  -  small  a function  v  that  J  ^ F(v)  Further  ^  cl4  f^*}  as the s e t  energy i t i s possible  Vz i s  c o m  P  1 e t e  i n  t o show  Flv) - Flv)  <  £/<L  where To s e e t h i s n o t e  Flu) = [ u , u ] - e (u,/j = [u-uJ,  =. | a- UJ  - I  |  UJ \  Whence F(v)  - F ( v)  The t r i a n g l e i n e q u a l i t y  =  |/-UJ|  gives  2  -  |v-u)|  u-u;]  - Qw.cuJ  28 Thus  -  F(?)  ^  F(V)  (I  + I y-vX)  V-UJ\  By t h e c o m p l e t e n e s s o f t h e s e t fcj> | fe  such  M  |  v-  / |  c a n be c h o s e n  that | V  where c  - V  £/  I <  c  i s t o be c h o s e n  and Choose  c such that («?  |/|  + 2 |u>» +  r  / c ) '/c  '/a  <  Thus  J «  F(?)  ^  F (v) +•  L e t u3 be a f u n c t i o n c o n s t r u c t e d  <  d + £  by t h e R a y ! e i g h - R i t z  method.• Then c/4 Letting Thus  F(w)  4  FlW  £-> o implies  OR  f [&)  Fie) d ** - I u> |  4  d+ *  29  Therefore | U> - UJ I —> A  0  as  Thus the R a y l e i g h - R i t z  M-?oo  (3  approximation  converges,  in the s e n s e of the e n e r g y norm, to the e x a c t s o l u t i o n i f the c o - o r d i n a t e  f u n c t i o n s are complete in  H  •  A  In p r a c t i c e i t i s n o t s t r i c t l y n e c e s s a r y co-ordinate  f u n c t i o n s t o be c o m p l e t e i n  H  ft  .  f o r the  It is  s u f f i c i e n t t h a t t h e y be c o m p l e t e w i t h r e s p e c t t o  any  subset of H  In  that contains  ft  respect recall  the e x a c t s o l u t i o n .  this  t h a t c o n s i d e r a t i o n i n t h i s t h e s i s has  been  r e s t r i c t e d to those f u n c t i o n s t h a t are s q u a r e summable. S p e c i f i c a l l y , t h e r i g h t h a n d s i d e o f Eq. 2.6 m u s t be a function.  Thus (AUJ)  If  A  Eq. 3.4  c?m.  and  i m p l i e s t h a t the Therefore  02  =  dsi  has o r d e r  continuous.  such  A si  if ^ Sm~\  i t may  <  oo  (3  i s a bounded f u n c t i o n d e r i v a t i v e s o f UJ  be c o n c l u d e d  of f u n c t i o n s t h a t have continuous  2m-|  that the  then are  space  derivatives  and  f u r t h e r , s a t i s f y a l l the boundary c o n d i t i o n s , c o n t a i n s  the  exact s o l u t i o n .  Thus i t i s s u f f i c i e n t t h a t the  co-ordinate  f u n c t i o n s be c o m p l e t e w i t h r e s p e c t t o t h i s s p a c e w h i c h w i l l be known as ensured  .  Convergence, however, i s s t i l l  i n the s e n s e of the e n e r g y norm.  only  30  The Ray 1 e i g h - R i t z p r o c e d u r e may  a l s o be e m p l o y e d f o r t h e  d e t e r m i n a t i o n o f t h e e i g e n v a l u e s o f Eq. 2.23.  The  necessary  development i s presented i n M i k h l i n (9).but w i l l not repeated here.  I t i s worthy of note t h a t the  eigenvalues  o b t a i n e d a r e b o u n d e d b e l o w by  so  be  approximate their  respective exact values. 3.2  G a l e r k i n ' s Method G a l e r k i n ' s method s p e c i f i e s t h a t the r e s i d u a l  o b t a i n e d by s u b s t i t u t i n g t h e a p p r o x i m a t e Eq. 2.6  i s made o r t h o g o n a l , t h r o u g h o u t  of the c o - o r d i n a t e f u n c t i o n s ^  .  solution u the domain, to  This procedure  into each  is  a p p l i c a b l e t o any o p e r a t o r t h a t i s p o s i t i v e d e f i n i t e . Thus i t i s r e q u i r e d t h a t  ' ^ (A  ui - -f )  da  -  {)  fydft= 0  0  j = l,...n  (3.5a)  si  w h i c h may  be w r i t t e n  (M£ VM"" This system the  j  =!,... M  (3.5b)  o f l i n e a r e q u a t i o n s has a u n i q u e s o l u t i o n f o r i f the  are l i n e a r l y independent  throughout il .  Note, however, t h a t the e v a l u a t i o n of t h i s e q u a t i o n i n the form g i v e n i s o n l y p o s s i b l e i f the c o - o r d i n a t e f u n c t i o n s have  31 2m-\  continuous  derivatives.  In f a c t c o n v e r g e n c e i s  e n s u r e d , as w i l l be p r o v e d i n t h i s s e c t i o n , i f t h e c o ordinate f u n c t i o n s are complete i n H  , i . e . they have  A  2 m -1  continuous conditions.  d e r i v a t i v e s and s a t i s f y . a l l t h e b o u n d a r y  H o w e v e r , as w i l l a l s o be d e m o n s t r a t e d , i t may  be p o s s i b l e t o e x p r e s s E q . 3.5b i n a f o r m s u c h t h a t t h e c o o r d i n a t e f u n c t i o n s n e e d o n l y be c o m p l e t e i n a s p a c e e q u i v a lent to H  to ensure  A  convergence.  The g e n e r a l c h a r a c t e r i s t i c s o f a c l a s s o f p r o b l e m s f o r w h i c h t h e G a l e r k i n p r o c e d u r e i s known t o c o n v e r g e  will  f i r s t be d i s c u s s e d . T h e o r e m s w i l l t h e n be p r e s e n t e d t h a t form the b a s i s o f the subsequent convergence  investigation.  In t h i s d i s c u s s i o n t h e c o n d i t i o n s t h a t t h e c o - o r d i n a t e f u n c t i o n s must s a t i s f y t o e n s u r e c o n v e r g e n c e , t h e type o f convergence o b t a i n e d , and t h e r e l a t i o n s h i p between t h e R a y l e i g h - R i t z and G a l e r k i n p r o c e d u r e s a r e s p e c i f i c a l l y d e a l t wi t h . A c l a s s o f equations f o r which the G a l e r k i n p r o c e d u r e i s known t o c o n v e r g e i s c h a r a c t e r i z e d by t h e e q u a t i on UJ  -  > T  =  e  where U J i s t h e r e q u i r e d element and e l e m e n t o f some H i l b e r t s p a c e H .  SL  i s the given  T i s some c o m p l e t e l y  (3.  32 c o n t i n u o u s o p e r a t o r i n H , and A i s a n u m e r i c a l  parameter.  The g e n e r a l p r o p e r t i e s o f s u c h e q u a t i o n s w i l l f i r s t b e established. Assume t h a t A c a n assume any f i x e d v a l u e w i t h m o d u l u s n o t e x c e e d i n g some c o n s t a n t Then, as T  R , so that  IA I ^ R  i s c o m p l e t e l y c o n t i n u o u s i t may b e e x p r e s s e d  in the form  T where  - T '+ T "  I i s degenerate  H ^  ||T"  (3.7) and  W**  ( 3  -  8 )  I t c a n t h e n b e shown ( 9 , p. 4 6 3 ) t h a t t h e l i n e a r o p e r a t o r ( £ - > T " ) ' .where £ i s t h e i d e n t i t y o p e r a t o r , e x i s t s and is bounded.  E q . 3.6 c a n b e w r i t t e n i n t h e f o r m (3.9)  UJ  w h i c h when m u l t i p l i e d b y ( E - > T " ) " ' g i v e s LO  F u r t h e r as  - >(E->T")-VW T  l  i s degenerate  { ( e - M T ' - F>  i t i s p o s s i b l e to w r i t e  (3.10)  33 w h e r e t h e s e t o f e l e m e n t s <J> a n d a l s o t h e s e t i j ^ c a n be e t  considered  as l i n e a r l y i n d e p e n d e n t . -I _ I  f E - y r T T ' uid,  Then  - f(a,,^)U, =  (3.12)  ) k  where U^,  = ( ->T")"(|) £  are l i n e a r l y independent. the  (3.13)  k  E q . 3.10 may now b e w r i t t e n i n  form U> -  >2 C  k V k  = F,  (3-14)  = ( UJ , 'ij> )  where  (3.15)  k  Forming t h e i n n e r p r o d u c t o f each term i n Eq. .3.14.with %  where  gives  ^  fr  =l  T h u s t h e s o l u t i o n o f E q . 3.6 c a n b e o b t a i n e d i f E q . 3.16 c a n be s o l v e d f o r t h e unknown  from Eq. 3.14 .  34 T h e c o n d i t i o n s u n d e r w h i c h E q . 3.16 w i l l h a v e a s o l u t i o n c a n be e x a m i n e d by i n v e s t i g a t i o n o f t h e m a t r i x o f c o e f f i c i e n t s of the  .  Writing the equation  i n matrix  form gives  , -Aa  I - Aa,, ,  c,  (3  <  •A,  -Aa  , I-AQ  Denote the determinant coefficients  Cl ^  functions of A plane. £  j  o f t h e a b o v e m a t r i x by DR(M  and c o n s e q u e n t l y  m  N M  i n the c i r c l e  IAI^  D (A) a r e  o f the complex  This c o n t i n u i t y , together with the fact  ( ) _ |  implies  0  D ( A ) ^ 0 and t h a t t h e R  The  continuous  R  R  •  that  determinant"  has o n l y a f i n i t e number o f r o o t s i n - t h e c i r c l e IA I ^ R If  D (A) = 0 t h e n t h e h o m o g e n e o u s s y s t e m R  obtained  f r o m E q . 3.16 by r e p l a c i n g t h e r i g h t h a n d s i d e by z e r o s has a non t r i v i a l s o l u t i o n . homogeneous  Then i t f o l l o w s t h a t the  equation id  -  "X  I  UJ  =  0  (3.  35 has a n o n - t r i v i a l s o l u t i o n a n d t h e \ q u a n t i t y which T  .  considered i s a  i s the r e c i p r o c a l of the eigenvalue of  Such  X  a r e a l s o known as c h a r a c t e r i s t i c v a l u e s .  If  D  I >i ) =f= o  R  has a u n i q u e operator  solution.  t h e n E q . 3 . 1 6 , a n d h e n c e E q . 3.6, In t h i s c a s e t h e r e f o r e t h e  ( £ - > " ! " ) ' e x i s t s a n d w i l l be d e n o t e d  Those values of  ))  f o r which  by  e x i s t s a r e known as  regular values. Thus t h e e x i s t e n c e o f s o l u t i o n s o f e q u a t i o n s o f t h e f o r m o f E q . 3.6 c o n t a i n i n g a c o m p l e t e l y operator i s proven. Fredholm's  continuous  The f o l l o w i n g a l t e r n a t i v e ( c a l l e d  alternative) holds:  e i t h e r the non-homogeneous  e q u a t i o n i s s o l u b l e a n d u n i q u e l y so f o r a n y term  ^  independent  and then t h e c o r r e s p o n d i n g homogeneous  has o n l y t h e t r i v i a l  equation  s o l u t i o n , o r the non-homogeneous n  e q u a t i o n i s n o t s o l u b l e f o r some v a l u e o f -tthe c o r r e s p o n d i n g solution  homogeneous e q u a t i o n has a n o n - t r i v i a l  The f i r s t p a r t o f t h e a l t e r n a t i v e h o l d s i f A  .  i s a r e g u l a r v a l u e and t h e s e c o n d istic  and then  i f )\ i s a c h a r a c t e r -  value. The f o l l o w i n g theorems  the d i s c u s s i o n of the convergence c a n now be  proven.  which  form the b a s i s f o r  of the Galerkin  procedure  36 Theroem 3.1. operators  Let  [Tvj  be a s e t o f c o m p l e t e l y  i n some H i l b e r t s p a c e  completely  continuous T"  Further l e t space which  T  {"fn}  as  O  b e  3  s e t  which  T  operator  II ~ ^  n  H  o f  ->  f  that  «  elements  t e n d t o some e l e m e n t  t e n d t o some  i n the sense ti  continuous  .  (3.19)  o f t h e same I f >l  Is a r e g u l a r  value of the equation - >Tco  then f o r s u f f i c i e n t l y  =  j  (3.20)  3  l a r g e n, ,  A  will  a l s o be a r e g u l a r  value f o r the equation (3.21)  a n d t h e s o l u t i o n o f E q . 3.21 w i l l Eq. 3.20 as  n. —*  oo  tend to the s o l u t i o n o f  .  proof: Consider V - >\ T  where  9  the equation a  V  =  J  i s an a r b i t r a r y e l e m e n t  (3.22)  from W  This  equation  37  may  be w r i t t e n V  ->7V  - > (Trv-  By h y p o t h e s i s  - ( E->T)  T) •  3  -  (3.23)  e x i s t s and a p p l y i n g  1  i t to  b o t h s i d e s o f Eq. 3 . 2 3 g i v e s - )> £ ( T  V  F T  -T)V  =  P> J  (3.24)  Now IU?|T -T)||  4  R  li | l T - T | i  (3.25)  n  w h i c h f r o m t h e c o n d i t i o n s o f t h e t h e o r e m c a n be made a s s m a l l a s r e q u i r e d f o r s u f f i c i e n t l y l a r g e n . C h o o s e Tt so 1 a r g e t h a t IIK fT„-T)|| h  It then follows  (9,  <  i/  e  p . 467) t h a t t h e o p e r a t o r (E -  >(J ( T ^ T ) ) " '  e x i s t s , i s d e f i n e d f o r t h e w h o l e s p a c e and i t s norm d o e s not exceed 2. From Eq. 3 . 2 4 t h e s o l u t i o n o f Eq. 3 . 2 2 i s  V Thus the  -  ( E ->r (T.-T)-'(j  operator  >  3  38  - ( -> * ' * E  T  r  e x i s t s f o r the given value of a v a l u e o f ^\  (^-Tir'l]  (f-> A  r  •  i s r e g u l a r f o r Eq.  (3.26)  Therefore the given  3.22.  To e s t a b l i s h t h e s e c o n d p a r t o f t h e t h e o r e m i t H P  w i l l f i r s t be shown t h a t  - P || -?  B = > FJ ( T - T  Define  a  0  )  R  Note t h a t  whence k = o  Thus making use o f Eq. CO  3.26 i  fc = o  nX  >  /_j  a  ^  te = i  and m a k i n g u s e o f E q . 3.25  l-IIBJI  as  n-^  oo  39 Thus  hence  By v i r t u e o f E q . 3.27 t h e f i r s t t e r m on t h e r i g h t hand t e n d s t o z e r o and by p o s t u l a t i o n s o d o e s t h e s e c o n d  side  term.  Thus || u;^  - to |] —>  and t h e t h e o r e m has b e e n T h e o r e m 3.2.  n  co  (3.28)  proved.  II T - T ))  If  o  n  completely continuous the  as  o  w h e r e T and \  are  o p e r a t o r s then the e i g e n v a l u e s of  equation UJ - ) | T u >  =  O  (3.29)  a r e o b t a i n e d by t h e l i m i t p r o c e s s as n. -*a> f r o m t h e e i g e n values of the equation «4i - >T  n  uJ  a  =  0  (3.30)  40 proof: The  p r o o f u t i l i z e s t h e f a c t t h a t as  t h e c o e f f i c i e n t m a t r i x o f Eq. 3.17  corresponding  is denoted by ^ R I A )  whose d e t e r m i n a n t  m a t r i x shown i n Eq. 3.17  T  T  n  to  c o n v e r g e s to  whose d e t e r m i n a n t  the  i s denoted by  D U) R  L e t ^ b e any 0  IAI^ R  the c i r c l e X  Surround  t)^(A)  besides  select  be i t s m u l t i p l i c i t y .  =  N  ~h  such  that  there  w i t h i n or on a c i r c l e  0  I D  mm  Define R  I A )  I  ,  I )\ - X 0 I < €  Q  >  o  a >N  such t h a t f o r  By R o u c h e ' s t h e o r e m ( 1 6 , p. 89) i n the c i r c l e  t  R  I A - Ao I = £ . q  within  In p a r t i c u l a r D C M i s non-zero on the  with this r a d i u s .  Now  and l e t p  by a c i r c l e of r a d i u s  0  i s no r o o t o f circle  r o o t o f DRD (M w h i c h l i e s  D lM R  has p e q u a l  . D e n o t e them b y  roots  Ch^i  whence  X . ~ J\ I < * , j = n  Since  <?  0  I,  f in  7 N  c a n b e c h o s e n as s m a l l a s r e q u i r e d , t h i s i s  e q u i v a l e n t to the  statement  41  n-f  (3.31) oo  T h e o r e m s 3.1 a n d 3.2 e n a b l e t h e q u e s t i o n o f t h e c o n v e r g e n c e o f t h e G a l e r k i n p r o c e d u r e t o be c o n s i d e r e d when a p p l i e d t o e q u a t i o n s o f t h e f o r m o f E q s . 3.6 a n d 3.18. Consider f i r s t the G a l e r k i n equations that are a p p l i c a b l e t o E q . 3.6. UJ  where T  -  )l  T  u)  =  -  1  i s c o m p l e t e l y c o n t i n u o u s i n some H i l b e r t s p a c e H .  A s s u m i n g an a p p r o x i m a t e s o l u t i o n i n t h e u s u a l m a n n e r t h e Galerkin equations  become  A  32) 52  Si.  where UJ  Eq. 3.32 may be r e w r i t t e n M  A l s o w i t h o u t any l o s s o f g e n e r a l i t y t h e c o - o r d i n a t e f u n c t i o n s ^  , w h i c h a r e l i n e a r l y i n d e p e n d e n t , may be c o n s i d e r e d  t o be o r t h o n o r m a l i z e d i n t h e s p a c e  H  .  That i s  42  = o fe=fj whence Q;  -  The c o n v e r g e n c e governed  ^ J T ^ J ) - (^fj) of the approximate  j.|,.  (3.33)  s o l u t i o n so o b t a i n e d i s  by t h e f o l l o w i n g t h e o r e m .  T h e o r e m 3.3.  The a p p r o x i m a t e  by t h e G a l e r k i n p r o c e d u r e  s o l u t i o n o f E q : 3.6  converges  constructed  to the exact s o l u t i o n  ( i n t h e norm o f H) i f a)  E q . 3.6 h a s o n l y one s o l u t i o n i n H  b)  the operator T i s completely  c)  the c o - o r d i n a t e f u n c t i o n s form a complete  continuous  in H set in H  proof: As t h e s e t o f c o - o r d i n a t e cornplete  n  H  functions  is  i t i s possible to write  Define (3.34)  43  Then  || ^ -  J 11  as w i l l now be  -7 0  as n->co  1 1  )| < 2/2  p o s i t i v e number.  continuous  T= T 1T"  be e x p a n d e d i n t o a sum |l T  ||T  f t  -Tll-?o  proven.  As T i s c o m p l e t e l y and  . Also  1  , where  s  where  i n H i t can T  1  is  degenerate  i i s an a r b i t r a r i l y  small  Then CO  (T-T )u> B  (T-T.)u>  T L U -T u> ft  = £  (Tfc,^)^  » E f T ' u , ^ ) < k , + g (T" ,< L U „ w  Recal 1  whence I  as t h e  ej^  a r e assumed n o r m a l i z e d  i n H . Now by B e s s e l ' s  i n e q u a l i t y t h e r i g h t h a n d s i d e o f E q . 3.36 d o e s n o t  Hence  exceed  44  I2(TV4V)(M<  <» M  < 3 , 3 7 )  C o n s i d e r now t h e f i r s t t e r m i n E q . 3 . 3 5 . o p e r a t o r T ' U J may b e w r i t t e n  where H  s  The d e g e n e r a t e  i n the form  , Uj  i s a f i n i t e number a n d  a r e members o f  . Thus oo  a>  <  II  2. 1 ( u -  The s e r i e s f i c i e n t of large a  i T C j i ^ / f  2  4>f? ) I  || cu (|  , say  2  S  -  Kuj,^r)  (3  converges.  38)  Hence t h e coef-  w i l l be l e s s than £ / 2 f o r s u f f i c i e n t l y  a > N . T h u s f r o m E q s . 3.35 , 3.37 a n d 3.38  || ( T - T j w  || <  «f Ml  n  >M  whence ' T - T  a  || —? 0  as  a  —>  e»  45 w h i c h was t o b e p r o v e d ! The o p e r a t o r T continuous.  ^  i s d e g e n e r a t e and h e n c e c o m p l e t e l y  A  T h u s f r o m T h e o r e m 3.1 t h e e q u a t i o n a  - >T„ uv -  f„  <- > 3 39  has a u n i q u e s o l u t i o n f o r s u f f i c i e n t l y l a r g e a a n d - uj || Substituting w  for t  a  —7>  and J  » - ^ ( ( / . ^ ) ^ + > f T  where  /A^=  Substituting  0  W  f -f i T \ ) +  as  n  oo  i n E q . 3.39 g i v e s  n  A*  . , ^ ) ^ ) W T ^ f f e )  the value of •w  ft  -  < 3  ( 3  4 0 )  4 1  )  f r o m E q . 3.40 i n t o E q . 3.41  gives - >2  A (T^,  ) -  f f  <fc)  ( 3  From t h e p r e c e e d i n g s t a t e m e n t s t h e c o n s t a n t s c a l c u l a t e d  -  4 2 )  from  this equation ensure that the approximate s o l u t i o n as given by E q . 3.40 c o n v e r g e s t o t h e c o r r e c t s o l u t i o n .  B u t E q . 3.42  i s i d e n t i c a l t o E q . 3.33 o b t a i n e d f r o m t h e G a l e r k i n Thus t h e G a l e r k i n  procedure.  approximation converges to the exact  s o l u t i o n i n t h e norm o f H.  46 S i m i l a r l y by r e p e a t i n g t h e a b o v e a r g u m e n t s t h e c a t i o n of the G a l e r k i n method to the problem of f i n d i n g e i g e n v a l u e s of the id  —  \  the  equation T  UJ  =  c  (  3  c a n be shown t o be e q u i v a l e n t t o f i n d i n g t h e e i g e n v a l u e s the  appli-  of  equation  From w h i c h i t f o l l o w s by T h e o r e m 3.2 Eq. 3.43  t h a t the e i g e n v a l u e s  are the l i m i t s of the c o r r e s p o n d i n g e i g e n v a l u e s  Eq. 3.44.  T h u s t h e f o l l o w i n g t h e o r e m may  T h e o r e m 3.4.  of  be s t a t e d .  The a p p l i c a t i o n o f t h e G a l e r k i n m e t h o d t o t h e  problem of s e e k i n g e i g e n v a l u e s of e q u a t i o n s of the lu  -  >\T  leads to a convergent  uJ  =  form  O  process i f  a)  T is completely continuous  b)  t h e c o - o r d i n a t e f u n c t i o n s f o r m a c o m p l e t e s e t i n H.  in H  Note t h a t the above theorems have been w i t h r e s p e c t t o a g i v e n H i l b e r t s p a c e H.  developed  In g e n e r a l , when  dealing with d i f f e r e n t i a l equations, this space w i l l with the  of  Hk  space developed  coincide  i n the previous s e c t i o n .  47 The b a s i c e q u a t i o n s c o n s i d e r e d i n t h i s t h e s i s b e e n a s s u m e d t o be e x p r e s s i b l e i n t h e f o r m o f E q .  A w =  have  2.6.  |  In o r d e r t o s e e how  t h i s corresponds to equations of the  f o r m o f E q . 3.20, w h i c h h a v e b e e n c e n t r a l t o t h e p r e c e e d i n g d i s c u s s i o n , and a t t h e same t i m e t o i l l u s t r a t e t h e r e l a t i o n s h i p b e t w e e n t h e R a y 1 e i g h - R i t z and G a l e r k i n p r o c e d u r e s , c o n s i d e r t h a t the o p e r a t o r A =  A  R  In t h i s e q u a t i o n  +  R  has t h e f o r m  K  (3.45)  i s a symmetric  b e l o w o p e r a t o r o f o r d e r 2m  and  K  and p o s i t i v e  i s any o p e r a t o r s u c h  t h a t i t s f i e l d o f d e f i n i t i o n encompasses fcu  w h e n e v e r Ru  has a m e a n i n g  general  A  that of R  is meaningful.  n e e d n o t be s y m m e t r i c .  bounded , that is  Thus i n  Substituting for  A  i n E q . 2.6 g i v e s  R  +  UJ  k  (3.46)  UJ  w h i c h c a n be e x p r e s s e d i n t h e f o r m o f Eq. 3.20 W  or-  +  K UJ  =  R"  1  f  as  48  UJ  +  T - "R k  where  (3.47)  TuJ  and  Define a space  H  a s i n C h a p t e r 3.1 i n w h i c h  t h e i n n e r p r o d u c t i s g i v e n by (3.48) Si  in which  D  i sa differential  operator o f orderm  T h e o r e m 3.3 e n s u r e s t h e c o n v e r g e n c e approximate continuous  terms  of the  s o l u t i o n o f E q : 3.47 i f T=R"'K i s c o m p l e t e l y i n some H i l b e r t s p a c e  f u n c t i o n s a r e complete continuous  .  in H  R  o f t h e energy  H  and i f the co-ordinate  i n H . Thus i f T  i s completely  r e p e a t i n g t h e p r o o f o f T h e o r e m 3.3 i n product i n H  R  gives the equivalent of  Eq. 3.42 a s (3.49) in  which  and the  have been assumed orthonormalized in H  i.e.  49  = 1 =  Further  0  the theorem states I UJ -  UJ |  —^  O  as  M-><»  R  1  Consider  now a p p l y i n g  d i r e c t l y t o E q . 3.46.  the Galerkin  The r e q u i r e d e q u a t i o n s  procedure are  w h i c h may b e w r i t t e n  g a ^ R ^ ) * ^ .  ^ J . )  =  (3.50)  K has been s p e c i f i e d t o be such t h a t i t s f i e l d o f d e f i n i t i o n encompasses t h a t o f R . A s s u m e f u r t h e r i t i s defined f o r every element of H  R  that  . T h e n E q . 3.50 c a n  be w r i t t e n  £ s{ c 4...+j ]„ + w h i c h t h e n has m e a n i n g f o r a n y w r i t t e n by noting f i r s t that the orthonormalized relations  hold  in  H  R  =  <j> e H R fe  qb  and s e c o n d l y  fe  (fH)  (3  -  51)  . E q . 3.51 may b e r e -  have been assumed that the f o l l o w i n g  50  T h u s Eq. 3.51  may  be  written  This equation  c o i n c i d e s w i t h Eq: 3.49.  t i o n of the G a l e r k i n p r o c e d u r e  Thus the a p p l i c a -  t o Eq. 3.46  provides  an  a p p r o x i m a t e s o l u t i o n t h a t c o n v e r g e s i n t h e norm o f H t h e e x a c t s o l u t i o n o f Eq. 3.47. Eq. 3.46  C l e a r l y any s o l u t i o n o f  i s a s o l u t i o n o f Eq. 3.47. D  no s o l u t i o n e x i s t s i n  Eq. 3.47  and  procedure  i n t h i s way  w i l l be c o n s i d e r e d T h u s Eq. 3.51 t o Eq. 3.46  solution i f  T - R' K  the c o - o r d i n a t e  1  H o w e v e r i t may  be  that  but a s o l u t i o n does e x i s t i n  R  In t h i s c a s e t h e s o l u t i o n o f Eq. 3.47 s o l u t i o n o f Eq. 3.46  to  R  i s the  generalized  any s o l u t i o n o f  a s o l u t i o n o f Eq.  3.46.  o b t a i n e d by a p p l y i n g t h e ensures  Galerkin  convergence to the  is completely  continuous  f u n c t i o n s are complete in H  R  exact  in .  The  o r d i n a t e f u n c t i o n s t h e r e f o r e need not s a t i s f y the boundary c o n d i t i o n s of the problem i . e . those conditions i n v o l v i n g d e r i v a t i v e s of order  and conatural  boundary  m. o r  higher.  H  R  51 If convergence  K  H  i s not d e f i n e d f o r a l l f u n c t i o n s i n  R  c a n be o b t a i n e d by c h o o s i n g c o - o r d i n a t e f u n c 0  H  t i o n s from  £ m - I  i . e . functions with continuous  R  d e r i v a t i v e s t h a t s a t i s f y a l l the boundary c o n d i t i o n s . i s where ' K  A p a r t i c u l a r c a s e o f Eq. 3.46 n u l l o p e r a t o r , i n which  s i t u a t i o n /\=R  and p o s i t i v e b o u n d e d b e l o w .  Eq. 3.51  and i s  k  k  R  symmetric  then reduces  £ a C4 , ^ 3 = ff, i) t  Rayleigh-Ritz procedure  under  Thus f o r a symmetric,  (3-  r  o b t a i n e d by  conditions.  to  j.. ..M  T h i s e q u a t i o n i s i d e n t i c a l t o Eq. 3.2 and c o n v e r g e s  i s the  the  identical  p o s i t i v e bounded below  o p e r a t o r t h e two m e t h o d s l e a d t o t h e same e q u a t i o n s and governed  by t h e same c o n v e r g e n c e  procedure Ritz  thus appears  The G a l e r k i n  as a g e n e r a l i z a t i o n o f t h e R a y l e i g h -  procedure. The G a l e r k i n p r o c e d u r e  energy  criteria.  are  can thus e n s u r e  convergence  norm f o r e q u a t i o n s o f t h e t y p e g i v e n i n Eq. 3.46  when  t h e c o - o r d i n a t e f u n c t i o n s do n o t s a t i s f y t h e n a t u r a l b o u n d a r y c o n d i t i o n s of the problem. d e r i v a t i v e s o f i7J  This convergence  up t o o r d e r m  as t o t h e m a n n e r o f c o n v e r g e n c e  i n v o l v e s the  b u t p r o v i d e s no i n f o r m a t i o n of higher order d e r i v a t i v e s .  The f o l l o w i n g p r e s e n t a t i o n i l l u s t r a t e s t h e m a n n e r i n w h i c h these higher order d e r i v a t i v e s converge.  52 A s s u m e CJ i s some f i x e d e l e m e n t f r o m H . R  (Rio  - J! , ej ). = ^ R ( C D - w ) + K ( u i - w ) , g )  +  Thus |( Rw + kuo  , ^ )|«  |  W - U J |  R  |  gl  R  + ( K (cu~u)),cj)  The s e c o n d t e r m on t h e r i g h t hand s i d e o f t h e e q u a t i o n i s a bounded  linear function in  H  and may t h e r e f o r e be  R  e x p r e s s e d as an i n n e r p r o d u c t on t h i s s p a c e ( 9 ) . f f t ( u5 - co), j ) = [ w - tu, Tp] ^ where  ij/ i s a f i x e d e l e m e n t o f H  R  Thus  1 uJ - OL» 1 11|»l  R  . Thus E q . 3.54 may  be w r i t t e n  However the G a l e r k i n p r o c e d u r e e n s u r e s  T h u s Eq. 3.55  '(R*  implies + kuJ  - / )  J  d s i  °  J  6  SI  I f A i s s y m m e t r i c and p o s i t i v e b o u n d e d i.e.K O s  a s t r o n g e r c o n v e r g e n c e c a n be p r o v e n .  below To see  (3.54)  53 t h i s assume there exists  cj  6. L ( n . ) 2  (j e H 1  A  f A C D - /, 5 ) =  . Then s i n c e H i s dense i n ft  such that  L (si) 5  || g'- g || <. 6. . Now  9-3')  C A - -f, g' ) + ( A f i - f i  Further I(Ao-/,g')l  <  f J A c II +  II/l}  t  <  C £ C = Consf.  and f o r M  sufficiently  large  Hence  Whence a s £ c a n be made a r b i t r a r i l y  n  f f\ uj - I ) 0  dsi  -> o  small cj L L fa) £  54  CHAPTER  4  THE F I N I T E ELEMENT PROCEDURE The f i n i t e e l e m e n t p r o c e d u r e was  originally  d e v e l o p e d by e n g i n e e r s on t h e b a s i s o f p h y s i c a l  intuition  f o r a p p l i c a t i o n i n the a n a l y s i s of complex s t r u c t u r a l  systems.  The r e v i e w by Z i e n k i e w c z ( 1 9 ) o u t l i n e s t h e d e v e l o p m e n t o f the  m e t h o d and c o n t a i n s a c o m p r e h e n s i v e l i s t o f r e f e r e n c e s .  R e c e n t l y the mathematical framework  o f t h e p r o c e d u r e has  come u n d e r c l o s e s c r u t i n y and i t i s t h e p u r p o s e o f t h i s c h a p t e r to d i s c u s s t h i s a s p e c t of the method. In 1969 Oden ( 1 1 , 1 2 ) p o i n t e d o u t t h a t t h e f o r m u l a t i o n o f a f i n i t e e l e m e n t model  of a function i s a p u r e l y  t o p o l o g i c a l c o n s t r u c t i o n and has n o t h i n g t o do w i t h v a r i a tional principles.  T h e f i n i t e e l e m e n t m e t h o d , as w i l l  be  shown i n t h i s c h a p t e r , i s i n f a c t a means o f c o n s t r u c t i n g an a p p r o x i m a t e s o l u t i o n f o r m t o a g i v e n e q u a t i o n .  This  a p p r o x i m a t i o n i s e x p r e s s e d i n t e r m s o f known c o - o r d i n a t e f u n c t i o n s and unknown p a r a m e t e r s .  This approximate  solution  f o r m may be u s e d i n c o n j u n c t i o n w i t h a number o f t e c h n i q u e s to d e t e r m i n e t h e unknown p a r a m e t e r s .  55 The p r o b l e m c o n s i d e r e d i s t h a t o f o b t a i n i n g an approximate conditions.  s o l u t i o n t o E q . 2.6 u n d e r h o m o g e n e o u s  boundary  As was d i s c u s s e d i n C h a p t e r 3 t h e r e a r e a  number o f t e c h n i q u e s a v a i l a b l e t h a t a r e b a s e d upon t h e idea of assuming  an a p p r o x i m a t e  s o l u t i o n i n the form  M *  £  =•  Q  *T\  I t w i l l f i r s t be d e m o n s t r a t e d  t h a t the f i n i t e element  pro-  c e d u r e g e n e r a t e s s u c h an a p p r o x i m a t i o n , a n d t h e n t h e c o n v e r g e n c e r e s u l t s t h a t have been p r e s e n t e d f o r the G a l e r k i n p r o c e d u r e w i l l be i n t e r p r e t e d i n t e r m s o f a f i n i t e  element  a p p r o x i m a t i on. 4.1  G e n e r a t i o n of a F i n i t e Element  Approximation  The b a s i c s t e p s t h a t c h a r a c t e r i z e t h e e l e m e n t p r o c e d u r e w i l l be p r e s e n t e d .  finite  A rigorous discussion  o f t h e f o l l o w i n g p o i n t s has b e e n p r e s e n t e d by Oden ( 1 1 ) . The f i r s t s t e p i s to r e p l a c e t h e domain o f d e f i n i t i o n o f t h e p r o b l e m Si  by Si.  s u c h t h a t SL  may be e x a c t l y  s u b d i v i d e d i n t o a n u m b e r , s a y E, o f non o v e r l a p p i n g s u b domains c a l l e d elements. w i l l be d e n o t e d by 52  e  The d o m a i n o f a t y p i c a l  and s u c h d o m a i n s a r e g e n e r a l l y  chosen to have a s i m p l e g e o m e t r i c a l form. are  element  s p e c i f i e d t o h a v e a common b o u n d a r y .  Adjacent Thus  elements  56  where  ©  i s t h e empty s e t , a n d  «  5.*  u 5.  <->  €  42  _ #  The e l e m e n t s a r e c h o s e n , i f p o s s i b l e , s u c h t h a t 51  coincides  w i t h SL , b u t i f n o t , i n s u c h a m a n n e r t h a t t h e e r r o r —t  involved is acceptable.  I t w i l l be a s s u m e d t h a t t h e SI  have  b e e n t h u s c h o s e n a n d t h e n o t a t i o n SL , SI w i l l be u s e d t o represent  SL  , SI  The s e c o n d  s t e p i n the method i n v o l v e s t h e assump-  t i o n o f an a p p r o x i m a t e s o l u t i o n f o r UJ e l e m e n t s t h a t c a n be e x p r e s s e d  i n each o f the  i n the form (4.3,  where the iz*  <j)*  and t h e  are co-ordinate at  functions defined only i n  are the values of  u5  €  o r one o f i t s  R  d e r i v a t i v e s a t c e r t a i n nodal t h e b o u n d a r y o f 51* . corresponds ordinates  F o r e x a m p l e i f 0.  to the value of then  p o i n t s g e n e r a l l y s i t u a t e d on e  n = I,  N  a t t h e node w i t h c o -  57  Such a d e f i n i t i o n ensures t h a t the independent throughout  are l i n e a r l y  51  I t i s p o s s i b l e , by a l i n e a r t r a n s f o r m a t i o n , t o e x p r e s s any  w*  containing N l i n e a r l y independent  i n t h e f o r m o f Eq. 4.3.  In p a r t i c u l a r a p o l y n o m i a l  be so e x p r e s s e d , w h i c h means t h a t t h e a p p r o x i m a t e may  terms may solution  be e x p r e s s e d i n p o l y n o m i a l f o r m , w h i c h i s o f t e n  c o n v e n i e n t , and t h e n t r a n s f o r m e d i n t o t h e f o r m o f Eq. 4.3. In t h e a p p l i c a t i o n o f t h e f i n i t e e l e m e n t  procedure  i t i s o n l y n e c e s s a r y to assume c o - o r d i n a t e f u n c t i o n s d e f i n e d over i n d i v i d u a l elements to o b t a i n a s o l u t i o n . in order to demonstrate  However,  t h a t s u c h a p p r o x i m a t i o n s c a n be  c o n s i d e r e d t o be o f t h e f o r m o f Eq. 3.1  i t is convenient  to i n t r o d u c e o t h e r f u n c t i o n s which are d e f i n e d i n terms the  i n the f o l l o w i n g manner. Consider functions  domain  of  i i  ^  R  d e f i n e d over the whole  such t h a t  (4.5)  58 where the the assumed domain  Si  X;,  r e p r e s e n t s a p o i n t i n the domain.  a p p r o x i m a t i o n f o r LU  Then  throughout the whole  may be w r i t t e n t  N  (4.6)  On i n t e r e l e m e n t b o u n d a r i e s w h e r e n o d e s o f a d j a c e n t e l e m e n t s c o i n c i d e i t i s n a t u r a l to s p e c i f y t h a t these nodal values s h o u l d be t h e same.  Assume t h a t t h e r e a r e  g l o b a l d e g r e e s o f f r e e d o m i n SI by  .  M  independent  w h i c h w i l l be d e n o t e d  Then the element d e g r e e s o f freedom a r e r e l a t e d  t o t h e g l o b a l d e g r e e s o f f r e e d o m by t h e r e l a t i o n s h i p (4.7) where =  1  i f node a  0  o t h e r w i se  R  c o i n c i d e s w i t h Oj  Then  define (4.8)  59  E q . 4.6 may t h e n be w r i t t e n M  in which the  <j).  are l i n e a r l y independent throughout the  J  d o m a i n. Thus a f i n i t e e l e m e n t a p p r o x i m a t i o n has t h e f o r m o f E q . 3.1.  The e s s e n t i a l f e a t u r e o f the method  l i e s i n f o r m u l a t i n g an a p p r o x i m a t e  solution that i s defined  o v e r the whole domain i n terms o f a p p r o x i m a t i o n s t h a t a r e non-zero only over  subdomains.  A r e f i n e d a p p r o x i m a t i o n i s o b t a i n e d by r e s u b d i v i d i n g the domain  SI  i n t o a l a r g e r number o f e l e m e n t s .  The same a p p r o x i m a t e  s o l u t i o n i s a s s u m e d i n e a c h o f t h e new  e l e m e n t s and t h e r e f o r e t h e f i n a l a p p r o x i m a t i o n i s a g a i n i n t h e form o f Eq. 3.1.  T h e c o - o r d i n a t e f u n c t i o n s j>j a r e  r e f i n e d i n s u c h a way t h a t t h e y h a v e t h e same s h a p e b u t are  d e f i n e d t o be n o n - z e r o o v e r a s m a l l e r r e g i o n o f SI  than  their predecessors. It i s also p o s s i b l e to r e f i n e the approximation by l e a v i n g t h e number o f e l e m e n t s c o n s t a n t a n d i n c r e a s i n g t h e number o f c o - o r d i n a t e f u n c t i o n s p e r e l e m e n t . The unknown p a r a m e t e r s may be e v a l u a t e d , f o r e x a m p l e ,  i n the approximate  solution  by s o l v i n g t h e e q u a t i o n s  g i v e n by t h e G a l e r k i n p r o c e d u r e :  60  This equation i s in p r a c t i c e generated assembling  the r e l a t i o n s o b t a i n e d from i n d i v i d u a l  W r i t i n g E q . 4.9  f o r each of the elements  where the s u p e r s c r i p t  g i v e s the r e l a t i o n s h i p between to determine  in turn  elements. gives  i n d i c a t e s that the inner  a r e e v a l u a t e d o v e r t h e s u b d o m a i n s 51 employed  by  .  and ^  S o l v i n g Eqs.  4.10  which can then  the r e l a t i o n s h i p between  m a k i n g u s e o f E q s . 4.7 and  products be  and |j  by  4.8.  An i m p o r t a n t f e a t u r e o f t h e f i n i t e e l e m e n t  pro-  cedure t h a t f o l l o w s from the above c o n s t r u c t i o n of the g o v e r n i n g e q u a t i o n i s the m a t r i x t h a t may  banded  nature of the c o e f f i c i e n t  be o b t a i n e d by s u i t a b l y o r d e r i n g t h e  Such a f e a t u r e i s i m p o r t a n t i n the n u m e r i c a l s o l u t i o n of problems 4.2  w i t h a l a r g e number o f d e g r e e s o f  General  Remarks  It is worthwhile f i n i t e element  freedom.  to note the analogy between a  a p p r o x i m a t i o n o f a f u n c t i o n and a F o u r i e r  .  61 s e r i e s a p p r o x i m a t i o n o f a f u n c t i o n . One i m p o r t a n t d i f f e r e n c e , however, l i e s in the f a c t t h a t i n a F o u r i e r s e r i e s the c o - o r d i n a t e f u n c t i o n s h a v e c o n t i n u o u s d e r i v a t i v e s t o any o r d e r t h r o u g h o u t the whole domain, whereas a f i n i t e c o - o r d i n a t e f u n c t i o n j>j  . g e n e r a l l y has  element  discontinuities  in i t s lowest d e r i v a t i v e s at element boundaries.  Another  d i f f e r e n c e i s that refinement of a f i n i t e element a p p r o x i m a t i o n i s e f f e c t e d by a r e d e f i n i t i o n o f t h e c o - o r d i n a t e f u n c t i o n s as o p p o s e d  t o s i m p l y a d d i n g e x t r a f u n c t i o n s as  i s common i n F o u r i e r s e r i e s a p p r o x i m a t i o n s .  One  advantage  of a g i v e n f i n i t e element a p p r o x i m a t i o n i s i t s f a c i l i t y to approximate v a r i o u s boundary as t h e b o u n d a r y  conditions. This is possible  c o n d i t i o n s a r e h a n d l e d by p r e s c r i b i n g  v a l u e s t o t h e g e n e r a l i z e d ' c o - o r d i n a t e s Oj  l o c a t e d on t h e  boundary. A c e n t r a l q u e s t i o n i n the a p p l i c a t i o n of a f i n i t e element a p p r o x i m a t i o n concerns the c o n d i t i o n s t h a t the assumed s o l u t i o n w i t h i n each element must s a t i s f y to e n s u r e c o n v e r g e n c e as more and m o r e e l e m e n t s a r e t a k e n . p a r t i c u l a r , two r e l a t e d q u e s t i o n s m u s t be (i)  answered:  on w h a t b a s i s s h o u l d t h e a p p r o x i m a t i o n i n e a c h  be c h o s e n , (ii)  In  element  and  w h a t c o n t i n u i t y o f uJ  and i t s d e r i v a t i v e s s h o u l d be  e n s u r e d a t t h e nodes and a c r o s s e l e m e n t  boundaries?  62 T h e s e q u e s t i o n s c a n be a n s w e r e d by i n v e s t i g a t i n g t h e  condi-  t i o n s under which the p a r t i c u l a r method employed to  evaluate  t h e unknown p a r a m e t e r s  i s known t o c o n v e r g e .  It is c l e a r  t h a t s u c h e v a l u a t i o n c a n be e f f e c t e d by a number o f ent t e c h n i q u e s .  Thus i t i s a l s o c l e a r t h a t the  differ-  finite  e l e m e n t " m e t h o d " n e e d n o t be a s s o c i a t e d w i t h any p a r t i c u l a r technique.  S p e c i f i c a l l y i t i s not a c c u r a t e to s t a t e t h a t  the f i n i t e element method i s a R a y l e i g h - R i t z The f i n i t e e l e m e n t m e t h o d s i m p l y g e n e r a t e s  an  procedure. approximate  s o l u t i o n f o r m , d e f i n e d i n t e r m s o f unknown p a r a m e t e r s ,  that  may  to  be u s e d i n c o n j u n c t i o n w i t h a number o f t e c h n i q u e s  o b t a i n an a p p r o x i m a t e  answer to the given  equation.  It i s n a t u r a l to i n v e s t i g a t e the r e l a t i v e advantages of the d i f f e r e n t s o l u t i o n t e c h n i q u e s a v a i l a b l e . T r a d i t i o n a l l y v i r t u a l work or R a y l e i g h - R i t z have been used  i n f i n i t e element work.  been used  The G a l e r k i n p r o c e d u r e  i n a number o f s p e c i f i c c a s e s  (17,18,20).  cases c o u l d , however, have been a n a l y s e d u s i n g the Ritz procedure. procedure  m e t h o d i s n o t a p p l i c a b l e , and a t t h e same t i m e  Rayleigh-  Rayleigh-Ritz ensuring  does not appear to have been p r e v i o u s l y  As was procedure  These  The p o s s i b i l i t y o f a p p l y i n g t h e G a l e r k i n  to a c l a s s of problems to which the  convergence,  has  pointed out i n Chapter  explored.  3 the G a l e r k i n  i s a g e n e r a l i z a t i o n of the R a y l e i g h - R i t z method  63 and h e n c e i n g e n e r a l p r e f e r a b l e .  In p a r t i c u l a r ,  t h a t a r e c h a r a c t e r i z e d by n o n - s y m m e t r i c  problems  o p e r a t o r s may  be  amenable to the G a l e r k i n p r o c e d u r e , whereas they cannot be h a n d l e d by R a y l e i g h - R i t z .  A l s o , as i s i l l u s t r a t e d i n  C h a p t e r 6, t h e G a l e r k i n p r o c e d u r e i s a p p l i c a b l e t o a l l those problems of s t r u c t u r a l mechanics  that virtual  c a n be u s e d f o r , w i t h t h e a d d e d a d v a n t a g e . t h a t v i r t u a l w o r k G a l e r k i n has p r o v e n c o n v e r g e n c e On t h e b a s i s o f t h e s e r e m a r k s  work  unlike criteria.  the Galerkin  pro-  c e d u r e w i l l be c h o s e n f o r t h e d e t e r m i n a t i o n o f t h e generalized  c o - o r d i n a t e s ctj and f o r t h e i n v e s t i g a t i o n o f  those s u f f i c i e n t conditions  t h a t the element  approximation  — e  _  tu m u s t s a t i s f y i n o r d e r t o e n s u r e c o n v e r g e n c e o f OJ the c o r r e c t answer. 4.3  Convergence  Criteria  In C h a p t e r 3 c o n d i t i o n s  that ensured  of the G a l e r k i n procedure f o r a wide c l a s s of were p r e s e n t e d .  convergence problems  In t h i s s e c t i o n t h e s e r e s u l t s w i l l  be  u t i l i z e d to p r o v i d e s u f f i c i e n t convergence c r i t e r i a f o r a f i n i t e element  approximation.  T h e o r e m 3.3 a s s e r t s t h a t t h e c o n v e r g e n c e  of  t h e G a l e r k i n a p p r o x i m a t i o n i s e n s u r e d when a p p l i e d t o e q u a t i o n s of the form  to  64  -  UJ  >Tu> -  /  i f t h e s o l u t i o n i s u n i q u e i n some H i l b e r t s p a c e H, t h e operator T i s completely continuous in H co-ordinate f u n c t i o n s are complete  in H  , and i f t h e .  The i n t e r p r e -  t a t i o n o f these c o n d i t i o n s i n terms o f a f i n i t e  element  a p p r o x i m a t i o n w i l l be p r e s e n t e d by means o f a p a r t i c u l a r example. Consider the problem o f determining the e q u i l i b r i u m c o n f i g u r a t i o n o f a u n i f o r m beam when s u b j e c t e d t o both a normal  l o a d and a l o a d t h a t i s p r o p o r t i o n a l t o t h e  s l o p e o f t h e beam.  S u c h non c o n s e r v a t i v e l o a d s a r e e n -  countered i n the study of a e r o e l a s t i c i t y .  The g o v e r n i n g  equations are EI  (j"" +  u;(o) where c  U(L)  c cj' -  =  if"(o)  i s a c o n s t a n t and  j  x  £ [o,L]  (4.11a)  - y"(i_) - 0 i s t h e normal  (4.11b) force.  The  o p e r a t o r i n t h i s e q u a t i o n c a n r e a d i l y be s e e n t o be u n symmetric.  T h i s e q u a t i o n c o r r e s p o n d s t o E q . 3.46 i n w h i c h  65  EI d  R -  ;  4  K  S  As i n C h a p t e r 3.2 c o n s t r u c t a s p a c e  I Functions  U |  u  8  = /"[U,U]  that are in H  R  C  J  HR  (4.12)  i n which  = /• f ' f l u V J x Jo  R  (4-13)  must then s a t i s f y the c o n d i -  tion EI  The s p a c e  ( u " - U " ) dx  0  2  ft  asn-^oo',  i^fi D  (4.14)  R  c o n t a i n s the e x a c t s o l u t i o n and, a s i s  v e r i f i e d i n C h a p t e r 7, T tinuous in this space.  - R"' ^  i s c o m p l e t e l y con-  T h u s T h e o r e m 3.3 e n s u r e s c o n v e r -  gence i f the c o - o r d i n a t e f u n c t i o n s are complete i n H  R  .  A f i n i t e element approximation i s o b t a i n e d by d i v i d i n g t h e beam i n t o  F_ s e c t i o n s and w i t h i n e a c h  section  assuming a s o l u t i o n of the form  The e l e m e n t s t i f f n e s s e q u a t i o n s a r e , o n t h e b a s i s o f Eq. 4.10, g i v e n b y  66  i< f(«w<tf)'+ ciwti)*  up*  -  j- 1  n  <  4  -  i  6  )  where the r e l a t i o n (4.17)  has b e e n u s e d .  E q s . 4.16 c a n t h e n b e a s s e m b l e d i n t o t h e  form ^  f  L  it  L  and t h e a p p r o x i m a t e  (4.18)  solution (4.19)  converges t o the c o r r e c t answer i f the in  H  R  . The c o n d i t i o n s  assumed s o l u t i o n w i t h i n that the co-ordinates plete in  H  R  ^  are complete  t h a t m u s t be s a t i s f i e d b y t h e each element i n o r d e r to ensure  d e f i n i n g t h e t o t a l s o l u t i o n b e com-  must t h e r e f o r e  be e s t a b l i s h e d .  67  In t h i s e x a m p l e , Eq. 4.14 in  H  must have g e n e r a l i z e d  R  continuous  Thus the f i n i t e  Further  r e q u i r e d to s a t i s f y these Before  c o m p l e t e n e s s c a n be o b t a i n e d  plete in tion  H  the  f+  in  R  the c o n d i t i o n s  under which  the d e f i n i t i o n of complete-  in terms of a f i n i t e  element  A f i n i t e element approximation  i s com-  i n some s t a t e d norm i f f o r a r b i t r a r y  V £ H and any  £>0  the domain c o r r e s p o n d i n g  are  conditions.  considering  n e s s w i l l be r e p e a t e d  across  i t must a l s o s a t i s f y  f o r c e d b o u n d a r y c o n d i t i o n s as a l l f u n c t i o n s  hence  element  must e n s u r e the c o n t i n u i t y of s l o p e  element boundaries.  approximation.  functions  s e c o n d d e r i v a t i v e s and  first derivatives.  approximation  shows t h a t  func-  there e x i s t s a subdivision to  M  of  degrees of freedom such  that M J  The  conditions  that a f i n i t e element approxima-  t i o n must s a t i s f y f o r c o m p l e t e n e s s have been by O l i v e i r a ( 1 3 ) . t h a t has  c  provided  T h e y w i l l be q u o t e d f o r a f u n c t i o n components  to,, u> , .. . t u 2  c  .  It is  a s s u m e d t h a t t h e e n e r g y norm i s b a s e d upon a s y m m e t r i c energy product  that contains  d e r i v a t i v e s of each component  68 iP  T  o f maximum o r d e r  m  .  T  It i s f u r t h e r assumed t h a t  the exact s o l u t i o n i s such t h a t the d e r i v a t i v e s of i t s components o f o r d e r element.  na + I T  are c o n t i n u o u s w i t h i n each  D i s c o n t i n u i t i e s of the m  tives are s t i l l  and  T  m  T  + I  deriva-  a l l o w e d a t p o i n t s w h i c h a l w a y s r e m a i n on  e l e m e n t b o u n d a r i e s as t h e s i z e o f t h e e l e m e n t i s p r o g r e s sively  reduced. O l i v e i r a then proves t h a t completeness w i l l  obtained i f c o n t i n u i t y of the  m  -I  T  be  derivatives is  e n s u r e d t h r o u g h o u t t h e d o m a i n and i f t h e a p p r o x i m a t i o n u) *  for w  w i t h i n e a c h e l e m e n t i s b a s e d upon a  T  polynomial of degree not l e s s than m  T  , a l l the terms  o f w h i c h a r e a f f e c t e d by i n d e p e n d e n t a r b i t r a r y  coefficients.  T h e s e c o n d i t i o n s a r e o f t e n e x p r e s s e d by s t a t i n g t h a t t h e e l e m e n t s m u s t be c o n f o r m i n g a n d t h a t t h e y m u s t be a b l e t o represent constant strain. Thus i n the example b e i n g c o n s i d e r e d the p l e t e n e s s r e q u i r e m e n t means t h a t  ij  com-  m u s t be b a s e d upon a  c o m p l e t e p o l y n o m i a l o f o r d e r n o t l e s s t h a n two.  Satisfaction  o f b o t h t h e c o n f o r m i n g and c o m p l e t e n e s s c o n d i t i o n s c a n be o b t a i n e d by a s s u m i n g an a p p r o x i m a t i o n w i t h i n e a c h  element  of the form  if  e  =  a  0  e  +  a, x e  +  fl V + a£x a  3  (4.20)  69  and c h o o s i n g d e g r e e s o f f r e e d o m c o r r e s p o n d i n g t o t h e d i s p l a c e m e n t and r o t a t i o n a t each end o f the e l e m e n t . N o t e t h a t i t was p o s s i b l e t o c h o o s e an a p p r o x i mation in  H  R  as  K = c d/dx  ordinate functions in w  1  R  H  R  .  i s d e f i n e d f o r a l l the coCo-ordinate functions in K  h a v e c o n t i n u o u s f i r s t d e r i v a t i v e s and  is defined  for a l l such f u n c t i o n s . S a t i s f a c t i o n of the i n d i c a t e d c o n d i t i o n s thus ensures energy convergence of the f i n i t e element t i o n t o t h e c o r r e c t s o l u t i o n L| . e  t"I  ( u" - «fi )* dx  approxima-  Specifically o  os  M  CD  (4.21)  I t s h o u l d be n o t e d t h a t c o n f o r m i t y i s o n l y a s u f f i c i e n t c o n d i t i o n f o r p r o v i n g the convergence of a f i n i t e element approximation.  O l i v e i r a ( 1 3 ) has a l s o  s t u d i e d the q u e s t i o n of non-conforming elements i n the c o n t e x t o f t h e R a y l e i g h - R i t z p r o c e d u r e and c o n c l u d e d t h a t under c e r t a i n c o n d i t i o n s they too can ensure c o n v e r g e n c e to t h e c o r r e c t s o l u t i o n . not  Non-conforming  be d i s c u s s e d i n t h i s t h e s i s .  elements  will  However i t i s p o s s i b l e  t o i n t r o d u c e a w e l l d e f i n e d norm i n t o t h e s p a c e o f f u n c t i o n s t h a t s a t i s f y t h e f o r c e d b o u n d a r y c o n d i t i o n s and h a v e c o n tinuous p r i n c i p a l d e r i v a t i v e s o n l y at nodal p o i n t s of the  70 domain.  I n t h i s way  the work of t h i s t h e s i s can be  extended  to provide a s y s t e m a t i c study of the  p r o p e r t i e s of non-conforming The d e v e l o p m e n t s r e s p e c t t o Eq. 4.11  elements. t h a t have been p r e s e n t e d  c a n b e p a r a l l e l e d f o r any  t h a t can be e x p r e s s e d  convergence with  equation  i n t h e f o r m o f Eq. 3.20.  Consider,  in g e n e r a l , the problem  of o b t a i n i n g a f i n i t e  element  s o l u t i o n f o r Eq. 3.46.  W r i t i n g t h i s equation out i n  indicia! notation gives  +  -  K  ».»-'.-<  -  (4 22)  Assume a s o l u t i o n (JJ  n  =  T  ( w, , LU, ,  uJ >"  (4.23a)  t  where W  T  » Z  Q  rk #  T - ..... c  The G a l e r k i n e q u a t i o n s f o r Eq. 4.22 (K ^ Tn  t k^ r  m  - fr)  £  t h e n become  J  a  = O  T-I....C fc-l,...L  If  K  i s d e f i n e d f o r a l l the c o - o r d i n a t e f u n c t i o n s i n  t h i s e q u a t i o n may b e e x p r e s s e d  (4.33b)  i n the  form  (4.24)  71  t  Q £  where J (R  T m  T  , ^  J  w ) <J3^" ^52.  i s the symmetric form of  . W r i t i n g E q . 4.25 i n e l e m e n t f o r m  m  '51  gives  e -  1  £  where  S i m i l a r l y e i g e n v a l u e problems that are e x p r e s s i b l e i n t h e f o r m o f E q . 3.18 may b e s o t r e a t e d .  Consider, for  example, the problem  *****  +  >  fr-  =  0  <4  The r e q u i r e d f i n i t e e l e m e n t e q u a t i o n s a r e  R  e-i  E  -  28)  72 Thus a p p l i c a t i o n o f the G a l e r k i n p r o c e d u r e e n a b l e s t h e e q u a t i o n s and c o r r e s p o n d i n g c o n v e r g e n c e c r i t e r i a o f a f i n i t e e l e m e n t s o l u t i o n t o be s e t down f o r a w i d e of problems.  class  CHAPTER 5 A P P L I C A T I O N OF THE GALERKIN.PROCEDURE MIXED AND NONHOMOGENEOUS BOUNDARY  TO PROBLEMS WITH CONDITIONS  A t t e n t i o n h a s b e e n c o n f i n e d so f a r i n t h i s  thesi  to problems w i t h unmixed homogeneous boundary c o n d i t i o n s . In t h i s c h a p t e r a p r o b l e m w i t h m i x e d h o m o g e n e o u s  boundary  c o n d i t i o n s w i l l be t r e a t e d a n d t h e m o d i f i c a t i o n s n e c e s s a r y to d e a l w i t h nonhomogeneous b o u n d a r y c o n d i t i o n s p r e s e n t e d . For i l l u s t r a t i o n purposes the equations governing the e q u i l i b r i u m c o n f i g u r a t i o n of a linear e l a s t i c continuum w i l l be c o n s i d e r e d . 5.1  Homogeneous Mixed Boundary C o n d i t i o n s The g o v e r n i n g e q u a t i o n s i n t h i s c a s e a r e (5  (5  O  73  74  $j Oj  (T . . + c a ^ = L  J  where  0"^  O  ^  S  T  0  ^  5  n  -  n  (5.id)  J  the d e n s i t y , X the  i s the s t r e s s .tensor,  body f o r c e p e r u n i t mass, U ponent o f u n i t outward a constant.  (5.1c)  5  ,  U  S  t h e d i s p l a c e m e n t , flj t h e com-  L  normal  to the boundary,  , and S  T  those p o r t i o n s of the boundary s t r e s s e s , and mixed  L  and  c  is  represent respectively  M  on w h i c h  the displacements,  conditions, are specified.  In o r d e r t o f o r m u l a t e t h i s p r o b l e m the displacement v e c t o r  U  i n terms o f  the g r a d i e n t of the s t r e s s  L  t e n s o r i n E q . 5.1a i s r e l a t e d t o t h e d i s p l a c e m e n t by means of the c o n s t i t u t i v e r e l a t i o n  %  -  ^  ^  +  >  ^  (5  -  2)  and t h e k i n e m a t i c e q u a t i o n (5.3) w h e r e j*. a n d  )\ a r e Lame's c o n s t a n t s ,  i s the Kronecker  75 d e l t a , and  £-  i s the s t r a i n tensor.  L  Carrying out the  J  indicated substitutions gives (5.4) w h i c h may be p l a c e d i n t o c o r r e s p o n d e n c e  with the general  e q u a t i on  V  - f f\ -  In t h i s c a s e  is a matrix of d i f f e r e n t i a l  L  o f t h e s e c o n d o r d e r and i s s y m m e t r i c .  For present  A^w-  i t i s convenient not to express  operators purposes  by means o f E q . 5.4  but i n s t e a d to use the r e l a t i o n  V  (5.6)  Eq. 5.6 c a n now be u s e d t o i l l u s t r a t e t h e s y m m e t r y o f A L  J  Thus r  i  i*  —*  1  "  /  .  «  52.  where  ^.  displacement  denotes  the s t r e s s tensor corresponding  vector  U  L  .  Now  5Z  to the  76  As t h e v a r i a b l e s c o n s i d e r e d a r e i n t h e f i e l d o f  definition  of the o p e r a t o r they must s a t i s f y a l l the b o u n d a r y c o n d i t i o n s o f E q s . 5.1.  Therefore  u • u" d *si.  |  = j 12^  r  1  "  "  L  Whence t h e o p e r a t o r  + >  L  r  = in the space  f-  A-  di  Ui"  ^- ) da  kk  +  c a- u-' ds  1  1  •  Z  c  u  O  i s symmetric  and t h e e n e r g y  product  i s g i v e n by '  1  "  c  J 5  ul  (5.7)  ds  M  In t h i s c a s e t h e m i x e d b o u n d a r y c o n d i t i o n g i v e s rise t o a s u r f a c e i n t e g r a l i n the energy equations governing  'j  The G a l e r k i n  the s o l u t i o n of t h i s problem  o b t a i n e d f r o m Eq. 4.25 \(^.  product.  by e q u a t i n g  i s the n u l l o p e r a t o r .  A^- and R- -  and  L  J  The e q u a t i o n s  are  can  J  be assuming  77  [ f f r . f i  ]  -  A  (f .  4;)  T  ( 5  '  8 )  w h e r e an a p p r o x i m a t e s o l u t i o n has b e e n a s s u m e d i n t h e form _  Q.  =  < a, , u  _  , u  2  T 5  >  (5.9) T  W r i t i n g Eq. 5.8  'Tfc  in which  out in f u l l  SI  ty  +  3  1,2,  =  gives  <j>jJs - ( V M j ^  I  { 5 , 1 0 )  cm.T  (7 , i s t h e s t r e s s t e n s o r d e r i v e d f r o m  the approx-  Tk  imate s o l u t i o n  U  .  i  This equation  ensures  convergence  of the a p p r o x i m a t e s o l u t i o n i f the c o - o r d i n a t e are complete in  H  ft  d e f i n e d by Eq. 5.7  .  By n o t i n g t h a t t h e e n e r g y  contains  approximation  may  element boundaries.  product  f i r s t d e r i v a t i v e s of the  p l a c e m e n t i t i s s u f f i c i e n t t h a t the assumed f i e l d be c o n t i n u o u s  functions  throughout  dis-  displacement  51. . T h u s a f i n i t e  element  g i v e use to s t r e s s d i s c o n t i n u i t i e s a t F u r t h e r i t c a n be n o t e d  b o u n d a r y c o n d i t i o n s p r e s c r i b e d on 5  r  and  S  M  o r d e r d e r i v a t i v e s and a r e t h e r e f o r e n a t u r a l .  that  the  contain The  f u n c t i o n s t h e r e f o r e need not s a t i s f y t h e s e b o u n d a r y  first  co-ordinate conditions  78 I f the c o - o r d i n a t e f u n c t i o n s are chosen H  i . e . i f they have c o n t i n u o u s f i r s t  from  derivatives,  A  w h i c h e n s u r e s a c o n t i n u o u s s t r e s s f i e l d , and the boundary  SI  E q . 5.10  satisfyall  c o n d i t i o n s , then  ^  "  L  d  Si  cu\ fds  +  r  (5.11)  J  c a n t h e n be w r i t t e n  Thus i f the c o - o r d i n a t e f u n c t i o n s are i n H  ft  i t i s not  n e c e s s a r y to express the energy product i n i t s symmetric form. In g e n e r a l E q . 5.12 may  be w r i t t e n  0  A  where (5.13b)  79 5.2  Nonhomogeneous Boundary  Conditions  The g o v e r n i n g e q u a t i o n s a r e (5.14a) Hi  =  ffijrtj  =  a^. The a p p r o a c h a way  * <r^.  U  T°  6  S  x  i  €  5  M  L  =  * 5  c  (5.14b). (5.14c) (5.i4d)  i n t h i s case i s to change v a r i a b l e s i n such  t h a t t h e p r o b l e m i s r e d u c e d t o one w i t h h o m o g e n e o u s  boundary  c o n d i t i o n s . Thus, assume t h a t t h e r e e x i s t s a  function  IT , whose components have c o n t i n u o u s  first  d e r i v a t i v e s , such t h a t U-  =  ul  ffytlj  =  T l  (5.15b)  i  (5.15c)  Olj/lj +  CU  ;  =  '  (5 15a)  L  D e f i n e a new v a r i a b l e U Ui S u b s t i t u t i n g Eq. 5.16 E q s . 5.15  gives  ~  U  L  L  such t h a t  - u\  i n t o E q s . 5.14  (5.16) and t a k i n g n o t e o f  80  fc:.:  =  .V. +  -  L  /»"  £ *  (5.17a)  £ S„  (5.17b)  J  U;  II  ,=  II  =  ^ " j ^] j n  0  +  0  ^  °  CU  T h u s i n t e r m s o f a"  5^  e  ^  (5.17c) (- )  Sm  t h e p r o b l e m has h o m o g e n e o u s  5  17d  boundary  c o n d i t i o n s , a n d t h e o n l y d i f f e r e n c e f r o m E q s . 5.1 i s t h e i n t r o d u c t i o n of the term  0>. • . I f t h i s t e r m w e r e known MM  i t would be p o s s i b l e to o b t a i n an approximate s o l u t i o n f o r u"  and c o n v e r g e n c e w o u l d be e n s u r e d i f t h e c o - o r d i n a t e  f u n c t i o n s were complete w i t h r e s p e c t to the  space  c o r r e s p o n d i n g t o t h e e n e r g y p r o d u c t g i v e n i n Eq.  5.7.  A s s umi ng U ' T  =  l^if],  (5-18)  t h e r e q u i r e d G a l e r k i n e q u a t i o n s , f r o m E q . 4.25,  [V,  4>J ] A  Thus a s o l u t i o n f o r  u"  I Ui" " Ui I  f / r ' - *J)  T  are  ; ' i,2  3  (5 i:  -  i9)  c o u l d be o b t a i n e d s u c h t h a t  0  (5.20)  81 H o w e v e r , a s o l u t i o n may without  U'  a c t u a l l y knowing  by a s s u m i n g an a p p r o x i m a t i o n the  be o b t a i n e d .  directly for  This is  accomplished  f o r the components of  in  form  J tf  =• ^  T  k  where the s e t of f u n c t i o n s space considered conditions.  a  TH  ^  T  tt  / ^}  with r e s p e c t to the nonhomogeneous boundary  such I £  a  _  (fi-  +  U-  i t follows that  I —^  )  - - (/ u. - "+  [  - (a* «  (  Q  t  )  1  L  f o r c e d b o u n d a r y v a l u e s and  + e  2  the space c o n s i d e r e d , values.  satisfies  L  such  there  os  0  C  ^  oo  { 5  2 2 )  Pi  F u r t h e r the f u n c t i o n  0^  + u  L  that  '  where  complete in the  1S  T h e n as t h e f u n c t i o n  .  (5.21)  1,213  = l  nonhomogeneous boundary c o n d i t i o n s exist  *  ;\ has  homogeneous  t h u s Eq. 5.22  e-  ul),  ]  implies  =  A  (5.23)  o  i s an a r b i t r a r y f u n c t i o n  Q *y 3  with homogeneous f o r c e d  in  boundary  Thus i n p a r t i c u l a r [u T  (u-;  w i ,  ft  j  =  A  o  T-i,,.,  p  =  I, • •  • L.  82  or [*r.  41  ]  A  -  [ U T  u' ,ft]  (5.24)  T  Eq. 5.19 c a n t h e r e f o r e b e w r i t t e n  ta ,^ ] T  T  Now  from Eq.  A  (/;,4> ). T  [u;,^ ] r  (5. 5)  A  2  5.17a si  U s i n g Eqs. 5.15 a n d the f a c t t h a t  j)* = 0  on 5  U  gives  J  (5.26)  T h u s E q s . 5.26 a n d 5.7 e n a b l e E q . 5 . 2 5 t o be w r i t t e n  which are the G a l e r k i n equations that govern a s o l u t i o n in  H  ft  when t h e b o u n d a r y c o n d i t i o n s a r e n o n h o m o g e n e o u s .  83 Written out i n f u l l fl  u s i n g E q . 5.7 t h e s e e q u a t i o n s r-  become  5M  SI  (5.28) A  3  T  i n which i t has been r e q u i r e d t h a t t h e a p p r o x i m a t e s o l u t i o n for  g i v e n by E q . 5.21 s h o u l d s a t i s f y t h e n o n h o m o g e n e o u s  f o r c e d boundary c o n d i t i o n s and t h a t the c o - o r d i n a t e (j).  functions  s h o u l d s a t i s f y homogeneous f o r c e d boundary c o n d i t i o n s .  J  5.  The f o r c e d b o u n d a r y c o n d i t i o n s a r e t h o s e o c c u r i n g on  U  In g e n e r a t i n g a f i n i t e e l e m e n t s o l u t i o n t h e r e i s no n e c e s s i t y t o i n t r o d u c e d i f f e r e n t sponding to the  IJJ a n d  §  in  tions associated with the degreesof 5  U  corre-  co-ordinate functions.  f i n i t e element approximation l i e on  approximations  the co-ordinate  For a func-  f r e e d o m t h a t do n o t  s a t i s f y h o m o g e n e o u s c o n d i t i o n s on 5  .  U  Eqs.  5.28 a r e s o l v e d by f i r s t s p e c i f y i n g t h e v a l u e s o f t h a t l i e on equations.  S  u  and e l e m i n a t i n g t h e s e  The remaining  a^ T  from the  co-ordinate functions  satisfy  h o m o g e n e o u s f o r c e d b o u n d a r y c o n d i t i o n s and t h u s o n l y one s e t o f c o - o r d i n a t e f u n c t i o n s n e e d be A solution in  H  A  introduced.  c a n be a c h i e v e d by  choosing  co-ordinate functions that s a t i s f y a l l the nonhomogeneous boundary c o n d i t i o n s and have  continuous  84 first derivatives.  I n t h i s c a s e E q . 5.28 may be w r i t t e n  or (5.30) w h i c h i s t h e same a s E q . 5 . 1 3 a o b t a i n e d eous boundary c o n d i t i o n  f o r t h e homogen-  problem.  It remains to demonstrate that the approximate solution  u.^ o b t a i n e d  c o r r e c t answer.  f r o m E q . 5.28 c o n v e r g e s t o t h e  U s i n g E q . 5.23 i t i s p o s s i b l e t o w r i t e (5.31)  A l s o E q . 5.16 may be u s e d t o w r i t e (5.32) S u b t r a c t i n g E q . 5.31 f r o m E q . 5.32 g i v e s A  A  Using the Cauchy Bunyakovsky  inequality gives  85 Eq. 5.20 t h e n  Q-  B u t as  implies  i s an a r b i t r a r y f u n c t i o n w i t h  L  forced boundary values,  homogeneous  i t may be s e t e q u a l  to  Ui-u  c  .  Then [ ULi -  , U  U  L  -U  L  L  ]  0  —^  whence I  ~  «i  l  0  A  (5.33)  i . e . t h e a p p r o x i m a t e s o l u t i o n to t h e nonhomogeneous c o n d i t i o n p r o b l e m c o n v e r g e s i n t h e norm o f exact H  A  to the  solution. The  in  j-J  boundary  ft  equations  governing  a f i n i t e element s o l u t i o n  may, on t h e b a s i s o f E q . 5 . 2 7 , be w r i t t e n a s  5?  in which of element  5  ,  r  £  5  M  represent  that p o r t i o n of the boundary  that c o i n c i d e s w i t h  5  T  , 5  M  respectively.  86  CHAPTER 6 BOUNDARY RESIDUAL CONCEPT AND In t h i s c h a p t e r  V I R T U A L WORK •  the equations  G a l e r k i n p r o c e d u r e as h a v e b e e n p r e s e n t e d pared with those concept obtained 6.1  obtained  generated  the  h e r e i n a r e com-  from the boundary r e s i d u a l  as a p p l i e d t o t h e G a l e r k i n p r o c e d u r e and t o  those  u s i n g v i r t u a l work p r i n c i p l e s .  Boundary Residual  Concept  An a l t e r n a t e i n t e r p r e t a t i o n o f t h e p r o c e d u r e t h a t has b e e n p r e s e n t e d (4) c o n s i d e r s with  by  Galerkin  by F i n l a y s o n and  that a boundary r e s i d u a l term i s  the domain r e s i d u a l i n the g o v e r n i n g  Scriven  included  equation.  The unknown p a r a m e t e r s a r e t h e n d e t e r m i n e d s u c h t h a t sum  of these  two  residuals is minimized.  the boundary r e s i d u a l c o n s i d e r e d  Specifically,  is that occuring  p o r t i o n o f t h e b o u n d a r y on w h i c h t h e n a t u r a l conditions  are s p e c i f i e d .  boundary conditions  are  the  on  that  boundary  Thus i f the e q u a t i o n  and  natural  87  Aij Uj  =  -Ii  £ SI ; i,j = 1,2,3  (6.1a)  B;J Uj  =  pi.°  £ 5*  (6.1b)  t h e r e q u i r e d e q u a t i o n s become  (hr)Uj  A* +  ~  ^Bg "j  -p °)^^S  where the components of the approximate  r  =0, T=-|,<> 3_(6.2) (  s o l u t i o n u.  have  been assumed to be U  = T  Z ^ r f e $k  (6.3)  and t h i s a p p r o x i m a t i o n s a t i s f i e s t h e f o r c e d  boundary  c o n d i t i ons . I t w i l l be shown t h a t s u c h an a p p r o a c h  leads  t o t h e same e q u a t i o n s as h a v e b e e n d e v e l o p e d h e r e i n f o r a s o l u t i o n i n H° b u t t h a t d i f f e r e n t e q u a t i o n s a r e o b t a i n e d in  i f the f o r c e d boundary  c o n d i t i o n s are not  homogeneous. C o n s i d e r a g a i n the nonhomogeneous problem d e f i n e d by E q s . 5.14. of the boundary  The e q u a t i o n s o b t a i n e d by t h e a p p l i c a t i o n residual concept are  88  si f(j .nj + c u T  T  -  C°  j f  T  k  J  s  F i r s t c o n s i d e r the g e n e r a t i o n .  As was  mentioned in Chapter  = o  ,  T* 1,2,3  r  of a s o l u t i o n i n  5 the  co-ordinate  f u n c t i o n s m u s t t h e n be s u c h t h a t t h e s t r e s s e s a r e throughout  U n d e r t h e s e c o n d i t i o n s Eq. 6.4 52.  'SL  g i v e s t h e same e q u a t i o n s  Thus the boundary r e s i d u a l as d o e s t h e  i f the s o l u t i o n i s s o u g h t i n n  concept Eq. 6.4.  +  Galerkin  .  A  If a s o l u t i o n i s required in H equations  to (6.5)  w h i c h c o i n c i d e s w i t h Eq. 5.29. procedure  reduces  are  =  'JL  concept  continuous  t h e d o m a i n and a l l t h e b o u n d a r y c o n d i t i o n s  satisfied.  (6.4)  A  the  required  d e r i v e d on t h e b a s i s o f t h e b o u n d a r y r e s i d u a l c a n be o b t a i n e d This  by a p p l y i n g G a u s s ' t h e o r e m t o  gives  (6.6)  89 Therefore n ^  ft  J  +  sz  C/  T  fdX f d  Ji  <j£  SI  T k  (6.7)  '52.  4  cr nu ° ds T  Sa  which corresponds conditions  are  t o Eq. 5.28  d  '  o n l y i f the f o r c e d  boundary  homogeneous.  The b o u n d a r y r e s i d u a l c o n c e p t ,  as  presented  h e r e i n , w o u l d a p p e a r t o be an a t t e m p t t o e x p r e s s Galerkin equations from  in a form in which co-ordinate  functions  a r e a d m i s s a b l e by a more i n t u i t i v e a p p r o a c h  has b e e n p r e s e n t e d  in this thesis.  i s n o t e n t i r e l y c o r r e c t and s h o u l d The  the  required modifications  s o l u t i o n i n E q . 6.2  However the be m o d i f i e d  w o u l d be t o e x p r e s s  i n the  than  approach somewhat. the  assumed  form (6.8)  tt'l  This approximation  i s r e q u i r e d to s a t i s f y the  nonhomogeneous  f o r c e d boundary c o n d i t i o n s , whereas the c o - o r d i n a t e <j>^  should  s a t i s f y homogeneous f o r c e d boundary  T h e n i n t e g r a t i n g Eq. 6.2 equation  by p a r t s r e s u l t s i n t h e  as g i v e n by Eq. 5.28.  i n t e r p r e t a t i o n o f Eq. 6.2  In t h i s c a s e t h e  functions conditions.  correct physical  i s that the domain r e s i d u a l  t h e r e s i d u a l on t h a t p o r t i o n o f t h e b o u n d a r y  plus  corresponding  90 to the n a t u r a l boundary c o n d i t i o n s  a r e made o r t h o g o n a l  t o a f i n i t e number o f d i s p l a c e m e n t s t h a t s a t i s f y homogeneous f o r c e d boundary 6.2  conditions.  V i r t u a l Work The p r i n c i p l e o f v i r t u a l w o r k s t a t e s t h a t f o r  any s y s t e m i n e q u i l i b r i u m t h e i n t e r n a l a n d t h e e x t e r n a l work p e r f o r m e d  by t h e e x i s t i n g s t r e s s s y s t e m a s t h e b o d y  moves t h r o u g h a n y c o m p a t i b l e v i r t u a l d i s p l a c e m e n t be e q u a l .  should  A c o m p a t i b l e d i s p l a c e m e n t i s one t h a t s a t i s f i e s  homogeneous f o r c e d boundary c o n d i t i o n s  and e n s u r e s  con-  t i n u i t y of the p r i n c i p a l d e r i v a t i v e s i . e . those d e r i v a t i v e s o f o r d e r l e s s t h a n m. i f t h e g o v e r n i n g e q u a t i o n h a s d e r i v a t i v e s o f maximum o r d e r 2m function is allowable in H  A  .  Such a d i s p l a c e m e n t  , a n d f u r t h e r , as was  presented  i n C h a p t e r 4 . 3 , a s e t o f s u c h f u n c t i o n s c a n be c o m p l e t e i n  A p p l y i n g t h e v i r t u a l w o r k p r i n c i p l e t o E q s . 5.14 gives  (6.9)  J  +  T=  /, s,  3  91 where t h e body has been g i v e n a c o m p a t i b l e ment  Su w h e n c e i t i s c o n t i n u o u s SUT = O  sents  on 5  C  . In t h i s e q u a t i o n  U  N o t e t h a t as  -^(°TjSu j  +  Tf  field  The p r i n c i p l e o f v i r t u a l  s t a t e s t h a t E q . 6.9 m u s t h o l d f o r a n y £ u  repre-  Tj  the equilibrium stress f i e l d i . e . that stress  t h a t s a t i s f i e s E q s . 5.14.  J  displace-  throughout the whole  T  domain and  virtual  work  .  T  0" j i s s y m m e t r i c T  0>jSiij )dsi =•  U K J S U T . J  fT  ST.  +  C T  T  J  S U  T  dsi  , J )  J 52-  T h u s E q . 6.9 c a n b e w r i t t e n  (<r. 8u T|  J  Tfj  - / a ^ S O dsi  -  T °Su ds T  J  I  J  (- cu  T  + Cr)  5  §u ds  (6.10)  T  =  r  T  O  w h e r e t h e f a c t t h a t t h e s t r e s s f i e l d s a t i s f i e s E q . 5.14d has b e e n u s e d . As  SU  is arbitrary l e t  r  T  T  =  1,2,3  whence and t h e  cj>£ m u s t b e c o n t i n u o u s  forced boundary c o n d i t i o n s . I,....  LT  represented  of the system then  and s a t i s f y  homogeneous  If the co-ordinate  functions  a l l p o s s i b l e degrees of freedom  i n E q . 6.10 i t w o u l d b e a l l o w a b l e  to  92 replace the still  Su  T  with  cj)^ a n d t h e s t r e s s s y s t e m  correspond t o the exact one.  would  However as t h e co-  o r d i n a t e f u n c t i o n s i n g e n e r a l do n o t r e p r e s e n t a l l p o s s i b l e degrees o f freedom stress field  cr j T  o f the system an approximate  i s o b t a i n e d which  cu si  5  T  $  i s d e f i n e d by =  d s  M  T I T  f X^ldsi.  +  "a  T °cj) J s T  R  +  (6.  c f* i e  15  T  ='i2.3  T  T h i s e q u a t i o n i s t h e same a s E q . 5.28.  Thus  for the type o f equations considered the equations developed by v i r t u a l  work c o r r e s p o n d t o t h o s e  by t h e G a l e r k i n p r o c e d u r e .  generated  93  CHAPTER 7 FINITE  ELEMENT S O L U T I O N OF A NON-SYMMETRIC PROBLEM  The p r o b l e m c o n s i d e r e d i s t h a t o f p a n e l as a n a l y s e d b y O l s o n ( 1 4 ) u s i n g f i n i t e e l e m e n t s .  flutter This  a n a l y s i s was a c h i e v e d u s i n g v i r t u a l w o r k p r i n c i p l e s a n d a l t h o u g h no c o n v e r g e n c e  p r o o f was p r e s e n t e d t h e a c c u r a c y  o f t h e r e s u l t s was j u s t i f i e d by t h e a g r e e m e n t o b t a i n e d i n comparison  w i t h known s o l u t i o n s . T h e c o n c e p t s ' p r e s e n t e d  herein enable the convergence p r o v e n , as w i l l be d e m o n s t r a t e d  o f h i s a n a l y s i s t o be i n the f o l l o w i n g d i s -  cussion . The p r o b l e m c o n c e r n s t h e b e h a v i o u r o f a p a n e l over which /|  a supersonic a i r stream flows i n the p o s i t i v e  direction.  The s p e c i f i c q u e s t i o n i s t h a t o f d e t e r m i n i n g  the c o n d i t i o n s under which t h e motion o f t h e panel becomes unstable.  T h e p a n e l i s a s s u m e d t o be a s t r e s s f r e e p l a t e  and o n l y o n e d i m e n s i o n a l d e f o r m a t i o n s a r e c o n s i d e r e d . N e g l e c t i n g the e f f e c t of the a i r entrapped below the panel  94 the d i f f e r e n t i a l e q u a t i o n and b o u n d a r y c o n d i t i o n s i n f i n i t e s i m a l motion  governing  o f a simply supported panel a r e (14)  5.2Y + & t i z i k 3V + rn_L £ V  £!v  +  Yfr)  = Yfij - YVO) = Y'Y')  +  * O  (7.1a)  =0  (7.1b)  t A  [  *" 3  w h e r e m. i s t h e p a n e l mass p e r u n i t a r e a , g_ t h e d y n a m i c pressure, U the freestream v e l o c i t y , M Mach n u m b e r , D t h e p l a t e b e n d i n g x~ QJL-  the freestream  r i g i d i t y and  y~ <W-»  , B~ <?q_ L JD(-^~i)'^ ' a r e t h e non d i m e n s i o n a l 3  d e f l e c t i o n , streamwise parameter,  ffl  c o - o r d i n a t e , and  aerodynamic  respectively. Assuming a s o l u t i o n i n the form  Y =  (7-2)  w h e r e , i n g e n e r a l , cK i s a c o m p l e x number  oc -  +  ,  Eq. 7.1 b e c o m e s ////  Be/'tf .(f . •+ Q  Au - A ij = 0  (7.3a)  95  yM= f(°)* tfO) = O  where  A  (7.3b)  i s an e i g e n v a l u e o f the form )\ = - B f L / u ) n M ; - 2 ) / f M - l ) j o i w  Thus the problem  reduces  (WL'/D)**  (7.4)  tothe determination  o f t h e e i g e n v a l u e s o f t h e n o n - s y m m e t r i c E q s . 7.3 u s i n g a f i n i t e element  approximation.  F i r s t consider the simplified equation  that  i s o b t a i n e d i f B= O . E q s . 7.3 t h e n b e c o m e if'"  1  - > Lj  In t h i s c a s e t h e  A  =  O  -  i/'Vo)  (7.5a)  s  </"(') = 0  (7.5b)  o p e r a t o r A i s g i v e n by  =  i  (7.6)  4  <LK+  U s i n g E q . 7.5b i n c a n r e a d i l y b e v e r i f i e d  that  /Au,r) = Cu,/J = f ' u V i * L>,o7  ( 7  '  7 )  96  whence t h e o p e r a t o r i s s y m m e t r i c .  I t i s also positive  b o u n d e d b e l o w as c a n be s e e n f r o m t h e f o l l o w i n g  develop-  ment.  If  A £ XI  »  2  fuYx,)- u'ta.))' = ( J " i . u"ft) A ) Ju X  J  4  The same i n e q u a l i t y r e s u l t s i f u ( x f +  X, ^ X  l i f e ) - aa'(x,)uYx ) 2  o  £  u  . Thus  £  j'uVO^t  Jo J o  o  As t h e f u n c t i o n  e  i s i n B.  u(o) = u(l) = o  Jo  , whence (7.8)  J  Jo  o  Now Jo  whence  U.M  2  = ( j \ a70dt)'  97  x dx  u(x) dx 4 2  uW2^X  E q s . 7.8 and 7.9  *  a llh) 2dt  f  (7.9)  imply  f'u(x) cJx 2  Recall  J.  ^ i  fa,u j = | u d x  f u'M^x  and f r o m E q . 7.7  a  f Au u) = f ( u / f d x t  o  Jo  Thus f«,<*)  «  i (Au,u)  and t h e o p e r a t o r i s s e e n t o be p o s i t i v e b o u n d e d  below.  The e q u a t i o n s g o v e r n i n g a f i n i t e e l e m e n t o f E q s . 7.5 a r e g i v e n by E q . 4.29 operator.  solution  i f K i s the i d e n t i t y  F o r t h i s p r o b l e m t h e y become  latf(titi'->$it))dx~o, k'l  j..,-"  .  1 0 )  Jo  i n w h i c h an a p p r o x i m a t e s o l u t i o n f o r ij w i t h i n e a c h has b e e n a s s u m e d _  ( 7  €  H  element  t o be ft  i«.  = k=7 1 f e <f* Q  (7.11)  98 E q . 7.10  c o r r e s p o n d s to t h a t d e v e l o p e d i n (14) u s i n g  v i r t u a l work p r i n c i p l e s .  The r e q u i r e d e i g e n v a l u e s a r e  o b t a i n e d by s e t t i n g t h e c o e f f i c i e n t m a t r i x o f Eq.  7.10  equal to z e r o . From t h e r e s u l t s o f C h a p t e r 4 c o n v e r g e n c e t h e e i g e n v a l u e s and e i g e n v e c t o r s i n H  i s ensured i f the  ft  co-ordinate f u n c t i o n s are complete in H  .  A  product in H  fi  o f o r d e r two.  of  The  energy  i s g i v e n by Eq. 7.7 and i n v o l v e s d e r i v a t i v e s Thus f u n c t i o n s i n  H  A  must have g e n e r a l i z e d  s e c o n d d e r i v a t i v e s and h e n c e c o n t i n u o u s f i r s t  derivatives.  The f i n i t e e l e m e n t a p p r o x i m a t i o n m u s t t h e r e f o r e e n s u r e s l o p e c o n t i n u i t y a c r o s s e l e m e n t b o u n d a r i e s and be s u c h that the f o r c e d boundary  c o n d i t i o n s are s a t i s f i e d .  Such  elements w i l l then give a s o l u t i o n that converges to the c o r r e c t answer i f they are complete i n H  A  condition f o r completeness  .  A sufficient  ( s e e C h a p t e r 4) i s t h a t t h e  e l e m e n t a p p r o x i m a t i o n s be b a s e d upon a c o m p l e t e of order not l e s s than  two.  In t h e a b o v e m e n t i o n e d  paper these c o n d i t i o n s  a r e a l l s a t i s f i e d by c h o o s i n g an e l e m e n t that i s based  upon  polynomial  approximation  99 and by c h o o s i n g d e g r e e s o f f r e e d o m t h a t c o r r e s p o n d t o d e f l e c t i o n s and r o t a t i o n s a t e a c h end o f t h e e l e m e n t .  Thus c o n v e r g e n c e i n h  fi  i s e n s u r e d and t h e  r e s u l t s obtained i l l u s t r a t i n g the monotonic  convergence  a c t u a l l y f o u n d a r e r e p r o d u c e d i n T a b l e 7.2.  Such con-  v e r g e n c e c o u l d have been p r e d i c t e d from the c o n v e r g e n c e p r o p e r t i e s of the R a y l e i g h - R i t z procedure. However i n the c a s e where the a e r o d y n a m i c B  i s n o n - z e r o t h e o p e r a t o r i n E q . 7.3 A  +  =  which i s non-symmetric not can  becomes  B i  (7.13)  and t h e R a y l e i g h - R i t z p r o c e d u r e i s  a p p l i c a b l e , w h e r e a s t h e G a l e r k i n p r o c e d u r e may  be u s e d .  parameter  In o r d e r t o be e n s u r e d o f c o n v e r g e n c e E q .  be e x p r e s s e d i n a f o r m t h a t c o r r e s p o n d s t o E q .  still 7.3a 3.43,  f o r w h i c h c o n v e r g e n c e c r i t e r i a a r e known, i n t h e f o l l o w i n g manner. L7  -  >Ttj  =  O  (7.14)  1 00  where  T  r l*  =  + B  A  i  (7.15)  Further, i n order to apply the results developed  with  r e s p e c t t o E q . 3.43 i t i s n e c e s s a r y t o show t h a t T completely continuous. U  +  E>  To do t h i s c o n s i d e r t h e e q u a t i o n (7.16)  U  under t h e b o u n d a r y c o n d i t i o n s 7.3b. function of X  is  jj?  i s some a r b i t r a r y  .• Now i f (7.17)  a = then  G(x,£)  i n g t o Eq. 7.16.  i s known a s t h e Green's f u n c t i o n c o r r e s p o n d If  G(x,i)  i s such t h a t  ° f 6(x,f)* dx di then  T  <  i s completely continuous i n  (7.18) L (JI) 2  (9) s i n c e  U =  (7.19)  I t w i l l be d e m o n s t r a t e d t h a t E q . 7.18 i s v a l i d a n d t h u s that  T  i s completely  continuous.  101 The e x i s t e n c e o f a Green's  f u n c t i o n i s ensured  i f E q . 7.16 h a s a u n i q u e s o l u t i o n f o r a r b i t r a r y J£ , w h i c h i s t h e case i f the homogeneous e q u a t i o n has a unique solution.  The g e n e r a l s o l u t i o n o f t h e homogeneous  c a n be f o u n d by  equation  assuming C^JL  U. =  (7.20)  w h i c h when s u b s t i t u t e d i n t o t h e e q u a t i o n g i v e s fm -r B)m  -  3  m=0,  0  a(i+L/3), 2  -a, £  where Thus u  = C, + C e  +  d  C £  +  3  Ce 4  w h i c h may be w r i t t e n ii -  D,  + D  3  e  -ax  _ /2 _ _ ax/s rIXC cosv/lax + ILC siaf3ax  (7 2 1 )  ax  +  K  1  w h e r e t h e D's a r e c o n s t a n t s a n d a r e c o m b i n a t i o n s o f t h e Cs %  .  A p p l y i n g the boundary  c o n d i t i o n s leads to a s e t  of f o u r s i m u l t a n e o u s e q u a t i o n s f o r the D  , which  was  shown n u m e r i c a l l y t o have a u n i q u e s o l u t i o n f o r a l l B interest.  of  Table 7.1 i l l u s t r a t e s t h e r e l a t i o n s h i p b e t w e e n  102  AERODYNAMIC  PARAMETER  DETERMINANT VALUE  B 614.12 512.00 421.87 343.00 274.62 216.00 166.37 125.00 91.12 64.00 42.87 27.00 8.00 1 .00 0.12  T A B L E 7.1 DETERMINANT VALUE VERSUS PARAMETER  -69,121 . -41 ,487. -24,690. -14,573. - 8,556. - 5,034. - 3,006. - 1 ,850. - 1 ,190. 800. 556. 388. 167 . 42. 10.  AERODYNAMIC  103  the d e t e r m i n a n t of the m a t r i x  B  and  .  T h u s Eq. 7.16  a u n i q u e s o l u t i o n f o r a r b i t r a r y J? ; w h e n c e t h e function exists.  Green's  F u r t h e r the Green's f u n c t i o n f o r  the  problem i s constructed,  by d e f i n i t i o n , s u c h t h a t i t  s a t i s f i e s the f o l l o w i n g  conditions.  d6  +  (x,r)  4  6(0,f)  =  6  dG(x,s)  G (o,f)  =  n  •+  B  =  O,  o<x<i  (7.22a)  (7.22b)  O  ^_GU±l)  =  f<x<\  °7  5>x  3X'  has  (7.22c) (7.22d)  Further tinuous  G (x, f)  at  9  X*  €  G'(x,f),  ~r  o  m u s t be con-  f)  and 6 fx,  6  6"(x,  dx  i)  -  3 6(x,f) dx  =  3  2  3  The c o n t i n u i t y o f t h e g e n e r a l  1  s o l u t i o n g i v e n i n Eq.  (7.22e)  7.21  and t h e a b o v e d e f i n i t i o n s e n s u r e t h e c o n t i n u i t y o f  the  Green's f u n c t i o n f o r the problem.  continuous  Therefore  as a  f u n c t i o n d e f i n e d on a c l o s e d b o u n d e d i n t e r v a l i s b o u n d e d i t f o l l o w s t h a t Eq. 7.18 completely  continuous  holds.  T  is  that  are  Thus the o p e r a t o r  i n the space of f u n c t i o n s  104 s q u a r e summable o v e r continuous in  H  A  £ 0, IJ  , and thus i t i s c o m p l e t e l y  .  Choosing c o - o r d i n a t e f u n c t i o n s that are complete in  H , as O l s o n d i d , t h u s e n s u r e s t h e c o n v e r g e n c e o f t h e A  e i g e n v a l u e s a n d e i g e n v e c t o r s o f E q s . 7.3 when t h e g o v e r n i n g e q u a t i o n s a r e g e n e r a t e d by t h e G a l e r k i n o r v i r t u a l  work  procedures. The s o l u t i o n o f t h e p r o b l e m i s c h a r a c t e r i z e d by t h e f a c t t h a t t h e two l o w e s t e i g e n v a l u e s a r e r e a l a n d distinct f o r a l l values of B  B  l e s s than the c r i t i c a l  which f i r s t causes f l u t t e r ; but they approach  C R  o t h e r and c o a l e s c e t o  A  when  CR  E> = B  .  C R  C R  and ) t  i t i s noted that the convergence  c R  .  each  T a b l e 7.3  r e p r o d u c e s t h e r e s u l t s o b t a i n e d and d e m o n s t r a t e s convergence obtained f o r B  value  the  In t h i s  case  i s not monotonic.  The r e q u i r e d G a l e r k i n e q u a t i o n s a r e o b t a i n e d d i r e c t l y from a p p l i c a t i o n o f t h e p r o c e d u r e t o Eq. 7.3a. In t e r m s o f a f i n i t e e l e m e n t a p p r o x i m a t i o n t h e e q u a t i o n s are £  <  - > > # # J < / X  =  O,  w h i c h c o r r e s p o n d t o t h o s e d e r i v e d by O l s o n on t h e b a s i s o f v i r t u a l work  principles.  (  7  -  2  3  )  105  Total Number of Elements 1 2 3 4 Exact Value  First  Eigenvalue  Eigenvalue  % Error  X  120 98 97 97  Second  .00 .18 .57 .46  23.6 0.79 0.16 0.05  % Error 2520.00 1920.00 1 595 .61 1570.87  61 .5 23.1 2.38 0.79  1558.55  97.41  T A B L E 7.2, E I G E N V A L U E R E S U L T S WHEN B=0  Total Number of Elements 1 2 3 4 Exact V a l ue  CR  'CR B CR  % Error  453.56 398.54 340.72 342.34  32.10 16.07 - 0.77 - 0.30  343.36  A  CR  1320.00 1206.31 1027.85 1043.46 1051.80  T A B L E 7.3, E I G E N V A L U E C O A L E S C E N C E  RESULTS  % Error 25.50 14.69 - 2.28 - 0.79  1 06  CHAPTER 8 A FINITE  ELEMENT SOLUTION OF THE L I N E A R VISCOUS FLOW PROBLEM  In t h i s c h a p t e r a f i n i t e e l e m e n t  solution will  be d e v e l o p e d f o r t h e two d i m e n s i o n a l f l o w o f an s i b l e v i s c o u s f l u i d i n which neglected i n comparison  incompres-  i n e r t i a l e f f e c t s may be  to viscous e f f e c t s .  The d e v e l o p -  ment f u r t h e r i l l u s t r a t e s t h e a p p l i c a t i o n o f t h e G a l e r k i n procedure to problems  i n v o l v i n g m o r e t h a n one d e p e n d e n t  variable. A programme i s w r i t t e n t h a t g e n e r a t e s a s o l u t i o n in the  space, which  requirements.  Comparison  i s b a s e d u p o n minimum  w i t h known s o l u t i o n s a r e p r e s e n t e d  that i l l u s t r a t e the convergence 8.1  convergence  obtained.  Generation of Equations Governing a F i n i t e Element S o l u t i o n The e q u a t i o n s g o v e r n i n g t h e f l o w o f an  sible viscous fluid are  incompres-  107  (i)  Equilibrium  =  Oij.flj  (ii)  Constitutive  (iii)  Compatibility 4j  in which tive, p  V  = ^  V;  -  V-  €  ^(Vi.j + V ^ )  - v . £  TJt '  6  L  S  the material  8  deriva-  the s t r a i n rate tensor,  the and  pressible. and p r e s s u r e  >  (- )  y  E q . 8.3b s p e c i f i e s t h a t t h e f l o w be i n c o m -  Formulating  B  (8.3b)  t h a t p o r t i o n o f t h e b o u n d a r y on w h i c h t h e v e l o c i t y  is s p e c i f i e s .  L  £52  c o e f f i c i e n t of v i s c o s i t y , p the h y d r o s t a t i c pressure, Sy  8  (8.3a)  V°  D/ D t  (  £ SI  0  i s the v e l o c i t y ,  t h e d e n s i t y , d^.  ST  the problem i n terms of the v e l o c i t y  g i v e s t h e w e l l known N a v i e r - S t o k e s  equations:-  Dfc  3c  108 together with the c o n s t r a i n t o f i n c o m p r e s s i b i 1 i t y . I f the r a t i o o f the i n e r t i a ! f o r c e s to the v i s c o u s f o r c e s i s small compared to u n i t y then the system o f e q u a t i o n s t o be s o l v e d c a n be r e d u c e d t o  -M W/ V  ) +  iu,  +v  P..  =  - Vj.j  0-..) a,-1 L  V  £  p>*i  6  5 2  (  '  8  5  a  )  =  O  a SI  (8.5b)  =  T°  6 5  (- )  =  V-'  o  8  T  e S  5c  (8.5d)  v  A d e t a i l e d d i s c u s s i o n o f the range o f a p p l i c a b i l i t y o f t h e s e e q u a t i o n s may be f o u n d i n S c h l i c h t i n g ( 1 5 ) . E q s . 8.5 may be p l a c e d i n t o c o r r e s p o n d e n c e  with  the general system o f e q u a t i o n s . (8.6) , uj >  =  <  , v  , p >  where  = < u>,, w  Throughtout  t h i s s e c t i o n r e p e a t e d s u b s c r i p t s n , m. w i l l be  £  3  V |  t  summed 1, 2, 3 a n d r e p e a t e d s u b s c r i p t s ^ , j 1, 2.  (8.7)  w i l l be summed  109 N o t e t h a t E q . 8.5b i s i n c l u d e d as o n e o f t h e e q u a t i o n s t o w h i c h t h e G a l e r k i n p r o c e d u r e i s t o be a p p l i e d . In t h i s c a s e t h e r e s i d u a l i s a v e c t o r  w i t h t h r e e com-  ponents which a r e v  ' 'ma  rn  jm  The G a l e r k i n p r o c e d u r e t h e n s p e c i f i e s t h a t t h i s r e s i d u a l v e c t o r s h o u l d be made o r t h o g o n a l t o a r b i t r a r y v a r i a t i o n s o f t h e assumed v e c t o r  where  OJ^ .  The r e q u i r e d e q u a t i o n s a r e  u ; ^ i s a r b i t r a r y a n d t h e r e f o r e t h i s e q u a t i o n may be  written A  ( V , Z  r  r  )  =  ( Kn.^^- {')  Aa = O T -  where of  U)^  LO  t  (8.10) 1,2,3  i s an a r b i t r a r y v a r i a t i o n o f t h e T t h c o m p o n e n t  .  Once a g a i n i t i s c o n v e n i e n t n o t t o i n t e r p r e t txJ„  i n t e r m s o f E q s . 8.5 b u t i n s t e a d t o u s e  ™  T  J'J  - ~ io v  (8  '  llb)  no  The e n e r g y p r o d u c t f o r t h i s m a t r i x o f o p e r a t o r s w i t h r e s p e c t to homogeneous boundary  conditions  is (8.12)  in which  si  (7.. i s t h e s t r e s s t e n s o r c o r r e s p o n d i n g t o U>  and  a  J  (8.13)  If m  R e a r r a n g i n g , a p p l y i n g Gauss' t h e s y m m e t r y o f 0^-  theorem  and m a k i n g u s e o f  gives SI  Thus the o p e r a t o r i s symmetric  and t h e r e f o r e a v a r i a t i o n a l  f o r m u l a t i o n e x i s t s , as has b e e n d e v e l o p e d by J o h n s o n However the i n t r o d u c t i o n of such a f u n c t i o n a l i s  (5).  unnecessary  as t h e f o l l o w i n g d e r i v a t i o n i l l u s t r a t e s . From t h e s y m m e t r y p r o o f t h e e n e r g y p r o d u c t o f the o p e r a t o r i s  Ill  5i  T h e a p p r o p r i a t e f i n i t e e l e m e n t e q u a t i o n s c a n be d e r i v e d f r o m Eq. 5.27 by a s s u m i n g t h a t t h e a p p r o x i m a t e s o l u t i o n w i t h i n each element i s UJ  where  hi  - e  T  (8.15a)  =1,2.  (8.15b)  - €  T h e n u s i n g E q . 8.14  i t c a n be s e e n t h a t (8.16a) 51*  (8.16b) .si.  The r e q u i r e d e q u a t i o n s , f r o m E q . 5.27, may be w r i t t e n  (h Kj  +  V ) Cj  - P*  - [ptrfil*  +  (  T  T * ^ -  T -  I , *  (  8  - '  7  a  )  112  Convergence  o f t h e a p p r o x i m a t e s o l u t i o n g e n e r a t e d by  E q s . 8.17 i s e n s u r e d i f t h e c o - o r d i n a t e f u n c t i o n s a r e complete i n  H  .  A  From E q . 8.14 i t c a n be s e e n t h a t t h e  e n e r g y p r o d u c t i n v o l v e s d e r i v a t i v e s o f v, a n d v one, and z e r o o r d e r d e r i v a t i v e s o f maximum o r d e r d e r i v a t i v e o f oJ  p  .  (the T  T  of order  a  Denote the component o f t h e  unknown f u n c t i o n u> ), o c c u r i n g i n t h e e n e r g y p r o d u c t by m  T  ; t h e n m,= m  e  = I ,  proves completeness  rn = 0  .  3  O l i v e i r a ' s work  i f the elements ensure c o n t i n u i t y o f  t h e m - l d e r i v a t i v e s a n d t h e a p p r o x i m a t i o n f o r e a c h comT  p o n e n t w i t h i n e a c h e l e m e n t i s b a s e d upon a c o m p l e t e p o l y nomial o f degree not less then m  T  . Thus i t i s s u f f i c i e n t  t h a t t h e c o - o r d i n a t e f u n c t i o n s f o r tu, = V, a n d uJ = K be a  4  b a s e d upon a f i r s t o r d e r p o l y n o m i a l a n d e n s u r e c o n t i n u i t y of the v e l o c i t y .  The c o - o r d i n a t e f u n c t i o n s f o r uu =p  o n l y be b a s e d upon a z e r o o r d e r p o l y n o m i a l .  3  need  Thus t h e  p r e s s u r e a p p r o x i m a t i o n w i t h i n e a c h e l e m e n t may be a c o n s t a n t , and need n o t e n s u r e any c o n t i n u i t y between  elements.  F u r t h e r i t c a n be n o t e d t h a t b o u n d a r y c o n d i t i o n s that involve the pressure or f i r s t d e r i v a t i v e of the v e l o c i t y are t h e n a t u r a l boundary c o n d i t i o n s . Hence t h e boundary  11 3  c o n d i t i o n on  Sy i s t h e o n l y f o r c e d b o u n d a r y  The a p p r o x i m a t e components  solution V  V, a n d  forced boundary  uJ  z  , o r more p r e c i s e l y t h e  a  , must t h e r e f o r e s a t i s f y  ft  this  condition.  If a complete s e t of functions H  condition.  i s chosen  from  t h e n as i n C h a p t e r 5.2 i t i s n o t n e c e s s a r y t o e x p r e s s  the energy product i n i t s symmetric  form.  In t h i s  case  f r o m E q . 8.12 i t c a n be s e e n KJ./I'-P'O.JW  (8.18)  si  si  whence T  A  ,  T  J ' J  Y  = I,  (8.19a)  2.  k  e.3 w  >  0  (8.19b) A  The r e q u i r e d e q u a t i o n s f o r a s o l u t i o n i n  are then  o b t a i n e d f r o m E q . 5.30:  te--  O  I,--  £»(,...£"  -  N  (8.20b)  114  For  a f i n i t e element s o l u t i o n i n  H*  the e l e m e n t s must  e n s u r e c o n t i n u i t y o f t h e p r e s s u r e and o f t h e f i r s t d e r i v a t i v e s of the v e l o c i t y .  The d e v e l o p m e n t o f an  e l e m e n t t o s a t i s f y t h e p r e s s u r e r e q u i r e m e n t p r e s e n t s no p r o b l e m ; t o d e r i v e an e l e m e n t t h a t s a t i s f i e s t h e v e l o c i t y r e q u i r e m e n t s i s s o m e w h a t more c o m p l i c a t e d b u t s u c h an e l e m e n t has b e e n d e v e l o p e d f o r u s e i n t h e a n a l y s i s o f p l a t e bending problems ( 1 ) .  Thus the g e n e r a t i o n o f a s o l u t i o n  is c e r t a i n l y f e a s i b l e i n this case.  However such a  s o l u t i o n w i l l n o t be g e n e r a t e d i n t h i s 8.2  Development  of a F i n i t e Element  thesis. Model  In t h i s s e c t i o n a f i n i t e e l e m e n t m o d e l i s developed in  [\^  a s s u m i n g z e r o body f o r c e s .  The  r e q u i r e d e q u a t i o n s a r e g i v e n i n Eqs. 8.17, o m i t t i n g t h e body f o r c e  term. T h e minimum c o n d i t i o n s f o r c o m p l e t n e s s p r e s e n t e d  i n t h e p r e v i o u s s e c t i o n a r e s a t i s f i e d by a s s u m i n g  >1 C  €  (const)  W h i c h i n t h e n o t a t i o n o f E q s . 8.15 i s  T-  \,i  (8.21a) (8.21b)  115 T*  p«  where }f  V*^  ^  _>  1,2  (const.)  e  i s the v e l o c i t y i n d i r e c t i o n T  a t node k, and  i s the p r e s s u r e i n e l e m e n t € , assumed c o n s t a n t .  The r e q u i r e d  condition  of c o n t i n u i t y  by a s s u m i n g a t r i a n g u l a r  c a n t h e n be  satisfied  element with s i x degrees of  f r e e d o m f o r t h e v e l o c i t y and one d e g r e e o f f r e e d o m f o r the p r e s s u r e . T h i s  e l e m e n t i s shown  below.  N o t e t h a t t h e l o c a t i o n o f t h e n o d e s and t h e l i n e a r i t y o f the assumed  v e l o c i t y f i e l d ensure that  the  v e l o c i t y i s c o n t i n u o u s a c r o s s e l e m e n t b o u n d a r i e s and t h u s t h r o u g h o u t t h e w h o l e d o m a i n as  required.  Although the element would appear i t i s i n f a c t not s u i t a b l e  satisfactory,  f o r problems i n which a  116 r e l a t i v e l y l a r g e number o f v e l o c i t i e s a r e  prescribed.  The d i f f i c u l t y s t e m s from t h e i n c o m p r e s s i b i 1 i t y which c o n s t r a i n s the nodal If the domain i s s u b d i v i d e d  velocities  condition  i n each element.  into E elements there  are  t h e n E c o n s t r a i nt? e q u a t i ons f o r t h e 2 ( E + 2 ) v e l o c i t y of freedom.  T h u s t h e r e a r e E+4  degrees of freedom remain-  ing to s a t i s f y the e q u i l i b r i u m c o n d i t i o n .  Therefore  t h e number o f p r e s c r i b e d v e l o c i t i e s e x c e e d s (E+4) s o l u t i o n can  degrees if  no  exist.  E q u i v a l e n t l y , t h e d i f f i c u l t y c a n be s e e n c o n s i d e r i n g the e q u a t i o n s the r e q u i r e d nodal are obtained the  t h a t m u s t be s o l v e d t o  velocities.  f r o m E q s . 8.17  The a s s e m b l e d  and t h e y may  by obtain  equations  be e x p r e s s e d  form  2L  E  V 11  R 0  K  o o  in  117 where the  K  a r e unknown.  correspond to p r e s c r i b e d v e l o c i t i e s I f F v e l o c i t i e s m u s t be p r e s c r i b e d  s a t i s f y the f o r c e d boundary  conditions  o f e q u a t i o n s may  to  be r e d u c e d  then t h i s  and to system  (8.22) V  >7 where B = 2L-F.  0  I  In o r d e r f o r a u n i q u e s o l u t i o n t o  f o r t h e n o d a l v e l o c i t i e s and e l e m e n t K defined  R  exist  p r e s s u r e s the m a t r i x  by  K  must have a non-zero  2  i  ,  K i  1  0  determinant.  2  Thus a l l i t s rows must  be l i n e a r l y i n d e p e n d e n t , w h i c h means B - E i . e . 2 L - F > E . B u t L=E+2 w h e n c e E+4>F b e c o m e s a n e c e s s a r y c o n d i t i o n f o r a unique s o l u t i o n to e x i s t . c o n c l u s i o n t h a t was presented  above.  W h i c h o f c o u r s e i s t h e same  o b t a i n e d by t h e m o r e p h y s i c a l  argument  118 To a v o i d t h i s d i f f i c u l t y i t i s a d v i s a b l e t o modify the element.  T o do t h i s a s s u m e a v e l o c i t y  field  t h a t i s based upon a s e c o n d o r d e r p o l y n o m i a l , i . e . (8.23) and a t r i a n g u l a r e l e m e n t w i t h n o d e s a t t h e mid p o i n t o f t h e s i d e s and a t t h e c o r n e r s .  The p r e s s u r e a s s u m p t i o n  remains  unchanged.  if  In t h i s c a s e E e l e m e n t s r e s u l t i n 6 ( E + 1 ) v e l o c i t y  degrees  o f f r e e d o m a n d t h e number o f p r e s c r i b e d v e l o c i t i e s m u s t e x c e e d 5E+1 f o r no s o l u t i o n t o e x i s t .  Once a g a i n c o n t i n u i t y  o f v e l o c i t y i s e n s u r e d , as t h e t h r e e n o d a l v e l o c i t i e s on each s i d e o f the element u n i q u e l y determine the q u a d r a t i c d i s t r i b u t i o n assumed, but c o n t i n u i t y of v e l o c i t y g r a d i e n t is not obtained.  119 The a s s u m e d v e l o c i t y f i e l d i s e x p r e s s e d  directly  i n terms o f a r e a c o - o r d i n a t e s (3) which a r e c h a r a c t e r i z e d by t h e p r o p e r t y t h a t t h e y a u t o m a t i c a l l y s a t i s f y E q .  4.4.  Thus  where  V , _  (8.25)  1  i n w h i c h A i s t h e a r e a o f the t r i a n g l e and c o - o r d i n a t e s o f node N.  (* ,^ ) are the N  F i g . 8.1 f u r t h e r d e f i n e s  N  these  co-ordi nates. E q s . 8.17 t h e r e f o r e t a k e t h e f o r m  T=l,2  (8.26a)  1 20  Fig.8.1  DEFINITION OF A R E A  COORDINATES.  121  (8.26b)  in which  t h e - )|  €  has been c a n c e l l e d from Eq. 8.26b.  E q s . 8.26 may now be a s s e m b l e d v e l o c i t i e s and p r e s s u r e s . is i n c l u d e d i n Appendix  8.3  Comparison  a n d s o l v e d f o r t h e unknown  The programme used f o r t h i s  1.  w i t h Known S o l u t i o n s  Two p r o b l e m s  are s o l v e d f o r which  s o l u t i o n s a r e known a n d t h e c o n v e r g e n c e element  solution is illustrated.  the exact  of the f i n i t e  The f i r s t  concerns the flow o f a v i s c o u s f l u i d between  example parallel  stationary plates of i n f i n i t e extent that are a distance c2d  a p a r t a n d t h e s e c o n d , t h a t s i t u a t i o n when o n e o f t h e  p l a t e s i s moving with r e s p e c t to the o t h e r a t c o n s t a n t velocity (Coueette flow).  The e x a c t s o l u t i o n f o r both  t h e s e c a s e s i s p r e s e n t e d by S c h l i c h t i n g ( 1 5 ) a n d t h e s e s o l u t i o n s are summarized below.  In both c a s e s t h e m o t i o n  i s s u c h t h a t t h e a c c e l e r a t i o n t e r m i n E q . 8.4 i s i d e n t i c a l l y zero.  122 (1)  Parallel  Flow,  S  /  <<  ^  ^  s-  _^  s.  s  s  I  /  /  / i/  / ' / s <> ' S ' { L »i  J — /  / / '  The b o u n d a r y c o n d i t i o n s a r e =  The s o l u t i o n i s  ^  (ii)  =  0  C o u e t t e Flow  .HI  /  /  /  i S  1  /  7  7  7—7  /  /  j /  /  /  O  The b o u n d a r y c o n d i t i o n s  are  P ( • H) * p.  P  0  (^H)  ^  pa  The s o l u t i o n i s  -  V(x,u)  Ji.  (u+d)  dp - O  - J-  £^  (d'-Y)  dp . P*_lP' dx  In t h e e x a m p l e c o n s i d e r e d  u the f o l l o w i n g  values  were used: ji  =  20  X I0~  6  |t|  sec  / ^ t  i  The a p p r o x i m a t e s o l u t i o n s c o r r e s p o n d i n g  to  d i f f e r e n t s u b d i v i s i o n s of the domain were c a l c u l a t e d . s u b d i v i s i o n s e m p l o y e d a r e i l l u s t r a t e d i n F i g . 8.2  and  three The  124 involve  t h e u s e o f 2, 8, and 32 e l e m e n t s .  F i g s . 8.3,  and 8.5  show t h e r e s u l t s o b t a i n e d f o r t h e p a r a l l e l  problem  f o r the d i f f e r e n t s u b d i v i s i o n s i n F i g . 8.6.  f o r each s u b d i v i s i o n  Two  flow  of the domain.  The r e s u l t i n g v e l o c i t i e s a r e c o m p a r e d w i t h t h e solution  8.4,  exact  values of v e l o c i t y are  plotted  c o r r e s p o n d i n g to the d i f f e r e n t  values  o b t a i n e d f r o m n o d e s a t t h e same h e i g h t o f t h e same triangle.  In e a c h c a s e i t c a n be s e e n t h a t t h e s e  span the e x a c t s o l u t i o n elements  and w i t h i n c r e a s i n g  both values converge  to the exact  values  number o f solution.  Similar  r e s u l t s were o b t a i n e d f o r the C o u e t t e  problem  and F i g . 8.7  flow  i l l u s t r a t e s the c o n v e r g e n c e  obtained.  In b o t h c a s e s t h e e x a c t v e l o c i t y s o l u t i o n quadratic distribution. within  each element  T h u s as t h e a s s u m e d v e l o c i t y  i s q u a d r a t i c i t w o u l d be e x p e c t e d  an a c c u r a t e a n s w e r w o u l d be o b t a i n e d . t h e p r e s s u r e t o be a c o n s t a n t w i t h i n the exact s o l u t i o n  is a  However, each element  i s a l i n e a r d i s t r i b u t i o n would  a more r e a l i s t i c t e s t o f t h e c o n v e r g e n c e  of the  F i g . 8.8 c o m p a r e s t h e e x a c t s o l u t i o n w i t h t h e  field that  assuming whereas appear solution.  pressures  o b t a i n e d on t h e a s s u m p t i o n  that the pressure acts at the  centroid  The p r e s s u r e d i s t r i b u t i o n f o r  of i t s t r i a n g l e .  t h e p a r a l l e l and  Couette  f l o w was  f o u n d t o be t h e same,  125 and e x c e l l e n t a g r e e m e n t was e v e n w i t h t h e two e l e m e n t  o b t a i n e d with the exact subdivision.  solution  126  a)  Fig. 8 . 2  2  Elements  b)  8  Elements  c)  32  Elements  ASSUMED DOMAIN SUB-DIVISIONS .  127  Fig. 8.3  2 E L E M E N T P A R A L L E L FLOW SOLUTION.  1 28  Fig. 8.4  8  E L E M E N T P A R A L L E L FLOW SOLUTION .  129  f.OI  Velocity in f.p.s. O P r e s s u r e p.s.f. x I0  Fig. 8.5  32 E L E M E N T P A R A L L E L FLOW  4  SOLUTION.  Fig. 8.6  V E L O C I T Y DISTRIBUTION IN CHANNEL FOR PARALLEL FLOW  Velocity in x direction ( f. p. s )  Fig. 8.7 V E L O C I T Y DISTRIBUTION IN CHANNEL FOR C O U E T T E  FLOW .  I  0  I  I  .5  Distance  Fig.8.8  PRESSURE  1.0 a l o n g Channel  DISTRIBUTION  I 1.5  (feet)  IN C H A N N E L .  2.0 _ CO  ro  CHAPTER 9 SUMMARY A mathematical  framework f o r the f i n i t e  element  p r o c e d u r e has b e e n p r e s e n t e d t h a t shows t h e m e t h o d t o be a t e c h n i q u e f o r g e n e r a t i n g an a p p r o x i m a t e  solution for a  g i v e n e q u a t i o n i n t e r m s o f known f u n c t i o n s p i e c e - w i s e d e f i n e d over the domain,  and unknown p a r a m e t e r s .  The  unknown p a r a m e t e r s , w h i c h a r e u s u a l l y n o d a l v a l u e s o f t h e r e q u i r e d f u n c t i o n s o r one o f i t s d e r i v a t i v e s , may f o r i n a number o f d i f f e r e n t w a y s .  be s o l v e d  Any t e c h n i q u e t h a t  g i v e s a s o l u t i o n t h a t converges to the c o r r e c t answer with i n c r e a s i n g number o f e l e m e n t s  is equally valid.  The  R a y l e i g h - R i t z p r o c e d u r e and v i r t u a l w o r k p r i n c i p l e s  have  commonly been used to g e n e r a t e the r e q u i r e d e q u a t i o n s . However, the R a y l e i g h - R i t z method i s l i m i t e d to those problems whose s o l u t i o n c o r r e s p o n d s to the s t a t i o n a r y v a l u e o f a known f u n c t i o n a l and t h e v i r t u a l w o r k a p p r o a c h d o e s y i e l d convergence  not  criteria.  U s i n g r e s u l t s o b t a i n e d by M i k h l i n ( 9 ) , i t has b e e n shown t h a t t h e G a l e r k i n p r o c e d u r e e n a b l e s 133  convergence  134  c r i t e r i a t o be s t a t e d f o r a w i d e c l a s s o f p r o b l e m s . those  problems to which the R a y l e i g h - R i t z  method i s  a p p l i c a b l e the G a l e r k i n method c o i n c i d e s with the Ritz procedure,  For Rayleigh-  b u t t h e r e e x i s t many p r o b l e m s f o r w h i c h  G a l e r k i n i s a p p l i c a b l e but Ray1eigh-Ritz  is  not.  It i s o f t e n assumed t h a t G a l e r k i n ' s r e q u i r e s t h a t the assumed c o - o r d i n a t e  method  functions  satisfy  a l l t h e b o u n d a r y c o n d i t i o n s ; h o w e v e r , i t i s shown t h a t suitably formulating  the problem i t i s o f t e n o n l y  to s a t i s f y the p r i n c i p a l boundary  has b e e n shown t o  t h a t a r e t h e same as t h o s e d e v e l o p e d  the G a l e r k i n p r o c e d u r e i f the f o r c e d boundary are homogeneous. concept  of the boundary r e s i d u a l concept i t t o be e q u i v a l e n t equations  A corrected  is presented  w h i c h shows  to a p p l i c a t i o n of the v i r t u a l  Galerkin's  work the  the e q u i l i b r i u m c o n f i g u r a t i o n of a l i n e a r  c o n t i n u u m u s i n g v i r t u a l w o r k l e a d s t o t h e same as a r e o b t a i n e d  the version  It is further demonstrated that deriving governing  by  conditions  If t h i s c o n d i t i o n i s not s a t i s f i e d  leads to i n c o r r e c t e q u a t i o n s .  principle.  necessary  conditions.  The b o u n d a r y r e s i d u a l c o n c e p t l e a d to e q u a t i o n s  by  equations  by s o l v i n g t h e e q u i l i b r i u m e q u a t i o n s  using  method. The c o n v e r g e n c e p r o o f o f O l s o n ' s (14)  e l e m e n t a n a l y s i s of the panel  f l u t t e r p r o b l e m has  finite been  135 presented.  T h i s proof i l l u s t r a t e s the a p p l i c a t i o n of the  G a l e r k i n p r o c e d u r e to a problem f o r which the R a y l e i g h R i t z method i s i n a p p l i c a b l e .  The a p p l i c a t i o n o f the G a l e r k i n  p r o c e d u r e t o p r o b l e m s i n v o l v i n g more t h a n one  dependent  v a r i a b l e has b e e n i l l u s t r a t e d by g e n e r a t i n g a f i n i t e s o l u t i o n f o r the l i n e a r v i s c o u s flow problem.  element  The d e v e l o p -  ment i n d i c a t e s t h a t e v e n i f a v a r i a t i o n a l f o r m u l a t i o n o f the problem does e x i s t the R a y l e i g h - R i t z p r o c e d u r e i s not n e c e s s a r i l y t h e m o s t a d v a n t a g e o u s way t o d e v e l o p t h e required equations.  N u m e r i c a l r e s u l t s o b t a i n e d show  e x c e l l e n t a g r e e m e n t w i t h known s o l u t i o n s . A t t e n t i o n i n t h i s t h e s i s has b e e n c o n f i n e d t o the c o n s i d e r a t i o n o f c o n f o r m i n g e l e m e n t s , however  the  approach adopted enables the study of non-conforming t o be u n d e r t a k e n i n a s y s t e m a t i c m a n n e r .  elements  Further, adopting  the p o i n t of view proposed e n a b l e s the convergence of a f i n i t e e l e m e n t a p p r o x i m a t i o n t o be i n v e s t i g a t e d f o r a f a r w i d e r c l a s s o f problems than have been c o n s i d e r e d h e r e i n . 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M e l o s h , R.J., "Development of the S t i f f n e s s Method t o D e f i n e B o u n d s on E l a s t i c B e h a v i o u r o f S t r u c t u r e s , " Ph.D. T h e s i s , U n i v e r s i t y o f W a s h i n g t o n , S e a t t l e , 1962. M i k h l i n , S.G., V a r i a t i o n a l M e t h o d s i n M a t h e m a t i c a l P h y s i c s , The M a c m i l l a n Co., New Y o r k , 1964. 136  137  10.  M i k h l i n , S.G., The P r o b l e m o f t h e Minimum o f a Q u a d r a t i c F u n c t i o n a l , H o l d e n Day, I n c . , San F r a n c i s c o , 1965.  11.  O d e n , J . T . , "A G e n e r a l T h e o r y o f F i n i t e E l e m e n t s I Topological Considerations," International Journal f o r N u m e r i c a l M e t h o d s i n E n g i n e e r i n g , V o l . 1, No. 3, 1969, pp. 2 0 5 - 2 2 1 .  12.  O d e n , J . T . , "A G e n e r a l T h e o r y o f F i n i t e E l e m e n t s I I A p p l i c a t i o n s , " I n t e r n a t i o n a l Journal f o r Numerical M e t h o d s i n E n g i n e e r i n g , V o l . 1, No. 3, 1969, pp. 247-260.  13.  O l i v e i r a , E.R.A., " T h e o r e t i c a l F o u n d a t i o n s o f t h e F i n i t e Element Method," I n t e r n a t i o n a l Journal of S o l i d s S t r u c t u r e s , V o l . 4, 1968, pp. 9 2 9 - 9 5 2 .  14.  O l s o n , M.D., "On A p p l y i n g F i n i t e E l e m e n t s t o P a n e l F l u t t e r , " National Research Council of Canada, A e r o n a u t i c a l R e p o r t L R - 4 7 6 , O t t a w a , M a r c h 1967.  15.  S c h l i c h t i n g , H., B o u n d a r y L a y e r T h e o r y , M c G r a w - H i l l , New Y o r k , 1960. Smirnov, V.I., A Course of Higher Mathematics, Vol. I l l , P a r t I I , Permagon P r e s s , O x f o r d , 1964. S z a b o , B.A. and L e e , G.C., " S t i f f n e s s M a t r i x f o r P l a t e s By G a l e r k i n ' s M e t h o d , " J o u r n a l o f t h e E n g i n e e r i n g M e c h a n i c s D i v i s i o n , A S C E , V o l . 95, No. EM3, J u n e 1 9 6 9 , pp. 5 7 1 - 5 8 5 .  16. 17.  18.  S z a b o , B.A. and L e e , G.C., " D e r i v a t i o n o f S t i f f n e s s M a t r i c e s f o r P r o b l e m s i n P l a n e E l a s t i c i t y by G a l e r k i n ' s Method," I n t e r n a t i o n a l Journal f o r N u m e r i c a l M e t h o d s i n E n g i n e e r i n g , V o l . 1, pp. 3 0 1 - 1 0 , 1969.  19.  Z i e n k i e w i c z , O.C., "The F i n i t e E l e m e n t M e t h o d : from I n t u i t i o n to G e n e r a l i t y , " A p p l i e d Mechanics R e v i e w s , V o l . 23, 1970, pp. 2 4 9 - 2 5 6 .  20.  Z i e n k i e w i c z , 0 . C , and P a r a k h , C . J . , " T r a n s i e n t F i e l d P r o b l e m s - Two - and T h r e e - D i m e n s i o n a l Analysis by I s o - P a r a m e t r i c F i n i t e E l e m e n t s , I n t e r n a t i o n a l J o u r n a l f o r Numerical Methods i n E n g i n e e r i n g , V o l . 2 , 1970 , pp. 61-71 .  NO.  777057  UNIVERSITY  OF  B C COMPUTING  CENTRE  M T S i A N 1 2 0 ) 1&8  ,**#*•************ T H I S JOB S U B M I T T E D THROUGH FRONT DESK READER * * * * * * * * * * * * * * * * * * * *G OSGH ^ AST SIGNON WAS: 1 4 : 3 3 : 2 7 C8-18-71 iER "OSGH" S I G N E D ON AT 1 4 : 3 7 : 2 8 ON 0 8 - 1 8 - 7 1 ;ST * S O U R C E * PROGRAM H2 FOR SLOW V I S C O U S FLOW OF I N C O M P R E S S I B L E NEWTON I A N F L U I D 0 1 U S I N G QUADRATIC D I S P L A C E M E N T F I E L D AND 6 NODE T R I A N G U L A R ELEMENTS C D I M E N S I O N N P 1 ( 1 5 0 ) , N P 2 ( 1 5 0 ) , X ( 1 5 0 ) ,Y (1 5 0 ) » NODE 1 { 1 5 0 ) , N 0 D E 2 ( 1 5 0 ) , N O 3 1 D E 3 < 1 5 0 ) , N 0 D E 4 ( 1 5 0 ) , N 0 D E 5 ( 1 5 0 ) , N 0 D E 6 ( 15 0 ) , T O A ( 100) , S E ( 1 3 , 1 3 ) , S K ( 9 0 4 1 , 2 5 0 ) , S { 9 0 0 0 ) , V K ( 9 9 ) , M U ( 1 0 0 ) , M C f 1 3 ) , D E L T A ( 1 6 0 ) , X 1 ( 1 5 0 ) , X 2 ( 150 ) ,X 5 1 3 ( 1 5 0 ) , X 4 ( 1 5 0 ) , X 5 ( 1 5 0 ) , X 6 { 1 5 0 ) ,Y1( 150) , Y 2 ( 1 5 0 ) , Y 3 ( 1 5 0 ) , Y 4 ( 1 5 0 ) , Y 5 ( 6 1 1 5 0 ) , Y 6 ( 1 5 0 ) , N 0 D E 7 ( 1 5 0 J . A l l 1 0 0 ) , A 5 ( 1 0 0)» A3( 100)» B 1 ( 1 0 0 ) , B 3 < 1 0 0 ) , 8 5 ,7 li100) ,R( 1 0 0 ) , R K ( 3 0 0 ) , R K l ( 1 5 0 ) ,RK2 < 150 ) 8 • REAL MU 9 t DOUBLE P R E C I S I O N S , D E L T A , RAT 10 , SUM, DE 10 COMMON/ZDET/DE,NCN 11 12 COMMON/ZCON/COND c WHEN NUMBERING S R T U C T U R E D F I R S T OR L A S T DEGREE OF FREEDOM SHOULD . l l 14 c NOT CORRESPOND TO P R E S S U R E t 2 ) P R E S S U R E DEGREE OF FREEDOM SHOULD NOT BE THE S M A L L E S T NUMBER IN I T S T R I A N G L E .GT. M F I X .BOTH L E A D TO A 15 -c V BLOW UP OF BAND! 16 c 17 NODES NO. 1 2 3 4 5 6 NOT 1 4 2 5 3 6 c NO. FOR M I N . BAND WIDTH IGNORING F I X E D D E G R E E S OF FREEDOM id c 19. NFIX=NO.CF P R E S C R I B E D VEL.. NOT = TO Z ERO+ANY P R E S C R I B E D P R E S S U R E c c NFIXO=NO. OF V E L O C I T I E S P R E S C R I B E D = TO 0 20 I F NO P R E S C R I B E D FORCES NFORCE=0,OTHERWISE NFORCE=1.UNKNOWN F O R C E S 21 c c READ I N AS ZERO F O R C E S 22 READ(5,2)NELEM,NNODE,NFIX,NFIXO,NFORCE 23 24 2 FORMAT (5 1 1 0 ) W R1TE(6,4)NELEN,NNODE,NFIX, NFIXG,NFORCE 25 4 26 FORMAT ( 'NELEM = ' , 1 1 0 , 'NNODE = M I C ' N F I X = ' , 1 1 0 , ' N F I X G ' , 1 1 0 , 1' NFORCE ',110) 27 28 D06 1=1,NELEM 29 R E A D ( 5 , 5 ) NODE 11 I ),N0DE2( I ) ,N0DE3( I ) , N 0 D E 4 ( I ) , N O D E 5 ( I ) , N 0 D E 6 ( I ) , N O D 1E7( I ) 30 5 FORMAT (7 1 1 0 ) 3 I _ 6 32 CONTINUE C NODE 7 = P R E S S U R E I N ELEMENT (CANNOT BE S P E C I F I E D AS 1 ) 33 C S P E C I F Y I N G A N E G A T I V E P R E S S U R E CORRESPONDS TO A C O M P R E S S I V E S T R E S S V- '34 WRITE(6,8) 35 .•** 8 F O R M A T ( ' E L E M E N T NO. NODE(l) N0DE12) N0DE(3) N0DE(4) NODE(5) 36 1 N 0 D E ( 6 ) , N 0 D E 7 ( I ) ') 3X D 0 1 0 1 = 1, NELEM 38 W R I T E ( 6 , 9 ) I , N O D E 1 < I ) , N 0 D E 2 ( I ) , N 0 D E 3 ( I ) , N 0 D E 4 ( I ) ,N0DE5 <I) , N 0 D E 6 ( I ) , 39 1N0DE7(I) -f 40 FORMAT! 1 7 , 1 11 , 6 1 9 ) 9 41 10 CONTINUE 42 C ONLY I N T E G E R V E L O C I T I E S ALLOWED 43 44 D 0 1 2 J = l,NNODE R E A D ( 5 , 1 1 ) N P 1 ( J ) , N P 2 ( J ) , X ( J ) ,Y(J)»RK1(J),RK2{J) 45 11 FORMAT(2 I 1 0 , 2 F 1 0 . 3 , 2 E l 5.4) 46 CONTINUE 12 47 ONLY NEED S P E C I F Y X,Y FOR NODES 1,3,5 A S 2 , 4 , 6 ARE MID P O I N T S C 48 49__ WRTTF(6.14) 14 F O R M A T ( • NODE NO. NP1 NP2 X Y X-FO 50 1RCE Y-FORCE' ) 51 D016 J=l,NNODE 52  53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74  15 16 C C C •  • 161  162 163  >• 7 5  > 76 77 78 - 79 80 " 81 82 83 84 85 86 87 , 88 89 90 . 9 i 92  17 18 19 20  2 01  >_  202  9z 1  "  • •>. 9^ 9f " 9^ , 91 9? •  9  (  "102 ,10.  104 lOf > 406 107 "10 J  iic  ii]  A12 >  21  I  -rlOC I 101  .105  203 .  22  23 25  27  28 29 291  WRITE(6,15)J,NP1(J),NP2<J),X(J},Y(J),RK1{J),RK2<J> F 0 R M A T I I 6 I 1 1 , 111 , F8 .2 , F8 .2 , 2 E15 .4 ) CONTINUE I F NODE IS F I X E D AT 0 NP = 0, I F NODE VEL.=V N P = V , I F 1NP=1 FOR P R E S S U R E D E G R E E S OF FREEDOM NP1=1,NP2=-1 MFIX = NF'.IXO + N F I X N=0 NFIX1=NFIX+1 . NU=NFIX+NF IXO NUU=NFIXO D025 J = 1 , N N 0 D E I F t N P l ( J ) . E Q . O ) G O TO 1 6 1 IF{ N P K J ) .EQ. 1 ) G 0 TO 18 GO TO 1 6 3 N=N +1 VK(N)=NP1(J) NP1(J)=N WR.ITE(6,162)J,N F O R M A T ( • NP 1 ( ' »I 3 » * ) - ' , 1 4 ) GO TO 20 NUU=NUU+1 VK(NUU)=NP1(J) N P l { J ) = NUU WRITE(6,17)JtNUU FORM AT ( ' N P K 1 3 , ' )=' , 1 4 ) GO TO 20 NU=NU+1 NPliJ)=NU WRITE(6,19)J,NU FORMAT( • N P K » , 1 3 , ' ) = » , 1 4 ) I F ( N P 2 ( J ) - E Q . - l ) GO TO 2 5 I F ( NP2 ( J ) . E Q . 0 ) G O TO 2 0 1 IF ( N P 2 ( J ) .EQ. D G O TO 2 2 GO TO 20 3 t  N=N+ 1  VK(N)=NP2(J) NP2<J)=N WRITE-16,202) J , N FORMAT<' N P 2 ( « , 1 3 , • )=• , 1 4 ) GO TO 25 NUU=NUU+1 V K ( N U U ) =NP 2 ( J ) NP2 1 J ) = N U U WRITE(6,21)J,NUU FORMAT ( • N P 2 ( » , 13, * )= • , I 4 ) GO TO 2 5 NU=NU+1 NP2(JJ=NU WRI TE (6 ,23 ) J ,NU FORMA T( ' N P 2 ( ',I3,» )=« , 1 4 ) CONTINUE IF(MFJX.EQ.O)GO TO 2 9 1 WRITE(6,27) FORMAT (» VK » ) DO 2 9 K=1,NUU WRITE(6,28)VK(K) F0RMAT(lX F12.2) CONTINUE WRITE(6,30)NU t  1  NODE  I S FREE  3  9  119 120 121 122 123 124  30  FORMAT ( TOTAL NO. OF DEGREES OF FREEDOM = S I 6 ) DO 3 0 1 J = l , 9 0 0 0 , S{J)=0 CONTINUE DO 32 1=1,90 DO 31 J = l ,250 SK(I,J)=0 CONT INUE CONTINUE I F < M F I X-90) 3 2 5 , 3 2 5 , 3 2 3 IF(NU-250)326,326,323 WRITE(6,324) FORMAT( ' D I M E N S I O N OF SK EXCEEDED ) GO TO 7 0 CONTINUE , . MBAND I S ASSUMED MAX. BAND WIDTH - I F E X C E E D E D PROGRAM P R I N T S OUT THE FACT AND STOPS MBAND=50 FACTOR-.00001 FACTOR= C O E F F . OF V I S C O S I T Y D I V I D E D BY 2 D050 I=1,NELEM MAX = 0 MIN=1000 DO 3 2 2 L = l , 1 3 DO 3 2 1 K = l , 13 SE(K,L)=0 CONTINUE CONTINUE X l { I )=X( N0DE1 ( I ) ) X2( I )=X(N0D£2( I ) ) X3(I ) =X(N0DE3(I) ) X4(I)=X(N0DE4{I)) _X3JJJ_5X-(.fiaD^E5_.LLLL X6(I)=X(N0DE6<I)) Yl(I)= Y(NODEKI)) Y2 ( I ) = Y ( N 0 D E 2 ( I ) ) Y31 I 5 = Y( N0DE3{ I ) ) Y4( I ) = Y ( N 0 D E 4 { I ) ) J^5Ji)_EX.(JiQD_E_L(-UJ_ Y6(I)=Y( N 0 D E 6 U ) ) A K I )=X5 ( I >-X3( I ) A3(I)=X1(I)-X5(I) A 5 U )=X3( I ) - X l ( I ) B l ( I > = Y 3 ( I ) -Y 5 ( I ) 1  & n  301  31 3.2 325 323 324 326 C C  321 322  1  B5( I } = Y K I ) - Y 3 ( I ) TOA( I ) = Y 1 ( I ) * ( X 5 ( I )-X3 ( I ) ') + Y 3 ( I ) * ( X 1 ( I ) -X 5 ( I ) ) +Y51 I ) *( X3< MU( I ) = F A C T O R / T 0 A ( I ) MU = C O E F F OF V I S C . D I V I D E D BY 4A SE( 1 , 1 ) = ( 2 * B 1 ( I . ) * * 2 + A1 ( I ) * * 2 ) * M U ( I ) j5EAA^2jJE±UJJJ&UJJjmX)lJJ S E U , 3 )= 4 . / 3 . * ( 2 * B 1 ( I ) * B 3 ( I ) + A l { I ) * A 3 ( I ) ) * M U ( I ) SEC 1 , 4 ) = A1II)*B3(I)* M U(I)*4./3. S E ( 1 1 5 )= - <2*B1 ( I )*B3( I ) + A K I )*A3( I ) )*MU( I ) * l . / 3 . SE( 1,6) = - A K I ) * B 3 U ) *MU< I )*1 . /3 S E ( 1,7 ) = 0 SE(1.8)=0 SE{ 1, 9 ) = - ( 2 * B 1 ( I ) * B 5 U ) + A l ( I )*A5{ I ) )*MU( I )*1 . / 3 , SE(1,10)= - A l ( I ) * B 5 ( I } * M U ( I 3*1./3. SEC 1 , 1 1 ) = ( 2 * B 1 ( I ) * B 5 ( I >+Al m * A 5 < I ) )*MUt I ) * 4 . / 3 .  S E ( 1 1 2 )= A l < I )*B5 ( I )*MU < I ) * 4 . / 3 . 141 SE( 2,2) =( 2*A1(.I ) * * 2 + B L ( I )**2.)*MUl I ) SE(2,3) = B l ( I ) * A 3 ( I )*MU( I ) * 4 . / 3 . SE(2,4) = (2*Al(I)*A3{I)+Bl{I)*B3(I))*MU(I)*4./3. SE(2,5) = - A 3 ( I ) * B 1 U )*MU<I ) * l . / 3 . SE(2,6) = - ( 2 * A 3 ( I ) * A 1 ( I ) + B 3 ( I ) * B 1 ( I ) )*MU( I ) * 1 . / 3 . SEC 2,7) =0 SEC 2,8) =0 SEC 2 , 9 ) = - A 5 { I ) * B 1 ( I ) * M U ( I ) *1*/3 SE(2,10)= -<2*A1(I)*A5(I)+Bl(I)*B5(I ) )*MU(I)*l./3. SE(2,11)= B U I )*A5( I)*MU-( I ) * 4 . / 3 . S£( 2,12? = (2»A1(I)*A5(I) + 8 1 U ) * B 5 ( 1 ) ) * M U ( I ) * 4 . / 3 . S E ( 3 ,3 ) = < 2 * ( B 3 ( I ) * * 2 + B K I )*B3( I ) + B 1 ( I ) * * 2 ) + A 3 ( I ) * * 2 + A 3 ( I ) * A K I 1 ) + A 1 U ) * * 2 ) *MU< I ) * 8 . / 3 . _&E13i£±-=. ( A3 ( I ) * B 3 l I ) + 0 . 5*1 A K I ) * B 3 ( I )+A3( I )*B1 ( I ) ) + A l { I ) *B1 1))*MU(I)*8./3. SE(3,5) = {2*B3(I)*BlU)+A3{I)*Al(I))*MU(I)H./3. S£(3,6) = B3( I ) * A 1 ( I )*MU-( I ) * 4 . / 3 . SE(3,7} = { 2 * ( B 3 C I ) * B 5 < I ) + B 3 ( I ) * * 2 + 2 * B l ( I ) * B 5 ( I ) + B 1 ( I J*B3( I ) ) + 1A3C I.)*A5<I) + A3< I ) * * 2 + 2 * A H I ) * A 5 ( I ) + A 1 ( I )*A3( I ) ) *MU< I ) * 4 . / 3 . SEC 3,8) = ( A3 { r ) * B 5 ( I ) + 2*A1 ( I )»B5( I ) -+A3 ( I )*B3 ( I ) + A l ( I )*B3( lU(I)*4./3. S E ( 3 , 9 ) =0 S E ( 3 1 0) = 0 SE 13 ,11 )= ( 2 * <2*B3 < I J * B 5 ( D + B l U )*B5( I ) + B l ( I ) * B 3 ( D + B l ! I !**2) + 1 2 * A 3 ( I ) * A 5 ( I ) + A l ( I ) * A 5 ( I ) + A l ( I ) * A 3 C I l + A l l I )**2 ) * M U ( I } #4 . / 3. SE(3,12)= ( 2 * A3 ( I ) * B 5 ( I H A H T )*B5( I 1 + A 3 U ) *B1< I ) + A1 t I ) * B 1 ( I ) )* lU(l)*4./3. SE(4,4) = ( 2 * ( A 3 ( I ) * * 2 + A l < I ) * A 3 ( I ) + A 1 { I ) * * 2) + B 3 ( I ) * * 2 + B 3 ( I ) * B l { 11 ) + B H I ) **2 ) * M U ( I )*8 ./3 . SEC 4,5) = A3CI)*B1U)*MU(I)*4./3. SEC4,6) = ( 2 * A 3 ( I )*A1< I ) + 8 3 ( I )*B1( I ) ) *MU-( I ) * 4 . / 3 . SE(4,7) = ( B 3 ( I ) * A 5 ( I ) + 2 * B U I ) * A 5 ( I ) + A 3 ( I ) * B 3 ( 1 ) + B l ( I )»A3(I) )*M lU(I)*4./3. SE(4,8) = (2=M A3 <I ) * A 5 ( I ) + A 3 < I ) * * 2 + 2 * A l ( I ) * A 5 ( I ) + A l ( I )*A3( I ) ) + 1B3( I ) * B 5 ( I )+B3( I )**2 + 2 * B l ( I ) *B 5 { !.) + B 1 ( I ) * B 3 t I ) )*MU ( I ) * 4 . / 3 . S £ ( 4 , 9 ) =0 SE ( 4 ,1 0) = 0 ____A_1JJ= C 2 * B 3 ( I )*A5( I l t R K I )»A5( I )+B3( I ) » A H I ) + A l ( I ) *B1 11 ) ) »M lU(I)*4./3. SEi 4 , 1 2 ) = ( 2*( 2 * A 3 ( I ) * A 5 ( I ) + A l ( 1 )*A5 ( I 1+A1 ( I )*A3 ( I ) + A l ( I ) * * 2 ) + 12»B3 { I )*B5 ( I ) +B1 ( I ?*B5( I ) + B 1 ( I ] * B 3 ( I ) + B K I ) **2) *MU( I ) * 4 . / 3 . _ SEC 5,5) = ( 2 * B 3 ( I ) * * 2 + A 3 ( I } * * 2 ) * M U ( I ) SEC 5 , 6 ) =B3( I )*A3( 1 )*MU( I 1 SE ( 5 , 7 ) = (2«B3 { I )*B5 ( I ) + A3 C I ) * A 5 ( I ) ) »HU( I ) » 4 . / 3 . SFJ5»8)= A3( I ) *B 5 ( I ) *MU ( I ) * 4 . / 3 . SE(5,9) = - ( 2 * B 3 ( I )* B5( I )+A3( I ) * A 5 ( I ) ) * M U ( I ) * 1 • / 3 . SEC 5 , 1 0 ) = -A3(I)*B5(1)*MU(I)*1./3. SE(5,11)=0 SE(5,12)=0 SE( 6,6) =( 2 * A 3 ( I )**2+B3< I ) * * 2 ) * M U ( I ) . , SE(6,7) = B3 C I ) * A 5 ( I )*MU( I ) * 4 . / 3 . SE(6,8) = ( 2 * A 3 ( I ) * A 5 ( I ) +B3( I,)*B5 ( I ) J*MU( I )*4 . / 3 . SE(6,9)= -A5(D*B3(I)*MU( I) * l . / 3 . SE(6,10)= - ( 2* A3 ( I ) * A5 ( I ) + B3 ( I ) *B5 ( I ) ) *MU ( I ) * 1 . / 3 . SE(6,11)=0 S E ( 6 ,12 )=0 . SEC 7,7) = < 2 * < B 5 ( I ) * * 2 + 8 3 C I ) * B 5 ( I ) + B 3 ( I ) * * 2 ) + A 5 ( I ) * * 2 + A3( I )*A5( 1 I ) + A3( I } * * 2 ) * M U ( I ) * 8 . / 3 . SE(7,8) = ( A 5 ( I ) * B 5 ( I ) + 0 . 5 * ( A 3 ( I ) * B 5 ( I ) + A 5 ( I ) *B 3 ( I ) ) + A 3 ( I ) t  1  233  1 ) )*MU( I  234  S E ( 7 , 9 )  23  5  23 6 23 7  ) * 8 . / 3 . =  ( 2 * B 3 •( I ) * B 5 ( I ) + A 5 ( I ) * A 3 { I ) ) * M U ( I ) * 4 . / 3 .  S E ( 7 , 1 0 ) =  B 5 ( I ) * A 3 ( I ) * MU (I  )* 4 • / 3 .  SE<7,11)=  ( 2 * ( B5 t I ) * * 2 + B 3 (  I ) * B 5 ( I ) + B 1 ( I ) * S 5 ( I  15(1)**2+A3( I ) * A 5 ( I 3  23 8  S E ( 7 , 1 2 J =  239  l U ( I ) * 4 . / 3 .  240  SE(8,8)  .24,3  -  242  +A1(I  ) * A 5 < I ) + 2 * A l ( I ) * A 3 ( I  ( B5 ( I ) * A 5 ( I ) +85(  I)*A3(I)+B1(  ( 2 * ! A5 ( I ) * * 2 + A 3 m * A 5  =  (I  1  ) + 2 * B l ( I •) * B 3 U  1)*MU(I  I ) * A 5 ( I  S E ( 8 , 9 )  +244 245  )+ 2 * B 1 (  I ) * A 3 ( I ) ) * M  ) +A3( I >**2)+B5 ( I )**2+B3(  =  B3< I 5 * A 5 ( I ) * M U ( I  ( 2 * A 5 ( I ) * A 3 ( I ) + B 5 < I ) * B 3 < I ) ) * MU ( I ) * 4 . / 3 .  S E ( 8 , 11 )=  ( A 5 ( I ) * B 5 ( I ) + A 5 ( I ) * B 3 ( I ) +A 1 ( I ) * B 5 ( I ) + A 1 ( I ) * B 3 ( I (2*1A5{ I )**2+A3(  1 5 ( 1 ) * * 2 + B 3 ( I)*B5  248  S E ( 9 , 9 )  '249  (I  ) + B 1 (I  I ) * A 5 ( I ) + A 1 (  =( 2 * B 5 ( I ) * * 2 + A 5 ( I ) * * 2  )*MU( I )  SE( 9 , 1 1 ) =  ( 2*B5( I ) * B 1 ( T )+A5(  S E ( 9 , 1 2 ) =  A 5 ( I ) * B l ( I  252  )*MU{  I ) * A 1 ( I)  S E t 1 0 . 1 1 )=  )* M U ( I ) * 4 . / 3 .  I ) * 4 . / 3 .  S E < 1 0 , 1 0 ) = < 2 * A 5 ( I ) * * 2 +B 5 ( I ) * * 2 ) *MU  _2.5_3  B5 ( I ) * A 1{ I )*MU( I ) * 4 .  254  S E ( 1 0 , 1 2 ) =  (2*A5(I  S E ( 1 1 , 11 )=  ( 2 * ( B 5 ( I ) * * 2 + B l ( I ) * B5(I) + B l ( I ) * * 2 ) + A 5 ( I ) * * 2  ^256  1(I)+A1(  257  ) * B 1 ( I ) ) * M U (  I ) * 4 . / 3 . + A l ( I ) * A 5  n * 8 . / 3 .  I ) * * 2 ) * M U (  S E ( 1 1 , 1 2 ) =  '258  )*A1(I}+B5(I  (I) / 3 .  ^2 5 5  (A5{I  ) *B5(I  )+0 . 5 * ( A l ( I ) * B 5 ( I )  + A 5 ( I ) * B 1 U )  )+Al(  I)*B1<  1 I ) ) *MU( I )*8 . / 3 .  •J2.59  S£JLL2_02JL=  260  K I l + B K I C  ( 2 * ( A 5 (I  )**2)  ASSEMBLE  »2 6 2  ) * * 2 + A l ( I ) * A 5 ( I ) + A1 ( I ) * * 2 ) + B 5 (  I )**2+Bl(  *MU(I ) * 8 . / 3.  INCOMPRESSI BIL ITY  CONDITION  MULTIPLIED  BY  .00006  A=. 00001  263  SE(13,  264  S E < 1 3 , 2 ) = A1 ( I  _2.6_5  1) = B 1 ( I ) * A  SE( 13.3) =4*(  )* A  B l ( I )+B3(  I ) )* A  266  S E ( 1 3 , 4 ) = 4 * ( A 1 ( I ) + A 3( I ) ) * A  267  S E ( 1 3 , 5 ) = B3 ( I ) * A  -26 8  SE( I 3 , 6 ) = A 3 (  269  I ) * A  S E ( 1 3 , 7 ) = 4 * ( B 3 ( I )+B5(I)  "270  )*A  J + A 5 U ) )*A _S.EJ.A.3J-9J.=A5..UJ_*A _ SE( 1 3 , 8 ) =4* ( A3 ( I  2.7.1  _  272  S E ( 1 3 , 1 0  27 3  SE( 1 3 , 1 1 ) = 4 * ( B l ( I  ^274  )=A5( I ) * A  SE( 13,12) = 4 * ( A l (  275  ) + 85 ( I ) ) * A I  SE(13,13)=0  276  DO  23  33  K = l , 1 2  ^JlLILiJ^i^£JL13jJLl 33  CONTINUE DO  27')  il80  35  K = l , 1 2  DO 3 4 L = l , 1 2  .28!.  S E ( L , K ) = S E ( K , L )  '*2 8;>  34  .28:1  35  284  CONTINUE CONTINUE DO  285  '288  )+B  )*MU( I ) * 4 . A 3 .  S E ( 9 » 1 0 ) = B5 ( I ) * A 5 ( I ) * M U ( I )  251  287  ) * 2 ) * M  I ) * A 5 ( I) + 2 * A 1 ( I ) * A 3 < I )  ) * B 5 ( I ) + 2 * B l ( I ) * B 3 ( I)  r25 0  ,286  I )*B5(  ) * 4 . / 3 .  SE( 8 , 1 0 ) =  S E ( 8 f 1 2 )=  24 7  2 T f|3  ) )+A  ) * 4 . / 3 .  l U ( I ) * 4 . / 3 .  246  v  2  UJ.±&3JJJ*l2J±mLU*&^JJb^  '"243  •'261  4  341  K = l , 1 3  MC(K)=0 341  CONTINUE MC(1)=NP1(N0DE1(I MC( 2 )= N P 2 ( N 0 D E 1 (  I  283  MC(3 ) = NP1 (N0DE2 ( I  290  MC ( 4 ) = N P 2 ( N 0 D E 2 ( I  291  MC(5)=NP1(N00E3(I  *29;>  M C ( 6 ) = NP2 ( N 0 D E 3 ( I  ) + A 5 ( I ) ) * A  :  I ) * B 5  29 3  M C ( 7 ) = N P 1 ( N 0 D E 4 ( I) )  294  MC(8) = NP2{M0DE4( MC(9) =  . 295 29ft  MC(10)=NP2(NODE5(  >297  )  143  I)  )  MC( 11 ) = NP1 ( N 0 0 E 6 ( I ) )  , 29 8  MC( 1 2 ) = N P 2 ( N 0 P E 6 ( I)  )  299  MC(13) = NP1(N0DE7(I ) J  300  DO  302  351  "303 v-304 305  37  K=l  ,13  IF< M C ( K ) - M F I X ) 3 7 , 3 7 , 3 5 1  ,._30 1  y  I)  N P K N 0 D E 5 ( I ) )  DO  36  L=  1,13  IF{MC(L)-MFIX)36 35 2 353  3 0 6  IF(  TO  36  307  354  IF(MC(L)  308  36  CONTINUE  "309  37  . L T . M I N )MIN=MC( L)  ;  CONTINUE N A N D= M A X - M I N  +1  IF(MBAND.GT.NAND) WRITE< 6 , 3 7 1 ) 1 371  ,352  I F ( M C ( L ) . GT . M A X ) M A X = M C { L ) GO  .•310  ,36  MC( K ) - M C ( L ) ) 3 5 3 , 3 6 , 3 5 4  FOR M A T ( GO  TO  372  WIDTH  EXCEEOED  IN  ELEMENT  ' , 1 4 , '  70  C  GENERATE  3 72  DO  48  BAND  1  GOTO  ,NAND  S(<NU-MFI  X ) * * 2 )  ,SK(MFI X*(NU-NF1XO) )  L = l , 1 3  I F ( M C { L ) - N F I X 0 ) 4 8 , 4 8 , 3 8 38  I F ( M C ( L J - M F IX ) 4 2 , 4 2 ,  3 9  DO  41  IF(  MCI K ) - M F I X ) 4 0 1  40 323  IF(MC(L)..LT.MC(K ) I KK=MC( K J - M F I X  01,40 GO  TO  41  M={KK-1)*(MBAND-1)+LL S(M)=S<M) + S E ( K - , L )  .32 5 326  GO 401  j32l3 '?  . ,4  LL=MC( L ) - M F I X  324  "327  39  K=l ,13  TO  41  MCL=MC(L)-NFIXO S K { M C ( K) , M C L ) = S K ( M C ( K ) , M C L ) + S E ( K , 1 )  41 42  CONTINUE GO  TO  48  DO  44  K= l  .13  IF<MC(K)-MFIX 43  SK(MC(K) 44  ) 4 3 , 4 3 , 4 4  M C L = M C ( L ) - N F IXO ,MCL)=SK(MC(K) ,MCL)+S E ( K , L )  CONTINUE  48  CONTINUE  50  CONTINUE DO  49  1=1,300  RK<I)=0 49  CONTINUE IF{NFORCE.EQ.0)G0 DO  501  TO  502  J=l,NNODE  RK(NP1(J))=RK1(J) IF{NP2< J ) . E Q . - l ) R K ( N P 2 ( J ) ) = RK 2 ( J ) 501  CONTINUE DO  504  J=1,NU  W R I T E ( 6 , 5 0 3 }RK( J ) 504  CONTINUE  503  F O R M A T ( I X , E 1 6 . 7 )  502  MFIX1=MFIX+1 N UF 0 = N U - NF I XO  GO  TO  501  BAND  WIDTH=  « , I 4 )  I F ( M F I X . E O . O ) G O TO 5 4 1 I F ( N F I X . E O . O ) G O TO 541 DO 54 L = N F I X l , N U F O SUM=0 NFIXO1=NFIXO+1 DO 5 1 K=NFIXQ.1,MFIX SUM=SUM + S K ( K , L ) * V K { K ) CONTINUE  51  144  te=L=HEJX ,  54 541  543  545 3J*& 544  _  J = L+NF I X O DELTA(M) = R M J ) - S U M CONTINUE GO TO 5 4 4 DO 5 4 3 J = l , N U F O H=J+NF IXO DELTA(J)=RK(M) .CONTINUE DO 5 4 6 K = l , N U F O WRITE(6 » 5 4 5 ) D E L T A ( K ) FQRMAT(1X,E16.7) CONTmLE NE T=NU—MF I X RAT 10= 1 .E-6 C A L L CBAND I ( S , P E L T A,NET,MBAND, l RAT 1 0 ) RNE T=NE T D= CON D/RN ET _WJLI_IE.(_6„,-5_5J_D_ FORMAT( ' R A T I O OF C O N D I T I C N NO. TO ORDER OF S WRITE(6,56)DE,NCN F ORMAT ( ' D E T E R M I N A N T ^ D16 .7 , ' * 1 . E ' 13 ) WRITE{6,58) FORMAT { * NODE VEL-PRESS.V) I F P R E S S U R E TERM =-A THEN ACTUAL P R E S S U R E IS V . 0 0 0 6 * A C O M P R E S S I V E D 0 6 0 1=1,NET L=MFIX +I WRI T E ( 6 , 5 9 ) L , D E L T A ( I ) FORMAT(I5,D18.8) CONTINUE t  55 56 58 X  59 60  6 1 611  l  't '5  62 63 64  65 66  ..  JJEJJ^Jt^^JUJ^-JILJUL  DO 6 3 K = 1 , M F I X SUM=0 I F ( N F I X . E Q . O ) GO TO 6 1 1 DO 6 1 L = 1 , N F I X M=L+NF IXO SUM=SUM+SK(K,L)*VK(M) CONTINUE NFIX1=NFIX+1 NETO=NU-NFIXO DO 6 2 L = N F I X 1 , N E T O M=L-NFIX _S.U.M^JJM+.SK.(-l<j-L.)JiD.E.L_TA-(.Ml__ CONTINUE R ( K ) = SUM CONTINUE WRITE(6,64) FORMAT(• NODE FORCE Q0__^6_J^L,HEI_X WRITE(6,65) I,R(I ) FORMAT(I 5,E18.8) CONTINUE  ')  413 414 D OF  70 FILE  STOP END  1  4  5  

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