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A numerical investigation of two boundary element methods Quek, Mui Hoon 1984

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NUMERICAL INVESTIGATION OF TWO BOUNDARY ELEMENT METHODS by MUI HOON QUEK B.Sc. ( H o n o r s ) , C o n c o r d i a U n i v e r s i t y , 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES De p a r t m e n t o f Computer S c i e n c e 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 t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December 1984 (c) Mui Hoon Quek, 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of COMPUTE Sct&N<i5-The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 2/ 2><?c , (9<?4-11 ABSTRACT T h i s t h e s i s i n v e s t i g a t e s t h e v i a b i l i t y of two boundary e l e m e n t methods f o r s o l v i n g s t e a d y s t a t e p r o b l e m s , t h e c o n t i n u o u s l e a s t s q u a r e s method and t h e G a l e r k i n m i n i m i z a t i o n t e c h n i q u e . In c o n v e n t i o n a l b oundary element methods, t h e s i n g u l a r i t i e s of t h e f u n d a m e n t a l s o l u t i o n i n v o l v e d a r e u s u a l l y l o c a t e d a t f i x e d p o i n t s on t h e boundary o f t h e p r o b l e m ' s domain or on an a u x i l i a r y b o u n d a r y . T h i s l e a d s t o some d i f f i c u l t i e s : when t h e s i n g u l a r i t i e s a r e l o c a t e d on t h e p r o b l e m domain's b o u n d a r y , i t i s not e a s y t o e v a l u a t e t h e s o l u t i o n f o r p o i n t s on or nea r t h a t boundary whereas i f t h e s i n g u l a r i t i e s a r e p l a c e d on an a u x i l i a r y b o undary, t h i s a u x i l i a r y boundary would have t o be c a r e f u l l y c h o s e n . Hence t h e methods s t u d i e d h e r e a l l o w t h e s i n g u l a r i t i e s , i n i t i a l l y l o c a t e d a t some a u x i l i a r y b o u n d a r y , t o move u n t i l t h e b e s t p o s i t i o n s a r e f o u n d . These p o s i t i o n s a r e d e t e r m i n e d by a t t e m p t i n g t o m i n i m i z e t h e e r r o r v i a t h e l e a s t s q u a r e s or t h e G a l e r k i n t e c h n i q u e . T h i s r e s u l t s i n a h i g h l y a c c u r a t e , a d a p t i v e , b u t n o n l i n e a r method. We s t u d y v a r i o u s methods f o r s o l v i n g s y s t e m s of n o n l i n e a r e q u a t i o n s r e s u l t i n g f r o m t h e G a l e r k i n t e c h n i q u e . A h y b r i d method has been implemented, w h i c h i n v o l v e s t h e o b j e c t i v e f u n c t i o n f r o m t h e l e a s t s q u a r e s method w h i l e t h e g r a d i e n t i s due t o t h e G a l e r k i n method. N u m e r i c a l examples i n v o l v i n g L a p l a c e ' s e q u a t i o n i n two d i m e n s i o n s a r e p r e s e n t e d and r e s u l t s u s i n g t h e d i s c r e t e l e a s t s q u a r e s method, t h e c o n t i n u o u s l e a s t s q u a r e s method and t h e i i i G a l e r k i n method a r e compared and d i s c u s s e d - The c o n t i n u o u s l e a s t s q u a r e s method a p p e a r s t o g i v e t h e b e s t r e s u l t s f o r t h e sample p r o b l e m s t r i e d . i v TABLE OF CONTENTS Ab s t r a c t i i L i s t of Tables v L i s t of F i g u r e s v i Acknowledgements v i i 1 . I n t r o d u c t i o n 1 1.1 Boundary Element Method 1 1.2 D i f f e r e n t Approaches of the Boundary Element Method 2 1.2.1 I n d i r e c t boundary element method 3 1.2.2 D i r e c t boundary element method 4 1.2.3 Methods using a u x i l i a r y boundary 5 1.3 O b j e c t i v e s of the T h e s i s 12 2. P r e l i m i n a r i e s 14 2.1 E l l i p t i c Equation 14 2.2 Fundamental S o l u t i o n 16 3. Formulations of the Least Squares and G a l e r k i n Method 18 4. Methods f o r S o l v i n g Nonlinear A l g e b r a i c Equations 28 4.1 Newton's Method f o r systems of Nonlinear equations 28 4.2 The Unconstrained M i n i m i z a t i o n Using a Model-Trust Region Approach 31 4.3 Marquardt's Mehtod 33 5. A l g o r i t h m and Numerical R e s u l t s f o r the G a l e r k i n Technique .36 5 . 1 A l g o r i t h m 36 5.2 Numerical R e s u l t s 38 5.2.1 D i r i c h l e t Boundary Value Problem 39 5.2.2 Neumann Boundary Value Problem 42 5.2.3 Mixed Boundary Value Problem 45 6. Al g o r i t h m f o r M i n i m i z i n g the Boundary C o n d i t i o n s 48 6.1 D i r i c h l e t Boundary C o n d i t i o n s 49 6.2 Neumann and Mixed Boundary C o n d i t i o n s 51 7. Co n c l u s i o n s 57 References 58 LIST OF TABLES Table 5.1.1 L o c a t i o n s of i n i t i a l and f i n a l s i n g u l a r i t i e s and t h e i r a s s o c i a t e d c o e f f i c i e n t s f o r the T o r s i o n problem 40 Table 5.1.2 The numerical s o l u t i o n s vs. exact s o l u t i o n s f o r the T o r s i o n Problem 40 Table 5.1.3 Accuracy and e f f i c i e n c y f o r the d i f f e r e n t methodsfor the T o r s i o n problem 4.1 Table 5.2.1 L o c a t i o n s of i n i t i a l and f i n a l s i n g u l a r i t i e s and t h e i r a s s o c i a t e d c o e f f i c i e n t s f o r the flow problem 43 Table 5.2.2 The computed pressure vs. exact pressure f o r the flow Problem 43 Table 5.2.3 Accuracy and e f f i c i e n c y f o r the d i f f e r e n t methodsfor the flow problem 44 Table 5.3.1 L o c a t i o n s of i n i t i a l and f i n a l s i n g u l a r i t i e s and t h e i r a s s o c i a t e d c o e f f i c i e n t s f o r the heat c o d u c t i n g problem 46 Table 5.3.2 The numerical s o l u t i o n s vs. exact s o l u t i o n s f o r the heat conducting Problem 46 Table 5.3.3 Accuracy and e f f i c i e n c y f o r the d i f f e r e n t methodsfor the heat c o d u c t i n g problem 47 Table 6.1.1 L o c a t i o n s of i n i t i a l and f i n a l s i n g u l a r i t i e s and t h e i r a s s o c i a t e d c o e f f i c i e n t s f o r the T o r s i o n problem 50 Table 6.1.2 The numerical s o l u t i o n s vs. exact s o l u t i o n s f o r the T o r s i o n Problem 50 Table 6.1.3 Accuracy and e f f i c i e n c y f o r the d i f f e r e n t methodsfor the T o r s i o n problem 51 Table 6.2.1 L o c a t i o n s of i n i t i a l and f i n a l s i n g u l a r i t i e s and t h e i r a s s o c i a t e d c o e f f i c i e n t s f o r the flow problem 53 Table 6.2.2 The numerical s o l u t i o n s vs. exact s o l u t i o n s f o r the flow Problem 53 Table 6.2.3 Accuracy and e f f i c i e n c y f o r the d i f f e r e n t methodsfor the flow problem 54 Table 6.3.1 L o c a t i o n s of i n i t i a l and f i n a l s i n g u l a r i t i e s and t h e i r a s s o c i a t e d c o e f f i c i e n t s f o r the heat c o d u c t i n g problem 55 Table 6.3.2 The numerical s o l u t i o n s vs. exact s o l u t i o n s f o r the heat conducting Problem 55 Table 6.3.3 Accuracy and e f f i c i e n c y f o r the d i f f e r e n t methodsfor the heat conducting problem 56 v i LIST OF FIGURES F i g u r e 1.1 Boundary nodes and s i n g u l a r i t i e s f o r d i r e c t b o u ndary e l e m e n t method 6 F i g u r e 1.2 Boundary nodes and s i n g u l a r i t i e s f o r K u p r a d z e ' s method 7 F i g u r e 1.3 Boundary nodes and s i n g u l a r i t i e s f o r O l i v e i r a ' s method 8 F i g u r e 5.1 T o r s i o n p r o b l e m 41 F i g u r e 5.2 P o t e n t i a l f l o w p a s t a c i r c u l a r c y l i n d e r 44 F i g u r e 5.3 Heat t r a n s f e r p r o b l e m 45 v i i ACKNOWLEDGEMENTS Over the years, many people, too many to name, have helped me i n one way or another. In p a r t i c u l a r , I would l i k e to acknowledge Dr. R.L. Johnston, who was my s u p e r v i s o r i n i t i a l l y d u r i n g h i s v i s i t to UBC. His keen i n t e r e s t i n boundary element methods motivated me to work i n t h i s a r ea. I would e s p e c i a l l y l i k e to thank Dr. U. Ascher, my s u p e r v i s o r , f o r a l l h i s time and e f f o r t spent g u i d i n g me through the f i n a l stages of t h i s t h e s i s . Last but not l e a s t , I am g r a t e f u l to a l l my f r i e n d s f o r the moral support they have given me. 1 CHAPTER 1 INTRODUCTION 1.1 Boundary E l e m e n t Method V a r i o u s n u m e r i c a l t e c h n i q u e s have been p r o p o s e d f o r t h e s o l u t i o n o f e l l i p t i c b o u ndary v a l u e p r o b l e m s . Most common methods use d i s c r e t i z a t i o n of t h e domain on w h i c h t h e p r o b l e m i s d e f i n e d . The f i n i t e e l e m e n t method i s one s u c h method. T h i s t e c h n i q u e d i s c r e t i z e s t h e domain of t h e p r o b l e m under c o n s i d e r a t i o n i n t o a number of e l e m e n t s : t h e g o v e r n i n g e q u a t i o n s of t h e p r o b l e m a r e t h e n a p p r o x i m a t e d o v e r t h e r e g i o n by f u n c t i o n s w h i c h s a t i s f y t h e b o u n d a r y c o n d i t i o n s . A method o f i n c r e a s i n g p o p u l a r i t y i s t h e b o u n d a r y e l e m e n t method. The t e r m "boundary e l e m e n t s " i s u s e d t o i n d i c a t e t h e d i s c r e t i z a t i o n of t h e boundary r a t h e r t h a n t h e domain. One of t h e most d e s i r a b l e f e a t u r e s of t h e b o u n d a r y e l e m e n t method i s t h a t much s m a l l e r s y s t e m s o f e q u a t i o n s a r e r e q u i r e d t o s o l v e a p r o b l e m . T h a t i s , f o r a g i v e n ( m o d e r a t e ) a c c u r a c y , fewer b o u n d a r y e l e m e n t s ( w i t h a h i g h e r c o m p u t a t i o n a l c o s t p e r e l e m e n t , though) a r e r e q u i r e d . T h e s e a d v a n t a g e s a r e more marked i n two- and t h r e e - d i m e n s i o n a l p r o b l e m s . The p u r p o s e of t h i s t h e s i s i s t o i n v e s t i g a t e t h e v i a b i l i t y of two b o u n d a r y e l e m e n t methods, t h e c o n t i n u o u s l e a s t s q u a r e s method and t h e G a l e r k i n m i n i m i z a t i o n t e c h n i q u e r e c e n t l y p r o p o s e d by Han, O l s o n and J o h n s t o n [ 1 ] , We compare t h e i r n u m e r i c a l 2 p r o p e r t i e s and, i n p a r t i c u l a r , c o n s i d e r a l g o r i t h m s f o r s o l v i n g t h e r e s u l t i n g s y s t e m s o f n o n l i n e a r e q u a t i o n s . We a l s o d i s c u s s t h e e x t e n t t o w h i c h t h e s e methods a c h i e v e t h e i r g o a l s . B e f o r e i n t r o d u c i n g t h e methods under i n v e s t i g a t i o n , some b a c k g r o u n d i s p r e s e n t e d . The b o u n d a r y e l e m e n t method can be n a t u r a l l y d i v i d e d i n t o t h r e e d i f f e r e n t b u t c l o s e l y r e l a t e d c a t e g o r i e s , namely, i n d i r e c t b o u n d a r y e l e m e n t method, s e m i - d i r e c t b o u n d a r y e l e m e n t method and d i r e c t b o u n d a r y e l e m e n t method ( s e e [ 2 ] ) . In t h e n e x t s e c t i o n , t h e i n d i r e c t and t h e d i r e c t b o u n d a r y e l e m e n t methods w i l l be d e s c r i b e d b e c a u s e t h e y a r e g e n e r a l l y much more u s e f u l t h a n t h e s e m i - d i r e c t a p p r o a c h . Boundary e l e m e n t methods t h a t use an a u x i l i a r y b o u n d a r y w i l l be d i s c u s s e d h e r e as w e l l . T h i s t h e s i s c o n c e n t r a t e s on two methods o f t h e l a t t e r t y p e . 1.2 D i f f e r e n t A p p r o a c h e s o f t h e Boundary E l e m e n t Method L e t us c o n s i d e r t h e D i r i c h l e t b o u n d a r y v a l u e p r o b l e m : (1.1) Au(P) = 0 , P e D (1.2) u ( P ) = f ( P ) , P e T where D i s a bounded domain i n t h e p l a n e w i t h smooth b o u n d a r y T, A d e n o t e s t h e L a p l a c i a n and f i s a p r e s c r i b e d f u n c t i o n on t h e b o u n d a r y . L a t e r on we w i l l g e n e r a l i z e ••( 1 .2) t o r e a d (1.3) Bu(P) = f (P) where B i s some bo u n d a r y o p e r a t o r . But t o d e s c r i b e t h e b a s i c a p p r o a c h e s , t h e s i m p l e r (1.2) i s u s e d . 3 1.2.1 I n d i r e c t boundary e l e m e n t method T h i s i s t h e l e a s t s o p h i s t i c a t e d b o u n d a r y t e c h n i q u e . I t s t a r t s w i t h a s o l u t i o n t h a t s a t i s f i e s t h e g o v e r n i n g e q u a t i o n s i n t h e domain but w h i c h has some unknown c o e f f i c i e n t s . T h e s e c o e f f i c i e n t s a r e t h e n d e t e r m i n e d by e n f o r c i n g t h e b o u n d a r y c o n d i t i o n s a t a number o f p o i n t s . E q u a t i o n (1.1) c a n be r e p r e s e n t e d as a s i m p l e l a y e r p o t e n t i a l , namely, (1.4) u(P) = ; r o(Q) l o g r ( P , Q ) d S Q P e D U T where r ( P , Q ) d e n o t e s t h e d i s t a n c e between t h e p o i n t s P and Q, and o i s t h e unknown s o u r c e d e n s i t y f u n c t i o n , w h i c h i s t o be d e t e r m i n e d so t h a t (1.2) i s s a t i s f i e d . Thus one o b t a i n s (1.5) / r o(Q) l o g r ( P , Q ) d S Q = f ( P ) , P e T w h i c h i s a Fredholm integral equation of the first kind. ( s e e [ 3 ] , [ 4 ] ) The s o l u t i o n can a l s o be r e p r e s e n t e d as a d o u b l e l a y e r p o t e n t i a l , namely, (1.6) u(P) = / R M(Q) g|- l o g r ( P , Q ) d S Q P e D U T where ^ — d e n o t e s t h e o u t w a r d n o r m a l d e r i v a t i v e a t Q e r and M i s 3 n Q t h e unknown s o u r c e d e n s i t y f u n c t i o n t o be d e t e r m i n e d by t h e Fredholm integral equation of the second kind ( s e e [ 3 ] ) 4 (1.7) -fft f ( P ) + / R M(Q) 9n Q l o g r ( P , Q ) d S Q = f ( P ) , P e V T h i s method i s c a l l e d " i n d i r e c t " b e c a u s e a f t e r t h e d e n s i t y f u n c t i o n s a r e d e t e r m i n e d , a q u a d r a t u r e i s r e q u i r e d t o c a l c u l a t e t h e s o l u t i o n u and i t s d e r i v a t i v e s a t a p o i n t i n t h e domain o r on t h e b o u n d a r y . 1.2.2 D i r e c t b o undary e l e m e n t method The d i r e c t method i s g e n e r a l l y p r e s e n t e d b a s e d on G r e e n ' s i d e n t i t y . I t u s e s t h e r e c i p r o c a l t h e o r e m and a f u n d a m e n t a l s o l u t i o n t o r e f o r m u l a t e t h e b o u n d a r y v a l u e p r o b l e m as an i n t e g r a l e q u a t i o n d e f i n e d on t h e boundary of t h e domain. In t h e c a s e o f ( 1 . 1 ) , t h i s c o m b i n a t i o n y i e l d s Green's third identity, (1.8) where c ( P ) u(P) r 9n, 2TT , l o g r ( P , Q ) - 1^- l o g r ( P , Q ) ] d S Q c ( P ) = < v , P e D, p e r, o t h e r w i s e . Hence, one needs t o d e t e r m i n e , from (1.8) w i t h P e T, t h e r e m a i n i n g boundary d a t a . F o r t h e D i r i c h l e t b o u n d a r y v a l u e p r o b l e m , s e t t i n g u(P) = f ( P ) , P e r, e q u a t i o n (1.8) becomes a Fredholm integral equation of the first kind f o r t h e unknown d a t a 9u 9n Q , namely 5 ( 1 " 9 ) / r ^ r ( P ' Q ) d S ° -= Sr f(Q) l o g r(P,Q) d S Q - TT f(P) , P e T The boundary T i s d i s c r e t i z e d i n t o a r c s V. = Q~. "To. , y j - 1 j j=1,...,N, where = and a set of nodes P^. e r. , i = 1,...,N, i s chosen, as shown i n f i g . 1.1. On each arc r , u and are chosen to be constants u. and u r e s p e c t i v e l y . Then (1.9) J becomes N (1.10) Z u n Jr l o g r ( P ,Q) d S Q j = l J J N = L U i h 3n~~ L O G R ( P / ' Q ) dS - TT U / , i = 1,...,N j = i j Q where P^. = Q^. i s m the mid-point of 'T , i = 1,...,N. On approximating the i n t e g r a l s by a p p r o p r i a t e quadrature formulae, we o b t a i n an N x N l i n e a r system f o r the unknowns u r i=1,...,N. (see [5]) n. 1.2.3 Methods using a u x i l i a r y boundary N o t i c e that i f P. and Q are taken to be the same p o i n t , the i n t e g r a n d of the l e f t hand s i d e of (1 .10) w i l l become s i n g u l a r . Hence care must be taken when a p p l y i n g a quadrature r u l e to approximate the corresponding i n t e g r a l s . To a v o i d t h i s problem, s e v e r a l people have proposed to use an a u x i l i a r y boundary which e n c l o s e s the o r i g i n a l boundary. 6 Fig.1.1 Boundary nodes and singularities for direct boundary element method Kupradze's method Kupradze and h i s co-workers have developed the method of c a n o n i c a l f u n c t i o n a l equations f o r s o l v i n g boundary value problems f o r v a r i o u s equations of mathematical p h y s i c s (see [ 6 ] , [ 7 ] ) . They c o n f i n e the o b s e r v a t i o n - p o i n t s P^ . to l i e on an a u x i l i a r y curve Tfl which i s a r b i t r a r i l y chosen and completely surrounds the curve T without touching i t , as shown i n f i g . 1.2. Hence (1.10) becomes (1.11) N I u n / r l o g r(P.,Q) d S Q r. 3 r ^ l o 9 r<p/'Q> d SQ' P. e r a N A f t e r _3 9n Q l o g r(P.,Q) i s determined from (1.11) one can o b t a i n 7 the value of u at a p o i n t i n s i d e r by s o l v i n g ( 1 . 8 ) . Fig.1.2 Boundary nodes and singularities for Kupradze's method The d i f f i c u l t i e s with t h i s approach, as p o i n t e d out by C h r i s t i a n s e n ( [ 8 ] and [ 9 ] ) , are (1) the a u x i l i a r y boundary TQ i s not a l l that a r b i t r a r y . I t has to be chosen c a r e f u l l y i n order to obt a i n a c c u r a t e r e s u l t s ; and (2) t h i s method of s o l v i n g the problem may not always give a unique s o l u t i o n . O l i v e i r a ' s method O l i v e i r a [10] proposed what turned out to be the converse of Kupradze's method i n plane e l a s t o s t a t i c s . In the simple l a y e r p o t e n t i a l r e p r e s e n t a t i o n , the i n t e g r a t i o n i s done with respect to 8 the a u x i l i a r y boundary Tfl , i . e . , (1.12) u(P) = ; r a (Q) l o g r(P,Q) dSQ , P e D U T a a ^ Here the unknown d e n s i t y a i s d e f i n e d on r but i t i s determined J a a by the boundary c o n d i t i o n (1.2) s p e c i f i e d on r . To so l v e (1.12), one needs to choose the boundary nodes P^ . e r , i=1,...,N, s u b d i v i d e the a u x i l i a r y boundary by the p o i n t s Q , j=1,...,N, with r = Q. ,Q. (see f i g . 1.3) and so l v e the N x N system of " > J J * J l i n e a r equations N (1.13) Z a / r l o g r(P,Q) d S n = f ( P . ) , i=1,...,N i = l ° ' J  1 n i y ' The advantage of t h i s approach i s that one can choose T q to be p o l y g o n a l , thus s i m p l i f y i n g the i n t e g r a t i o n r e q u i r e d . Heise [11] p o i n t s out that the a u x i l i a r y boundary has to be chosen c a r e f u l l y , as has C h r i s t i a n s e n with re s p e c t to Kupradze's 9 method. The method of fundamental s o l u t i o n s For the d e s c r i p t i o n of t h i s method, l e t us consid e r the problem with more general boundary c o n d i t i o n s (1.3). The method of fundamental s o l u t i o n s (MFS) i s , i n c e r t a i n r e s p e c t s , a g e n e r a l i z a t i o n of O l i v e i r a ' s approach. From both Kupradze's and O l i v e i r a ' s methods, one n o t i c e s that the l o c a t i o n of the a u x i l i a r y boundary i s important, hence i t i s very n a t u r a l to l e t the " s i n g u l a r i t i e s " Q move u n t i l the best l o c a t i o n i s found. The method i s d i s c u s s e d i n Mathon and Johnston [12]. . The d e s i r e d s o l u t i o n u i s approximated by a f u n c t i o n u^ of the form N (1.14) u^(c,Q;P) = I c.K(P,Q.), P e D U T where [ - l o g r(P,Q.), m=2 K (P, Q . ) = < , J J Y [r(P,Cj.)] , m>2 i s the fundamental s o l u t i o n of (1.1), c = ( c ^ , . . . , c ^ ) T and Q i s a 2N-vector g i v i n g the c o o r d i n a t e s of the s i n g u l a r i t i e s Q , Q \ D U r. T h i s can be c o n s i d e r e d as a p a r t i c u l a r , simple d i s c r e t i z a t i o n of the i n t e g r a l s i n (1.12), (1.13). Since u^ s a t i s f i e s the d i f f e r e n t i a l equation (1.1), one needs to determine c and Q so that u^ approximates (1.3) as w e l l as p o s s i b l e . Mathon and Johnston [12] suggested doing t h i s by minimizing the l e a s t squares sum of r e s i d u a l s , . i . e . , JIB (1.15) | \ B u N ( c * , 2 * r •) - f | | 2 2 M = min | | Z J B U V ( C , Q ; P . ) - f ( P . ) | | c,Q /=/ ' 2 f o r some ( p r e s c r i b e d ) boundary p o i n t s P. on T. In (1.15), 5 i s the boundary operator of (1.3); f o r the case of (1.2), B i s the i d e n t i t y o p e r a t o r . The norm ||«|| 2 i s d e f i n e d on r. The usual necessary c o n d i t i o n f o r a minimum i n (1.15) y i e l d s a system of 3N n o n l i n e a r equations (the "normal equations") M 9[£u„(c,Q,P ) ] (1.16) Z BuN(c,QrP.) • ~ _ £ = o, k=1, i =1 k where = ( c^-fQ^.) T i s the v e c t o r of the unknowns. We w i l l r e f e r to t h i s method as a " D i s c r e t e l e a s t squares" MFS. Ho-Tai, Johnston and Mathon [13] have implemented a code f o r s o l v i n g L a p l a c e ' s equation i n both two and three dimensions u s i n g t h i s method. T h i s software package uses the Gauss-Newton-Marquardt process with l i n e search to minimize (1.15). I t i s found that u s i n g the normalized fundamental s o l u t i o n [ l o g (1.17) K(P,Q) = < r(P,Q) P + r(Q) [ I r ( P , Q ) ] ' 1 + [p + r ( Q ) ] - \ m=2 m>2 can improve the rate of convergence of the n o n l i n e a r l e a s t squares a l g o r i t h m . In (1.17), p i s the r a d i u s of the c i r c u m s c r i b e d c i r c l e of the region D and r(Q) i s the d i s t a n c e of Q from the o r i g i n . 11 The main advantage of t h i s method i s that the number of terms i n the sum (1.14) can be much smaller than the number of boundary nodes with no d e t e r i o r a t i o n in accuracy. However, the p r i c e one pays f o r t h i s f e a t u r e i s that the system of a l g e b r a i c equations w i l l be n o n l i n e a r with the n o n l i n e a r i t y o c c u r r i n g when determining the c o o r d i n a t e s of the s i n g u l a r i t i e s . Continuous l e a s t squares method The d i s c r e t e l e a s t squares method can be viewed as r e s u l t i n g from a p a r t i c u l a r d i s c r e t i z a t i o n of a continuous l e a s t squares method. In the l a t t e r , the o b j e c t i v e f u n c t i o n to be minimized i s M d.18) z ; [BUN(C,Q,P) ] 2 dr i = 1 i where r ^ , . . . , r ^ form a p a r t i t i o n of the boundary r. The necessary c o n d i t i o n s f o r a minimum are M (1.19) Z Sr i=l i 2?u„(c ,Q,P) d[BuN(c,QrP)] 'AT-'*'4 7 3z, k dr = o, k=1 , . . . ,N To a c t u a l l y implement the method, the i n t e g r a l s i n (1.18), (1.19) need to be approximated n u m e r i c a l l y . For t h i s we use the 2-point Gauss quadrature r u l e . The r e s u l t i n g o b j e c t i v e f u n c t i o n d i f f e r s from (1.15) ( i f the same p o i n t s P. are chosen) i n that the equations here have d i f f e r e n t weights a c c o r d i n g to the quadrature r u l e . 12 G a l e r k i n m i n i m i z a t i o n method Han, Olson and Johnston [1] proposed to use the G a l e r k i n m i n i m i z a t i o n technique f o r s o l v i n g Laplace's equation. T h i s method i s very s i m i l a r to the continuous l e a s t squares method d(BuN) d e s c r i b e d above. Instead of using — ^ — as the weighting duN z k f u n c t i o n , i t uses - r — , the d e r i v a t i v e s of the approximate 0 Z k s o l u t i o n with respect to the s i n g u l a r i t y c o o r d i n a t e s and c o e f f i c i e n t s , as the weighting f u n c t i o n . Hence, the system of 3N no n l i n e a r equations i s (1.20) Z ; M r 9u„(c,Q,P) 2?u„(c,Q,P) dr = 0 , k=1,...,N with the same 2-point Gauss quadrature a p p l i e d to (1.20). 1.3 O b j e c t i v e s of the t h e s i s The main o b j e c t i v e s of t h i s t h e s i s are to i n v e s t i g a t e (1) the continuous and d i s c r e t e l e a s t squares methods f o r Laplace's equation, (2) the G a l e r k i n m i n i m i z a t i o n technique f o r Laplace's equation, i n p a r t i c u l a r , the extent to which the e r r o r i n the boundary c o n d i t i o n s i s minimized, given (1.20) and (3) a l g o r i t h m s f o r s o l v i n g the r e s u l t i n g system of n o n l i n e a r equations (1.20). In chapter 2, we give some general background concerning e l l i p t i c boundary value problems which i s needed l a t e r on. In chapter 3, the f o r m u l a t i o n of the continuous l e a s t squares and 13 the G a l e r k i n m i n i m i z a t i o n method f o r the two dimensional L a p l a c e ' s equation are presented. Chapter 4 d e s c r i b e s the v a r i o u s a l g o r i t h m s f o r s o l v i n g systems of n o n l i n e a r equations that we have t r i e d . The a l g o r i t h m and the corresponding r e s u l t s f o r the numerical examples f o r the G a l e r k i n technique are d i s c u s s e d i n chapter 5 . Chapter 6 focuses on the m i n i m i z a t i o n of the r e s i d u a l of the boundary c o n d i t i o n s . Corresponding r e s u l t s f o r /both d i s c r e t e and continuous l e a s t squares methods and the G a l e r k i n technique are presented. I t turns out that the numerical r e s u l t s i n chapter 6 are b e t t e r than those i n chapter 5 . The continuous l e a s t squares method looks p a r t i c u l a r l y good. Our main r e s u l t s and c o n c l u s i o n s are summarized i n chapter 7 . 14 CHAPTER 2 PRELIMINARIES 2.1 E l l i p t i c E q u a t i o n s C o n s i d e r t h e l i n e a r s e c o n d o r d e r e q u a t i o n o f t h e form m m (2.1) Lu = Z a. . (x). u + Z b . ( x ) u + c (x ) u = g (x) I ,J=1  J I J 1=1 I where a i j ' ^/' c a n o - 5 a r e s u i t a b l y d i f f e r e n t i a b l e r e a l v a l u e d f u n c t i o n s o f t h e i n d e p e n d e n t v a r i a b l e s x^,...,x and where i t may be assumed t h a t a . . = a . . . U J1 The e q u a t i o n (2.1) i s s a i d t o be elliptic i n some m - d i m e n s i o n a l domain D i f t h e r e e x i s t c o n s t a n t s M > 0 and K su c h t h a t m m 2 (2.2) • Z a . . ( x ) X. X. > M Z X. i  1J  1 J i i i , j - l J  J i=l f o r a l l r e a l m - t u p l e s (X^,X^,...,X ) a t e a c h p o i n t of D and |b£. |, |c| < K i n D. The Dirichlet boundary value problem, a l s o c a l l e d t h e f i r s t b o u n d a r y v a l u e p r o b l e m , a s k s f o r a s o l u t i o n u of (2.1) i n D which t a k e s on p r e s c r i b e d v a l u e s (2.3) u = f a t t h e b o u n d a r y r of t h a t r e g i o n . 15 The Neumann boundary value problem, a l s o known as t h e s e c o n d b o u n d a r y v a l u e p r o b l e m , r e q u i r e s t h e d e t e r m i n a t i o n i n some r e g i o n D of a s o l u t i o n u t h a t p o s s e s s e s p r e s c r i b e d n o r m a l d e r i v a t i v e s (2.4, on t h e bo u n d a r y T. Here, n s t a n d s f o r t h e u n i t n o r m a l d i r e c t e d i n t o t h e i n t e r i o r of D and we assume t h a t r i s smooth enough t o have a m e a n i n g f u l n o r m a l . F i n a l l y , t h e mixed boundary value problem imposes g i v e n of a l i n e a r c o m b i n a t i o n o f u and , r a t h e r t h a n of e i t h e r o f them s e p a r a t e l y , a l o n g T. T h i s i s a l s o known as t h e t h i r d b o u n d a r y v a l u e p r o b l e m . Under v a r i o u s h y p o t h e s e s a b o u t t h e geometry of T and t h e b e h a v i o u r o f t h e c o e f f i c i e n t s a . . , b., c and q, t h e most i j i  3 e s s e n t i a l o f w h i c h s t a t e s t h a t c ^ 0, i t i s p o s s i b l e t o e s t a b l i s h t h e e x i s t e n c e , u n i q u e n e s s and c o n t i n u o u s dependence on b o u n d a r y d a t a of t h e s o l u t i o n of t h e s e b o u n d a r y v a l u e p r o b l e m s . F o r s i m p l i c i t y , we c o n s i d e r o n l y t h e t w o - d i m e n s i o n a l L a p l a c e ' s e q u a t i o n i n t h i s t h e s i s , namely, v a l u e s (2.5) 3u 9n + au = f (2.6) Au = 16 2.2 F u n d a m e n t a l S o l u t i o n Now l e t us c o n s i d e r t h e homogeneous e l l i p t i c e q u a t i o n : m m (2.7) Lu = I a..u + Z b.u + c u = 0 i, j =1  J i j i = 1 i Suppose R i s t h e s q u a r e of t h e d i s t a n c e between two p o i n t s P and Q w i t h r e s p e c t t o t h e m e t r i c g i v e n by t h e q u a d r a t i c f o r m m d s 2 = Z a. .dx.dx . i  1J  1 J i , j=l J J ( s e e [ 1 4 ] ) . Def i n i t i o n K(P,Q) i s s a i d t o be a fundamental solution o f (2.7) i f i t p o s s e s e s a t t h e p a r a m e t e r p o i n t Q, a s i n g u l a r i t y c h a r a c t e r i z e d by t h e r e p r e s e n t a t i o n (2.8) K(P,Q) = UR _ / + V l o g R + W where U, V and W a r e r e g u l a r f u n c t i o n s of P i n a n e i g h b o u r h o o d o f Q, w i t h U * 0 a t Q, and where the e x p o n e n t / s h o u l d have t h e s p e c i f i c v a l u e / = \ (m-2) In t h e c a s e of two d i m e n s i o n s (m=2), l e t r ( P , Q ) be t h e E u c l i d e a n d i s t a n c e between P and Q ( c f . [ 1 4 ] ) . Then (2.9) K(P,Q) = V l o g r ( p f Q ) + W 1 7 where V and W a r e r e g u l a r f u n c t i o n s of P i n a n e i g h b o u r h o o d o f Q and V * 0 a t Q so t h a t K has a s i n g u l a r i t y of l o g a r i t h m i c t y p e a t P = Q. Example C o n s i d e r t h e L a p l a c e ' s e q u a t i o n , m (2.10) Au = L u = 0 . x . x . i=l i i The f u n d a m e n t a l s o l u t i o n of t h e e q u a t i o n i s of t h e form \ l o g —, m = 2 (2.11) K(P,Q) = < r r , m > 2 ( s e e [ 1 5 ] ) Mathon and J o h n s t o n [12] i n t h e MFS f o u n d t h a t by t a k i n g t h e n o r m a l i z e d f u n d a m e n t a l s o l u t i o n , I" l o g (2.12) K(P,Q) = < r l p - Q l i •2 m = 2 P + | Q | [|P-Q|- 1 + (p+|Q|)- 1 , m > 2 where p i s t h e r a d i u s of t h e c i r c u m s c r i b e d c i r c l e of t h e r e g i o n of i n t e r e s t , i t can improve t h e r a t e of c o n v e r g e n c e when s o l v i n g t h e s y s t e m of n o n l i n e a r e q u a t i o n s . 18 CHAPTER 3 FORMULATIONS OF THE LEAST SQUARES AND GALERKIN METHODS The boundary v a l u e p r o b l e m t h a t we a r e l o o k i n g a t i s t h e f o l l o w i n g : (3.1) L u ( P ) = Au(P) = u + u = 0 , P e D xx yy (3.2) £ u ( P ) = 0 , P e T where t h e L a p l a c e o p e r a t o r L i s a l i n e a r e l l i p t i c d i f f e r e n t i a l o p e r a t o r , D i s a bounded r e g i o n w i t h b o u n d a r y Y and B i s an o p e r a t o r of one of t h e f o l l o w i n g f orms (/') J?u(P) = o ( P ) + u(P) ( D i r i c h l e t p r o b l e m ) (3.3) (//) £ u ( P ) = a(P) + 9 g n P ) (Neumann p r o b l e m ) (///) 2?u(P) = a ( P ) + | 3 ( P ) u ( P ) + 7 ( P ) ^jp-( M i x e d p r o b l e m ) where a, 0, and 7 a r e t h e p r e s c r i b e d f u n c t i o n s . The a p p r o x i m a t e s o l u t i o n w h i c h s a t i s f i e s e q u a t i o n (3.1) i s w r i t t e n a s N (3.4) u J P ) = Z c. K ( P , Q . ) , Q./i D U T where c . , Q. a r e unknown c o e f f i c i e n t s and l o c a t i o n s o f t h e J J s i n g u l a r i t i e s r e s p e c t i v e l y , K i s a f u n d a m e n t a l s o l u t i o n of (3.1) as d e s c r i b e d i n s e c t i o n 2.2 and N i s t h e number of t e r m s u s e d f o r t h e a p p r o x i m a t i o n o f t h e o v e r a l l p r o b l e m . From our e x p e r i m e n t s , N 19 need n o t be t o o l a r g e , r e l a t i v e t o M, t h e t o t a l number of b o u n d a r y p o i n t s . I t was f o u n d e x p e r i m e n t a l l y t h a t a u s e f u l r e l a t i o n s h i p between N and M i s r o u g h l y M = 9 * N. S i n c e u ^ s a t i s f i e s ( 3 . 1 ) , i t r e m a i n s t o s e l e c t t h e c o e f f i c i e n t s c and t h e l o c a t i o n s o f t h e s i n g u l a r i t i e s Q, i n o r d e r t o a p p r o x i m a t e th e b o u n d a r y c o n d i t i o n (3.2) as c l o s e l y as p o s s i b l e . L e t t h e b o u n d a r y r e s i d u a l be d e n o t e d as /? D . B o t h n u m e r i c a l is t e c h n i q u e s d e s c r i b e d h e r e y i e l d ( c f . ( 1 . 1 9 ) , ( 1 . 2 0 ) ) (3.5) Sr RB wk dr = 0, k = 1,. .. ,N where i s a w e i g h t f u n c t i o n . F o r t h e G a l e r k i n t e c h n i q u e , i s g i v e n by t h e g r a d i e n t of u^, w i t h r e s p e c t t o z^, w h i l e f o r l e a s t s q u a r e s , W^ i s t h e g r a d i e n t of Bu^ w i t h r e s p e c t t o z^. The 3 - d i m e n s i o n a l v e c t o r z^ c o n s i s t s of t h e c o o r d i n a t e s o f t h e s i n g u l a r i t i e s and t h e c o r r e s p o n d i n g c o e f f i c i e n t c ^ . I f t h e c o o r d i n a t e s o f Q^ . a r e d e n o t e d as ( a ^ , b ^ ) , we have z, = and 9u da 3u 3b~ Su 3cT N N N 20 f o r t h e G a l e r k i n t e c h n i q u e . T h i s i s compared t o t h e w e i g h t f u n c t i o n f o r t h e l e a s t s q u a r e s method, w h i c h i s d(BuN) 3a, d(BuN) k d(BuN) 9c7~ as d e s c r i b e d i n s e c t i o n 1.2 ( s e e a l s o [ 1 2 ] ) P a r t i t i o n i n g t h e b o u n d a r y i n t o M e l e m e n t s , w i t h T. d e n o t i n g t h e b o u n d a r y f o r t h e /" ' ^  e l e m e n t , we o b t a i n f o r b o t h methods M (3.6) z ; r [ RB wk ] dr = o, i =1 i k = 1 , . . . ,N I . e, M (3.7) Z / p [ BuN Wk ). dr = 0, i = 1 i k = 1 , ...,N, The f o r m u l a t i o n i s g i v e n f o r t h e most g e n e r a l b o u n d a r y c o n d i t i o n s ( 3.3///) and (2.12) i s t a k e n t o be t h e f u n d a m e n t a l s o l u t i o n . Hence, (3.8) N Uw(P) = Z c l o g / = 1 ( x - a . ) 2 + ( y - b . ) 2 p + ^ a 2+b 2 j J 21 and (3.9) N Bu„ = a + 2 c. N i=l 1 0 l o g ( x - a / ) ^ ( y - b ^ . ) V - 2 . v 2 p + ' a.1+b. 2(»-.,)§S * 2 ( y - b , . ) f * (x-a ) 2 + ( y - b . ) 2 where ( x , y ) d e n o t e s t h e c o o r d i n a t e s of t h e b o u n d a r y nodes P, and , a r e t h e d i r e c t i o n c o s i n e s a l o n g t h e x and y - a x e s r e s p e c t i v e l y . D i f f e r e n t i a t i n g (3.8) and (3.9) w i t h r e s p e c t t o ak, b^ and c k, we o b t a i n t h e f o l l o w i n g : 9 u w ( 3 ' 1 0 ) 3 i 7 = c * -2 ( x - a ^ ) 2a (x-ak)2+(y-bk)2 { p + ^ k * + h k * ) ^ k 2 + b k 2 9b, °k - 2 ( y - b . ) 2b, U-ak)2+(y-bk)2 ( p + v ^ 2 + b ^ ) ^ k 2 + * k 2 du 9c" N = l o g ( x - a ) f c ) 2 + ( y - b j t ) 2 4.K 2 2 2 (3.11) d(BuN) da = p c -2U-ak) 2a + 7 c. _ ? 9x ^9n ( x - a , ) 2 + ( y - b , ) 2 2 ( x - a , ) [ 2 ( x - a , ) | f + 2 ( y - b , ) | * ] [ ( x - a ^ ) 2 + ( y - b ^ ) 2 ] 2 S U u ^ ) ~3b~; 0 c - 2 ( y - b . ) 2b, ( x - a j t ) 2 + ( y - b ) t ) 2 ( p + / a ^ + b ^ 2 ) + 7 C - 2 91 9n ( x - a ^ ) 2 + ( y - b A ; ) 2 2 ( y - b , ) [2(x-a f r>|* + 2 ( y - b p § £ ] k 9n  [ ( x - a , ) 2 + ( y - b , ) 2 ] 2 23 d(BuN) = p l o g (x-ak)2+(y-bk) 2 ( x - a ^ + 2 ( y - b j f c ) | j ( x - a ^ ) 2 + ( y - b ^ . ) 2 A p p l y i n g t h e Gauss q u a d r a t u r e r u l e t o t h e i n t e g r a l s i n (3.7) and s u b s t i t u t i n g (3.10) i n t o i t , we o b t a i n t h e s y s t e m o f n o n l i n e a r e q u a t i o n s f o r t h e G a l e r k i n t e c h n i q u e : (3.12) M g I L i=l 1=1 N | 1 k L.W. [ a ( P / ) + L c .HZ . ] < V1. , i l l . , i l l i i k J=l J J / k = 0, k = 1 , . . . ,N where U l. . = p(P l.) l o g ^(x.-a.)2 + ( y / - b . ) 2 i J i J 7(P ) p + l /a. 2+b J 3x' 2 ( x ' - a . ) - ^ - + 2 ( y -b. ) ( x ' - a . ) 2 + ( y ' - b . ) 2 i j * y 3n 24 and ^ i k  c k -2U\-ak) 2a ^ i k  C k - 2 ( y J - b , ) (p+ ^ a ^ + b ^ ) ^ a ^ + b ^ 2 2b : (p+ ^ a ^ + b ^ 2 ) ^ a ^ + b ^ 2 W = l o g ' ( x j - a , ) 2 + ( y | - b , ) p+ ^ a 2+b, g i s t h e number of Gauss p o i n t s ; L. = ~- / ( x -x . ) 2 + ( y . - y . ) 2 i s t h e f a c t o r f o r t h e Gauss Q u a d r a t u r e ; t h Wj i s t h e / Gauss w e i g h t ; I I t h {x.,y.) i s t h e c o o r d i n a t e s of t h e / g l o b a l Gauss t h p o i n t f o r t h e i b o u n d a r y e l e m e n t , xt = \ - + 1  U i + r x i ) ( V 1 } Y- - y, + \ < y / + 7 - y f ) ( t / + D where t^ i s t h e / ' Gauss p o i n t . To o b t a i n t h e s y s t e m of n o n l i n e a r e q u a t i o n s f o r t h e c o n t i n u o u s l e a s t s q u a r e s , we a p p l y t h e Gauss q u a d r a t u r e r u l e t o t h e i n t e g r a l s i n ( 3 . 7 ) , and s u b s t i t u t e (3.11) i n t o i t . The e q u a t i o n s a r e s i m i l a r t o (3.12) e x c e p t t h a t U, V and W a r e s l i g h t l y d i f f e r e n t from t h e a b o v e : 25 U / k = p c ~2(xrak) 2a ( x . - a ^ ) 2 + ( y r b ^ 2 ( p + • a ^ + b ^ ) ' V + b ^ 7 c -2 9x i an / , 9x. , 9y. + 2( X;-a,) [ 2 ( x ; - a , ) H i - + 2(y{ - b j t ) ^ 3 - 2 ( y ; - b , ) 2b, ( x . - a p 2 + ( y - b ^ 2 ( p + v ^ 2 ^ 2 ) •'a^+b^ + 7 C -2 •a 1 Hi 9n 9x 9y + 2 ( y ; - b , ) [ 2 ( x ; - a , ) ^ + 2 ( y J - b j k ) ^ - ] [ ( x ^ - a ^ ) 2 + ( y j - b ^ ) ) 2 ] 2 26 W j k = P l o g :+b, 2(V. 3x;. + 2<* -b*>*r ( x j - a , ) 2 + (y -b^) E q u a t i o n (3.12) i s a s y s t e m of 3N n o n l i n e a r e q u a t i o n s w i t h unknowns a, b and c . We w i l l d i s c u s s i n t h e n e x t c h a p t e r how t o s o l v e t h i s n o n l i n e a r s y s t e m . I f t h e p r o b l e m i s s y m m e t r i c , we c a n r e d u c e t h e s i z e of t h e s y s t e m . F o r t h e c a s e t h a t t h e p r o b l e m i s symmetric a b o u t t h e x - a x i s , we o n l y need t o s u p p l y t h e e l e m e n t s on t h e h a l f p l a n e . Hence t h e s o l u t i o n becomes (3.8' ) N u w ( P ) = Z c. l o g ^ ( x - a ^ . ) 2 + ( y - b . ) 2 ^ ( x - a . ) 2 + (y + b^.) 2 a / 2 + b / 2 a .2 + b . 2 / i and t h e s y s t e m of n o n l i n e a r e q u a t i o n s i s c h a n g e d a n a l o g o u s l y . The s o l u t i o n f o r t h e p r o b l e m w h i c h i s sy m m e t r i c a b o u t b o t h 27 28 CHAPTER 4 METHODS FOR SOLVING NONLINEAR ALGEBRAIC EQUATIONS T h e r e a r e a l a r g e number of p a p e r s d e a l i n g w i t h t h e n u m e r i c a l s o l u t i o n of n o n l i n e a r a l g e b r a i c e q u a t i o n s . T h i s c h a p t e r d i s c u s s e s t h e p r o s and c o n s of t h e d i f f e r e n t methods t h a t we have t r i e d . The n u m e r i c a l examples f o r t h e s e methods a r e p r e s e n t e d i n t h e n e x t c h a p t e r . Our s y s t e m of n o n l i n e a r e q u a t i o n s i s d e f i n e d as (4.1) g i v e n F : R n --> R n where F i s c o n t i n u o s l y d i f f e r e n t i a b l e , f i n d x* c R n s u c h t h a t F ( x ^ ) = 0. 4.1 Newton's Method f o r Systems of N o n l i n e a r E q u a t i o n s Newton's method comes s i m p l y f r o m (4.2) F ( X £ + p ) = F(xk) + f J ( z ) dz *k where J i s t h e J a c o b i a n o f F, J ( x ) = F ' ( x ) . The i n t e g r a l i n (4.2) i s a p p r o x i m a t e d by t h e l i n e a r t erm, (4.3) S J ( z ) d z ^ J ( X A . ) - p Xk Thus t h e a p p r o x i m a t i o n t o F a t n e a r b y p o i n t s x,+p of x, i s g i v e n 29 by (4.4) F ^ + p ) = F < x £ > + J(xk)-P Then t h e Newton s t e p t h a t makes F(x^+p) = 0 i s as f o l l o w s : (a) J U ^ p = -FUk) (4.5) ( b ) * k + l = *k  + P S i n c e  x k+j i - s n o t e x p e c t e d t o be e q u a l t o x*, but o n l y t o be a b e t t e r e s t i m a t e t h a n x^ , we a p p l y t h e Newton i t e r a t i o n (4.5) r e p e a t e d l y f r o m a s t a r t i n g g u e s s x^ u n t i l c o n v e r g e n c e i s a c h i e v e d . L o c a l C o n v e r g e n c e of Newton's Method  D e f i n i t i o n 4.1 I f t h e r e e x i s t c o n s t a n t s c > 0 and k' > 0 s u c h t h a t t h e se q u e n c e ( x ^ J = {x^ , x ^  , x-^, . . .} c o n v e r g e s t o x* and f o r a l l k > k' , I \xk+1 ~ | | < c | \xk+1 - x j | 2 , t h e n (x^} i s s a i d t o be quadrat i calI y convergent t o x*. D e f i n i t i o n 4.2 G i v e n a v e c t o r norm w e d e f i n e N ( x , r ) a s t h e open n e i g h b o u r h o o d of r a d i u s r a r o u n d x, i . e . N ( x , r ) = { x' e R n : I I x' - x I I < r } 30 D e f i n i t i o n 4.3 A f u n c t i o n g i s Lipschitz conti nuous w i t h c o n s t a n t 7 i n a s e t X, w r i t t e n g e L//?^(X), i f f o r e v e r y x,y e X, |g(x) - g ( y ) | < 7 |x - y | . Theorem 4.4 ( c f . [ 1 6 ] ) L e t F : R n — > R n be c o n t i n u o u s l y d i f f e r e n t i a b l e i n an open c o n v e x s e t D c R n. Assume t h a t t h e r e e x i s t s x* e R n and r, / 3 > 0, s u c h t h a t N ( x * , r ) <=• D, F ( x * ) = 0, J ( x * ) _ ' e x i s t s w i t h | | J ( x * ) ~ 1 | | < / 3 , and J e L/p ( N ( x # , r ) ) . Then t h e r e e x i s t s e > 0 s u c h t h a t f o r a l l x^ e N ( x * , e ) , t h e s e q u e n c e x^,x^,... g e n e r a t e d by Xk+1 = *k ~ J ( x A ; ) " 1 F ( x ^ ) ' k = 0 , l , . . . i s w e l l d e f i n e d , c o n v e r g e s t o x*, and obeys I \ x k + ] - ** I I ^ 0 7 Mxj. - x j | 2 , k=0,1,... The above t h e o r e m shows t h a t Newton's method i s l o c a l l y c o n v e r g e n t f r o m good g u e s s e s i f J ( x ^ ) i s n o n s i n g u l a r . But i t may g e n e r a l l y d i v e r g e i f t h e g u e s s e s a r e not good! T h i s i s one o f t h e d i s a d v a n t a g e s of Newton's Method. A n o t h e r r e l a t e d p r o b l e m , i s t h a t t h e J a c o b i a n m ight be n e a r l y s i n g u l a r o r , i n o t h e r words, t h e s y s t e m of l i n e a r e q u a t i o n s (4.5a) might be i l l - c o n d i t i o n e d . In t h i s c a s e t h e s o l u t i o n of t h e l i n e a r s y s t e m would not be r e l i a b l e . S i n c e Newton's method has t h e s e d i s a d v a n t a g e s , we p r o c e e d t o l o o k a t o t h e r methods. 31 4.2 The U n c o n s t r a i n e d M i n i m i z a t i o n U s i n g a M o d e l - T r u s t R e g i o n  A p p r o a c h S i n c e t h e Newton s t e p (4.5) i s g e n e r a l l y not g l o b a l l y c o n v e r g e n t , a q u e s t i o n a r i s e s , how would one d e c i d e whether t o a c c e p t *k+i a s t n e n e x t i t e r a t e i f x^. i s not c l o s e t o any s o l u t i o n x* o f ( 4 . 1 ) ? A r e a s o n a b l e answer i s t h a t ||F(x^. + ^ ) | | s h o u l d be l e s s t h a n | |F(x^.) | | f o r some norm | | • | | , a c o n v e n i e n t c h o i c e b e i n g t h e / 2 norm | | F ( x ) | I 2 = F ( x ) T F ( x ) . 2 R e q u i r i n g t h a t our s t e p r e s u l t i n a d e c r e a s e of | | F ( x ) | | 2 i s t h e same as we would r e q u i r e i f we were t r y i n g t o f i n d a minimum o f t h e f u n c t i o n | | F ( x ) | | 2 . T h u s , we have i n e f f e c t t u r n e d our a t t e n t i o n t o t h e c o r r e s p o n d i n g m i n i m i z a t i o n p r o b l e m : (4.6) min f ( x ) = j F ( x ) T F ( x ) x e R n it ^  it . where t h e i s added f o r l a t e r a l g e b r a i c c o n v e n i e n c e . D. Gay i n [17] has i m p l e m e n t e d a code f o r u n c o n s t r a i n e d m i n i m i z a t i o n u s i n g a m o d e l - t r u s t r e g i o n a p p r o a c h . M o d e l - T r u s t R e g i o n A p p r o a c h (See [ 1 6 ] ) The m o d e l - t r u s t r e g i o n a p p r o a c h i s b a s e d on e s t i m a t i n g t h e r e g i o n i n w h i c h t h e l o c a l model, u n d e r l y i n g Newton's method, can be t r u s t e d t o a d e q u a t e l y r e p r e s e n t t h e f u n c t i o n , and t a k i n g a s t e p t o a p p r o x i m a t e l y m i n i m i z e t h e model i n t h i s r e g i o n . 32 Now s u p p o s e t h a t we have x^, t h e c u r r e n t i t e r a t e and some e s t i m a t e 6 c o f t h e maximum l e n g t h of a s u c c e s s f u l s t e p we a r e l i k e l y t o be a b l e t o t a k e from x^. T h i s r a i s e s t h e q u e s t i o n : how can we b e s t s e l e c t a s t e p of maximal l e n g t h 5 £ from x c ? A n a t u r a l answer e x i s t s i f we c o n s i d e r a q u a d r a t i c m o d e l . I f we add t h e i d e a o f b o u n d i n g t h e maximal s t e p l e n g t h by § c > 0, t h e n t h e c o r r e s p o n d i n g answer t o t h e q u e s t i o n i s t o t r y t h e s t e p s t h a t s o l v e s min m (x +s) = f ( x ) + g T ( x ) s + ^ s T H s c c c 3 c 2 c (4.7) s u b j e c t t o | | s | | 2 ^ 6 c• where g ( x ) = J T ( x ) F ( x ) i s t h e g r a d i e n t of f and H = J T ( x ) 3 c c 3 c c J ( x ^ ) i s t h e Gauss-Newton a p p r o x i m a t i o n f o r t h e H e s s i a n of f . ( I f J i s s i n g u l a r , p e r t u r b H^ t o H c = J T ( x c ) J(^c) + \/n * e • | | i J T (x c ) »J (x ) | | -i • I , where e d e n o t e s t h e machine p r e c i s i o n ) P r o b l e m (4.7) i s t h e b a s i s o f t h e " m o d e l - t r u s t r e g i o n " a p p r o a c h t o m i n i m i z a t i o n . The name comes f r o m v i e w i n g 6^ as p r o v i d i n g a r e g i o n i n w h i c h we c a n t r u s t m c t o a d e q u a t e l y model f . The method c h o o s e s a t r i a l i t e r a t e , x, = x + s, t h a t t c a p p r o x i m a t e l y m i n i m i z e s t h e model m c of f o v e r a r e g i o n i n w h i c h m^ i s c u r r e n t l y t r u s t e d . The d e c i s i o n of how t o a d j u s t t h e t r u s t r e g i o n and w h e t h e r t o a c c e p t x ? a s t h e n e x t i t e r a t e i s by c o m p a r i n g how w e l l f and m c a g r e e a t x^ . I f n e c e s s a r y , t h e r e g i o n i s r e p e a t e d l y s h r u n k and x{ i s rec o m p u t e d u n t i l t h e a c c e p t a b l e new i t e r a t e i s f o u n d . In t h e r o u t i n e i m p l e m e n t e d by Gay, he computes- an x ? s a t i s f y i n g (H + XI) (x^ - x £ ) = -g where X > 0. 33 The n u m e r i c a l examples i n c h a p t e r 5 a r e done by u s i n g t h e r o u t i n e HUMSL by Gay. Note t h a t e v e r y s o l u t i o n t o (4.1) i s a s o l u t i o n t o ( 4 . 6 ) , but t h e r e may be l o c a l m i n i m i z e r s o f (4.6) t h a t a r e n o t s o l u t i o n s t o ( 4 . 1 ) . T h e r e f o r e t h i s i s one o f t h e d i s a d v a n t a g e s of t h i s method. On t h e o t h e r hand, t h i s method u s e s t h e g l o b a l a p p r o a c h , hence i t i s s t i l l w o r t h t r y i n g . 4.3 M a r q u a r d t ' s Method D. M a r q u a r d t [18] s u g g e s t e d a maximum n e i g h b o u r h o o d method f o r s o l v i n g s y s t e m s o f n o n l i n e a r e q u a t i o n s . Our t h i r d method a p p l i e s M a r q u a r d t ' s i d e a w i t h t h e h e l p of l i n e s e a r c h . L e t us f i r s t c o n s i d e r t h e term l i n e s e a r c h and t h e n d i s c u s s M a r q u a r d t ' s a l g o r i t h m . L i n e S e a r c h The i d e a of a l i n e - s e a r c h a l g o r i t h m i s s i m p l e : G i v e n a d e s c e n t d i r e c t i o n p^., i . e . a d i r e c t i o n s a t i s f y i n g p^ Tg(x^.) < 0, we t a k e a s t e p i n t h a t d i r e c t i o n t h a t y i e l d s an " a c c e p t a b l e " x ^ + ^ . T h a t i s , a t i t e r a t i o n k, c a l c u l a t e a d e s c e n t d i r e c t i o n p^., (4.8) s e t x £ + j := x^ + X^ p ^ f o r some X^ . > 0 t h a t makes an a c c e p t a b l e n e x t i t e r a t e . The t e r m " l i n e s e a r c h " r e f e r s t o a p r o c e d u r e f o r c h o o s i n g X^ . i n (4.8) ( s e e [ 1 6 ] ) . The f o l l w i n g a l g o r i t h m ( s e e [19]) i s a m o d i f i c a t i o n of t h a t g i v e n by D a v i e s , Swann and Campey: 34 (1) A r e g i o n c o n t a i n i n g t h e minimum i s f i r s t b r a c k e t e d by a s y s t e m a t i c s e a r c h t e c h n i q u e . I n i t i a l l y f ( x ) and f(x+Xg) a r e c a l c u l a t e d where, f ( x ) i s t h e o b j e c t i v e f u n c t i o n , t h a t i s f ( x ) = F ( x ) T F ( x ) , and X i s t h e i n i t i a l s t e p l e n g t h , u s u a l l y X i s s e t t o 1. Then, d e p e n d i n g on whether f(x+Xg) < f ( x ) o r n o t , e i t h e r f ( x + 2 / X p ) o r f(x-2 zXp_) a r e worked o u t f o r i = 1,2,... u n t i l a v a l u e of i i s f o u n d f o r which f (x + 2* " 7Xp_)< f (x + 2* X£) o r f ( x - 2 f "7Xp_)_< f ( x - 2 ' X 2 ) p r o v i d e d t h a t 2lX S m, where m i s t h e upper bound t o t h e s t e p l e n g t h w h i c h i s s e t t o be 200 h e r e . The f i n a l t h r e e p o i n t s i n t h e c o m p u t a t i o n c o n s t i t u t e t h e b r a c k e t . I f 2l X > m, t h e n x+mp o r x-mp i s c h o s e n as t h e minimum. (2) T h i s i s r e f i n e d by • f i t t i n g a q u a d r a t i c i n t e r p o l a t i o n p o l y n o m i a l t o t h e t h r e e p o i n t s say a, b, c making up t h e b r a c k e t . The minimum of t h i s p o l y n o m i a l ( b 2 - c 2 ) f (x+ap_) + ( c 2 - a 2 ) f (x+bp_) + ( a 2 - b 2 ) f (x+cp) (4.9) d = \ (b -c )f(x+ap_) + (c -a )f(x+bp_) + (a -b )f(x+cp_) i s u s e d t o p r o d u c e a new b r a c k e t . The p o i n t d i s compared w i t h a, b and c and i f i t i s w i t h i n t h e r e q u i r e d a c c u r a c y o f one o f them, d i s c h o s e n as t h e minimum, o t h e r w i s e r e p e a t t h e q u a d r a t i c i n t e r p o l a t i o n ( 4 . 9 ) . M a r q u a r d t ' s a l g o r i t h m M a r q u a r d t ' s a l g o r i t h m f o r t h e n o n l i n e a r l e a s t - s q u a r e s m i n i m i z a t i o n i s a m o d i f i c a t i o n of Newton's method. He adds t h e 35 p a r a m e t e r p t o t h e d i a g o n a l t e r m s o f t h e H e s s i a n m a t r i x H (H J T J , where J i s t h e J a c o b i a n m a t r i x of F) t o make t h e m a t r i x p o s i t i v e d e f i n i t e . T h a t i s , a t t h e k i t e r a t i o n t h e e q u a t i o n ( 4 . 1 0 ) < H ( x P + ui) Pk = a t * * ) i s c o n s t r u c t e d where q_ = J T F . T h i s e q u a t i o n i s t h e n s o l v e d f o r p^.. The new v e c t o r x k + l = xk ~ Pyt w i l l l e a d t o a new f ( x ^ + ^ ) . Some form of t r i a l and e r r o r i s r e q u i r e d t o f i n d a v a l u e p. w h i c h w i l l make f ( x ^ + ^ ) < f ( x ^ . ) . The d e v e l o p m e n t o f s u c h a s t r a t e g y i s n o t e a s y but we do n o t have t o w o r r y a b o u t t h i s a s p e c t h e r e , s i n c e i t has a l r e a d y been done by t h e d e s i g n e r s of H a r w e l l VA07AD ( s e e [ 2 0 ] ) . L e t us summarize t h e a l g o r i t h m h e r e : (1) Compute H(x^) = j T ( x f c ) J ^ X / t ) a n d ^ x k ^ = j T ^ x P - ( x P ' (2) Choose t h e M a r q u a r d t p a r a m e t e r p. (3) S o l v e (H(x^) + u I) pk = 3.Uk). (4) Choose t h e s t e p s i z e X^ by t h e l i n e s e a r c h a l g o r i t h m . (5) S e t x k + 1 = x k - \ k pk. 36 CHAPTER 5 ALGORITHM AND NUMERICAL RESULTS FOR THE GALERKIN TECHNIQUE In t h i s c h a p t e r , we f i r s t d e s c r i b e t h e a l g o r i t h m f o r s o l v i n g t h e b o u n d a r y v a l u e p r o b l e m , g i v e n t h e G a l e r k i n f o r m u l a t i o n i n c h a p t e r 3. Then n u m e r i c a l r e s u l t s f o r t h r e e examples a r e p r e s e n t e d . 5. 1 A l g o r i t h m F i r s t o f a l l , we have t o p r o v i d e t h e i n i t i a l g u e s s of t h e l o c a t i o n s o f t h e s i n g u l a r i t i e s . Then t h e c o r r e s p o n d i n g c o e f f i c i e n t s c a n be computed by s o l v i n g t h e s y s t e m of N l i n e a r e q u a t i o n s . H a v i n g t h u s c o n s t r u c t e d a f u l l i n i t i a l g u e s s , we can now p r o c e e d t o s o l v e t h e s y s t e m of 3N n o n l i n e a r e q u a t i o n s . L e t us r e v i e w t h e a l g o r i t h m f o r s o l v i n g t h e s y s t e m o f l i n e a r e q u a t i o n s . To o b t a i n t h e c o e f f i c i e n t s , we have t o s o l v e t h e l i n e a r s y s t e m , (5.1 ) Gc where G kj M Z i=l g z 1 = 1 L . W H. W / k' R k M Z I =1 i k' c , c w ) T t h e unknown c o e f f i c i e n t s , L, W, H and W a r e same as i n c h a p t e r 3. 37 The w e l l - k n o w n a l g o r i t h m of LU d e c o m p o s i t i o n and back s u b s t i t u t i o n a l g o r i t h m i s a p p l i e d i n t h e l i n e a r s o l v e r . The a l g o r i t h m u s e s t h e well-known Gauss e l i m i n a t i o n w i t h a p a r t i a l p i v o t i n g p r o c e d u r e . I t f a c t o r i z e s a p e r m u t a t i o n of t h e m a t r i x G i n t o a p r o d u c t of a l o w e r t r i a n g u l a r m a t r i x and an upper t r i a n g u l a r m a t r i x , hence PG = LU where P i s c a l l e d a p e r m u t a t i o n m a t r i x . Now, (5.1) becomes LUc = PR, w h i c h c a n be w r i t t e n as (5.2) L d = PR where t h e v e c t o r d i s d e f i n e d by (5.3) Uc = d Hence, t h e unknown c c a n be f o u n d by f i r s t s o l v i n g (5.2) f o r d, and t h e n u s i n g t h e back s u b s t i t u t i o n t o f i n d c i n ( 5 . 3 ) . A f t e r t h e c o e f f i c i e n t s a r e o b t a i n e d , we c a n now s o l v e t h e s y s t e m of 3N n o n l i n e a r e q u a t i o n s ( 3 . 1 2 ) . I t i s g e n e r a l l y known t h a t n o n l i n e a r e q u a t i o n s a r e much h a r d e r t o s o l v e t h a n l i n e a r e q u a t i o n s e s p e c i a l l y when t h e J a c o b i a n i s v e r y c l o s e t o b e i n g s i n g u l a r . In t h e G a l e r k i n m i n i m i z a t i o n t e c h n i q u e w i t h b o u n d a r y e l e m e n t s we e n c o u n t e r s u c h n e a r l y s i n g u l a r J a c o b i a n s . The p r e v i o u s c h a p t e r d i s c u s s e d t h e v a r i o u s methods us e d f o r s o l v i n g t h e n o n l i n e a r s y s t e m . We f i r s t t r i e d s o l v i n g t h e s y s t e m of 3N n o n l i n e a r e q u a t i o n s (3.12) by d i r e c t l y a p p l y i n g t h e t h r e e methods d e s c r i b e d . However, due t o t h e i l l - c o n d i t i o n i n g , Newton's method f a i l e d t o c o n v e r g e and t h e model t r u s t r e g i o n method c o n v e r g e d 38 v e r y s l o w l y . Hence we use a p a r t i c u l a r i t e r a t i v e method where we d i v i d e t h e sys t e m of 3N n o n l i n e a r e q u a t i o n s i n t o a s y s t e m o f N l i n e a r e q u a t i o n s f o r t h e c o e f f i c i e n t s and a s y s t e m of 2N n o n l i n e a r e q u a t i o n s f o r t h e c o o r d i n a t e s o f t h e s i n g u l a r i t i e s . T h e s e two systems of e q u a t i o n s a r e s o l v e d back and f o r t h f o r a number of i t e r a t i o n s t o o b t a i n t h e f i n a l r e s u l t s . The methods of c h a p t e r 4 were a p p l i e d t o t h e s y s t e m s of 2N e q u a t i o n s . F o r t h e M a r q u a r d t ' s method, s i n c e i t t a k e s c a r e of t h e c a s e when t h e J a c o b i a n i s b e i n g s i n g u l a r , t h e s o l u t i o n was f o u n d t o c o n v e r g e n i c e l y w h i l e s o l v i n g t h e 3N n o n l i n e a r e q u a t i o n s d i r e c t l y . The n u m e r i c a l r e s u l t s from a p p l y i n g t h e s e t h r e e methods a r e p r e s e n t e d i n t h e n e x t s e c t i o n . N o t i c e t h a t our o b j e c t i v e i s t o s e l e c t t h e c o e f f i c i e n t s and t h e l o c a t i o n s of t h e s i n g u l a r i t i e s so t h a t t h e b o u n d a r y c o n d i t i o n s (3.2) can be a p p r o x i m a t e d as c l o s e l y a s p o s s i b l e , t h a t i s , we a r e t r y i n g t o make Bu^ c l o s e t o z e r o . But s o l v i n g t h e s y s t e m o f n o n l i n e a r e q u a t i o n s (3.12) may n o t a l w a y s make Bu^ c l o s e t o z e r o , hence i n c h a p t e r 6 we d i s c u s s t h e method o f m i n i m i z i n g t h e r e s i d u a l i n t h e bo u n d a r y c o n d i t i o n s (3.2) by u s i n g t h e G a l e r k i n f o r m u l a t i o n s . 5.2 N u m e r i c a l R e s u l t s L e t us l o o k a t t h e t h r e e s i m p l e examples w h i c h r e p r e s e n t e a c h t y p e of boundary p r o b l e m . F o r e a c h p r o b l e m , t h e r e a r e t h r e e t a b l e s t a b u l a t i n g t h e r e s u l t s . The f i r s t t a b l e shows t h e l o c a t i o n s of t h e i n i t i a l and f i n a l s i n g u l a r i t i e s and t h e i r a s s o c i a t e d c o e f f i c i e n t s from t h e t h r e e a l g o r i t h m s f o r n o n l i n e a r 39 e q u a t i o n s . The n u m e r i c a l s o l u t i o n and t h e a b s o l u t e e r r o r as compared t o t h e e x a c t s o l u t i o n f o r some c h o s e n b o u n d a r y nodes a r e t a b u l a t e d i n t h e s e c o n d t a b l e . The t h i r d t a b l e summarizes t h e a c c u r a c y and e f f i c i e n c y f o r e a c h o f t h e methods. The f i r s t row o f t h e t a b l e p r e s e n t s t h e r o o t mean s q u a r e e r r o r s computed from t h e r e s i d u a l s of t h e b o u n d a r y c o n d i t i o n s . The s e c o n d row shows t h e r o o t mean s q u a r e e r r o r s of t h e a b s o l u t e e r r o r between t h e n u m e r i c a l and t h e e x a c t s o l u t i o n . T h e n , t h e l2 norms of t h e f u n c t i o n ( 3 . 1 2 ) , t a b u l a t e d i n t h e t h i r d row, show us how w e l l t h e s o l u t i o n s a t i s f i e s t h e o b j e c t i v e f u n c t i o n f o r t h e p a r t i c u l a r method. The number of f u n c t i o n e v a l u a t i o n s a r e shown i n t h e l a s t row. A l l c o m p u t a t i o n s were done i n d o u b l e p r e c i s i o n on an Amdahl 470. 5.2.1 D i r i c h l e t Boundary V a l u e P r o b l e m C o n s i d e r t h e f o l l o w i n g T o r s i o n P r o b l e m : Au(P) = 0, P e D (5.4) u(P) = ( x 2 + y 2 ) / 2, . P e T w i t h a r e c t a n g u l a r domain w i t h s i d e s S^ . = 2, = 1 as shown i n f i g . 5.1. T h i s i s a s t a n d a r d example t h a t has been c o n s i d e r e d by many a u t h o r s . O b s e r v e t h a t t h e p r o b l e m i s s y m m e t r i c about t h e x and y - a x e s , hence we c o n s i d e r o n l y t h e f i r s t q u a d r a n t . I t was d i s c r e t i z e d t o 9 b o u n d a r y e l e m e n t s a l o n g t h e x - a x i s and 4 a l o n g t h e y - a x i s and t h e 2 - p o i n t Gauss q u a d r a t u r e r u l e was u s e d . 3 INITIAL FINAL SINGULARITIES s NGULAR TIES New on's Metf iod Minim zation Ms jthod Marqi. jarclt' s Ms sthod X y coef f. X y coeff . X y coeff. X y coeff. 1 . 10 1.15 0.40 0. 10 0. 65 0.60 -0.0043 -0.1149 0.0474 0.3871 1.1985 0.2659 2.1 125 0.6951 0.9322 0.1178 -0.1723 0.0322 1.3168 1.2294 0. 2914 . 73E-5 0.6128 0.7661 0.0085 -0.1327 0.0535 1.1027 1.1546 0.2786 0. 1028 0. 5952 0. 7860 -0.0022 -0. 1119 0.0509 Table 5.1.1 Locations of the i n i t i a l and f i n a l s i n g u l a r i t i e s and their associated c o e f f i c i e n t s for the Torsion problem Boundary Nodes x y Exact Solution Newton's Method Minimization Method Marquardt's Method Numerical Absolute Solution Error Numerical Absolute Solution Error Numerical Absolute Solution Error • 1.00 0.00 1.00 0.25 1.00 0.50 0.56 0.50 0.00 0.50 0.5000 0.5313 0.6250 0.2793 0.1250 0.5012 . 124E-2 0.5304 .860E-3 0.6256 .611E-3 0.2798 .495E-3 0.1254 .403E-3 0.4969 .307E-2 0.5333 .203E-2 0.6183 .665E-2 0.2850 .564E-2 0.1312 .616E-2 0.5015 .145E-2 0.5289 .233E-2 0.6285 .353E-2 0.2836 .423E-2 0.1294 .442E-2 Table 5.1.2 The numerical solutions vs Exact solutions for the Torsion problem 41 Newton's Method Min. Method Marquardt's Method RMSE (from boundary condition) RMSE (compared with exact soln) MMI # function evaluations . 548E-3 .589E-3 . 128E-5 41 .443E-2 .477E-2 . 228E-3 99 .292E-2 . 298E-2 . 779E-4 54 Table 5.1.3 Accuracy and e f f i c i e n c y for the d i f f e r e n t methods for the Torsion problem y 2 A U = Q -> X Fig.5.1 Torsion problem 42 s i n g u l a r i t i e s were p l a c e d not t o o f a r away from t h e domain as shown i n t h e t a b l e . The e x a c t s o l u t i o n s f o r p o i n t s on t h e b o undary i s u = ( x 2 + y 2 ) / 2. We r a n a l l 3 programs f o r 20 i t e r a t i o n s w i t h t h e same l o c a t i o n s of t h e i n i t i a l s i n g u l a r i t i e s and t h e r e s u l t s a r e t a b u l a t e d i n t a b l e s 5.1.1 t o 5.1.3. 5.2.2 Neumann Boundary V a l u e P r o b l e m As an example u s i n g Neumann bo u n d a r y c o n d i t i o n s , t h e p o t e n t i a l f l o w p a s t a c i r c u l a r c y l i n d e r i s c h o s e n . The f l o w o v e r s u c h a c y l i n d e r i n an i n f i n i t e domain i s a p p r o x i m a t e d by a f i n i t e domain as shown i n f i g . 5.2. E m p l o y i n g t h e f a c t t h a t i t i s s y mmetric a b o u t t h e x - a x i s , we l o o k o n l y a t t h e u p p e r h a l f p l a n e . The d i s c r e t i z a t i o n scheme i s 18 b oundary e l e m e n t s a l o n g t h e x - a x i s , 7 e l e m e n t s a l o n g t h e y - a x i s and 17 e l e m e n t s on t h e s e m i - c i r c l e . 10 s i n g u l a r i t i e s a r e p l a c e d c a r e f u l l y a r o u n d t h e domain as shown i n t h e f i g u r e . S i n c e o n l y Neumann bo u n d a r y c o n d i t i o n s a r e p r e s c r i b e d , t h e f u n c t i o n u computed i s a c c u r a t e t o an a r b i t r a r y c o n s t a n t . Hence t o g e t m e a n i n g f u l r e s u l t s , p a r a m e t e r l i k e p r e s s u r e w h i c h i s o b t a i n e d by d i f f e r e n t i a t i n g t h e p o t e n t i a l f u n c t i o n w i l l be computed. The e x a c t p r e s s u r e d i s t r i b u t i o n c o r r e s p o n d s t o t h e i n f i n i t e domain s o l u t i o n and i s g i v e n by P(8) = (1 - 4 s i n 2 0 ) / 2 ( s e e [ 1 ] ) . The 2 - p o i n t Gauss q u a d r a t u r e r u l e i s a p p l i e d . INITIAL FINAL SINGULARITIES S NGULAR TIES New :on's Metr nod Minim zation Me sthod Marc ^uardt's f lethod X y coeff. X y coeff . X y coeff. X y coeff. 11 0 3 0 -0 703 t 13 282 3.000 -0.6962 13.576 3.114 -0 7350 16 827 4.618 - 1.3430 11 0 1 1 0 -1 331 1 13 494 10.057 -1.3401 13.608 10.347 - 1 3590 13 .465 12.858 - 1.2073 5 0 1 1 O- -0 1 174 7 .493 12.160 -0.3311 7 . 338 12.312 -0 3142 5 .981 13.054 -O.1705 - 5 0 1 1 0 0 1 174 - 7 493 12.160 0.331 1 - 7.338 12.312 0 3142 - 5 .981 13.054 0.1705 - 1 1 0 1 1 0 1 331 1 -13 494 10.057 1.3401 -13.608 10.347 1 3590 - 13 465 12.858 1.2073 - 1 1 0 3 0 0 7031 -13 282 3.000 0.6962 -13.576 3.114 0 7350 - 16 827 4.618 1 .3430 - 0 7 0 2 0 0840 - 0 685 0. 162 0.0838 - 0.684 0.161 0 0831 - 0 595 0. 149 0.0975 - 0 4 0 4 0 1678 - 0 396 0.402 0.1715 - 0.394 0.399 0 1738 - 0 325 0. 343 0.207 1 0 4 0 4 -0 1678 0 396 0.402 -0.1715 0. 394 0. 399 -0 1738 0 .325 0. 343 -0.2071 0 7 0 2 -0 0840 0 685 0. 162 -0.0838 0.684 0.161 -o 083 1 0 595 0. 149 -0.0975 Table 5.2.1 Locations of the i n i t i a l and f i n a l s i n g u l a r i t i e s and their associated c o e f f i c i e n t s for the flow problem Boundary Nodes x y Exact So 1ut i on Newton's Method Minimization Method Marquardt's Newton Numerical Absolute Solution Error Numerical Absolute Solution Error Numerical Absolute Solution Error 0.10 0.00 0.85 0.53 0.45 0.90 0.09 0.99 0.500 -0.054 - 1.103 -1.483 0.4996 .432E-3 -0.0504 .381E-2 -1.1627 .601E-1 -1.4604 .226E-1 0.499S .416E-3 -0.0510 .324E-2 -1.1620 .594E-1 -1.4629 .201E-1 0.4999 .947E-4 -0.0616 .996E-2 -1.1323 .296E-1 -1.4940 .110E-1 Table 5.2.2 The computed pressure vs. Exact pressure for the flow problem Newton's Method Min. Method Marquardt's Method RMSE (from boundary condition) RMSE (compared with exact soln) 1 1 F 1 1 n function evaluations .623E-1 . 162E-1 .901E-2 21 .584E-1 . 161E-1 .180E-1 49 .458E- 1 .145E-1 .361E-2 45 Table 5.2.3 Accuracy and e f f i c i e n c y for the d i f f e r e n t methods for the flow problem > X Fig.5.2 Potential flow past a circular cylinder 45 T a b l e s 5.2.1 t o 5.2.2 show t h e c o m p a r i s o n o f t h e p e r f o r m a n c e o f t h e 3 methods a f t e r 10 i t e r a t i o n s . 5.2.3 M i x e d B o u n d a r y V a l u e P r o b l e m H e a t c o n d u c t i o n i n a r e c t a n g u l a r p r i s m ( s e e [ 2 1 ] ) i s u s e d a s an example of a p r e s c r i b e d m i x e d b o u n d a r y c o n d i t i o n p r o b l e m . The p r o b l e m i s d e f i n e d i n f i g . 5.3 t o g e t h e r w i t h t h e l o c a t i o n s of t h e i n i t i a l s i n g u l a r i t i e s . Symmetry a b o u t t h e x - a x i s i s e x p l o i t e d . We d i s c r e t i z e t h e b o u n d a r y a l o n g t h e x - a x i s i n t o 28 b o u n d a r y e l e m e n t s and 13 b o u n d a r y e l e m e n t s f o r t h e b o u n d a r y a l o n g y - a x i s . The e x a c t s o l u t i o n f o r p o i n t s on t h e b o u n d a r y i s u = 50(3 - x ) . A g a i n we a p p l y t h e 2 - p o i n t G a u s s q u a d r a t u r e r u l e so t h a t t h e number o f p o i n t s i s 9 t i m e s t h e number o f unknowns i n t h e s y s t e m o f n o n l i n e a r e q u a t i o n s . T a b l e s 5.3.1 t o 5.3.2 show t h e c o m p a r i s o n o f t h e r e s u l t s f r o m t h e t h r e e methods a f t e r 20 . i t e r a t i o n s . y U =300 A U = 0 u=o U n = 0 Fig.5.3 Heat transfer problem INITIAL FINAL SINGULARITIES S NGULAR TIES New :on's Metf nod Minim zation Me Jthod Marqi jardt's Me 3 t h o d X y coeff . X y coeff. X y coeff. X y coeff. 3 . 50 3 . 50 1 . 50 -1 .50 -3 . 50 -3 .50 1 . 50 3 . 50 3 . 50 3 . 50 3 . 50 1 .50 0.0240 0.0135 -0.0148 -0.0306 -0.1005 -0.0720 4.9204 .739E4 -5.6309 -5.0037 4.5976 -10.371 0.0372 .529E3 9.2809 12.0771 0.0702 4.8111 0.0226 78.3130 0. 1310 -0.4382 -0.0162 -0.2959 5 . 2576 5.3605 0.1114 -2.0192 -3.7196 -4.1167 1.6219 1.7023 4.4315 3.9353 3.3098 1.1221 -0.0568 0.1060 -0.0330 -0.0342 -0.0933 -0.0718 4.2662 3.7029 1.1200 -1.3240 -4.2617 -4.5005 1.3078 3.3583 3.9783 4 . 5945 3.9737 1.1967 0.0278 0.0137 -0.0081 -0.0397 -0. 1384 -0.0703 Table 5.3.1 Locations of the i n i t i a l and f i n a l s i n g u l a r i t i e s and their associated c o e f f i c i e n t s for the Heat conducting problem Newton's Method Minimization Method Marquardt's Method Boundary Nodes Exact Numer i ca1 Absolute Numer i ca1 Absolute Numer i ca1 Absolute X y So 1ut i on Solut ion Error Solut ion Error Solution Error 3 .00 0.00 0.0000 0.0956 .956E-1 0.0093 .935E-2 0.6922 .692E+0 3.00 1 . 50 0.0000 -0.0731 .731E-1 -0.1017 .102E+0 -0.5692 . 569E+0 3 .00 3 .00 0.0000 0.1746 .175E+0 1.4548 .145E+1 0.1892 . 189E+0 1 .80 3 .00 60.0000 59.9784 .216E-1 59.4613 .539E+0 60.2592 .259E+0 0. 20 3.00 140.0000 140.0622 .622E-1 14 1.0457 .105E+1 139.6369 . 363E+0 - 1 . 80 3 .00 240.0000 240.0501 .501E-1 240.9951 .995E+0 239.9191 .809E-1 -3.00 3.00 300.0000 300.0192 .192E-1 304.1322 .413E+1 301.0805 .108E+1 -3 .00 1 . 50 300.0000 300.0446 .446E-1 300.3968 .397E+0 300.3944 .394E+0 -3 .00 0.00 300.0000 299 .9295 .706E-1 298.4519 .155E+1 299.3188 •681E+0 Table 5.3.2 The numerical solutions vs. Exact solutions for the Heat conducting problem Newton's Method Mln. Method Marquardt's Method RMSE (from boundary condition) RMSE (compared with exact soln) | | F | | # function evaluations .2S1E-3 .713E- 1 .246E-6 4 1 .438E-2 . 1 11E+1 .153E-3 99 . 239E-2 .459E+0 .379E-4 65 Table 5.3.3 Accuracy and e f f i c i e n c y for the d i f f e r e n t methods for the Heat conducting problem 48 CHAPTER 6 ALGORITHM FOR MINIMIZING THE BOUNDARY CONDITIONS We have seen i n t h e l a s t c h a p t e r t h a t none of t h e t h r e e methods f o r s o l v i n g t h e n o n l i n e a r e q u a t i o n s t h a t were t r i e d g i v e us v e r y good r e s u l t s w i t h t h e G a l e r k i n m i n i m i z a t i o n t e c h n i q u e . M o r e o v e r , t h e r e s u l t s w i t h t h e d i s c r e t e l e a s t s q u a r e s method of Mathon and J o h n s t o n [12] a r e much b e t t e r . In t h i s c h a p t e r , l e t us l o o k a t m i n i m i z i n g t h e b o u n d a r y c o n d i t i o n s r e s i d u a l more d i r e c t l y . One method w h i c h m i n i m i z e s t h e b oundary r e s i d u a l d i r e c t l y i s t h e c o n t i n u o u s l e a s t s q u a r e s method, p r e s e n t e d i n c h a p t e r 3. Here we compare i t a g a i n s t t h e d i s c r e t e l e a s t s q u a r e s method o f [ 1 2 ] . S i n c e t h e d i f f e r e n c e between t h e two methods i s o n l y i n t h e w e i g h t s p u t on t h e e q u a t i o n s , t h e s o f t w a r e p a c k a g e of H o - T a i , J o h n s t o n and Mathon [13] was m o d i f i e d t o h a n d l e b o t h c a s e s . To r e c a l l , t h i s p a c k a g e u s e s a M a r q u a r d t t e c h n i q u e f o r t h e m i n i m i z a t i o n o f t h e o b j e c t i v e f u n c t i o n . S i n c e i t t u r n s out t h a t t h e r e s u l t i n g a p p r o x i m a t i o n s a r e b e t t e r t h a n t h o s e i n t h e p r e v i o u s c h a p t e r (as documented b e l o w ) , we have a l s o i m plemented a h y b r i d t e c h n i q u e , where t h e o b j e c t i v e f u n c t i o n i s t h e one f o r l e a s t s q u a r e s ( 1 . 1 8 ) , w h i l e t h e g r a d i e n t i s due t o t h e G a l e r k i n method ( 3 . 1 2 ) . Thus, we s o l v e t h e n o n l i n e a r e q u a t i o n s o f t h e G a l e r k i n method w i t h t h e l e a s t s q u a r e s o b j e c t i v e f u n c t i o n . T h i s c o m b i n a t i o n i s r e f e r r e d t o h e r e as 49 " G a l e r k i n m i n i m i z a t i o n " . 6.1 D i r i c h l e t B oundary C o n d i t i o n s When B i s i d e n t i t y , t h a t i s , we c o n s i d e r t h e D i r i c h l e t b o u ndary v a l u e p r o b l e m s , t h e two methods of G a l e r k i n m i n i m i z a t i o n and c o n t i n u o u s l e a s t s q u a r e s c o i n c i d e . Our o b j e c t i v e f u n c t i o n t o be m i n i m i z e d i s M ( 6 . 1 ) <t> = 0 ( C , Q ) = i ; R [ U A , ( C , Q , P ) ] 2 dr / = 1 i The g r a d i e n t f u n c t i o n i s (6.2) 9z, k u^ y , k — 1,...,N wh i c h i s t h e same as our r e s u l t i n g f u n c t i o n i n ( 3 . 1 2 ) . The Gauss-Newton a p p r o x i m a t e H e s s i a n f u n c t i o n i s (6.3) H. . = 9u^ - i T 9u 9z . i 9z . J l , ] = 1,...,N. We t a k e t h e same example as i n s e c t i o n 5.2.1; t h e d i s c r e t i z a t i o n scheme i s t h e same; t a b l e s 6.1.1 t o 6.1.3 t a b u l a t e t h e r e s u l t s a f t e r 20 i t e r a t i o n s . C o m p aring t h e r e s u l t s w i t h t a b l e s 5.1.1 t o 5.1.3, we see t h a t m i n i m i z i n g t h e r e s i d u a l i n t h e bo u n d a r y c o n d i t i o n s g i v e s us much b e t t e r r e s u l t s . Note t h a t t h e r e s u l t s p r o d u c e d by a l l t h r e e methods, i . e . t h e d i s c r e t e l e a s t s q u a r e s m i n i m i z a t i o n , t h e c o n t i n u o u s l e a s t s q u a r e s m i n i m i z a t i o n and t h e G a l e r k i n m i n i m i z a t i o n a r e a l m o s t t h e same f o r t h e D i r i c h l e t b o u n d a r y c o n d i t i o n s . INITIAL SINGULARI TIES I n i t i a l Coeff. from Least Squares I n i t i a l Coeff . from Ga1erk i n FINAL SINGULARITIES Discrete Least Squares Continuous Least Squares Galerkin Minimization X y X y coeff. X y coeff. X y coeff . 1 . 10 0. 10 -0.0043 -0.0042 1.7409 0.4787 0.0353 1.7630 0.2303 0.0216 1.7935 0. 3279 0.0264 1.15 0.65 -0.1149 -0.1151 1.2460 0.7239 -0.2064 1.2352 0.7207 -0.1928 1 . 2389 0.7222 -0. 1975 0.40 0.60 0.0474 0.0475 0.3388 1.3300 0.1236 0. 3677 1.3182 0.1218 0.3374 1.3250 0. 1227 Table 6.1.1 Locations of the i n i t i a l and f i n a l s i n g u l a r i t i e s and their associated c o e f f i c i e n t s for the Torsion problem Boundary Nodes x y Exact Solut i on Discrete Least Squares Continuous Least Squares Galerkin Minimization Numerical Absolute Solution Error Numerical Absolute Solution Error Numerical Absolute Solution Error 1.00 0.00 1.00 0.25 1 .00 0.50 0.56 0.50 0.00 0.50 0.5000 0.5313 0.6250 0.2793 0.1250 0.5003 .286E-3 0.5309 .318E-3 0.6241 .946E-3 0.2794 .354E-4 0.1250 .264E-4 0.5003 .280E-3 0.5310 .236E-3 0.6241 .928E-3 0.2794 .488E-4 0.1250 .336E-2 0.5003 .285E-3 0.5310 .235E-3 0.6240 .957E-3 0.2794 .313E-4 0.1250 .180E-2 Table 6.1.2 The numerical solutions vs. Exact solutions for the Torsion problem o 51 D1screte Least Squares Cont1nuous Least Squares Galerk1n Minimizat i on RMSE (from boundary condition) RMSE (compared with exact soln) l | F | | # function evaluations . 151E-3 .290E-3 . 145E-5 68 .338E-4 .281E-3 . 372E-5 63 .350E-4 .289E-3 . 197E-4 65 Table 6.1.3 Accuracy and e f f i c i e n c y for the d i f f e r e n t methods for the Torsion problem 6.2 Neumann and M i x e d B o u n d a r y C o n d i t i o n s S i n c e we o b t a i n b e t t e r r e s u l t s f r o m t h e G a l e r k i n a l g o r i t h m when o b s e r v i n g t h e m i n i m i z a t i o n o f -5u^, l e t us u s e t h e same method t o s o l v e f o r t h e c a s e when 2? * I . T h u s we c o n s i d e r t h e m i n i m i z a t i o n o f M (6.5) <)> = L J [2?u^(c,Q,P) ] 2 dr i =1 i w i t h t h e g r a d i e n t f u n c t i o n (6.2) a p p r o x i m a t e d by M duN (6.6) .g^ = 2 J [ BuN.-rf- ] dr , k=1,...,N.. i = 1 / k The H e s s i a n f u n c t i o n o f f i s a g a i n a p p r o x i m a t e d by ( 6 . 3 ) . We use t h e same e x a m p l e s as i n s e c t i o n s 5.2.2 and 5.2.3. E v a l u a t i n g t h e r e s u l t s a g a i n s t t h e ones from t h e d i s c r e t e l e a s t s q u a r e s method o f [12] and t h e c o n t i n o u s l e a s t s q u a r e s method, we 52 f i n d t h a t t h e y a r e c o m p a r a b l e , w i t h t h e c o n t i n u o u s l e a s t s q u a r e s r o u t i n e p r o d u c i n g t h e b e s t r e s u l t s . In f a c t t h e s o l u t i o n s from t h e G a l e r k i n r o u t i n e a r e s l i g h t l y b e t t e r t h a n t h a t from t h e d i s c r e t e l e a s t s q u a r e s r o u t i n e . A l s o we can see t h a t t h e s e r e s u l t s a r e f a r b e t t e r t h a n t h o s e shown i n t h e l a s t c h a p t e r even t h o u g h t h e g r a d i e n t f u n c t i o n does not c o r r e s p o n d t o t h e o b j e c t i v e f u n c t i o n . T a b l e s 6.2.1 t o 6.3.3 show t h e r e s u l t s from b o t h t h e Neumann and t h e mixed t y p e b o u n d a r y c o n d i t i o n s e x a m p l e s . • INITIAL SINGULARIT ES I n i t i a l Coeff. from Least Squares I n i t i a l Coeff. from Ga1erk i n FINAL SINGULARITIES D i screte .east Squares Cont i nuous Least Squares Galerkin Minimization X y X y coef f. X y coeff. X y coeff. 11.0 3 0 -0.G077 -0.7031 19 .5767 3 .0902 -0.2524 20 4356 2 0999 -0 1799 15 . 57 12 3 .3021 -0.6366 11.0 11 0 - 1 . 1988 - 1 . 331 1 31 . 5518 21 .5662 -5. 1488 37 1839 24 9923 -6 2538 17 . 2049 13 .2127 -2.1874 5.0 11 0 -0.4493 -0.1174 5 . 3954 18 . 2338 -0.0746 3 0382 21 4707 -0 2009 6 .4219 16 .1231 -0.2726 - 5.0 11 0 0.4493 0. 1 174 - 5 . 3954 18 . 2338 0.074G - 3 0382 21 4707 0 2009 - 6 .4219 16 1231 0.2726 -11.0 11 0 1 . 1988 1 .331 1 -31 .5518 21 . 5662 5.1488 -37 1839 24 9923 6 2538 - 17 . 2049 13 2127 2.1874 -11.0 3 0 0.6077 0.7031 -19 . 5767 3 .0902 0.2524 -20 4356 2 0999 0 1799 - 15 .5712 3 302 1 0.6366 - 0.7 0 2 0.0705 0.0840 - 0 .3011 0 .0686 0.0639 - 0 3014 0 6765 0 0420 - 0 . 1340 -0 0557 0.3198 - 0.4 0 4 -0.1414 -0.1678 - 0 1223 0 0986 -0.8519 - 0 1121 0 0845 -0 9971 - 0 1401 0 0786 -0.5791 0.4 0 4 -0.1414 -0.1678 0 1223 0 0986 -0.8519 0 1121 0 0845 -O 9971 0 1401 0 0786 -0.5791 0.7 0 2 -0.0705 -0.0840 0 301 1 0 0686 -0.0639 0 3014 0 6765 -0 0420 0 1340 -0 0557 -0.3198 Table 6.2.1 Locations of the i n i t i a l and f i n a l s i n g u l a r i t i e s and their associated c o e f f i c i e n t s for the flow problem Boundary Nodes x y Exact Solut i on Discrete Least Squares Continuous Least Squares Galerkin Minimization Numerical Absolute Solution Error Numerical Absolute Solution Error Numerical Absolute Solution Error 0.10 0.00 0.85 0.53 0.45 0.90 0.09 ' 0.99 0. 500 -0.054 -1.103 -1 .483 0.4999 .717E-4 -0.0469 .735E-2 -1.0815 .21 1 E- 1 -1.45G9 .261E-1 0.4999 .682E-4 -0.0554 . 1 12E-2 -1.1062 .352E-2 -1.4830 .448E-2 0.5000 .945E-5 -0.0530 .130E-2 -1.0877 .149E-1 -1.4575 .255E-1 Table 6.2.2 The numerical solutions vs. Exact solutions for the flow problem C O D i screte Least Squares Cont i nuous Least Squares Ga1erk i n M i n i m i zat i on RMSE (from boundary condition) RMSE (compared with exact soln) | | F | | ff function evaluations .446E- 1 . 157E-1 .267E-1 4 1 . 136E-1 . 263E-2 .1G0E-1 42 . 145E- 1 .125E-1 .739E+0 51 Table 6.2.3 Accuracy and e f f i c i e n c y for the d i f f e r e n t methods for the flow problem INITIAL SINGULARITIES I n i t i a l Coeff. from Least Squares I n i t i a l Coeff. from Ga1erk i n FINAL SINGULARITIES Discrete Least Squares Continuous Least Squares Galerkin Minimization X y X y coeff . X y coeff. X y coeff . 3 . 50 1 . 50 0.0244 0.0240 6.7194 1.5394 0.0389 6 .6528 1.5350 0.0389 6.0902 1.2337 0.0257 3 . 50 3 . 50 0.0087 0.0135 5.7385 4 . 4923 0.0309 5.6352 4 .4311 0.0295 6.0946 4.2459 0.0434 1 . 50 3 . 50 -0.0098 -0.0148 0.4672 6.7853 -0.0198 0.4935 6.6436 -0.0186 0.6781 6.3838 -0.0130 - 1 . 50 3 . 50 -0.0245 -0.0306 -2.4609 6.8200 -0.0429 -2.3860 6.7359 -0.04 15 -2.5108 7.1916 -0.0632 -3 . 50 3 . 50 -0.1052 -0.1005 -6.9170 6.0324 -0.2206 -6.8473 6.0299 -0.2202 -6.8889 5.6536 -0.2143 -3 . 50 1 . 50 -0.0720 -0.0720 -7.4194 1.6608 -0.1006 -7.3751 1.6615 -0.1010 -7.0232 1 .5129 -0.0861 Table 6.3.1 Locations of the i n i t i a l and f i n a l s i n g u l a r i t i e s and their associated c o e f f i c i e n t s for the heat conducting problem D i screte Least Squares Continuous Least Squares Ga1erk i n M i n i m i zat i on Boundary Nodes Exact Numerical Absolute Numer i cal Absolute Numer i ca1 Absolute X y Solut i on Solut ion Error Solut i on Error Solut ion Error 3.0 0.00 0.0000 ' ' 0.0026 . 258E-2 0.0028 . 284E-2 0.0034 .341E-2 3.0 1 . 50 0.0000 -0.0014 . 138E-2 -0.0016 .159E-2 -0.0006 .629E-3 3.0 3.00 0.0000 -0.0043 .428E-2 -0.0051 .51 1E-2 -0.0061 .608E-2 1 . 8 3 .00 60.0000 59.9987 . 128E-2 59.9986 . 137E-2 59.9972 . 280E-2 0.2 3 .00 140.0000 139.9997 . • .349E-3 140.0001 .632E-4 139.9980 .202E-2 -1.8 3.00 240.0000 239.9992 .764E-3 240.0002 . 17 1E-3 239.9992 .757E-3 -3.0 3.00 300.0000 300.0018 .182E-2 300.0032 .324E-2 300.0035 .346E-2 -3.0 1 . 50 300.0000 300.0002 .233E-3 300.0011 .110E-2 300.0019 .187E-2 -3.0 0.00 300.0000 299.9962 .385E-2 299.9965 .345E-2 299.9963 .367E-2 Table 6.3.2 The numerical solutions vs. Exact solutions for the heat conducting problem D i screte Least Squares Cont i nuous Least Squares Ga1erk i n M i n i m i zat i on RMSE (from boundary condition) RMSE (compared with exact soln) | | F 1 | ft function evaluations . 796E-5 . 199E-2 .251E-G 64 .413E-5 .200E-2 .810E-4 63 .120E-4 .262E-2 .981E-4 66 Table 6.3.3 Accuracy and e f f i c i e n c y Tor the di f f e r e n t methods for the Heat conducting problem 57 CHAPTER 7 CONCLUSIONS In t h e two p r e c e d i n g c h a p t e r s , we have seen how t h e G a l e r k i n method and t h e c o n t i n u o u s l e a s t s q u a r e s method p e r f o r m on examples r e p r e s e n t i n g t h e t h r e e t y p e s of b o u n d a r y p r o b l e m s . From t h e r e s u l t s , we can make s e v e r a l o b s e r v a t i o n s . Our f i r s t a p p r o a c h i n v o l v e d u s i n g t h e t h r e e a l g o r i t h m s f o r n o n l i n e a r e q u a t i o n s p r e s e n t e d i n c h a p t e r 4 and s e c t i o n 5.1 t o s o l v e t h e e q u a t i o n s d e r i v e d from t h e G a l e r k i n t e c h n i q u e . But t h e s e d i d not y i e l d b e t t e r r e s u l t s t h a n t h e e x i s t i n g d i s c r e t e l e a s t s q u a r e s m i n i m i z a t i o n method. Thus, we i m p l e m e n t e d a h y b r i d t e c h n i q u e , where t h e o b j e c t i v e f u n c t i o n i s t h e one f o r t h e l e a s t s q u a r e s method, w h i l e t h e g r a d i e n t i s due t o t h e G a l e r k i n method. T h i s method was f o u n d t o p e r f o r m b e t t e r t h a n our f i r s t a p p r o a c h and t h e r e s u l t s a r e c o m p a r a b l e t o t h o s e from t h e d i s c r e t e l e a s t s q u a r e s m i n i m i z a t i o n . However, c o m p a r i n g t h e r e s u l t s f r o m t h e G a l e r k i n method w i t h t h o s e from t h e c o n t i n u o u s l e a s t s q u a r e s method, i t was f o u n d t h a t t h e l a t t e r g i v e s even b e t t e r r e s u l t s . The r e s u l t s of t h i s i n v e s t i g a t i o n o f t h e G a l e r k i n and t h e l e a s t s q u a r e s methods l e a d us t o c o n c l u d e t h a t t h e c o n t i n u o u s l e a s t s q u a r e s method i s one of t h e b e t t e r c h o i c e s , among t h e methods c o n s i d e r e d , f o r s o l v i n g b o u n d a r y v a l u e p r o b l e m s . 58 REFERENCES 1. Han, P.S., O l s o n , M.D. and J o h n s t o n R.L., "A G a l e r k i n B oundary E l e m e n t F o r m u l a t i o n w i t h Moving S i n g u l a r i t i e s " , t o a p p e a r i n Engineering Computations. 2. B r e b b i a , C.A. and W a l k e r , S., " I n t r o d u t i o n t o Boundary E l e m e n t Methods", Recent Advances in Boundary Element Methods, B r e b b i a , C.A ( E d ) , pp.1-43, P e n t e c h P r e s s , London, 1 978. 3. Jaswon, M.A. and Symm, G.T., Integral Equation Methods in Potential Theory and El asl osl al i cs , A c a d e m i c P r e s s , London, 1 977. 4. F a i r w e a t h e r , G. and J o h n s t o n , R.L., "Method of F u n d a m e n t a l S o l u t i o n s " , Proceedings of the LMS Durhum Symposium on the Numerical Solution of Integral Equations, 1982. 5. F a i r w e a t h e r , G., R i z z o , F . J . and S h i p p y , D . 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M a r q u a r d t , D.W., "An A l g o r i t h m f o r L e a s t - S q u a r e s E s t i m a t i o n of N o n l i n e a r P a r a m e t e r s " , /. Soc. Indust. Appl. Math. , 11, 1 963. 19. K o w a l i k , J . and O s b o r n e , M., Methods for Unconstrained Opt i mi z at i o n Pr obi ems, Amer. E l s e v i e r , N.Y., 1968. 20. Hopper, M . J . ( e d ) , H a r w e l l S u b r o u t i n e L i b r a r y C a t a l o g u e , T h e o r e t i c a l P h y s i c a l D i v i s i o n , A.E.R.E., H a r w e l l , England., 1973. 21. B r e b b i a , C.A., The Boundary Element Method for Engineers, P e n t e c h P r e s s , London, 1978. 

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