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7/30/2019 Taylor Expansion
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8 T h i r d o r d e r e x p a n s i o n s
8 . 1 G e n e r a l
A s s o m e t h i n g n e w w e w i l l i n t r o d u c e t h i r d o r d e r t e r m s i n t h e b o u n d a r y e l e -
m e n t m o d e l . S o f r o m h e r e l e t t h e p e r t u r b a t i o n e x p a n s i o n s o f a n d b e o f
t h e f o r m
= "
1
+ "
2
2
+ "
3
3
+ O "
4
8 . 1
= "
1
+ "
2
2
+ "
3
3
+ O "
4
8 . 2
T h e s e r e d e n i t i o n s o f t h e e x p a n s i o n s o f a n d d o n o t c o n t r a d i c t t h e f o r m e r
d e n i t i o n s 2 . 4 4 a n d 2 . 4 4 . T h u s t h e r e s u l t s o b t a i n e d i n t h e p r e v i o u s p a r t s
o f t h i s w o r k a r e s t i l l v a l i d b u t t h e y m a y b e i m p r o v e d b y t h e i n t r o d u c t i o n o f
t h i r d o r d e r t e r m s .
I t i s e a s i l y s h o w n t h a t t h e t h i r d o r d e r p o t e n t i a l
3
m u s t s a t i s f y t h e L a p l a c e -
e q u a t i o n a n d t h u s t h e b o u n d a r y i n t e g r a l e q u a t i o n 2 . 4 9 i s v a l i d f o r t h e t h i r d
o r d e r p o t e n t i a l . A n a l y t i c a l l y t h i s i s w r i t t e n a s
r
2
3
= 0 8 . 3
+
Z
,
0
3
@ G
@ n
0
, G
@
3
@ n
0
d s =
3
x 8 . 4
I n d i s c r e t e f o r m t h e b o u n d a r y i n t e g r a l e q u a t i o n s o b t a i n e d f o r
3
w i l l h a v e
t h e s a m e c o e c i e n t s a s t h e e q u a t i o n s f o r
1
a n d
2
a s d e s c r i b e d i n s e c t i o n
4 . 5 . T h i s m e a n s t h a t t h e c a l c u l a t i o n t i m e p e r t i m e s t e p w i l l b e i n c r e a s e d b y
r o u g h l y 5 0 b y i n t r o d u c i n g t h i r d o r d e r t e r m s . F o r v e r y w e a k l y n o n l i n e a r
w a v e s i t m a y n o t b e w o r t h t h e e x t r a c a l c u l a t i o n t i m e t o o b t a i n t h e t h i r d
o r d e r p o t e n t i a l b u t f o r w a v e s o f i n t e r m e d i a t e n o n l i n e a r i t y t h e t h i r d o r d e r
t e r m s m a y m a k e t h i s m o d e l f e a s i b l e a n d t h i s m o d e l w i l l s t i l l b e a l o t f a s t e r
t h a n a f u l l n o n l i n e a r B E M .
8 . 2 F r e e s u r f a c e a n d o p e n b o u n d a r y c o n d i t i o n s
8 . 2 . 1 G e n e r a l
T h e m o s t d e m a n d i n g t a s k b y i n t r o d u c i n g t h i r d o r d e r t e r m s i n t h e B E M i s
t h e e v a l u a t i o n o f n e w b o u n d a r y c o n d i t i o n s b a s e d o n t h e d y n a m i c 3 . 3 1 a n d
k i n e m a t i c 3 . 3 2 f r e e s u r f a c e b o u n d a r y c o n d i t i o n s m o d i e d b y d i s s i p a t i v e
t e r m s f o r a s p o n g e l a y e r f o r i n s t a n c e g i v e n b y 3 . 4 2 .
6 8
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T a y l o r e x p a n d i n g t h e k i n e m a t i c b o u n d a r y c o n d i t i o n f r o m z = 0 w e g e t
@
@ t
,
@
@ z
+
@
@ x
@
@ x
+ x
+
@
@ z
@
@ t
,
@
@ z
+
@
@ x
@
@ x
+ x
+
2
1
2
@
2
@ z
2
@
@ t
,
@
@ z
+
@
@ x
@
@ x
+ x
+
3
1
6
@
3
@ z
3
@
@ t
,
@
@ z
+
@
@ x
@
@ x
+ x
= O
4
a t z = 0 8 . 5
+
@
@ t
,
@
@ z
+
@
@ x
@
@ x
+ x
,
@
2
@ z
2
+
@
2
@ x @ z
@
@ x
,
1
2
2
@
3
@ z
3
+
1
2
2
@
3
@ x @ z
2
@
@ x
+
3
1
6
@
3
@ z
3
@
@ z
,
@
@ t
,
@
@ x
@
@ x
= O
4
a t z = 0 8 . 6
S i m i l a r l y f r o m t h e d y n a m i c b o u n d a r y c o n d i t i o n w e g e t
@
@ t
+ g +
1
2
r
2
+ x
+
@
@ z
@
@ t
+ g +
1
2
r
2
+ x
+
2
1
2
@
2
@ z
2
@
@ t
+ g +
1
2
r
2
+ x
+
3
1
6
@
3
@ z
3
@
@ t
+ g +
1
2
r
2
+ x
= O
4
a t z = 0 8 . 7
+
@
@ t
+ g +
1
2
r
2
+ x
+
@
2
@ t @ z
+
1
2
@
@ z
,
r
2
+ x
@
@ z
+
1
2
2
@
3
@ t @ z
2
+
1
4
2
@
2
@ z
2
,
r
2
+
1
2
x
2
@
2
@ z
2
+
3
1
6
@
3
@ z
3
@
@ t
+ g +
1
2
r
2
+ x
= O
4
a t z = 0 8 . 8
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P e r t u r b a t i o n e x p a n d i n g a n d i n 8 . 6 a n d k e e p i n g o n l y t e r m s o f t h i r d
o r d e r i n " w e g e t
@
3
@ t
,
@
3
@ z
+ x
3
= ,
@
1
@ x
@
2
@ x
,
@
2
@ x
@
1
@ x
+
1
@
2
2
@ z
2
+
2
@
2
1
@ z
2
,
1
@
2
1
@ x @ z
@
1
@ x
+
1
2
2
1
@
3
1
@ z
3
8 . 9
+
@
3
@ t
,
@
3
@ n
0
+ x
3
= ,
@
1
@ x
@
2
@ x
,
@
2
@ x
@
1
@ x
,
1
@
2
2
@ x
2
,
2
@
2
1
@ x
2
,
1
@
1
@ x
@
@ x
@
1
@ n
0
,
1
2
2
1
@
2
@ x
2
@
1
@ n
0
8 . 1 0
S i m i l a r l y p e r t u r b a t i o n e x p a n d i n g 8 . 8 a n d k e e p i n g o n l y t e r m s o f t h i r d o r d e r
w e g e t
@
3
@ t
+ g
3
+ x
3
= ,
@
1
@ x
@
2
@ x
,
@
1
@ z
@
2
@ z
,
1
@
2
2
@ t @ z
,
2
@
2
1
@ t @ z
,
1
2
1
@
@ z
,
r
1
2
, x
1
@
2
@ z
, x
2
@
1
@ z
,
1
2
2
1
@
3
1
@ t @ z
2
,
1
2
x
2
1
@
2
1
@ z
2
8 . 1 1
+
@
3
@ t
+ g
3
+ x
3
= ,
@
1
@ x
@
2
@ x
,
@
1
@ n
0
@
2
@ n
0
,
1
@
@ t
@
2
@ n
0
,
2
@
@ t
@
1
@ n
0
,
1
@
1
@ x
@
@ x
@
1
@ n
0
+
1
@
1
@ n
0
@
2
1
@ x
2
, x
1
@
2
@ n
0
, x
2
@
1
@ n
0
+
1
2
2
1
@
@ t
@
2
1
@ x
2
+
1
2
x
2
1
@
2
1
@ x
2
8 . 1 2
T h e e q u a t i o n s 8 . 9 a n d 8 . 1 1 s h o w t h e t e r m s a s t h e y w i l l l o o k d i r e c t l y a f t e r
t h e e x p a n s i o n w h i l e t h e e q u a t i o n s 8 . 1 0 a n d 8 . 1 2 a r e r e w r i t t e n t o f o r m s
r e a d y f o r i m p l e m e n t a t i o n i n a c o m p u t e r p r o g r a m .
8 . 2 . 2 U p d a t i n g - A p p l y i n g a n i m p l i c i t A d a m s m e t h o d
A s w e h a v e s e e n i n t h e p r e v i o u s s e c t i o n 8 . 2 . 1 t h e t h i r d o r d e r e x p r e s s i o n s
f o r t h e d y n a m i c a n d k i n e m a t i c f r e e s u r f a c e b o u n d a r y c o n d i t i o n s a r e q u i t e
7 0
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c o m p l i c a t e d a n d t h u s t o a v o i d c o m p l i c a t i n g t h i n g s e v e n m o r e w e w i l l n o t
T a y l o r e x p a n d t h e s e e q u a t i o n s i n t i m e . I n s t e a d w e w i l l u s e a n e x p l i c i t A d a m s -
B a s h f o r t h m e t h o d f o r t h e u p d a t i n g o f
k
a n d t h e n u s e t h e k i n e m a t i c b o u n d -
a r y c o n d i t i o n f o r n d i n g n e w v a l u e s o f
@
k
@ t
a s d e s c r i b e d i n s e c t i o n 4 . 3 . 3 .
W e h a v e c h o s e n a n e x p l i c i t A d a m s - B a s h f o r t h m e t h o d o f h i g h e r o r d e r t h a n
b e f o r e k
a
= 3 a c c o r d i n g t o 4 . 2 5 w i t h t h e c o e c i e n t s A
0
= 5 5 = 2 4
A
1
= , 5 9 = 2 4 A
2
= 3 7 = 2 4 a n d A
3
= , 9 = 2 4 . S o s t i l l l e t t i n g i n d i c e s m ; j "
r e f e r t o t i m e t = t
m
a n d s p a c e x = x
j
t h e n e w u p d a t i n g o f w i l l b e
k m + 1 j
=
k m j
+ t
5 5
2 4
@
k
@ t
m j
,
5 9
2 4
@
k
@ t
m , 1 j
+
3 7
2 4
@
k
@ t
m , 2 j
,
9
2 4
@
k
@ t
m , 3 j
; k 2 f 1 ; 2 ; 3 g 8 . 1 3
A f t e r s o m e c o n s t r u c t i v e c o n v e r s a t i o n s w i t h J e s p e r S k o u r u p w e c h o s e t o
u p d a t e o n t h e f r e e s u r f a c e b y a n i m p l i c i t A d a m s m e t h o d . T h i s p r o c e -
d u r e i s o f t e n a t t r i b u t e d t o A d a m s - M o u l t o n a n d t h i s w i l l a l s o b e d o n e h e r e
e v e n t h o u g h t h e f o r m u l a i s d u e t o A d a m s . S i m i l a r l y t o t h e e x p l i c i t A d a m s -
B a s h f o r t h m e t h o d s a n i m p l i c i t A d a m s - M o u l t o n m e t h o d i s o f t h e f o r m
y
m + 1
= y
m
+ t B
0
f
m + 1
+ + B
k
a
f
m + 1 , k
a
8 . 1 4
C h o o s i n g k
a
= 3 w e h a v e t h e c o e c i e n t s B
0
= 9 = 2 4 B
1
= 1 9 = 2 4 B
2
=
, 5 = 2 4 a n d B
3
= 1 = 2 4 . T h e m e t h o d i s o f t e n u s e d a s t h e c o r r e c t o r s t e p i n a
p r e d i c t o r - c o r r e c t o r m e t h o d b u t w e u s e t h e m e t h o d f o r p r e d i c t i o n o f
k
o n l y .
T h e d y n a m i c f r e e s u r f a c e b o u n d a r y c o n d i t i o n w i t h F
k
a s r i g h t h a n d s i d e
c a n f o r k 2 f 1 ; 2 ; 3 g b e w r i t t e n a s
@
k
@ t
m + 1 j
+ g
k m + 1 j
+ x
k m + 1 j
= F
k m + 1 j
8 . 1 5
I n s e r t i n g
@
k
@ t
m + 1 j
i n t h e e q u a t i o n f o r t h e i m p l i c i t A d a m s - M o u l t o n m e t h o d
w e g e t
k m + 1 j
=
1
1 + t B
0
x
"
k m j
+ t
B
0
,
, g
k m + 1 j
+ F
k m + 1 j
+ B
1
@
k
@ t
m j
+ B
2
@
k
@ t
m , 1 j
+ B
3
@
k
@ t
m , 2 j
8 . 1 6
7 1
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W h e n
k m + 1 j
a n d
k m + 1 j
a r e k n o w n t h e d y n a m i c s u r f a c e b o u n d a r y
c o n d i t i o n s w i l l p r o v i d e
@
k
@ t
m + 1 j
t o b e u s e d i n t h e f o l l o w i n g t i m e s t e p s .
A f t e r h a v i n g s o l v e d t h e b o u n d a r y i n t e g r a l e q u a t i o n s t h e k i n e m a t i c f r e e s u r -
f a c e b o u n d a r y c o n d i t i o n s s u p p l y
@
k
@ t
m + 1 j
a t ,
f 0
T h e m e t h o d d e s c r i b e d a b o v e h a s b e e n i m p l e m e n t e d a n d i s h e r e a f t e r u s e d
i n t h e t h i r d o r d e r m o d e l a l s o f o r t h e r s t a n d s e c o n d o r d e r t e r m s . T h e
m e t h o d s e e m s s t a b l e a n d r e l i a b l e w h e n t h e C o u r a n t n u m b e r i s b e l o w u n i t y
s a y C r = 0 8 . S i n c e t h e m o d e l m a y t u r n o u t t o b e u n s t a b l e w h e n t h e C o u r a n t
n u m b e r i s g r e a t e r t h a n u n i t y c a r e s h o u l d b e t a k e n t o e n s u r e t h a t t h i s d o e s
n o t h a p p e n .
I t s h o u l d b e n o t e d t h a t t h e A d a m s - B a s h f o r t h - M o u l t o n m e t h o d s a r e n o r -
m a l l y u s e d i n p r e d i c t o r - c o r r e c t o r m e t h o d s a n d s i n c e o u r m e t h o d r e p r e s e n t s
t h e p r e d i c t o r s t e p o n l y s t a b i l i t y c a n n o t b e g u a r a n t e e d . T o i m p r o v e s t a b i l i t y
a c o r r e c t o r s t e p c a n b e i n c l u d e d i n t h e m o d e l b u t s i n c e t h i s r o u g h l y w o u l d
d o u b l e t h e c a l c u l a t i o n t i m e a n d n o t i m p r o v e t h e a c c u r a c y b y m u c h w e h a v e
n o t i m p l e m e n t e d a c o r r e c t o r s t e p .
8 . 3 P i s t o n b o u n d a r y c o n d i t i o n
A p i s t o n t y p e w a v e - m a k e r s e e s e c t i o n s 3 . 3 a n d 3 . 4 c a n i n m a t h e m a t i c a l
t e r m s b e d e s c r i b e d b y a b o u n d a r y c o n d i t i o n o n t h e f o r m 3 . 1 3 a n d
@
@ z
= 0
i . e .
@
@ n
0
= ,
@
@ x
T a y l o r e x p a n d i n g 3 . 1 3 f r o m x = 0 w e g e t
,
@
@ x
+
@
@ x
,
@
@ x
+
1
2
2
@
2
@ x
2
,
@
@ x
+
1
6
3
@
3
@ x
3
,
@
@ x
+ O
4
= V
n
; a t ,
r 1 0
8 . 1 7
P e r t u r b a t i o n e x p a n d i n g a n d a n d k e e p i n g o n l y t e r m s o f t h i r d o r d e r w e
g e t
,
@
3
@ x
,
1
@
2
2
@ x
2
,
2
@
2
1
@ x
2
,
1
2
2
1
@
3
1
@ x
3
= V
n 3
8 . 1 8
7 2
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U s i n g t h e L a p l a c e - e q u a t i o n
@
1
@ n
0
= V
n 1
a n d
@
@ z
= 0 w e g e t
@
3
1
@ x
3
=
@
@ x
,
@
2
1
@ z
2
= ,
@
2
@ z
2
@
1
@ x
=
@
2
@ z
2
@
1
@ n
0
=
@
2
@ z
2
V
n 1
=
@
2
@ z
2
,
@
1
@ t
= ,
@
2
@ z @ t
@
1
@ z
= 0
; a t ,
r 1 0
8 . 1 9
T h u s 8 . 1 8 c a n b e r e d u c e d t o
@
3
@ n
0
= V
n 3
+
1
@
2
2
@ x
2
+
2
@
2
1
@ x
2
= V
n 3
,
1
@
2
2
@ z
2
,
2
@
2
1
@ z
2
; a t ,
r 1 0
8 . 2 0
7 3