The Calculation Methods for Solving a Kind of Boundary Value Problems in Elasticity

) 7 1 0 2 A T S M M ( s n o it a c il p p A d n a s e i g o l o n h c e T n o it a l u m i S d n a g n il l e d o M , s c it a m e h t a M n o e c n e r e f n o C l a n o it a n r e t n I 7 1 0 2 8 7 9 : N B S

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, y ti s r e v i n U g n o t o a i J g n i q g n o h C , s c it s it a t S d n a s c it a m e h t a M f o l o o h c S a n i h C . R P , 4 7 0 0 0 4 , g n i q g n o h C : s d r o w y e

K Nonilnear boundary value condiiton, Elasitctiy, extrapolaiton algortihm, Mechanica l e r u t a r d a u

q methods, Boundaryintegra lequaitons.

t c a r t s b

A . Thispaperwil lstudyt henumerica lsolutionsforequationswithakindofboundaryvalue n

o

c ditions .The equaitons wil lbe converted into nonlinear boundary integra lequaitons by the , y r o e h t l a i t n e t o

p in which logarithmic singularity and Cauchy singularity are calculated e r u t a r d a u q l a c i n a h c e M . y l s u o e n a t l u m i

s methods(MQMs) are presented to solve the nonlinear r h t e r a s n o i t u l o s e h t f o y c a r u c c a e h t t a h t s n o it a u q

e e eorder .Accordingtotheasymptotica lcompac t c i t o t p m y s a s r e w o p d d o n a h t i w s r o r r e e h t , y r o e h t e c n e g r e v n o

c expansion isobtained .Someresults

l a c i r e m u n e h t y b s m e l b o r p r o f s n o i t a m i x o r p p a e s e h t g n i d r a g e r n w o h s e r

a example.

n o it c u d o r t n I e d l l i w r e p a p s i h

T scribet hei solatedelasitcequationsonaboundedplanarregions

Ω

int heplane : s n o it i d n o c e u l a v y r a d n u o b r a e n il n o n h t i w

(

) ( )

{

, 0,

, 2 , 1 , , , , j

ji in

j i n o x f x g p σ η Ω = = Γ + − = ) 1 ( e r e h

W _Ω⊂R2_{is a connected domain with a smooth closed curve Γ ,the stress tensors are}

2 1, )

( ,

ji n n n

σ = is the uni toutward norma lvector on Γ ,the tractor vector is assumed given

2 1, )

(_{p p} T

p= w ith pi =(σi1n1+σi2n2) , f(x)=(f1(x), f2(x))T and continuous on Γ , and

2 1( , ), ( , ))

( ) ,

(_{x g x g x} T

g η = η η tha t

g

_i

(

x

,

η

)

is a nonlinear function corresponding to η . r o t c e v g n i w o l l o

F computationa lrules,t herepeatedsubscriptsi mplyt hesummationfrom1t o2. e c n a t s n i e h t d n a s m e l b o r p g n i r r a l u c r i c e h t . g . e , s n o it a c i l p p a y n a m n i d e i d u t s e r a m e l b o r p e h T o r p n e e b e v a h s d o h t e m e m o S . ] 2 2 [ s r a b c i t a m s i r p f o g n i d n e b e h t d n a , ] 8 1 ; 4 1 [ s m e l b o r

p posed for

d e x i m g n i m r o f n o c n o n e h T . y t i c i t s a l e e h t g n i v l o

s finiteelemen tmethodsareestablishedbyHuand t s a e l e h T . y r a d n u o b r a e n i l a h t i w y t i c it s a l e e h t e v l o s o t ] 2 1 [ i h

S -squaremethodsarei ntroducedbyCa i

r a l u g n i s r e p y h e h t d e v l o s ] 5 [ g n o H d n a n e h C . s m e l b o r p c i t s a l e e h t f o n o i t u l o s e h t g n i n i a t b o r o f ] 4 [ .l a t e

integra lequationsapplyingt hedua lboundaryelemen tmethodsi nelasitctiy .Kuoe ta.l[15]solvet rue ] 6 1 [ e i N d n a i L . s d o h t e m l a u d d e s u s m e l b o r p y t i v a c r a l u c r i c e h t f o s n o i t u l o s n e g i e s u o i r u p s d n a i x a d e s s e r t s e h t d e h c r a e s e

r -symmetricrodsproblemsbyahigh-orderi ntegrationfactormethod .And v o l k e t S e v l o s o t ] 6 [ .l a t e g n e h C y b d e t p o d a e r a ) s M Q M ( s d o h t e m e r u t a r d a u q l a c i n a h c e m e h t e h t o t n i d e t r e v n o c e r a ) 1 ( . s q E e h T . s n o i t u l o s y c a r u c c a h g i h n i a t b o o t y ti c it s a l e n i s n o i t u l o s n e g i e ; 3 [ s n o i t a u q e l a r g e t n i y r a d n u o

b 6 ;21] (BIEs)wtih theCauchy and loga-rtihmicsingularitiesby the . d o h t e m l a n o i t a i r a v * * 2

1

,

)

,

(

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)

(

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,

(

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(

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,

(

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(

2

ji x j ji x j ji

j

y

k

y

x

x

d

s

h

y

x

p

x

d

s

y

y

y

σ

η

η

Γ

Γ

=

=

∈

Γ

+

∫

∫

* . . * . . . .

1 _{[ (3 4 ) ln ,]}

) 1 ( 8 1 .] ) ( ) 2 1 ( ) 2 ) 2 1 ( ( [ ) 1 ( 4 j i ji ji i j j i j i ji ji r r r v h v

r _{v r r v n r n r}

k n r v σ µ π σ π + − − = − ∂ − − + + − = ∂ −







s i o i t a r n o s s i o P e h T . ] 3 [ s n o i t u l o s l a t n e m a d n u f s ' n i v l e K e r a s l e n r e k e h

T v with v=λ/[2(λ+µ)] ,

.j

r

meansthederivativeto x_j ,and 2 2

2 2 1

1 ) ( )

(y x y x

r= − + − isthedistanceof x and

y

. Theparts *_{( , ) ( )}

x j

ji y x x ds

k η

Γ

∫

fo theEqs.(2)aretheCauchysingularityandtheparts *_{( , ) ( )}

x j

ji y x x ds

h η

Γ

∫

are

. y t i r a l u g n i s c i m h t i r a g o l e h t e l a r g e t n i r a e n i l n o n e h

T quationsare obtained after theboundary conditions aresubstituted into : ) 2 ( . s q E * * *

(

,

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(

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(

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(

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2

ji x j ji x j ji x j ji

j

y

k

y

x

x

d

s

h

y

x

g

x

d

s

h

y

x

f

x

d

s

σ

η

η

η

Γ Γ

Γ

=

+

+

∫

∫

∫

) 3 ( d n a n e n i a l a s t o u R . r e p a p y n a m n i d e s s u c s i d n e e b e v a h s n o i t a u q e l a r g e t n i r a e n i l n o n f o d n i k s i h T d n a l d n e

W [20 ;21] approximated the solution of nonlinear integra lequations by spline Galerkin's t e l e v a W e h T . s d o h t e

m -Galerkin'smethodsconstructedfromLegendrewavele tfunctionsareusedby h C d n a n o s n i k t A . n o i t u l o s e h t e t a m i x o r p p a o t ] 7 1 [ n a r a g s e M d n a d a j e n k e l a

M andler[2]presented

o w t d n a s d o h t e m e l u r s ' n o s p m i S t c u d o r

p -grid methods to solve the nonilnear equations . e t e r c s i d r a e n i l n o n e h t o t s n o i t u l o s f o e c n e t s i x e e h t r o f h c r a e s e r l a c i t e r o e h t e h t e v a g ] 9 1 [ z e u g i r d o R u s h ti w s m e l b o r p e u l a v y r a d n u o

b fficien tcondtiions .Abels[1] discussed the convergence in the n i l n o n f o s m e t s y

s earboundarycondiitonswithHölderspace.

, y t i r a l u g n i s y h c u a C e h t d n a y t i r a l u g n i s c i m h t i r a g o l e h t g n i d u l c n i m e t s y s r a e n i l n o n a s i s i h t e c n i S p a s n o i t a u q e e t e r c s i d e h t n i a t b o o t s i y t l u c i f f i d e h

t propriately. Thedisplacemen tvector and stress n i r o s n e

t

Ω

canbecalculated[8 ;11].

g l a l e l l a r a p e v i t c e f f e y t t e r p e r a ) s A E ( s m h t i r o g l a n o i t a l o p a r t x e n o s d r a h c i

R orithmwhicharebased

p m y s a n

o toticexpansionabou terrors .Thesolutions ,whicharesolvedoncoarsegrid andfinegrid, t c u r t s n o c o t d e s u e r

a highaccuracysolutions .Its 'notifiedasgoodstabilityandopitma lcomputationa l ] 7 [ .l a t e g n e h C . y t i x e l p m o

c harnessed extrapolation algorithms to obtain high accuracy order for t

Seklov eigenvalue of Laplace equaitons. Huang and Lü established extrapolaiton algorithms to l o s y c a r u c c a h g i h n i a t b

o utions for solving Laplace equationson arcs[13] and Steklov eigenvalue . ] 0 1 [ y t i c i t s a l e n i m e l b o r p l a c i n a h c e

M QuadratureMethods

y r a d n u o b n o s r o t a r e p o l a r g e t n i e h t f o n o i t a t o n e h t e n i f e D y l t s r i f e

W Γ asfollows to simplify the

: s n o it a u q e

{

* * , 2 , 1 , , ) ( ) , ( ) ( ) ( . 2 , 1 , , ) ( ) , ( ) ( ) ( x ji ji x ji ji j i y s d x x y k y K j i y s d x x y h y H η η η η Γ Γ = Γ ∈ = = Γ ∈ =

∫

∫

) 4 ( il p m i s e b n a c ) 2 ( . s q E o

S fiedast hefollowingoperatorequations:

1

2 1 1 1

0 1 11 12 1 1

2 1 2 2 1 2

2 2 2

2 2 0 1

2 2

)

(

)

(

)

(

,)

(

I _K+K _IK_{+K η}η

=

_HH _HH g_g +₊f_f

) 5 ( e r e h

e s o p p u

S _C2m_{[0,2π] bet hese tof2mtimesdifferenitableperiodicf unctionsi nwhicht heperiodic}

l l a f

o functionsare 2π . Aregularparametermapping: [0,2π]→Γ isi ntroduced ,sot hesmoothclosed e

v r u

c (x₁(s),x₂(s)) satisfying x('s)2 = x'₁(s)2+ x'₂(s)2 >0 with _{( )} 2m_{[0,2 ], 1,2.}

i s C i

x ∈ π −

f o y t i r a l u g n i s e h

T thekernels *_, *

ji ji k

h wil lbeanalyzedas t→s beforet heapplicationofdiscrete .

s d o h t e m

n o i s s e r p x e e h t e c n i

S .

.

.

(

1

)

1

(

)

,

,

)

(

)

(

i j j

i

r

n

r

i

n

o

t

s

i

j

s

t

o

s

t

r

−

+

−

≠

−

=

−

+

−

_{( 6)}

f o y t i r a l u g n i s y h c u a C e h t o

s *

ji

k comes from the par t (n_ir_.j−n_jr_.i)/r. Moreover ,the logarithmic f

o y t i r a l u g n i

s *

ji

h comes from thecomponen t logx−y and the other parts of the operators are s

r o t a r e p o e h t o S . h t o o m

s H_i,j and K_i,j wil lbedividedi ntosevera lparts. r

o t a r e p o r a l u g n i s c i m h t i r a g o l e h

T H_i,j wil lbedividedi ntot hreepartson _C2m_{[0,2π] asfollows:}

2

0

)

(

)

₀

(

,

)

(

)

('

)

,

(

A

η

t

=

∫

π

a

_o

t

τ

η

τ

x

τ

d

τ

h

t i

w 1/2

0

0ln 2 sin 2 , (3 4 )/[8 (1 )] )

,

( t

o t c e c v v

a _{τ =} − −τ _{=− − πµ − , da n}

2

0

)

(

)

₀

(

,

)

(

)

('

)

,

(

B

η

t

=

∫

π

b

_o

t

τ

η

τ

x

τ

d

τ

h

t i

w 1/2

0ln ( ) () ln 2 sin 2 )

,

( t

o t c x t x t e

b _{τ = − −} − −τ _,and

2

0

(

,

)

(

)

('

)

,

)

(

)

(

B

_ji

η

t

=

∫

π

b

_ji

t

τ

η

τ

x

τ

d

τ

h

t i

w b_ji(t,τ)=c₁r_.ir_.j,c₁=1/[8πµ(1−v)]. r

a l u g n i s y h c u a C e h

T operator K_i,j wil lalsobedividedi ntot hreepartson 2

] 2 , 0 [

m

C π asfollows:

2

0

)

(

)

₀

(

,

)

(

)

('

)

,

(

C

η

t

=

∫

π

c

_o

t

τ

η

τ

x

τ

d

τ

h

t i

w c_o(t,τ)=c₂(n₁r_.2−n₂r_.1)/r,c₂=−(1−2v)/[4π(1−v)] ,and

2

0

(

,

)

(

)

('

)

,

1

,

2

,

)

(

)

(

M

_ii

η

t

=

∫

π

m

_ii

t

τ

η

τ

x

τ

d

τ

i

=

h

t i

w (, ) ₃ r[(1 2 ) 2_{. .} ]/ , ₃ 1/[4 (1 )]

i i

ii t c n v rr r c v

m τ ∂ π

∂ − + =− −

= ,and

2

0

(

,

)

(

)

('

)

,

1

,

2

,

,

)

(

)

(

M

_ji

η

t

=

∫

π

m

_ji

t

τ

η

τ

x

τ

d

τ

i

=

i

≠

j

h t i

w (, ) ₃ r(2_{. .} )/

j i

ji t c n rr r

m τ ∂

∂

= .

t a h t d n i f n a c e

W B₀,B_i,j,M_i,j aresmoothoperators, A₀ isal ogarithmicsingularoperator ,and

0

C i saCauchysingularoperator.

: s a n w o h s e b l li w ) 4 ( . s q E f o n o i t a u q e t n e l a v i u q e e h t n e h T

1

,

)

(

)

(

)

(

e r e h

w _η(_t)₌(_η1(_x(_t)),_η2(_x(_t))), _f(_t)₌(_f1, _f2)T _{w tih}

,j j( ( )), ( ()) ( ( ), ( )) i

i H f x t g t g xt t

f = η = η

d n a

0 0

0

0 0

0 0

0

0 0

0

)

,

(

)

,

(

)

,

(

O

A C

I

A C

I

C

A

I

=

=

−

=

1 1

0 12 11 12

2 2 1 2 2

2 0 1

2

)

,

(

)

.

(

B_BB _BB_B

M

M_M M_M

B

+

+

=

=

y

r a d n u o b e h t e s o p p u

s Γ wil lbedivided into 2n,(n∈N) equa lparts ,so themesh width wli lbe

/n

h=π a ndthenodeswil lbe t_j =τ_j = jh,(j=0,1,..,.2n−1). Thesmoothintegra loperatorswith o

i r e p e h

t d 2π canbeeasily toobtain high accuracyNyström'sapproximaitons[23] .Forexample , r

o t a r e p o n o it a m i x o r p p a e h

t _B0h _fo

0

B canbeapproximatedasfollows:

1 2

0 0

0

.)

(

)

,

(

n h

j j j

t

b

h

B

η

−

τ

η

τ

=

=

∑

) 8 ( r

t s y N e h

T öm'sapproximation Bh_ji of B_ji and Mh_jiof M_ji canbeapproximatedsimliarly. e

t a m i x o r p p a o t r e d r o n

I thel ogarithmicsingularoperator A₀,t hecontinuousapproximationkernel )

, (

p t

a τ canbeDefinedasfollows:

{

0 2 / 1

0 2

, ,

) , (

. ,

n l

)

,

(

a t for_ht h

p

t

_{ch e fort h}

a

π τ τ

τ

τ

−

≥ −

< −

=

) 9 ( e

r u t a r d a u q s 'i d i S y

B rules[22],t heapproximationoperatorcanbeconstructedasfollows:

1 2

0

0

.

)

('

)

(

)

,

(

)

(

)

(

h n

j j j p j

x

t

a

h

t

A

η

−

τ

η

τ

τ

=

=

∑

( 01 ) r

o t a r e p o r a l u g n i s y h c u a C e h t r o f d e n i a t b o e b n a c r o t a r e p o n o i t a m i x o r p p a e h

T C₀,C₀h can be

: s a d e n i f e d

1 2

, 1 2 0

0

,

)

('

)

(

)

2

/

)

(

(

t

o

c

)

(

2

)

(

)

(

h _{i i j} n _{j i j j ji}

j

t

x

t

t

t

h

t

t

a

c

t

C

η

−

η

ε

=

−

=

∑

) 1 1 ( e

r e h w

{

1, ,

, ,

0

r e b m u n d d o s i j i fi

ji fi i jisevennumber

ε

−

−

=

: s w o l l o f s a e t i r w e r e b n a c ) 7 ( . s q E e h t s u h T

1

,

)

(

)

(

)

(

2

h h h

h

h h

h

A

B

g

f

M

C

I

+

+

η

−

+

η

+

) 2 1 ( e

r e h

w Ah,Bh,Ch and _Mh _{are theapproximatematrixescorresponding to theoperatorsA , B ,C}

d n

a M, respectively.

n o i t a l o p a r t x e e h t t c u r t s n o c d n a , n o i t u l o s e h t r o f s r o r r e f o n o i s n a p x e c i t o t p m y s a e h t e v i r e d e w o S

m h t i r o g l

a toobtainhigheraccuracyordersolutions. .

2 m e r o e h

T Considert heasymptoitcpropertyand _{x(t), f(t)∈V}2m_{[0,2π] ,thereexsit saf unciton}

2 2

1 V m [0,2π]

5 3

1

(

)

)

(

j

j

t t

h

h

ω

_{t t}

o

h

η

η

₌

=

+

=

−

) 3 1 (

] 9 [ s m h t i r o g l a n o i t a l o p a r t x e e h t t a h t s e i l p m i ) 2 2 ( . q E n i s r o r r e e h t t u o b a n o i s n a p x e c i t o t p m y s a n A

r e d r o y c a r u c c a h g i h e h T . r e d r o e t a m i x o r p p a e h t e v o r p m i o t ) 2 ( . s q E f o n o i t u l o s e h t o t d e i l p p a e b n a c

5₎

(h

o can beobtainedbycomputing somecoarsegridsandfinegridson Γ inparallel .TheEAs :

s w o ll o f s a d e b i r c s e d e r a

l u c l a c o t s d i r g e n ¯ d n a s d i r g e s r a o c t a s e u l a v e h t e s

U atet heapproximatevaluesa t t_i.

*

2 /

1

.

)

)

(

)

(

8

(

)

(

7

h i h i

i

h

t

η

t

η

t

η

=

−

) 4 1 ( l

a c i r e m u

N Example :

1 e l p m a x

E Wefirsltyi ntroducesomedenotefor 1,2: h( ) ( ) ( )

i h

i

i P P P

e

i= =η −η ist heerroroft he ;t

n e m e c a l p s i

d h_{( )} h_{( )/} h/2_{( )}

i i

i P e P e P

r = is the error ratio; h_{( )} *_{( )} *_{( )}

i h

i

i P P P

e =η −η is the error after e

n

o -stepEAs ;and 1 2 /

7 ( ) ( )

) (

h

h i h

i

i P P P

p = η −η isaposterior ierroresitmate. e

s o p p u

S Ω isanisotropicelilpitca lbodywtihtheaxis a=0.3,b=0.5 intheplanedomain .The r

e t e m a r a

p formulaefort heboundary Γ wli lbedescribedas x=0.3cos(t), y=0.5sin(t),t∈[0.2π].

f o o i t a r s r o r r e , s r o r r e e h T . 1 e l b a

T η_1h(P) a tpoints P=P₁,P₂.

n 1 6 3 2 6 4 1 28 2 56 5 12 1

1h(P)

e 8.466E-4 1.040E-4 1.296E-5 1.618E-6 2.022E-7 2.527E-8

1 1h(P)

r 8.137 8.028 8.011 8.002 8.000

1 1h(P)

e 4.12E-0 7 1.29E-0 8 4.03E-1 0 1.24E-1 1 3.93E-1 3 1

1h(P)

r 31.92 31.99 32.41 31.68

2 1h(P)

e 6.085E-4 7.501E-5 9.292E-6 1.160E-6 1.450E-7 1.813E-8 2

1h(P)

r 8.112 8.073 8.008 8.000 8.000

2 1 ( )

h _P

e 3.91E-0 7 1.24E-8 4.00E-1 0 1.30E-1 1 4.03E-1 3

2 1 ( )

h _P

r 31.47 31.09 30.85 32.16

s n o i t u l o s l a c i r e m u n e h t e t a l u c l a c y l t s r i f e

W ( ₁ , ₂ )T

h h

h η η

η = on the boundary Γ following f

o s e u l a v e t a m i x o r p p a e h t s t s i l 1 e l b a T . ) 6 1 ( . s q

E η_1h(p) a tpoints P₁₌(acosπ₈,bsinπ₈) _and

2 (acos4,bsin 4)

P ₌ π π _{Table 2 lists the approximate values of}

2h(p)

η a t points

1 (acos8,bsin8)

P ₌ π π _and

2 (acos4,bsin4)

P ₌ π π _.

f o o i t a r s r o r r e , s r o r r e e h T . 2 e l b a

T η_2h(P) a tpoints P=P₁,P₂.

n 1 6 3 2 6 4 1 28 2 56 5 12 1

2h(P)

e 4.169E-4 5.100E-5 6.309E-6 7.805E-7 9.753E-8 1.219E-8

1 2h(P)

r 8.175 8.083 8.018 8.003 8.000

1 2h(P)

e 7.67E-0 6 2.44E-0 7 7.80E-9 2.45E-1 0 7.75E-1 2

1 2h(P)

r 31.44 31.29 31.77 31.65

2 2h(P)

e 9.338E-4 1.147E-4 1.420E-5 1.774E-6 2.217E-7 2.771E-8

2 2h(P)

r 8.138 8.080 8.005 8.001 8.000

2 2h(P)

e 8.30E- 60 2.75E-7 8.75E-9 2.79E- 01 8.83E- 21

) (

h _P

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