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1

2

1

3

1

2

1

2

3

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R2

u

(u) + b= 0

(u) n= t

N

(u) n= 0

C

u= u D

N D C

= N

D

C N

D

C = b t N n u D

uV

vV a(u,v) =l(v),

V

V ={v|vH1(),v|D(x) =0,v C}

a(u,v) :=

(u) : (v)d =

(u) : S : (v)d

l(v) :=

b

vd +

N

t

vd,

S

u1(r, ) = KI

2

r

2cos

2( cos ) + KII

2

r

2sin

2(2 + + cos )

u2(r, ) = KI

2

r2

sin 2

( cos ) + KII2

r2

cos 2

(2 cos )

11(r, ) = KI

2rcos

2

1 sin

2sin

3

2

KII

2rsin

2

2 + cos

2cos

3

2

22(r, ) = KI

2rcos

2

1 + sin

2sin

3

2

+

KII2r

sin

2cos

2cos

3

2

12(r, ) = KI

2rsin

2cos

2cos

3

2 +

KII2r

cos

2

1 sin

2sin

3

2

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r KI KII

E

= E

2 (1 + ), =

3 4

3 1 +

x

uh(x) =iI

Ni(x) i+jJ

Nj(x)H(x) j +mM

Nm(x)

4=1

F(x) m

Ni i i

j

cm

I

M

J

H

F

{F(r, )}

rsin

2, cos

2, sin

2sin , cos

2sin

5 5

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re

G

KI KII

e=u uh

ees

e2 ees2 =

hTD1 h d

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ees2 =

hTD h d,

h

h

D = D

N

(x) =

ne

i=1

Ni(x)i (xi),

ne i (xi)

i

i i,j = pa

j = xx,yy,xy i

p

a

x

x

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(x) =

nvi=1

Ni(x)i (x),

Ni nv

smo sing

= smo+ sing .

smo

sing

= smo+

sing .

sing

KI KII

sing

sing

hsmo

sing

hsmo =

h sing .

smo

hsmo

smo

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u(x) =iNx

i(x)uhi

(x) =

iNx

s(i(x)uhi),

uhi i(x)

i

x

Nx nx x

s

u

xi i

i p(xi) nx

x

(i(x))1inx = (A1(x)p(x))Tp(xi)i(x)

A(x) =nx

i=1 i(x)p(xi)p(xi)T

x

xi i

i(s) =f4(s) =

1 6s2 + 8s3 3s4 |s| 10

|s|> 1

s xi

x

s

i

x

x

s=x xC + xC xi

di

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di

p

p= [1,x,y, [F1(r, ), F2(r, ), F3(r, ), F4(r, )]]

=

es

.

D

e

D

e

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D D

e

e

D= e 1 if e 1D= 1 1

e if e

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b = 4

Ix(x, y) = |t|

x cos

2 y sin

2

+ y

a2

|t|2

m sin

2 n cos

2

Iy(x, y) = |t|

x cos

2 y sin

2

ya

2

|t|2

m sin

2 n cos

2

Ixy(x, y) = y a2

|t|2

|t|

m cos

2+ n sin

2

IIx (x, y) = |t|

2

y cos

2+ x sin

2

ya

2

|t|2

m cos

2+ n sin

2

IIy (x, y) = y a2

|t|2|t|

m cos

2+ n sin

2

IIxy (x, y) = |t|

x cos

2 y sin

2

+ y

a2

|t|2

m sin

2 n cos

2

x

y

t

m

n

t= (x + iy)2 a2 = (x2 y2 a2) + i(2xy) =m + inm=

(t) =

(z2 a2) =x2 y2 a2n= (t) = (z2 a2) = 2xy=

(t) =

(m in)

[, ] , i2 =1

KI,ex =

a KII,ex=

a

= 100, = 0 = 0, = 100

= 30, = 90

= 0.5

re

re

rq = 0.9 E= 107

= 0.333 22

5 5

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,

Mesh 1

288 elem.

Mesh 2

512 elem.

Mesh 3

800 elem.

Mesh 4

3200 elem.

Mesh 5

12800 elem.

D

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Mesh1

273 elem.

Mesh 2

525 elem.

Mesh 3

818 elem.

Mesh 4

3272 elem.

Mesh 5

13088 elem.

0

1

2

-1

-2

a. SPR-CX b. XMLS

D

D = 0

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sing

smo

es

=uu

O(hp) h p

O( p/2)

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D

D = 0

D

m (

|D

|) (D)

= 1

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103

104

105

104

103

102

101

dof

ees

SPR-CX s=0.48

XMLS s=0.57

SPR-X s=0.50

SPR-C s=0.26

e s=0.48

103

104

105

105

104

103

102

101

dof

ees

SPR-CX s=0.73

XMLS s=0.72

SPR-X s=0.77

SPR-C s=0.26

e s=0.48

ees

ees

D

m

m

m(|D|)

(D)

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103 104 105

2

4

6

8

10

dof

SPR-CXXMLS

SPRX

SPRC

103 104 1050.8

1

1.2

1.4

dof

(zoomedy-axis)

SPR-CXXMLS

SPRX

103 104 105

102

101

dof

m(|D|)

SPR-CX

XMLS

SPRX

SPRC

103 104 105103

102

101

100

dof

(D)

SPR-CX

XMLS

SPRX

SPRC

m (|D|) (D)

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D

0

4

-4

a. SPR-CX b. XMLS

2

-2

D

D= 0

m (|D|)

(D)

D

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103

104

105

103

102

101

dof

ees

SPR-CX s=0.48

XMLS s=0.54

e s=0.48

103

104

105

104

103

102

101

dof

ees

SPR-CX s=0.59

XMLS s=0.71

e s=0.48

ees

ees

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103 104 1050.8

1

1.2

1.4

dof

SPR-CXXMLS

103 104 105

102

101

dof

m(|D|)

SPR-CX

XMLS

103 104 105102

101

100

dof

(D)

SPR-CX

XMLS

m (|D|) (D)

0

1

-1

a. SPR-CX b. XMLS

0.5

-0.5

D

D= 0

D

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103

104

105

104

103

102

dof

ees

SPR-CX s=0.54

XMLS s=0.57

e s=0.48

103

104

105

104

103

102

dof

ees

SPR-CX s=0.77

XMLS s=0.70

e s=0.48

ees

ees

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103 104 1050.8

1

1.2

1.4

dof

SPR-CXXMLS

103 104 105

102

101

dof

m(|D|)

SPR-CX

XMLS

103 104 105102

101

100

dof

(D)

SPR-CX

XMLS

m (|D|) (D)

m (|D|)

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G

G

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