This is an Open Access document downloaded from ORCA, Cardiff University's institutional

repository: http://orca.cf.ac.uk/102950/

This is the author’s version of a work that was submitted to / accepted for publication.

Citation for final published version:

Peng, Xuan, Atroshchenko, E., Kerfriden, Pierre and Bordas, Stephane Pierre Alain 2017.

Isogeometric boundary element methods for three dimensional static fracture and fatigue crack

growth. Computer Methods in Applied Mechanics and Engineering 316 , pp. 151-185.

10.1016/j.cma.2016.05.038 file

Publishers page: http://dx.doi.org/10.1016/j.cma.2016.05.038

<http://dx.doi.org/10.1016/j.cma.2016.05.038>

Please note:

Changes made as a result of publishing processes such as copy-editing, formatting and page

numbers may not be reflected in this version. For the definitive version of this publication, please

refer to the published source. You are advised to consult the publisher’s version if you wish to cite

this paper.

This version is being made available in accordance with publisher policies. See

http://orca.cf.ac.uk/policies.html for usage policies. Copyright and moral rights for publications

made available in ORCA are retained by the copyright holders.

s♦♦♠tr ♦♥r② ♠♥t ♠t♦s ♦r tr ♠♥s♦♥

t r r♦t

❳ P♥1 tr♦s♥♦2 P rr♥1 P ♦rs3,1,4¶

1♥sttt ♦ ♥s trs ♥ ♥ ♥tr♥ r ❯♥rst②

❯

2♣rt♠♥t ♦ ♥ ♥♥r♥ ❯♥rst② ♦ ♥t♦

3❯♥rsté ①♠♦r té s ♥s ♥♦♦ t ♦♠♠♥t♦♥

r r ♦♥♦r ①♠♦r sr ❯♥t ♥ ♥♥r♥

♥ ♠♣s rr

4♦♥♦rr② Pr♦ss♦r ♥t♥t ②st♠s ♦r ♥ ♦rt♦r② ♦♦ ♦ ♥ ♥

♠ ♥♥r♥ ❯♥rst② ♦ ❲str♥ str tr♥ ②

r② ❲ str

s♦♦♠tr ♦♥r② ♠♥t ♠t♦ s ♦♥ ❯ s ♦♣t t♦ ♠♦

rtr ♣r♦♠ ♥ ❯ ss ♥t♦♥s r s ♥ ♦t r r♣rs♥tt♦♥

♥ ♣②s q♥tt② ♣♣r♦①♠t♦♥ st qrtr s♠ ♦r s♥r ♥trt♦♥ s

♣r♦♣♦s t♦ ♥♥ t r♦st♥ss ♦ t ♠t♦ ♥ ♥ t ② st♦rt ♠♥t

♦♥r♥ st② ♥ r ♦♣♥♥ s♣♠♥t s ♣r♦r♠ ♦r ♣♥♥②s♣ r ♥

♣t r ♦ ②s t♦ ①trt strss ♥t♥st② t♦rs s t ♦♥t♦r M ♥tr

♥ rt r ♦sr ♥tr r ♠♣♠♥t s ♦♥ t r♠♦r ♦ ♥tr

qt♦♥s ♥ ♦rt♠ s ♦t♥ ♥ t t♦ st ♦r t r r♦t t♥s

t♦ t s♠♦♦t♥ss ♥♦t ♦♥② ♥ r ♦♠tr② t s♦ ♥ strsss s♦t♦♥ r♦t ②

②❲♦rs s♦♦♠tr ♥②ss ❯ ♥r st rtr ♦♥r② ♠♥t ♠t♦

r r♦t

¶♦rrs♣♦♥♥ t♦r ①♠ rssst♣♥♦rs♠♥♦rtstr♥

♥tr♦t♦♥

s♠t♦♥ ♦ t t rtr ♣r♦♣t♦♥ s ♥♦t ♦♥② r t♦ ♣r♦r♠ t ♠

t♦r♥ ssss♠♥t ♦r ♣rt t sr♥ ♦ ♠♥ ♦♠♣♦♥♥t t s♦ t♦ ♣ t

♥rst♥ ♦ t ♠♥s♠ ♦ strtrs r ♥ ♥♥r♥ ❱rst ♥♠r ♠t♦s

♥ tt♠♣t ♦r ♠♦♥ t rtr t t ♦♣♠♥t ♦ ♦♠♣tt♦♥

♠♥s ♥ ♣st s ♥s ♦r ♥♠r rtr ♠♦♥ ♣r♠r② ♥ t

♠s♥r♠s♥ ♣r♦r s t ♥tt♦♥ ♦ t rs r s② t♦ ♦r ♠♦r ♦rrs

s♠r ♥ ♦♠tr② s③ t♥ t ♦♠♣♦♥♥ts r♥t s ♦ t rs ♥ ♦♠♣♦♥♥ts

rqrs ♦ r♥ ♠s ♦r t ♥t♦♥ ♦ t ts ♥ r♠s♥ ♦♠ ♥ssr②

♦r rt s♠t♦♥ ♥ t rs ♣r♦♣t

♥t ♠♥t ♠t♦ ♥ ♣♣ t♦ s♠t t r ♣r♦♣t♦♥ rt② t

rt♥ ♣t r♠s♥ ♦♣rt♦♥ ❬❪❬❪❬❪ ♦♠ s♦tr ♣s ♥ ♦♣

s ♦♥ ts ❬❪❬❪ ♥ r ♣♣r ♥ rrr ❬❪ rtss t r♠s♥

♦♠s ♠rs♦♠ ♦r ♠trs ♦r ♦r r② ♦♠♣t ♦♠♣♦♥♥ts s t ♦♠♣①t② s

♥rs t♦ t ♣rs♥ ♦ rs s ♥tr♥ ♦♥rs

♦ ♣rtt♦♥ ♦ ♥t② P❯ ♥r♠♥t s ♥ ♣r♦♣♦s t♦ rs t ♠s r♥

♥ rtr ♠♦♥ ❬❪ t♦ t t♦♥ ♥r♠♥t ♥t♦♥s t s♦♥t♥ts r

♥tr♦ ♥t♦ t ♠♦ ♥ t r♣rs♥tt♦♥ ♦ t r ♦♥② ♠s ♦r ♥tt♥ t ♥

r♠♥ts ♠s t r ♠s ♥♣♥♥t r♦♠ t ♦♠♣♦♥♥ts ♠s ①t♥

♥t ♠♥t ♠t♦ ❳ ❬❪ s② ♦♣ t t st ♥t♦♥s s ♥ ♠♣t

r♣rs♥tt♦♥ ♦ t r s ♥ ♠♣♠♥t ♦r r r♦t ♣r♦♠ ❬❪❬❪❬❪❬❪

s s ♦r ♥str ♣♣t♦♥s ❬❪❬❪❬❪ ♠sr ♠t♦s s♦ ♥ ♣r♦

♣♦s t t ♠ ♦ rtr r♥ t ♠s r♥ ♦r ♥st♥ t ♠♥tr r♥

❬❪ ♥ t ①t♥ ❳ ❬❪❬❪❬❪ ♦r ♠♦r ts t rrs ♦

rr t r ♣♣r ② ②♥ t ❬❪

rtr ♠♦♥ ② t ♦♥r② ♠♥t ♠t♦ ①ts ♠♦r ♥ts t♥

② ♥ tr♠s ♦ ♠sr♠s ♦rts s ♦♥② t ♦♥r② srt③t♦♥ s rqr ♥

♥ ♦rr t♦ ♣♣r♦①♠t t q♥tt② ♦ ♥trst ❲♥ rs ♦ ♦♥② t ♦♥r②

srs r ♣t ♥st ♦ r♥rt♥ t ♦♠ ♠s ♥ ♦rr t♦ r♠♥t t

s♥r s②st♠ s ② t ♦♣s srs ♥ rtr ♦♥ ♥ ♥ ❬❪ ♣r♦♣♦s

t ♦♥r② ♥tr r♣rs♥tt♦♥s ② ♥tr♦♥ t ②♣rs♥r qt♦♥ r

r♦♠ t s♦♥r② ❬❪ s ♦ ♦♥r② ♥tr qt♦♥ ♠s t r

♣r♦♣t♦♥ s♠t♦♥ ♠♦r t tr♦ s♥ ♦♠♥ ♥ t ♦rrs♣♦♥♥

s ssq♥t② ♠♣♠♥t ♦r ♥ rtr ❬❪❬❪❬❪ ♥ s ①t♥ t♦

♠tr♥♦♥♥r rtr ❬❪❬❪ ♥ ②♥♠ r ♣r♦♣t♦♥ ❬❪ ♦♠♠r ♣s

s ♦♥ r ❨ ❬❪ ♥ ❬❪ ♣rt r♦♠ t s ♦♥ t

♦♦t♦♥ ♠t♦ t r♥ ♥ ♣rtr t s②♠♠tr r♥

s s♦ r♥ tt♥t♦♥ ♥ t ♣♣t♦♥ ♦r rtr ♥②ss ❬❪❬❪❬❪ s②♠♠tr

♠tr① s②st♠ ♦ s♦ tts t ♦♣♥ t ❬❪❬❪

s♦♦♠tr ♥②ss s rst ♥tr♦ ② s t ❬❪ s ♦

s t♦ s t s♠ s♣♥ ss ♥t♦♥s t♦ r♣rs♥t t ♦♠trs ♥ ♣♣r♦①

♠t t ♣②s q♥tts ♦ ♥trst ♥ t ♥stt♦♥ ♦♥ t ♦♥t ♦ ♥

s ♥rs♥② r♥ tt♥t♦♥ r♥t② s♥ ♦♥② t ♦♥r② r♣rs♥tt♦♥ ♦

t ♦♠tr② s rqr ♥ tts t ♥trt♦♥ ♦ s♥ ♥ ♥②ss

s r② ♥ ♣♣ ♥ ♠♥② s ❬❪❬❪❬❪❬❪❬❪❬❪❬❪❬❪ ♥ s

♥ rtr ♦♣ t ♠♦r ♥♠r s♣ts s s t P❯ ♥r♠♥t ❬❪❬❪ t

tr♠♠ ❯ ❬❪❬❪ t st s♦t♦♥ ❬❪ t r♥ ♦r♠ ❬❪❬❪ t ♥t ♦

s♠♦♦t♥ss t♦ ♦♥r② ♥trs s r♦t ② s ♥stt ♥ ❬❪

s♦♦♠tr ♥②ss s ♥ ♣♣ t♦ rtr ♥ ♦r♣♦rt♦♥ t ❳ ❬❪❬❪❬❪❬❪

❱r♦♦s t ♣rs♥t s♠ t♦ ♠♦ ♦s r ♣r♦♣t♦♥ ② s♥ s♣♥s t♦

♥rt t ♦ s♦♥t♥ts ❬❪ ②♥ t ♣♣ t s♣♥ s t♦ s♠t

t ♥ ♠♥t♦♥ ♥ ♦♠♣♦sts ❬❪ s♣ s♥stt② ♥②ss ♦ strss ♥t♥

st② t♦rs ♦r r rs s ♣r♦r♠ ② ♦ ♥ ♦ ❬❪ ♠t t ♣r♦♣♦s ♥

♥r s ♦♥ t ♥s♣r r ♣rtt♦♥ ♦ ♥t② ♦♠♣♦st♦♥s

♥ t ♠t♦ ♥ts r♦♠ r♦st ♥ ♥♦♥trt ♥♠r st♥ ♦♥strt♦♥

❬❪❬❪ ♦♥ t ♣r♦♣♦s ♦♠tr ♠♣♣♥ ② ♣s♦rrs ♦ s♣♥s r♦♠

t ♣r♠tr s♣ ♥t♦ t ♣②s s♣ s tt t s♥rt② ♦ t②♣ r1/2 ♥ ♣

tr ♥ ♥r st rs ❬❪❬❪ tr♥ t ♥♥ t s♦♦♠tr ♥②ss ②

t s ♦♥r② ♥t ♠♥t ♠t♦ ♥rts ♦t ♥ts ♦ ♥

rt♥ ss♦♥ ♦ t ♦♠♥ ♥s t♦ ♦♥ ♦r ♦♠♣t ♦♠tr② ♥

♦rr t♦ ♦t♥ t s♥ ♥tr ❬❪ ♣♦t t♣s ♦ t ♣♣t♦♥ ♥ rtr ♥ t

r♠♦r ♥ ♦♥ s

r♦rr ♦♥t♥t② ♠♣r♦s t r② ♦ t strss ♥r t r t♣

s r t♦ rtr ♥②ss ♥ t rs ♦ r♦♠ s r ♦♠♣r t♦ t C0

r♥ ss

rtr t♥♥t ♥ ♥♦r♠ t♦rs r ①t② rt♥ ♥ ♦ t♥s t♦

t ①t r♣rs♥tt♦♥ ♦ t r rs

♦ r t♣ r♦♥t s②st♠ ♥ ♦♥strt rt② s ♦♥ t s♣♥s

r ♦r srr♣rs♥t rs ♣s t♦ rt② t t rtr ♣r♠trs

♦♥♣t ♦ ♥trt♦♥ tr♦ s♥ t♦ ♥②ss ts t ♠♥strtr

s♥ s ♦♥ t t rtr ♥②ss

♥ ts ♦r t ♣♣t♦♥ ♦ ♥ rtr ♥②ss ♥ t r r♦t

①♣♦r ss s♥ t ♦♥♥t♦♥ ♦♥r② ♥tr qt♦♥ s ♦r stt②

t ②♣rs♥r ♥tr qt♦♥ s ♥tr♦ t♦♥② ② ①♣♦t♥ t s♠♦♦t♥ss

♦ ❯ ♦♠trs ♥ ♦ s♥rt② r♠♦♥ t♥q ♣r♦♣♦s ② ♥ ❬❪

s ♣♣ ♦♥ t r♦s ♦rrs ♦ s♥r ♥trs ♣ t♦ ②♣rs♥r O(1/r3) ♥ ts

♠♣r♦ rs♦♥ t♦r t♦ st♦rt ♠♥ts ♦r t s♣t rt♦ ♦♠♠♦♥② rs

♥ s♦♦♠tr s ♠t♦s s ♦r♠t r s ①♣t② r♣rs♥t ② ❯

sr s ♥tr♥ ♦♥r② ♥ ♥ ♦rt♠ s ♦t♥ t♦ sr t r ♣r♦♣t♦♥

s tt t s♠♦♦t♥ss ♥ ♦♠tr② r♦t ② ♥ ♥ s♦t♦♥ r♦t ② s ②

♥stt ♦r ①trt♥ t strss ♥t♥st② t♦rs ♥ r r♦t

rst ♦ t ♣♣r s ♦r♥③ s ♦♦s t♦♥ rs t ♦♥r② ♥tr qt♦♥s

s tt ♣♣ ♥ rtr ♠♦♥ t♦♥ strts t ❯ ss ♥t♦♥s ♦♥

srs ♥ t ♦♦t♦♥ s♠ t♦♥ ♦t♥s t ♠♣r♦ s♥r ♥trt♦♥

s ♦♥ t s♥rt② strt♦♥ t♥q ❬❪ r r♦t rt ♦r s t

♥ st♦♥ ♥♥ ♣t♥ t r sr ♦♠tr② ♦♠♣t♥ t strss ♥t♥st②

t♦rs ♥ t t rtr r t Prs ♠r ①♠♣s ♦r ♦t stt rtr

♥②ss ♥ r r♦t r ♥ ♥ st♦♥ ❲ ♦♥ ♦r ♦r ♥ ♣r♦♣♦s t tr

rsr ♦ ♥trst ♥ t st st♦♥

♦♥r② ♥tr qt♦♥s ♦r r ♠♦♥

♦♥sr ♥ rtrr② ♦♠♥ Ω ♦♥t♥s r s ♥ r ♦♥r② ♦ t

♦♠♥ ∂Ω = S + Sc+ + Sc− r S s ♦♠♣♦s ♦ Su r rt ♦♥r② ♦♥t♦♥s

r ♣rsr ♥♦♥ s♣♠♥t u St r ♠♥♥ ♦♥r② ♦♥t♦♥s r ♣rsr

♥♦♥ trt♦♥ t s♣♠♥t s ♥ ② ♥♥ u ♥ t s tt

cij(s)uj(s) =

∫

∂ΩUij(s,x)tj(x)S(x)−−

∫

∂ΩTij(s,x)uj(x)S(x),

r t Uij Tij r ♥♠♥t s♦t♦♥s ♥ ♦r ♥r stt②

Uij(s,x) =1

16πµ(1− ν)r[(3− 4ν)δij + r,ir,j ] ,

Tij(s,x) = − 1

8π(1− ν)r2

∂r

∂n[(1− 2ν)δij + 3r,ir,j ]− (1− 2ν)(r,inj − r,jni)

,

r µ = E/[2(1 + ν)] E s ❨♦♥s ♦s ♥ ν P♦ss♦♥s rt♦ s s t s♦r ♣♦♥t ♦r

♦♦t♦♥ ♣♦♥t s t♦ tr♠s s ♥tr♥② ♥ t r♠♥r ♣rt ♦ ts

♣♣r −∫

♥♦ts t ♥tr s ♥tr♣rt ♥ t ② Pr♥♣ ❱ s♥s trt♦♥

s ♦t♥ ② r♥tt♦♥ ♦ t s♣♠♥t t rs♣t t♦ s ♥ ♠t♣t♦♥

② t st t♥s♦r Eijkl

cij(s)tj(s) = −∫

∂ΩKij(s,x)tj(x)S(x)−=

∫

∂ΩHij(s,x)uj(x)S(x),

Hij(s,x) = Eikpq∂Tpj(s,x)

∂sqnk(s), Kij(s,x) = Eikpq

∂Upj(s,x)

∂sqnk(s),

r =∫

♥♦ts t ♠r ♥t Prt ♥tr

♦ t ♦♥r② ♠♥t ♠t♦ s t♦ srt③ t ♦♥r② ♦♠tr② ♥ t

♣②s s s♥ sts ♦ ss ♥t♦♥s sq♥t② t s♦r ♣♦♥t s ♣ t t

♦♦t♦♥ ♣♦♥ts ♥ t s♣♠♥t s tr♥s♦r♠ ♥t♦ t s②st♠ ♦ ♥r r

qt♦♥s ♦r ♥ t ♦♠♥ ♦♥t♥s r t ♦♦t♦♥ ♣♦♥ts ♦♥ t ♦r♣♣♥

srs Sc+ ♥ Sc− ♦ ♦♥ rr t♦ r ♥ t♥ t s②st♠ ♠tr①

♦♠s s♥r ♦ ②s t♦ t ts ♣r♦♠ r r ♥ t ♦♦♥ st♦♥s

r r ♠♦

qt♦♥s

t② s ② t ♦♣s r srs s r♠♥t tr♦ t s ♦

qt♦♥s ② ♣rsr♥ t s♣♠♥t qt♦♥ ♦♥ ♦♥ r sr Sc+ ♥

♦♥ t rst ♦ t ♦♥r② S ♦r t ♦♦t♦♥ ♣♦♥t s+ ♦♥ t r sr Sc+ qt♦♥

♥ rrtt♥ s

cij(s+)uj(s

+) + cij(s−m)uj(s

+) =

∫

SUij(s

+,x)tj(x)S(x)−∫

STij(s

+,x)uj(x)S(x)

−−∫

Sc+

Tij(s+,x+)uj(x

+)S(x)−−∫

Sc−

Tij(s−m,x

−)uj(x−)S(x)

+

∫

Sc+

Uij(s+,x+)tj(x

+)S(x) +

∫

Sc−

Uij(s−m,x

−)tj(x−)S(x).

♥ ♥♦♦s② t trt♦♥ qt♦♥ ♦♥ t ♦tr r sr Sc− ♥ r

♦♠s

cij(s−)tj(s

−) + cij(s+m)tj(s

−) =

∫

SKij(s

−,x)tj(x)S(x)−∫

SHij(s

−,x)uj(x)S(x)

−=

∫

Sc−

Hij(s−,x−)uj(x

−)S(x) + =

∫

Sc+

Hij(s+m,x

+)uj(x+)S(x)

+−∫

Sc−

Kij(s−,x−)tj(x

−)S(x)−−∫

Sc+

Kij(s+m,x

−)tj(x+)S(x).

s−m ♥♦ts t ♠rr♦r ♣♦♥t ♦ s+ ♦♥ t Sc− ♠♥s s−m ♥ s

− sr t s♠ ♣②s

♥ ♣r♠tr ♦♦r♥t t t ♥♦r♠ t♦rs t r ♦♣♣♦st st t♦ tr♠s ♦

♦t qt♦♥s ♥ t ♥ s ♦ qt♦♥ r ♦♠tt t♦ t rt♦♥r r

t♦ t ♦♣s ♦♥r② ♥ rtr ♣r♦♠ t♦ ♠♣ tr♠s rs ♥ ♥

♦♣rt♦r ♥♦t ♦♥② ①ts s♥rt② ♦♥ t r sr r t ♦♦t♦♥ ♣♦♥ts ♦t

t s♦ ♦♥ t ♦♥ r t ♠rr♦r ♣♦♥ts ♦ t ♦♦t♦♥ ♣♦♥ts ♦t

r ♦♣♥♥ s♣♠♥t qt♦♥

♦♥r② ♥tr qt♦♥ ♦r r ♣r♦♠ ♥ s♦ r♦r♠t ② stt♥ t

♦♥r② q♥tt② s r ♦♣♥♥ s♣♠♥t ♦r ♦♣ ♦ r srs t t s♦r

♣♦♥t ♣♣r♦ t♦ s♥ r sr ♦r ①♠♣ Sc = Sc+ ♥ ♥♦t tt n = n+ = −n

−

cij(s+)uj(s

+) + cij(s−)uj(s

−) =

∫

SUij(s

+,x)tj(x)S(x)−−∫

STij(s

+,x)uj(x)S(x)

+

∫

Sc

Uij(s+,x+)(tj(x

+) + tj(x−))S(x)

−−∫

Sc

Tij(s+,x+)(uj(x

+)− uj(x−))S(x).

♦rrs♣♦♥♥ trt♦♥ s

cij(s+)tj(s

+)− cij(s−)tj(s

−) = −∫

SKij(s

+,x)tj(x)S(x)−=

∫

SHij(s

+,x)uj(x)S(x)

+−∫

Sc

Kij(s+,x+)(tj(x

+) + tj(x−))S(x)

−=

∫

Sc

Hij(s+,x+)(uj(x

+)− uj(x−))S(x).

qt♦♥ ♥ s ♦♥ ♦♥② t ♣rs♥t s t ♥♥♦♥ ♦r t

rtr ♣r♦♠ ♦r t s♣♠♥t ♥s t♦ ♥♦♥ ♦♥ t r srs

qt♦♥ s♦ s♦ s♦ t S → ∞ ♥ ♥♦t tt trt♦♥r r srs r

ss♠ rr t

0 = t∞j (s)−=

∫

Sc

Hij(s,x)Juj(x)KS(x).

Juj(x)K = uj(x+)−uj(x−) s t r ♦♣♥♥ s♣♠♥t t ssr♣ts ❵ r ♦♠tt

s♥ t ♥tr s ♦♥② ♦r s♥ r sr t∞ s ♥tr♣rt s t s♦t♦♥ ♥ t ❵♥♦

r s♣

❯ srt③t♦♥ ♥ ♦♦t♦♥

❯ ss ♥t♦♥s r t ♥r③t♦♥ ♦ s♣♥ ♥t♦♥s tt ♦s ❵♣r♦t♦♥

r♦♠ sqr ♥ ♦♠♥s t♦ ♦r♠ ♦♠♣① ♦♠trs ♦ t s ♦♥♣t ♦ s♣♥ s

rst ♦t♥ s♣♥ ss ♥t♦♥s r ♥ ♦r ♥♦t t♦r s ♥♦♥rs♥

sq♥ ♦ r ♥♠rs ♥ ♥ t ♣r♠tr s♣ ♥♦t t♦r s ♥♦t s Ξ =

ξ1, ξ2, ..., ξn+p+1 r ξi ∈ R s t it ♣r♠tr ♦♦r♥t ♥♦t p s t ♦rr ♦ t

♣♦②♥♦♠ ♥ s♣♥ ss ♥t♦♥s n s t ♥♠r ♦ t ss ♥t♦♥s ♦r ♥ ♦rr

p t s♣♥ ss ♥t♦♥s Ni,p t 1 6 a 6 n r ♥ ② t ♦① ♦♦r rrs♦♥

Ni,0(ξ) =

1 ξi 6 ξ < ξi+1

0 ♦trs

t♥ ♦r p > 0

Ni,p(ξ) =ξ − ξiξi+p − ξi

Ni,p−1(ξ) +ξi+p+1 − ξ

ξi+p+1 − ξi+1Ni+1,p−1(ξ).

♦♥t♥t② ♦ s♣♥ ss ♥t♦♥s t ξi ♥ rs ② r♣t♥ t ♥♦t sr

t♠s ξi s ♠t♣t② k ξi = ξi+1 = ... = ξi+k−1 t♥ t ss ♥t♦♥s r Cp−k

♦♥t♥♦s t ξi Prtr② ♥ k = p t ss s C0 ♥ k = p+1 s t♦ s♦♥t♥t②

t ξi t rst ♥ st ♥♦t k = p + 1 t ♥♦t t♦r s ♥ ♦♣♥ ♥♦t t♦r

♦r ts ♥ rrr ♥ ❬❪

♥ t ♥♦t t♦rs Ξ = ξ1, ξ2, ..., ξn+p+1 ♥ H = η1, η2, ..., ηm+q+1 ♥ t ♦♥tr♦

♣♦♥ts ♥t Pi,j s♣♥ sr S(ξ, η) s ♥ ② t t♥s♦r♣r♦t ♦ s♣♥ ss

♥t♦♥s ♥ ♥ ♣r♠tr ♦♠♥ [ξ1, ξn+p+1]× [η1, ηm+q+1]

S(ξ, η) =n∑

i=1

m∑

j=1

Ni,p(ξ)Mj,q(η)Pi,j ,

r Ni,p(ξ)Mj,q(η) r t s♣♥ ss ♥t♦♥s ❯ ss ♥t♦♥s ♥

♦♥strt ② rt♦♥③♥ t t♥s♦r♣r♦t s♣♥ ss ♥t♦♥s s

Ri,j(ξ, η) =Ni,p(ξ)Mj,q(η)wi,j

∑nk=1

∑ml=1Nk,p(ξ)Ml,q(η)wk,l

,

r t sr r wi,j s t t ss♦t t t ♦♥tr♦ ♣♦♥t Pi,j ♦r ♥trt♦♥

♣r♣♦s t ❯ ss ♥t♦♥s r s② t ♥ t ♠♥t ♥ ② t

♥♦♥③r♦ ♥♦t ♥trs [ξi, ξi+1]× [ηj , ηj+1] r t ss♥ r ♥ ♣♣ ❬❪

r sss s ♥ s t♦ ♥rt t ♦♦t♦♥ ♣♦♥ts ♦r ♦s ♦♠♥

♦♠♣♦s ② tr♠ss ♥ ♦♠♣t ❯ ♣ts t ♥♠r ♦ ♦t♥ ♦♦t♦♥

♣♦♥ts ② t r sss s ♥t t♦ t ♥♠r ♦ ♦♥tr♦ ♣♦♥ts ♦r ss ♥t♦♥s

♠♥s ♦♥ ♦♦t♦♥ ♣♦♥t s ss♦t t ♦♥ ♦♥tr♦ ♣♦♥t ♦r t♦s ♦♦t♦♥

♣♦♥ts ♥ t sr♣ s ♦r ♦r♥rs ♦r ♥ s♦♥t♥♦s ss ♥t♦♥s r ♥ ts

♦♦t♦♥ ♣♦♥ts ♦st r♦♠ t ♦r♥ ♣ ②

ξs,i = ξs,i + α(ξs,i+1 − ξs,i), ♦r

ξs,i = ξs,i − α(ξs,i − ξs,i−1), α ∈ (0, 1).

♦t tt ♥ ts s t ss♦t ♦♥tr♦ ♣♦♥ts s♦ ♦ s tt t s♦♥t♥

♦s ss ♥t♦♥s r ♦t♥ ♦r t s ♦♥ t ♦st ♦♦t♦♥ ♣♦♥ts s♦ ♠r

♥t♦ ♦♥ s tt t ♥♠r ♦ qt♦♥s ♥ ♥♥♦♥s ♣s ♦♥sst♥

♠r ♥trt♦♥

t♦ t s♥rts ♥ s tr s♥r ♥trt♦♥ ♥ ♥♦♥s♥r ♥trt♦♥

tr srt③t♦♥ ♦r t ♠♥t ♦♥t♥♥ t ♦♦t♦♥ ♣♦♥t s♥r ♥trt♦♥ s

♣r♦r♠ ♥ t ♠♥t ♦♥s t♦ s♥r ♠♥ts ♠♥ts ① t ♦♦t♦♥

♣♦♥t r ♥♦♥s♥r ♠♥ts s♥r ♥trt♦♥ ♥s t♦ r② trt ♥

❱r♦s ♥♠r ♠t♦s ♥ ♣r♦♣♦s ♥ ♣st s ♥ ♦♥ ♥ rr t♦

r ♦r ♥ ❬❪ r♦st t♥q ♦♣ ♥ ❬❪ ♥ ♣♣ t♦ rr③ t

s♥r tr♠s ♥t♦ ② s♥r t s ♦ s♠♣ s♦t♦♥ ♦ rr③t♦♥

t♥q s ♦♥ s♠♣ s♦t♦♥s s ♥ ♣♣ ♦r rtr ♦♣ ♥ t r♠♦r

♦ ❬❪❬❪❬❪ ♦r ts ♠t♦ s ♥ ♥ t ♦♣♥ srs s s

rs ❬❪ ♥ t ♣rs♥t ♦r s t s♥rt② strt♦♥ t♥q ♣r♦♣♦s

② ♥ ❬❪❬❪ t♦ r♠♦ t s♥rts rs ♥ ♦t s s ♥t ♠t♦

♦r t trt♠♥t ♦ s♥r ♥trs rrss ♦ ♠s rt③t♦♥ ♥ ♣r♦ t♦ ♥t

♦r rtr ❬❪

♥rt② strt♦♥ t♥q ♦r s♥r ♥trs

tr♥s♦r♠s r♦s ♦rrs ♦ s♥r ♥trt♦♥ ♥t♦ ② s♥r ♦♥ s ♦♥

t ♥tr♥s ♦♦r♥t s②st♠ ♦ t s♥r ♠♥t tr srt③t♦♥ ♥ t ②

s♥r ♥trt♦♥ tr♥s t♦ rr t qrtr s ♣r♦r♠ ♥ t ♣♦r ♦♦r♥ts

② ①♣♥♥ t ♥tr♥ ♥t♦ srs t rs♣t t♦ t ♥tr♥s ♦♦r♥t t s♥rt②

♥ r♣rs♥t ①♣t② ♥ t s♥r tr♠s r strt r♦♠ t ♥tr♥ ♥

t r♠♥♥ t♦ rr ♦r rr ss♥ r s ♣♣ t strt tr♠s r

s♠♥②t② ss♠ tt t ♦♦r♥t ♦ ♣♦♥t ♦ ♥trst s x(xi = x, y, z)

♥ ♣②s s♣ ξ(ξi = ξ, η) ♥ ♣r♠tr s♣ ♦ ❯ ss ♥t♦♥s ξ(ξi = ξ, η) ♥

♣r♥t s♣ [−1, 1]× [−1, 1] ♦r t ②♣rs♥r ♥tr ♦ t ♦r♠

I = =

∫

SH(s,x(ξ))R(ξ)J(ξ)S,

r H(s,x(ξ)) s t ②♣rs♥r r♥ R(ξ) s t ❯ ss ♥t♦♥ ♥ J(ξ) s t

♦ tr♥s♦r♠t♦♥ r♦♠ ♣r♥t s♣ t♦ ♣②s s♣ r ♣♦r ♦♦r♥ts

ρ(ρ, θ) ♥tr t t s♦r ♣♦♥t r ♥tr♦ ♥ t ♣r♥t s♣ ♣r♥t ♦♠♥ s

s ♥t♦ tr♥s ♦r qrtr ♥tr② ♦r ♣♦♥t ξ ♥ t str♥s

ξ = ξs + ρ♦sθ,

η = ηs + ρs♥θ,

tr t ♣♦r ♦♦r♥t tr♥s♦r♠t♦♥ qt♦♥ ♦♠s

I = limε→0

∫ 2π

0

∫ ρ(θ)

α(ε,θ)H(ρ, θ)R(ρ, θ)J(ρ, θ)ρρθ,

r ρ(θ) = h/♦sθ h s t s♦rtst st♥ r♦♠ t s♦r ♣♦♥t t♦ t ♠♥t ♥

θ s t ♥ r♦♠ ♣r♣♥r rt♦♥ t♦ t ♣♦♥t s ♥ r ♥ θ0 s

t ♥ ♦ t ♣r♣♥r ♥ t♥ t ♥ t♦ t ♣♦♥t ♥ ♦♠♣t s

θ = θ + θ0.

r r♥s♦r♠t♦♥ t♥ ♦♦r♥t s②st♠ ♦r

♥tr♥ F (ρ, θ) = H(ρ, θ)R(ρ, θ)J(ρ, θ)ρ s ①♣♥ s

F (ρ, θ) =F−2(θ)

ρ2+F−1(θ)

ρ+ F0(θ) + F1(θ)ρ+ F2(θ)ρ

2 + · · · =∞∑

i=−1

Fi(θ)ρi.

rst t♦ s♥r tr♠s ♦♥ t rt ♥ s r strt ♥ s♠

♥②t② rst♥ ♥

I = I1 + I2,

I1 =

∫ 2π

0

∫ ρ(θ)

0

[

F (ρ, θ)− F−2(θ)

ρ2− F−1(θ)

ρ

]

ρθ,

I2 =

∫ 2π

0I−1(θ)♥

ρ(θ)

β(θ)θ −

∫ 2π

0I−2(θ)

[ γ(θ)

β2(θ)+

1

ρ(θ)

]

θ,

r I1 s rr ♥ I2 r rr ♥ ♥trs ♦t ♥ ♣♣ t ss♥ qr

tr r t♦♥ ♦ α(ε, θ) β(θ) ♥ γ(θ) s s t ♠t♥ ♣r♦ss r ♥ ♥

♣♣♥① ♥ ♠♦r ts ♥ rrr ♥ ❬❪

♦♥♦r♠ ♠♣♣♥ ♦r

t s ♥ r ② ♦♥ t ❬❪ tt t ①♣♥s♦♥ ♦♥ts Fi(θ) ♥ qt♦♥

①ts r♦s ♦rrs ♦ ♥rs♥rt② ♥ t ♥r θ rt♦♥ t♦ t s♥rt②

♥ t r ρ rt♦♥ s ♥ ② ♥ s ♥rs♥rt② s s♥st t♦ t s♣

♦ t ♠♥t ♥ ♦♠s sr ♥ t ♠♥t s ② st♦rt Fi(θ) ♥

r♣rs♥t s

Fi(θ) =Fi(θ)

Ap(θ)=

Fi(θ)

[0.5(|ms1|2 + |ms

2|2)(ωs♥(2θ + ϕ) + 1)]p/2,

r Fi(θ) r t rr tr♦♥♦♠tr ♥t♦♥s ♥ ♥tr ❵p s t ♦rr ss♦t t

❵i r♥r ss t♦rs msi = mi|ξ=ξs

, (i = 1, 2) ♥ r t s

m1 =[∂x

∂ξ,∂y

∂ξ,∂z

∂ξ

]

,

m2 =[∂x

∂η,∂y

∂η,∂z

∂η

]

.

♥tr♦♥ t♦ ♣r♠trs

λ = |ms1|/|ms

2|,

♦sψ = ms1 ·ms

2/|ms1||ms

2|,

s tt

ϕ = rt♥λ2 − 1

2λ♦sψ,

ω =

√

1− 4s♥2ψ

(λ+ λ−1)2< 1.

♥ t ♥ ♦♥ tt ♥ t ♠♥t s♣t rt♦ s r ♦r ♥ t♥ t♦

ss t♦rs t♥s t♦ 0 ♦r π s♥ψ → 0 A(θ) t♥ t♦ 0 rst♥ t ♥rs♥rt② ♦

Fi(θ) ♦t s♥r♦s ♥t st♦rt s♣ ♦ t s♥r ♠♥t r ♦♠♠♦♥

♣♥♦♠♥♦♥ ♥ s♦♦♠tr ♥②ss

♦♥ t ❬❪ ♦♥strt t ♦♥♦r♠ ♠♣♣♥ r♦♠ t ♣r♥t s♣ (ξ, η) t♦ ♥

♣r♠tr s♣ (ξ, η) r t t♦ r ♥r ss t♦rs ♥ t ♥ ♣r♠tr s♣

r ♦rt♦♦♥ ♥ ♥t ♥t t♦ ♦tr

ms1 · ms

2 = 0,

|ms1| = |ms

2|.

♥ A(θ) ♦♠s ♦♥st♥t ♠s t ♥trt♦♥ ♥♦♥s♥st t♦ t ♠♥t s♣

t srs s ①♣♥ ♥ t ♥ s♣ qrtr ♦r t s♥r ♥tr tr♥s t♦

st rrss t st♦rt ♠s

♠♣♣♥ ♣r♦♣♦s ② ♦♥ t s t♦r ♦r tr♥r ♠♥t ♥ ts ♦r ①t♥

t ♥t♦ t qrtr ♠♥t r ♥ ❬❪ t ♦♥ tr♥s♦r♠t♦♥ ♠tr① T

r♦♠ (ξ, η) t♦ ♥ ♣r♠tr s♣ (ξ, η) s

T =

1 δ1

0 δ2

, s♦ tt ξ = Tξ,

r δ1 = ♦sψ/λ δ2 = s♥ψ/λ ♥ t ♥ ss t♦rs

[

ms1 m

s2

]

=

[

ms1 m

s2

]

T−1 =

[

ms1 −(δ1/δ2)m

s1 + (1/δ2)m

s2

]

sts② t rt♦♥ ♥ qt♦♥s ♥r ♥tr♣♦t♦♥ s s r♦♠ (ξ, η) t♦ t

♥ ♣r♠tr s♣ (ξ, η) ♦r qrtr ♠♥t

ξ =

4∑

I=1

NI(ξ)ξI,

N1 = 0.25(ξ − 1)(η − 1),

N2 = 0.25(ξ + 1)(η − 1),

N3 = 0.25(ξ + 1)(η + 1),

N4 = 0.25(ξ − 1)(η + 1).

♦♠♥♥ qt♦♥s ♥ t ♥♦ ♦♦r♥ts ξI♥ ♦t♥ s ξ

1(1+ δ1, δ2)

ξ2(−1+ δ1, δ2) ξ

3(−1− δ1,−δ2) ♥ ξ

4(1− δ1,−δ2) t s♦ ♥♦t tt s♥ 0 < ψ < π

δ2 > 0 t qrtr ♠♥t s r♥t t♦ ♣♦st r ♦♥ ♣♦ss ♣♦t s s♦♥

♥ r s rqrs t s♦r ♣♦♥t s♦ ♥♦t ♦t ♥ t ♥rt ♣♦♥t

♥ t ♦♠tr② r |msi | 6= 0

t ♥ rrr r♦♠ r tt t s♣ ♦ t ♠♥t ♥ ♦♥♦r♠ s♣ s ♦♥tr♦

② t ♦♥ts δ1 ♥ δ2 s ♠♥s tt λ rt ♠♥t s♣t rt♦ ♥ ♦sψ

rt ♠♥t st♦rt♦♥ t r♦♠ t ♦♥♦r♠ ♠♥t s s rst

♥ str♥s t θ ♣♣r♦s t♦ ±π/2 t ♣♦♥t ♦s t♦ t s ♥t t♦ t

s♦r ♣♦♥t ♦ t str♥s r s ρ(θ) = h/♦sθ s ♥♦t t rt②

♦ t ts ♥r s♥rt② ♥ ρ(θ) t ♦♦♥ ♠♦ tr♥s♦r♠t♦♥ s ♣♣ ♥

t ♥r rt♦♥ s tt t ♥trt♦♥ ♣♦♥ts str t♦ t s r t

♥rs♥rt② s sr ♣t② ♦r♥ t♦ t θ ❬❪

w(θ) =1

π

(

θ +π

2

)

, θ ∈ (−π2,π

2), z ∈ (0, 1)

z = z(s) = w(θ1) +1

2(s+ 1)(w(θ2)− w(θ1)), s ∈ (−1, 1), z ∈ (z(θ1), z(θ2)) ⊂ (0, 1)

f(z) =zm

zm + (1− z)m,

θ = πf(z)− π

2,

J−1(θ) =∂θ

∂s=π[w(θ2)− w(θ1)]mf(z)

m−1

2(f(z)m + (1− f(z))m)2,

r s s t ss ♣♦♥t r♦♠ ♥tr (−1, 1) t rt♦♥ ♦ θ ♥ θ ♥ ♦♥ ♥ qt♦♥

♠r qrtr

♥ ♥♠r ♠♣♠♥tt♦♥ ss♥ r s ♣♣ ♥ ♦t r ♥ ♥r rt♦♥

ss ♣♦♥ts r s ♥ t r rt♦♥ ss ♣♦♥ts r s ♥ ♥r rt♦♥ ♦

str♥ ♦r ♦♥♦r♠ ♥ss s♣ ♣rtr② ♦r ♥♦♥s♥r ♠♥t

♥ ♣t ss♦♥ s♠ s s ♦r♥ t♦ t rt st♥ t♥ t ♠♥t

♥ ♦♦t♦♥ ♣♦♥t t rs r s ♠♣r② t♦t ♥② rr♦r ♦♥tr♦ ♦rt♠

r r♦t

♣♣r♦s s t♦ r♣rs♥t ♥ tr t r ♣r♦♣t♦♥ ♥ ss ♥t♦ t♦

♠♥♦s t ♠♣t ♠t♦ ♥ t ①♣t ♠t♦ t②♣ ♣♣t♦♥ ♦ t ♦r♠r

♠t♦ ♦ t st ♠t♦ ❬❪ s ♦♣ ♥ t ❳ t♦ r♣rs♥t

♥ ♦ t s♦♥t♥t② ❬❪❬❪ st ♥t♦♥ s s♥ st♥ ♥t♦♥ t♦

t r sr ♥ ♦♥ t ♥r②♥ ♠s ♦ ♦♥sst♥t t t ♠s

srt③t♦♥ ♦ t ♣r♦♠ ♦r ♥♣♥♥t strtr ♠s ♥ t rs r ♦♣♥

srs ♦♥ ♠♦r st ♥t♦♥ s♦ ♥ ♣r♣♥r t♦ t r sr

s rqr ♥ ♦rr t♦ sr t r r♦♥t qt② t♦ r♣rs♥t t r sr

♣♥s ♦♥ t rs♦t♦♥ ♦ t ♥r②♥ ♠s rt② sr♥ t r sr

s② ♥tr♦s t♦♥ ♦♠♣tt♦♥ ①♣♥s ❬❪ t♦♥t②♣ qt♦♥s s♦

s♦ s♦ s t♦ ♣t t r r♦♥t ♥ t r ♦s ❬❪ ♥rss t

♦♠♣tt♦♥ ♦rt ♦♣♣ ♥ ♠r ❬❪ ♣r♦♣♦s t st ♠r♥ ♠t♦ t♦ ♣t

t r r♦♥t ♦t♦♥ ts tt♥ t ♣r♦ss ♦ ♣t♥ t r sr ❬❪ rs

♥ ②♦♥ ❬❪ ♣r♦♣♦s ♥ ♠♣t①♣t ♠t♦ ♥ t st r♣rs♥t r

s ①♣t② rt③ ② tr♥r ts sr ♥♦♦s ♣r♣♦s ♦ t t♦r

st ♠t♦ ❬❪ s ♠t♦s t ♥t ♦ t st r♣rs♥tt♦♥ ♦r t P❯

♥r♠♥t ♦♥ t♦ ♣t t r sr ② s♦♥ t qt♦♥s t♦♥②

sr♣ tr♥s ♥ ♥s ♥ rt♥ ② s ♦ ①♣t r srs rtr t♥ ♣r

sts

ttr ♠t♦ ss sts ♦ tr♥r ♦r qrtr ts t♦ srt③ t r sr

rt② ♦r ♥t ♠♥t s ♠t♦s t r ♦t♦♥ ♣r♦ss s s② ♦♠♣♥

t t♦♠t r♠s♥ ♦♣rt♦♥ ♦r ❳ ♣♣t♦♥s t ss♦♥ ♦ t

s♦ ♠♥ts ♥s t♦ ♣r♦r♠ ♦r t ♥trt♦♥ ♣r♣♦s ♦t r② ♦♥

♦♣ ♠s♥r♠s♥ ♣s ❬❪❬❪ ①♣t r♣rs♥tt♦♥ ♦ r srs

② tr♥t♦♥ s ♥ s ♥ ♠sr ♠t♦s s ❬❪ t s♦ ♥♦t tt ts

r♣rs♥tt♦♥ ♠t♦ s② rsts ♥ C0 r sr ♥ t r r♦♥ts r ♦♠♣♦s

♦ ♥ s♠♥ts s t♦ t st t♦ s♦rt♦♠♥s t r r♦♥t s ♥♦t ♣

tr ①t② ♥ ♦♠♣tt♦♥ s♣t ♥ ts ♥r② ♥ t ①trt♦♥ ♦ t

rtr ♣r♠trs ♦r ①♠♣ t s r♦♠ t ♥♠r s♦t♦♥ ♥ ♦♠tr② ♣♣r♦①

♠t♦♥ rr♦r ♠t t t r r♦t t ♦ r r♦♥t ♦♦r♥t

s②st♠ s ♥♦t ♥ ♥ t t♦♥ ♥ s s♦♥t♥♦s rst♥ ♥ t ♥♦♥♥q

r♥ ♥r♠♥t ♦r s♦♠ ♦ t♣ ♥tr ♦♥ t r r♦♥t ♥ss t r♥ ♥r♠♥t

s ♥♦♥ ❬❪ s r♠② t r r♦♥ts ♥ t♦ s♠♦♦t tr♦ s♦♠ ♥♠r

t♥qs ❬❪❬❪ ♠r s♥r♦ ♦rs ♥ r♥ s ♦r rtr ♠♦♥

ss Ps③♥② ♥ ❩♠♠r♠♥ ❬❪ ♣♦♥t ♦t tt r ♥♠rs ♦ ts r ♥ ♥ ♦rr

t♦ ♠♦r rt② r♣rs♥t t r sr ♥ t st♦r ♥rss r♣② t rs♣t t♦

t r ♦ t r sr ♥ r ♣r♦♣ts ♥ t② ♣r♦♣♦s t s ♦ ♣r♠t

r sr t ❯ ♣t t♦ sr t r ♣r♦♣t♦♥ ♥ tr ♣♣r♦ t

r r♦t s r③ ♦♠tr② ② ♦r♠♥ t ❯ sr tr♦ t ♠r♥

r ♦rt♠ ❬❪ t♦ ♠♦ t ♦♥tr♦ ♣♦♥ts t♦ t ♣r♠tr③t♦♥ ♦ t ❯

♣t t r t♣ ♥ s♠♣ ♥②r ♦♥ t r r♦♥t ts t st♦r ♦r r

r r r♦♥t ♣t♥ C(ξ) s t ♦ r r♦♥t r C′(ξ) s t ♥ r r♦♥tr tr trt♦♥

srt③t♦♥ ♥rss ♠② ♥ t ♦ r r♦♥t ♦♦r♥t s②st♠ s sts

♦♥ t s♠♦♦t ♦♠tr② ♦r ts ♠t♦ s ♦♥ r♠s♥ t ♥t ♠♥ts

♠s r♥rt♦♥ ♥s t♦ r② ♦♥tr♦ t♦ ♥sr t ♠s qt② ♥t② ♠t

t ♣r♦♣♦s ♥ ①♣t② r♣rs♥t ♦r♠♥s♦♥ ♦♠tr② trs ② ❯ ❬❪❬❪

tr♦ t ♣rtt♦♥ ♦ ♥t② ♣♣r♦①♠t♦♥ ♥st ♦ s♥ sts t ♦r♠♥s♦♥

trs s s rs r rt② sr tr♦ t t♦♥ ♦ t st♥ ♥

♥ ♥t ♥♦♥trt ② ♣r♦♥ ♣r♦♠s♥ tr♥t t♦ ♦ s♦♥t♥t② ♥ t

r♠♦r ♦r ♠♦r st ♥♠r qrtr s♠ s sr ♥ ♦rr

t♦ ② ①♣♦t t ①t r♣rs♥tt♦♥ ♥ ♦♠tr②

♥ t ♣rs♥t ♦r s ❯ ♣ts t♦ srt③ t r srs r r♦♥t

s ①t② sr ♥ t ♦ r t♣ s②st♠ s ♥ ♥tr② ♥ ♥q② s ♦♥

t ❯ ♣t ♥ t ♣②s q♥tts r s♦ ♣♣r♦①♠t ② t ❯

ss ♥ t s♣rt ♦ s♦♦♠tr ♥②ss ♦♠♥♥ t t s♠♦♦t♥ss ♥ ♦♠tr②

♥ strss s♦t♦♥ s ② ①♣♦t t♦ t t rtr ♣r♠trs ♥ ♦ t r

♥ st ♠♥♥r

r sr ♣t♥ ♦rt♠

r ♣r♦♣t♦♥ s r③ ♦♠tr② ② ♥♥ t r r♦♥t s♦ tt t ♥ r

r♦♥t r C′(ξ) s ♣ss tr♦ t ♥ ♣♦st♦♥s ♦ t s♠♣ ♣♦♥ts ♦♥ t ♦ r r♦♥t

r C(ξ) s ♣r♠tr③ ② t ♥♦t t♦r Ξ = ξ1, ξ2, ..., ξn+p+1 n s t ♥♠r

♦ ss ♥t♦♥s ❲ ♥ t s♠♣ ♣♦♥ts ♦♥ C(ξ) t♦ Mj = M(ξj), j = 0, 1, ..., N − 1

♥ t st ♦ ♦rrs♣♦♥♥ ♥ ♣♦st♦♥s t♦ M ′j N ♥ ♣♦st♦♥s ♦ s♠♣ ♣♦♥ts

r sr s t ♦♥str♥ts r♥ t ♦r♠t♦♥ ♣r♦ss r♦♠ C(ξ) t♦ C′(ξ) ♥ st

N = n r ❲ ♦♣t t ♦rt♠ sr ♥ ❬❪ ♥ ts ♦rt♠ t ♦r♠t♦♥ ♦

r ♥r ♠t♣ ♦♥str♥ts s trt♦♥ ♣r♦ss ♦r tt trt♥ st♣ ♥ t

rr♦r t♦r s

ej,t =−−−−→Mj,tM

′j .

‖et‖ < tol t trt♦♥ ss ♥ t ♥ r r♦♥t r s ♦t♥ tol = 1.e − 4 ♥

ts ♦r

♦ ♣t t ♦♥tr♦ ♣♦♥ts Pi, i = 0, 1, ..., n− 1 ♥ ♠♦♠♥t t♦r m s tt ♥

tt trt♥ st♣

Pi,t = Pi,t−1 +mi,t

♠♦♠♥t t♦r mt ♥ ♦♠♣t s

mi,t =1

N

N−1∑

j=0

fijej,t−1,

r fj = f(ξj) r t ♥♥ ♥t♦♥s ♦rrs♣♦♥♥ t♦ ♦♥str♥t M ′j ❲ ♦♦s

t ♥♥ ♥t♦♥s t♦ t ❯ ss ♥t♦♥s r s t♦ sr t r

fj = Rj ♦ ♠ sr t ♥♥ ♥t♦♥s fij = fi(ξj) ss♦t t ♦♥str♥t

M ′j r ♥r② ♥♣♥♥t s♦ tt t ♦♥str♥t s t t♦ t ♦r♠t♦♥ ♦ t r

t ♣r♠tr ♦♦r♥t ξj ♦ Mj s♦ sts② ξj ∈ [ξi, ξi+p+1] s s t r

sss t♦ ♥rt t s♠♣ ♣♦♥ts

♥② t rr♦r t♦r s t ♥ rrs ②

ej,t = ej,t−1 −1

N

N−1∑

k=0

〈Rj , fk〉ek,t−1

ts ♦r ♣t♥ t r r♦♥t s ♥ ♥ ♦rt♠ ♥ t ♥ r r♦♥t

r s ♦t♥ t ♥ r srs ♥ ♥rt ② ♦t♥ ♦♥ t r ①t♥s♦♥

rt♦♥ r♦♠ t ♦ r t♦ t ♥ r ♥rt r srs s ♠r

♥t♦ t ♦ r srs t C0 ♦♥t ♦r C1 ♦♥t ♥ ts ♦r C0 ♠r s ♦♣t

♦rt♠ r r♦♥t ♣t♥ ♦rt♠

t ♦ r r♦♥t r C(ξ) s♠♣ ♣♦♥ts Mj ♥ ♣♦st♦♥s ♦ s♠♣ ♣♦♥ts M ′j

st ♥ r r♦♥t r tt ♣sss tr♦ M ′j

t = 0tol = 1.e− 4

ej,0 =−−−−−→Mj,0M

′j

‖et‖ > tol ♦t = t+ 1mi,t =

1N

∑N−1j=0 fijej,t−1

Pi,t = Pi,t−1 +mi,t

ej,t = ej,t−1 − 1N

∑N−1k=0 〈Rj , fk〉ek,t−1

♥

♦♠♣tt♦♥ ♦ strss ♥t♥st② t♦rs

r♥ ♦r ♦r t ♦t♦♥ ♦ t rtr s rtr③ ② s♦♠ rtr ♣

r♠trs s s t strss ♥t♥st② t♦rs s ♥ ①trt r♦♠ t ♥♠r

s♦t♦♥ t rtr ♣r♠trs r ♦♠♣t s ♦♥ t ♣♦♥ts t♣s ♦♥ t r

r♦♥t ♥♣♥♥t② t ♥ rr s ♦ ♣♣r♦ ② t♦r t♦ ♦♠♣t

rt② t s ♥ t ♦ ♣♣r♦ s t♦ ♦ t srt③t♦♥ ♥ ♣t ♣♥♥ s

♠ s ♣♦ss ❱r♦s ♠t♦s ♥ ♦♣ t♦ ①trt t s ♥ t r♠♦r

♦ ♥ s♣♠♥t ♦rrt♦♥ ♠t♦ ❬❪ t ♦r t♦t t rt♣

s♥r ♠♥t s s♠♣ ♥ st ② ♦r ts ♠ rtss ♣t ♣♥♥ ♦

ts ♠t♦ ♥ ♥♦t ♥t ♥ ♥ ①tr♣♦t♦♥ t♥q s ♣r♦r♠ ♣♦♥ r♦♣ ♦

t♦♥s rt r ①t♥s♦♥ ♠t♦ ❱ ❬❪❬❪ s ♣♣ t♦ ♦♠♣t s

s ♦♥ t t♦♥ ♦ t rs str♥ ♥r② ♣r rt r ①t♥s♦♥ ♦r♥

❱ rs ♦♥ t ♦♥strt♦♥ ♦ strtr ♠s ♦♥ t r r♦♥t ♥rss t

♠s r♥ ♦r t s♦ ♥♦t tt t rt♦♥ ♦r♠ ♦ t str♥ ♥r②

♥♦s t ♥r② rs rt ♥ t r ①t♥s♦♥ s ♥ ♣♣ ♦r t♦♠t r

r♦t ❬❪❬❪ r t r ①t♥s♦♥ s ss♥ t ♣②s ♥tr♣rtt♦♥ s ♠t♦

♠♥♠③ t str♥ ♥r② ♥ ♦ s♥s ts ♥ t♦ s♥♥t r♥ r♦♠ ♦

♣♣r♦ ♥ s r♥t② ♥ ♥stt ♥ t r♠♦r ♦ ❳ ❬❪

rt r ♦sr ♥tr ❱ ♠t♦ s ♣r♦♣♦s s ♦♥ t rt r ①

t♥s♦♥ s ♥♦tr tr♥t t♦ ①trt s ♥ ♥r st rtr t♦ t s♠♣t②

♥ r② t ❱ s ♥ ② s ♥ ♥ ❬❪ ❲ t s♦ ♥♦t

r r t♣ ♦♦r♥t s②st♠

tt ts ♠t♦ rqrs t ♠♥t ♥r t r r♦♥t ♥ ♥ ♦♥sst♥ t t ♥♦r

♠ rt♦♥ ♦ t r r♦♥t ♣t♥♣♥♥t J ♥tr ♣r♦♣♦s ② ❬❪ s ♥

ttrt ♠t♦ t♦ ts r♦st♥ss rr♥ t rt ♥♣♥♥ ♥ srt③t♦♥

♥ ♣t ♦ ♥tr ♠t♦ s♦♦♥ s ①t♥ ♥t♦ ♠♥② r♥s ♥ ♦♦

rsts ♥ ♦t ♥ ❬❪❬❪❬❪❬❪❬❪ ♦♥t♦r J ♥tr s s t♦ st ♥t♦

t q♥t ♦♠♥ ♥tr ♦r♠ ♥ s t ♥♦ strss s ♥♦t strt♦rr s♦t♦♥

♥ rqr♠♥t ♥ ♠s srt③t♦♥ s r① ❲ ♥ t ♦♥t♦r ♥t♦♥ ♥

♦♣t rt② ❬❪ ♥ ♦rr t♦ ①trt ♠① ♠♦ s r♥t t♥qs r ♦♣

Jx ♥trs x = 1, 2, 3 s t ♦♠♣♦♥♥ts ♦ J ♥tr ♥ rt② s t♦ t

t s ♦r t t♦♥ ♦ J2 ♥ J3 ♦r GIII ①ts ♥♠r t② t♦

t s♥rt② ❬❪ J1 ♥tr ♦r J ♥tr ♥ s♦ ♥ s t♦ ①trt ♠① ♠♦

s t s♦♠ ①r② ♦♣rt♦♥ ♥ ② s t♦ ♦♠♣♦s t s♣♠♥t ♥ strss

s ♥t♦ s②♠♠tr ♥ ♥ts②♠♠tr ♣♦rt♦♥s t strtr ♠s ♦♥ t r r♦♥t

t♥ tr ♠♦s ♦ J ♥tr ♥ t rt② ❬❪❬❪❬❪❬❪ ♦tr ♠t♦

♥♠ M ♥tr ♦r ♥trt♦♥ ♥r② ♥tr s ♦♣ ② ♥tr♦♥ s②♠♣t♦t

s s t ①r② s♦t♦♥ ❬❪ s ♥ ①t♥ ♥ ❳ ❬❪❬❪ ♥ ❬❪

♦t t ❱ ♥ ♦♥t♦r M ♥tr ♥ ♥stt ♦r t t♦♥ ♦ s ♥

t rtr ♥②ss s♦♦♠tr ♣♦♥ts r t♣ ♦♦r♥t s②st♠ s

sts ♦♥ t r r♦♥t s ♥ r ♣②s q♥tts r ♥ t r t♣

♦ ♦♦r♥t s②st♠ ts t s♣rsr♣t ❵♦ s ♦♠tt ♥ ts st♦♥

♦♥t♦r M ♥tr

♥t♦♥ ♦ Jk ♥tr st♠s r♦♠ t♦ ♠♥s♦♥s s

Jk := limΓǫ→0

∫

Γǫ

(Wδjk − σijui,k)njΓ = limΓǫ→0

∫

Γǫ

PkjnjΓ, k = 1, 2

r Pkj s t s② t♥s♦r W = 1/2σijǫij s t str♥ ♥r② ♥st② Γǫ s s♠

♦♥t♦r t rs R ♥tr t r t♣ ❵o ♥ t ❵xo − yo ♣♥ ♥ nj s t ♥t ♦tr

♥♦r♠ ♦ Γǫ

t ♥ ①t♥ t♦ tr ♠♥s♦♥ ♣♦♥ts ♥t♦♥ ② t♥ tr sr r♦♥

t r r♦♥t ❲♥ t ♦♥t♦r Γǫ s s♠ ♥♦ t ♣♥ str♥ ♦♥t♦♥ s ♣♣r♦①

♠t② sts ❲ ♦ s t ♦♥t♦r ♥t♦♥ rt② ♦♥ t ♣r♠s tt s♠

♦♥t♦r s ss♠

t s ♥♦♥ tt t J ♥tr J1 ♥ t s t rt♦♥s♣

J = GI +GII +GIII =1− ν2

EK2

I +1− ν2

EK2

II +1

2µK2

III ,

r Gi ♥ Ki i = I, II, III r t ♥r② rs rts ♥ s ♦r t tr ♠♦s ♦

rtr

② ♣♣②♥ t J ♥tr ♥r t♦ stts ♦♥ t r stt ♥♦t t s♣rsr♣t ❵

t ♦tr t ①r② stt s♣rsr♣t ❵ t♥ ♥ t♠ t♦tr t ♠① tr♠ M

♥ ♦t♥

J (1+2) =

∫

Γǫ

[

0.5(σ(1)ij + σ

(2)ij )(ǫ

(1)ij + ǫ

(2)ij )δ1j − (σ

(1)ij + σ

(2)ij )

∂(u(1)i + u

(2)i )

∂x1

]

njΓ

rr♥♥ t t♦ stt tr♠s s

J (1+2) = J (1) + J (2) +M (1,2)

r

M (1,2) =

∫

Γǫ

[

W (1,2)δ1j − σ(1)ij

∂u(2)i

∂x1− σ

(2)ij

∂u(1)i

∂x1

]

njΓ

W (1,2) = σ(1)ij ǫ

(2)ij = σ

(2)ij ǫ

(1)ij

♦♠♥ t qt♦♥ t ♦♦♥ rt♦♥s♣ ♥ ♦t♥ ♦r t M ♥tr

M (1,2) =2(1− ν2)

E(K

(1)I K

(2)I +K

(1)II K

(2)II ) +

1

µK

(1)IIIK

(2)III .

♥ s ♥ ①trt rs♣t② ♦r ①♠♣ t stt t ♣r ♠♦ III s②♠♣

t♦t s t K(2)I = 0 K

(2)II = 0 K

(2)III = 1 ♥ KIII ♥ r stt 1 ♥ ♦♥ s

K(1)III = µM (1, ♠♦ III)

KI ♥ KII ♥ ♥ ♥ s♠r s♦♥ r t rst ♦rr s②♠♣t♦t s♣♠♥t ♥

strss s♦t♦♥s s ♣♣♥① r st s t ①r② s

❱rt r ♦sr ♥tr

♥ t ❱ t str♥ ♥r② rs rt s q t♦ t ♦r ♦♥ ② ♦s♥ t rt

r ①t♥s♦♥ r ♠♦s ♦ t str♥ ♥r② rs rt r ♥ ②

GI =1

2R

∫ R

0σyy(x)Juy(R− x)Kx,

GII =1

2R

∫ R

0σxy(x)Jux(R− x)Kx,

GIII =1

2R

∫ R

0σyz(x)Juz(R− x)Kx,

r OP ′ = R s t rt r ♥ ♦r t t♦♥ ♦ Juj(R − x)K ♦♥ PO t

♣♦♥t ♥rs♦♥ ♦rt♠ ♥s t♦ ♣r♦r♠ ♥ ♦rr t♦ ♥ t ♣r♠tr ♦♦r♥t ♥

t r ♠♦ ② ❯ sr ❬❪ ♦♠♥s ♦ ts ♥trs OP ′ ♥ PO r

rt③ ② s♥ ♥r ♠♥t ❬❪ ♥ t R s ♥t ♦r t r t♣s ♥ KI

KII ♥ KIII ♥ ♦♠♣t ♦r♥ t♦ qt♦♥

Prs

Prss s ♥ s t♦ sr t st② stt r r♦t ♥ t t

r ♣r♦ss ❲ s t ♦r♥ Prs s ♦♦s

a

N= C(∆K)m,

r N ♥♦ts t ♥♠r ♦ ♦ ②s C ♥ m r t ♠tr ♣r♠trs ∆K s t

r♥ ♦r ♠① ♠♦ r t K s t♥ s t q♥t Keq s ♥ s

❬❪

Keq =√

K2I +K2

II + (1 + ν)K2III

❲ s♣② t ♠①♠♠ r ♥ ∆amax s♦ tt t ♥♠r ♦ ♦ ②s ♥

♦t♥ ♦r♥ t♦ t t s♦ ♥♦t tt t r ♣r♦♣t♦♥ ♦t② ♦

r ♦r t r t♣s ♦♥ t r♦♥t ♥ s♥ ♣r♦♣t♦♥ st♣ t r ♥ ♦r

r t♣ s rr③ ② t ∆amax

∆ai = C(∆Kieq)

m ∆amax

C(∆Kmaxeq )

= ∆amax( ∆Ki

eq

∆Kmaxeq

)m.

♠①♠♠ ♦♦♣ strss rtr♦♥ s s t♦ tr♠♥ t rt♦♥ ♦ r ♣r♦♣t♦♥

❲ ss♠ tt t r ♣r♦♣ts ♥ t rt♦♥ θc s tt t ♦♦♣ strss s ♠①♠♠

s ♥ ② t ♦♦♥ ①♣rss♦♥ ❬❪

θc = 2rt♥

[

−2(KII/KI)

1 +√

1 + 8(KII/KI)2

]

.

♠r ①♠♣s

♥ ts st♦♥ ♥♠r ①♠♣s ♦t t ♣♥♥②s♣ r ♥ ♣t r

♥ ♠♦ ♥ ♥♥t ♦♠♥ t t qt♦♥ ♥ ♥ ♥t ♦♠♥ t t

qt♦♥s ♥ ♦♥r♥ ♥ s ♦♠♣r ♥ t ♥♠r s r

t ② ♦t ❱ ♥ M ♥tr ♥ t r r♦t ♦rt♠ s

❨♦♥s ♠♦s E = 1000 ♥ P♦ss♦♥s rt♦ ν = 0.3 ♦r t ss

P♥♥②s♣ r

♣♣♦s ♣♥♥②s♣ r s st t♦ t r♠♦t t♥s♦♥ σ0 t∞ = (0, 0, σ0)

rs ♦ r s a ♥♥t♦♥ ♥ s ϕ ♥ rr ♥ θ s ♥ ♥ t r ♣♥

Oxy s ♥ r ♥②t s♦t♦♥ ♦ s rs

KI =2

πσ0

√aπ♦s2ϕ,

KII =4

π(2− ν)σ0

√aπ♦sϕs♥ϕ♦sθ,

KIII =4(1− ν)

π(2− ν)σ0

√aπ♦sϕs♥ϕs♥θ.

r ♦♠tr② ♦r ♣♥♥②s♣ r a = b ♥ ♣t r a 6= b

♥ ♣rtr ♥ t r ♣♥ s ♦r③♦♥t ϕ = 0 t ♥②t ♥♦r♠ s♣♠♥t

s ♥ s

uz(r, θ, 0) =2(1− ν)σ0

πµ

√

a2 − r2, r 6 a.

♥r ♥trt♦♥ tst

♣r♦♠ s ♠♦ ② qt♦♥ s♦ tt ♦♥② s♥ ❯ ♣t s s t♦

r♣rs♥t t r ♥ t ♥♠r ♦♠♣r t t ♥②t s♦t♦♥

♦♦t♦♥ ♣♦♥ts r ♠♦ s r♦♠ t ♣♦ ♥ ♦rr ♥♦t t♦ ♦t t t ♥rt ♣♦♥t

❯ ss ♥t♦♥s ss♦t t t ♣♦ ♦r r ♥♦r t♦ C0 tr♦ t

♦rrs♣♦♥♥ ♦♥tr♦ ♣♦♥ts sr♥ t s♠ rs ♦ r♦♠ s r♦♠ ts ♠♦

♦♦t♦♥ ♣♦♥ts r ♠r t♦ ♦♥ qt♦♥

❲ ♥♦t tt t s♦t♦♥ ♦♥② rs ♥ r rt♦♥ ♣♥ t s♠ ♥ ♥r

rt♦♥ ts ♠♥ts r s ♥ ♥r rt♦♥ s t♦ s♣t rt♦ ♦

♠♥t t t r♥♠♥t ♥ r rt♦♥ r ♦♠♣rs t L2 ♥♦r♠ rr♦r

♥ ♦r ϕ = 0 ❵♥♣❴s ♥♦ts t ♥♠r ♦ ss ♣♦♥ts ♥ ♥r rt♦♥ ♥

str♥ ♦r♥ ♠♥s rt s ♦ t ♠t♦ ♥ ♠♣r♦ ♥♦ts t

t ♦♥♦r♠ ♠♣♣♥ t ♥ ♦sr tt

• ♥ ♥♣❴s= 30 t ♦r♥ ♥ ♠♣r♦ t ♦♠♣r rr♦r ♦r

t rr♦r r♦♠ ♦r♥ s ♥♦♥♥♦r♠② strt st t ♠♣r♦ ts

♠♦r ♥♦r♠ rr♦r strt♦♥

• ♥ ♥♣❴s= 18 t rr♦r r♦♠ ♦r♥ ts r eL2=

♠♣r♦ ♣s t s♠ r② s t r♦♠ ♥♣❴s= 30

• t rr♦r ♦♠s ♥r t r r♦♥t s s t♦ t r t♣ s♥rt②

s ♦♥ tt ♦r♥ ♥ ♠♦r ss ♣♦♥ts ♥ ♦rr t♦ t rs♦♥

r② ♠♦ t ♥♦t η = 0.875 ♥①t t♦ t r r♦♥t ♥ r rt♦♥ ♦sr t♦

t r r♦♥t η = 0.94 ♥ r♣t ♦♠♣rs♦♥ s ♥ r ❲ ♥ tt ♥ ♥♣❴s= 30

♦r♥ st s rr♦r t ♠♣r♦ ♠t♦ s♦s r r② t♥

♦r ❲ ♥ rr tt t♦ t r t♣ s♥rt② r♥ ♠s ♥r t r r♦♥t

s♦ ttr r② ♥ t t ♦r♥ s s♥st t♦ t ♠♥t st♦rt♦♥

♥ s r rsts ♠♣r♦ ♣rs♥ts r♦st ♣♣t♦♥ ♦r ts ♥ ♦

♠s ♦♥rt♦♥

♦r♥ ♥♣❴s eL2= ♠♣r♦ ♥♣❴s eL2

=

♦r♥ ♥♣❴s eL2= ♠♣r♦ ♥♣❴s eL2

r rr♦r ♥ r ♦♣♥♥ s♣♠♥t ♦r ♣♥♥② r ❵♥♣❴s ♥♦ts t ♥♠r♦ ss ♣♦♥ts ♥ ♥r rt♦♥ ♥ str♥ ♥♦t t♦rs ♥r rt♦♥ξ❬❪ r rt♦♥ η❬❪

♦r♥ ♥♣❴s eL2 ♠♣r♦ ♥♣❴s eL2

♦r♥ ♥♣❴s eL2 ♠♣r♦ ♥♣❴s eL2

r rr♦r ♥ r ♦♣♥♥ s♣♠♥t ♦r ♣♥♥② r ♥♦t t♦rs ♥r rt♦♥ξ❬❪ r rt♦♥ η❬❪

r ❯p = q = 2 r♣rs♥t r sr ♠ss t ♥ ♥♦r♠r♥♠♥t ♥ r rt♦♥ ♦♦ ② r r♥ ♠♥ts t s ♦s t♦r r♦♥t ♦ts r ♦♦t♦♥ ♣♦♥ts

♦♥r♥ tst

❯♥♦r♠ ♠s r♥♠♥t ♥ t ♣r♠tr s♣ s ♣r♦r♠ ♥ t t ♠♥t

s③ h s h =√

Smaxe r Smax

e ♥♦ts t ♠①♠♠ r ♦ t ♠♥t ♦♥r♥

r s ♣♦tt ♥ r r ♦♠♣r ♦t t qrt ♥ ❯ ss

♥t♦♥s t ♥ ♦♥ tt t r t♦♥ ♣s t♦ ♠♣r♦ t r② t

t ♦rr ♦ ♦♥r♥ rt oc ♣s ♠♦st t s♠ oc = 1 tr♦t oc s

t♦ t ♣②s s♥rt② ♦♥ t r r♦♥t

s stt ♥ t ♦ st♦♥ t ♥♦r♠ r♥♠♥t s ♥♦t ♥ ♥t ② t♦ ♠♣r♦ t

r② ♦r ♣♥♥② r s ♠s ♦♥rt♦♥s r s♥ tr♦ ♣♥ t

♠♥t ♥♠r ♥ ♥r rt♦♥ t ♠s s ♥♦r♠② r♥ t t ♠♥t ♥♠r

♥ ♥ r rt♦♥ t♥ t ♠♥t t r r♦♥t s rtr r② r♥

♦♥st ♥♦t ♥srt♦♥ ♥ ♦rr t♦ r t rr♦r s ② r t♣ s♥rt②

r s♦s ♠s ♥ r ♣♦ts t rst ♦r ♦♥r♥ st② t ♥

s♥ tt t r② s ♠♣r♦ ♠♦st ② ♦♥ ♦rr ♥ t ♥ st♠t ♦♥r♥ rt

s t♦ t♠s r t♥ t ♥♦r♠ r♥♠♥t s ♥ts t ♥② ♦ ♥

t ♣♣t♦♥ ♦ rtr s♠t♦♥

trss ♥t♥st② t♦r tst

♥ ts sst♦♥ t ♦♠♣tt♦♥ ♦ s s ♥st ♦ s♥ qt♦♥ t♦ ♠♦

t ♣♥♥②s♣ r ♥ t ♥♥t ♦♠♥ ♣t t♦ ♦r♣♣ r srs ♥

10−1

100

10−2

10−1

100

h

L2

no

rm e

rro

r in

CO

D

p=2

p=3

1

0.99

1

1.00

r L2 ♥♦r♠ rr♦r ♦ ♦r ♣♥♥②s♣ r

102

103

10−3

10−2

10−1

100

Degrees of freedom

L2

no

rm e

rro

r in

CO

D

p=2, uniform

p=3, uniform

p=2, graded

0.60

1.41

1

1

0.59

1

r L2 ♥♦r♠ rr♦r ♦ ♦r ♣♥♥②s♣ r

0.02 0.03 0.04 0.05 0.06 0.07 0.08−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

distancegtogcrackgfrontg(R/a)

err

or

VCCI

Mgintegral

r Pt ♥♣♥♥ ♦r ❱ ♥ M ♥tr

t s③ L = 200a s tt ♦ ♦♠♣r t ♥♠r s t t ♥②t s♦t♦♥

♦r ♥♥t ♦♠♥ qt♦♥s r s ♦r ts s

r ♥stts t ♣t ♥♣♥♥ ♦ t ♥tr ♥ ❱ ♦r ♠♦ I ♣♥♥②

s♣ r r ❵R ♥♦ts t rt r ♥ ♥ ❱ ♥ t rs ♦ t ♦♥t♦r

♥ M ♥tr t ♥ s♥ tt ♥ R/a s r♦♠ 0.02 t♦ 0.08 ♦t ♠t♦s s♦ ♣t

♣♥♥t ♦r ♦r M ♥tr t rr♦r rs t♥ ❲♥ t rs ♦ ♦♥t♦r s

s♠ KI ♦♥rs t♦ ♥②t ♥rs♥ R s♥ t strss ♦r t r

t♣ s ♥♥ ② ♦tr t♣s ♥ t r r♦♥t ♣♥ str♥ ♦♥t♦♥ s ♥♦t sts ♣r♦♣r②

t ♠t♦ ♦♠s ♥rt ♦r ❱ t rr♦r rs t♥ ♥ ♥r② s♠

rt r ♥ s ♥ ♦r R s t♦♦ s♠ t② ♥ ♥♠r t♦♥

♦ strss ♥ ♦s t♦ r r♦♥t rs t♦ t ♥r② ♦ KI r♦♠ t

r ♥ s♦ rr tt M ♥tr ♣rs♥ts s♠r rt♦♥ ♥ rr♦r t♥ ❱

r ♦♠♣rs t s ♦t♥ r♦♠ M ♥tr t R = 0.02a ♥ ❱ t R =

0.04a ♦r t ♠① ♠♦ ♣♥♥②s♣ r t ♥♥t♦♥ ♥ ϕ = π/6 t s s♥ tt

♦t ♠t♦s r t t ♥②t s♦t♦♥ KIII r♦♠ M ♥tr s♦s t♦♥

♥r θ = π/2 ♥ 3π/2 ♣rs♥ts t rr♦r t θ = 0 π/4 ♥ π/2 t ♥ ♦sr

tt t rr♦r ♦ KI ♥ KII s t♥ ② ♦t ♠t♦s t♥ ♦r KIII ② M

♥tr ♥ ♦♥ tt t ♥ ♣r♦ rt s ♥ t ♥♠r

0 1 2 3 4 5 6

−0.4

−0.2

0

0.2

0.4

0.6

θ (rad)

SIF

s

KI

KIII

KII

Analytical

M integral

VCCI

r trss ♥t♥st② t♦rs ♦r ♣♥♥② r t ϕ = π/6

s ♦♥ r r♦♥t s qt s♠♦♦t t♦ t ♦♥② ♠♥ts ♥ ♥r rt♦♥ ♥

t♦t ♥② s♠♦♦t♥ss ♦♣rt♦♥ s s t ♣r♠s ♦r st ♦t♦♥ ♦r t r

r♦t s♠t♦♥

KII KIII

❱ M ♥tr ❱ M ♥tr

θ = 0

θ = π/4

θ = π/2

rr♦r ♦ s ♦r ♣♥♥②s♣ r t ϕ = π/6

♣t r

♣♣♦s ♥ ♣t r s st t♦ t r♠♦t t♥s ♦♥ σ0 ♥ t ♥♦r♠ rt♦♥

t∞ = (0, 0, σ0) s♠♠♦r ①s s a s♠♠♥♦r ①s b ♥♥t♦♥ ♥ s ϕ

♥ ♣t ♥ θ s ♥ ♥ t r ♣♥ s ♥ r ♥②t s♦t♦♥ ♦

s rs

KI =σ02(1 + ♦s2ϕ)

√bπf(θ)

E(k),

KII =σ02s♥2ϕ

√bπk2(b/a)♦sθ

f(θ)B(k),

KIII =σ02s♥2ϕ

√bπk2(1− ν)s♥θ

f(θ)B(k),

k2 = 1− b2

a2,

f(θ) = (s♥2θ +b2

a2♦s2θ)1/4,

B(k) = (k2 − ν)E(k) + νb2

a2K(k),

r K(k) ♥ E(k) r ♣t ♥trs ♦ t rst ♥ ♥ s♦♥ ♥ rs♣t②

K(k) =

∫ π/2

0

1√

1− k2s♥2θθ,

E(k) =

∫ π/2

0

√

1− k2s♥2θθ.

♥ ♣rtr ♥ ϕ = 0 t s♣♠♥t ♥ t r ♥♦r♠ rt♦♥ rs

uz(x, y, 0) =2(1− ν)σ0

µ

b

E(k)

√

1− x2

a2− y2

b2.

r♥ ♦ t ♣t r ♥ ♣♥♥② r s tt t ♠♦ s ♥♦t ♦♥st♥t

t♦ t rt♦♥ ♦ rtr ♦♥ t r r♦♥t ♣r♦♠ s ♠♦ ②

qt♦♥ ♥ ♠s ♦♥rt♦♥ ♥ ♦♦t♦♥ s ♥♦♦s t♦ ♣♥♥②s♣ r

t♣ ♥ ♥♠r s♣t s ♦r ♣t r t ♠♥ts ♠♥t s♣t rt♦ s

s ♥♦♥♦rt♦♦♥ ss t♦rs r s♦s tt ♦r♥ ♣rs♥ts rr♦♥♦s

rst t ss ♣♦♥ts ♥ ♥r rt♦♥ ❲ t ♠♣r♦ s rs♦♥

♥ rr♦r strt♦♥

♦r t ♦♥r♥ st② rst t rst ♦ ♥♦r♠ r♥♠♥t ♥ ♣r♠tr s♣

♥ r ♥ t s♠ r ♠s ♦♥rt♦♥s ♦r ♣t r r ♥rt s

♦♥ ♦r ♣♥♥② r s ♥ r r ♦♠♣rs t rst t♥ ♥♦r♠ ♠s ♥

r ♠s ♦♥r♥ tr s ♠♦st t s♠ s tt ♦ ♣♥♥② r ♥ ♥

♦♥ tt t s♦ sts ♦r ♠♦♥ ♣t r

♦r t tst ♦ s ♦♠♣tt♦♥ ♣t t♦ ♦r♣♣ r srs ♥ t s③

♦r♥ ♥♣❴s eL2 ♠♣r♦ ♥♣❴s eL2

r rr♦r ♥ r ♦♣♥♥ s♣♠♥t ♦r ♣t r ♥♦t t♦rs ♥r rt♦♥ ξ❬❪ r rt♦♥ η❬❪

r ❯p = q = 2 r♣rs♥t r sr ♠ss t ♥ ♥♦r♠r♥♠♥t ♥ r rt♦♥ ♦♦ ② r r♥ ♠♥ts t s ♦s t♦r r♦♥t ♦ts r ♦♦t♦♥ ♣♦♥ts

10−1

10−2

10−1

100

h

L2

no

rm e

rro

r in

CO

D

p=2

p=3

1

0.96

1

0.98

r L2 ♥♦r♠ rr♦r ♦ ♦r ♣t r

102

103

10−3

10−2

10−1

100

Degrees of freedom

L2

no

rm e

rro

r in

CO

D

p=2, uniform

p=3, uniform

p=2, graded

0.58

0.58

1.28

1

1

1

r L2 ♥♦r♠ rr♦r ♦ ♦r ♣t r

L = 200a s tt ♦ ♦♠♣r t ♥♠r s t t ♥②t s♦t♦♥ ♦r

♥♥t ♦♠♥ qt♦♥s r s r ♦♠♣rs t s ♦t♥ r♦♠ M

♥tr t R = 0.02b ♥ ❱ t R = 0.02b ♦r t ♠① ♠♦ ♣t r t

♥♥t♦♥ ♥ ϕ = π/6 ♣rs♥ts t rr♦r t θ = 0 ♥ π/2 ♦r t ♥ tr

♠♦s t ♥ s♥ tt t rr♦r ♦r t s s t♥ ♥ t s ♦♥ t r

r♦♥t s s♠♦♦t ❲ ♥♦t tt t s r② ♦ ♣t r s t ♦rs t♥ ♣♥♥②

r s t♦ t r rtr ♦♥ t r r♦♥t ♥ t ① R s s t

s♥rt② t t s♠♣ ♣♦♥ts ♥r t s♠♠♦r ♥ s♠♠♥♦r ①s ♦ r♥t

t♦ ♥r② ♥ s t♦♥ ♦r st ② t♦ st♠t t s ♦r

♣t r ♦ ♦♥ ♦ t tr ♦r

KI KII KIII

❱ M ♥tr ❱ M ♥tr ❱ M ♥tr

θ = 0

θ = π/2

rr♦r ♦ s ♦r ♣t r t ϕ = π/6

t r r♦t

♥ ts st♦♥ t r sr ♣t♥ ♦rt♠ s tst ♦♠♥ t t Prs

❲ st t r r♦t ♦ t ♦r③♦♥t ♣♥♥② r ♥r t ♥♦r♠ t♥s♦♥ r♦♠

0 1 2 3 4 5 6−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

θ (rad)

SIF

s

KI

KIII

KII

Analytical

M integral

VCCI

r trss ♥t♥st② t♦rs ♦r ♣t r t ϕ = π/6

st♦♥ t ♣r♠trs m = 2.1 ♥ t s♣ ∆amax = 0.2a ❲ ♣r♦♣t

st♣s ♥ ♠ ♦♠♣rs♦♥ t t ①t rst ♥ t rst r♦♠ t ❳

st ♠t♦ ❬❪ s ♥ r ♥ t ♥ ♦sr tt t r r♦♥t ♦r

st♣ rs t ①t s♦t♦♥ ② t r r♦♥t ts r② r♦♠

t ①t s♦t♦♥ t t r r♦t ② ❳ st ♠t♦ t♦ t t tt t

st ♠t♦ s rstrt ♥ sr♥ t r r♦♥t ①t② ♥ ts ♥r②

♠t st♣ ② st♣ ❲ t♥ ♦♠♣t t r ♣r♦♣t♦♥ ♦r m = 5 ♥ t rst

s ♣rs♥t ♥ r ❲ ♥ tt t ♥♠r r r♦♥t st rs t t

①t r♦♥t t♦ t ♥① s s♣♣♦s t♦ ①rt t rr♦r ♦ ♦ts ♦

t s♠♣ ♣♦♥ts s tst s♦s t ♣r♦♣♦s r ♣r♦♣t♦♥ s♠ ♦♥s t t② t♦

♦ t r ♥ st ♠♥♥r t♥s t♦ t s♠♦♦t♥ss ♥ t ♥♠r strss ♥ s

s♦t♦♥ ♥ ①t r♣rs♥tt♦♥ ♥ r ♦t♦♥ ♥② s♠t t r r♦t ♦r

♣t r t ♥♥t♦♥ ♥ ϕ = π/6 ♠♦ ② t qt♦♥s ♥ ♥t ♦♠♥

t♥ r♦♠ st♦♥ r strts t ♥ t ♦ t ♣r♦♣t♦♥ st♣

♦♥s♦♥s

♦r♠t♦♥ ♥ ♠♣♠♥tt♦♥ ♦ s♦♦♠tr ♦♥r② ♠♥t ♠t♦s

♦r s♠t♥ t rtr ♣r♦♠ r ♦t♥ ♥ ts ♣♣r s♠ ❯ ss

♥t♦♥s r s ♦r t srt③t♦♥ ♦ ♦♠tr②r ♥ t ♣♣r♦①♠t♦♥ ♦ s♣

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

y

Exact

IGABEM

m = 2.1 ❳ m = 2.1 ♠r t

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

y

Exact

IGABEM

m = 5

r t r r♦t ♦ t rst st♣s ♦ ♣♥♥② r

t♣ t♣ t♣

r t r r♦t s♠t♦♥ ♦ ♥ ♣t r

♠♥ttrt♦♥ ♥ t s♦♦♠tr r♠♦r s♥rt② strt♦♥ t♥q

♣r♦♣♦s ♥ ❬❪ ♦r t trt♠♥t ♦ ②♣rs♥r ♥trs ♥ ♠♣r♦

❬❪ s ♥ ①t♥ t♦ qrtr ♠♥t s tt t ♥ ♣♣ t♦ t♥s♦r♣r♦t

❯ ss ♥t♦♥s ♦t t ♦r♠ ♥ qt♦♥s ♦ ♥ r

③ ♦r t r ♠♦♥ ♥ t t♦ ②s t♦ ①trt s t ♦♥t♦rs M ♥tr

♥ ❱ r ♦♠♣r ♥ ♦rt♠ t♦ ♣r♦♣t t ❯r♣rs♥t r sr s

♣rs♥t ♥ t ts ♦ ts ♦r ♥

♣r♦♣♦s s♥r ♥trt♦♥ s♠ ♥ ♣rsr t qrtr r② ♦r ②

st♦rt ♠♥ts ①st ♦♠♠♦♥② ♥ s t ♥s r♦st ♠♣

♠♥tt♦♥

② st♥ t r ♠s r♥♠♥t ♥ t rt♦♥ r t r t♣ s♥rt②

rs t ♦♥r♥ rt ♥ ♠♣r♦ ② t♠s ♥ r② ♥ ♠♣r♦ ② ♦♥

♦rr t♥ t ♥♦r♠ r♥♠♥t s s♦s t ♥② ♦ ♥ t ♣♣t♦♥

♦ rtr ♣r♦♠

♦ r t♣ s②st♠ s st♣ ♥tr② ♥ ♥q② t♥s t♦ t ❯ r♣r

s♥tt♦♥ ♦ t r sr ♦♠♥♥ t t ♦♥t♥t② ♥ strss s♦t♦♥ ♥ t

♦t♥ s ♦♥ t r r♦♥t r s♠♦♦t ♥ rt

♣r♦♣♦s ♦rt♠ ♦r r ♣r♦♣t♦♥ s t t♦ st ♥ ♦r ♥①

♥ Prs t♦ t s♠♦♦t♥ss ♥ r r♦♥t ♦♠tr② ♥ ♥♠r s

tr ♦r ♦s ♦♥ t sr rs ♣r♦♠ r t r ♥trst♦♥

t t ♦② ♦♠tr② ♥ t st s♦t♦♥ s ♣r♦♣♦s ♥ ❬❪ s s♦ ♣♦♥t ♦

♥trst

♥♦♠♥ts

rst ♥ st t♦rs ♦ t♦ ♥♦ t ♥♥ s♣♣♦rt ♦ t r♠♦r

Pr♦r♠♠ ♥t r♥♥ t♦r ♥♥ ♥r r♥t ♥♠r ❵♥trt♥ ♠r

♠t♦♥ ♥ ♦♠tr s♥ ♥♦♦② P ♦rs s♦ t♥s ♣rt ♥♥

♦r s t♠ ♣r♦ ② t ❯ ♥♥r♥ ♥ P②s ♥ sr ♦♥ P

♥r r♥t P ♥ srtst♦♥ strts ♦r ❵t♦♠st ♥♥♦ s♠

t♦♥ t P ♥r r♥t P ❵♥rs t② ♦r ♥str② ♥t

t♦♠t r r♦t ♠t♦♥ t t ❳t♥ ♥t ♠♥t t♦ ♥ t

r♦♣♥ sr ♦♥ trt♥ ♥♣♥♥t sr r♥t t r♥t r♠♥t ♦

♥tt ❵♦rs r t♠ ♠ts s♠t♦♥ ♦ tt♥ ♥ ♥♦♥♥r ♠trs

t ♣♣t♦♥s t♦ sr s♠t♦♥ ♥ ♦♠♣tr srr② tr♦s♥♦ s

♣rt② s♣♣♦rt ② ♦♥②t r♥t ♥♠r ♥tt ❵♦♥r② ♠♥t ♠♦♥

♦ r ♣r♦♣t♦♥ ♥ ♠r♦♣♦r ♠trs

♣♣♥①

♦ ①♣♥s♦♥s ♥

♣♣♦s♥ t ♣r♠tr ♦♦r♥t ξ(ξ, η) s ♥ t ♥♦t ♥tr [ξ1, ξ2]×[η1, η2] t ♠♣♣♥

t♥ ♣r♠tr ♦♦r♥t ♥ ♣r♥t ♦♦r♥t s

ξ =1

2(ξ2 − ξ1)ξ +

1

2(ξ2 + ξ1),

η =1

2(η2 − η1)η +

1

2(η2 + η1).

♥ t ♦♥ tr♥s♦r♠t♦♥ ♦r ts ♦

Jξ =∂ξ

∂ξ=

1

2(ξ2 − ξ1),

Jη =∂η

∂η=

1

2(η2 − η1),

J(ξ) = JξJη.

②♦r ①♣♥s♦♥ ♦ xi − si t rs♣t t♦ t s♦r ♣♦♥t ♥ t ♣r♥t s♣ ♦

xi − si =[∂xi∂ξ

∣

∣

∣

ξ=ξs

(ξ − ξs) +∂xi∂η

∣

∣

∣

ξ=ξs

(η − ηs)]

+[∂2xi∂ξ2

∣

∣

∣

ξ=ξs

(ξ − ξs)2

2

+∂2xi∂ξ∂η

∣

∣

∣

ξ=ξs

(ξ − ξs)(η − ηs) +∂2xi∂η2

∣

∣

∣

ξ=ξs

(η − ηs)2

2

]

+ · · ·.

♦t tt∂xi∂ξ

=∂xi∂ξ

∂ξ

∂ξ=∂xi∂ξ

Jξ,

∂xi∂η

=∂xi∂η

∂η

∂η=∂xi∂η

Jη,

∂2xi∂ξ2

=∂2xi∂ξ2

(∂ξ

∂ξ

)2=∂2xi∂ξ2

J2ξ ,

∂2xi∂η2

=∂2xi∂η2

(∂η

∂η

)2=∂2xi∂η2

J2η ,

∂2xi∂ξ∂η

=∂2xi∂ξ∂η

∂ξ

∂ξ

∂η

∂η=

∂2xi∂ξ∂η

JξJη

♦ t ♣♦r ♦♦r♥ts ρ(ρ, θ) ♥tr t t s♦r ♣♦♥t r ♥tr♦ ♥ t ♣r♥t s♣

s ♥ r ♣r♥t ♦♠♥ s s ♥t♦ ♦r tr♥s ♦r qrtr ♥tr②

tr♥ s rr s ♥rt sqr [−1, 1]×[−1, 1] t t♦ ♣♦♥ts ♦♥t t♦tr

♣♣♦s♥ ♣♦♥t ρ(ρ, θ) ∈ [0, ρ(θ)]×[θ1, θ2] ♥ t tr♥ ♥r ♠♣♣♥ t♥ t ♣♦r

♦♦r♥ts ♥ t sqr ♦♦r♥ts s②st♠ ξ(ξ, η) s ♣r♦r♠ s

ρ =1

2(η + 1)ρ(θ),

θ =1

2(θ2 − θ1)ξ +

1

2(θ2 + θ1).

♥ t ♦♥ tr♥s♦r♠t♦♥ ♦r ts ♦

Jρ =∂ρ

∂η=

1

2ρ(θ),

Jθ =∂θ

∂ξ=

1

2(θ2 − θ1),

J(ρ) = JρJθ.

qt♦♥ ♦♠s

xi − si = ρ[∂xi∂ξ

∣

∣

∣

ξ=ξs

♦sθ +∂xi∂η

∣

∣

∣

ξ=ξs

s♥θ]

+ ρ2[∂2xi∂ξ2

∣

∣

∣

ξ=ξs

♦s2θ

2+∂2xi∂ξ∂η

∣

∣

∣

ξ=ξs

♦sθs♥θ +∂2xi∂η2

∣

∣

∣

ξ=ξs

s♥2θ

2

]

+O(ρ3)

: = ρAi(θ) + ρ2Bi(θ) +O(ρ3).

♥ ♥

A :=

(

3∑

k=1

[Ak(θ)]2

)

1

2

,

C :=3∑

k=1

Ak(θ)Bk(θ).

rts ♦ r = |x− s| r

r,i =xi − sir

=Ai

A+

(

Bi

A−Ai

C

A3

)

ρ+O(ρ2)

:= di0 + di1ρ+O(ρ2).

tr♠ 1/r3 s1

r3=

1

A3ρ3− 3C

A5ρ2+O(

1

ρ)

:=S−2

ρ3+S−1

ρ2+O(

1

ρ).

❯ ss ♥t♦♥ s s♦ ①♣♥ s

Na(ξ) = Na(ξs) + ρ[∂Na

∂ξ

∣

∣

∣

ξ=ξs

Jξ♦sθ +∂Na

∂η

∣

∣

∣

ξ=ξs

Jηs♥θ]

+O(ρ2)

:= Na0 +Na1(θ)ρ+O(ρ2).

♦r t sr ♣♦♥t ξ(ξ, η) ♥ t ♥♦t ♥tr [ξ1, ξ2] × [η1, η2] ♥ t♦ t♥♥t

t♦rs ♦♥ t ξ ♥ η rt♦♥s rs♣t② s

m1 =[∂x

∂ξ,∂y

∂ξ,∂z

∂ξ

]

,

m2 =[∂x

∂η,∂y

∂η,∂z

∂η

]

.

♥ ♥ t t ♥♦r♠ t♦rs tr♦

n = m1 ×m2 =[∂y

∂ξ

∂z

∂η− ∂z

∂ξ

∂y

∂η,∂z

∂ξ

∂x

∂η− ∂x

∂ξ

∂z

∂η,∂x

∂ξ

∂y

∂η− ∂y

∂ξ

∂x

∂η

]

.

♦♥ ♦r tr♥s♦r♠t♦♥ r♦♠ ♣r♠tr s♣ t♦ ♣②s s♣ s t ♥t ♦ t

♥♦r♠ t♦r n

J(ξ) =[(∂y

∂ξ

∂z

∂η− ∂z

∂ξ

∂y

∂η

)2+(∂z

∂ξ

∂x

∂η− ∂x

∂ξ

∂z

∂η

)2+(∂x

∂ξ

∂y

∂η− ∂y

∂ξ

∂x

∂η

)2]1/2

: =[

3∑

k=1

J2k (ξ)

]1/2

♥t ♥♦r♠ t♦r n ♦ ①♣rss s

n(ξ) =n

J(ξ).

♦♠♣♦♥♥t Ji(ξ) ♥ ①♣♥ t t s♦r ♣♦♥t ♦r ♥st♥

J1(ξ) = J1(ξs) + ρ[∂J1∂ξ

∣

∣

∣

ξ=ξs

Jξ♦sθ +∂J1∂η

∣

∣

∣

ξ=ξs

Jηs♥θ]

+O(ρ2)

: = J10 + J11(θ)ρ+O(ρ2),

∂J1∂ξ

=∂

∂ξ

(∂y

∂ξ

∂z

∂η− ∂z

∂ξ

∂y

∂η

)

.

♦ ♥ ♦t♥ Ji(ξ) s

Ji(ξ) = Ji0 + Ji1(θ)ρ+O(ρ2).

♦♠♥♥ t qt♦♥ rr t

ni(ξ) =1

J(ξ)[Ji0 + Ji1(θ)ρ+O(ρ2)].

♦ t tr♠s r ♣r♣r ♦r t ①♣♥s♦♥ ♦ t ♥tr♥ ts t s♠♣ ①♠♣

I = =

∫

S

r,ini(ξ)Na(ξ)

r3S.

tr srt③t♦♥

I =

∫ 2π

0

∫ ρ(θ)

0

r,iniNa

r3J(ξ)J(ξ)ρρθ,

r J(ξ) s r♦♠ ♣r♥t t♦ ♣r♠tr s♣ ♥ ♥ qt♦♥ J(ξ) r♦♠ ♣r♠tr

t♦ ♣②s s♣ ♥ ♥ qt♦♥ ρ(θ) s t ♣♣r ♦♥ ♦ ρ ♥ ♥ s♥ ♥

r

♥ ssttt qt♦♥s ♥t♦ t srt③t♦♥

I =

∫ 2π

0

∫ ρ(θ)

0

[

di0 + di1ρ+O(ρ2)] 1

J(ξ)

[

Ji0 + Ji1ρ+O(ρ2)][

Na0 +Na1ρ+O(ρ2)]

[S−2

ρ3+S−1

ρ2+O(

1

ρ)]

J(ξ)J(ξ)ρρθ

=

∫ 2π

0

∫ ρ(θ)

0

[

di0Ji0Na0 + (di1Ji0Na0 + di0Ji1Na0 + di0Ji0Na1)ρ+O(ρ2)]

[S−2

ρ2+S−1

ρ+O(1)

] 1

ρJ(ξ)J(ξ)J(ξ)ρρθ

=

∫ 2π

0

∫ ρ(θ)

0

(I−2

ρ2+I−1

ρ+O(1)

)

J(ξ)ρθ,

r I−2 I−1 r ♦♥② ♥t♦♥s ♦ θ

I−2 = S−2di0Ji0Na0,

I−1 = S−1di0Ji0Na0 + S−2(di1Ji0Na0 + di0Ji1Na0 + di0Ji0Na1).

trt♥ t ①♣t s♥r ♣rt ♥ t ♦r♥ ♥tr♥ ♥ qt♦♥ t rr

♥tr ♦t♥

Ir =

∫ 2π

0

∫ ρ(θ)

0

[r,iniNa

r3J(ξ)ρ− I−2

ρ2− I−1

ρ

]

J(ξ)ρθ,

s ♦ ♥tr ♥ t s♥ ♥♦r♠ ss♥ r ♥ t ①♣t ♣rt t♥

♥ trt ♥ s♠♥②t ② ♦r t s♦r ♣♦♥t ♦t ♥ t

s♥r ♠♥t s♠ r s rt t♦ ① t s♦r ♣♦♥t rs ε ♥ ♣②s

s♣ ❲♥ ♠♣♣♥ t r ♥t♦ t ♥tr♥s ♣♦r ♦♦r♥t t r st♦rt

♥r② ♣♦r ♦♦r♥t ρ s r♣rs♥t t rs♣t t♦ ε s

ρ := α(ε, θ) = εβ(θ) + ε2γ(θ) +O(ε3).

♦ t t ♦♥ts β ♥ γ t rs ♦ t r s ♥ s t ②♦r ①♣♥s♦♥ ♥

♥tr♥s ♣♦r ♦♦r♥ts s

ε = ρA(θ) + ρ2C(θ)

A(θ)+O(ρ3).

rrs♦♥ ♦ ts srs s

ρ = α(ε, θ) = ε1

A− ε2

C

A4+O(ε3).

s t

β =1

A,

γ = − C

A4,

r ♦♥② ♥t♦♥s ♦ θ ♥ ts rst ♦♦ t t ①♣t str♦♥ s♥r ♣rt ♥ ♥

t ♠t ♦r♠ s

limε→0

∫ 2π

0

∫ ρ(θ)

α(ε,θ)

I−1(θ)

ρJ(ξ)ρθ

= limε→0

∫ 2π

0

∫ ρ(θ)

α(ε,θ)

I−1(θ)

ρJ(ξ)J(ρ)ηξ

= limε→0

∫ 2π

0I−1(θ)J(ξ)Jθ

[

∫ ρ(θ)

α(ε,θ)

1

ρJρη

]

ξ

= limε→0

∫ 2π

0I−1(θ)J(ξ)Jθ

[

∫ ρ(θ)

α(ε,θ)

1

ρρ]

ξ

= limε→0

∫ 2π

0I−1(θ)J(ξ)Jθ[♥ρ(θ)− ♥α(ε, θ)]ξ

=

∫ 2π

0I−1(θ)J(ξ)♥ρ(θ)Jθξ − lim

ε→0

∫ 2π

0I−1(θ)J(ξ)♥α(ε, θ)Jθξ

=

∫ 2π

0I−1(θ)J(ξ)♥ρ(θ)Jθξ − lim

ε→0

∫ 2π

0I−1(θ)J(ξ)♥εβ(θ)Jθξ

=

∫ 2π

0I−1(θ)J(ξ)♥

ρ(θ)

β(θ)Jθξ − J(ξ)♥ε lim

ε→0

∫ 2π

0I−1(θ)θ

=

∫ 2π

0I−1(θ)J(ξ)♥

ρ(θ)

β(θ)Jθξ,

r J(ρ) s r♦♠ ♣♦r t♦ sqr ♦♦r♥ts ♥ ♥ qt♦♥ ♦t tt t st

tr♠ s ♥ s♥∫ 2π

0I−1(θ)θ = 0.

tr ♥trt♥ t s♥r tr♠ t rs♣t t♦ ρ ♥②t② ♥ t t s ♦ qt♦♥s

t ①♣t str♦♥ s♥r ♥tr♥ s tr♥srr s rr ♦♥♠♥s♦♥ ♥

tr ♥ ♥♦r♠ ss♥ r t♥ ♥ ♣♣ ♠r trt♠♥t ♣♣s t♦ t ①♣t

②♣rs♥r tr♠ t♦♥ ♦r qt♦♥ s ♦t♥

I = Ir +

∫ 2π

0I−1(θ)J(ξ)♥

ρ(θ)

β(θ)Jθξ −

∫ 2π

0I−2(θ)J(ξ)

[ γ(θ)

β2(θ)+

1

ρ(θ)

]

Jθξ

①r② s♣♠♥t ♥ strss s ♥ M ♥tr

①r② strss σ(2)ij ♥ s♣♠♥t u

(2)j r ♥ s

σxx =K

(2)I√2πr

♦sθ

2

(

1− s♥θ

2s♥

3θ

2

)

− K(2)II√2πr

s♥θ

2

(

2 + ♦sθ

2♦s

3θ

2

)

σyy =K

(2)I√2πr

♦sθ

2

(

1 + s♥θ

2s♥

3θ

2

)

+K

(2)II√2πr

s♥θ

2♦s

θ

2♦s

3θ

2,

τxy =K

(2)I√2πr

s♥θ

2♦s

θ

2♦s

3θ

2+

K(2)II√2πr

♦sθ

2

(

1− s♥θ

2s♥

3θ

2

)

,

τyz =K

(2)III√2πr

♦sθ

2,

τzx = − K(2)III√2πr

s♥θ

2,

τzz = ν(σxx + σyy),

ux =KI

2µ

√

r

2π♦s

θ

2

(

κ− 1 + 2s♥2θ

2

)

+(1 + ν)KII

E

√

r

2πs♥

θ

2

(

κ+ 1 + 2♦s2θ

2

)

,

uy =KI

2µ

√

r

2πs♥

θ

2

(

κ+ 1− 2♦s2θ

2

)

+(1 + ν)KII

E

√

r

2π♦s

θ

2

(

1− κ+ 2s♥2θ

2

)

,

uz =2KIII

µ

√

r

2πs♥

θ

2.

r (r, θ) r t r t♣ ♣♦r ♦♦r♥ts ♥ µ = E2(1+ν) κ = 3− 4ν

①r② str♥ ♥ ♦t♥ ② r♥tt♥ uj t rs♣t t♦ t ♣②s

♦♦r♥t

r♥s

❬❪ rt P ❲r③②♥ ♥ ♥r rtrr② r r♣rs♥tt♦♥ s♥

s♦ ♠♦♥ ♥♥r♥ t ♦♠♣trs

❬❪ ❱ rtt♥ ❱ ♦♦ ♥ ② ♥ r♠s♥ t♥qs ♦r ♦♠♣①

r ♣r♦♣t♦♥ ♥

❬❪ ♦ ♥ ♦s♠ ♥ ③ r r♦t ♠♦♥ t♦

♠t ♣t ♠s r♥♠♥t s ♦♥ ♠♦P t♥q ♣♣ t♠t

♦♥

❬❪ ö♠♥♥ ♥ ♥ r ♦♣♠♥t ♦ ♥ s♦tr ♦r ♣t

r r♦t s♠t♦♥s ♥ strtrs ♥♥r♥ rtr ♥s

♥r②

❬❪ ♥♦ rt♥♠ ♥ ♥ ❲♠s tr♠♥s♦♥

♥♠r st② ♦ t r r♦t s♥ r♠s♥ t♥qs ♥♥r♥ rtr

♥s

❬❪ r♥♦ ❱ ♥t♥s ♥ ♦st r ♦♥ ♣t r♠s♥ t♥qs

♦r r r♦t ♠♦♥ ♥♥r♥ rtr ♥s ♥

❬❪ ♦ës ♦♦ ♥ ②ts♦ ♥t ♠♥t ♠t♦ ♦r r r♦t t♦t

r♠s♥ ♥tr♥t♦♥ ♦r♥ ♦r ♠r t♦s ♥ ♥♥r♥

❬❪ ♠r ♦ës ♦r♥ ♥ ②ts♦ ①t♥ ♥t ♠♥t ♠t♦

♦r tr♠♥s♦♥ r ♠♦♥ ♥tr♥t♦♥ ♦r♥ ♦r ♠r t♦s ♥

♥♥r♥

❬❪ ♠r ♦♣♣ ét ♥ ♦ës r♠♥s♦♥ ♥♦♥♣♥r r

r♦t ② ♦♣ ①t♥ ♥t ♠♥t ♥ st ♠r♥ ♠t♦ ♥tr♥t♦♥

♦r♥ ♦r ♠r t♦s ♥ ♥♥r♥

❬❪ r♦ ♦ës ♥ ②ts♦ ♦♥♣♥r r r♦t ② t ①t♥

♥t ♠♥t ♥ stsPrt st ♣t ♥tr♥t♦♥ ♦r♥ ♦r ♠r

t♦s ♥ ♥♥r♥

❬❪ ♦ës r♦ ♥ ②ts♦ ♦♥♣♥r r r♦t ② t ①t♥ ♥t

♠♥t ♥ stsPrt ♥ ♠♦ ♥tr♥t♦♥ ♦r♥ ♦r ♠r

t♦s ♥ ♥♥r♥

❬❪ t♦s t③ P ♦rs ♥ ss ♦♥t♦♥ ♥ ♦♣t♠②

♦♥r♥t ❳ ♦r ♥r st rtr ♥tr♥t♦♥ ♦r♥ ♦r ♠r

t♦s ♥ ♥♥r♥ ♣s ♥♥

❬❪ ♦rs ♥ ♦r♥ ♥r ♥t ♠♥ts ♥ sts ♦r ♠ t♦r♥ s

sss♠♥t ♦ ♦♠♣① strtrs ♥♥r♥ rtr ♥s

❬❪ P ♦rs ♥ ♦t rt r♦r② ♥ ♣♦str♦r rr♦r st♠t ♦r ①

t♥ ♥t ♠♥ts ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥

❬❪ ❲②rt ♦t ♦♦♥ P rt♥② Pr♦♥ ♠ ♥ ♥

strtr♥ ❳ ♣♣r♦s ♣♣ t♦ tr♠♥s♦♥ r ♣r♦♣t♦♥ ♦r♥

♦ ♦♠♣tt♦♥ ♥ ♣♣ t♠ts

❬❪ P r②s ♥ ②ts♦ ♠♥t r r♥ ♠t♦ ♦r ②♥♠ ♣r♦♣t♦♥

♦ rtrr② rs ♥tr♥t♦♥ ♦r♥ ♦r ♠r t♦s ♥ ♥♥r♥

❬❪ ❱♥tr ❳ ❳ ♥ ②ts♦ t♦r st ♠t♦ ♥ ♥ s♦♥t♥t②

♣♣r♦①♠t♦♥s ♦r r r♦t ② ♥tr♥t♦♥ ♦r♥ ♦r ♠r t♦s

♥ ♥♥r♥

❬❪ ③ ♦rs ♥ ❩ ♥ tr♠♥s♦♥ ♠♦♥ ♦ r r♦t s♥

♣rtt♦♥ ♦ ♥t② ♠t♦s ♦♠♣trs trtrs ♠r

❬❪ P ♦rs ③ ♥ ❩ r♠♥s♦♥ r ♥tt♦♥ ♣r♦♣t♦♥

r♥♥ ♥ ♥t♦♥ ♥ ♥♦♥♥r ♠trs ② ♥ ①t♥ ♠sr ♠t♦ t♦t

s②♠♣t♦t ♥r♠♥t ♥♥r♥ rtr ♥s

❬❪ ❱ P ②♥ ③ ♦rs ♥ ♦t sss ♠t♦s r ♥

♦♠♣tr ♠♣♠♥tt♦♥ s♣ts t♠ts ♥ ♦♠♣trs ♥ ♠t♦♥

❬❪ ♦♥ ♥ ♥ rt♦♥s ♦ ♥tr qt♦♥s ♦ stt② ♦r♥ ♦ ♥♥r

♥ ♥s

❬❪ ♥ ♥ ♦♥ ♦ ♦♥r② ♠♥t ♠t♦s t ♠♣ss ♦♥

②♣rs♥r ♥trs ♥ r♥t srs ♣♣

❬❪ P♦rt ♥ P ♦♦ ♦♥r② ♠♥t ♠t♦

t ♠♣♠♥tt♦♥ ♦r r ♣r♦♠s ♥tr♥t♦♥ ♦r♥ ♦r ♠r t♦s ♥

♥♥r♥

❬❪ ❨ ♥ ♦♥r② ♠♥t ♠t♦ ♦r tr♠♥s♦♥ rtr

♠♥s ♥②ss ♥♥r♥ ♥②ss t ♦♥r② ♠♥ts

❬❪ P s♥♦ ♥ ♦♥r② ♠♥t ssss♠♥t ♦ tr♠♥s♦♥

t r r♦t ♥♥r♥ ♥②ss t ♦♥r② ♠♥ts

❬❪ P s♥♦ ♥ r♠♥s♦♥ ♦♥r② ♠♥t ♥②ss ♦ t

r r♦t ♥ ♥r ♥ ♥♦♥♥r rtr ♣r♦♠s ♥♥r♥ rtr ♥s

st

❬❪ ❱ tã♦ ♥ P ♦♦ ♦♥r② ♠♥t ♦r♠t♦♥ ♦r

st♦♣st rtr ♠♥s ♥tr♥t♦♥ ♦r♥ ♦r ♠r t♦s ♥ ♥

♥r♥

❬❪ P ♥s ♥ P ♦♦ ♦♥r② ♠♥t ♠t♦ ♥

②♥♠ rtr ♠♥s ♥♥r♥ ♥②ss t ♦♥r② ♠♥ts

♥r②

❬❪ ♥s ♥ ❲ ②♥♠ t♦♠t t r r♦t ♥

Prssr ❱sss ♥ P♣♥ ♦♥r♥ ♣s ♠r♥ ♦t② ♦ ♥

♥♥rs

❬❪ rtr P ❲r③②♥ ♥ ♥r t♦♠t r r♦t s♠t♦♥

♥tr♥t♦♥ ♦r♥ ♦r ♥♠r ♠t♦s ♥ ♥♥r♥

❬❪ r ♥ ❳♦ ②♠♠tr ♦r♠ ♥tr qt♦♥ ♠t♦ ♦r tr

♠♥s♦♥ rtr ♥②ss ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥

❬❪ r♥ rtr ♣r♦♣t♦♥ ♥ ② t s②♠♠tr r♥ ♦♥r② ♠♥t

♠t♦ ♥tr♥t♦♥ ♦r♥ ♦ rtr

❬❪ r♥ ♦t ♣r♥tt ♥ ♦③③ rtr ♥②ss ② t s②♠♠tr

r♥ ♦♠♣tt♦♥ ♥s

❬❪ P s♦ Pr ♥ tr tr♥t♥ ♠t♦ ♦r ♥②③♥

♥♦♥♣♥r rs ♥ tr r♦t ♥ strtr ♦♠♣♦♥♥ts ♦♠♣tr ♦♥ ♥

♥♥r♥ ♥s

❬❪ r♥ ♥ ♦t ♦♣♥ ♦r rtr ♠♥s ♣♣t♦♥s

♦♠♣tt♦♥ ♥s

❬❪ s ♦ttr ♥ ❨ ③s s♦♦♠tr ♥②ss ♥t ♠♥ts

❯ ①t ♦♠tr② ♥ ♠s r♥♠♥t ♦♠♣tr t♦s ♥ ♣♣ ♥s

♥ ♥♥r♥

❬❪ ♦tt ♠♣s♦♥ ♥s ♣t♦♥ P ♦rs s ♥

❲ rr s♦♦♠tr ♦♥r② ♠♥t ♥②ss s♥ ♥strtr s♣♥s

♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥

❬❪ ♠♣s♦♥ P ♦rs r②♥ ♥ ③ t♦♠♥s♦♥ s♦♦

♠tr ♦♥r② ♠♥t t♦ ♦r st♦stt ♥②ss ♦♠♣tr t♦s ♥ ♣♣

♥s ♥ ♥♥r♥

❬❪ ♠♣s♦♥ ♦tt s ♦♠s ♥ ♥ ♦st s♦♦♠tr

♦♥r② ♠♥t ♥②ss ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥

♣ ♣t

❬❪ ❩♥ ♥ s♦♦♠tr ♥②ss ♥ ♦r ♣♦t♥t ♣r♦♠

♥♥r♥ ♥②ss t ♦♥r② ♠♥ts

❬❪ ♥ ❳ ♥ s♦♦♠tr ♥②ss ♥ s♣ ♦♣t♠③t♦♥ ♦♥r② ♥tr

♦♠♣tr s♥

❬❪ P♦ts ♥♥s P s sss ♥ rr ♥ s♦♦♠tr ♦r

①tr♦r ♣♦t♥t♦ ♣r♦♠s ♥ t ♣♥ ♥ ♦♥t ♦♥r♥ ♦♥

♦♠tr ♥ P②s ♦♥ P ♣s ❨♦r ❨ ❯

❬❪ ♥♥s ❱ ♦sts P♦ts P s sss P r♦stts

♦tt ♥ s s♦♦♠tr ♦♥r②♠♥t ♥②ss ♦r t

rsst♥ ♣r♦♠ s♥ s♣♥s ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥

♥r♥ ♣t♠r

❬❪ ❱ ♦sts ♥♥s P♦ts ♥ P s ♣ s♣ ♦♣t♠③t♦♥ t

s♣♥ s s♦♦♠tr s♦r ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥

♥♥r♥ rr②

❬❪ P r②♥ ♥ ♦ts ①t♥ s♦♦♠tr ♦♥r② ♠♥t ♠t♦

❳ ♦r t♦♠♥s♦♥ ♠♦t③ ♣r♦♠s ♦♠♣tr t♦s ♥ ♣♣

♥s ♥ ♥♥r♥

❬❪ P r②♥ ♥ ♦ts ①t♥ s♦♦♠tr ♦♥r② ♠♥t

♠t♦ ❳ ♦r tr♠♥s♦♥ ♠♠ ♦st sttr♥ ♣r♦♠s ♦♠

♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥ rr②

❬❪ r rss ❩♥r ♥sr ♥ P rs ♦♥r② ♠♥t ♥②ss

t tr♠♠ ❯ ♥ ♥r③ ♣♣r♦ r①♦r

❬❪ ❨ ❲♥ ♥s♦♥ ♥ P ② ♠t♣t ♥♦♥s♥r s♦♦♠tr ♦♥r②

♠♥t ♠t♦ s♥ tr♠♠ ♠♥ts ♦♠♣tt♦♥ ♥s

❬❪ rss ❩♥r r ♥ P rs st s♦♦♠tr ♦♥r② ♠♥t

♠t♦ s ♦♥ ♥♣♥♥t ♣♣r♦①♠t♦♥ ♦♠♣tr t♦s ♥ ♣♣

♥s ♥ ♥♥r♥

❬❪ s ♥t♥r ♥ Prt♦rs ♥ ♥t ♣♦str♦r rr♦r st♠

t♦♥ ♦r ♣t ④⑥ ♦♥r② ♠♥t ♠t♦s ♦r ②s♥r ♥tr qt♦♥s

♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥

❬❪ ♠ ♥t ♠♣♦ ♥ st♥ s♦♠tr ♥②ss ♥ s②♠♠tr

r♥ ♥♠r st② ♣♣ t♠ts ♥ ♦♠♣tt♦♥ ♣s

❬❪ s ♦♥ ♥ s s♦♦♠tr ♥②ss ♦ ♦♥r② ♥tr

qt♦♥s r♣♦rt

❬❪ ②r ♥s♦♥ ②ts♦ ❨ ③s ♥ s ❳ ♥ s♦♦♠t

r ♥②ss ♦r ♥r rtr ♠♥s ♥tr♥t♦♥ ♦r♥ ♦r ♠r t♦s

♥ ♥♥r♥

❬❪ ♦rs ❱③ ♥ ♦♠♠ ①t♥ s♦♦♠tr ♥②ss ♦r s♠

t♦♥ ♦ stt♦♥r② ♥ ♣r♦♣t♥ rs ♥tr♥t♦♥ ♦r♥ ♦r ♠r t♦s

♥ ♥♥r♥

❬❪ ❱ P ②♥ ♥ts P ♦rs ♥ ③ s♦♦♠tr ♥②ss ♥

♦r ♥ ♦♠♣tr ♠♣♠♥tt♦♥ s♣ts t♠ts ♥ ♦♠♣trs ♥ ♠

t♦♥ ♥

❬❪ ②♥♥ ❱③ ②♥ ②♥❳♥ ❳ ❩♥ P rs

❩ ❨ ③s ♦r♥③s ♥ ③ ♥ ①t♥ s♦♦♠tr t♥ s

♥②ss s ♦♥ r♦♦ t♦r② ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥

♥♥r♥ rr②

❬❪ ❱ ❱r♦♦s ♦tt ♦rst ♥ s ♥ s♦♦♠tr ♣♣r♦

t♦ ♦s ③♦♥ ♠♦♥ ♥tr♥t♦♥ ♦r♥ ♦r ♠r t♦s ♥ ♥♥r♥

❬❪ ❱ P ②♥ P rr♥ ♥ P ♦rs ♦ ♥ tr♠♥s♦♥ s♦♦♠tr

♦s ♠♥ts ♦r ♦♠♣♦st ♠♥t♦♥ ♥②ss ♦♠♣♦sts Prt ♥♥r♥

♣r

❬❪ ♦ ♥ ♦ s♦♦♠tr s♣ s♥ s♥stt② ♥②ss ♦ strss ♥t♥st②

t♦rs ♦r r r ♣r♦♠s ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥

♣t♠r

❬❪ ♠t ♥ r②♥ s♦♦♠tr ♥r ♣♣r♦①♠t♦♥s ♦♠♣tr

t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥ t♦r

❬❪ ❯♣rt ♦♥ ♠t ♥ r②♥ r st♥ st♠t♦♥s ♦r

♥r s♦♦♠tr ♥②ss ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥

t♦r

❬❪ ❲ ♦♥ ♥ ♥ ♠ ♣♣♥ t♥qs ♦r s♦♦♠tr ♥②ss ♦

♣t ♦♥r② ♣r♦♠s ♦♥t♥♥ s♥rts ♦♠♣tr t♦s ♥ ♣♣

♥s ♥ ♥♥r♥ rr②

❬❪ ♠ ♥ ❲ ♦♥ ♥r s♦♦♠tr ♥②ss ♦ ♣t ♦♥r②

♣r♦♠s ♥ ♦♠♥s t rs ♥♦r ♦r♥rs ♥tr♥t♦♥ ♦r♥ ♦r ♠r

t♦s ♥ ♥♥r♥ ♥r②

❬❪ tr♥ ❲♥ ♦♥ ♥ r s♦♦♠tr ♥②ss ♥♥ ② t s

♦♥r② ♥t ♠♥t ♠t♦ ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r

♥ ♥r②

❬❪ ♥ ♦r♠t♦♥ ♥ ♥♠r trt♠♥t ♦ ♦♥r② ♥tr qt♦♥s t

②♣rs♥r r♥s ♥r ♥trs ♥ ♦♥r② ♠♥t ♠t♦s ♣s

❬❪ P ♥ ❲ r ❯ ♦♦ s♣r♥r

❬❪ ❨ ♥ ♦♣ ♦♠ ♥tts ♦r ♥♠♥t s♦t♦♥s ♥ tr ♣♣t♦♥s

t♦ ②s♥r ♦♥r② ♠♥t ♦r♠t♦♥s ♥♥r♥ ♥②ss t ♦♥r②

♠♥ts

❬❪ ❨ ❲♥ ♥ ♥s♦♥ t♣t ♥♦♥s♥r s♦♦♠tr ♦♥r② ♠♥t

♥②ss ♥ ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥

st

❬❪ ♥ rs♥s♠② ♦♣ ♥ ③③♦ ♥r ♦rt♠ ♦r t

♠r ♦t♦♥ ♦ ②♣rs♥r ♦♥r② ♥tr qt♦♥s ♦r♥ ♦ ♣♣

♥s

❬❪ ♦♥ ❲♥ ♥ ❳♦ ♥② ♠♣r♦♠♥t ♦ t ♣♦r ♦♦r♥t tr♥s♦r♠t♦♥

♦r t♥ s♥r ♥trs ♦♥ r ♠♥ts ♥♥r♥ ♥②ss ❲t

♦♥r② ♠♥ts

❬❪ sr ♥ st ♠t♦s ♥ ②♥♠ ♠♣t srs ♦♠

♣r♥r ♥ s♥ss

❬❪ ♦ës ♦r P rtr ♥ ♠ ♦♠♣tt♦♥ ♣♣r♦ t♦ ♥

♦♠♣① ♠r♦strtr ♦♠trs ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥

♥♥r♥ ②

❬❪ rt ♥ t♥ ♠r ♠s ♦r t ♠t♦♥♦ ♥ t

qt♦♥s ♦♥ r♥t ♦♠♥s ♦r♥ ♦ ♦♠♣tt♦♥ P②ss

♣t♠r

❬❪ ♦♣♣ ♥ ♠r t r ♣r♦♣t♦♥ ♦ ♠t♣ ♦♣♥r rs

t t ♦♣ ①t♥ ♥t ♠♥tst ♠r♥ ♠t♦ ♥tr♥t♦♥ ♦r♥ ♦

♥♥r♥ ♥ ②

❬❪ P rs ♥ ②♦♥ r ♣r♦♣t♦♥ t t ①t♥ ♥t ♠♥t ♠t♦

♥ ②r ①♣t♠♣t r sr♣t♦♥ ♥tr♥t♦♥ ♦r♥ ♦r ♠r

t♦s ♥ ♥♥r♥

❬❪ Ps③♥② ♥ ❲ ❩♠♠r♠♥ ♠r s♠t♦♥ ♦ ♠t♣ rtr ♣r♦♣

t♦♥ s♥ rtrr② ♠ss ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥

rr②

❬❪ r③♦♥ P r rt ♥❲ ttr ♠♣r♦♠♥ts ♦ ①♣t r sr

r♣rs♥tt♦♥ ♥ ♣t t♥ t ♥r③ ♥t ♠♥t ♠t♦ t ♣♣t♦♥

t♦ tr♠♥s♦♥ r ♦s♥ ♥tr♥t♦♥ ♦r♥ ♦r ♠r t♦s ♥

♥♥r♥

❬❪ Ps③♥② ♥ ❲ ❩♠♠r♠♥ ♠r rtr r♦t ♠♦♥ s♥ s♠♦♦t

sr ♦♠tr ♦r♠t♦♥ ♥♥r♥ rtr ♥s

❬❪ r ♥ ♥ ♦ ♦r♠t♦♥ ♦ ❯ rs t♠t

♠t♦s ♦r rs ♥ srs r♦♠s♦ ♣s

❬❪ ♥ ♥ ❲ ❲s♦♥ ♥ t ♥t ♠♥t ♠t♦ ♥ ♥r rtr

♠♥s ♥♥r♥ rtr ♥s ②

❬❪ ♥ ♥ t ♠t♦ ♦ rt r ①t♥s♦♥s ♥tr♥t♦♥ ♦r♥ ♦r ♥♠r

♠t♦s ♥ ♥♥r♥

❬❪ ♥ P ❲r③②♥ ♥ ♥r ♥ t rt r ①t♥s♦♥ ♠t♦

♦r t♥ t rts ♦ ♥r② rs rts ♦r ♣♥r r ♦ rtrr②

s♣ ♥r ♠♦ ♦♥ ♥♥r♥ rtr ♥s ②

❬❪ s P ❲r③②♥ ♥ ♥r s♠t♦♥ ♦ rtrr② r

r♦t s♥ ♥ ♥r②s ♦r♠t♦♥ Prt P♥r r♦t ♥♥r♥ rtr

♥s

❬❪ s P ❲r③②♥ ♥ ♥r ♠t♦♥ ♦ rtrr② ①♦

r r♦t ❯s♥ ♥ ♥r②s ♣♣r♦ ♥ ② rr♦ ♥ ♠♥t ②

t♦rs rtr t r ♥ ♠ ♦t♦♥ ❱♦♠ ♦♥r♥ Pr♦

♥s ♦ t ♦t② ♦r ①♣r♠♥t ♥s rs ♣s ♣r♥r ♥tr♥t♦♥

Ps♥

❬❪ t P ♦rs P rr♥ ♥ rt♠② ♦ ♥r② ♥♠③t♦♥

♦r tr r♦t ♥ ♥r st rtr s♥ t ①t♥ ♥t ♠♥t

t♦s t ❲♦r ♦♥rss ♦♥ ♦♠♣tt♦♥ ♥s ❲ ❳ r♦♥

❬❪ rr ❱rt r ♦sr t♥q st♦r② ♣♣r♦ ♥ ♣♣t♦♥s ♣♣

♥s s

❬❪ ♣t ♥♣♥♥t ♥tr ♥ t ♣♣r♦①♠t ♥②ss ♦ str♥ ♦♥♥tr

t♦♥ ② ♥♦ts ♥ rs ♦r♥ ♦ ♣♣ ♠♥s

❬❪ tr♥ r ♥ ♥♠ ♦♥t♦r ♥tr ♦♠♣tt♦♥ ♦ ♠①♠♦

strss ♥t♥st② t♦rs ♥tr♥t♦♥ ♦r♥ ♦ rtr

❬❪ ❨ ❲♥ ♥ ♦rt♥ ♠①♠♦ r ♥②ss ♦ s♦tr♦♣ s♦s

s♥ ♦♥srt♦♥ s ♦ stt② ♦r♥ ♦ ♣♣ ♥s

❬❪ P ❲♥ ♥ P ♦♦ ♦♥t♦r ♥tr ♦r t t♦♥ ♦ strss

♥t♥st② t♦rs ♣♣ ♠t♠t ♠♦♥

❬❪ ♥ r ♣♦st♣r♦ss♥ ♣♣r♦ ♥ t ♥t ♠♥t ♠t♦ Prt

t♦♥ ♦ strss ♥t♥st② t♦rs ♥tr♥t♦♥ ♦r♥ ♦r ♥♠r ♠t♦s

♥ ♥♥r♥

❬❪ P Prr ♥ rt ①trt♦♥ ♦ strss ♥t♥st② t♦rs r♦♠ ♥r③ ♥t

♠♥t s♦t♦♥s ♥♥r♥ ♥②ss t ♦♥r② ♠♥ts ♣r

❬❪ ② ♥ ①♠♦ ♥tr ♠t♦ ♦r ♥②ss ♦ rtr

♣r♦♠s s♥ ♥♥r♥ ♥②ss t ♦♥r② ♠♥ts

❬❪ ♥ ♥ ❲ trss ♥t♥st② t♦r ♦♠♣tt♦♥ ♦♥ ♥♦♥♣♥r r

r ♥ tr ♠♥s♦♥s ♥tr♥t♦♥ ♦r♥ ♦ ♦s ♥ trtrs

❬❪ P s♦ ♥ tr t♦♥ ♦ rtr ♠♥s ♣r♠trs ♦r ♥ r

trr② tr♠♥s♦♥ r ② t ❵q♥t ♦♠♥ ♥tr ♠t♦ ♥tr♥t♦♥

♦r♥ ♦r ♠r t♦s ♥ ♥♥r♥

❬❪ ♠r ♥ ♥ q♥t ♦♠♥ ♥tr ♠t♦ ♦r tr

♠♥s♦♥ ♠①♠♦ rtr ♣r♦♠s ♥♥r♥ rtr ♥s

❬❪ r ♥ ♥ ♥ t ♦♠♣♦st♦♥ ♦ t ♥tr ♦r r

♣r♦♠s ♥tr♥t♦♥ ♦r♥ ♦ rtr

❬❪ ② ♥ ♦♠♣♦st♦♥ ♦ t ♠①♠♦ ♥trrst

♥tr♥t♦♥ ♦r♥ ♦ ♦s ♥ trtrs

❬❪ ♦s③ ♥ ♦r♥ ♥ ♥trt♦♥ ♥r② ♥tr ♠t♦ ♦r ♦♠♣tt♦♥ ♦ ♠①

♠♦ strss ♥t♥st② t♦rs ♦♥ ♥♦♥♣♥r r r♦♥ts ♥ tr ♠♥s♦♥s ♥♥r

♥ rtr ♥s

❬❪ P s♥♦ ♥ rt③ ♦♥r② ♠♥t ♥②ss ♦ tr♠♥s♦♥ ♠①♠♦

rs t ♥trt♦♥ ♥tr ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥

♥r♥ r

❬❪ r ♥ ❲r♦ ♦♥r② ♠♥t ♠t♦ ♦r ①s②♠♠tr r

♥②ss ♥tr♥t♦♥ ♦r♥ ♦ rtr

❬❪ r♦♥ ♥ ♥ t r①t♥s♦♥ ♥ ♣ts ♥r ♣♥ ♦♥ ♥ tr♥srs

sr ♦r♥ ♦ s ♥♥r♥

❬❪ ♠r ♦♣♣ ♥ ♦r♥ ①t♥ ♥t ♠♥t ♠t♦ ♥ st ♠r♥

♠t♦ ♦r tr♠♥s♦♥ t r ♣r♦♣t♦♥ ♥♥r♥ rtr ♥s

♥r②