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QO |oORTwk RwQ@ u=mt= CyH� =yu; Q]k Q@=Q@ 2 5 =y`tW RmQt x@ RmQt |xrY=iC@Uv '50MPa u; |a=HDQ= pwOt 'xHQO 30 l =N l =m]Y= |x} w=R '�l =Nw ?trwm � Qywt pOt =@ pt=m |Q}tN � |a=HDQ= l =N Q=DiQ '0 3 xU=t uwU=wB"CU= |]N � |a=HDQ= |xOwOLt QO =y`tW Kr=Yt Q=DiQ w x=Qty=v Kr=Yt u=} QHw �CU= lWN |}xU=t l =N� OQ=Ov OwHw l =N QO |}xQiL ?; Q=Wi xm =Hv; R='OOvw}B|tv wkwx@ |m}t=v}O |=yVvQm O=H}= QF= QO |}xQiL ?; |=yQ=Wi xi=[=pmW x@� xU=t VvQm � VvD Q=DiQ u=}@ |=Q@ xO}J}B |Q=DiQ |=ypOt R= xO=iDU=QO R}v ?trwm � Qywt pOt Ovv=ty xO=U |=ypOt w OQ=Ov |DQwQ[ �Z=@kv= w `=UD=pL |OL p}rLD |=yVwQ QO '|iQ] R= "Ovvm|t C}=im |m}D=DU= p}rLDV}B R=i�Q=oR=U� x=Qty u=} QH uwv=k Cr=L R= '�?}W uwQO `tW h}OQ� Q[=L |xr�Utu=} QH w u; p@=kt |x]kv Q[=L VywSB QO xm [4]'CU= xOW xO=iDU= Kr=Yt |=Q@R= p@k |m =N Kr=Yt xmv}= ZQi Cra x@ u}vJty "CU= xDiQo Q=Qk Q_vOt x=Qty=vhqDN= p}rO x@� OvwW|t |Q}tN w |]NQ}e Q=DiQ |xOwOLt OQ=w `tW Kr=YtQ=DiQ '|r@k |OOa C=ar=]t Ovv=ty '�xOW QmP Kr=Yt wv wO |DNU QO O=} RxHwD =@ 'p=L Qy x@ "CU= xOW ZQi |]N � |a=HDQ= wv R= =y`tW |m}v=mtwO Qy xm ji= K]U x@ C@Uv ?}W |x} w=R w l =N |rN=O l =m]Y= |x} w=R x@=@ 'Q=O}=B � VRer |xv=DU; QO |m}D=DU= Cr=L QO ?}W 'Ov=xOW ?=NDv= xHQO 30|m}D=DU= Cr=L QO ?}W `k=w QO "CU= 1 pO=at xv}tm |Q=O}=B u=v}t]= ?} Q[?} Q[ |xv}tm ?=NDv= x@ xHwD =@ u}k} x@ |}xRQr |Q=PoQ=@ Cr=L QO w Q=O}=B

"OW Oy=wN Q=O}=B=v 'u=v}t]=

\@=wQ w |m} R}i |R=UpOt R= pY=L |Uw=o |WtN Qovr |=y|vLvt "8 pmW[10]"xrRrR Q=@ CLD lWN |}xU=t ?}W � `tW h}OQ |xr�Ut QO |r}rLD

pL w Sw}i} QDvU |R=UpOt R= pY=L |Uw=o |WtN Qovr |=y|vLvt "9 pmW[23]"|}xU=t l =N QO |Q=OQ@Owo Qw=Ht Qw=vW `tW |xr�Ut QO |OOa

"CU= p � y OQ@ Q=mQB |=y|vLvt w M(z) |WtN Qovr |=y|vLvt |xar=]tQwmPt pmW QO xJv=vJ "CU= xOW x�=Q= 10 pmW QO Q_vOQwt |xr�Ut |rm pOt|}xR=U Q=DiQ QO xrRrR |}xRQr |Q=PoQ=@ |=yCyH QF= |UQQ@ hOy 'OwW|t xOy=Wt|vD@ |xOW?Yv `tW h}OQ �p�y |=y|vLvt� |m}vmDw�S w �M |WtNQovr�"CU= xHQO 30 ?}W QO QDt 10 pw] w QDt 1 Q]k x@ T=}ktnQR@ |OOa pOt

55

"""Q=DiQ

OQw;Q@|

=Q@O}OH

|]@=wQ

|x�=Q=

CLD ?}W |Oa@xU |OOa pOt QO nQR@ VvQm |ki= u=mtQ}}eD V}=tv "13 pmW�"OvDUy u}J\N pmW x@ =y`tW� T@] |xrRrR Q=@ ?}mQD wO

xOy=Wt 13 pmW QO nQR@ VvQm |R=UpOt j]vt QO 'T@] l}ORv |xRwLOwyWt =y`tW CWB w u}@ QO 3l =N |oORTwk 'QwmPt pmW QO "OwW|tQ_vOt |}xU=t ?}W |}xRQr |R=UQ=O}=B QO C@Ft |}xO}OB u=wvax@ xm CU=|xO}OB� CU=y`tW u}@ R= Kr=Yt 4uDi=} u=} QH p@=kt |x]kv xm OQ}o|t Q=Qk

"�?QNt

Qovr `@=D R= p(z) w V (z) 'M(z) |x@U=Lt "5xDi=}VR=Q@ pmW|Uw=o |WtN

|xrO=at j@=]t Q[=L Q=DWwv s=tD QO 'M(z) jta =@ |WtN Qovr `} RwD `@=D'|Uw=o `@=D |rY= C=}YwYN R= "CU= xOW x�=Q= 7 |x]@=Q QO GQOvt |Uw=o|rm Cr=L QO xm CU= ��1� pO=at xQ=wty �7 |x]@=Q QO c ?} Q[ |va}� e |=tvl =N Q=Wi� swO |x@DQt w �|WQ@ |wQ}v� pw= |x@DQt C=kDWt w QwmPt |xrO=at

%OvwW|t x�=Q= 9 w 8 \@=wQ CQwYx@ �`tW pw] OL=w QO

M(z) = �:ec(az+b)2

(7)

V (z) = dM(z)dz

= �: 2ac(az + b)| {z }S:F:

ec(az+b)2| {z }

M(z)

=

) V (z) = M(z):(S:F:) (8)

p(z) = d2Mdz2

= �0�2ca2ec(az+b)

2+ 2ca(az + b)ec(az+b)

2 � (2ca(az + b)�

p(z) = �:ec(az+b)2| {z }

M(z)

(2ca2 + 4c2a2(az + b)2)| {z }p:F:

�V � s�=k w � T �|[Qa '�L� |rw] |ki= |=yxir-wt `} RwD V}=tv "11 pmW"T@] |xrRrR

� V � s�=k w �T �|[Qa '�L� |rw] |ki= |=yxir-wt `} RwD V}=tv "12 pmW"lwy}O |xrRrR

K} QY |Oa@xU OwOLt p[=iD VwQ ltm x@ |Oa@xU |OOa |R=UpOt|m}D=DU= p}rLDV}B |xrLQt wO pt=W 'xr�Ut |OOa |R=UpOt "CU= xOW s=Hv=|}=yv p}rLD w pOt hm QO Q=OQ}o w pOt ?v=wH QO |m]re |RQt \}=QW =@QO O=R; u=O}t |=yRQt w pOt |@v=H |=yRQt QO um =U |RQt \}=QW =@ |m}t=v}Ow |rw] |ki= |=yOQwmQ R= xrRrR |Q=PoQ=@ CyH "CU= |Oa@xU pOt pm uwt=Q}B|xrRrR� T@] x=oDU}= QO T@] |=yxrRrR l}ORv |xRwL ?=DW s�=k w |[QaxOW xO=iDU= �lwy}O |xrRrR� lwy}O x=oDU}= QO T@] |xrRrR OQwmQ w �T@]12 w 11 |=ypmW QO ?}DQD x@ =yu; |Ot=U@ w |v=tR `} RwD V}=tv xm 'CU=C=YDNt |=yQwLt CyH xU QO |Q=PoQ=@ u=mt= ?ULQ@ "OwW|t xOy=Wt|Q=PoQ=@ CyH V w TV ' T ' LV ' LTV ' LT 'L ptDLt ?}mQD 7 '|Oa@xU'p=Ft |=Q@ Ov=xOW xDiQo Q_vQO =y`tW |WtN Qovr |=y|HwQN G=QNDU= wxrRrR Qy V w T 'L |=yxir-wt xm CU=vat u}O@ LTV |}xRQr Q=@ ?}mQDp=ta= |Oa@xU pOt �x; y; z� C=YDNt |=DU=Q 3 QO ?}DQD x@ sy =@ u=tRsy

"Ov=xOWQ=@ ?}mQD wO CLD |}xU=t ?}W |ki= u=mtQ}}eD |=y|HwQN R= |}xvwtv|xrRrR LT |[Qa w |rw] |ki= |=yxir-wt u=tRsy QF= w L |rw] |ki=

56

2/2|

xQ=tW'

35�2

|xQwO'

�1398

u=DU@=D�

h}QW

u=Qta|

UOvyt OQwt xDiQo CQwY C=@U=Lt CLY '19 |x]@=Q |}=yv |xH}Dv j@=]t TB

pmW x@ |WtN Qovr `@=D 'Twmat Ov}=Qi R= xO=iDU= =@ ,=OOHt xm =QJ CU= O}}-=D"OW pY=L x}rw= |Uw=o `@=D

R= y(z) l =N � `tW |@Uv |@v=H pmWQ}}eD |x@U=Lt "2"5xDi=} VR=Q@ |Uw=o |WtN Qovr `@=D |Q}op=QoDv=

`tW |@v=H |@Uv pmWQ}}eD '�3 |r= 1 \@=wQ� u}W}B |xOW KQ]t \@=wQ j@=]t%O};|t CUOx@ 21 |r= 18 |=yx]@=Q R= y(z) l =N �

y(z) =ZZ

M(z)EpIp

dz =Z Z

(�:e�(az+b)2)

EpIp:dz

!:dz (18)

dy(z)dz

= �EpIp

:�p

�erf(az + b)2a

�+ c1 (19)

y(z) = �EpIp

Z �p�erf(az + b)

2a + c1�:dz (20)

y(z) = �EpIp

�p�

2a ( (az + b)erf(az + b)a

+e�(az+b)2

ap�

�+ c1z + c2 (21)

OQi |a@=D xm CU= |Uw=o |=]N `@=D Q@=Q@ erf(u) 'Q}N= \@=wQ |t=tD QOw |Uw=o |=]N `@=D 'C=}[=} Q QO w �OQ=O uQ=kD C=YDNt -=O@t x@ C@Uv� CU=

%OvwW|t h} QaD 23 w 22 |=yx]@=Q j@] u; stDt

erf(u) = 1p�

uZ�u

e�t2dt = 2p

�

uZ0e�t

2dt (22)

erfc(u) = 1�0@ 1p

�

uZ�u

e�t2dt = 2p

�

uZ0e�t

2dt

1A (23)

�K} QY |x]@=[ uDW=Ov Cra x@� |rY=Q}e |a@=D 'C=}[=} Q QO =]N `@=D"CU= 14 pmW j@=]t u; stDt w =]N `@=D V}=tv w CU=

|QUm |rwrPy `@=D |=Q@ c2 w c1 |Q}op=QoDv= |=yC@=F |x@U=Lt "1"2"5l}vwDwvwt |m}D=DU= |Q=PoQ=@ QO

xm CU= xr�Ut QO |Y=N |RQt \}=QW x@ R=}v c2 w c1 |=yC@=F uOQm =O}B |=Q@OwHwt �l}vwDwvwt/|m}D=DU=� |QUm |rwrPy p�y `@=D uOQm\@DQt j} Q] R= \ki

"|UQQ@OQwt Q}eDt ?UL Q@ u; stDt w =]N `@=D `} RwD "14 pmW

) p(z) = M(z):(p:F:) = (9)

VR=Q@ |=yC@=F b w a 'c '� |=yQDt=Q=B |O=OQ=Qk h} Q=aD j@=]t '=yu; QO xm'`tW pw] QO |WQ@ |wQ}v V (z) ' M(z) pmW|Uw=o |WtN Qovr `@=Dp:F: VQ@ ?} Q[ S:F: `tW pw] OL=w QO l =N ptar=Tma =} Q=Wi p(z)u}vJty CU= �Q=O?}W =} K]Ut� u}tR K]U R= jta z w l =N Q=Wi ?} Q[Q=QkQ@ xQ=wty Q[=L Q=DWwv uDt s=tD QO 'QwmPt |=yQDt=Q=B R= |rai h} Q=aD

"OvDUy

`@=D R= pY=L p(z) `@=D R= M(z) w dM=dz |x@U=Lt "1"5�pQDvm � |Uw=o |xDi=}VR=Q@ M(z) |WtN Qovr

jta QO |WtN Qovr `@=D ,=OOHt u=wD|t 'u}W}B \@=wQ CLY pQDvm CyHQO �l =N |@v=H Q=Wi� ptar=Tma `@=D R= |Q}op=QoDv= Q=@ 2 ?UL Q@ =Q M(z)

CUOx@ 12 |r= 10 \@=wQ KQW QO 9 |x]@=Q ltm x@ =Q p(z) `@=D |va} 'jta%OQw;

M(z) =ZZ

p(z)dz =Z �Z

pdz�dz + c0i (10)

)M(z) =ZZ

�:e�(az+b)2(�2a2 + 4a2(az + b)2)dz (11)

) dM(z)dz

= (�� � 2a2)��p

�erf(az + b)2a

�+4�a2 �

p�erf(az + b)

4a � (az + b)e�(az+b)2

2a!

+ c01 (12)

|=yC@=F O}=@ M(z) uOQw; CUOx@ |=Q@ 'swO Q=@ p=QoDv= |x@U=Lt R= TBC=@U=Lt OvwQ "OQw; CUOx@ xr�Ut |RQt \}=QW ltm x@ =Q c02 w c01 |Q}op=QoDv=

%CU= 14 w 13 \@=wQ pmW x@ QwmPt) dM(z = 0)

dz= (�� � 2a2)

�p�erf(b)

2a�

+4�a2 p

�erf(b)4a � (b)e�(b)2

2a!

+ c01

= (�� � 2a):(a(z = 0) + b)e�(a(z=0)+b)2

= �� � 2a(b)e�(b)2(13)

c01 = (�� � 2a(b)e�(b)2) + (4�a2)

be�(b)2

2a!

= 0 (14)

QiY jta QO Q@=Q@ M(z = 0) = �e�b2 Q@=Q@ Qovr \QW T=U=Q@ c02 C@=F w%CU= 17 |r= 15 \@=wQ j@=]t �u}tR K]U QO CUQO |va}�

M(z) =Z

((�� � 2a2)��p

�erf(az + b)2a

�+4�a2 �

p�erf(az + b)

4a � (az + b)e�(az+b)2

2a!

)dz

= �2�a(az + b)e�(az+b)2(15)

)M(z) =Z

(�2�a(az + b)e�(az+b)2)dz (16)

)M(z) = �e�(az+b)2+ c02 ) c02 = 0 (17)

57

"""Q=DiQ

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=Q@O}OH

|]@=wQ

|x�=Q=%Ow@ Oy=wN 37 |r= 34 |=yCQ=@a pO=at '19 |x]@=Q QO

dpdz

= dpdy:dydz) : dp

dz

����z=0

= dpdy

����z=0

: dydz

����z=0

(34)

dpdz

����z=0

= �:ecb2 h4c2a3b(3 + 2cb2)

i= Ki:

dydz��z=0 (35)

dydz

����z=0

= 1Ki� �:ecb2 h4c2a3b(3 + 2cb2)

i(36)

dy(z)dz

����z=0

= �EpIp

:�p

�erf(b)2a

�+ c1 (37)

xOW ?=NDv= |Uw=o |xrO=at 'Q[=L VywSB |=yVR=Q@ |t=tD QO u}vJtyTB "CU= �1 Q@=Q@ xQ=wty c Q=Okt xm CU= |}xvwo x@ M(z) |WtN Qovr |=Q@&OwW|t QDxO=U 38 |x]@=Q pmW x@ c1 C@=F |x@U=Lt |=Q@ Q}N= \@=wQ sy R=@CUOx@ 38 |x]@=Q R= 37 w 36 |x]@=Q wO |w=UD R= c1 C@=F Q=Okt xH}Dv QO

%O};|t

c1 =�:e�b2

�4a3b(3� 2b2)�Ki

� �EpIp

:�p

�erf(b)2a

�(38)

|x]@=Q QO |Q=Po|=H |=Q@ c2 |Q}op=QoDv= swO C@=F |x@U=Lt OvwQ 'xt=O= QO'c2 C@=F |x@U=Lt |=Q@ "CU= xOW x�=Q= ha=[t |Q}op=QoDv= R= pY=L y(z)

|=Q@ '|@ QHD |=yV}=tR; R= pY=L |rwrPy p � y |=y|vLvt QO xm =Hv; R=|vLvt QO u}vJty 'CU= xOw@ p = 0 Q=Okt z = 0 |x]kv QO |}xU=t l =NCU= QiY ,=Q=@H= R}v p(z) Q=Okt 'CU= y(z) = 0 xm |}=H QO '-=O@t QO p� yG=QNDU= |rwrPy p � y `@=D R= p(z) ?UL Q@ =Q y(z) u=wD|t |iQ] R= wpY=L c2 C@=F w xO=iDU= �42 |r= 39 \@=wQ� |RQt \QW wO R= u}=Q@=v@ 'OQm

%OwW|t

y(z = 0) = �EpIp

:

0@ (az+b)erf(az+b)a

+ e�(az+b)2ap�

1A+ c1z + c2 =

p(z = 0) = y(z = 0) = 0 : (p� y) (39)

y(z = 0) = �EpIp

�0@p�2a ( (az+b)erf(az+b)

a

+ e�(az+b)2ap�

1A+ c1z + c2 (40)

) p(z = 0):Ki

1� (p(z = 0)=pu)

= �EpIp

p�

2a ( b:erf(b)a

+ e�(b)2

ap�

)

!+ c2 = 0 (41)

c2 = � �EpIp

p�

2a ( berf(b)a

+ e�(b)2

ap�

)

!(42)

|x@U=Lt |=Q@ |}=yv |x]@=Q 'c2 w c1 |Q}op=QoDv= C@=F wO |x@U=Lt =@ TBpmWQ}}eD `@=D Q=Okt "OwW|t KQ]t 21 |x]@=Q pmW x@ M(z) ?UL Q@ y(z)

`@=D w M(z) |Uw=o |WtN Qovr `@=D wO j}irD pY=L 21 |x]@=Q |@UvxOW QmP C=@U=Lt pmW x@ u; |=yC@=F w |m}D=DU= |QUm |rwrPy p � y

"CU=

|rwrPy p� y `@=D ?UL Q@ M(z) w dM=dz |x@U=Lt "3"5|m}D=DU= |QUm

`@=D ?UL Q@ =Q M(z) |WtNQovr u=wD|t C=@U=Lt R= Q[=L VN@ |=yDv= QO

%�26 |r= 24 \@=wQ� OvDUy x@U=Lt p@=k y w p Q}O=kt x@

p(z) = y(z)(1=Ki) + y(z)=pu(z)

(24)

dpdz

= dpdy:dydz

(25)

dpdz

= �ec(az+b)2��� 2a(az + b)� (�2a2

+4a2(az + b)2)�

+ (4a2(2a)(az + b)��

(26)

|QUm |rwrPy `@=D ?UL Q@ dp=dy |v=}t jDWt Q=Okt |x@U=Lt u}vJty28 w 27 \@=wQ pmW x@ |Q}HvR C=kDWt |x]@=Q QO |Q=Po|=H |=Q@ '|m}D=DU=

%OQ}o|t CQwYdpdy

= 1:(1=Ki + y=pu)� (1=pu)(y)(1=Ki + y=pu)2 = (27)

) dpdy

= 1=Ki

(1=Ki + y=pu)2 (28)

|va}� u}tR K]U QO xOW xO=U dp=dz Q=Okt '9 |xrO=at j@=]t |iQ] R=%O};|t CUOx@ 29 |x]@=Q R= �z = 0 QO

) dpdz��z=0 = �:ecb

2 h4c2a3b(3 + 2cb2)i

= Cte:1 (29)

y ?UL Q@ 30 |x]@=Q j@=]t QwmPt `@=D '|QUm |rwrPy p� y |x]@=Q QO%OwW|t |U} wvR=@ ,=OOHt

y(z) = p:(1=Ki)(1� p=pu)

) dydp

= (1=Ki + y=pu)2

1=Ki(30)

|Q=Po|=H Q}N= pO=at |x]@=Q 'y Q}eDt |=H x@ dp=dy |x@U=Lt |x]@=Q QOxH}Dv QO 'OwW|t TmaQ@ w xO=U 30 |x]@=Q |xH}Dv TBU 'OwW|t xO=U w

%O};|t CUOx@ 31 |x]@=Q R= Q_vOQwt jDWt Q=Oktdpdy

= (pu � p)2

(1=Ki):p2u

= (dydp

)�1 (31)

|}=yDv= |ki= VN@� CU= QiY dp=dy Q=Okt 'p = pu xm |v=tR |iQ] R=31 Twmat |x]@=Q QO xm �1 pmW j@=]t 'p � y |rwrPy |=yQ=Owtv QO'�29 |x]@=Q� dp=dz C=kDWt Q}O=kt |x@U=Lt =@ uwvm = "Ovm|t jOY R}vp(z) |x]@=Q ?UL Q@ 31 |xrO=at QO p(z) jta `@=D Q=Okt |Q=Po|=H =@p@k QO 9 |xrO=at QO |Uw=o |WtN Qovr `@=D R= ha=[t |Q}ojDWt R= pY=L32 |x]@=Q j@=]t pY=L C=Q=@a z = 0 |x]kv QO 31 |x]@=Q dp=dy w

%CU=

p(z) = �:e�(az+b)2 ��2a2 + 4a2(az + b)2� = (32)

dpdy

����(p=0;y=0)

= Ki = (dydp

)�1 (33)

c1 C@=F |x@U=Lt |=Q@ xOWpY=L |x]@=Q 'dy(z)=dz |x@U=Lt QO TB

58

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u=DU@=D�

h}QW

u=Qta|

UOvyt ��:e�(b)2

= pu(02

2 )| {z }0

� p2u

yK

0BB@ (Ky(0) + pu)(ln(����Ky(0) + pu

����)� 1)

Ky

1CCA+

2664�� � 2a(b)e�(b)2+

0BB@p2u ln(

����pu����)yK

1CCA3775 :(0)

| {z }C1:z=0

+C2 (50)

C2 = p3u

K2y2 ((ln(����pu����))� 1) + (�:e�(b)2

) (51)

Q}N= |xOW x@U=Lt \@=wQ ?UL Q@ Ov=wD|t y `tW |@v=H pmWQ}}eD `@=D"OwW u} Ro}=H |QUm |rwrPy p� y |vLvt |=Q@

|rm}U |rwrPy p� y `@=D R= p(z) w y(z) G=QNDU= "4"5"CU= xOW x�=Q= Q[=L VN@ QO Cv=Sv=D |rm}U |rwrPy p � y `@=D C=@U=Lt

[21w20]%CU= xOW x�=Q= |rY= `H=Qt QO 52 |x]@=Q pmW x@ QwmPt `@=Dp(z) = p = A:pu tanh

�K:zApu

:y�; (A = 0 9) (52)

Qovr |kta `} RwD |wQ |[=} Q |r}rLD C=}rta =@ 'p� y `@=D ?UL Q@ p=Ll =N � `tW |@Uv pmWQ}}eD w p(z) l =N ptar=Tma Q}O=kt 'M(z) |WtN\ki uwJ p(z) |kta `@=D Q=Okt xm OwW|t u=Wv Q]=N "CU= xOW x@U=Lt y(z)

CLY =t= 'CU= 9 |x]@=Q QO |r@k `@=D u=ty 'CU= M(z) |WtN Qovr `@=D|x]@=Q� |rm}U Cv=Sv=D |xrO=at |=Q@ O}=@ '21 |x]@=Q QO xOWCi=} y(z) Q=Okt"OwW|t x@U=Lt KQW u}= x@ Q}N= |xOt;CUOx@\@=wQ T=U=Q@ xm 'OwW pQDvm �52%O};|t CUOx@ 56 |r= 53 \@=wQ CQwYx@ y(z) ?UL Q@ p(z) |xrO=at =OD@=p(z)Apu

= tanh�K:zApu

:y�) tanh�1(tanhu) = u

Ki = K:z : (F=L2) (53)

) tanh�1�p(z)Apu

�=�Ki

Apu:y�

(54)

) y =�ApuKi

�tanh�1

�p(z)Apu

�(55)

) dy(z)dz

= dydp:dpdz

= �EpIp

:�p

�erf(az + b)2a

�+ c001 (56)

x@U=Lt 57 |x]@=Q pmW x@ dy=dp Q=Okt 'y |=Q@ '55 |x]@=Q T=U=Q@ uwvm =%OwW|t

dydp

= ApuKi

: (1=Apu)1� (p=Apu)2 = 1

Ki:(1� (p=Apu)2)

�1 < (p=Apu) < 1 (57)

Q@=Q@ p@k R= 'pmW|Uw=o |WtNQovr |x]@=Q j@=]t dp=dz jDWt Q=Okt wjDWt Q=Okt |x@U=Lt |=Q@ "CU= xOt; CUOx@ z = 0 |x]kv QO 29 |x]@=Qxmv; pw= %CiQo Q_vQO O}=@ =Q syt |xDmv OvJ 'z = 0 |RQt |x]kv QO dy=dp

xDiQo Q=Qk Q_vOt Qw_vt u}= |=Q@ 43 |x]@=Q "OQw; CUOx@ |QUm |rwrPy p� y%CU=

p = puyKi

pu +Kiy= puy(K:z)pu + y(K:z)

;Ki = K:z : (F=L2) (43)

%�44 |x]@=Q� |Q}op=QoDv= Q=@ 2 pY=L =@ CU= Q@=Q@ M(z) Q=Okt w

M(z) =ZZ

puy(K:z)pu + y(K:z)

:dz (44)

%�45 |x]@=Q� OwW|t s=Hv= |Q}op=QoDv= |xrLQt u}rw= =OD@=

dmdz

= puyK

0BB@ zKy�pu ln(

����Kyz + pu����)

y2K2

1CCA+ C1 (45)

%Ot; Oy=wN CUOx@ 46 |x]@=Q KQW x@ z = 0 |x]kv QOC1 C@=F Q=Okt w

dM(z = 0)dz

=�� 2ac(b)ec(b)2�

= puyK

0BB@ 0Ky�pu ln(

����Ky(0) + pu����)

y2K2

1CCA+ C1 (46)

%O};|t CUOx@ 47 |x]@=Q R= C1 'pw= |xrLQt |Q}op=QoDv= C@=F TB

) C1 = �� 2a(b)e�(b)2+

0BB@p2u ln(

����pu����)yK

1CCA (47)

49 w 48 \@=wQ R= xO=iDU= =@ |WtN Qovr Q=Okt 'swO Q=@ |Q}op=QoDv= =@ w%O};|t CUOx@

M(z) =Z "

pu

z � pu ln(

��Kyz + pu��)

yK

!+ C1

#dz = (48)

M(z) = pu(z2

2 )

� p2u

yK

0BB@ (Kyz + pu)(ln(����Kyz + pu

����)� 1)

Ky

1CCA+�� � 2a(b)e�(b)2

+�p2

u(ln(����pu����)� 1)

yK

��| {z }

C1

:z + C2 (49)

50 \@=wQ R= xO=iDU= =@ 'M(z = 0) = �e�(b)2 ZQi =@ C2 C@=F Q=Okt w%O};|t CUOx@ 51 w

59

"""Q=DiQ

OQw;Q@|

=Q@O}OH

|]@=wQ

|x�=Q=w M(z) |WtNQovr |x]@=Q QO Cv=Sv=D |rwrPy p(z) |x]@=Q |Q=Po|=H =@

%O};|t CUOx@ 63 CQ=@a 'QwmPt \@=wQ R= |Q}op=QoDv=dM(z)dz

=ZA:pu tanh

�K:zApu

:y�dz (63)

%OwW|t x@U=Lt 64 |x]@=Q CQwYx@ 63 |x]@=Q p=QoDv= pY=LdM(z)dz

= A:pu�ApuK:y

�ln�

cosh�K:zApu

:y��

+ c0001 (64)

pmW x@ '|Uw=o M(z) |x]@=Q R= dM(z)=dz |Q=Po}=H =@ c0001 C@=F Q=Okt%O};|t CUOx@ z = 0 |x]kv QO 65 |x]@=Q

V (z = 0) = dM(z = 0)dz

= � � 2ac(b)ec(b)2

= A:pu�ApuK:y

�0BBB@ln cosh�K:(z = 0)Apu

:y�

| {z }1 0

1CCCA| {z }0

+c0001 (65)

CUOx@ 66 |x]@=Q j@=]t c = �1 ZQi =@ c0001 |Q}op=QoDv= C@=F TB%O};|t

c0001 = �2�a(b)e�(b)2(66)

w 67 \@=wQ j@=]t M(z) Q=Okt ,=t}kDUt '|Q}op=QoDv= Qo}O Q=@ l} =@ uwvm =%Ot; Oy=wN CUOx@ 68

M(z) =ZA:pu

�ApuK:y

�ln�

cosh�K:zApu

:y��

dz = (67)

M(z) = A2p2u

K:y

Zln�

cosh�K:zApu

:y��

dz +�c000:z1

�=�A2p2

u

K:y

�:I +

�c000:z1

�(68)

|rwrPy Twv}Um |a}@] sD} Q=or p=QoDv= Q=Okt 'Q}N= p=QoDv= |x@U=Lt |=Q@%OwW|t x@U=Lt 69 |x]@=Q pmW x@

I =Z

ln�

cosh�K:yApu

:z��

dz = z ln�

cosh(K:yApu

:z)�

��K:yApu

���2(ky=Apu)z ln(e2(ky=Apu)z + 1) + Li2(�e2(ky=Apu)z)

2�kyApu

�2

��z2

2��

+ c0002 (69)

|r=QoDv= h} QaD ?UL Q@ wO |x@DQt sD} Q=or |rB =} sD} Q=or |O Li2(z) 'u; QO xm%CU= 70 |x]@=Q QO �u}at p=QoDv=�

Li2(z) = �zZ

0

ln(1� t)t

:dt = �1Z

0

ln(1� zt)t

:dt (70)

QiY ,=@} QkD u}tR K]U QO CUQO w |}xU=t |=yl =N QO p(z = 0) = 0 Q=Okt\QW Q=J=v x@ 'CU= p = 0 xm |}=H =yvD QO p� y |=y|vLvt QO ,=}v=F w CU=|=y|vLvt sUQ C=YDNt -=O@t 'QwmPt |x]kv xm =QJ 'CU= Q=QkQ@ R}v y = 058 |x]@=Q j@=]t 'xOW QmP\}=QW|rm |xH}Dv u}=Q@=v@ [1]'OyO|tV}=tv =Q p�y

%Ow@ Oy=wNdydp

����z=0

= 1Ki:(1� (p(0)=Apu)2) = 1

Ki(58)

\}=QW QO p � y |=y|vLvt [20w14'1]'l}Uqm |r}rLD |v=@t j@=]t 'u; QO xmp R= QDnQR@ xQ=wty pu Q=Okt xm =QJ 'OW Oy=wNv QiY Q@=Q@ pu Q=Okt 'p = 0l} OL QO Ov=wD|t pu Q=Okt 'CU= QiY p Q=Okt xm |v=tR |iQ] R= "CU=|Q}op=QoDv= C@=F Q=Okt u=wD|t q=L "OW=@ QiY hr=Nt w p R= QDnQR@ uwr}UB=p@k Cr=L j@=]t dy=dp ?}W uwJ 'OQw; CUOx@ z = 0 |x]kv QO =Q c001,=OOHt c001 C@=F TB '�l}vwDwvwt l}D=DU= |QUm |rwrPy p� y |vLvt� OwW|tswO C@=F |x@U=Lt '|vwvm |xrLQt QO "OwW x@U=Lt Ov=wD|t 38 |x]@=Q Q@=Q@y(z = 0) Q=Okt 'c002 C@=F |x@U=Lt |=Q@ "CU= Q_vOQwt c002 |va} |Q}op=QoDv=60 w 59 \@=wQ j@=]t y(z = 0) Q=Okt w 'OwW|t xO=U G}=Dv w |Q=Po|=H

%OwW|t p}mWD

y(z = 0) =�ApuKi

�tanh�1

2664p(z = 0)Apu| {z }

0

3775 (59)

tanh�1(0) = tanh(0) = 0) y = 0 = �EpIp p

�2a ( (b)erf(b)

a+ e�(b)2

ap�

!+ c002 (60)

%OwW|t x@U=Lt 61 |x]@=Q j@=]t c002 TB

c002 = �EpIp

p�

2a ( (b)erf(b)a

+ e�(b)2

ap�

!(61)

xOy=Wt pL=Qt j@=]t '(c002 w c001 ) |Q}op=QoDv= |=yC@=F |x@U=Lt R= TB|Uw=o |WtN Qovr `@=D wO x@ xDU@=w y(z) |@Uv pmWQ}}eD `@=D Q=Okt 'xOWG=QNDU= ,=OOHt 21 |xrO=at pmW x@ 'Cv=Sv=D |rm}U |rwrPy p � y `@=D wCUOx@ y(z) `@=D Q=Okt xm CU= |i=m \ki p(z) G=QNDU= |=Q@ u}vJty "OwW|tQO &OwW u} Ro}=H y |=H x@ '52 |}=OD@= |x]@=Q QO p� y |rY= `@=D QO 'xOt;Oy=wN pmW |Uw=o |WtN Qovr `@=D x@ xDU@=w w z jta R= |a@=D R}v y CQwYu;

"OW

|rm}U |rwrPy p� y `@=D R= M(z) |x@U=Lt "5"5`@=D ?UL Q@ y(z) w p(z) Q}O=kt '�4"5 VN@� Q}N= VN@ C=@U=Lt j@=]t|r}rLD \@=wQ ltm x@ u; c002 w c001 |Q}op=QoDv= |=yC@=F w |rm}U |rwrPy p� y'CU= xOW x@U=Lt xOW QmP pL=Qt Tma OvwQ 'xt=O= QO "Ot; CUOx@ |[=} Q|x]@=Q� O};|t CUOx@M(z) Q}O=kt '|rm}U |rwrPy p� y `@=D ?UL Q@ |va}

%�62M(z) =

ZZp(z)dz =

ZZ �A:pu tanh

�K:zApu

:y��

dz (62)

60

2/2|

xQ=tW'

35�2

|xQwO'

�1398

u=DU@=D�

h}QW

u=Qta|

UOvyt K]Ut u}tR QO =yxU=t |=Q@ 76 w 75 \@=wQ R= xO=iDU= =@ Ov=wD|t w Ow@ Oy=wN

%OwW x@U=Lt �?}W QF= uwO@�pu(z) = (C1:z + C2:D): 0:z (75)

pu(z) =�C3:D: 0

�:z (76)

|}xU=t l =N QF-wt XwYNt uRw 0 '�m� `tW |=vyB =} Q]k D '=yu; QO xmpw] OL=w QO pu l =N|}=yv Q=Wi |x@U=Lt?}=Q[C3 wC2 'C1 w (kN=m3)l =N |rN=O l =m]Y= |x} w=R ?ULQ@ 'API |x}YwD j@=]t xm 'OvDUy `tW|x@U=Lt \@=wQ "Ov=xOW sUQ 16 pmW |=yQ=Owtv j@=]t |Q=Owtv pmW x@ �

[21w20'1]%OvDUy 79 |r= 77 \@=wQ KQW x@ C3 w C2' C1 ?}=Q[C1 = ( K0 tan� sin�

tan(� � �) cos�+ tan�

tan(� � �)(tan� tan�)

+K0 tan�(tan� sin� � tan�)) (77)

C2 = tan�tan(� � �)

�Ka (78)

C3 = Ka(tan8 � � 1) +K0 tan� tan4 � (79)

Cr=L w k0 uwmU Cr=L Q=Wi |@v=H ?}=Q[ w � w � |=}=wR Q=Okt '=yu; QO xm%Ov=xOW h} QaD 82 |r= 80 \@=wQ pmW x@ Ka lQLt

� = �2 ; � = 45 + �

2 (80)

K0 = 1� sin�) (K0 � 0 4) (81)

Ka = tan2(45� �2 ) (82)

K]Ut u}tR |=Q@ '76 w 75 \@=wQ QO pu |x@U=Lt \@=wQ 'OW xQ=W= xJv=vJ|=y?}W |va}� Q=O?}W u}tR |xr�Ut OQwt QO OQ@ Q=m p@=k ,=t}kDUt w OvDUyp Q}O=kt 'K]Ut u}tR QO 'l =N |@v=H |=yQ=Wi Q}O=kt T=U=Q@ "OvDU}v �|m =NK}LYD 'QwmPt pmWt |xQ=J x=Q "OvDUy Q=O?}W u}tR R= QDnQR@ xQ=wty pu w?}=Q[ w \@=wQ |x�=Q= ltm x@ 'Q=O?}W u}tR QF= uOQw; ?=UL x@ |=Q@ pu Q}O=ktVy=m |=Q@ =Q 83 |xOvy=m |x]@=Q [13]'�2008� u=QoWywSB |NQ@ "CU= K}LYDx�=Q= |rm}U |Q=PoQ=@ Cr=L QO A |xOvy=m ?} Q[ Q@ xwqa pu Q=Okt |x} wv=F

[13]%Ov=xO=OR = 0 74 + 0 0378(z=D)� 0 6315(S);R � 1 (83)

Q@ |}xU=t ?}W |x} w=R S w QDt x@ `tW Q]k D 'QDt ?UL Q@ jta z 'u; QO xm0 66 =D 0 50 |xOwOLt QO ?}W |=}=wR |=Q@ 83 |x]@=Q "CU= u=}O=Q ?UL84 |x]@=Q CQwYx@ pu Q=Okt 'O}OH Cr=L QO u}=Q@=v@ "CU= OQ@ Q=m p@=k u=}O=Q

[13]%OwW|t xDiQo Q_v QO ?}W QF= |=Q@ Q}N= \@=wQ |x}rm QOpuslope = (A = 0 9)�R� pu (84)

|=yjta |=Q@ 76 |x]@=Q Q=Okt w |L]U j=ta= |=Q@ 75 |x]@=Q Q=Okt� Qovr |vLvt xm =Hv; R= "CU= z jta V}=Ri= u}L QO w `tW pw] QO QDu}}=B?}W- `tW h}OQ |OOa |m}t=v}O p}rLD R= pY=L M(z) � z pO=at jtaVvD pmW x@ �u; C}akwt w 'Ct=N[ 'xUOvy %pt=W� VRer K]U Q=F; '|}xU=tQ=Okt R= z jta Qy |=Q@ u}=Q@=v@ 'OvQ=O OwN QO =Q |}xRQr � |m}t=v}O VvQm �

Cr=L QO C \rDNt O=Oa= |xawtHt w[a \rDNt OOa l}� z sD} Q=or |O|rB h} QaD pO=at xm �CU= R w[a |k}kL OOa l} Cq=L |NQ@ QO w |rmLi(z) u}vJty "CU= xOW h} QaD 'OW=@ 2 Q@=Q@ u; s |x@DQt xm |v=tR 'sD} Q=or\rDNt OOa l} Q@=Q@ 'CU= �\rDNt |rm Cr=L QO =}� |ivt OOa l} z xm |v=tRx@ M(z = 0) = �e�b2 |RQt \QW ltm x@ c0002 |Q}op=QoDv= C@=F "CU=

%CU= |@=}CUO p@=k |DL=Q x@ 72 w 71 \@=wQ pmWc0002 = �

�A2p2

u

K:y(�K:yApu

)� Li2(�1)2(ky=Apu)2

�+ �e�b

2(71)

c0002 =�A3p3

u

2k2y2Li2(�1)�

+ �e�b2

(72)

\@=wQ T=U=Q@ 'Q[=L VywSB QO M(z) |WtN Qovr |=yQ=Owtv |x@U=Lt'CU= xOw@ 5?rDt w Excel |=yxt=vQ@ ltm x@ |OOa |=yVR=Q@ R}v w xDi=}\U@QwDUO pmW x@ ?rDt |xt=vQ@ QO Li2(x) sD} Q=or|O `@=D h} QaD xm |Qw] x@'=yu; Q}eDt h} QaD QO OwHwt hqDN= x@ O}=@ xm 73 |x]@=Q 'CU= di log(x)

%CW=O pwP@t |i=m xHwDdi log(x) = Li2(1� x) (73)

R= sy Q}_v w Cw=iDt h} QaD wO xm � Li2(x)w di log(x) h} QaD u}vJty`@=D l}Uqm w swUQt h} QaD =@ Cw=iDt |mOv= xm � ?rDt |xt=vQ@ QO '�OvDUy `@=D

%CU= 74 |x]@=Q pmW x@ '�CU= Li2 sD} Q=or |O

di log(x) = +xZ

1

ln(t)1� tdt) Li2(x) = +

0Zx

ln(1� t)t

dt (74)

|xt=vQ@ QO sD} Q=or|O |[=} Q Q}O=kt 'OwW|t xOy=Wt Q}N= \@=wQ QO xJv=vJX=wN R= xO=iDU= x@ O}=@ '23 w 22 \@=wQ x@ C@Uv w OwW|t x@U=Lt ?rDt"OQm xHwD \@=wQ QO ZwQit C@Ft Ctqa w |Q}op=QoDv= OwOL Q}}eD w u}at p=QoDv=QiY Q@=Q@di log(1) Q=Okt 15 pmW QO di log(z) `@=D Q=Owtv j@=]t u}vJty

"CU=

z jta ?UL Q@ pu |}=yv Q=Okt |x@U=Lt "6"5'CU= xOW |Q=Po|=H pu pO=at xm p(z) |}=yv Q=Okt 'xOWQmP \@=wQ |x}rm QOpu(z) |rm pmW x@ z jta R= |a@=D ' w[wt =@ \@DQt l}Uqm `H=Qt j@=]t

"?rDt |xt=vQ@ QO 10 w QiY |xOwOLt u}@ z sD} Q=or |O `@=D Q}O=kt "15 pmW

61

"""Q=DiQ

OQw;Q@|

=Q@O}OH

|]@=wQ

|x�=Q=

[21]" (Pu)|}=yv l =N Q=Wi Q}O=kt p}mWD |=Q@ K pwOt wC3 w C2 'C1 ?}=Q[ |x@U=Lt |=Q@ API |O=yvW}B |=y|vLvt "16 pmW

|=y|R=UpOt R= pY=L|Uw=o|WtN Qovr |xv}W}@|kta `} RwD G}=Dv "17 pmW"|}xU=t ?}W QO T@] |xrRrR Q=@ ?}mQD 7 w `tW 7 O=OaD |=Q@ |OOa

w OvOt; CUOx@ VR=Q@ |=yC@=F TBU "CU= xOW VR=Q@ |Uw=o |WtN QovrQwmPt |=yx]@=Q R= Qovr Q}O=kt u}vJty "OvOW |Q=Po|=H 68 w 49 \@=wQ QOOvv=ty� OvOW s}UQD xU}=kt CyH OL=w Q=Owtv l} QO |oty w OvOt; CUOx@

"�20 w 19 |=ypmW|=yp}rLD pY=L nvQ |@; |HwQN |=y|vLvt '20 w 19 |=ypmW j@=]t|t=tD u}ov=}t |vLvt |=Q@ �3D-FDA� |Oa@xU K} QY OwOLt p[=iD |OOaxrRrR |rw] |xir-wt pt=W |=yp}rLD \UwDt |=Q@ '�ave. all� =yp}rLD G}=Dv|xir-wt |=Q@ w �ave. all, LCC� Q=Okt wO u}= \UwDt |=Q@ '6�ave. LCC�xDW=Po V}=tv x@ L Component |}=yvD x@ '?}W |rw] VRer |=DU=Q QO |rw]Q=@ 2 x@ \w@ Qt 'nvQRtQk |HwQN |=y|vLvt 'QwmPt |=ypmW QO "CU= xOW|=y|vLvt w �49 |x]@=Q� l}vwDwvwt l}rw@ QB}=y p � y `@=D R= |Q}op=QoDv=|x]@=Q� Cv=Sv=D |rwrPy p � y `@=D R= |Q}op=QoDv= Q=@ 2 x@ \w@ Qt nvQ|mWtx@ Cv=Sv=D |rm}U p � y `@=D R= pY=L Qovr |=y|HwQN QO "OvDUy �68

`=iDQ= uDiQo Q_vQO x@ |R=}v w OwW|t xO=iDU= QwmPt |x]@=Q wO u}@=t �xv}tm�"CU}v xOW QmP \@=wQ QO z |=H x@ zslip pmW x@ |}xU=t ?}W VRer K]U|rm}U =} |m}D=DU= p� y |=y|vLvt |rm \@=wQ QO s=y@= |xOvvm O=H}= |xDmv`iQ p@=k �|O=a@= w |O=tv� 85 |x]@=Q pmW x@ x}rw= |DNU sy@t h} QaD QF= QO�

%CU=Ki = (kpy = K):z

: F=L2 = (F=L3):(L)) kN=m2 = (kN=m3):m (85)

|}=m} w �16 pmW j@=]t�CU= QDU@ ptar=Tma pwOtK =} kpy Q=Okt 'u; QO xm|rwrPy p� y |=y|vLvt |}=OD@= VN@ |DNUKi =t= 'OQ=O kN=m3 TvH R=

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R= pY=L |WtN Qovr |=y|HwQN |UQQ@ "6K} QY |Oa@xU OwOLt p[=iD |OOa |=y|R=UpOt

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=yCWwv=B1. pile-tip2. pile-head3. arching4. owing5. Matlab6. longitudinal contained components7. sum of square errors8. root mean squares of errors

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3. Erfani Joorabchi, A., Liang, R.Y., Li, L. and et al. \Yieldacceleration and permanent displacement of a slope re-inforced with a row of drilled shafts", Soil Dynamics andEarthquake Engineering, 57, pp. 68-77 (2014).

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