!"# $"

" !%&' () * "

$+, -* .' /+0**

12 !/+(+" '. '3 /+(+" +

) :"#/#/%&'() *+) :%/"/%,(

!456 ! "#$%&'( $ )*+ ,- .# / 0 1 2 $$ 3 . 5! %67 "8 $ 9 :6 "+ '; $ ( "0#$ -< = '' $ > #?@ SV

#PA 9% %+" - 0B # %9 3 ,6"<9# $ 'C *(%A 0; / D $ <*8%&'#$ )*+ ' 0"%+$.

$7, : ,- .#D 0E )*+ 8 F"%&'#$ )6,' G? "

Transient Site Response Analysis of Nonhomogeneous Two-dimensional Topographic Features Using BEM

M. Kamalian and A. Sohrabi Bidar International Institute of Earthquake Engineering and Seismology

Abstract: This paper presents the complete algorithm of site response analysis of nonhomogeneous topographic structures using transient two-dimensional boundary element method (BEM). Seismic behaviour of various topographic features including canyon, half plane, sedimentary filled valley and ridge sections, subjected to incident SV and P waves are analysed. The analysis shows the efficiency of the proposed algorithm and its advantage over common transformed domains methods in forming a basis for extension to non-linear behaviour. Keywords: Boundary element method, Time domain, Site effect, Nonhomogeneous, Two-dimensional topography effects,

Scattering, Amplification

*& **&./ 0123)

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

bi4.5) 0.657 08

c1,2 '9.: 4 '1; <1 :.

Gij and Fij => ?@ 4 A6.55B@ ' 08

GijN+1-n and FijN+1-n +8 .55B@ C5 1D 08 4 A6

?@

M1(τ) and M2(τ)AE 'FG =6 HI1@

NmA =6 HI1@

r4 C.J@ K) L5I MN O).5P

ti?@ .I

t,∆t')E QP AE

uiA6 .55B@

λ and µ D R.9

ρ'.S 5+)

86/9

.T TU T 6V U OE4.8WGT X'=T

4PY5(0 0) 45OE.T ZT7 .I .IV 8 [0\FT

E5E1@ 4 LI.G H5(0) 'E E 5L]@ OE.5.I[') .

8 C)E OWG'.] _> =I` 81G T T/

aI AV1; 4> <1<1 OE. 0TI][.T2)V E'

/ +)U. O 4J'E d5L) OE.54. E/#@#[@.8 4

e85O 4J L:. <1OE. 0D80JFT 'E/#

@"/5I .1M )] .5B5OE. ). 0f@ 84O T8'T/

0 aI / (8 @ . (8 5TI .1TMT8 E T1:

1I 81G .] ()V .

RMgV5L) 808 hTg TI 1S1 [E T5 T8

4 (@ OWGPY5(0 0O .` S1@ 1 AV T) .

MN Ri')OWCaI0V5L) T8 .T TI 8WGT X'

TTU GTT 1TTi)')1TT` E 5OE.TT TT .TTI hUTT> L0

(8WG0TaI j 0T8 [I AT E 1TG TU T 0

R0.I MJ@ 5 =56') ()V 'T O .k.T@ Tj

.P171@ l 1: X1@ hS( <1'7O075e5T O

=> U5 U mE O T Q[M+T AV .TN TI Q1T@ 4

(4 0 / : 0 .

(4 0 : 0MJ@ 5OE. =0(8WG 0TaI j 0TI

+@ =I` O: O4.P 5hT) :(T4 0T2> 'n4 T)

[S0 4TJ )FEM((T4 0E.T 0[TS n4 T) 0

E.0)BEM((4 4 0RTU. )HYBRID ( TU U.@ E T5R

TT2> n4 4 'E.TT 407TTTT 'VTT.TTI('TTU TT

i1.0.8 E Co (4 L0Ti1. I AE 0

o AV0i@ = ATJS .AV AT6 T(@ T) Tj

@ @ 81G h8.)E e*'M6 I (*7 'i;5Ta'4

+5h 4.I E 6MI )1 507A1TI T@ TV 81G

g =p+5FG.') 5[2@ 1 [MJ@ 4 5 .` =.

(TT4 .I TTU0TT2> 'Mp+TT ' 4TTJ) TTaI TTU

@63 I ) ' O.8 .? j TI0RTS1 QT2>

TFG 3T) AV HTi@ TI 4 T8E. <1T qrU s6a)

J ..5 1I 81G X.G.I (@'T tTG uDT>

U'6a) <1 A1@'(@ I qrU 5@'T8E. A1Tj 0

v.) qrS08E. 0Q V TT8E. 04TwxT> E16+ .T/

]4.[(4 0 i` E .W5() n4 =0I'() Ty [TI

(1M n4 0I PE '()) Cj1U 5 =T> I U [TL

G .7 =63 I )5. =i10oT 0Ti@ =T

=p+ => 56'g5FG.'+T8 A1@) .[TS n4 0

E.0 T: [I 0.]1T 00.TI MJ@ T5 =T56'J5T(F0

4J) 4 4J')+3/ TFG '=p+T =T> TU T

IxS <1 3)5T)4. ' .?T D4 Tj TI0TI

) 4 TT8E.5 TTaI TT2 4TTJ hUTT> uD TTa OWTT

')] zU5) 4 9 H3a3@ K. 5) 25E+T I E0

MJ@ 4 OE1>5 =56'J5(F0TI'() T) 4 5T

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

"

I'() =`> I ') .

[TS n4 .I TU I FI 0E.T 0MJ@ T5ZT7 =

OE. 0(8WG0TaI4 0T2)V T@ 'T1 TU T; A

8 U ) 0U'8oT @T: TU .P Q2) 0

i@=+8 ]&|.[ 1T]~[T)V 4][T45L

)+U'U ) 1I 1TW C[TS h0E.T 0 QTP TI QTP

o0+T8 E O I AE T80TaI4 0T1. xTP

) .U .2)V E'+8 U7 ?@ 5 (307 ()V 5e5TI (@ 4 O

TT9 u 1TTN'[TTS uD TTa 0E.TT 0 1TTI .TT9>

:0E0 1W L@63 I h' 1I Q1@ .p.T 8TaI5=

S.)I 4']&[@ i:'O .N 4 .@.I \0+8 O

I 1( I U ) .U <.*5 1W .3[TS h0E.T 0

=p+ => 1D 54 CaI0o 0O T =I` AE

I z 1I u i: > L9'I O j 5T u'T`

(` 0) 4 QED 5.I 20 TJ .Wg AT5O T =TI` .

1TTI .GTT5U .TT5 A][U TT5 A TT68 4 A]4[4

P5.0A 68 4]4|[T: A A3T) L9 4 T` Q

+8 N 80G5 1W RT` T()V .T(0[TS 0

E.0U.@ 5[TS R0E.T 0 4TJ .TN: 40.TI =T>

J <1 3) =p+5(F0FG'4')+3/G5.0TC

) .` O 1 OE 4 4.

1W[S h0E. 0aI4 0o 0T; j.P AE '

(0G5.G.I => ' =p+ 56'IT T.P U

OE1> I E18 AV .I U MJ@5OE.T =0(8WGT0

aI 4 0) n.+P 5 .U@5S1 4 ' 4 ]~[Tj.P

[S n4 0E. 0aI4 0 0.TI MJ@ T5OE.T =0O T80

G'.IV 4'o 0 ) .I U I AE i1. 0T()V

JN E < G hS( <1 I R)SH (_@5 1TI O h

` TTUTT5 JTTN A4 <1TT TTI s)SV 4P(75eTT5P'

@iJ'e8 45.TU .I U L0 .U T5 A T68 4 A

]4[4 E5)+U L' 1TW .I TU TU +8 [TS h0

E.0aI4 0o 0I AE MJ@ OE1> 5Z7 =OE.T 0

.P171@ l 1:'4 TaI0hST( <1T .TI.I SV 4P

a@5) h. I 1W > LTI T(@ ()V X1@ O p h

=p+ 1I 4J LW8 .TT L 1TWMJ@ =TU h T5 =

OE. Z7 0.P1T71@ l 1T: '4 TaI0.TI.I LW8T)

ST( <1T hSV 4PoTT 0[TTS n4 XT1@ ATE 0

E.0 O . . TU A A3T) 1_ I'T` 4

1WhOE. O 0.P171@ M* 9 : j '

TTG O uDTT> TTU'TT) 5hoTTTT.IV O '.TTI TTf@ 4

'P5]@ J@ .5JTN <1T .0SV 4P[2@ 1T T4

MJ@5 .P .` =.

:6'' ;<

a+).5 a@ .I h/> =5'J5FT(0A+T3/

LW8 4 A+8 B@ O 4TJ 55(M6T.0XT1@ CTj1U

aEI .5 A'1 :

)(( )2 2 21 2 j,ij 2 i, jj i ic c u c u b u 0− ⋅ + ⋅ + − =&&

uiI5.W)B@554 A6.biI5) .W)54.075.60T G .c14

c2(:. 01; <1 '9.: 4'T A3) G 'TU T8

@.@ TTITT5 E RuD TTa c12=(λ+2µ)/ρ4c2

2=µ/ρTTI TT

'V.λ4µ.9 4 D Rρ+) 5.S 'G ).

aE. .W) 0J .I h/> 5F(0A+3/ 4 A+T8

TI LW8 T: I n4 5OT) T80T)E4 '][T a .TI

+).5=)(mIF E. I'V:

( ) ( ) ( ) ( )( )ij i ij i ij ic u , t G t x, t F u x, t dΓ

ξ ⋅ ξ = ∗ − ∗ ⋅ Γ∫ )(

Gij 4Fij(*TT7 0TT ' TT a +TT).5=)(I 4TT5 .W)

1 80i8 .I Q 0B@55E.T ?T@ 4 AT6.0TF) x

_Jt: F4 I U +8 E1T >4 [U. I C0

1JjF) ξJ 4T_ t≤τ7 TOTV T) .u Ti:

Gij*ti4Fij*ui(.W)0/ L11)+8 L.ti?@ O 4

4 .I0E. .I s \F ΓI T5 T A' .Cij(ξ).T9R

7) O G5W1'F) ξ+8 A 1I . E /

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

,

Fij) '' 1 ..9 L4 O 1I E. 8 HI@ (@ R.8

xP I 4 0@ 56' 456' )+6' .

@ 1N 'J U5 .I 1 X'<1T Q128 l.a

OE. 0P .` 5E.T .W) a .0T hUT> 'TI A1T@

E .N . ./]4|:[

( ) ( )( ) ( )( ) ( )

ij iinc.

ij i ij i j

c u , t

G t x , t F u x , t d u , tΓ

ξ ⋅ ξ =

∗ − ∗ ⋅ Γ + ξ∫

)(

ujinc.B@ 55 hS( <1 E MN> A6.I 5 A' .

=6"') "

=>uD a)(T)(B TU T AV Q[M+T5T8.0

M+ ES A6 4 AE OE1> 4 .8 0)1 ..I0TL

E =6 I 1_ =: .' 1:

=6>6"') $"

.I0ES 0)E 'I AE 1J N4+ OEI 0+T@ 5h

O)1P I0U a TN=N∆tI .`.I .l. I

B@55FG u.')E OEI .8 ' 'B A1@58.0B@554 A6.

E. ?@0E u 1N I I .5A./ :

)(( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )i i

i i

n n 1i 1 2

n n 1i 1 2

u x, M u x M u x

t x, M t x M t x

−

−

τ = τ ⋅ + τ ⋅

τ = τ ⋅ + τ ⋅

M1(τ)4M2(τ)FG =6 HI1@'u 1TN TI TU +8 AE

E.a@ .O ):

)(( ) ( )

n 1 n

n 1 n1 2

T T :T T

M & Mt t

−

−

ττ − − τ

τ = τ =∆ ∆

p p

I S1@ I a)( 'A1@ a)(E =6T TI T .

o0ES AE 0 1) :

( )

( ) ( )( ) ( )

( ) ( )( ) ( )

Nij i

N 1 n n N 1 n nN ij1 i ij1 iN 1 n n 1 N 1 n n 1n 1 ij2 i ij2 i

c u

G r t x F r u xd

G r t x F r u x

+ − + −

+ − − + − −= Γ

⋅ ξ =

⋅ + ⋅ + − ⋅ Γ ⋅ ⋅

∑ ∫

)|(

Gij1,2N+1-n 4Fij1,2

N+1-n @.@ I 5+T8 RT80 T56'B@T55 .

?@ 4 A6 E . I U ).a@ .O ):

)~(( ) ( )

( )

( ) ( )( )

n tN 1 nijk ij k

n 1 t

n tN 1 nijk ij k

n 1 t

G r G r, t M ( ) d

F r F r, t M ( ) d

⋅∆+ −

− ⋅∆

⋅∆+ −

− ⋅∆

= − τ ⋅ τ ⋅ τ

= − τ ⋅ τ ⋅ τ

∫

∫

a )|( 'E .@ O =6 I A1@) .51)EI [+'./ :

( ) ( ) ( ) ( ) ( )( )NN N 1 n n N 1 n nij i ii ij ij

n 1c u G r t x F r u x d+ − + −

= Γ⋅ ξ = ⋅ − ⋅ ⋅ Γ∑ ∫

)(

AV U:

)(( ) ( ) ( )

( ) ( ) ( )

N 1 n N 1 n N nij ij1 ij2N 1 n N 1 n N nij ij1 ij2

G r G r G r

F r F r F r

+ − + − −

+ − + − −

= +

= +

[ a)(.I a)|(U.@ RiT TI TU AV T5 R

+8 .8 . uS 80+8 <1 (iS 1S1 80

i@. U ?@ 0.I.I r-1/2 OT wx> ) T) .u Ti: TI

+8 .W 80 56')(+8 I i+) 80 T56')~(

.i 0) .+8 U .Ur A T80G1ij 4F1

ij

I 4 .'+8 g 85 . .).e85+8 LT80 .T

U ) 1G.I .) i@. 4 1) A8 E O +T8 T80

_)5@1D .5C.+8 80N 1 nijG + −4N 1 n

ijF + −HTS.

][O p ).

=6:6"') $

.I0ES 0)6 '() E 0. 74[4 TS C

O O . .P56U u* h+')η(HI1@ E

=6Nk(η) 1 O 'A1@ a)(o 0A6

E u 1N IES .0 ./:

)#(

( )

( ) ( )

( ) ( )q

q

QNNij i

n 1q 1

n N 1 nik ij k

n N 1 nik ij k

c u

T G r N J d

U F r N J d

= =

+ −

Γ

+ −

Γ

⋅ ξ =

⋅ ⋅ η ⋅ ⋅ η−

⋅ ⋅ η ⋅ ⋅ η

∑ ∑

∫

∫

QI5() =U a@ .W)0E. 04|J| I5I1Uv .W)5i@ L =.

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

)'.J@ F) U.I C6'O.P E 80TI HT`4 AV

g 4 .5 .L : u 1N0) T5T O' 1T .T(.W)0

a)(> I'4 .I s1P w a O:` E O I 0

()0 : 0iJ =I` ).T() 0 .T 4.TI AV

(.W) E 0 a)(4> U0+8 T80G1ij 4F1

ij

e8 45_) =6 HI@ L5.J@ F) .C(@ )5` u5.T0

RM;'U:

4> . .W)0+8 G1ij:AV .T) TU .W) L

a95) 5 O'() .W) )8 10 : 0OT:` TI

iJ =I` s1P w a .T A 6)V .I K4.3

(S1@ =I` a@ I'() 0+@ .6j1U 5=TN> T@ 1 h

HI@ +8 q.9 I1Uv HI1@ =6 54 .TI L0E .T8 T.

A0R I .

4> . .W)0+8 F1ij :TAV .T) TU .TW) L

1`0) 5 O'E O I 1g n4 C5+ .5.TI TU h

i RMN h+S U.> Q1('mIF a ET=TI` .

iJ:

)(( )

q

q q

1ij k

static 1 staticij k ij ij k

F N J d

F N J d F F N J d

Γ

Γ Γ

⋅ ⋅ ⋅ η =

⋅ ⋅ ⋅ η+ − ⋅ ⋅ ⋅ η

∫

∫ ∫

(.W)04+@ Q4 4 4 0@.@ I k1 5 . R

TT: 40TT).TT> .TT .TTW) ATT8 TTU 4 .TTW)

@1D5> I C'g n4 E O I T5+T .5h

i' iJ RMN h+S U.> .I '1 .) Q4 .TW) T5 [

TTI TTUTT5 =1TT) A 1TTI A+TT6+TT8 .TT) TTi@. 4TT80

@TT1D5 1TTD 4 C5 TT: C0TT TT)8 .TTW

(.W)0 : 0s1P w a O:` CU I 4.I ' 1T .

4'6@ TU 1 T5 ) 4 RMTN h+TS TU.> C5T2

a )(. (@F'T8E. TU T O T =I` 0

M+ I +I .+ Mp'J I U 5(F0)5I '() .

) U 4'() E O I I A1@0ET2 0]&

4[?TT E.TT TTI0iTTJ TTI fTT 4 +TTI OTT

(.W)01` . 0QT` .T/ .TI('T() TU T 0

E20.N U 0.I TJ L+I 5E O T 4 XT a)(

6I T O .I ' EV uTS Ta@ )1T0M+T [T ?

81*).

=6=6+ # ?&

.P a)(.I0.8E. K) E C04 1T T1)

P .` h8 U uD a :125E uD a OW ). T .

7V 81G :

)(( ) 0UFTGN

1n

nn1Nnn1N =⋅−⋅∑=

−+−+

@ ) I'a uS 4+@ I Q1M0 'A1T@

a)(E =6 I 1)EI .+' ./:

)(ΝΝ1Ν1 ΖΤGUF +⋅=⋅AV U:

( ) N.cni1N

1n

nn1Nnn1NN UUFTGΖ +⋅−⋅=∑−

=

−+−+ )(

E 6)j a)( .I'V .I ZN @ B@ e*55AT6.

xP I u 1N 4 =i` u_J ?@ 40OE. 0B@55AT6.

_J)1U ') ' = hS( <1 E ' 1 .

@6.+A

=6)(MU 8'.P171@ 9 : C'TI LW8)

@ 1;5 A3) C'8 .2> <1'kT: E TU T[E

751 I L0E \F 5T TU.> L'TJ TI TU5T) X5h

.; E 4 JNJ AV m5(F0)`1 +I '@ ]@ J5 .T` .

'8 .J5) X5h+T ?IT@ l.a U JN5<1T h

OE. hS( 0X1@ .`a E. .W)0)(.8 4

J E C5F(0)`1 +I 'E.T .TW) a X1@0)(

MJ@5 =')1T .3T@ A 1TI .TN 4 .TI (0T8E. 04 EV

PE0B@55)6. a@ 4 ( 4 .I (3@ 0T8E. 0.3T

. XTE.0+TM63@ 5 ='8 .I(T'P U T.

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

3>6 B /' C' D (Hybrid

J5(@ X) C5hE. .W) a I LW8 JN0)(

.I0=>M+U .U 81G .

1TW[TTS h0E.TT 0 [TT Q.TT) RTT` .UxT k1TT

4 aI085.i.I U 0MJ@ 5 =T56'J5T(F0TFG '4

g5FG.'U.@ E O I i 4 C3G 5(T4 R0[TS 0

E.0[TS 40T>.; 4TJ 'TTI O OT T.P TU

].[G [ Q.) T5 .TI T T` .0M+T .T8 =T>

1G. .8 AV > T(T4 E C0[TS 0E.T 0[TS 0

U.@ 4 4J58 .` O 1 ()V R.?I0?*I

[S0E. 0 E O I ) T(0. 74[ T T CT8.P'

Q2)'P5 ..

(0 : 0T)1P I ?*I L0T>.; 'OT T@ T)

U' 1W ` 4[S h0E. 0MJ@ .UxT k1 T5 =

56'.P171@ l 1: '4 aI0o LW8) 0ATE

) I) xP ?.G O 9 : (j .8') 5hoT T.IV O '

.P .` S1@ 1 f@ 4) .@ hS( <1'1T) E (

a U O q*) .6 0E . I .:

)(2p 0( f (t t ))2

p 0f (t) 1 2 ( f (t t )) e− π⋅ ⋅ − = − ⋅ π ⋅ ⋅ −

8. 70fp4t0@.@ I54 Rg )U. R_) ATE T5 T .

./> I 5 A') .I(' U a k1 t

.I0.84 K) E C0) E. 5h:.T sT .TI JTN

T_J <1 (iS E F) MN 4 hS( <1t=0 TN

'1 .=6)( @)E e*'; 45 1 <1 .T6

I5 A' .

0.I .I 1( ') ITI AE OE1> OV

4 4 U.> .U> 0.) 5E I 4 O[0TaI A4I AE

)Τ=tc2/2b ()O O ?) .) )U. OE1> 5) [TI

J R` OV 5T(0TP [I 'E TI 40)TU.

aI A4I)Ω=ωb/πc2(O p ) . LuD a tω4b

@.@ I54E )U. AE R0) 45(7 h0I 9 : 5 A'T) .

4'U 1 Q1(5[6'i+T) AT8 TaI A4TI )U.

(70.I <1 1; I 9 : 'J5) X5hJN.

@6>6+E ! FG9 ?!

w8T)1@ TU T AV T L0 1TW [TS h0

E.0 p TMJ@ OT5 =56'7 TOE. Z0O C

/&- /# :/# /&->

2

/# ?/%H

X

Y

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

-0.5

0

0.5

1

0 1 2 3t/t0

Nor

mal

ized

Dis

plac

emen

t (ا.-)

0

0.5

1

0 1 2 3 4f/f0

Nor

mal

ized

Am

plitu

de(ب)

3:–J+ - K L " /MN-O / J P /

8 A3) LW8 .=6)(O 0) HF I 5 hT3T) O. A

'<1 ?I@ l.a U 8 SV 4PT T.P .T` .

) O 5 h:a O.'.I.I (r).I <1 :. '0.TI.I (c2)

.TT9 4)1TT17 R'.TTI.I / .TTJ5\FTT 4 O X

)5h@.@ I JN5E O I R|4~E.T AT0T

O . TU T .Ur A T T)4 XT1@ T L][

4 6+TT'+TT1 45A][+TT1 54 4 A6+TT']|[

4U][fU 4 [ [e) 451M][)5 .T` TS1@ 1 [

.P .TJATTJ OT 5J@ T)8 O XT5m

U .9>A+3/ 0N1G 5+TI u5aT9 5g T5.

A+3/ O .TU l. T) .T)T TI E TI OTV 0Tj

J R` aI A4I )U.5(0P [I 'O p ) .

J+ (?RB S

0(M)()(4)|(J 5(0P [I ' I OTV

TTJ@ LTT5 TT TTI mHTTS. XTT1@ OTT TTp .

]&|4[E TI 0(+TT)U. 0TTaI A4TI //4/

1 O .` + . T6)j T T O'4 .T8 1T

hpTT` hSTT( <1TT TT>SV 4P)1TT*8 'iTT 'TT5 A

4 hp` 1 4 .8 (I1S'B@55 A6 . O' 1T .

10U) wG U '5) AJ T(I1S N1G A0

I )4J 1S4 A.W5(0P [I 'T I OTV

J@ L5 E m5) ) J .WO .U 1i: A .

K( S

hS( <1P4E I Q128 |RTg )U. I 4 S

aI A4I0.I.I |/4 .I 0I@ O 5 O O .0(M

)~(4)(J5(0P [I ' IJ@ OV 5T9> m.

I HS. X1@ O p .]&|[E I 0(+T)U. 0

aI A4I/4/ O .U + .AT8 TU 1T; T O

'T 4 hpT` 1 4 .8 1'B@T55 )1T*8 AT6 .'

i'5 (I1S A O' 1 .=6)(T1 T804 hpT`

TT' @ )TTE TTe*'B@55) ATT6.TTF 0a`1 TTI TT5

)~/x/r=- ( E =i` u_J 5O k. <1 A

.I 1 O X1@' O .` .A8 =6T U 1;

O')1*8 1'+I5I1G 'T5T1 A T80iTJ

B@ O551 4 A6.80MJ@ 5TM'TE \FT EV TU.> 5L

]4[ 1S4 . 4' U 1 80hpT` T1 4

4'E EV U.> 5@.@ TI L T5 R~/4|/T .TI.I

hS( <1P ].[

@6:6&T ! FG9 ?!

w8T)1@ TU T AV T L0 1TW [TS h0

E.0MJ@ O p 5 =56'OE.T ZT7 0O T80

.IV'4 LW8 )8 A3) LW8 .=6)(T.IV O 'TI

) HF5 h3) O.T A'8 Ta U TTI@ l.T ?T<1

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

%

40r

20r

r

X

Y

3r

NزPQQRS NPن هPVWا

3=6 FG9 L+E ! /' ?!

Incident P- waveIncident Angle 90

0

1

2

3

4

Am

plifi

catio

n

Incident SV- waveIncident Angle 90

0

1

2

3

4

-2 -1 0 1 2

X/R

Ampl

ifica

tion

Hrz., Present Study Vrt., Present StudyHrz., Wong [14] Vrt., Wong [14]Hrz., Dravinski and Mossessian [15] Vrt., Dravinski and Mossessian [15]Hrz., Mossessian and Dravinski [16] Vrt., Mossessian and Dravinski [16]

3@6( " / LG' UV# L $ W V X/Z?RB S

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

Incident P- waveIncident Angle 90

0

1

2

3

4

Am

plifi

catio

n

Incident SV- waveIncident Angle 90

0

1

2

3

4

-2 -1 0 1 2

X/R

Am

plifi

catio

n

Hrz., Present Study Vrt., Present StudyHrz., Wong [14] Vrt., Wong [14]

3X6( L" / LG' UV# $ W VZ/>?RB S

Incident P- waveIncident Angle 90

0

1

2

3

4

Ampl

ifica

tion

Incident SV- waveIncident Angle 90

0

1

2

3

4

-2 -1 0 1 2

X/R

Ampl

ifica

tion

Hrz., Present Study Vrt., Present StudyHrz., Wong [14] Vrt., Wong [14]Hrz., Dravinski and Mossessian [15] Vrt., Dravinski and Mossessian [15]Hrz., Mossessian and Dravinski [16] Vrt., Mossessian and Dravinski [16]

3[6 $ W V " / LG' UV# L( Z/:?RB S

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

#

Incident P-Wave Incident Angle 60

0

1

2

3

-2 -1 0 1 2

X/R

Am

plifi

catio

n

Hrz., Present Study Vrt., Present StudyHrz., Wong [14] Vrt., Wong [14]Hrz., Dravinski and Mossessian [15] Vrt., Dravinski and Mossessian [15]Hrz., Mossessian and Dravinski [16] Vrt., Mossessian and Dravinski [16]

3\6" / LG' UV# L( $ W V X/Z/" S [Z?). P /) P

Incident P-Wave Incident Angle 60

0

1

2

3

-2 -1 0 1 2

X/R

Ampl

ifica

tion

Hrz., Present Study Vrt., Present StudyHrz., Wong [14] Vrt., Wong [14]

3]6# L( $ W V " / LG' UVZ/>/" S [Z?). P /) P

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

5 5.5 6 6.5 7 7.5 8 8.5

Normalized Time

Nor

mal

ized

Dis

plac

emen

tNumerical Solution (Hrz.)Numerical Solution (Vrt.)Analytical Solution (Hrz.)Analytical Solution (Vrt.)

3^69 /-_ L " /MN/G9 $ ;`&N / a)\>@=/>Z6x/r= ( UV# "T $

P S " /H#P" Dc /[Z/)

80r

3r

X

Y

B(-r,0) A(0,0)

[\]^_

`abc def

NزPRS NPن هPVWا

40r

3>Z6 FG9 L&T ! /' ?!

-2

-1

0

1

2

3

0 1 2 3 4 5 6

Normalized Time

Nor

mal

ized

Dis

plac

emen

t

Numerical SolutionAnalytical Solution

3>>69 /-_ L " /MN/G9 L9 $ A3 " )>Z( UV# "T $

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

-4

-2

0

2

4

6

0 2 4 6 8 10 12 14

Normalized Time

Norm

aliz

edDi

spla

cem

ent

dt=0.223dt=0.167

3>:6 _ L " /MN/G9 L9 $ A3 " )>Z(J&N L " D

-2

-1

0

1

2

3

0 2 4 6 8 10 12 14

Normalized Time

Norm

aliz

edD

ispl

acem

ent

dt=0.223 (Hrz.)dt=0.223 (Vrt.)dt=0.167 (Hrz.)dt=0.167 (Vrt.)

3>=6 _ L " /MN/G9 $ B3 " )>Z(" D J&N L

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

"

Incident P- wave

012345678

Ampl

ifica

tion

Incident SV- wave

0

1

2

3

4

5

6

7

8

-3 -2 -1 0 1 2 3

X/R

Ampl

ifica

tion

Hrz., Present Study Vrt., Present StudyHrz., Dravinski and Mossessian [15] Vrt., Dravinski and Mossessian [15]

Hrz., Mossessian and Dravinski [16] Vrt., Mossessian and Dravinski [16]

3>@6( " / LG' UV# L $ W V X/Z

hp` SV 4PI RTg )U. /T T.P .T` .TJ5X

J 4 .IV5) X5h@.@ TI JTN T5 E O T TI R|4~

E. A0 O .4 LW8 > 4 )._) LW8

O .P . T 4'TU 1T uD Ta)(4)(TI

@.@5OE. R0O 4J 80T) 4 T.IV 5hJTN I T5 A

') .

(LW8 >

TJ 4 .IV \ U AV .I l. > L5X

N1TG A8 E O 5T) 1G.I =Ti` T u .=6T)(

1 80 4 hp` ' @ )TE e*'B@55T) AT6.5JTN h

) EV U.> ( T I MJ@ .5TM'AV][aT94 5<1T

hS(SV 1 O .` +. 6)jT O' 1T

)1*8'+I5I1G '5) A@iTJ 'MJ@ 45TM' 1TS4

_) U )1W)8 4 'B@ 4 55hpT` T1 A6 .

.I ()E 8 .N .I ..I0a94 5hS( <1 P)5[

)(I3 'IS I S'1 T80B@T55 T TI AT6 .

'V.

q(>)LW8

G > 5.9 U AV .I l. .A117 R

.I <1 :.'.I 4 4'.IV \ '@.@ I 5R

4+04 ) h3 C_) .5J .5+8 O X.

LM+1@ T4 X+T6'1 4T+5A][+1 45T4 A

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

,

Incident P- wave

0

1

2

3

4

5

6

7

8

Ampl

ifica

tion

Incident SV- wave

0

12

3

4

56

7

8

-3 -2 -1 0 1 2 3

X/R

Am

plifi

catio

n

Hrz., Present Study Vrt., Present StudyHrz., Dravinski and Mossessian [15] Vrt., Dravinski and Mossessian [15]

Hrz., Mossessian and Dravinski [16] Vrt., Mossessian and Dravinski [16]

3>X6( " / LG' UV# L $ W V \X/Z

-4

-2

0

2

4

6

0 2 4 6 8 10 12 14

Normalized Time

Norm

aliz

edDi

spla

cem

ent

BEM, dt=0.223Hybrid, dt=0.112Hybrid, dt=0.167Hybrid, dt=0.223

3>[69 /-_ L " /MN/G9 L9 $ A3 " )>Z(a" $+ * " !%&'

dV a.J&N L "

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

-2

-1

0

1

2

3

0 2 4 6 8 10 12 14

Normalized Time

Nor

mal

ized

Dis

plac

emen

t

BEM, dt=0.223Hybrid, dt=0.112Hybrid, dt=0.167Hybrid, dt=0.223

3>\69 /-N_ L " /M/G9 L9 $ B3 ")>Z(a" $+ * " !%&'

dV a.J&N L "

ghVل هjS :kWا

ghVهPf لjS :ب

2b

50b

25b

X

Y

h

NزPRS NPن هPVWا

3>]6' /e /' 3 L')J+ (+ # ?&)K(+ # ?&

0

1

2

3

4

-2 -1 0 1 2x/b

Am

plifi

catio

n

Homogeneous Scheme (Vrt.)Homogeneous Scheme (Hrz.)Non-Homogeneous Scheme (Vrt.)Non-Homogeneous Scheme (Hrz.)

3>^69 /-( a./ L+ " !T U'+ # ?& # ?&

() * / $ W V a" / a" Z/>?). P SV

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

0

1

2

3

4

-2 -1 0 1 2x/b

Am

plifi

catio

n

Homogeneous Scheme (Vrt.)Homogeneous Scheme (Hrz.)Non-Homogeneous Scheme (Vrt.)Non-Homogeneous Scheme (Hrz.)

3:Z69 /-( a.+ " !T U' / L+ # ?&# ?&

() * / $ W V a" / a" X/>P ?).SV

4 6+']|[)5 O => [.JATJ O 5O X

U =i` )8 A+3/ TI 5.'T +TI 5aT9 5

O .U l. ) .0(M)(4)(1 T80T 4 hpT` '

@)E e*'B@55F) 4 A6.A4B=6T E )(TI

E0)TTE QTTP 4 '/4|~/ 1TTTT O .TT` +

. 6)j O')1*8 1'+I5I1G '5T(I1S A

1TTS4 .0(MTT)(4)(J5TT(0TTP [I 'TTI

J@ OV 5 T TI .T9> mXT1@ OT Tp .

HS.][4]|[E TI 0(+T)U. 0TaI A4TI /4~/

O .U + . U 1F)8 O'T1 4 .T8 1

4 hp`'B@55)1*8 A6 .'+TI 5iT 'T5T(I1S A

O' 1 .0(M)|(4)~(@.@ TI T5 .T8 RTE C

K)A4B=6 E )(1 T80T ' @ )TE Te*'

B@55.[S n4 4 E O I U A60E. 0U.@ 4T5R

[S n4 I AV0 4J ][ I OV 1 ) T +

O .` .T)[TS n4 0E.T 0QTP E O T TI

)E'/) 4T(P E O I RU. n4 0)TE '

/|~/4/TT TTIOTTV TT) . TT6)jTT O

'> 1 'n4 U[S0TE.0E TI0)TE QTP '

I'aI/)5209 ?*I T I T 'T) T8

E I (@ RU. n4 E MN>0P (0)TE 'E .6j1TU

/ 0` '1i` =I` ). u i: I=> .W M+T TI

[S n4 E O 0E. 0` 5 Ta@ RU. n4 I s

(P0)E 'U.0) 5 81G E.

@6=6' FG9 /e '

w8T)1@ TU T AV T L0 1TW [TS h0

E.0MJ@ O p 5 =56'OE. Z7 0f@ A3) 8

8 .If@ 1_ L0) HF I 5h51'M6T i+T) TI '

)T) I @ 5T(7 h(.TI.I/N1TG AT8 45 T u0

O .P ._) 4 \ .?IT@ l.a f@

hp` <1SV Rg )U. I / .P .` ..I0=T>

M+ O LW8) 4 LW8 4 E O .T

LW8=6)& () 4 f@ :125hu 1TN I JN

J C5XO .P ._) j f6.I 40?T TI0E AV

a@~E. A 0T OT O .LW8T) T

=6)&q(J54 f@ X)5hE JN T.u 1TN TI AV L

O )PS .I 4 )0?I0@.@ TI ()V T5E R4

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

|E. A 0 O O .

0(M)(4)(J5(0P [I ') 4 K0f@ 4

75LAVk1 n4 4 E O I U IOV TI T)

E0(+)U. 0TaI A4I /4/ 1T TO .T` +

.A8 U 1; O'T 4 hp` 1 4 .8 1'

B@55)1*8 A6 .'+I5I1G '5) AT I E OTV

n4 4 O' 1 .

X6& /c

L 1WMJ@ =TU h T5 OE.T ZT7 =0l 1T:

.P171@'4 aI0A4 hS( <1 .I.I LW8) JN

SV 4Po 0[TS n4 XT1@ AE 0E.T 0Tp

O . : j => I0O T> T(j .T8 TU

G') 5ho .IV O 'f@ 4T .I 'PT5O A3T) .

U O .I TU [TS n4 0E.T 0TaI4 0

o0AE 'OE1T> I )1@MJ@ T5 OE.T =0(8WGT0

n.+P LW8)I ./.@ T5 1TW Rn4 TI OT Tp h

[S0E 4J 50OE. Z7 A1I @ .U 81G h8. 0

(8WG08) g I LW5TFG.'V OT[2@ T 4

MJ@5= ./.

F)

1. Bouchon, M., “Effect of Topography on Surface Motion,” Bull. Seismol. Soc. Am., Vol. 63, pp. 615-632, 1973.

2. Lysmer, J., and Kuhlemeyer, R. L., “Finite Dynamic Model for Infinite Media,” J. of Engineering Mechanics, ASCE, Vol. EM4, pp. 859-877, 1969.

3. White, W., Valliappan, S., and Lee, I. K., “Unified Boundary for Finite Dynamic Models,” J. of Engineering Mechanics, ASCE, Vol. 103(5), pp. 969-964, 1977.

4. Wong, H. L., “Effects of Surface Topography on the Diffraction of P, SV and Rayleigh Waves,” Bull. Seismol. Soc. Am., Vol. 72, pp. 1167-1183, 1982.

5. Dravinski, M., and Mossessian, T. k., “Scattering of Plane Harmonic P, SV, and Reyleigh Waves by Dipping Layers of Arbitrary Shape,” Bull. Seismol. Soc. Am., Vol. 77, pp. 212–235, 1987.

6. Mossessian, T. K., and Dravinski, M., “Application of a Hybrid Method for Scattering of P, SV, and Reyleigh Waves by Near-Surface Irregularities,” Bull. Seismol. Soc. Am., Vol. 77, pp. 1784-1803, 1987.

7. Kawase, H., “Irregular Ground Analysis to Interpret Time Characteristics of Strong Ground Motion Recorded in Mexico City during 1985 Mexico Earthquake,” In Ground Motion and Engineering Seismology, Ed. Cakmak A.S., Development in Geotechnical Engineering, Vol. 44, pp. 467-476, 1987.

8. Kawase, H., “Time-Domain Response of a Semi-Circular Canyon for Incident P, SV and Rayleigh Waves Calculated by the Discrete Wavenumber Boundary Element Method,” Bull. Seismol. Soc. Am., Vol. 78, pp. 1415-1437, 1988.

9. Kawase, H., and Aki, K., “A Study of the Response of a Soft Basin for Incident S, P and Rayleigh Waves with Special Reference to the Long Duration Observed in Mexico City,” Bull. Seismol. Soc. Am., Vol. 79, pp. 1361-1382, 1989.

10. Sanchez-Sesma, F. J., and Campillo, M., “Diffraction of P, SV and Rayleigh Waves by Topographic Features: a Boundary Integral Formulation,” Bull. Seismol. Soc. Am., Vol. 81, pp. 2234-2253, 1991.

11. Sanchez-Sesma, F. J., and Campillo, M., “Topographic Effects for Incident P, SV and Rayleigh Waves,” Tectonophysics, Vol. 218, pp. 113-125, 1993.

12. Sanchez-Sesma, F. J., and Luzon, F., “Seismic Response of Three Dimensional Alluvial Valleys for Incident P, SV and Rayleigh Waves,” Bull. Seismol. Soc. Am., Vol. 85, pp. 269-284, 1995.

13. Papageorgiou, A. S., and Kim, J., “Propagation and Amplification of Seismic Waves in 2D Valleys Excited by Obliquely Incident P- and SV- Waves,” Int. J. Earthq. Eng. and Struct. Dyn., Vol. 22, pp. 167-182, 1993.

14. Pedersen, H. A., Sanchez-Sesma, F. J., and Campillo, M., “Three-Dimensional Scattering by Two-Dimensional Topographies,” Bull. Seismol. Soc. Am., Vol. 84, pp. 1169-1183, 1991.

15. Reinoso, E., Wrobel, L. C., and Power, H., “Two-Dimensional Scattering of P, SV and Rayleigh Waves: Preliminary Results for the Valley of Mexico,” Int. J. Earthq. Eng. and Struct. Dyn., Vol. 26, pp. 595-616, 1997.

16. Hadely, P. K., Askar, A., and Cakmak, A. S., “Scattering Of Waves By Inclusions In A

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020

, "%, %

Nonhomogeneous Elastic Half Space Solved By Boundary Element Methods,” Technical Report NCEER-89-0027, 1989.

17. Mansur, W. J., “A Time-Stepping Technique to Solve Wave Propagation Problems Using the Boundary Element Method,” Ph.D. Thesis, Southampton University, 1983.

18. Antes, H., “A Boundary Element Procedure for Transient Wave Propagation in Two-Dimensional Isotropic Elastic Media,” Finite Element Analysis Design, Vol. 1, pp. 313-322, 1985.

19. Israil, A. S. M., and Banerjee, P. K., “Advanced time Domain Formulation of BEM for Two-Dimensional Transient Elastodynamics,” Int. J. for Num. Methods in Eng., Vol. 29, pp. 1421-1440, 1990.

20. Israil, A. S. M., and Banerjee, P. K., “Two- Dimensional Transient Wave Propagation by Time Domain BEM,” Int. J. Solids and Structures, Vol. 26, pp. 851-864, 1990.

21. Israil, A. S. M., and Banerjee, P. K., “Advanced Development of Boundary Element Method for Two-Dimensional Dynamic Elasto-Plasticity,” Int. J. Solids and Structures, Vol. 29, pp. 1433-1451, 1992.

22. Kamalian, M., “Time Domain Two-Dimensional Hybrid FEM / BEM Dynamic Analysis Of Non-Linear Saturated Porous Media,” Ph.D. Thesis, Tehran University, 2001,(In Farsi).

23. Kamalian, M., Gatmiri, B., and Sohrabi, A., “On Time-Domain Two-Dimensional Site Response Analysis of Topographic Structures by BEM,” Journal of Seismology and Earthquake Engineering,Vol. 5(2), pp. 35-45, 2003.

24. Kamalian, M., Jafari, M. K., Dehghan, K., Sohrabi, A., and Razmkhah, A., “Two-Dimensional Hybrid Response Analysis Of Trapezoidal Shaped Hills In

Time Domain,” Advances in Boundary Element Techniques IV, Ed. R. Gallego, and M.H. Aliabadi, pp. 231-236, 2003.

25. Gatmiri, B., Kamalian, M., “Time Domain Two-Dimensional Hybrid FEM / BEM Dynamic Analysis Of Non-Linear Saturated Porous Media,” 2nd Canadian Specialty Conference On Computing In Geotechnique, pp. 216-221, 2002.

26. Gatmiri, B., and Kamalian, M., “Combination of Boundary Element and Finite Element Methods for Evaluation of Dynamic Response of Saturated Porous Media,” 5th European Conference on Numerical Methods in Geotechnical Engineering, pp. 947-955, 2002.

27. Takemiya, H., and Fujiwara, A., “SH-Wave Scattering And Propagation Analysis At Irregular Sites By Time Domain BEM,” Bull. Seism. Soc. Am., Vol. 84, pp. 1443-1455, 1994.

28. Brebbia, C. A., and Dominguez, J., Boundary Elements, An Introductory Course, Computational Mechanics Publications, Southampton, Boston, 1989.

29. Ahmad, S., and Banerjee, P. K., “Multi-Domain BEM for Two-Dimensional Problems of Elastodynamics,” Int. J. for Num. Methods in Eng., Vol. 26, pp. 891-911, 1988.

30. Wolf, J. P., Dynamic Soil-Structure Interaction,Prentice Hall, 1985.

31. Kamalian, M. Jafari, M.K. Sohrabi-Bidar, A. Razmkhah, A. and Gatmiri, B. Time-Domain Two-Dimensional Site Response Analysis of Non-Homogeneous Topographic Structures by A Hybrid FE / BE Method, Soil Dyn. Earthquake Eng., Accepted, 2005.

Dow

nloa

ded

from

jcm

e.iu

t.ac.

ir at

8:2

4 IR

DT

on

Wed

nesd

ay A

pril

1st 2

020