Transmitting Boundary Conditions Suitable for Analysis

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Transmitting boundary conditions suitable for analysis

of dam-reservoir interaction and wave load problems

M. Cetin a, Y. Mengi b,*

a Machinery Laboratory, Turkish Standards Institute, Ankara, Turkeyb

Department of Engineering Sciences, Middle East Technical University, Ankara 06531, TurkeyReceived 28 July 2001; received in revised form 4 November 2002; accepted 9 January 2003

Abstract

Transmitting boundary conditions (tbcs) are developed, suitable for both boundary and finite element

analyses, for the radiation of waves propagating in horizontal direction along a compressible inviscid fluid

layer. The derivation of the proposed tbcs uses a completely continuum approach and is based on the

spectral properties of radiating waves. The formulation is presented in Fourier space and accommodates

the effects due to surface waves, as well as, due to the radiation of waves in viscoelastic foundation. The

proposed tbcs may be used in the analyses of both dam-reservoir systems and wave load problems offloating or submerged bodies. Here, for assessment, they are used in the analyses of two simple benchmark

problems. The results indicate that, when used in conjunction with boundary element analysis, the pro-

posed tbcs not only improve the prediction of BEM, but also reduce the computational load of the

analysis.

2003 Elsevier Science Inc. All rights reserved.

Keywords: Transmitting boundary conditions; Compressible inviscid fluid; Two dimensional; Radiation of waves

1. Introduction

To explain the object of the study, we refer to Fig. 1 showing a dam-reservoir-foundationsystem, which is under the influence of a seismic environment. In the figure, the interface with

foundation is designated by L and, to make the solution region finite, the reservoir is divided intonear and far fields through the use of a vertical truncation surface V. We assume that the near fieldcontains geometrically irregular part of the system, and the far field is regular and has a uniform

* Corresponding author. Fax: +90-312-2101269.

E-mail address: [email protected] (Y. Mengi).

0307-904X/03/$ - see front matter 2003 Elsevier Science Inc. All rights reserved.doi:10.1016/S0307-904X(03)00048-9

Applied Mathematical Modelling 27 (2003) 451470

www.elsevier.com/locate/apm
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depth. In the analysis, we retain the part of the system containing the near field and perform theanalysis by considering the influence of the far field on near field. In this study, some conditions

(equations), transmitting boundary conditions (bcs), on V representing the influence of the farfield on near field (or, more precisely, the influence of radiation of waves propagating away fromV on near field) are developed in a form suitable for performing the analysis in near field by

boundary or finite element method. Of course, it goes without saying that, in boundary element(BE) analysis, it is possible to keep the sides of the solution region open, but, as it will be seen

Nomenclature

p fluid pressurevi velocity components for fluidqf0 initial mass density of fluidc speed of soundx angular frequencyk; a wave numbersg gravitational accelerationcp dilatational wave velocityz hysteretic damping ratioqs mass density of foundationb

qf0

qscpa parameter of foundation

P pressure amplitudeVi fluid velocity amplitudeaj mode participation factorsav vertical ground accelerationp1, p0 homogeneous and inhomogeneous solutions for fluid pressure

Fig. 1. Division of reservoir into two regions.

452 M. Cetin, Y. Mengi / Appl. Math. Modelling 27 (2003) 451470


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from the example problems considered in the study, the use of transmitting bcs (tbcs) mentioned

above improves the prediction of boundary element method (BEM) considerably and reduces thecomputational load of the analysis.

Some tbcs are already proposed in literature, see [15], for the vertical truncation boundary Vin Fig. 1 accommodating in the analysis the influence of radiation of waves in infinite fluid do-main. In these works, two-dimensional tbcs are derived for a compressible inviscid fluid through

the use of a continuum solution in the far field coupled with one-dimensional finite elementdiscretization employed in vertical direction along V (see Fig. 1). The influence of surface waves

was disregarded in the derivation. The formulation stated above requires, when carried out inFourier or Laplace space, the solution of a discrete eigenvalue problem, which determines themodes to be considered by tbcs. The tbcs proposed in [15] are in general suitable when the

analysis in near field is performed by finite element method (FEM); but, the same bcs are also usedin [2], even though in that work, BEM is employed for near field. This is accomplished by carrying

the field variables, through a transformation, from finite to boundary element nodes. It is obviousthat such a transformation would involve some approximations and be a new source of errors innumerical results. Other techniques, instead of using tbcs, are also employed in literature to in-clude in the analysis the radiation damping due to unboundedness of fluid domain. For example,

for this purpose, in [6] fluid hyperelements and in [7] infinite elements are used. Both of theseelements are suitable for performing the analysis by FEM.

As mentioned previously, in the present work, a new type of tbcs is proposed (in Fourier space),representing in the analysis the effect of radiation of waves in fluid domain. The proposed tbcs

could be used in conjunction with performing the analysis in near field by BEM or FEM. Thederivation of these tbcs is based on spectral theory of waves propagating in horizontal directionalong the fluid layer and involves no finite element approximation present in tbcs proposed in [1

5]. In the formulation, the surface wave bcs are taken into account at the top surface of fluid layer;special absorbing bcs proposed in [1] are used at the bottom surface to account for the influence ofviscoelastic foundation in the analysis. The tbcs are presently developed for two-dimensional case

and when the fluid is compressible inviscid. The extension of these spectral-theory-based tbcs tothree-dimensional and viscous cases will be the subject of a future work.

The proposed tbcs may be used effectively in dynamic interaction analyses of dam-reservoir-

foundation systems and wave load problems by BEM or FEM. In present work, they are assessedby applying them to BE analysis of some benchmark problems involving dam-reservoir systems.

The applications to wave load analysis of submerged bodies will be given in a separate paper.The paper is organized as follows. In Section 2, we establish the spectrum for the waves

propagating in horizontal direction along a fluid layer. This section also contains the orthogo-nality conditions for the mode shapes induced by surface waves, which are needed in the for-mulation of tbcs. In Section 3, we develop the proposed tbcs using the spectral properties

established in previous section. The last section contains some benchmark problems.

2. Spectral properties of surface waves

As stated in Section 1, the object of this study is to develop some tbcs on V (see Fig. 1), suitable

for two-dimensional analysis, accounting for the influence of radiation of waves in the analysis of

M. Cetin, Y. Mengi / Appl. Math. Modelling 27 (2003) 451470 453


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dam-reservoir-foundation or wave load problems. As the derivation of these tbcs will be based onspectra of surface waves, we study here the spectral properties of these waves together with the

orthogonality conditions.To this end, we refer to Fig. 2 showing fluid layer of depth h, which represents the far field in

Fig. 1. The fluid layer, which is referred to x1x2 rectangular coordinate system as shown in Fig. 2,

is under the influence of linear surface waves propagating to the right. We assume that the fluid iscompressible inviscid and undergoes small disturbances with respect to initial state. We further

assume that the fluid behaves barotropically and thus, pressure would be the function of densityalone. Under these conditions, the the field equations of flow would be

Momentum equation

ovi

ot 1

qf0

op

oxi1

Continuity equation

oqf

ot qf0

ovj

oxj 0 2

State equation

p c2qf 3where t is time; p, qf and vi denote, respectively, the dynamic values of pressure, mass densityand velocity (i.e., the deviations of these variables from their initial values); qf0 is the initial massdensity and c is the speed of sound.

Eqs. (1) and (2) are written in indicial form, where the range of indices should be taken from 1to 2 and the repeated index implies summation over its range. Through the eliminations in Eqs.(1)(3), one obtains

r2p 1c2

o2p

ot2 0 4

or, in Fourier space,

r2pF x2

c2pF 0 5

Fig. 2. Fluid layer under the influence of surface waves.



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which is the governing equation for pressure. Here, r2 is two-dimensional Laplacian operator, xdesignates Fourier transform parameter (angular frequency) and the superscript F denotes the

Fourier transform. To simplify the notations, we drop, from now on, the superscript F in the

equations written in Fourier space. When the pressure p is determined from Eq. (5), the velocityfield may be found from (in Fourier space), in view of Eq. (1),

vj 1ixqf0

op

oxj6

where i is the imaginary number.To derive the dispersion relation for the surface waves propagating along the fluid layer, for the

pressure p (defined in Fourier space) we put

p Peikx1 7which correspond to pressure waves propagating in x1 direction. Here, P is the amplitude of thedynamic pressure pand function ofx2 only, kdenotes the wave number in x1 direction. The wavenumber is related to the wavelength k by

k 2pk

Insertion of Eq. (7) into (5) yields

d2P

dx22 k2 x

2

c2

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}a2

P 0 8

where a denotes the wave number in x2 direction. Solution of Eq. (8) is

P Aeax2 Beax2 9The constants A and B in Eq. (9) are to be found from bcs at x2 0 and x2 h. The free

surface wave bc at x2 0 is, in Fourier space,op

ox2 x

2

gp

which, in terms of the pressure amplitude P, becomes

P0 x2

gP at x2 0 10

where the prime designates the derivative with respect to x2 and g is gravitational acceleration.On the other hand, the base of the fluid layer lies on an elastic foundation (see Fig. 2). The in-fluence of the radiation of waves in elastic foundation on the wave pattern in fluid layer could be

taken into account approximately by the condition at x2 h, in the absence of ground motion,op

ox2 bixp 11



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which is proposed in [1]. Here, b is defined by

b qf0qscp

12

with qs and cp ( ffiffiffiffiffiffiffiffik2Gqsq ; k, G: Lames and shear moduli) being respectively the mass density anddilatational wave velocity of elastic foundation. If desired, viscoelastic effects in the foundation

may be introduced by using hysteretic mechanism [8], or more precisely, by replacing the dila-tational wave velocity cp by, in Fourier space,

cpffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2izp

where z is hysteretic damping ratio. We now express the bc in Eq. (11) in terms of the pressure

amplitude P. For that, we insert Eq. (7) into (11), which gives

P0 bixP at x2 0 13

It may be noted that this equation holds in the absence of ground motion. However, as will beseen in the next section, the effect of ground motion may be included in the formulation of tbcsthrough the use of a particular solution involving ground excitation.

We now proceed with the derivation of dispersion relation of surface waves. For that, we insertthe solution for P in Eq. (9) into the bcs in Eqs. (10) and (13), which gives

a x2g

a x2

g

a bixeah a bixeah

" #A

B

! 0

0

!14

To have a nonzero solution, the determinant of the coefficient matrix in Eq. (14) should vanish,which yields

tanhah a x2

g abixa2 bix x2

g

15

with

a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 x

2

c2

rwhich is the desired dispersion relation relating the frequency x to the wave number k ofsurface waves propagating in horizontal direction along the fluid layer. This relation gives thespectrum of surface waves, which can be described geometrically by spectral lines on x kplane governed by Eq. (15).

As an example, the spectrum when the base is rigid is given in Fig. 3; for this case, we note thatb 0, therefore, the dispersion relation is given by, in view of Eq. (15),

a tanhah x2

g16

From the figure we note that, for a fixed frequency (say, for x x), there exists infinite numberof k, each corresponds to a different mode. The modes are designated by the number 1, 2,

etc. in the figure. For the modes, the wave number might be real or imaginary. In the figure,xi i 1; 2; . . . denote the cut-off (free vibration) frequencies at k 0, which are governed by theequation



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tanx

c cx

gIn the figure, the dispersion curves for imaginary k are plotted as dashed lines. The modes withreal and imaginary k represent, respectively, the propagating and decaying modes.

The spectrum for incompressible case is given in Fig. 4 when the base is rigid, for which the

speed of sound c is infinite, therefore, a k and the dispersion relation becomes, in view of Eq.(16),

x2 gktanhkh 17From the figure we see that there is only one propagating mode in this case.

We now present the mode shape for the pressure amplitude P associated with a point (x, k) on a

dispersion line of spectrum. It can be determined by inserting the solution of Eq. (14) into (9),which results in

P C a coshax2

x2

gsinhax2

18

where C is a constant. C might be related, if desired, to the displacement amplitude w of thewaves at the top surface of fluid layer (see Fig. 2) by

C qf0gwa

Fig. 3. Spectrum for compressible case (when the base is rigid, b 0).



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The mode shapes for the velocity amplitudes Vi i 1; 2 defined byvi Vix2eikx1

may be obtained through the use of Eq. (18) in (7) and (6), which gives

V1 kqf0x

P; V2 1qf0ix

P0 19

2.1. Orthogonality conditions

Here, we derive the orthogonality conditions for the mode shapes of pressure induced bysurface waves, which will be used later in the derivation of tbcs.

To this end, we first write the governing equation of the pressure amplitude P, Eq. (8), in theform

LP a2P 0 20where the operator L is defined by L d

dx22

and P satisfies the bcs

Fig. 4. Spectrum for incompressible case (when the base is rigid, b 0).



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P0 x2

gP at x2 0

P0bixP at x

2 h

21

It is very easy to show that, for a fixed x, the operator L is self-adjoint (for the definition of self-

adjoint operators, see [9]), that is, it satisfies

hLP;Qi hP;LQi 22if Q satisfies the same bcs as P, that is, the bcs in (21). In Eq. (22), h. . . ; . . .i denotes the innerproduct defined by

hf;gi Z0

hfx2gx2dx2

We now consider the modes corresponding to a fixed x and number them as 1, 2, etc. (seeFig. 3). From Eq. (18) it follows that the mode shape of P for the ith mode is given by, in nor-

malized form,

Pi ai coshaix2 x2

gsinhaix2 23

where

ai ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2i

x2

c2rwith ki being the wave number of the ith mode. It is obvious that Pi satisfies Eq. (20) with a ai,that is,

LPi a2iPi 24The inner product of Eq. (24) with Pj (which is the jth mode ofP) gives

hLPi;Pji a2i hPi;Pji 25We now interchange i and j in Eq. (25), which yields

hL

Pj;

Pii

a2j hPj;P

ii 26

Finally, subtraction of Eqs. (25) and (26) results in, in view of self-adjoint property of the operatorL (that is, in view of Eq. (22)),

a2i a2j hPi;Pji 0which implies that if the a value of two different modes are distinct, then the corresponding modeshapes would be orthogonal, that is

hPi;Pji 0 for i 6 j 27



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For i j, one may compute hPi;Pii in view of expression given for Pi in Eq. (23). The result is

bi h

Pi;Pii

1

8aiai

x2

g 2

1

e2aih

1

8aiai

x2

g 2

1

e2aih

h

2

a2i x4

g2 28It should be noted that the orthogonality condition in Eq. (27) holds for a fixed frequency x x,that is between the modes along a horizontal line x x in Fig. 3.

3. A new type of transmitting boundary conditions (tbcs) based on spectral theory

Here, we formulate, in Fourier space and in a form suitable for boundary or finite elementanalysis, a new type of tbcs for the boundary V (Fig. 1). In the formulation, we use the spectral

theory of surface waves presented in Section 2. We assume that the geometrically irregular part of

the solution region is contained in near field and the far field has a uniform depth h. We furtherassume that the foundation is elastic. Viscoelastic effects in foundation may also be considered in

the formulation through the use of hysteretic damping [8]. It may be noted that the rigid foun-dation case is a special case of the formulation, presented here, for which one should put b 0,where b is defined in Eq. (12).

Tbcs will be derived first in the absence of ground motion; then, they will be extended to thecase when the base excitation is present.

3.1. TBCS in the absence of ground motion

A semi-infinite fluid region with constant depth h (the far field in Fig. 1) is considered. The

far field has the pressure ~ppx;x2 at x1 0, and also under the influence of surface waves at the topsurface. We expand the pressure in terms of the shape functions Pi defined in Eq. (23), which gives(if we consider the waves propagating in x1 direction),

pXmj1

ajPjeikjx1 29

where kj is jth wave number corresponding to a fixed x (which represents jth mode), m is thenumber of modes considered, and aj is the participation factor for jth mode. When kj is real, it

should be taken as positive in Eq. (29); when complex its imaginary part should be taken as

negative so that Eq. (29) gives the pressure values decaying with the distance away from V.We note that the shape functions Pi are orthogonal as verified in Section 2. Writing Eq. (29) atx1 0, we get

~ppXmj1

ajPj 30

Using the orthogonality condition, Eq. (27) in Eq. (30), for the constant aj one gets

aj 1bj

h~pp;Pji 31



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with

bj hPj;Pjiwhose expression being given in Eq. (28).

Now, we consider the nodes at the boundary V x1 0. Let the number of nodes be n.Further, let xk2 k 1; 2; . . . ; n be the x2 coordinates of the nodes on V. To proceed with theformulation of tbcs, h~pp;Pji in Eq. (31) is to be integrated numerically over V. This numericalintegration should be consistent with the method of formulation employed in the analysis of nearfield. For example, if BEM with constant elements [10] is used in the analysis, the proper inte-

gration procedure for h~pp;Pji would be rectangular rule, which gives

h~pp;Pji Xnk1

Ajk~ppk

or, when inserted into (31),

aj 1bj

Xnk1

Ajk~ppk 32

with the definitions

Ajk DhkPjxk2; ~ppk ~ppxk2 33where Dhk is the length of kth boundary element on V. For the sake of proceeding with the

formulation of tbcs, here we will use the expression in Eq. (32) for aj, implying that we employconstant boundary elements for the analysis in near field.

We continue the formulation of tbcs by taking the derivative of Eq. (29) with respect to x1,which yields

op

ox1Xmj1

ajPjikjeikjx1 34

Writing Eq. (34) at x1 0 and x2 xs2 (that is, at the node ofsth element on V), and inserting Eq.(32) into (34), we get

op

ox1

s

Xmj1

Xnk1

BsjAjk~ppk 35

with the definitions

op

ox1

s

opox1

xs2x10

Bsj ikjbjPjxs2

36

Eq. (35) may be written in matrix form as

err1p BA~pp 37



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where

err1pop

ox1 1...

op

ox1

n

2666666437777775

Eq. (37) is the tbcs to be used on V in the absence of ground motion. We note that these tbcshave nonlocal form.

3.2. TBCS in the presence of ground motion

In this case, the tbcs on V may be obtained by using a procedure similar to that given above.Since the fluid is inviscid, it can sustain only vertical ground motion at the base L (see Fig. 1);

therefore, the bc at x2 h would beop

ox2 bixp qf0av 38

where av is free field value of vertical acceleration on L defined in the absence of fluid layer. Thepositive direction ofav in Eq. (38) should be taken in x2 direction. We assume that av is generated

by a vertical waves in foundation, implying that it is uniform over L. At the top surface x2 0, wesatisfy the free surface wave bc, that is

op

ox2 x

2

gp

For the formulation of tbcs on V, we write, for the pressure,

p p1 p0

where p1 satisfies homogeneous boundary conditions, that is,

op1

ox2 x

2

gp1 at x2 0

op1ox2

bixp1 at x2 h 39

while p0 satisfies inhomogeneous boundary conditions:

op0

ox2 x

2

gp0 at x2 0

op0

ox2 qf0av at x2 h

40

where both p1 and p0 satisfy, in Fourier space, Eq. (5).



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In view of the boundary conditions ofp0 in Eq. (40), it is obvious that p0 may be taken as the

function on x2 only; then, from Eq. (5), one gets for p0:

p0 A sinx

c x2 B cos xc x2 41

where the constants A and B may be found from the boundary conditions of p0, which yields

B qf0av

x2

gcos x

ch

xc

sin xch

A cx

gB

42

To derive the tbcs on V, we note that p1 p p0 satisfy the homogeneous bcs in Eq. (39);therefore, we may expand

p

p0

in terms of shape functions Pj, that is

p p0 Xmj1

ajPjeikjx1

Using exactly the same procedure which was employed in the absence of ground motion, oneobtains the tbcs in the presence of ground motion aserr1p BA~pp p0 43where the load vector p0 p0k with p0k p0xk2 contains the inhomogeneous pressure values atthe nodes along V. The definitions of other variables and parameters in the above equation are

given previously.

4. Assessment of the proposed TBCS

For that, the proposed tbcs are installed into a boundary element computer program (devel-

oped by the present authors for fluid-solid interaction analysis), and they are checked by applyingthem to the analysis of two simple benchmark problems for which the analytical solutions areavailable in literature.

Problem 1. Rigid dam with finite and infinite reservoirs, and horizontal reservoir base (subjectedto horizontal harmonic earthquake excitation).

A rigid dam with a constant reservoir depth h 100 m is considered (see Fig. 5). Reservoirfoundation is taken as rigid. The dam-reservoir system is subjected to a horizontal earthquake(EQ) excitation at foundation level. Here, the response of the dam-reservoir system to this EQinput is studied in Fourier space; that is, EQ acceleration is taken as harmonic having the formage

ixt (ag: acceleration amplitude, x: frequency) and the response spectrum is determined byvarying the frequency x. In this example problem, the hydrodynamic pressure on the upstream

face of the dam is chosen as the response quantity, it is determined by



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taking the reservoir as finite L 2h with zero velocity on V, taking the reservoir as infinite with open boundary (on the right side, see Fig. 5), using TYPE1 tbcs on V,

using TYPE2 tbcs on V,

where TYPE1 tbcs correspond to those proposed in this study and TYPE2, to viscous type of tbcs

having form

op

on ix

~ccp 44

where n is outer normal axis at a point of V with respect to near field (see Fig. 1). ~cc is thecharacteristic velocity generally taken as speed of sound in the fluid; but, it may be also found

from the related dispersion relation if surface waves have primary importance. Here, it is taken as~cc c. It may be noted that the tbc in Eq. (44) has a local form.

In computations, the effect of surface waves is ignored. We ignore it because the analyticalsolution presented in [11] is obtained under this condition. The water is treated as inviscid fluid and

velocity of sound is taken as 1438 m/s. The angular frequency is normalized with respect to angularfrequency of the first mode, x1 (the modal frequencies are given by xn 2n1pc2h with n 1; 2; . . .).

Figs. 6 and 7 contain boundary element results for the hydrodynamic pressure distributions onthe upstream face of the dam, obtained respectively for the frequencies x=x1 0:5 and 1.85. Itmay be noted that one of these frequencies is below the first natural frequency x1 and the other

above it. From the study of these figures it may be concluded that

when x < x1, the results obtained from various reservoir models (finite and infinite models,and the models involving transmitting boundary conditions) are very close, and coincide almostexactly with the analytical result;

when x > x1, the picture changes drastically; the model with the transmitting boundary con-ditions TYPE1 gives the best result while the finite reservoir model, the worst.

The reason for this drastic change may be explained as follows. It may be shown that when theeffect of surface waves is disregarded, the spectrum for the waves propagating in horizontal di-rection in the reservoir is given by

Fig. 5. Rigid dam-reservoir system.



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x2

c2 2n 1

2hp

2 k2 n 1; 2; . . .

where each n represents a different mode. This equation implies that for the frequency rangebelow the first cut-off (free vibration) frequency, that is for x < x1 p2h c, the spectrum containsno propagating modes; this is why good results are obtained in Fig. 6 without using transmittingboundary conditions. On the other hand, for x > x1, the use of transmitting boundary conditions

Fig. 6. Hydrodynamic pressure distribution on upstream face of the dam xx1

0:5.

Fig. 7. Hydrodynamic pressure distribution on upstream face of the dam xx1 1:85.



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improves the results considerably as shown in Fig. 7, since this range of frequencies containspropagating modes and these modes are accounted for by TYPE1 tbcs. It may be noted that the

best results are obtained by this type of tbcs, even with a small value of truncation lengthL 0:3h.

The total hydrodynamic force Fdyn acting on upstream face of the dam (normalized withrespect to total hydrostatic force Fstat

qf0gh2

2 ) is shown Fig. 8 for infinite reservoir model, ob-tained using TYPE1 and TYPE2 tbcs and open boundary. The agreement of the results obtainedusing the tbc TYPE1 with the analytical is excellent.

As stated previously, the results presented in this example problem are obtained by using BEM.As illustration, the boundary element network (with constant elements) employed in the analysis

of the reservoir model with TYPE1 tbcs is given in Fig. 9. n in the figure refers to the outernormal of boundaries.

Problem 2. Rigid dam with horizontal reservoir base (subjected to various time dependentloadings)

Fig. 8. Total force on upstream face of the dam for infinite reservoir.

Fig. 9. Boundary element network employed in the analysis of the reservoir model with TYPE1 tbc.



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The dam-reservoir problem (see Fig. 5) analyzed in Problem 1 is reconsidered; but, this time,

with various time dependent inputs, namely, with the horizontal ground acceleration given by

(i) ramp type time variation,(ii) El Centro EQ record.

The properties of the dam-reservoir system are

c (speed of sound): 1438 m/s,h (depth of reservoir): 180 m (uniform),water: treated as inviscid fluid.

The results are determined in time domain by using the infinite reservoir model with open

boundary, and the transmitting boundary conditions TYPE1 and TYPE2. They are comparedwith the analytical solutions. In the computations, the effect of surface waves is ignored for thereasons stated in Problem 1. The results in time domain are obtained by using the procedure

first, the transfer function relating a response quantity to the input is computed in Fourierspace (in this step, the boundary element program mentioned previously is used);

then, the response in time domain is determined through the use of inversion by FFT algorithm[1214].

The hydrodynamic pressure distribution on the upstream face of the dam is shown in Fig. 10where the time variation of ground acceleration is taken ramp type as shown in the same figure.

The result obtained by the tbcs TYPE1 coincides with the exact while others differ considerablyfrom the analytical. Hydrodynamic pressure distributions resulting from open boundary and

Fig. 10. Hydrodynamic pressure distribution on upstream face of the dam subjected to ramp acceleration (at t 2:0 s).



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TYPE2 tbcs are close to each other. Time history of the hydrodynamic pressure at the base of the

dam is given in Fig 11, where again, TYPE1 tbcs give excellent results.Now, we consider the case in which the horizontal ground acceleration time history is given by

El Centro EQ record (see Fig 12). Hydrodynamic pressure distribution on the upstream face ofthe dam in this case is shown in Fig 13. Time history of the pressure at the base of the dam isplotted in Fig. 14. The figures show that the TYPE1 tbcs give excellent results, for the reasons

discussed previously.

Fig. 11. Hydrodynamic pressure at the base of the dam subjected to ramp acceleration.

Fig. 12. Time history of El Centro earthquake acceleration record.



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References

[1] J.F. Hall, A.K. Chopra, Two dimensional dynamic analysis of concrete gravity and embankment dams including

hydrodynamic effects, Earthquake Engineering and Structural Dynamics 10 (1982) 305.

[2] J.L. Humar, A.M. Jablonski, Boundary element reservoir model for seismic analysis of gravity dams, Earthquake

Engineering and Structural Dynamics 16 (1988) 1129.

Fig. 13. Hydrodynamic pressure distribution on upstream face of the dam subjected to El Centro earthquake excitation

(at t 2:0 s).

Fig. 14. Hydrodynamic pressure at the base of the dam subjected to El Centro earthquake excitation.



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[3] C.S. Tsai, G.C. Lee, R.L. Ketter, A semi-analytical method for time-domain analyses of dam-reservoir

interactions, International Journal for Numerical Methods in Engineering 29 (1990) 913.

[4] R. Chandrashaker, J.L. Humar, Fluidfoundation interaction in seismic response of gravity dams, Earthquake

Engineering and Structural Dynamics 22 (1993) 1067.[5] R. Yang, C.S. Tsai, G.C. Lee, Explicit time-domain transmitting boundary for dam-reservoir interaction analysis,

International Journal for Numerical Methods in Engineering 36 (1993) 1789.

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Transmitting Boundary Conditions Suitable for Analysis

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