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KSCE Journal of Civil Engineering (2012) 16(1):103-106

DOI 10.1007/s12205-012-0870-8

103

www.springer.com/12205

Structural Engineering

Analysis of Dam-Reservoir Interaction by Using Homotopy Analysis Method

M. A. Karaca* and S. Kkarslan**

Received May 12, 2009/Revised December 5, 2010/Accepted April 12, 2011

Abstract

In this paper, dam-reservoir interaction for a vibrating structure in an unbounded and incompressible and inviscid fluid is analyzedby using homotopy analysis method. In the derivation of the hydrodynamic pressure variable, it is assumed that vibration of dam is inthe normal direction of dam-reservoir interface and this interface is vertical. Moreover, bottom of fluid is rigid and horizontal. Theresults are compared with finite element method and analytical ones. It is seen that the results are efficient and gives better valuesthan the previous published results and this method can be extendable for compressible fluid domains.

Keywords: dam reservoir interaction, Homotopy Analysis Method (HAM), Finite Element Method (FEM), hydrodynamic pressure

1. Introduction

In the earthquake seismic regions, for the design of dams, the

effect of the hydrodynamic pressures exerted on the face of the

dam is an important issue as a result of the earthquake ground

motions. Zienkiewicz et al. (1965) presented the finite element

formulation for analyzing the coupled response of the submerged

structures assuming water to be incompressible. Nath (1971)

analyzed the problem using the method of finite differences but

neglecting radiation damping. Chakrabarti and Chopra (1974)

have formulated the reservoir as a continuum of infinite length.

Two dimensional problem of the added-mass effect of horizontal

acceleration of a rigid dam with an inclined upstream face of

constant slope was solved analytically by Chwang and Housner

(1978) using a momentum balance approach.

In the finite element formulation, unbounded domain of reservoir

arise a problem in modeling. To overcome this difficulty, the un-

bounded domain should be truncated at a certain distance away

from the structure. The most commonly used boundary condi-

tion along the truncation surface is the Sommerfeld radiation

condition (Sommerfeld, 1949). Since this boundary condition

takes the form of that for a rigid stationary boundary, the behavior

of the reservoir domain is not truly represented. Another bound-

ary condition along the truncating surface for an unbounded and

incompressible fluid domain was developed by Sharan (1985).

Although this boundary condition is better than the Sommerfeld

radiation condition, it does not represent the behavior well when

truncation surface is very near to dam surface. Another boundary

condition along the truncating surface of an unbounded reservoir

domain was developed by approximating the analytical solution

of the hydrodynamic pressure (Kkarslan, 2005).

It is obvious that the effective and accurate results are depend-

ent on the far boundary condition due to nature of the unbounded

domain of the reservoir for finite element analysis. To avoid this

disadvantage that does not consider the far boundary condition, a

new and efficient method called Homotopy Analysis Method

(HAM) (Liao, 1992, 1999, 2003; Inc, 2007) is used for the first

time to get hydrodynamic pressures on the dam face.

2. Formulation of Unbounded Reservoir Domain

2.1 Analytical Formulation of the Hydrodynamic Pressure

For incompressible and inviscid fluid, the hydrodynamic pres-

surep resulting from the ground motion of a rigid dam (Fig. 1)

satisfies theLaplace equation in the following form:

(1)

The following boundary conditions are defined by assuming

the effects of the surface waves and viscosity of the fluid are

neglected :

At the fluid-solid interface (S1),

(2)

where agis the ground acceleration subjected on the dam face.

At the bottom of the fluid domain, if the bottom is rigid (S2),

one can write the following:

(3)

p2

0=

p

n------ ag=

p

n------ 0=

*Assistant Professor, Dept. of Mathematics, Istanbul Technical University, Istanbul 34469, Turkey (E-mail: karacam@itu.edu.tr)

**Associate Professor, Dept. of Civil Engineering, Istanbul Technical University, Istanbul 34469, Turkey (Corresponding Author, E-mail: kucukarslan@itu.

edu.tr)

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M. A. Karaca and S. Kkarslan

104 KSCE Journal of Civil Engineering

At the far end (S3) wherex coordinate is infinite,

(4a)

The Sommerfelds radiation condition for the truncated surface

is given by:

(4b)

and the Sharans boundary condition is given by

(4c)

At the free surface (S4) when neglecting the surface waves,

(5)

The analytical solution of Eq. (1) with the boundary conditions

(4) is:

(6)

where ,His the height of the fluid, ag is the

ground acceleration, andis the mass density of the fluid.

Solution of the Eq. (6) is obtained by assuming that: 1) the

fluid domain extends to infinity and its motions is two dimen-

sional, 2) fluid-structure interface is vertical, 3) the submerged

structure is rigid, 4) the bottom of fluid domain is rigid and

horizontal.

2.2 Homotopy Analysis Method

Equation (1) is considered as:

(7)

where Nis a linear operator for this problem, and p(x,z) is an

unknown function. By means of the HAM (Liao, 1992, 1999,

2003; Inc, 2007), one first constructs the zeroth-order deforma-

tion equation as:

(8)

where is a linear operator,p0(x,z)is an initial guess, is an

auxiliary parameter and is the embedding parameter.

Obviously, when r=0 and r=1, it gives:

and (9)

respectively. In the HAM (Liao, 1992; Liao, 1999; Liao, 2003;

Inc, 2007), the Taylor expansion of(x,z;r) about the embedding

parameter is obtained by:

(10)

where

(11)

The convergence of the series in equation (10) depends on the

auxiliary parameter . If it converges at r=1, one gets:

(12)

Lets define the vectors:

(13)

By differentiating the zeroth-order deformation Eq. (8) m-

times with respect to rand then dividing them by m! and at the

end setting r=1, one gets the mth-order deformation equation as:

(14)

where

(15a)

and

(15b)

It should be noted thatpm(x,z) form 1 is governed by the Eq.

(14).

2.3 Application of the Homotopy Analysis Method for Dam-

Reservoir Interaction Problem

In this section, the application of the HAM for the analysed

physical dam-reservoir interaction problem will be done and

approximate solutions will be obtained for current investigation.

One can select the linear operator as:

(16)

with the following property:

(17)

where c1 and c2 are constants. Also, one can define the following

operator as:

(18)

p 0=

p x z,( )

n

------------------ 0=

p x z,( )

n------------------

p

2H-------=

p 0=

p x z,( ) 2agH1( )

n 1+

n2-----------------

n 1=

exp nx

H

----

cos nz

H

----

=

n 2n 1( )( ) 2=

N p x z,( )[ ] 0=

1 r( ) x z r;,( ) p0 x z,( )[ ] rhN x z r;,( )[ ]=

h 0

r 0 1,[ ]

x z0;,( ) p0 x z,( )= x z1;,( ) p x z,( )=

x z r;,( ) p0 x z,( ) pm x z,( )rm

m 1=

+

+=

m x z,( )1

m!------

mx z r;,( )

rm

-------------------------

r 0=

=

h

x z,( ) p0 x z,( ) pm x z,( )m 1=

+

+=

n p0 x z,( ) p1 x z,( ) pn x z,( ), , ,{ }=

pm x z,( ) mpm 1 x z,( )[ ] hm pm 1( )=

m pm 1( )1

m 1( )!------------------

m 1

N x z r;,( )[ ]

rm 1

--------------------------------------

r 0=

=

m 0, m 11, m 1>

=

x z r;,( )[ ]

2x z r;,( )

x2

------------------------=

c1x c2+[ ] 0=

N x z r;,( )[ ]

2x z r;,( )

x2

------------------------

2x z r;,( )

x2

------------------------+=Fig. 1. Rigid Dam and Fluid

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Analysis of Dam-Reservoir Interaction by Using Homotopy Analysis Method

Vol. 16, No. 1 / January 2012 105

By using this above definition and Eqs. (8) and (9), one can get

the mth-order deformation as:

(19)

where

Finally, the solution of the mth-order deformation equations

form 1 becomes:

(20)

By selecting and usingH

= 5, ag=0.1, =1,one can get the following approximations for

the hydrodynamic pressure,

(21)

(22)

The selection of the initialp0(x,z) is very important to obtain the

converging results. In the selection of this initial function, the

curicial point should be related to the physics of the problem, i.e.,

it needs to satisfy the boundary conditions and the behaviour of

the pressure distribution.

3. Numerical Example

To test the accuracy of the proposed method, a comparison will

be done by using analytical and two-dimensional finite elements

having 4 nodes rectangular elements. The geometry of rigid dam

is shown in Fig. 2, in which dam is subjected to a horizontal

uniform acceleration ag. The infinite reservoir was analyzed for

four different locations of the truncation boundary, each resulting

in a different size lof an equivalent finite reservoir. Typical finite

element model is shown in Fig. 2.

The results for the hydrodynamic pressures for different loca-

tions were obtained by using analytical and homotopy analysis

methods in the Table 1.

The truncated boundary taken very close to the dam face is the

interest of this study, because taking the truncation boundary at a

far location can give good results when compared to other

available boundary conditions, but one loses the efficiency in

terms of number of unknowns.

In Figs. 3-5, the distribution of hydrodynamic pressure on dam

face is compared with Sommerfeld, Sharan and homotopy

pm x z,( ) mpm 1 x z,( )[ ] hm pm 1( )=

m pm 1( )

2pm 1 x z r;,( )

x2

-------------------------------

2m 1 x z r;,( )

z2

--------------------------------+=

m x z,( ) mpm 1 x z,( )[ ] h1

m pm 1( )[ ]+=

p0 x z,( ) 4ex 10( )

1 2z

2100( )( )( )

2=

1 x z,( )e

x

10------

ht2

2z

2600+( )

30000--------------------------------------------=

p2 x z,( )

e

x

10------

h1

6

---h4z

6250 3h 2+( )

2z

4300000 h 1+( )z

2+ +

15000000------------------------------------------------------------------------------------------------------------------------=

Fig. 2. A Typical Finite Element Mesh for a Ten Row in the Vertical

Direction and Two Columns in the Horizontal Direction

Table 1. The Comparison of Bottom Hydrodynamic Pressure (z =

0) with Exact Solution for DifferentxLocations (H=5, ag=0.1, =1 values are assumed)

xExact valuefor pressure

HAM

Pressure Absolute Error %

0.5 0.324 0.346 6.79

1 0.281 0.296 5.33

1.5 0.243 0.253 4.11

2 0.210 0.216 2.85

2.5 0.181 0.185 2.20

3 0.155 0.158 1.93

3.5 0.133 0.135 1.50

4 0.114 0.115 0.88

4.5 0.098 0.099 0.10

5 0.084 0.084 0.00

Fig. 3. Comparison of HAM and Sommerfelds and Sharans

Boundary Condition for Hydrodynamic Pressure on Dam

forl/H=0.1 and Mesh Size 101

Fig. 4. Comparison of HAM and Sommerfelds and Sharans

Boundary Condition for Hydrodynamic Pressure on Dam

forl/H=0.5 and Mesh Size 105

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M. A. Karaca and S. Kkarslan

106 KSCE Journal of Civil Engineering

analysis results for different truncated lengths and for different

mesh sizes. In these figures, C0 =p0/agHis used for dimension-

less plot in the horizontal direction.

In these plots (Figs. 3-5), one can see the convergence of the

HAM results to the Sharans result whenx coordinates becomes

larger as expected (Sharan, 1985).

To illustrate the effect of the asymptotic behavior ofpm(x,z),

three cases are studied in the Table 2. In this table, the case1,

case2 and case3 represent , and ,

respectively.

4. Conclusions

Dam-reservoir interaction for a vibrating structure in an un-

bounded and incompressible and inviscid fluid is analyzed by

using the homotopy analysis method. The HAM results were

compared with analytical ones and finite element method by

using Sommerfelds and Sharans boundary conditions. It is seen

that the proposed numerical method (HAM) is an efficient and

accurate method for the analyzed problem and this new method

can extendable for compressible fluid domains.

References

Chakrabarti, P. and Chopra, A. K. (1974). Hydrodynamic effects in

earthquake response of gravity dams.ASCE J. Stuct. Div., Vol. 100,

No. 100, pp. 1211-1224.

Chwang, A. T. and Hausner, G. W. (1978). Hydrodynamic pressures on

sloping dams during earthquakes. Part 1 : Momentum method. J.

Fluid Mech,. Vol. 87, No. 2, pp. 335-341.

n, M. (2007). On exact solution of Laplace Equation with Dirichlet

and Neumann boundary conditions by homotopy analysis method.

Physics Letters A, Vol. 365, No. 5, pp. 412-415.

Kkarslan, S. (2005). An exact truncation boundary condition for

incompressible-unbounded infinite fluid domains.Applied. Mathe-

matics and Computation., Vol. 163, No. 1, pp. 61-69.

Liao, S. J. (1992). The proposed homotopy analysis technique for the

solution of nonlinear problems1, PhD Thesis, Shanghai Jiao Tong

University.

Liao, S. J. (2003). An explicit, totally analytic approximate solution for

Blasius viscous flow problems.Int. J. Non-Linear Mech. Vol. 34,

No. 4, pp. 759-778.

Liao, S. J. (2003).Beyond perturbation: Introduction to the homotopy

analysis method, Champan & Hall/CRC Press, Boca Raton.

Liao, S. J. (2003) On the analytic solution of magneto hydrodynamicflows of non-Newtonian fluids over a stretching sheet. J. Fluid.

Mech. Vol. 488, No. 1, pp. 189-212.

Nath, B. (1971). Coupled hydrodynamic response of gravity dam.

Proc. Inst. Civ. Engng, Vol. 48, No. 2, pp. 245-257.

Sharan, S. K. (1985). Finite element analysis of unbounded and incom-

pressible fluid domains. Int. J. Numer. Meth, Vol. 21, No. 9, pp.

1659-1669.

Sommerfeld, A. (1949). Partial differential equations in physics,

Academic Press, New York.

Zienkiewicz, O. C., Irons, B., and Nath, B. (1965). Natural frequencies

of complex free or submerged structures by the finite element

method. Symp. Vibrations Civ. Engng, Butterworths, London.

p p0 p p0 p1+ p p0 p1 p2+ +

I

Fig. 5. Comparison of HAM and Sommerfelds and Sharans

Boundary Condition for Hydrodynamic Pressure on Dam

forl/H=1.0and Mesh Size 1010

Table 2. The Comparison of Approximated Hydrodynamic Pres-

sure (x=10) with Exact Solution for Different zLocations

(H= 5, ag=0.1, =1 values are assumed)

z Exact value Case1 Case2 Case3

0.5 0.01730 0.01730 0.01730 0.01730

1 0.01665 0.01578 0.01666 0.01665

1.5 0.01560 0.01362 0.01564 0.01560

2 0.01417 0.01060 0.01428 0.01417

2.5 0.01239 0.00671 0.01267 0.01238

3 0.01030 0.00196 0.01089 0.01027

3.5 0.00795 -0.00366 0.00906 0.00791

4 0.00543 -0.01014 0.00732 0.00531

4.5 0.00274 -0.01749 0.00584 0.00254

5 0. -0.02570 0.00479 0.0004