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Linear dynamic analysis of arch dams utilizing modified efficient fluid hyper-element
Engineering Structures 29 (2007) 2654–2661 www.elsevier.com/locate/engstruct
Linear dynamic analysis of arch dams utilizing modified efficient fluid hyper-element Ahmad Aftabi Sani ∗ , Vahid Lotfi Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran Received 10 September 2006; received in revised form 9 January 2007; accepted 10 January 2007 Available online 1 March 2007
Abstract The accurate dynamic analysis of concrete arch dams relies heavily on employing a three-dimensional semi-infinite fluid element. The usual method for calculating the impedance matrix of this fluid hyper-element is dependent on the solution of a complex eigenvalue problem for each frequency. In the present study, a modified efficient procedure is proposed, which simplifies this procedure significantly, and results in great computational time savings. Moreover, the accuracy of this technique is examined thoroughly and it is concluded that modified efficient procedure is also extremely accurate under all practical conditions. It is also shown that the original efficient method proposed in a previous study can be considered as a special case of the present more general procedure. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Dynamic analysis; Arch dams; Fluid hyper-element
1. Introduction There are different alternatives for the dynamic analysis of concrete dams [1–3]. However, the rigorous analysis of concrete arch dam–reservoir systems is based on the FE(FE–HE) method (i.e., Finite Element-(Finite Element–Hyper Element)). This means, the dam is discretized by solid finite elements, while, the reservoir is divided into two parts, a near field region (usually an irregular shape) in the vicinity of the dam and a far field part (assuming a uniform channel), which extends to infinity. The former region is discretized by fluid finite elements and the latter part is modeled by a threedimensional fluid hyper-element. The analysis is carried out in the frequency domain either by a direct approach [4], or by sub-structuring techniques [5,6]. Regardless of the option selected from among these rigorous techniques, a major portion of the numerical calculation time spent is on the solution of a complex eigenvalue problem related to fluid hyper-element, which must be solved at each frequency. To remedy this, an efficient procedure was proposed in a previous study [7] for the required impedance matrix calculations of the fluid
hyper-element, which greatly reduces the computational time. Although, the method was very accurate in most practical conditions, it was estimated that it would introduce errors (in the range of 10%) under certain circumstances. In this paper, a modified efficient procedure is proposed for the above-mentioned problem. This is envisaged to be an enhancement of the previously proposed efficient method [7]. The approach mainly relies on mode shapes corresponding to an arbitrary frequency. In the following sections, the analysis technique (i.e., FE-(FE–HE) method) is reviewed very briefly. The fluid hyper-element impedance matrix’s theoretical background is also described without going into unnecessary details. Subsequently, the modified efficient procedure is introduced and its formulation is presented. It is also shown that the original efficient method can be considered as a special case of the modified efficient procedure presented here. Thereafter, a special-purpose program is developed based on the proposed theory, and the response of the Morrow Point arch dam is obtained for various conditions. This is utilized to investigate the accuracy of the modified efficient procedure thoroughly. 2. Method of analysis
∗ Corresponding author. Tel.: +98 (0) 511 270 6867.
E-mail addresses: ahmad [email protected] (A. Aftabi S.), [email protected] (V. Lotfi). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.01.005
The analysis technique utilized in this study is based on the FE-(FE–HE) method, which is applicable for a general concrete
A. Aftabi S., V. Lotfi / Engineering Structures 29 (2007) 2654–2661
arch dam–reservoir system. It is worthwhile to mention that the procedure utilized can be easily generalized for a flexible foundation rock case. However, the foundation is assumed to be rigid to avoid the extra complexities that would otherwise arise. The formulation can be explained much more easily if one concentrates initially on a dam with a finite reservoir system (basically the same as a model of a dam and a reservoir nearfield), and subsequently add the effects of a reservoir far-field region for the general case. For this purpose, let us begin with this simpler formulation and then complete the formulation for the more general case on that basis. 2.1. Dam with finite reservoir system This is the problem which can be totally modeled by the finite element method. It can be easily shown that in this case, the coupled equations of the system may be written as [8]: M 0 r¨ C 0 r˙ K −BT r + + B G p¨ 0 L p˙ p 0 H −MJag = (1) −BJag M, C, K in this relation represent the mass, damping and stiffness matrices of the dam body, while G, L, H are the assembled matrices of the fluid domain. The unknown vector is composed of r, which is the vector of nodal relative displacements and the vector p that includes the nodal pressures. Moreover, J is a matrix with each of the three rows equal to a 3 × 3 identity matrix (its columns correspond to unit rigid body motion in cross-canyon, stream, and vertical directions) and ag denotes the vector of ground accelerations. Furthermore, B in the above relation is often referred to as interaction matrix. For harmonic ground excitations ag (t) = ag (ω)eiωt with frequency ω, displacements and pressures will all behave harmonic, and Eq. (1) can be expressed as: 2 −ω M + K(1 + 2βd i) −BT r −2 2 p −B ω (−ω G + iωL + H) −MJag = (2) −ω−2 BJag In this relation, it is assumed that the damping matrix of the dam is of hysteretic type. This means: 2βd K (3) ω where βd is the constant hysteretic factor of the dam body. Relation (2) is the coupled equations of a dam with a finite reservoir system in the frequency domain. It should be also noted that the system of equations is made symmetric by multiplying the lower partition matrices by a factor of ω−2 .
the present study. Apart from that, there are three possible boundary condition types for the reservoir near-field region. The condition of type I is considered for the contact of fluid with flexible solid, such as the dam–reservoir interface. The second type of condition is the so-called approximate boundary condition. This can be imposed at the reservoir bottom and sidewalls. The last type of condition (III) is referred to as Sommerfeld boundary condition. This is usually applied at the reservoir near-field upstream boundary (in cases in which the far-field region is not modeled), as a substitute for a precise transmitting boundary. However, when a fluid hyperelement is utilized, this condition is not required, and waves are transmitted exactly through that semi-infinite element. The exact relations and detailed explanation about these conditions can be found in other studies (e.g., Ref. [4]). 2.3. Fluid hyper-element As mentioned, the three-dimensional fluid hyper-element is utilized to model the reservoir far-field region for the more general case. This part of the water domain is assumed to be a uniform channel with an arbitrary geometric shape in the vertical plane which includes x, z-axes (see Fig. 1b for a typical discretization), and extends to infinity in the upstream direction (negative y-direction). Although, this is a three-dimensional semi-infinite fluid element, its discretization is performed in the vertical plane perpendicular to the channel axis, which is referred to as the reference plane (y = 0). Therefore, the element consists of several sub-channels which extend to infinity, and all nodes of the hyper-element are located on that reference plane. The formulation of this element is briefly reviewed as follows: Assuming water to be linearly compressible and neglecting its viscosity, the small amplitude, irrotational motion of water due to harmonic excitation is governed by the Helmholtz equation: ∂2 P ∂2 P ∂2 P ω2 + + + P=0 ∂x2 ∂ y2 ∂z 2 C2
The water surface boundary condition is easily applied, especially if surface waves are neglected, as is the case in
(4)
where P is the amplitude of the hydrodynamic pressure (in excess of hydrostatic pressure) and C is the velocity of pressure waves in water. By seeking solutions of the form eky in the stream direction, Eq. (4) becomes
C=
2.2. Reservoir near-field boundary conditions
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λ2 P +
∂2 P ∂2 P + =0 ∂x2 ∂z 2
(5)
with the following definition of λ. λ2 = k 2 +
ω2 C2
(6)
By applying the variational method on Eq. (5), a matrix relation is obtained for each sub-element (see [7] for details). Assembling the effects of different sub-elements and imposing the homogeneous boundary conditions (corresponding to zero
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ground acceleration) on the resulting matrix equation leads to the following eigenvalue problem: [−λ2j A + iωqLh + C]X j = 0
(7)
λ2j , X j are the jth eigenvalue and eigenvector of the fluid hyperelement. There also exist particular solutions (i.e., P`p ) which correspond to the uniform unit acceleration of the reservoir boundary in the `-direction (x, or z-direction). Therefore, the vector of pressure amplitudes for the hyper-element can be totally written as: Ph = Xh Γ + P p ag (ω)
(8)
by help of the following definitions for the reservoir hyperelement modal matrix, vector of participation factors and a matrix which includes particular solution vectors for different directions. Xh = [X1 , X2 , . . . , Xn ]
becomes: # " −ω2 M + K(1 + 2βd i) −BT ¯ h (ω)) −B ω−2 ((−ω2 G + iωL + H) + H " # −MJag r × = (12) ¯ p (ω)ag ) p ω−2 (−BJag + R
¯ p (ω) are the expanded forms of the Hh (ω) ¯ h (ω) and R where H and R p (ω) matrices which cover the entire fluid domain pressure degrees of freedom. Assuming that the pressure degrees of freedom related to the hyper-element are ordered first in the unknown pressure vector, these matrices have the following form: ¯ h (ω) = Hh (ω) 0 H (13a) 0 0 R p (ω) ¯ R p (ω) = (13b) 0
Γ = [γ1 , γ2 , . . . , γn ]T P p = Pxp 0 Pzp
The relation (12) is the equation to be used instead of (2) when the reservoir is extended to infinity and one is considering the direct approach in the frequency domain.
Moreover, the fluid y-direction (stream component) accelerations vector for the reference plane, becomes (see Ref. [7] for details);
2.5. Fluid hyper-element (modified efficient procedure)
¨ h = − 1 Xh Kh Γ U ρ
(9)
where Kh is a diagonal matrix with the jth diagonal element being equal to k j . Solving for the participation vector from (8), substituting in (9), and multiplying the resulting relation by −A, one obtains: Rh = Hh Ph − R p ag (ω)
(10)
by employing the following definitions: ¨h Rh = −AU 1 Hh = AXh Kh XTh A ρ R p = Hh P p
This element is basically formulated in the same manner as the usual fluid hyper-element. However, the main required matrix (i.e., Hh (ω)) is calculated based on the modified efficient method. It should be mentioned that the proposed technique is a modified version of the procedure originally referred to as the efficient method [7]. It will also be explained underneath how this modified technique transforms to the previously proposed efficient method. The fundamental concepts of this technique are presented as follows: Let us consider the formula which is utilized to define matrix Hh (ω):
(11a) (11b) (11c)
In relation (10), Rh represents a consistent vector equivalent to the integration of inward horizontal acceleration (the negative of the stream component) for the hyper-element, and this vector contains essentially similar quantities as the components of the right hand side vectors of the usual fluid finite elements [7]. 2.4. Dam–reservoir system The formulation for a dam with finite reservoir was already presented. For the case where the reservoir extends to infinity, a hyper-element must be used along with the fluid finite elements utilized for the reservoir near-field. Moreover, the governing relation for the hyper-element was derived in the previous section (relation (10)). Therefore, if the matrices of the hyperelement are assembled with the fluid finite elements, Eq. (2)
Hh (ω) =
1 AXh (ω)Kh (ω)XTh (ω)A ρ
(Repeated (11b))
The calculation of the above matrix is very time consuming and cumbersome due to its ingredients Xh (ω), Kh (ω), which are both frequency dependent. These matrices depend on the solution of the above-mentioned eigenvalue problem (7), which may be rearranged as: [C + iωqLh ]X j = λ2j AX j
(14)
In general, this eigenvalue problem must be solved repeatedly for the different required frequencies. However, let us assume that it is only solved for the unique case of ω = ω¯ (i.e., ω¯ being an arbitrary frequency), which can be written as: ¯ j = λ¯ 2j AX ¯j [C + iωqL ¯ h ]X
(15)
Based on the solution of this eigenproblem, one can form the following matrices: ¯ 1, X ¯ 2, . . . , X ¯ n] ¯ h = [X X
(16a)
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..
.
¯ = Λ
λ¯ 2j
..
.
(16b)
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¯ j is also the j-th eigenvector Therefore, it can be claimed that X of the following eigenproblem: ˆ h ]X ¯ j = λ2j AX ¯j [C + iωqL ¯ h + i(ω − ω)q ¯ L
(27)
It is worth to mention that the bar superscript denotes that these matrices are for the special case of ω = ω. ¯ Furthermore, the orthogonality relations in this case are:
which is very similar to the original relation (14), or its following alternative form:
¯ Th AX ¯h = I X ¯ Th (C + iωqL ¯h X ¯ h )X
The difference is mainly in the third term, which Lh is ˆ h . Obviously, this term replaced by an approximate form, L becomes negligible when the excitation frequency is close to ω. ¯ Moreover, ω¯ can be selected as any arbitrary value (e.g., the first natural frequency of the system, or wherever one is interested in decreasing the error introduced due to this approximation). The orthogonality relations corresponding to (27) may be written as:
(17a) ¯ =Λ
(17b)
Similarly, in the general case of (14), the orthogonality relations are written as: XTh AXh = I XTh (C + iωqLh )Xh
(18a) =Λ
(18b)
with the following matrix definitions: Xh = [X1 , X2 , . . . , Xn ] .. . λ2j Λ= .. .
(19a)
(19b)
¯ Th AX ¯h = I X
(29a)
¯ Th (C + iωqL ˆ h )X ¯h = Λ X ¯ h + i(ω − ω)q ¯ L
(29b)
∗ ¯ + i(ω − ω)qL Λ=Λ ¯ D
(30)
The equality (30) may be presented in a different form taking into account definitions (16b), and (19b):
or alternatively: ¯ Th (C + iωqL ¯ h = Diagonal Matrix (21) X ¯ h + i(ω − ω)qL ¯ h )X Let us investigate the conditions under which this is possible. By employing (17b), relation (21) can be written as: ∗ ¯ + i(ω − ω)qL ¯ Th (C + iωqL ¯h = Λ X ¯ h + i(ω − ω)qL ¯ ¯ (22) h )X
¯ is a diagonal matrix. However, this is not the case for L∗ Λ ¯ h ) in general. Thus, let us define a diagonal ¯ Th Lh X (i.e., L∗ = X ∗ matrix L D as below: L∗D = Diagonal L∗ (23) It is apparent that Eq. (21) would be satisfied now, if the second Lh appearing in that equation is replaced by Lˆ h , such that it yields: ∗ ¯ + i(ω − ω)qL ¯ Th (C + iωqL ¯h = Λ X ¯ h + i(ω − ω)q ¯ Lˆ h )X ¯ D (24)
Of course, this requires the following equality: (25)
ˆ h is not required to be defined explicitly, Although, the matrix L this can be achieved through relation (25) by employing (17a): ˆ h = AX ¯ h L∗D X ¯ Th A L
(28)
Comparing relation (24), and (29b), we get:
By comparing relations (17) and (18), it is easy to see that ¯ h , if the Xh (ω) could be frequency independent and equal to X following operation results in a diagonal matrix: ¯ Th (C + iωqLh )X ¯ h = Diagonal Matrix X (20)
¯ h = L∗D ¯ Th L ˆ hX X
2 [C + iωqL ¯ h + i(ω − ω)qL ¯ h ]X j = λ j AX j
(26)
λ2j = λ¯ 2j + i(ω − ω)q ¯ L ∗D j
(31)
Moreover, the wave number k j , can be found from (6) by employing (31): r kj = λ¯ 2j − ω2 /C 2 + i(ω − ω)q ¯ L ∗D j (32) Based on the above discussion and following the same procedure, which leads to (11b), one can obtain the following equation: Hh (ω) =
1 ¯ ¯ Th A AXh Kh (ω)X ρ
(33)
¯ h in this relation is frequencyIt must be emphasized that X independent, and corresponds to the modal matrix for the unique case of ω = ω. ¯ The only frequency-dependent matrix is Kh , which is a diagonal matrix and its elements are easily calculated using Eq. (32) for any frequency ω. This considerably simplifies the calculation of the main contributing matrix of the fluid hyper-element (i.e., Hh (ω)). In this manner, it is not required any more to solve a complex eigenvalue problem for each frequency. In other words, the procedure depends on the solution of the eigenproblem for an arbitrary frequency (i.e., the one corresponding to ω = ω). ¯ It can be also easily verified that the relations obtained herein for the modified efficient technique would be the same as the ones derived previously for the more simplified method referred to as the efficient method, if the arbitrary frequency ω¯ is chosen to be equal to zero. The main advantage of the original efficient method is that the relation (15) would become a standard eigenproblem in that case, and the solution can be obtained
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Fig. 1a. Finite element mesh of the dam body.
¯ h would be a modal by real number arithmetic. Therefore, X matrix with all its elements being real numbers. However, the main disadvantage in comparison to the modified efficient method could be the less accurate calculation of the hyperelement impedance matrix, which will ultimately transform into an increased error in the response of the dam to certain frequencies. This will be examined thoroughly in the next section.
Fig. 1b. Discretization of water domain (fluid finite elements (L/H = 0.2), and the fluid hyper-element).
3. Modeling and basic parameters A computer program [DACAD85] was developed based on the theory presented in the previous section. The program is based on the FE-(FE–HE) concept. This means the dam is treated as being composed of solid finite elements, while the reservoir is divided into two parts, the near-field region in the vicinity of the dam, which is discretized by fluid finite elements, and the far-field part, which is modeled by a three-dimensional hyper-element. The program has three options for calculating the fluid hyper-element impedance matrix. These are based on the exact method, the previous efficient approach or the modified efficient procedure explained above in detail.
Fig. 1c. Complete dam–reservoir system.
characteristics: Elastic modulus (E d ) = 27.5 GPa Poisson’s ratio = 0.2 Unit weight = 24.8 kN/m3
3.1. Models
Hysteretic damping factor (βd ) = 0.05
An idealized symmetric model of the Morrow Point arch dam is considered. The geometry of the dam may be found in Ref. [5]. The dam is discretized by 40 isoparametric 20-node solid finite elements (Fig. 1a). The water domain is divided into two regions (Fig. 1b). The near-field part is considered as a region, which extends to a specified length L = 0.2H (H being the dam height or maximum water depth in the reservoir), which is measured in upstream direction at the dam’s mid-crest point. The far-field region starts from that point and extends to infinity in the upstream direction. Both these regions combined are assumed to form a uniform reservoir shape as in a previous study (Fig. 1c) [7]. The near-field region is discretized by 80 isoparametric 20-node fluid finite elements, while the far-field region is modeled by a fluid hyper-element which itself is constructed from 40 isoparametric 8-node sub-elements.
The impounded water is taken as an inviscid and compressible fluid with unit weight equal to 9.81 kN/m3 , and pressure wave velocity C = 1440 m/s.
3.2. Material properties The dam concrete is assumed to be homogeneous with isotropic linearly viscoelastic behavior and the following main
4. Results of analyses The responses of dam crest are obtained due to upstream, vertical and cross-stream excitations for several values of the wave reflection coefficient α. In particular, α values equal to 0.0, 0.50, and 0.75 are considered. These values are selected similarly to the values considered in the previous study [7]. It should be mentioned that the response quantities which will be presented in this section are the amplitudes of the complex valued radial accelerations for two points located at dam crest (Fig. 1a). This is either the mid-crest point (θ = 0◦ ) selected for upstream or vertical excitations, or a point located at (θ = 13.25◦ ) which is used for the case of cross-stream excitations. This is due to the fact that radial acceleration is zero at the mid-crest for the cross-stream type of ground motion. It must be emphasized that these accelerations are obtained for unit amplitude ground accelerations in each case. Therefore,
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Fig. 2. Frequency-response functions due to different excitations (α = 0): efficient method [7].
Fig. 3. Frequency-response functions due to different excitations (α = 0): modified-efficient method.
they are actually dimensionless quantities, or it can be thought as normalized accelerations at the dam crest. In each case, the amplitude of the radial acceleration is plotted against the dimensionless frequency for a significant range. The dimensionless frequency for upstream and vertical excitation is defined as ω/ω1S , where ω is the excitation frequency and ω1S is the fundamental frequency of the dam on a rigid foundation with an empty reservoir for a symmetric mode. For the cross-stream excitation cases, the dimensionless frequency is defined as ω/ω1a , where ω1a is the fundamental resonant frequency of the dam on a rigid foundation with an empty reservoir for an anti-symmetric mode. As mentioned above, several values of α are considered, and results are obtained for different types of excitations. Analyses are carried out based on the modified efficient method proposed herein, and in each of these cases, the response is compared against the exact method. However, prior to this presentation, it is worthwhile to take a glance at a similar comparison for the original efficient method, which was obtained in the previous study [7]. This is provided in Fig. 2 only for the worst case (i.e.,
the maximum error in comparison to exact method) that was calculated for α = 0.0. Thereafter, the current study results for the modified efficient procedure are presented in Figs. 3– 5 for the three α values mentioned above (i.e., α equal to 0.0, 0.50, and 0.75). It is also worthwhile to mention that ω¯ = ω1S is chosen for all these cases. It should be noted that Fig. 3 is directly comparable with Fig. 2, and one can capture a feeling of what degree of improvement is yielded for the modified efficient procedure in comparison to the original efficient method. However, to make a more quantitative comparison, the percentage error is also calculated at the frequency corresponding to the first major peak of the response for both these methods (under this worst condition case of α = 0.0), and it is presented in Table 1. It is observed that the percentage error for the original efficient approach could reach up to about 10%, which occurs for the upstream excitation. However, this small amount of error is decreased even further to a maximum value of about 0.5% in the case of modified efficient procedure, which is a vast improvement.
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Fig. 4. Frequency-response functions due to different excitations (α = 0.5): modified-efficient method.
Fig. 5. Frequency-response functions due to different excitations (α = 0.75): modified-efficient method.
Table 1 Comparison of the percentage error at the first major peak of response between the original efficient method and the modified efficient procedure for α = 0.0
Table 2 Percentage error of the response at the first major peak for the modified efficient method
Method
Upstream
Vertical
Cross-stream
α
Upstream
Vertical
Cross-stream
Efficient [7] Modified efficient
9.79 0.17
5.03 0.55
1.65 0.08
0.0 0.5 0.75
0.17 0.54 0.10
0.55 0.18 0.33
0.08 0.23 0.12
Let us now concentrate purely on the results of the modified efficient method for different values of the wave reflection coefficient α (Figs. 3–5). It is observed that the response in each case matches very well with the exact response, such that it is hardly distinguishable from the corresponding exact curve. Once again, it is worthwhile to quantify the approximation involved. For this purpose, the percentage errors for these cases are also calculated at the frequencies corresponding to the first major peak of the response. These values are presented in Table 2. It is observed that the percentage errors for all the cases are very small, and the maximum error is about
0.5%. This proves that the modified efficient method is a very robust technique and it works extremely well under different conditions, some of which would be considered as highly challenging cases. 5. Conclusions The formulation based on FE-(FE–HE) procedure for the dynamic analysis of concrete arch dam–reservoir systems was explained briefly. A modified efficient procedure was proposed for the calculation of the impedance matrix of the three
A. Aftabi S., V. Lotfi / Engineering Structures 29 (2007) 2654–2661
dimensional fluid hyper-element. This was envisaged to be an enhancement of a previously proposed efficient method [7]. A computer program was developed based on this methodology, and the response of the Morrow Point arch dam was studied as a typical example for various values of the wave reflection coefficient α, and different excitation types. The results are always presented in comparison with the exact response, so that one can easily capture a feeling of the possible error due to the approximation introduced. Overall, the main conclusions obtained from the present study can be listed as follows: • Initially, the percentage error at the first major peak of the response is calculated for the modified efficient procedure, and it is compared against similar quantities obtained in a previous study for the original efficient method. This is carried out for α = 0.0 and three types of excitation. It is observed that the percentage error for the original efficient approach could reach up to about 10%, which occurs for the upstream excitation. However, this small amount of error is decreased even further to the maximum value of about 0.5% in the case of the modified efficient procedure, which is a vast improvement. • In the next stage, the concentration was purely on the results of the modified efficient method for different values of the wave reflection coefficient α. Based on these results, it is observed that response in each case matches very well with the exact response, to the extent that it is hardly distinguishable from the corresponding exact curve. The percentage errors for these cases are also calculated at the
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frequencies corresponding to the first major peak of the response. It is noted that the percentage errors for all the cases are very small, and the maximum error is about 0.5%. This proves that the modified efficient procedure is a very robust and accurate method, and that it works extremely well under different conditions, some of which would be considered as highly challenging cases. References [1] Camara RJ. A method for coupled arch dam–foundation reservoir seismic behavior analysis. Earthquake Engineering and Structural Dynamics 2000; 29:441–60. [2] Maity D, Bhattacharyya SK. Time-domain analysis of infinite reservoir by finite element method using a novel far-boundary condition. Finite Element in Analysis and Design 1999;32:85–96. [3] Maeso O, Aznarez JJ, Dominguez J. Effects of space distribution of excitation on seismic response of arch dams. Journal of Engineering Mechanics Division, ASCE 2002;128(7):759–68. [4] Lotfi V. Direct frequency domain analysis of concrete arch dams based on FE-(FE–HE)-BE technique. Journal of Computers & Concrete 2004;I(3): 285–302. [5] Hall JF, Chopra AK. Dynamic analysis of arch dams including hydrodynamic effects. Journal of Engineering Mechanics Division, ASCE 1983;109(1):149–63. [6] Fok K-L, Chopra AK. Frequency response functions for arch dams: Hydrodynamic and foundation flexibility effects. Earthquake Engineering and Structural Dynamics 1986;14:769–95. [7] Lotfi V. An efficient three-dimensional fluid hyper-element for dynamic analysis of concrete arch dams. Journal of Structural Engineering and Mechanics 2006;24(6):683–98. [8] Lotfi V. Seismic analysis of concrete dams using the pseudo-symmetric technique. Journal of Dam Engineering 2002;XIII(2):119–45.
Linear dynamic analysis of arch dams utilizing modified efficient fluid hyper-element Ahmad Aftabi Sani ∗ , Vahid Lotfi Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran Received 10 September 2006; received in revised form 9 January 2007; accepted 10 January 2007 Available online 1 March 2007
Abstract The accurate dynamic analysis of concrete arch dams relies heavily on employing a three-dimensional semi-infinite fluid element. The usual method for calculating the impedance matrix of this fluid hyper-element is dependent on the solution of a complex eigenvalue problem for each frequency. In the present study, a modified efficient procedure is proposed, which simplifies this procedure significantly, and results in great computational time savings. Moreover, the accuracy of this technique is examined thoroughly and it is concluded that modified efficient procedure is also extremely accurate under all practical conditions. It is also shown that the original efficient method proposed in a previous study can be considered as a special case of the present more general procedure. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Dynamic analysis; Arch dams; Fluid hyper-element
1. Introduction There are different alternatives for the dynamic analysis of concrete dams [1–3]. However, the rigorous analysis of concrete arch dam–reservoir systems is based on the FE(FE–HE) method (i.e., Finite Element-(Finite Element–Hyper Element)). This means, the dam is discretized by solid finite elements, while, the reservoir is divided into two parts, a near field region (usually an irregular shape) in the vicinity of the dam and a far field part (assuming a uniform channel), which extends to infinity. The former region is discretized by fluid finite elements and the latter part is modeled by a threedimensional fluid hyper-element. The analysis is carried out in the frequency domain either by a direct approach [4], or by sub-structuring techniques [5,6]. Regardless of the option selected from among these rigorous techniques, a major portion of the numerical calculation time spent is on the solution of a complex eigenvalue problem related to fluid hyper-element, which must be solved at each frequency. To remedy this, an efficient procedure was proposed in a previous study [7] for the required impedance matrix calculations of the fluid
hyper-element, which greatly reduces the computational time. Although, the method was very accurate in most practical conditions, it was estimated that it would introduce errors (in the range of 10%) under certain circumstances. In this paper, a modified efficient procedure is proposed for the above-mentioned problem. This is envisaged to be an enhancement of the previously proposed efficient method [7]. The approach mainly relies on mode shapes corresponding to an arbitrary frequency. In the following sections, the analysis technique (i.e., FE-(FE–HE) method) is reviewed very briefly. The fluid hyper-element impedance matrix’s theoretical background is also described without going into unnecessary details. Subsequently, the modified efficient procedure is introduced and its formulation is presented. It is also shown that the original efficient method can be considered as a special case of the modified efficient procedure presented here. Thereafter, a special-purpose program is developed based on the proposed theory, and the response of the Morrow Point arch dam is obtained for various conditions. This is utilized to investigate the accuracy of the modified efficient procedure thoroughly. 2. Method of analysis
∗ Corresponding author. Tel.: +98 (0) 511 270 6867.
E-mail addresses: ahmad [email protected] (A. Aftabi S.), [email protected] (V. Lotfi). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.01.005
The analysis technique utilized in this study is based on the FE-(FE–HE) method, which is applicable for a general concrete
A. Aftabi S., V. Lotfi / Engineering Structures 29 (2007) 2654–2661
arch dam–reservoir system. It is worthwhile to mention that the procedure utilized can be easily generalized for a flexible foundation rock case. However, the foundation is assumed to be rigid to avoid the extra complexities that would otherwise arise. The formulation can be explained much more easily if one concentrates initially on a dam with a finite reservoir system (basically the same as a model of a dam and a reservoir nearfield), and subsequently add the effects of a reservoir far-field region for the general case. For this purpose, let us begin with this simpler formulation and then complete the formulation for the more general case on that basis. 2.1. Dam with finite reservoir system This is the problem which can be totally modeled by the finite element method. It can be easily shown that in this case, the coupled equations of the system may be written as [8]: M 0 r¨ C 0 r˙ K −BT r + + B G p¨ 0 L p˙ p 0 H −MJag = (1) −BJag M, C, K in this relation represent the mass, damping and stiffness matrices of the dam body, while G, L, H are the assembled matrices of the fluid domain. The unknown vector is composed of r, which is the vector of nodal relative displacements and the vector p that includes the nodal pressures. Moreover, J is a matrix with each of the three rows equal to a 3 × 3 identity matrix (its columns correspond to unit rigid body motion in cross-canyon, stream, and vertical directions) and ag denotes the vector of ground accelerations. Furthermore, B in the above relation is often referred to as interaction matrix. For harmonic ground excitations ag (t) = ag (ω)eiωt with frequency ω, displacements and pressures will all behave harmonic, and Eq. (1) can be expressed as: 2 −ω M + K(1 + 2βd i) −BT r −2 2 p −B ω (−ω G + iωL + H) −MJag = (2) −ω−2 BJag In this relation, it is assumed that the damping matrix of the dam is of hysteretic type. This means: 2βd K (3) ω where βd is the constant hysteretic factor of the dam body. Relation (2) is the coupled equations of a dam with a finite reservoir system in the frequency domain. It should be also noted that the system of equations is made symmetric by multiplying the lower partition matrices by a factor of ω−2 .
the present study. Apart from that, there are three possible boundary condition types for the reservoir near-field region. The condition of type I is considered for the contact of fluid with flexible solid, such as the dam–reservoir interface. The second type of condition is the so-called approximate boundary condition. This can be imposed at the reservoir bottom and sidewalls. The last type of condition (III) is referred to as Sommerfeld boundary condition. This is usually applied at the reservoir near-field upstream boundary (in cases in which the far-field region is not modeled), as a substitute for a precise transmitting boundary. However, when a fluid hyperelement is utilized, this condition is not required, and waves are transmitted exactly through that semi-infinite element. The exact relations and detailed explanation about these conditions can be found in other studies (e.g., Ref. [4]). 2.3. Fluid hyper-element As mentioned, the three-dimensional fluid hyper-element is utilized to model the reservoir far-field region for the more general case. This part of the water domain is assumed to be a uniform channel with an arbitrary geometric shape in the vertical plane which includes x, z-axes (see Fig. 1b for a typical discretization), and extends to infinity in the upstream direction (negative y-direction). Although, this is a three-dimensional semi-infinite fluid element, its discretization is performed in the vertical plane perpendicular to the channel axis, which is referred to as the reference plane (y = 0). Therefore, the element consists of several sub-channels which extend to infinity, and all nodes of the hyper-element are located on that reference plane. The formulation of this element is briefly reviewed as follows: Assuming water to be linearly compressible and neglecting its viscosity, the small amplitude, irrotational motion of water due to harmonic excitation is governed by the Helmholtz equation: ∂2 P ∂2 P ∂2 P ω2 + + + P=0 ∂x2 ∂ y2 ∂z 2 C2
The water surface boundary condition is easily applied, especially if surface waves are neglected, as is the case in
(4)
where P is the amplitude of the hydrodynamic pressure (in excess of hydrostatic pressure) and C is the velocity of pressure waves in water. By seeking solutions of the form eky in the stream direction, Eq. (4) becomes
C=
2.2. Reservoir near-field boundary conditions
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λ2 P +
∂2 P ∂2 P + =0 ∂x2 ∂z 2
(5)
with the following definition of λ. λ2 = k 2 +
ω2 C2
(6)
By applying the variational method on Eq. (5), a matrix relation is obtained for each sub-element (see [7] for details). Assembling the effects of different sub-elements and imposing the homogeneous boundary conditions (corresponding to zero
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ground acceleration) on the resulting matrix equation leads to the following eigenvalue problem: [−λ2j A + iωqLh + C]X j = 0
(7)
λ2j , X j are the jth eigenvalue and eigenvector of the fluid hyperelement. There also exist particular solutions (i.e., P`p ) which correspond to the uniform unit acceleration of the reservoir boundary in the `-direction (x, or z-direction). Therefore, the vector of pressure amplitudes for the hyper-element can be totally written as: Ph = Xh Γ + P p ag (ω)
(8)
by help of the following definitions for the reservoir hyperelement modal matrix, vector of participation factors and a matrix which includes particular solution vectors for different directions. Xh = [X1 , X2 , . . . , Xn ]
becomes: # " −ω2 M + K(1 + 2βd i) −BT ¯ h (ω)) −B ω−2 ((−ω2 G + iωL + H) + H " # −MJag r × = (12) ¯ p (ω)ag ) p ω−2 (−BJag + R
¯ p (ω) are the expanded forms of the Hh (ω) ¯ h (ω) and R where H and R p (ω) matrices which cover the entire fluid domain pressure degrees of freedom. Assuming that the pressure degrees of freedom related to the hyper-element are ordered first in the unknown pressure vector, these matrices have the following form: ¯ h (ω) = Hh (ω) 0 H (13a) 0 0 R p (ω) ¯ R p (ω) = (13b) 0
Γ = [γ1 , γ2 , . . . , γn ]T P p = Pxp 0 Pzp
The relation (12) is the equation to be used instead of (2) when the reservoir is extended to infinity and one is considering the direct approach in the frequency domain.
Moreover, the fluid y-direction (stream component) accelerations vector for the reference plane, becomes (see Ref. [7] for details);
2.5. Fluid hyper-element (modified efficient procedure)
¨ h = − 1 Xh Kh Γ U ρ
(9)
where Kh is a diagonal matrix with the jth diagonal element being equal to k j . Solving for the participation vector from (8), substituting in (9), and multiplying the resulting relation by −A, one obtains: Rh = Hh Ph − R p ag (ω)
(10)
by employing the following definitions: ¨h Rh = −AU 1 Hh = AXh Kh XTh A ρ R p = Hh P p
This element is basically formulated in the same manner as the usual fluid hyper-element. However, the main required matrix (i.e., Hh (ω)) is calculated based on the modified efficient method. It should be mentioned that the proposed technique is a modified version of the procedure originally referred to as the efficient method [7]. It will also be explained underneath how this modified technique transforms to the previously proposed efficient method. The fundamental concepts of this technique are presented as follows: Let us consider the formula which is utilized to define matrix Hh (ω):
(11a) (11b) (11c)
In relation (10), Rh represents a consistent vector equivalent to the integration of inward horizontal acceleration (the negative of the stream component) for the hyper-element, and this vector contains essentially similar quantities as the components of the right hand side vectors of the usual fluid finite elements [7]. 2.4. Dam–reservoir system The formulation for a dam with finite reservoir was already presented. For the case where the reservoir extends to infinity, a hyper-element must be used along with the fluid finite elements utilized for the reservoir near-field. Moreover, the governing relation for the hyper-element was derived in the previous section (relation (10)). Therefore, if the matrices of the hyperelement are assembled with the fluid finite elements, Eq. (2)
Hh (ω) =
1 AXh (ω)Kh (ω)XTh (ω)A ρ
(Repeated (11b))
The calculation of the above matrix is very time consuming and cumbersome due to its ingredients Xh (ω), Kh (ω), which are both frequency dependent. These matrices depend on the solution of the above-mentioned eigenvalue problem (7), which may be rearranged as: [C + iωqLh ]X j = λ2j AX j
(14)
In general, this eigenvalue problem must be solved repeatedly for the different required frequencies. However, let us assume that it is only solved for the unique case of ω = ω¯ (i.e., ω¯ being an arbitrary frequency), which can be written as: ¯ j = λ¯ 2j AX ¯j [C + iωqL ¯ h ]X
(15)
Based on the solution of this eigenproblem, one can form the following matrices: ¯ 1, X ¯ 2, . . . , X ¯ n] ¯ h = [X X
(16a)
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..
.
¯ = Λ
λ¯ 2j
..
.
(16b)
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¯ j is also the j-th eigenvector Therefore, it can be claimed that X of the following eigenproblem: ˆ h ]X ¯ j = λ2j AX ¯j [C + iωqL ¯ h + i(ω − ω)q ¯ L
(27)
It is worth to mention that the bar superscript denotes that these matrices are for the special case of ω = ω. ¯ Furthermore, the orthogonality relations in this case are:
which is very similar to the original relation (14), or its following alternative form:
¯ Th AX ¯h = I X ¯ Th (C + iωqL ¯h X ¯ h )X
The difference is mainly in the third term, which Lh is ˆ h . Obviously, this term replaced by an approximate form, L becomes negligible when the excitation frequency is close to ω. ¯ Moreover, ω¯ can be selected as any arbitrary value (e.g., the first natural frequency of the system, or wherever one is interested in decreasing the error introduced due to this approximation). The orthogonality relations corresponding to (27) may be written as:
(17a) ¯ =Λ
(17b)
Similarly, in the general case of (14), the orthogonality relations are written as: XTh AXh = I XTh (C + iωqLh )Xh
(18a) =Λ
(18b)
with the following matrix definitions: Xh = [X1 , X2 , . . . , Xn ] .. . λ2j Λ= .. .
(19a)
(19b)
¯ Th AX ¯h = I X
(29a)
¯ Th (C + iωqL ˆ h )X ¯h = Λ X ¯ h + i(ω − ω)q ¯ L
(29b)
∗ ¯ + i(ω − ω)qL Λ=Λ ¯ D
(30)
The equality (30) may be presented in a different form taking into account definitions (16b), and (19b):
or alternatively: ¯ Th (C + iωqL ¯ h = Diagonal Matrix (21) X ¯ h + i(ω − ω)qL ¯ h )X Let us investigate the conditions under which this is possible. By employing (17b), relation (21) can be written as: ∗ ¯ + i(ω − ω)qL ¯ Th (C + iωqL ¯h = Λ X ¯ h + i(ω − ω)qL ¯ ¯ (22) h )X
¯ is a diagonal matrix. However, this is not the case for L∗ Λ ¯ h ) in general. Thus, let us define a diagonal ¯ Th Lh X (i.e., L∗ = X ∗ matrix L D as below: L∗D = Diagonal L∗ (23) It is apparent that Eq. (21) would be satisfied now, if the second Lh appearing in that equation is replaced by Lˆ h , such that it yields: ∗ ¯ + i(ω − ω)qL ¯ Th (C + iωqL ¯h = Λ X ¯ h + i(ω − ω)q ¯ Lˆ h )X ¯ D (24)
Of course, this requires the following equality: (25)
ˆ h is not required to be defined explicitly, Although, the matrix L this can be achieved through relation (25) by employing (17a): ˆ h = AX ¯ h L∗D X ¯ Th A L
(28)
Comparing relation (24), and (29b), we get:
By comparing relations (17) and (18), it is easy to see that ¯ h , if the Xh (ω) could be frequency independent and equal to X following operation results in a diagonal matrix: ¯ Th (C + iωqLh )X ¯ h = Diagonal Matrix X (20)
¯ h = L∗D ¯ Th L ˆ hX X
2 [C + iωqL ¯ h + i(ω − ω)qL ¯ h ]X j = λ j AX j
(26)
λ2j = λ¯ 2j + i(ω − ω)q ¯ L ∗D j
(31)
Moreover, the wave number k j , can be found from (6) by employing (31): r kj = λ¯ 2j − ω2 /C 2 + i(ω − ω)q ¯ L ∗D j (32) Based on the above discussion and following the same procedure, which leads to (11b), one can obtain the following equation: Hh (ω) =
1 ¯ ¯ Th A AXh Kh (ω)X ρ
(33)
¯ h in this relation is frequencyIt must be emphasized that X independent, and corresponds to the modal matrix for the unique case of ω = ω. ¯ The only frequency-dependent matrix is Kh , which is a diagonal matrix and its elements are easily calculated using Eq. (32) for any frequency ω. This considerably simplifies the calculation of the main contributing matrix of the fluid hyper-element (i.e., Hh (ω)). In this manner, it is not required any more to solve a complex eigenvalue problem for each frequency. In other words, the procedure depends on the solution of the eigenproblem for an arbitrary frequency (i.e., the one corresponding to ω = ω). ¯ It can be also easily verified that the relations obtained herein for the modified efficient technique would be the same as the ones derived previously for the more simplified method referred to as the efficient method, if the arbitrary frequency ω¯ is chosen to be equal to zero. The main advantage of the original efficient method is that the relation (15) would become a standard eigenproblem in that case, and the solution can be obtained
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Fig. 1a. Finite element mesh of the dam body.
¯ h would be a modal by real number arithmetic. Therefore, X matrix with all its elements being real numbers. However, the main disadvantage in comparison to the modified efficient method could be the less accurate calculation of the hyperelement impedance matrix, which will ultimately transform into an increased error in the response of the dam to certain frequencies. This will be examined thoroughly in the next section.
Fig. 1b. Discretization of water domain (fluid finite elements (L/H = 0.2), and the fluid hyper-element).
3. Modeling and basic parameters A computer program [DACAD85] was developed based on the theory presented in the previous section. The program is based on the FE-(FE–HE) concept. This means the dam is treated as being composed of solid finite elements, while the reservoir is divided into two parts, the near-field region in the vicinity of the dam, which is discretized by fluid finite elements, and the far-field part, which is modeled by a three-dimensional hyper-element. The program has three options for calculating the fluid hyper-element impedance matrix. These are based on the exact method, the previous efficient approach or the modified efficient procedure explained above in detail.
Fig. 1c. Complete dam–reservoir system.
characteristics: Elastic modulus (E d ) = 27.5 GPa Poisson’s ratio = 0.2 Unit weight = 24.8 kN/m3
3.1. Models
Hysteretic damping factor (βd ) = 0.05
An idealized symmetric model of the Morrow Point arch dam is considered. The geometry of the dam may be found in Ref. [5]. The dam is discretized by 40 isoparametric 20-node solid finite elements (Fig. 1a). The water domain is divided into two regions (Fig. 1b). The near-field part is considered as a region, which extends to a specified length L = 0.2H (H being the dam height or maximum water depth in the reservoir), which is measured in upstream direction at the dam’s mid-crest point. The far-field region starts from that point and extends to infinity in the upstream direction. Both these regions combined are assumed to form a uniform reservoir shape as in a previous study (Fig. 1c) [7]. The near-field region is discretized by 80 isoparametric 20-node fluid finite elements, while the far-field region is modeled by a fluid hyper-element which itself is constructed from 40 isoparametric 8-node sub-elements.
The impounded water is taken as an inviscid and compressible fluid with unit weight equal to 9.81 kN/m3 , and pressure wave velocity C = 1440 m/s.
3.2. Material properties The dam concrete is assumed to be homogeneous with isotropic linearly viscoelastic behavior and the following main
4. Results of analyses The responses of dam crest are obtained due to upstream, vertical and cross-stream excitations for several values of the wave reflection coefficient α. In particular, α values equal to 0.0, 0.50, and 0.75 are considered. These values are selected similarly to the values considered in the previous study [7]. It should be mentioned that the response quantities which will be presented in this section are the amplitudes of the complex valued radial accelerations for two points located at dam crest (Fig. 1a). This is either the mid-crest point (θ = 0◦ ) selected for upstream or vertical excitations, or a point located at (θ = 13.25◦ ) which is used for the case of cross-stream excitations. This is due to the fact that radial acceleration is zero at the mid-crest for the cross-stream type of ground motion. It must be emphasized that these accelerations are obtained for unit amplitude ground accelerations in each case. Therefore,
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Fig. 2. Frequency-response functions due to different excitations (α = 0): efficient method [7].
Fig. 3. Frequency-response functions due to different excitations (α = 0): modified-efficient method.
they are actually dimensionless quantities, or it can be thought as normalized accelerations at the dam crest. In each case, the amplitude of the radial acceleration is plotted against the dimensionless frequency for a significant range. The dimensionless frequency for upstream and vertical excitation is defined as ω/ω1S , where ω is the excitation frequency and ω1S is the fundamental frequency of the dam on a rigid foundation with an empty reservoir for a symmetric mode. For the cross-stream excitation cases, the dimensionless frequency is defined as ω/ω1a , where ω1a is the fundamental resonant frequency of the dam on a rigid foundation with an empty reservoir for an anti-symmetric mode. As mentioned above, several values of α are considered, and results are obtained for different types of excitations. Analyses are carried out based on the modified efficient method proposed herein, and in each of these cases, the response is compared against the exact method. However, prior to this presentation, it is worthwhile to take a glance at a similar comparison for the original efficient method, which was obtained in the previous study [7]. This is provided in Fig. 2 only for the worst case (i.e.,
the maximum error in comparison to exact method) that was calculated for α = 0.0. Thereafter, the current study results for the modified efficient procedure are presented in Figs. 3– 5 for the three α values mentioned above (i.e., α equal to 0.0, 0.50, and 0.75). It is also worthwhile to mention that ω¯ = ω1S is chosen for all these cases. It should be noted that Fig. 3 is directly comparable with Fig. 2, and one can capture a feeling of what degree of improvement is yielded for the modified efficient procedure in comparison to the original efficient method. However, to make a more quantitative comparison, the percentage error is also calculated at the frequency corresponding to the first major peak of the response for both these methods (under this worst condition case of α = 0.0), and it is presented in Table 1. It is observed that the percentage error for the original efficient approach could reach up to about 10%, which occurs for the upstream excitation. However, this small amount of error is decreased even further to a maximum value of about 0.5% in the case of modified efficient procedure, which is a vast improvement.
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Fig. 4. Frequency-response functions due to different excitations (α = 0.5): modified-efficient method.
Fig. 5. Frequency-response functions due to different excitations (α = 0.75): modified-efficient method.
Table 1 Comparison of the percentage error at the first major peak of response between the original efficient method and the modified efficient procedure for α = 0.0
Table 2 Percentage error of the response at the first major peak for the modified efficient method
Method
Upstream
Vertical
Cross-stream
α
Upstream
Vertical
Cross-stream
Efficient [7] Modified efficient
9.79 0.17
5.03 0.55
1.65 0.08
0.0 0.5 0.75
0.17 0.54 0.10
0.55 0.18 0.33
0.08 0.23 0.12
Let us now concentrate purely on the results of the modified efficient method for different values of the wave reflection coefficient α (Figs. 3–5). It is observed that the response in each case matches very well with the exact response, such that it is hardly distinguishable from the corresponding exact curve. Once again, it is worthwhile to quantify the approximation involved. For this purpose, the percentage errors for these cases are also calculated at the frequencies corresponding to the first major peak of the response. These values are presented in Table 2. It is observed that the percentage errors for all the cases are very small, and the maximum error is about
0.5%. This proves that the modified efficient method is a very robust technique and it works extremely well under different conditions, some of which would be considered as highly challenging cases. 5. Conclusions The formulation based on FE-(FE–HE) procedure for the dynamic analysis of concrete arch dam–reservoir systems was explained briefly. A modified efficient procedure was proposed for the calculation of the impedance matrix of the three
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dimensional fluid hyper-element. This was envisaged to be an enhancement of a previously proposed efficient method [7]. A computer program was developed based on this methodology, and the response of the Morrow Point arch dam was studied as a typical example for various values of the wave reflection coefficient α, and different excitation types. The results are always presented in comparison with the exact response, so that one can easily capture a feeling of the possible error due to the approximation introduced. Overall, the main conclusions obtained from the present study can be listed as follows: • Initially, the percentage error at the first major peak of the response is calculated for the modified efficient procedure, and it is compared against similar quantities obtained in a previous study for the original efficient method. This is carried out for α = 0.0 and three types of excitation. It is observed that the percentage error for the original efficient approach could reach up to about 10%, which occurs for the upstream excitation. However, this small amount of error is decreased even further to the maximum value of about 0.5% in the case of the modified efficient procedure, which is a vast improvement. • In the next stage, the concentration was purely on the results of the modified efficient method for different values of the wave reflection coefficient α. Based on these results, it is observed that response in each case matches very well with the exact response, to the extent that it is hardly distinguishable from the corresponding exact curve. The percentage errors for these cases are also calculated at the
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frequencies corresponding to the first major peak of the response. It is noted that the percentage errors for all the cases are very small, and the maximum error is about 0.5%. This proves that the modified efficient procedure is a very robust and accurate method, and that it works extremely well under different conditions, some of which would be considered as highly challenging cases. References [1] Camara RJ. A method for coupled arch dam–foundation reservoir seismic behavior analysis. Earthquake Engineering and Structural Dynamics 2000; 29:441–60. [2] Maity D, Bhattacharyya SK. Time-domain analysis of infinite reservoir by finite element method using a novel far-boundary condition. Finite Element in Analysis and Design 1999;32:85–96. [3] Maeso O, Aznarez JJ, Dominguez J. Effects of space distribution of excitation on seismic response of arch dams. Journal of Engineering Mechanics Division, ASCE 2002;128(7):759–68. [4] Lotfi V. Direct frequency domain analysis of concrete arch dams based on FE-(FE–HE)-BE technique. Journal of Computers & Concrete 2004;I(3): 285–302. [5] Hall JF, Chopra AK. Dynamic analysis of arch dams including hydrodynamic effects. Journal of Engineering Mechanics Division, ASCE 1983;109(1):149–63. [6] Fok K-L, Chopra AK. Frequency response functions for arch dams: Hydrodynamic and foundation flexibility effects. Earthquake Engineering and Structural Dynamics 1986;14:769–95. [7] Lotfi V. An efficient three-dimensional fluid hyper-element for dynamic analysis of concrete arch dams. Journal of Structural Engineering and Mechanics 2006;24(6):683–98. [8] Lotfi V. Seismic analysis of concrete dams using the pseudo-symmetric technique. Journal of Dam Engineering 2002;XIII(2):119–45.