**1. Introduction**

88 Hydrodynamics – Natural Water Bodies

Wasman, P. Retention versus export food chains: processes controlling sinking loss from

Dynamic responses of dam-reservoir systems subjected to ground motions are often a major concern in the design. To ensure that dams are adequately designed for, the hydrodynamic pressure distribution along the dam-reservoir interface must be determined for assessment of safety.

Due to the fact that analytical methods are not readily available for dam-reservoir systems with arbitrary geometry shape, numerical methods are often used to analyze responses of dam-reservoir systems. In numerical methods, dams are often discretized into solid finite elements through Finite Element Method (FEM), while the reservoir is either directly modeled by Boundary Element Method (BEM) or is divided into two parts: a near field with arbitrary geometry shape and a far field with a uniform cross section. The near field is discretized into acoustic fluid finite elements by using FEM or boundary elements by BEM, while the far field is modeled by BEM or a Transmitting Boundary Condition (TBC). Based on these numerical methods, several coupling procedures were developed.

A FEM-BEM coupling procedure was used to implement the linear and non-linear analysis of dam-reservoir interaction problems (Tsai & Lee, 1987; Czygan & Von Estorff, 2002), respectively, in which the dam was modeled by FEM, while the reservoir was modeled by BEM. A BEM-TBC coupling method was adopted to solve dam-water-foundation interaction problems and dam-reservoir-sediment-foundation interaction problems (Dominguez & Maeso, 1993; Dominguez et al., 1997). The dam and the near field of the reservoir were discretized by using BEM, while the far field of the reservoir was represented by a TBC. As a traditional numerical method, BEM has been popular in simulating unbounded medium, but it needs a fundamental solution and includes a singular integral, which affect its application. In order to avoid deriving a fundamental solution required in BEM, the TBC attracted some researchers' interests. A Sommerfeld-type TBC was used to represent the far field (Kucukarslan et al., 2005), while a Sharan-type TBC was proposed for infinite fluid (Sharan, 1987). The Sommerfeld-type and Sharan-type TBCs are readily implemented in FEM due to their conciseness, but a sufficiently large near field is required to model accurately the damping effect of semi-infinite reservoir. Except for the aforementioned TBCs, an exact TBC (Tsai & Lee, 1991), a novel TBC (Maity &

Hydrodynamic Pressure Evaluation of

as the dam-reservoir interaction problems.

Dam

Dam-reservoir interface

**2. Problem statement** 

Fig. 1. Dam-reservoir system

Reservoir Subjected to Ground Excitation Based on SBFEM 91

was well suitable for all frequencies and no additional computational costs were increased for low frequency analysis in comparison with for high frequency analysis. Its advantages were exhibited by analyzing the harmonic responses of dam-reservoir systems in the frequency domain. However in the time domain, its advantages are not as obvious as those in the frequency domain because integral convolutions still need evaluating. Although a Riccati equation and Lyapunov equations were presented to solve the integral convolutions (Wolf & Song, 1996b), solving them needed great computational costs, especially for large-scale systems, which limited the SBFEM applications in the time domain. To simplify the integral convolutions and save computational costs, some recursive formulations were proposed (Paronesso & Wolf, 1998; Yan et al., 2004), based on a diagonalization procedure and the linear system theory (Paronesso & Wolf, 1995). The integral convolution was transformed into an equivalent system of linear equations, named state-variable description which was represented by finite-difference equations. However, the coefficient matrix quaternion of finite-difference equations was calculated by using Hankel matrix realization algorithms, which complicated the analysis. Furthermore, the diagonalization procedure increased the order of the dynamic mass matrix, and some global lumped parameters, such as springs, dashpots and masses, used in the diagonalization procedure must be introduced at additional internal nodes corresponding to inner variables in the state-variable description, besides the nodes on the structure-medium interface. The number of global lumped parameters would become very large for large-scale systems. This weakened the feasibility of the diagonalization procedure. A new diagonalization procedure of the SBFEM for semi-infinite reservoir was proposed (Li, 2009), whose calculation efficiency was proven to be high, although it still included convolution integrals. With the improvement of the SBFEM evaluation efficiency in the time domain analysis, the SBFEM will show gradually its advantages and potential to solve problems including unbounded soil or unbounded acoustic fluid medium, such

Consider dam-reservoir interaction problems subjected to horizontal ground accelerations. The dam-reservoir system and its Cartesian coordinate system were shown in Fig.1. The

Free surface

y

Near field L H

Near-far-field interface

Far field

x

Reservoir bottom

Bhattacharyya, 1999) and a non-reflecting TBC (Gogoi & Maity, 2006) were proposed, respectively. These complicated TBCs gave better results even when a small near field was chosen, but their implementations in a finite element code became complex and tedious.

In this chapter, the scaled boundary finite element method (SBFEM) was chosen to model the far field. The SBFEM does not require fundamental solutions and is able to model accurately the damping effect of semi-infinite reservoir and incorporate with FEM readily, but the SBFEM requires the geometry of far field is layered (or tapered). Although BEM and some of TBCs can handle far fields with arbitrary geometry, far fields in most dam-reservoir systems are always chosen to be layered with a uniform cross section, which ensures the SBFEM can be used in dam-reservoir interaction problems.

Based on a mechanically-based derivation, the SBFEM was proposed for infinite medium (Wolf & Song, 1996a; Song & Wolf, 1996) which was governed by a three-dimensional scalar wave equation and a three-dimensional vector wave equation, respectively. A dynamic stiffness matrix and a dynamic mass matrix were introduced to represent infinite medium in the frequency domain and the time domain, respectively. The dynamic stiffness matrix satisfies a non-linear ordinary differential equation of first order, while the dynamic mass matrix is governed by an integral convolution equation. The SBFEM reduces spatial dimensions by one. Only boundaries need discretization and its solutions in the radial direction are analytical. Therefore, it can handle well bounded domain problems with cracks and stress singularities and unbounded domain problems including infinite soil or unbounded acoustic fluid medium. In analyzing crack and stress singularities problems, the SBFEM placed the scaling center on the crack tip and only discretized the boundary of bounded domain using supper elements except the straight traction free crack faces, which permitted a rigorous representation of the stress singularities around the crack tip (Song, 2004; Song & Wolf, 2002; Yang & Deeks, 2007). The response of unbounded domain problems was obtained by using the SBFEM alone or coupling FEM and the SBFEM. A FEM-SBFEM coupling procedure was used to analyze unbounded soil-structure interaction problems in the time domain (Ekevid & Wiberg, 2002; Bazyar & Song, 2008). For unbounded acoustic fluid medium problems, a FEM-SBFEM coupling procedure combined with acoustic approximations was proposed to evaluate the responses of submerged structures subjected to underwater shock waves in the time domain (Fan et al., 2005; Li & Fan, 2007). Results showed that the SBFEM was able to model accurately the damping behavior of the unbounded soil and infinite acoustic fluid medium, but it was computationally expensive because the evaluations of the dynamic mass matrix and dynamic responses need solving integral convolution equations. In the frequency domain, dynamic condensation and substructure deletion methods were used to evaluate the dynamic stiffness matrix, which avoid evaluating integral convolution equations, but evaluation errors increased with frequency increasing so that results at high frequencies were not acceptable (Wolf & Song, 1996b). To evaluate accurately high frequencies behaviors of the dynamic stiffness matrix, a Pade series was presented to analyze out-of-plane motion of circular cavity embedded in full-plane through using the SBFEM alone (Song & Bazyar, 2007). Good results were obtained at high frequencies, but results at low frequencies were inferior even if a high order Pade series was used. The high order Pade series was not only complex, and also increased computational cost. A simplified SBFEM formulation was presented through discovering a zero matrix and a FEM-SBFEM coupling procedure was used to analyze dam-reservoir interaction problems subjected to ground motions (Fan & Li, 2008). The simplified SBFEM

Bhattacharyya, 1999) and a non-reflecting TBC (Gogoi & Maity, 2006) were proposed, respectively. These complicated TBCs gave better results even when a small near field was chosen, but their implementations in a finite element code became complex and tedious. In this chapter, the scaled boundary finite element method (SBFEM) was chosen to model the far field. The SBFEM does not require fundamental solutions and is able to model accurately the damping effect of semi-infinite reservoir and incorporate with FEM readily, but the SBFEM requires the geometry of far field is layered (or tapered). Although BEM and some of TBCs can handle far fields with arbitrary geometry, far fields in most dam-reservoir systems are always chosen to be layered with a uniform cross section, which ensures the

Based on a mechanically-based derivation, the SBFEM was proposed for infinite medium (Wolf & Song, 1996a; Song & Wolf, 1996) which was governed by a three-dimensional scalar wave equation and a three-dimensional vector wave equation, respectively. A dynamic stiffness matrix and a dynamic mass matrix were introduced to represent infinite medium in the frequency domain and the time domain, respectively. The dynamic stiffness matrix satisfies a non-linear ordinary differential equation of first order, while the dynamic mass matrix is governed by an integral convolution equation. The SBFEM reduces spatial dimensions by one. Only boundaries need discretization and its solutions in the radial direction are analytical. Therefore, it can handle well bounded domain problems with cracks and stress singularities and unbounded domain problems including infinite soil or unbounded acoustic fluid medium. In analyzing crack and stress singularities problems, the SBFEM placed the scaling center on the crack tip and only discretized the boundary of bounded domain using supper elements except the straight traction free crack faces, which permitted a rigorous representation of the stress singularities around the crack tip (Song, 2004; Song & Wolf, 2002; Yang & Deeks, 2007). The response of unbounded domain problems was obtained by using the SBFEM alone or coupling FEM and the SBFEM. A FEM-SBFEM coupling procedure was used to analyze unbounded soil-structure interaction problems in the time domain (Ekevid & Wiberg, 2002; Bazyar & Song, 2008). For unbounded acoustic fluid medium problems, a FEM-SBFEM coupling procedure combined with acoustic approximations was proposed to evaluate the responses of submerged structures subjected to underwater shock waves in the time domain (Fan et al., 2005; Li & Fan, 2007). Results showed that the SBFEM was able to model accurately the damping behavior of the unbounded soil and infinite acoustic fluid medium, but it was computationally expensive because the evaluations of the dynamic mass matrix and dynamic responses need solving integral convolution equations. In the frequency domain, dynamic condensation and substructure deletion methods were used to evaluate the dynamic stiffness matrix, which avoid evaluating integral convolution equations, but evaluation errors increased with frequency increasing so that results at high frequencies were not acceptable (Wolf & Song, 1996b). To evaluate accurately high frequencies behaviors of the dynamic stiffness matrix, a Pade series was presented to analyze out-of-plane motion of circular cavity embedded in full-plane through using the SBFEM alone (Song & Bazyar, 2007). Good results were obtained at high frequencies, but results at low frequencies were inferior even if a high order Pade series was used. The high order Pade series was not only complex, and also increased computational cost. A simplified SBFEM formulation was presented through discovering a zero matrix and a FEM-SBFEM coupling procedure was used to analyze dam-reservoir interaction problems subjected to ground motions (Fan & Li, 2008). The simplified SBFEM

SBFEM can be used in dam-reservoir interaction problems.

was well suitable for all frequencies and no additional computational costs were increased for low frequency analysis in comparison with for high frequency analysis. Its advantages were exhibited by analyzing the harmonic responses of dam-reservoir systems in the frequency domain. However in the time domain, its advantages are not as obvious as those in the frequency domain because integral convolutions still need evaluating. Although a Riccati equation and Lyapunov equations were presented to solve the integral convolutions (Wolf & Song, 1996b), solving them needed great computational costs, especially for large-scale systems, which limited the SBFEM applications in the time domain. To simplify the integral convolutions and save computational costs, some recursive formulations were proposed (Paronesso & Wolf, 1998; Yan et al., 2004), based on a diagonalization procedure and the linear system theory (Paronesso & Wolf, 1995). The integral convolution was transformed into an equivalent system of linear equations, named state-variable description which was represented by finite-difference equations. However, the coefficient matrix quaternion of finite-difference equations was calculated by using Hankel matrix realization algorithms, which complicated the analysis. Furthermore, the diagonalization procedure increased the order of the dynamic mass matrix, and some global lumped parameters, such as springs, dashpots and masses, used in the diagonalization procedure must be introduced at additional internal nodes corresponding to inner variables in the state-variable description, besides the nodes on the structure-medium interface. The number of global lumped parameters would become very large for large-scale systems. This weakened the feasibility of the diagonalization procedure. A new diagonalization procedure of the SBFEM for semi-infinite reservoir was proposed (Li, 2009), whose calculation efficiency was proven to be high, although it still included convolution integrals. With the improvement of the SBFEM evaluation efficiency in the time domain analysis, the SBFEM will show gradually its advantages and potential to solve problems including unbounded soil or unbounded acoustic fluid medium, such as the dam-reservoir interaction problems.
