**2. Problem statement**

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

Fig. 1. Dam-reservoir system

Hydrodynamic Pressure Evaluation of

far field was expressed as the following SBFEM formulation.

y

<sup>L</sup> <sup>H</sup>

Fig. 2. SBFEM discretization model of layered far field

**3.1 SBFEM formulation in the frequency domain** 

the dynamic stiffness matrix of the far field and **V***<sup>n</sup>*

is written as

where **Φ**

 

**S** (Li, et al., 2008) satisfies

**3. SBFEM formulation** 

Reservoir Subjected to Ground Excitation Based on SBFEM 93

Fig.2 showed the SBFEM discretization model of the far field shown in Fig.1, which was a layered semi-infinite fluid medium whose scaling center was located at minus infinity. The whole semi-infinite layered far field was divided into some layered sub-fields. Each layered sub-field was represented by one element on the near-far-field interface, so the whole far field was discretized into some elements on the near-far-field interface. Based on the discretization, a dynamic stiffness or mass matrix was introduced to describe the characteristics of the far field in the SBFEM. The interaction between the near field and the

Near-far-field interface

Free surface

Layered sub-fields

On the discretized near-far-field interface, the SBFEM formulation in the frequency domain (Fan & Li, 2008; Li et al., 2008) for the far field filled with unbounded acoustic fluid medium

Reservoir bottom

 

denotes the column vector composed of nodal velocity potentials

 *e w*

*<sup>T</sup> i i* 1 01 12 0 <sup>2</sup> <sup>0</sup>

 

*n f n w e*

in which *<sup>n</sup> v* is the normal velocity; *w* denotes the near-far-field interface; **N** *<sup>f</sup>* is the shape function for a typical discretized acoustic fluid finite element; and *e* denotes an assemblage of all fluid elements on the near-far-field interface. The dynamic stiffness matrix

 

*T e*

 

**S EE S E E C M 0** (9)

 *v d*

satisfies

x

**V S Φ** (7)

**V N** (8)

 ; **S** is

*n* 

dam was subjected to a horizontal ground acceleration *<sup>x</sup> a* and the semi-infinite reservoir was filled with an inviscid isentropic fluid. To evaluate the response of the dam-reservoir system under a horizontal ground acceleration *<sup>x</sup> a* excitation, the semi-infinite reservoir was divided into two parts: a near field and a far field. The near field was located between the dam-reservoir interface and the radiation boundary (the near-far-field interface at *x L* ), while the far field was from *x L* to . Note that the geometry of the reservoir was chosen to be arbitrary for *x* 0 and flat for *x* 0 .

For an inviscid isentropic fluid (acoustic fluid) with the fluid particles undergoing only small displacements and not including body force effects, the governing equations is expressed as

$$
\nabla^2 \phi = \frac{1}{c^2} \ddot{\phi} \tag{1}
$$

where denotes velocity potential and *c* denotes the sound speed in fluid. Reservoir pressure *p* , the velocity vector **v** and the velocity potential have a relationship as follows:

$$\mathbf{v} = \nabla \phi \tag{2a}$$

$$p = -\rho \dot{\phi} \tag{2b}$$

where denotes fluid density. Boundary conditions of the near field for Eq.(1) are following. Along the dam-reservoir interface, one has

$$\mathbf{v} \bullet \mathbf{n} = \frac{\partial \phi}{\partial n} = \upsilon\_n \tag{3}$$

where the unit vector **n** is perpendicular to the dam-reservoir interface and points outward of fluid; *<sup>n</sup> v* is the normal velocity of the dam-reservoir interface. The boundary condition along the reservoir bottom is

$$\frac{\partial \phi}{\partial n} + q\dot{\phi} = \upsilon\_n \tag{4}$$

where *q* is defined as

$$q = \frac{1}{c} \left( \frac{1 - \alpha\_r}{1 + \alpha\_r} \right) \tag{5}$$

in which *<sup>r</sup>* denotes a reflection coefficient of pressure striking the bottom of the reservoir. By ignoring effects of surface waves of fluid, the boundary condition of the free surface is taken as

$$
\phi = 0 \tag{6}
$$

The boundary condition on the radiation boundary (near-far-filed interface) should include effects of the radiation damping of infinite reservoir and those of energy dissipation in the reservoir due to the absorptive reservoir bottom. To model these effects accurately, the SBFEM was adopted in this chapter.
