**3. SBFEM formulation**

92 Hydrodynamics – Natural Water Bodies

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

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

*c*

**v** 

*p* 

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

*<sup>n</sup> q v*

*r r*

*<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

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

*n* 

*q c* 1 1 1 

2 <sup>1</sup> 

denotes velocity potential and *c* denotes the sound speed in fluid. Reservoir

denotes fluid density. Boundary conditions of the near field for Eq.(1) are following.

*n v n* 

(1)

(2a)

(2b)

**v n** (3)

(4)

0 (6)

(5)

have a relationship as follows:

2

to be arbitrary for *x* 0 and flat for *x* 0 .

Along the dam-reservoir interface, one has

along the reservoir bottom is

where *q* is defined as

SBFEM was adopted in this chapter.

in which

taken as

pressure *p* , the velocity vector **v** and the velocity potential

expressed as

where

where  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 far field was expressed as the following SBFEM formulation.

Fig. 2. SBFEM discretization model of layered far field
