**3.1 SBFEM formulation in the frequency domain**

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 is written as

$$\mathbf{V}\_n(\boldsymbol{\alpha}) = \mathbf{S}^\boldsymbol{\sigma}(\boldsymbol{\alpha}) \boldsymbol{\Phi}(\boldsymbol{\alpha}) \tag{7}$$

where **Φ** denotes the column vector composed of nodal velocity potentials ; **S** is the dynamic stiffness matrix of the far field and **V***<sup>n</sup>* satisfies

$$\mathbf{V}\_n\left(\boldsymbol{\alpha}\right) = \sum\_{\boldsymbol{\epsilon}} \int\_{\Gamma\_w'} \mathbf{N}\_f^T \boldsymbol{v}\_n\left(\boldsymbol{\alpha}\right) d\Gamma\_w^{\boldsymbol{\epsilon}} \tag{8}$$

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

$$\left(\mathbf{S}^{\circ}\left(\boldsymbol{\phi}\right) + \mathbf{E}^{1}\right)\mathbf{E}^{0-1}\left(\mathbf{S}^{\circ}\left(\boldsymbol{\phi}\right) + \mathbf{E}^{1T}\right) - \mathbf{E}^{2} - i\boldsymbol{\phi}\mathbf{C}^{0} - \left(i\boldsymbol{\phi}\right)^{2}\mathbf{M}^{0} = \mathbf{0} \tag{9}$$

Hydrodynamic Pressure Evaluation of

far-field interface. The matrix *<sup>e</sup>*

the dynamic stiffness matrix

corresponding variables of **Φ**

an increment in time step.

where 

Song, 1996b)

 *i* <sup>2</sup> 

and the matrix <sup>2</sup> **B** is

Reservoir Subjected to Ground Excitation Based on SBFEM 95

*j j j*

1

2

inside the near-far-field interface, while for those adjacent to reservoir bottom, *<sup>e</sup>*

 

where the symbol *b* denotes the reservoir bottom of the near-far-field interface, i.e. the line

the global coefficient matrices <sup>0</sup> **E** , <sup>1</sup> **E** , <sup>2</sup> **E** , <sup>0</sup> **C** and **M**0 in Eq.(9). Details about them can be

For a vertical near-far-field interface as shown in Fig.2, as the matrix <sup>1</sup> **E** was a zero matrix,

*i* 2 0 2 0 01 0

 

The corresponding SBFEM formulation of Eq.(7) in the time domain is written as (Wolf &

 *<sup>t</sup> <sup>n</sup> ttd*

in which *t* **<sup>M</sup>** is the dynamic mass matrix of the far field; **Φ***t* and **V***<sup>n</sup> t* are the

 **S** forms a Fourier transform pair. Upon discretization of Eq.(17) with respect to time and assuming all initial conditions equal to zero, one can get the following equation

> *<sup>n</sup> n n <sup>j</sup> <sup>n</sup> nj nj <sup>j</sup>* 1 1 1 <sup>1</sup>

in which *n j* <sup>1</sup> *nj t* <sup>1</sup> **M M** , *<sup>j</sup>* **Φ Φ** *j t* and *<sup>n</sup>* **V V** *n n n t* where *t* denotes

 **V M <sup>Φ</sup>** (17)

**V M <sup>Φ</sup> M M <sup>Φ</sup>** (18)

0

and **V***<sup>n</sup>*

Applying the inverse Fourier transformation to Eq. (9) with <sup>1</sup> **E** 0 yields

*c* <sup>0</sup> 1 1 1 

*y* 0 as shown in the Fig.2. Assembling all elements' *<sup>e</sup>*

found in the literatures (Wolf & Song, 1996b; Li et al., 2008).

is an excitation frequency. The

**3.2 SBFEM formulation in the time domain** 

11

 

21 31

*j j d d j j d d j j* 12 13

22 23 32 33 

Note that Eqs.(10-14) are only the functions of nodal coordinates of elements inside the near-

 

*f*

*f f*

**N N**

*b r T e f f b r*

**S** in Eq.(9) can be re-written readily as

*H d*

**B** (14)

<sup>0</sup> **C** is a zero matrix for elements not adjacent to reservoir bottom

**C N <sup>N</sup>** (15)

<sup>2</sup> **E** , *<sup>e</sup>*

<sup>0</sup> **E** , *<sup>e</sup>* <sup>1</sup> **E** , *<sup>e</sup>*

**S E C ME E** (16)

**S** can be obtained by the Schur factorization.

in the time domain, respectively. *t* **M** and

<sup>0</sup> **C** satisfies

<sup>0</sup> **C** and *<sup>e</sup>* **M**0 can yield

**B N** (13)

where global coefficient matrices <sup>0</sup> **E** , <sup>1</sup> **E** , <sup>2</sup> **E** , <sup>0</sup> **C** and **M**0 only depend on the geometry of the near-far-field interface and the reflection coefficient *<sup>r</sup>* . They are obtained through assembling all elements' *<sup>e</sup>* <sup>0</sup> **E** , *<sup>e</sup>* <sup>1</sup> **E** , *<sup>e</sup>* <sup>2</sup> **E** , *<sup>e</sup>* <sup>0</sup> **C** and *<sup>e</sup>* **M**0 on the near-far-field interface. The matrices *<sup>e</sup>* <sup>0</sup> **E** , *<sup>e</sup>* <sup>1</sup> **E** , *<sup>e</sup>* <sup>2</sup> **E** , *<sup>e</sup>* <sup>0</sup> **C** and *<sup>e</sup>* **M**0 corresponding to each element can be evaluated numerically or analytically using the following equations.

$$\mathbf{E}\_e^0 = \int\_{-1}^1 \int\_{-1}^1 \mathbf{B}^{1T} \mathbf{B}^1 \left| \mathbf{J} \right| d\eta d\varphi \tag{10a}$$

$$\mathbf{E}\_e^1 = \int\_{-1}^1 \int\_{-1}^1 \mathbf{B}^{2T} \mathbf{B}^1 \left| \mathbf{J} \right| d\eta d\varphi \tag{10b}$$

$$\mathbf{E}\_e^2 = \int\_{-1}^1 \int\_{-1}^1 \mathbf{B}^{2T} \mathbf{B}^2 \left| \mathbf{J} \right| d\eta d\varphi \tag{10c}$$

$$\mathbf{M}\_{\varepsilon}^{0} = \int\_{-1}^{1} \int\_{-1}^{1} \frac{1}{c^{2}} \mathbf{N}\_{f}^{T} \mathbf{N}\_{f} \left| \mathbf{J} \right| d\eta d\varepsilon \tag{10d}$$

where the **N** *<sup>f</sup>* is defined in Eq.(8) and the others **J** , <sup>1</sup> **B** , <sup>2</sup> **B** are defined below. The matrix **J** is defined as

$$\mathbf{J} = \begin{bmatrix} H & 0 & 0 \\ \frac{d\mathbf{N}\_f}{d\eta}\mathbf{x} & \frac{d\mathbf{N}\_f}{d\eta}\mathbf{y} & \frac{d\mathbf{N}\_f}{d\eta}\mathbf{z} \\\\ \frac{d\mathbf{N}\_f}{d\xi}\mathbf{x} & \frac{d\mathbf{N}\_f}{d\xi}\mathbf{y} & \frac{d\mathbf{N}\_f}{d\xi}\mathbf{z} \end{bmatrix} \tag{11a}$$

where the symbol *H* denotes the water depth in the far field and **x** , **y** and **z** are element nodal coordinates column vectors. Due to the fact that *x* coordinates of all nodes inside the near-far-field interface (vertical surface) are same, the matrix **J** becomes

$$\mathbf{J} = \begin{bmatrix} H & 0 & 0 \\ 0 & \frac{d\mathbf{N}\_f}{d\eta}\mathbf{y} & \frac{d\mathbf{N}\_f}{d\eta}\mathbf{z} \\ 0 & \frac{d\mathbf{N}\_f}{d\boldsymbol{\xi}}\mathbf{y} & \frac{d\mathbf{N}\_f}{d\boldsymbol{\xi}}\mathbf{z} \end{bmatrix} \tag{11b}$$

Write the inverse of **J** in the following form

$$\mathbf{J}^{-1} = \begin{bmatrix} \dot{j}\_{11} & \dot{j}\_{12} & \dot{j}\_{13} \\ \dot{j}\_{21} & \dot{j}\_{22} & \dot{j}\_{23} \\ \dot{j}\_{31} & \dot{j}\_{32} & \dot{j}\_{33} \end{bmatrix} \tag{12}$$

The components *mn j m n*, 1,2,3 can be evaluated by using Eq.(11b). Therefore, the matrix <sup>1</sup> **B** is defined as

$$\mathbf{B}^1 = \begin{bmatrix} j\_{11} \\ j\_{21} \\ j\_{31} \end{bmatrix} \mathbf{N}\_f \tag{13}$$

and the matrix <sup>2</sup> **B** is

94 Hydrodynamics – Natural Water Bodies

where global coefficient matrices <sup>0</sup> **E** , <sup>1</sup> **E** , <sup>2</sup> **E** , <sup>0</sup> **C** and **M**0 only depend on the geometry of

*T <sup>e</sup> d d* 1 1 0 11 1 1

*T <sup>e</sup> d d* 1 1 1 21 1 1

*T <sup>e</sup> d d* 1 1 2 22 1 1

*T <sup>e</sup> f f d d c*

0 0

*fff*

**NNN J xyz**

**NNN xyz**

where the symbol *H* denotes the water depth in the far field and **x** , **y** and **z** are element nodal coordinates column vectors. Due to the fact that *x* coordinates of all nodes inside the

*ddd ddd ddd ddd*

*fff*

0 0

*d d d d d d d d*

**N N J y z**

> **N N y z**

*jjj jjj jjj*

The components *mn j m n*, 1,2,3 can be evaluated by using Eq.(11b). Therefore, the

11 12 13

21 22 23 31 32 33

*f f*

 

> 

**J** (12)

*f f*

where the **N** *<sup>f</sup>* is defined in Eq.(8) and the others **J** , <sup>1</sup> **B** , <sup>2</sup> **B** are defined below. The matrix

**E BBJ** (10a)

**E B BJ** (10b)

**E BBJ** (10c)

<sup>0</sup> **C** and *<sup>e</sup>* **M**0 corresponding to each element can be evaluated

 

 

 

 **M NN J** (10d)

<sup>0</sup> **C** and *<sup>e</sup>* **M**0 on the near-far-field interface. The

*<sup>r</sup>* . They are obtained through

(11a)

(11b)

the near-far-field interface and the reflection coefficient

numerically or analytically using the following equations.

<sup>0</sup> **E** , *<sup>e</sup>*

<sup>1</sup> **E** , *<sup>e</sup>*

<sup>2</sup> **E** , *<sup>e</sup>*

1 1 0 1 1 2 1

*H*

near-far-field interface (vertical surface) are same, the matrix **J** becomes

0

*H*

0

1

Write the inverse of **J** in the following form

matrix <sup>1</sup> **B** is defined as

assembling all elements' *<sup>e</sup>*

<sup>1</sup> **E** , *<sup>e</sup>*

<sup>2</sup> **E** , *<sup>e</sup>*

<sup>0</sup> **E** , *<sup>e</sup>*

matrices *<sup>e</sup>*

**J** is defined as

$$\mathbf{B}^2 = \begin{bmatrix} j\_{12} \\ j\_{22} \\ j\_{32} \end{bmatrix} \frac{d\mathbf{N}\_f}{d\eta} + \begin{bmatrix} j\_{13} \\ j\_{23} \\ j\_{33} \end{bmatrix} \frac{d\mathbf{N}\_f}{d\xi} \tag{14}$$

Note that Eqs.(10-14) are only the functions of nodal coordinates of elements inside the nearfar-field interface. The matrix *<sup>e</sup>* <sup>0</sup> **C** is a zero matrix for elements not adjacent to reservoir bottom inside the near-far-field interface, while for those adjacent to reservoir bottom, *<sup>e</sup>* <sup>0</sup> **C** satisfies

$$\mathbf{C}\_{e}^{0} = \frac{1}{c} \left( \frac{1 - \alpha\_{r}}{1 + \alpha\_{r}} \right) H \int\_{\Gamma\_{b}} \mathbf{N}\_{f}^{T} \mathbf{N}\_{f} d\Gamma\_{b} \tag{15}$$

where the symbol *b* denotes the reservoir bottom of the near-far-field interface, i.e. the line *y* 0 as shown in the Fig.2. Assembling all elements' *<sup>e</sup>* <sup>0</sup> **E** , *<sup>e</sup>* <sup>1</sup> **E** , *<sup>e</sup>* <sup>2</sup> **E** , *<sup>e</sup>* <sup>0</sup> **C** and *<sup>e</sup>* **M**0 can yield the global coefficient matrices <sup>0</sup> **E** , <sup>1</sup> **E** , <sup>2</sup> **E** , <sup>0</sup> **C** and **M**0 in Eq.(9). Details about them can be found in the literatures (Wolf & Song, 1996b; Li et al., 2008).

For a vertical near-far-field interface as shown in Fig.2, as the matrix <sup>1</sup> **E** was a zero matrix, the dynamic stiffness matrix **S** in Eq.(9) can be re-written readily as

$$\mathbf{S}^{\pi^{\circ}}(\phi) = \sqrt{\left(\mathbf{E}^{2} + i\alpha \mathbf{C}^{0} - \alpha^{2} \mathbf{M}^{0}\right) \mathbf{E}^{0-1}} \,\mathrm{E}^{0} \tag{16}$$

where is an excitation frequency. The **S** can be obtained by the Schur factorization.

#### **3.2 SBFEM formulation in the time domain**

The corresponding SBFEM formulation of Eq.(7) in the time domain is written as (Wolf & Song, 1996b)

$$\mathbf{V}\_n(t) = \int\_0^t \mathbf{M}^{\circ \circ}(t - \tau) \ddot{\boldsymbol{\Phi}}(\tau) d\tau \tag{17}$$

in which *t* **<sup>M</sup>** is the dynamic mass matrix of the far field; **Φ***t* and **V***<sup>n</sup> t* are the corresponding variables of **Φ** and **V***<sup>n</sup>* in the time domain, respectively. *t* **M** and *i* <sup>2</sup> **S** forms a Fourier transform pair. Upon discretization of Eq.(17) with respect to time and assuming all initial conditions equal to zero, one can get the following equation

$$\mathbf{V}\_{n}^{n} = \mathbf{M}\_{1}^{\circ \circ} \dot{\mathbf{O}}^{n} + \sum\_{j=1}^{n-1} \left( \mathbf{M}\_{n-j+1}^{\circ \circ} - \mathbf{M}\_{n-j}^{\circ \circ} \right) \dot{\mathbf{O}}^{j} \tag{18}$$

in which *n j* <sup>1</sup> *nj t* <sup>1</sup> **M M** , *<sup>j</sup>* **Φ Φ** *j t* and *<sup>n</sup>* **V V** *n n n t* where *t* denotes an increment in time step.

Applying the inverse Fourier transformation to Eq. (9) with <sup>1</sup> **E** 0 yields

Hydrodynamic Pressure Evaluation of

In the frequency domain, using Eqs.(7, 16, 25, 26) yields

11 12 13 1 21 22 23 2 31 32 33 3

 

For a harmonic response with an exciting frequency

solve the harmonic response of a reservoir, i.e.

**mmm mmm**

Eq.(29) can be solved for any frequency

2

geometry shape.

11 12 13

21 22 23

reservoir to solve the transient response of a reservoir, i.e.

**mmm Φ mmm Φ mmm Φ**

*i*

 

11 12 13

2 0 2 0 01 0 21 22 23 2

 

3 3 <sup>31</sup> <sup>32</sup> <sup>33</sup>

*i t e* 

31 32 33 1 1

**mmm Φ V Φ 0 kkk**

2 0 2 0 01 0 3 3

Substituting Eq.(28) into Eq.(27) leads to the FEM-SBFEM coupling equation of a reservoir to

.

11 12 13 1 1 21 22 23 2 1 2 31 32 33 3 3

**mmm Φ 000 Φ mmm Φ 0M 0 Φ mmm Φ Φ 000**

**k k E C ME E k kkk**

21 22 23 31 32 33

<sup>2</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup>

In the time domain, using Eqs.(17, 18, 25, 26) yields the FEM-SBFEM coupling equation of a

*n n n n n n*

 

*<sup>n</sup> <sup>n</sup>*

where the superscript *n* denotes the instant at time *t nt* . Note that a damping matrix appears on the left hand side of Eq.(30). It can be regarded as the damping effect derived from the far-field medium and imposed on the dam-reservoir system. As the near-field domain is modeled by FEM, Eqs.(29, 30) are suitable for a reservoir with any arbitrary

<sup>1</sup> 11 12 13 1

21 22 23 2 <sup>1</sup> <sup>1</sup>

**kkk Φ M**

**<sup>V</sup> kkk <sup>Φ</sup>**

31 32 33 3

**kkk Φ**

*<sup>n</sup> <sup>n</sup> <sup>n</sup>*

1

 

 *<sup>j</sup> n j*

 

**M Φ**

*n n*

**V**

3

*n j <sup>j</sup> <sup>n</sup>*

 

*i*

**kkk <sup>Φ</sup> <sup>V</sup> k k E C ME E k Φ 0 kkk Φ V**

Reservoir Subjected to Ground Excitation Based on SBFEM 97

*<sup>n</sup>*

, *n*

*n*

*n*

*i t*

**Φ V**

2

*e*

(27)

(29)

(30)

1 1

**Φ Φ** (28)

$$\int\_0^t \mathbf{m}^\alpha \left(t - \tau\right) \mathbf{m}^\alpha \left(\tau\right) d\tau - \frac{t^3}{6} \mathbf{e}^2 - \frac{t^2}{2} \mathbf{c}^0 - t \mathbf{m}^0 = 0 \tag{19}$$

where *t* is time and

$$\mathbf{m}^{\circ}\left(t\right) = \mathbf{U}^{-1T}\mathbf{M}^{\circ}\left(t\right)\mathbf{U}^{-1}\tag{20}$$

$$\mathbf{e}^2 = \mathbf{U}^{-1T} \mathbf{E}^2 \mathbf{U}^{-1} \tag{21}$$

$$\mathbf{m}^0 = \mathbf{U}^{-1T} \mathbf{M}^0 \mathbf{U}^{-1} \tag{22}$$

$$\mathbf{c}^{0} = \mathbf{U}^{-1T}\mathbf{C}^{0}\mathbf{U}^{-1} \tag{23}$$

in which **U** satisfies

$$\mathbf{E}^0 = \mathbf{U}^T \mathbf{U} \tag{24}$$

A procedure (Wolf & Song, 1996b) was presented to evaluate the dynamic mass matrix *t* **M** at different time *t* governed by the convolution integral Eq.(19). In that procedure, discretization of Eq.(19) with respect to time was implemented, and an algebraic Riccati equation for evaluating *t t* **M** at first time step and a Lyapunov equation for evaluating *t jt* **M** at other jth time steps were formed, respectively. The *t jt* **M** at any time was obtained by utilizing Schur factorization to solve these two types of equations. When the coefficient matrix <sup>0</sup> **c** 0 , a simple diagonal procedure (Li, 2009) can be adopted to evaluate the *t* **M** , which can avoid Schur factorization and solving Riccati equation and Lyapunov equation.

### **4. FEM-SBFEM coupling formulation of reservoir**

To obtain the response of dam-reservoir system, the near-field fluid domain is discretized into an assemblage of finite elements. The corresponding finite-element governing equation of Eq.(1) for the near-field domain can be expressed as

$$
\begin{bmatrix}
\mathbf{m}\_{11} & \mathbf{m}\_{12} & \mathbf{m}\_{13} \\
\mathbf{m}\_{21} & \mathbf{m}\_{22} & \mathbf{m}\_{23} \\
\mathbf{m}\_{31} & \mathbf{m}\_{32} & \mathbf{m}\_{33}
\end{bmatrix}
\begin{bmatrix}
\ddot{\boldsymbol{\Phi}}\_{1} \\
\ddot{\boldsymbol{\Phi}}\_{2} \\
\ddot{\boldsymbol{\Phi}}\_{3}
\end{bmatrix} + 
\begin{bmatrix}
\mathbf{k}\_{11} & \mathbf{k}\_{12} & \mathbf{k}\_{13} \\
\mathbf{k}\_{21} & \mathbf{k}\_{22} & \mathbf{k}\_{23} \\
\mathbf{k}\_{31} & \mathbf{k}\_{32} & \mathbf{k}\_{33}
\end{bmatrix}
\begin{bmatrix}
\boldsymbol{\Phi}\_{1} \\
\boldsymbol{\Phi}\_{2} \\
\boldsymbol{\Phi}\_{3}
\end{bmatrix} = 
\begin{bmatrix}
\mathbf{V}\_{n1} \\
\mathbf{V}\_{n2} \\
\mathbf{V}\_{n3}
\end{bmatrix} \tag{25}
$$

where the global mass matrix **m** , the global stiffness matrix **k** and the global vector **V***n* are treated in the standard manner as in the traditional FE procedures; the subscripts 1 and 2 refer to nodal variables at the dam-reservoir interface and the near-far-field interface, respectively, while the subscript 3 refers to other interior nodal variables in the near-field fluid. At the near-far-field interface, the near-field FEM-domain couples with the far-field SBFEM-domain. The kinematic continuity condition requires that both fields have the same normal velocity at the near-far-field interface. Hence, one has

$$-\mathbf{V}'\_{n2} = \mathbf{V}\_n \tag{26}$$

A procedure (Wolf & Song, 1996b) was presented to evaluate the dynamic mass matrix *t* **M** at different time *t* governed by the convolution integral Eq.(19). In that procedure, discretization of Eq.(19) with respect to time was implemented, and an algebraic Riccati equation for evaluating *t t* **M** at first time step and a Lyapunov equation for evaluating *t jt* **M** at other jth time steps were formed, respectively. The *t jt* **M** at any time was obtained by utilizing Schur factorization to solve these two types of equations. When the coefficient matrix <sup>0</sup> **c** 0 , a simple diagonal procedure (Li, 2009) can be adopted to evaluate the *t* **M** , which can avoid Schur factorization and solving Riccati

To obtain the response of dam-reservoir system, the near-field fluid domain is discretized into an assemblage of finite elements. The corresponding finite-element governing equation

> 11 12 13 1 11 12 13 1 1 21 22 23 2 21 22 23 2 2 31 32 33 3 31 32 33 3 3

where the global mass matrix **m** , the global stiffness matrix **k** and the global vector **V***n* are treated in the standard manner as in the traditional FE procedures; the subscripts 1 and 2 refer to nodal variables at the dam-reservoir interface and the near-far-field interface, respectively, while the subscript 3 refers to other interior nodal variables in the near-field fluid. At the near-far-field interface, the near-field FEM-domain couples with the far-field SBFEM-domain. The kinematic continuity condition requires that both fields have the same

**mmm Φ kkk Φ V mmm Φ kkk Φ V mmm Φ kkk Φ V**

  3 2

6 2

2 00

**m m e cm** (19)

0

*<sup>T</sup> t t* 1 1 **m UM U** (20)

2 12 1 *<sup>T</sup>* **e U EU** (21)

0 1 01 *<sup>T</sup>* **m U MU** (22)

0 101 *<sup>T</sup>* **c U CU** (23)

<sup>0</sup> *<sup>T</sup>* **E UU** (24)

*n n n*

*n n* <sup>2</sup> **V V** (26)

(25)

*<sup>t</sup> t t td t*

 

0

where *t* is time and

in which **U** satisfies

equation and Lyapunov equation.

**4. FEM-SBFEM coupling formulation of reservoir** 

of Eq.(1) for the near-field domain can be expressed as

normal velocity at the near-far-field interface. Hence, one has

In the frequency domain, using Eqs.(7, 16, 25, 26) yields

$$
\begin{bmatrix}
\mathbf{m}\_{11} & \mathbf{m}\_{12} & \mathbf{m}\_{13} \\
\mathbf{m}\_{21} & \mathbf{m}\_{22} & \mathbf{m}\_{23} \\
\mathbf{m}\_{31} & \mathbf{m}\_{32} & \mathbf{m}\_{33}
\end{bmatrix}
\begin{bmatrix}
\dot{\boldsymbol{\Phi}}\_{1} \\
\dot{\boldsymbol{\Phi}}\_{2} \\
\dot{\boldsymbol{\Phi}}\_{3}
\end{bmatrix} +
$$

$$
\begin{bmatrix}
\mathbf{k}\_{11} & \mathbf{k}\_{12} & \mathbf{k}\_{12} \\
\mathbf{k}\_{21} & \mathbf{k}\_{22} + \sqrt{\left(\mathbf{E}^{2} + i\alpha \mathbf{C}^{0} - \alpha^{2} \mathbf{M}^{0}\right) \mathbf{E}^{0-1}} \mathbf{E}^{0} & \mathbf{k}\_{23} \\
\mathbf{k}\_{31} & \mathbf{k}\_{32} & \mathbf{k}\_{33}
\end{bmatrix}
\begin{bmatrix}
\boldsymbol{\Phi}\_{1} \\
\boldsymbol{\Phi}\_{2} \\
\boldsymbol{\Phi}\_{3}
\end{bmatrix} = \begin{bmatrix}
\mathbf{V}\_{n1} \\
\mathbf{0} \\
\mathbf{V}\_{n3}
\end{bmatrix} \tag{27}
$$

For a harmonic response with an exciting frequency ,

$$
\blacksquare \Phi = \overline{\Phi} e^{i\alpha t} \tag{28}
$$

Substituting Eq.(28) into Eq.(27) leads to the FEM-SBFEM coupling equation of a reservoir to solve the harmonic response of a reservoir, i.e.

$$\begin{pmatrix} \begin{bmatrix} \mathbf{m}\_{11} & \mathbf{m}\_{12} & \mathbf{m}\_{13} \\ \mathbf{m}\_{21} & \mathbf{m}\_{22} & \mathbf{m}\_{23} \\ \mathbf{m}\_{31} & \mathbf{m}\_{32} & \mathbf{m}\_{33} \end{bmatrix} + \\\ \begin{bmatrix} \mathbf{k}\_{11} & & & & \\ & \mathbf{k}\_{12} & & & \mathbf{k}\_{13} \\ \mathbf{k}\_{21} & \mathbf{k}\_{22} + \sqrt{\left(\mathbf{E}^{2} + i\alpha \mathbf{C}^{0} - \alpha^{2} \mathbf{M}^{0}\right) \mathbf{E}^{0-1}} \mathbf{E}^{0} & \mathbf{k}\_{23} \\\ \mathbf{k}\_{31} & & & \mathbf{k}\_{32} \end{bmatrix} \begin{bmatrix} \overleftarrow{\mathbf{O}}\_{1} \\ \overleftarrow{\mathbf{O}}\_{2} \\ \overleftarrow{\mathbf{O}}\_{3} \end{bmatrix} e^{i\alpha t} = \begin{bmatrix} \mathbf{V}\_{n1} \\ \mathbf{0} \\ \mathbf{V}\_{n3} \end{bmatrix} \end{cases} \tag{29}$$

Eq.(29) can be solved for any frequency .

In the time domain, using Eqs.(17, 18, 25, 26) yields the FEM-SBFEM coupling equation of a reservoir to solve the transient response of a reservoir, i.e.

$$
\begin{aligned}
\begin{bmatrix}
\mathbf{m}\_{11} & \mathbf{m}\_{12} & \mathbf{m}\_{13} \\
\mathbf{m}\_{21} & \mathbf{m}\_{22} & \mathbf{m}\_{23} \\
\mathbf{m}\_{31} & \mathbf{m}\_{32} & \mathbf{m}\_{33}
\end{bmatrix}
\begin{bmatrix}
\dot{\Phi}\_{1}^{u} \\
\dot{\Phi}\_{2}^{u}
\end{bmatrix} + \begin{bmatrix}
\mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{M}\_{1}^{\alpha} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{bmatrix}
\begin{bmatrix}
\dot{\Phi}\_{1}^{u} \\
\dot{\Phi}\_{2}^{u}
\end{bmatrix} \\
+ \begin{bmatrix}
\mathbf{k}\_{11} & \mathbf{k}\_{12} & \mathbf{k}\_{13} \\
\mathbf{k}\_{21} & \mathbf{k}\_{22} & \mathbf{k}\_{23} \\
\mathbf{k}\_{31} & \mathbf{k}\_{32} & \mathbf{k}\_{33}
\end{bmatrix}
\begin{bmatrix}
\Phi\_{1}^{u} \\
\Phi\_{2}^{u}
\end{bmatrix} = \begin{bmatrix}
\mathbf{V}\_{n1}^{u} \\
\mathbf{0} \end{bmatrix}
\end{aligned} \tag{30}$$

where the superscript *n* denotes the instant at time *t nt* . Note that a damping matrix appears on the left hand side of Eq.(30). It can be regarded as the damping effect derived from the far-field medium and imposed on the dam-reservoir system. As the near-field domain is modeled by FEM, Eqs.(29, 30) are suitable for a reservoir with any arbitrary geometry shape.

Hydrodynamic Pressure Evaluation of

1994) corresponding to different

*y H* 0.6 corresponding to *<sup>r</sup>*

0.5

*y H* 1

0.5

*y H*

Reservoir Subjected to Ground Excitation Based on SBFEM 99

Results from different mesh densities were the same. The hydrodynamic pressure obtained by using 10 3-noded SBFEM elements and the corresponding analytical solutions (Weber,

Fig. 5. Hydrodynamic pressures on vertical dam-reservoir interface caused by different

1 2 

1 4 

<sup>1</sup> Analytical solution

*r* 0.95

0 2 4 6

*Cp*

<sup>0</sup> <sup>1</sup> <sup>2</sup>

*Cp*

*r* 0.75

SBFEM

1 1 

1 1 

*<sup>r</sup>* were plotted in Fig.5. The SBFEM solutions were the

0.8 was shown in Fig.6. The SBFEM solution and the

*Cp* figure of a point located at

*r*

Analytical solution

SBFEM

exact same to the analytical solutions. Furthermore, a

analytical solution (Weber, 1994) were the same.

1 4 

> 1 2
