# An Efficient Dynamic Coupling Calculation Method for Dam–Reservoir Systems Based on FEM-SBFEM

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

**α**(0 <

**α**≤ 1.0), which is simple and easy to implement. The suggested value of the reduction coefficient

**α**for the added mass matrix of the hydrodynamic pressure is selected to be 0.6 so as to ensure that the error of the maximum value of the dynamic response of the dam is limited within 5%, which is acceptable, and the elapsed time of calculation can be reduced to one twentieth of the accurate solution, which is a great jump in calculation efficiency. The proposed method provides a practical and effective process for the analysis of dam–reservoir dynamic interaction systems with a large computational scale and a fine grid scale.

## 1. Introduction

## 2. A Calculating Method for Hydrodynamic Pressure of Reservoir Based on SBFEM

#### 2.1. The Basic Equation and Boundary Conditions

^{2}p = 0

_{0}boundary condition of the reservoir water is

_{1}of the dam is as follows:

_{n}

_{2}between the reservoir and bottom and bank slope is as follows:

_{n}

**▽**

^{2}is the Laplace operator, p is the reservoir hydrodynamic pressure, n is the normal direction of the interface between the solid and fluid, ρ is the water density, and ü

_{n}and ϋ

_{n}are normal acceleration values of the dam–reservoir interface and the river-valley interface, respectively.

_{3}is automatically satisfied, which is illuminated below.

#### 2.2. Solution for Hydrodynamic Pressure Based on SBFEM

_{1}has the range [0, +∞]. As shown in Figure 2, ξ

_{1}= 0 is at the upstream face of the dam, and ξ

_{1}= +∞ is at the infinity of the reservoir. The circumferential local coordinates ξ

_{2}and ξ

_{3}have the range [–1, 1]. The coordinates (X

_{1}, X

_{2}, X

_{3}) of the global Cartesian coordinate system at any point in the reservoir area can be expressed in terms of the local coordinates (ξ

_{1}, ξ

_{2}, ξ

_{3}) of the scaled boundary. The radial local coordinate ξ

_{1}serves as a factor of proportionality, giving rise to the following set of equations:

_{1}, x

_{2}, x

_{3}) are the nodal coordinates of the elements on the interface between the reservoir and dam. $\left[N\left({\xi}_{2},{\xi}_{3}\right)\right]$ represents the shape function of the elements, which is only related to the circumferential local coordinates but unrelated to the radial coordinate ξ

_{1}.

**▽**can be expressed in terms of the scaled boundary coordinates through the Jacobian matrix [J] as follows:

^{−1}= [{b

^{1}} {b

^{2}} {b

^{3}}].

_{1})} is the hydrodynamic pressure at nodes of the fluid element.

_{2}, X

_{3}) plane.

^{0}], [E

^{1}], [E

^{2}], [C

^{0}], and [M

^{1}] are independent of the radial coordinate ξ

_{1}and can be straightforwardly obtained from geometry information of the grid on the dam upstream face. Then the coefficient matrices of elements can be integrated into the total SBFEM coefficient matrices, the process of which is similar to the FEM.

_{i}] is the diagonal matrix, and the real part of λ

_{i}is positive.

_{n}}, and that induced by the vibration of the river valley {ϋ

_{n}} surrounding the reservoir area.

## 3. An Efficient Dynamic Coupling Calculation Method for Dam–Reservoir Systems

#### 3.1. Conventional Dynamic Coupling Analysis Method for Dam–Reservoir Systems

_{s}], [C

_{s}], and [K

_{s}] are mass, damping, and stiffness matrices of the dam, respectively. {ü

_{r}(t)}, $\left\{{\dot{u}}_{r}\left(t\right)\right\},$ and {u

_{r}(t)} are the relative acceleration, velocity, and displacement, respectively. {ü

_{g}(t)} is the earthquake acceleration from the input system. [L

_{1}] is the conversion matrix for mapping global coordinates to the local coordinates of the dam upstream face.

^{u}] and [M

^{v}] are the dam upstream face and river valley components, respectively, of the hydrodynamic pressure additional mass matrix [M

_{p}]. [L

_{2}] is the conversion matrix for mapping global coordinates to the local coordinates of the bank slope (river valley). As long as the additional mass matrix [M

_{p}] is added to the mass matrix of the finite element dynamic equation of the dam, the hydrodynamic pressure due to the ground motion input in different directions can be considered.

#### 3.2. Simplification of Hydrodynamic Pressure Additional Mass Matrix

#### 3.2.1. Physical Meaning and Distribution Characteristics of Matrix

^{u}] is a full matrix, and all elements are non-zero, that is, when calculating the hydrodynamic pressure caused by the excitation of the dam upstream face, the hydrodynamic pressure acting on a certain node is related to the acceleration excitations {ü

_{n}} of all nodes on the dam upstream face below the water level. However, when calculating the hydrodynamic pressure caused by river valley excitation, the hydrodynamic pressure acting on a certain node on the dam upstream face is only related to the node acceleration excitations {ϋ

_{n}} at the boundary between the dam surface and the river valley. Therefore, the additional mass matrix [M

^{v}] is a very sparse matrix containing a large number of zero elements. Obviously, due to the existence of the additional mass matrix [M

^{u}], the additional mass matrix [M

_{p}] is a full matrix, which greatly increases the time consumption of solving the equivalent stiffness matrix in dynamic analysis.

_{p}] can be reduced by the simplifying matrix [M

^{u}]. When the additional mass matrix [M

^{u}] is of order n (i.e., there are n nodes below the water level line on the dam upstream face in total); the physical meaning of the element ${M}_{ij}^{u}$ is the hydrodynamic pressure acting on the node i caused by the unit normal acceleration excitation of the node j on the water upstream face. The i-th row elements in matrix [M

^{u}], that is ${M}_{i1}^{u}$ ${M}_{i2}^{u}$… ${M}_{ii}^{u}$… ${M}_{in}^{u}$, are selected, and then these row elements give the contribution of all nodes on the upstream face of the dam (below the water level) to the hydrodynamic pressure acting on node i.

^{u}] are analyzed using the example of a vertical dam upstream face in a rectangular valley. The height of the dam is 200 m, and the width of the river valley is 400 m. The grid division of the reservoir is shown in Figure 3, and the water depth in front of the dam is 200 m (full reservoir).

^{u}] are extracted, and the influence of all nodes at the dam upstream face on the hydrodynamic pressure of node A under unit normal acceleration excitation is shown in Figure 4 (normalized relative value). As shown in Figure 4, node A has the greatest impact on itself, and the closer the node is to node A, the greater the impact is on the hydrodynamic pressure of node A. Other nodes on the upstream face of the dam also conform to similar laws, so it is no longer described repetitively.

#### 3.2.2. Theoretical Analysis and Simplified Processing Method

^{u}], among all the elements in the i-th row, the element values corresponding to node i (${M}_{ii}^{u}$) and its adjacent nodes are relatively large, and the acceleration values of these closer nodes in the dynamic coupling analysis are not much different. Therefore, when calculating the hydrodynamic pressure acting on node i, the added mass element of the i-th row can be directly superimposed on ${M}_{ii}^{u}$, and the corresponding elements at the original position of the row are directly taken as 0, which means that the hydrodynamic pressure acting on node i is only related to the acceleration of node i, and it has nothing to do with the acceleration of other nodes on the dam upstream face. At the same time, the instantaneous acceleration distribution of the dam upstream face is not consistent, which contains both positive and negative values. Therefore, the above treatment method for the additional mass matrix will cause the amplification of hydrodynamic pressure. Considering the amplification effect of the superposition of additional mass elements on the hydrodynamic pressure, the appropriate reduction treatment is needed in the process of element superposition.

^{u}], a simple and easy simplification method is proposed by row processing, which is briefly described as follows:

^{u}]. (2) Let the diagonal elements of the matrix ${M}_{ii}^{u}=\mathit{\alpha}{{\displaystyle \sum}}_{j=1}^{n}{M}_{ij}^{u}$, where

**α**is the reduction coefficient (0 <

**α**≤ 1.0); at the same time, set the value of other non-diagonal elements to zero. (3) According to this method, the additional mass matrix [M

^{u}] is processed from the first row until the last row.

^{u}] is transformed into a new diagonal matrix [M

^{u}

^{α}]. The additional mass matrix [M

^{v}] does not need to be simplified. Further, the overall additional mass matrix [M

_{p}] is updated to [M

_{p}

_{α}] as follows:

#### 3.3. Efficient Dynamic Coupling Calculation Method Based on FEM-SBFEM

_{p}] with [M

_{p}

_{α}] in Equation (31), the efficient dynamic coupling calculation method for dam–reservoir systems based on the FEM-SBFEM is established:

**α**to realize the simplification of the additional mass matrix [M

_{p}] to a large extent, which is simple and easy to implement. The processed additional mass matrix [M

_{p}

_{α}] contains many zero elements, so the computational efficiency of dynamic coupling systems is greatly improved. The practice shows that the value of reduction coefficient

**α**has an obvious influence on the calculation accuracy, which is presented in Section 4.

## 4. Dynamic Coupling Analysis of Arch Dam and Reservoir Systems

**α**is carried out. Different reduction factors

**α**of additional mass matrix [M

_{p}

_{α}] are selected to calculate and analyze the dynamic response of the arch dam. By comparing with the exact solution, which means the unsimplified full additional mass matrix [M

_{p}] is utilized in dynamic analysis, the recommended value of reduction factor

**α**is discussed.

#### 4.1. Calculation Model

#### 4.1.1. Dam and Reservoir Model

#### 4.1.2. Material Parameters, Input Seismic Load, and Damping Methods

_{d}= 2.4 g/cm

^{3}, elasticity modulus E = 25 GPa, Poisson’s ratio $\nu $ = 0.167), which can fully show the advanced nature of the proposed efficient dynamic coupling calculation method. The density ρ

_{w}of the reservoir water is 1.0 g/cm

^{3}.

^{2}, and the PGA of vertical bedrock motion was 1.0 m/s

^{2}.

#### 4.2. Effect of Additional Mass Matrix Simplification

^{u}] is a full matrix of order 2397, which contains 2397 × 2397 = 5,745,609 elements of the matrix and could provide an accurate dynamic coupling calculation result of dam and reservoir systems.

**α**(0 <

**α**≤ 1.0) was selected for forming [M

^{u}

^{α}], in which

**α**= 1.0, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, and 0.3.

**α**= 1.0 indicates that the additional mass matrix is simplified to a diagonal matrix without reduction, and

**α**≈ 0 indicates that the added mass matrix [M

^{u}

^{α}] is close to the zero matrix, that is, the hydrodynamic pressure is almost not considered. When 0 <

**α**≤ 1.0 in this numerical case, the simplified added mass matrix [M

^{u}

^{α}] is a diagonal matrix, which contains only 2397 elements and can present an approximate analysis result of the dynamic interaction of the dam and reservoir. The simplification of the additional mass matrix can not only save a lot of memory occupied by the additional mass matrix of hydrodynamic pressure but also greatly reduce the computational time. However, only a reasonable value of the reduction coefficient

**α**can ensure high calculation accuracy, as discussed below.

#### 4.3. Results and Discussion

**α**= 1.0, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, and 0.3) conditions, the effects of the hydrodynamic pressure added mass matrix [M

^{u}

^{α}] ([M

_{p}

_{α}]) on the dynamic stress and acceleration extreme (maximum) value of the arch dam and their regularities of distribution and calculation time are studied. When calculating the error of dynamic stress and acceleration of the arch dam caused by the simplification of the additional mass matrix, the accurate calculation results of the unsimplified additional mass matrix [M

^{u}] ([M

_{p}]) are taken as the benchmark. The results of the unsimplified additional mass matrix were verified to be of high accuracy [2,21]. By comparing the calculation accuracy and calculation time, the recommended value of the reduction coefficient

**α**is given.

#### 4.3.1. Acceleration of Arch Dam

_{x}) and vertical (a

_{y}) directions under different reduction coefficients

**α**for simplification of additional mass matrix and unsimplified additional mass matrix conditions when the earthquake occurs. Table 2 shows the errors corresponding to the maximum dynamic acceleration caused by simplification of the added mass matrix of hydrodynamic pressure. It can be concluded from Table 1 and Table 2 that the additional mass matrix reduction coefficient

**α**has a big impact on the acceleration in the up-downstream direction (a

_{x}) but has relatively little impact on the vertical acceleration (a

_{y}). Compared with the accurate results in the condition of the unsimplified additional mass matrix, the maximum errors of acceleration, when the reduction coefficient

**α**= 0.6 and 0.7, are 0.4% and 4.6%, respectively, which are acceptable from the perspective of calculation accuracy since the max errors are limited to less than 5.0%. When the reduction coefficient

**α**< 0.6 and

**α**> 0.7, the max errors of acceleration become bigger, which are from 9.0% to 22.7%.

_{x}) on the upstream face of the arch dam. As shown in Figure 7, the distribution laws of the maximum acceleration along the up-downstream direction (a

_{x}) for each condition are similar, but the extent and area of the high acceleration response region are significantly different and gradually change with the decrease of the reduction coefficient

**α**. In addition, the maximum acceleration distribution is most consistent with the accurate analysis solution (from unsimplified additional mass matrix) when the reduction coefficient

**α**= 0.6, which also corresponds to the minimum error of acceleration in both vertical (a

_{y}) and up-downstream (a

_{x}) directions, as summarized in Table 2.

#### 4.3.2. Stress of Arch Dam

_{1}) and minor principal stress (σ

_{3}), that occurred during the earthquake are collected in Table 3. The compressive stress of the dam concrete is set to be positive. Table 4 lists the errors corresponding to the maximum dynamic stress when the added mass matrix of hydrodynamic pressure is simplified with different reduction coefficients

**α**. Table 3 and Table 4 demonstrate that the reduction coefficient

**α**has an obvious influence on both the major principal stress and minor principal stress of arch dam. As shown in Table 4, the maximum errors of dynamic stress are less than 5.0% when the reduction coefficient

**α**= 0.4, 0.6, 0.7, and 0.8, which means the calculation accuracy of dynamic stress is effectively controlled. Furthermore, the maximum errors of dynamic stress are 2.2% and 1.3%, respectively, for reduction coefficient

**α**= 0.6 and 0.8, the errors of which are relatively small in all reduction coefficients conditions. The maximum errors of dynamic stress vary from 5.0 to 11.2%, when the reduction coefficient

**α**= 0.3, 0.5, 0.9, and 1.0, which are relatively big.

_{3}) on the upstream face of the arch dam for every condition of hydrodynamic pressure added mass matrix is depicted in Figure 8. Because seismic loads on the dam will eventually be transferred to the arch abutment, there are high maximum minor principal stress values on the contour of the dam, as shown in Figure 8. As seen in Figure 8, the dynamic minor principal stress (σ

_{3}) distribution rules of each condition are basically consistent, whereas there are some differences in the location and range of the high stress zone of the arch dam. Although the calculation accuracy of the maximum dynamic stress is relatively high for the reduction coefficient

**α**= 0.8, as mentioned above, the scope and area of the high stress zone are obviously different from the accurate solution for the unsimplified added mass matrix. For the condition of the reduction coefficient

**α**= 0.4, the calculation error of maximum dynamic stress is restricted to less than 5%; however, the position of the high stress zone distinctly has a certain degree of deviation compared to the accurate result. When the reduction coefficient

**α**= 0.6, the distribution law of the maximum dynamic stress of the arch dam, including the location of the high stress zone and the range of the high stress zone, is the most similar to the accurate solution, and the computational accuracy for maximum dynamic stress is also high in all the simplification conditions of the additional mass matrix reduction coefficient

**α**, as seen in Table 4.

#### 4.3.3. Computational Efficiency

**α**are used to simplify the additional mass matrix, the consumption time of the dynamic coupling calculation is basically the same, which accounts for about 5% of that under the accurate solution condition. Table 5 shows that for the proposed efficient dynamic coupling calculation method of dam–reservoir systems, the elapsed time is sharply reduced, and the calculation efficiency has made a great jump. It can be seen that the additional mass matrix has a great influence on the computational efficiency of dynamic coupling analysis. Combining Table 2, Table 4, and Table 5, it can be seen that the dynamic coupling calculation efficiency of dam–reservoir systems has been greatly improved after the additional mass matrix is simplified, and the reasonable selection of reduction coefficient

**α**can effectively ensure high calculation accuracy.

#### 4.3.4. Suggested Value of Reduction Coefficient **α**

**α**of 0.6 cannot only ensure the accuracy of the distribution law of the maximum dynamic stress of the dam but also the high calculation accuracy of the maximum dynamic stress. At the same time, from the perspective of computational efficiency, different reduction coefficients almost do not affect the running time of the coupling calculation, and the calculation time can be reduced to about one-twentieth of the unsimplified condition of the additional mass matrix.

## 5. Conclusions

- An efficient 3D dynamic fluid–solid coupling calculation method for dam–reservoir systems based on the FEM-SBFEM is proposed by simplifying the hydrodynamic pressure additional mass matrix according to the physical meaning and distribution characteristics of the additional matrix. The proposed method not only ensures the high accuracy of the numerical calculation results but also greatly reduces the consumption time of the dynamic coupling calculation.
- The hydrodynamic pressure added mass matrix has a great influence on the computational efficiency of dynamic coupling analysis. The proposed method, which is simple and easy to implement, only needs to determine a reduction coefficient
**α**(0 <**α**≤ 1.0) to simplify the hydrodynamic pressure added mass matrix to a great extent and save a lot of memory occupied by the added mass matrix. - The suggested value of the reduction coefficient
**α**for the added mass matrix of the hydrodynamic pressure is selected to be 0.6 so as to ensure that the distribution law of the dynamic response of the dam is consistent with the accurate solution, which means the unsimplified additional mass matrix condition. The error of the maximum value of the dynamic response of the dam is limited to within 5%, which is acceptable, and the elapsed time of calculation can be reduced to one twentieth of the accurate solution, which is a great jump in calculation efficiency. - The proposed method provides an accurate and efficient approach for dynamic fluid–solid coupling analysis and seismic safety evaluation of dam and reservoir systems and makes the application of dam–reservoir systems and a fluid–solid coupling analysis method in fine analysis with large-scale DOFs technically feasible.
- The proposed dynamic coupling calculation method can also be further applied to the nonlinear numerical analysis of CFRD and the fine damage analysis of concrete dams under earthquake conditions. Furthermore, the additional mass matrix simplification method in the dynamic coupling analysis of dam and reservoir systems provided in this study is also applicable to the additional mass of hydrodynamic pressure calculated by other numerical methods (FEM, BEM, PSBFEM, etc.).

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Distribution of maximum acceleration along the up-downstream direction (a

_{x}) on upstream face of arch dam (m/s

^{2}).

**Figure 8.**Distribution of maximum minor principal stress (σ

_{3}) on upstream face of arch dam (MPa).

Added Mass Matrix | Unsimplified (Accurate) | α = 1.0 | α = 0.9 | α = 0.8 | α = 0.7 | α = 0.6 | α = 0.5 | α = 0.4 | α = 0.3 |
---|---|---|---|---|---|---|---|---|---|

a_{x} (m/s^{2}) | 6.583 | 5.518 | 5.596 | 5.989 | 6.280 | 6.554 | 7.209 | 7.926 | 8.075 |

a_{y} (m/s^{2}) | 2.285 | 2.287 | 2.339 | 2.364 | 2.327 | 2.282 | 2.339 | 2.376 | 2.419 |

Added Mass Matrix | α = 1.0 | α = 0.9 | α = 0.8 | α = 0.7 | α = 0.6 | α = 0.5 | α = 0.4 | α = 0.3 |
---|---|---|---|---|---|---|---|---|

Error of a_{x} | 16.2% | 15.0% | 9.0% | 4.6% | 0.4% | 9.5% | 20.4% | 22.7% |

Error of a_{y} | 0.1% | 2.4% | 3.5% | 1.8% | 0.1% | 2.3% | 4.0% | 5.9% |

Added Mass Matrix | Unsimplified (Accurate) | α = 1.0 | α = 0.9 | α = 0.8 | α = 0.7 | α = 0.6 | α = 0.5 | α = 0.4 | α = 0.3 |
---|---|---|---|---|---|---|---|---|---|

σ_{1} (MPa) | 1.567 | 1.643 | 1.520 | 1.588 | 1.615 | 1.564 | 1.471 | 1.497 | 1.458 |

σ_{3} (MPa) | −1.604 | −1.783 | −1.685 | −1.617 | −1.551 | −1.568 | −1.508 | −1.555 | −1.521 |

Added Mass Matrix | α = 1.0 | α = 0.9 | α = 0.8 | α = 0.7 | α = 0.6 | α = 0.5 | α = 0.4 | α = 0.3 |
---|---|---|---|---|---|---|---|---|

Error of σ_{1} | 4.9% | 3.0% | 1.3% | 3.1% | 0.2% | 6.1% | 4.5% | 7.0% |

Error of σ_{3} | 11.2% | 5.0% | 0.8% | 3.3% | 2.2% | 6.0% | 3.1% | 5.2% |

Added Mass Matrix | Unsimplified (Accurate) | α = 1.0 | α = 0.9 | α = 0.8 | α = 0.7 | α = 0.6 | α = 0.5 | α = 0.4 | α = 0.3 |
---|---|---|---|---|---|---|---|---|---|

consumption time (hours) | 117.799 | 5.804 | 5.810 | 5.861 | 5.897 | 5.835 | 5.801 | 5.822 | 5.856 |

time-consuming ratio | 100.0% | 4.9% | 4.9% | 5.0% | 4.9% | 5.0% | 4.9% | 4.9% | 5.0% |

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## Share and Cite

**MDPI and ACS Style**

Xu, H.; Xu, J.; Yan, D.; Chen, K.; Zou, D.
An Efficient Dynamic Coupling Calculation Method for Dam–Reservoir Systems Based on FEM-SBFEM. *Water* **2023**, *15*, 3095.
https://doi.org/10.3390/w15173095

**AMA Style**

Xu H, Xu J, Yan D, Chen K, Zou D.
An Efficient Dynamic Coupling Calculation Method for Dam–Reservoir Systems Based on FEM-SBFEM. *Water*. 2023; 15(17):3095.
https://doi.org/10.3390/w15173095

**Chicago/Turabian Style**

Xu, He, Jianjun Xu, Dongming Yan, Kai Chen, and Degao Zou.
2023. "An Efficient Dynamic Coupling Calculation Method for Dam–Reservoir Systems Based on FEM-SBFEM" *Water* 15, no. 17: 3095.
https://doi.org/10.3390/w15173095