An Efficient Dynamic Coupling Calculation Method for Dam–Reservoir Systems Based on FEM-SBFEM
Abstract
:1. Introduction
2. A Calculating Method for Hydrodynamic Pressure of Reservoir Based on SBFEM
2.1. The Basic Equation and Boundary Conditions
2.2. Solution for Hydrodynamic Pressure Based on SBFEM
3. An Efficient Dynamic Coupling Calculation Method for Dam–Reservoir Systems
3.1. Conventional Dynamic Coupling Analysis Method for Dam–Reservoir Systems
3.2. Simplification of Hydrodynamic Pressure Additional Mass Matrix
3.2.1. Physical Meaning and Distribution Characteristics of Matrix
3.2.2. Theoretical Analysis and Simplified Processing Method
3.3. Efficient Dynamic Coupling Calculation Method Based on FEM-SBFEM
4. Dynamic Coupling Analysis of Arch Dam and Reservoir Systems
4.1. Calculation Model
4.1.1. Dam and Reservoir Model
4.1.2. Material Parameters, Input Seismic Load, and Damping Methods
4.2. Effect of Additional Mass Matrix Simplification
4.3. Results and Discussion
4.3.1. Acceleration of Arch Dam
4.3.2. Stress of Arch Dam
4.3.3. Computational Efficiency
4.3.4. Suggested Value of Reduction Coefficient α
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
References
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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 |
---|---|---|---|---|---|---|---|---|---|
ax (m/s2) | 6.583 | 5.518 | 5.596 | 5.989 | 6.280 | 6.554 | 7.209 | 7.926 | 8.075 |
ay (m/s2) | 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 ax | 16.2% | 15.0% | 9.0% | 4.6% | 0.4% | 9.5% | 20.4% | 22.7% |
Error of ay | 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 |
---|---|---|---|---|---|---|---|---|---|
σ1 (MPa) | 1.567 | 1.643 | 1.520 | 1.588 | 1.615 | 1.564 | 1.471 | 1.497 | 1.458 |
σ3 (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 σ1 | 4.9% | 3.0% | 1.3% | 3.1% | 0.2% | 6.1% | 4.5% | 7.0% |
Error of σ3 | 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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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
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 StyleXu, 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