Transient Dynamic Response Analysis of Two-Dimensional Saturated Soil with Singular Boundary Method
Abstract
:1. Introduction
2. Governing Equations
3. Singular Boundary Method in Frequency-Domain
4. Exponential Window Method
- (1)
- Determine the total calculation time and the number of sampling frequencies , then to determine the angular frequency resolution with ;
- (2)
- Determine the shifting constant according to the numerical experiments and experience as
- (3)
- Construct a desired damping system with scaling the variables ( and ) with the scaling function as . Bring new variables into the governing equations, and a novel frequency-domain boundary value problem Equations (6) and (7) with is obtained.
- (4)
- Simultaneously, the boundary condition is scaled into , and the frequency-domain boundary condition can be obtained via discretized Fourier transform
- (5)
- Perform the SBM to evaluate the solutions of the frequency-domain problems at the frequencies . The remaining of results can be obtained through conjugate symmetric property as
- (6)
- Perform the IFFT with the inverse DFT with Hanning window function , and obtain the time-domain solutions asThe Hanning window function is used to alleviate the Gibbs oscillations.
- (7)
- Descale the time-domain solutions and obtain the solutions of the original problems as
5. Numerical Examples
5.1. Verification of the Proposed SBM-EWM Method
5.2. A Half-Space Problem Subjected to a Transient Load
5.3. A Tunnel Embedded in a Saturated Poroelastic Half-Space
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Detailed Derivations of the 2D Fundamental Solutions
- (1)
- Solid loads
- (2)
- Fluid load
Appendix B. The of OIFs for 2D Saturated Poroelastic Problems
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Liu, D.; Wei, X.; Li, C.; Han, C.; Cheng, X.; Sun, L. Transient Dynamic Response Analysis of Two-Dimensional Saturated Soil with Singular Boundary Method. Mathematics 2022, 10, 4323. https://doi.org/10.3390/math10224323
Liu D, Wei X, Li C, Han C, Cheng X, Sun L. Transient Dynamic Response Analysis of Two-Dimensional Saturated Soil with Singular Boundary Method. Mathematics. 2022; 10(22):4323. https://doi.org/10.3390/math10224323
Chicago/Turabian StyleLiu, Dongdong, Xing Wei, Chengbin Li, Chunguang Han, Xiaxi Cheng, and Linlin Sun. 2022. "Transient Dynamic Response Analysis of Two-Dimensional Saturated Soil with Singular Boundary Method" Mathematics 10, no. 22: 4323. https://doi.org/10.3390/math10224323
APA StyleLiu, D., Wei, X., Li, C., Han, C., Cheng, X., & Sun, L. (2022). Transient Dynamic Response Analysis of Two-Dimensional Saturated Soil with Singular Boundary Method. Mathematics, 10(22), 4323. https://doi.org/10.3390/math10224323