Dynamic Analysis of Dam-Reservoir Interaction by Euler-Lagrange Approach Using Perfectly Matched Layer (PML) in Radiation Boundary

Dynamic Analysis of Dam‐Reservoir Interaction by Euler‐LagrangeApproach Using Perfectly Matched Layer (PML) in Radiation BoundaryReza Tarinejad *, Sajjad PirboudaghiFaculty of Civil Engineering, University of Tabriz, Tabriz, 5166616471 IranReceived: 24 February 2013; Accepted: 09 September 2013Keywords:Dam-reservoir interaction, Euler-Lagrange, Perfectly matched layer (MPL), Radiation boundary, Finite elementmethod A key issue in the seismic analysis of a dam-reservoir system is the accurate and inexpensive modeling of the unbounded reservoir. The Perfectly matched layer (PML) is one of the state-of-the-art boundary conditions, introduced by Berenger for the absorption of radiated boundary waves. In this research,seismic analysis of damreservoir systemis carried out using the PML in the radiation boundary of the reservoir. For this purpose,the numerical implementation of the PML based on the finite element method is presented. A special purpose computer code is developed to carry out the dynamic analysis of the dam-reservoir systems using the MATLAB software. Sensitivity analyses corresponding to the PML layer parameters, the decay function and layer thickness, are carried out to achieve accurate results in less computing time. Coupling of dam and reservoir equations The dam-reservoir interaction is a classic coupled problem, which contains two harmonic differential equations in the frequency domain. Damreservoir analysis by using PML at far end of reservoir A computer code is developed to carry out the seismic analysis of the dam-reservoir interaction system using the PML layer in the radiation boundary of the reservoir. Verification of the developed code is carried out by the comparison of the results using Sommerfeld boundary condition by appropriate length of the reservoir, three times of the dam height, with the results of the previous investigations. Good agreement was obtained in time and frequency domain. Sensitivity analysis with PML’s length (LP), reservoir’s length (LR) and attenuation coefficient (σ*) are performed to achieve accurate results in less computing time. The convergence errors of the dam crest horizontal displacement (L2 error) are indicated versus different parameters in Figs. 1 and 2. The maximum responses of the dam-reservoir system obtained from two kinds of boundary conditions, the optimum PML layer and Sommerfeld, are presented in Table 2. Although the small length of the reservoir is modeled by using the PML layer in comparison with the classic Sommerfeld boundary condition, the results show good agreement for different kinds of responses. The PML was adopted in the radiation boundary of the reservoir in the seismic analysis of the dam-reservoir interaction system and the finite element implementation was presented. A computer code is developed and verified to carry out the dynamic analysis of the interaction system using the MATLAB software. Sensitivity analysis of the PML parameters, PML’s length (Lp), reservoir’s length (LR) and attenuation coefficient (σ*), was performed to achieve the accurate results with less computational time in comparison with the results obtained from the classic Sommerfeld boundary conditions. Results from this analysis demonstrated that for the same accuracy, the PML-type radiation boundary leads to 50% off in time consumption of the analysis compared to the Sommerfeld boundary condition.