Second-order scalar wave field modeling with a first-order perfectly matched layer

The forward modeling of a scalar wave equation plays an important role in the numerical geophysical computations. The finite-difference algorithm in the form of a second-order wave equation is one of the commonly used forward numerical algorithms. This algorithm is simple and is easy to implement based on the conventional grid. In order to ensure the accuracy of the calculation, absorption layers should be introduced around the computational area to suppress the wave reflection caused by the artificial boundary. For boundary absorption conditions, a perfectly matched layer is one of the most effective algorithms. However, the traditional perfectly matched layer algorithm is calculated using a staggered grid based on the first-order wave equation, which is difficult to directly integrate into a conventional-grid finite-difference algorithm based on the second-order wave equation. Although a perfectly matched layer algorithm based on the second-order equation can be derived, the formula is rather complex and intermediate variables need to be introduced, which makes it hard to implement. In this paper, we present a simple and efficient algorithm to match the variables at the boundaries between the computational area and the absorbing boundary area. This new boundary-matched method can integrate the traditional staggered-grid perfectly matched layer algorithm and the conventional-grid finite-difference algorithm without formula transformations, and it can ensure the accuracy of finite-difference forward modeling in the computational area. In order to verify the validity of our method, we used several models to carry out numerical simulation experiments. The comparison between the simulation results of our new boundary-matched algorithm and other boundary absorption algorithms shows that our proposed method suppresses the reflection of the artificial boundaries better and has a higher computational efficiency.

Hydrodynamic pressures on arch dam faces with irregular reservoir geometry

A scaled boundary finite element method is developed for the analysis of hydrodynamic pressures acting on arch dam faces with a reservoir of irregular geometry. Water compressibility and reservoir boundary absorption are considered simultaneously. The reservoir is idealized as composed of finite and infinite subdomains. Governing equations for evaluating hydrodynamic pressures of finite fluid domain have been established and the solution procedures are presented. In addition, the boundary conditions at the interface between finite and infinite subdomains are decided. Numerical examples validate the present method with high accuracy and show that the fairly irregular geometry of a reservoir has an important influence on the hydrodynamic pressures acting on the arch dam face. Frequency response functions of the hydrodynamic pressures in the stream and cross-stream directions are evidently influenced by reservoir geometry, while hydrodynamic pressures in the vertical direction are relatively less influenced. Another feature of the frequency response function of hydrodynamic pressures is that the amplitude of resonant peaks is affected much more by reservoir geometry as compared to the resonant frequency. Besides, peak value of hydrodynamic pressures in the arch direction and cantilever direction is also significantly influenced by reservoir geometry. Based on these facts, it can be concluded that the effect of reservoir geometry on hydrodynamic pressure should be considered for seismic design of arch dams.

Second-order Scalar Wave Field Modeling with First-order Perfectly Matched Layer

The forward modeling of a scalar wave equation plays an important role in the numerical geophysical computations. The finite-difference algorithm in the form of a second-order wave equation is one of the commonly used forward numerical algorithms. This algorithm is simple and is easy to implement based on the conventional-grid. In order to ensure the accuracy of the calculation, absorption layers should be introduced around the computational area to suppress the wave reflection caused by the artificial boundary. For boundary absorption conditions, a perfectly matched layer is one of the most effective algorithms. However, the traditional perfectly matched layer algorithm is calculated using a staggered-grid based on the first-order wave equation, which is difficult to directly integrate into a conventional-grid finite-difference algorithm based on the second-order wave equation. Although a perfectly matched layer algorithm based on the second-order equation can be derived, the formula is rather complex and intermediate variables need to be introduced, which makes it hard to implement. In this paper, we present a simple and efficient algorithm to match the variables at the boundaries between the computational area and the absorbing boundary area. This new boundary matched method can integrate the traditional staggered-grid perfectly matched layer algorithm and the conventional-grid finite-difference algorithm without formula transformations, and it can ensure the accuracy of finite-difference forward modeling in the computational area. In order to verify the validity of our method, we used several models to carry out numerical simulation experiments. The comparison between the simulation results of our new boundary matched algorithm and other boundary absorption algorithms shows that our proposed method suppresses the reflection of the artificial boundaries better and has a higher computational efficiency.