Application of Multiquadric Radial Basis Function method for Helmholtz equation in seismic wave analysis for reservoir of rigid dams

Abstract

The high costs of meshing, required problem dependent fundamental solutions, singularity, modeling all over the domain and … are the most important weaknesses in the common numerical mesh methods for solving continuum mechanics problems. In this study, aiming for eliminating some of these shortcomings, one of the well-known Radial Basis Functions (RBF) methods, Multiquadric (MQ), was developed for analyzing 2D seismic waves in reservoirs of rigid dams. In this regard, the Helmholtz equation and the governing complex boundary conditions are reproduced using MQ function in the frequency domain. The results showed that using the real and complex forms of the MQ function, the computational time will be optimized for frequencies smaller and larger than the natural frequency of the reservoir, respectively. Also, to determine the most important factor in accuracy and convergence of MQ method, the optimal shape parameter, firstly the inefficiency of some of the previously introduced methods was shown, then a new high-speed algorithm was presented. The results of this research show that the optimal shape parameter can be formulated in terms of the frequencies of earthquake loads. This advantage simplifies considerably application of MQ method in this particular problem and reduces its computational time. The high accuracy of the present method compared to the exact solutions was shown in two different examples with and without considering effects of sediment absorption. The achieved high accuracy is due to a continuous estimation function defined all over the domain and using an exact algorithm for finding the optimal shape parameter.