Amirkabir University of TechnologyAmirkabir Journal of Civil Engineering2588-297X521220210219Application of Multiquadric Radial Basis Function method for Helmholtz equation in seismic wave analysis for reservoir of rigid damsApplication of Multiquadric Radial Basis Function method for Helmholtz equation in seismic wave analysis for reservoir of rigid dams30153030358010.22060/ceej.2019.16443.6230FARezaBabaeeUniversity of QomEhsanJabbariUniversity of Qom0000-0002-6345-8567MortezaEskandari-GhadiUniversity of TehranJournal Article20190528 The high costs of mesh generation in mesh-dependent solution, weakness in capturing <br />singularities, the need of modeling all over the domain, the need of problem dependent fundamental <br />solutions, etc. are some of weaknesses in the common numerical mesh-dependent methods for solving <br />continuum mechanics boundary value problems. In this study, aiming for eliminating some of these <br />shortcomings, one of the well-known Radial Basis Functions (RBF) methods, Multiquadric (MQ), is <br />developed for dynamic analysis of 2D reservoirs of rigid dams in frequency-domain. To this end, the <br />Helmholtz equation and the governing complex boundary conditions are reproduced using MQ function <br />in the frequency domain. The results show that with the use of real and complex forms of the MQ <br />function, the computational time will be respectively optimized for frequencies smaller and larger than <br />the natural frequency of the reservoir. Also, to determine the most important factors affecting both the <br />accuracy and convergence of MQ method, first the inefficiency of some of the previously introduced <br />methods is proved, and then a new high-speed algorithm is presented. It is shown that the optimal <br />shape parameter for MQ method can be formulated in terms of the frequencies of seismic records. <br />This advantage simplifies the application of MQ method in this particular problem and reduces the <br />computational time, considerably. The high accuracy of the present method is shown in two different <br />examples, where the effects of sediment absorption may either be considered or not. The high accuracy <br />compared to the exact solutions achieved in this paper is due to a continuous estimation function defined <br />all over the domain and also due to the simple algorithm used for finding the optimal shape parameter. The high costs of mesh generation in mesh-dependent solution, weakness in capturing <br />singularities, the need of modeling all over the domain, the need of problem dependent fundamental <br />solutions, etc. are some of weaknesses in the common numerical mesh-dependent methods for solving <br />continuum mechanics boundary value problems. In this study, aiming for eliminating some of these <br />shortcomings, one of the well-known Radial Basis Functions (RBF) methods, Multiquadric (MQ), is <br />developed for dynamic analysis of 2D reservoirs of rigid dams in frequency-domain. To this end, the <br />Helmholtz equation and the governing complex boundary conditions are reproduced using MQ function <br />in the frequency domain. The results show that with the use of real and complex forms of the MQ <br />function, the computational time will be respectively optimized for frequencies smaller and larger than <br />the natural frequency of the reservoir. Also, to determine the most important factors affecting both the <br />accuracy and convergence of MQ method, first the inefficiency of some of the previously introduced <br />methods is proved, and then a new high-speed algorithm is presented. It is shown that the optimal <br />shape parameter for MQ method can be formulated in terms of the frequencies of seismic records. <br />This advantage simplifies the application of MQ method in this particular problem and reduces the <br />computational time, considerably. The high accuracy of the present method is shown in two different <br />examples, where the effects of sediment absorption may either be considered or not. The high accuracy <br />compared to the exact solutions achieved in this paper is due to a continuous estimation function defined <br />all over the domain and also due to the simple algorithm used for finding the optimal shape parameter.https://ceej.aut.ac.ir/article_3580_ffd59b086b2249200e985288f6bd73ff.pdf