Development of the Multiquadric mesh-less method for analyzing the dynamic interaction of dam-reservoir-foundation problems

Document Type : Article


1 D‌e‌p‌a‌r‌t‌m‌e‌n‌t o‌f C‌i‌v‌i‌l E‌n‌g‌i‌n‌e‌e‌r‌i‌n‌g U‌n‌i‌v‌e‌r‌s‌i‌t‌y o‌f Q‌o‌m

2 S‌c‌h‌o‌o‌l o‌f C‌i‌v‌i‌l E‌n‌g‌i‌n‌e‌e‌r‌i‌n‌g U‌n‌i‌v‌e‌r‌s‌i‌t‌y o‌f T‌e‌h‌r‌a‌n


The Multiquadric Radial Basis Function (MQ-RBF) method, despite its advantages, has not yet been developed to be used for Dam-Reservoir-Foundation Interaction (DRFI) problems. In this study, this mesh-less method was developed for solving the DRFI systems in the frequency domain. A new domain decomposition technique was also used for analyzing dynamic interaction problems for the first time in MQ-RBF. In this regard, the computational domain is divided into dam, reservoir, and foundation subdomains. Then, the MQ-RBF method is separately applied to each subdomain. For applying the dynamic interaction between two adjacent subdomains, two Multiquadric shape functions must be considered for each computational center on their interaction boundary. Besides, each shape function is also defined using the computational centers in the subdomain. One of the important challenging issues in RBFs is the determination of the Optimal Shape Parameter (OSP). Thereafter, some new relations in terms of the earthquake frequencies are proposed for the OSPs in different cases of the interaction systems. In this regard, a few frequency magnitudes were considered and, consequently, different relations were presented for all frequencies using the obtained OSPs. It is found that, the OSP does not depend on the shear modulus of neither the dam nor the foundation. Moreover, the OSP value are not sensitive to the fluid compressibility and do not depend on the number of subdomains. Apparently, these properties reduce the computational costs and facilitate the MQ-RBF application. In order to validate the capabilities of the approach, nine numerical examples are solved in which the Roots Mean Square Error (RMSE) criterion has been evaluated for comparing the results with those of the exact and FD methods. Results show that the proposed method is of acceptable accuracy, i.e. more accurate than FD even with much more FD computational nodes. Also, it is shown that the errors increase by increasing the earthquake frequency value while the FD errors seem to be unacceptable in frequency values close to the resonance frequency, unlike those of the MQ-RBF.


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