Using the Fundamental Solution Method and Discrete Wavelet Transform to Reduce the Computational Costs in the Analysis of Rectangular Water Tanks under the Effect of Earthquake Loading

Water tanks as vital structures play an important role in drinking water supply and safety after an earthquake. Therefore, studying and understanding the behavior of these structures against earthquake load in order to accurately design these structures is important to engineers. The use of numerical modeling to solve such problems has many applications. Despite the high accuracy of methods such as the finite element method, these methods impose a high computational cost on users. In fact, one of the main challenges in solving problems related to the vibration behavior of water tanks against earthquakes is the high cost of its calculations. In this paper, a method called the method of fundamental solutions with pressure formulation is used to analyze this category of problems. The method used in this paper has a much lower computational cost than the finite element method. Another feature of this method is the possibility of using a large time step in calculations. For this purpose, discrete wavelet transform, which has been proposed in recent years as a suitable method for the down-sampling of discrete waves, is used. This means that in this method, in tanks with real dimensions, sometimes the solution time step can be considered with acceptable accuracy up to 0.16 seconds. For this purpose, first, the proposed method for laboratory results is validated and then the structure of a tank with real dimensions under the load of 10 earthquake records is analyzed. In this regard, each earthquake wave is filtered up to five stages. At each stage of the filter, two waves of approximation and detail are obtained. The number of points of each wave of approximation and detail is half the wave of the previous stage. Due to the fact that previous studies have shown that the frequency content of the main wave is closer to the approximate wave, so at each stage of the filter, the wave of details is omitted. In this way, the number of records in each stage of the filter is half of the previous stage. This means that in the first to fifth stages, the number of discrete points is halved, a quarter, an eighth, a sixteenth, and a thirty-second, respectively. Based on the results obtained from the analysis of the tank with real dimensions, it is determined that the error of base shear of the tank (as an important parameter in the design) can be ignored in all approximate waves. This error was less than 7% for all earthquake records and all approximate waves. Also, based on the results of this article, it can be said that in order to obtain the maximum height of the fluid, care must be taken in using approximate waves with more than two levels of the wavelet filter. Because the error created for this parameter increases dramatically with the increase in the number of wavelet filters. However, this increasing trend in earthquake records is very different. Of course, the approximate wave with one filter stage introduces a negligible error in almost all earthquake records. Also, approximate waves are successful in predicting the change in fluid height. Therefore, it can be concluded that if parameter of the base shear is required for analysis, the wavelet method can work well with an error of less than 5% by reducing the calculations by 32 times. On the other hand, based on the results obtained from this article, it seems that the wavelet method has limitations in obtaining the fluid height, especially at high levels of decomposition (A2 to A5). Finally, to summarize, it can be said that if the base shear is the desired parameter from the analysis, the results presented in this paper show that the use of this method can reduce the cost of calculations with appropriate accuracy in some earthquake records by more than 90%.