exponential basis functions

In this paper, formulation of a recently proposed time-weighted residual method has been developed for the vibration analysis of Timoshenko beams under moving loads. Employing pre-integration relations as well as equilibrium equations, is the main idea of this method. In the first step of the proposed method, the time interval is subdivided into a number of sub-intervals and then the acceleration field in each sub-interval is defined as the combination of an unknown function and an exponential series with constant coefficients. Finally, the solution of the problem is estimated by the time-weighted residual method along with exact satisfaction of the initial and the boundary conditions at the two ends of the beam. Storing the information of solution at each time step on the exponential coefficients, is the most important advantage of this method, so that the solution is progressed in time without the need to discretize beams and only by using an appropriate recursive relation to update the exponential coefficients. In order to investigate the accuracy and efficiency of the proposed method, the results of solving three sample problems of constant and accelerated moving load on the beams with different boundary conditions, are compared with the results of finite element method. This comparison illustrates the speed and accuracy of the proposed method in estimating the internal shear forces and bending moments rather than those obtained by the finite element method.

Accurate determination of the response of structures under dynamic loads such as earthquake loads plays an important role in the safe and economical design of structures. The purpose of this paper is to utilize a novel solution method based on the use of exponential basis functions for dynamic analysis of Bernoulli beam subjected to different types of base excitations. This method was firstly introduced for solving scalar wave propagation problems, named as stepwise time-weighted residual method. The proposed method considers the solution as a series of exponential basis functions with unknown constant coefficients; and the problem is solved in time without the need for spatial discretization of the beam and by using an appropriate recursive relation to correct the coefficients of the exponential bases. In order to apply the earthquake excitation, first by using the central finite difference relation, the earthquake acceleration history is converted to displacement history. Moreover, the displacement history is applied to the beam as a time-varying boundary condition. In this study, the capabilities of the proposed method in solving several sample problems of vibration of single and multi-span beams under various stimuli such as earthquake acceleration variations are compared with the results of other existing methods.

Solitary waves are often applied for simulating tsunami phenomenon. In this article, a meshless method based on exponential basis functions (EBFs) is developed to simulate the propagation of solitary waves and run-up on the slope. This method is a boundary-type meshless method using exponential basis functions with complex exponents. The solution to governing equations is considered as a series of these basis functions and boundary conditions satisfied via a point-wise collocation approach. In this research, the simplified Navier-Stokes equations in the Lagrangian form for an incompressible inviscid fluid are employed. Governing equations and boundary conditions are established on pressure as a potential equation. A stable Lagrangian time marching algorithm is developed for tracking free surface on the beach. So, the solution process can be preceded by boundary nodes tracking, and the most important characteristics of a fluid, such as the displacement, velocity, and acceleration, are the results of pressure Laplace equation. Geometry updating is only performed by changing the location of boundary nodes, and there is no need to mesh generation or any integration in solution process. According to the Lagrangian formulation, the numerical solution is performed at a time marching approach using an implicit two-step algorithm. In this algorithm, pressure equation is solved twice a step, and position of boundary nodes is corrected at the end of each time step. Minimum calculation time, convenient performance, and high accuracy in the solution process are the advantages of this method. The results of the present numerical method in the prediction of solitary wave propagation and estimation of run-up are verified through the comparison with experimental data. Different wave amplitude cases are simulated, and the resulted run-up is compared with experiments. The results show that presented meshless method and developed time marching algorithm are capable of simulating the run-up under non-breaking condition quiet accurately.