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.

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.