This paper details the use of time series for solving finite element equations. It was aimed at eliminating the discretization of time intervals to reduce calculation time and to obtain accurate results using timedependent variables. For this purpose, the matrix form of the time series finite element formulation for both continuous and discontinuous time functions were extracted using Taylor and Fourier series for estimating time-dependent functions. Based on the extracted formulation, for solving elastodynamic problems, a computer code and algorithm was introduced. The efficiency and accuracy of the formulation was evaluated by solving different analytical examples. To reveal the power of time series for modeling real problems three practical examples such as settlement, flood protector structures and temporary cofferdams were modeled using time series approach and the results of analyses were presented. The results showed that having removed time discretization procedure, the time series based finite element method can be efficiently applied to decrease the calculation time and computational efforts. In addition, time-continuous responses resulting from this method, help reduce the calculation steps which is vital in many engineering problems.

Introduction

For more than fifty years, finite element analysis is a well-known approach for many researchers in different fields of science and industry [1-8]. Although the first developments of the discrete analysis go back to Newton [9] during the 16th century, the existence of finite element analysis was not considered before the development of the virtual work principle. The concepts of the virtual work principle return to the 17th and 18th centuries when Schellbach [10] made first analytical procedures, and Strutt [11] obtained numerical results for several problems. In elasticity, Ritz [12] showed a discrete approach for analyzing the Kirchhoff plate. The first use of triangular and rectangular elements for a two-dimensional Sant-Venant torsion problem was done by Courant [13]. The Modern shape of finite element method for analyzing structural and aeronautical problems was made by Turner et al. [14], Argyris [15] and Zienkiewicz [16, 17]. Over the last three decades, the development of computing devices has given a rapid growth to the use of finite element analysis for solving the engineering problems. In analyzing complex problems however, the slow solving process can be noted as a serious challenge. As an example, the dynamic analysis of the complex structures under earthquake loading usually takes hours to give accurate results [18]. To tackle this problem, several solutions have been proposed, among which the combination of finite element method with boundary element or discrete element methods can be cited [19-22].