This study investigates a time-domain method for modeling general transient elastodynamic problems using the Spectral-based finite element method (SEM) which is based upon a conforming mesh of two-dimensional quadrilaterals. Employing the Galerkin weighted residual method, detailed formulation of the SEM is derived in which various aspects involving in elastodynamic problems are discussed. The accuracy and efficiency of the method is fully demonstrated by comparing results obtained from the SEM with those reported in other studies. For this purpose, a set of wave propagation and structural dynamic problems, subjected to various load forms such as triangular load, Heaviside step load, sinusoidal impulsive load, and ramped load are modeled using the SEM. Furthermore, support motion boundary conditions are examined using the SEM. Each problem is successfully modeled using a very small number of degrees of freedom in comparison with other numerical methods. The numerical results agree very well with the analytical solutions as well the results from other numerical methods.

This work presented here is the solution of the one-dimensional Bratus problem. The nonlinear Bratus problem is first linearised using the quasi-linearization method and then solved by the Spectral element method. We use the Legendre polynomials for interpolation. Finally we show the results with a numerical example.

This paper deals with the detection of crack in frame structures based on Euler-Bernoulli beams theorem by the Spectral element method. The effect of cracking is modeled using Castigliano’s theorem and laws of fracture mechanics as mass-less rotational and translational springs which are embedded in different locations of the steel frame structure in both beam and column. The crack location is revealed precisely without prior knowledge of their positions in frame structure. This means that there is no necessity to know the location and the node number which is assigned to crack. Finally the effect of crack when it is embedded simultaneously in both beam and column members, is also studied.

In this paper an efficient analytical method is presented for calculating the eigenvalues of special matrices related to Finite element Meshes (FEMs) with regular topologies. In the proposed method, a skeleton graph is used as the model of a FEM. This graph is then considered as the Cartesian product of its generators. The eigenvalues of the Laplacian matrix of the entire graph are then easily calculated using the eigenvalues of its generators. An exceptionally fast method is also proposed for computing the second eigenvalue of the Laplacian of the graph model of a FEM, known as the Fiedler vector. After ordering the entries of the second eigenvector, the graph model is partitioned and the corresponding FEM is bisected.      

In this paper, the Galerkin Spectral element method (GSEM) is presented for solving the one-dimensional viscoelastic wave equation. The finite difference approximation is applied for temporal discretization, and the convergence order of the technique is demonstrated. An a priori error estimate is determined for the total-discrete system. The computational efficiency of the method is confirmed through a numerical example, which shows that the method is effectively applicable.

This article presents an analysis of free vibration of elastically supported Timoshenko beams by using the Spectral element method. The governing partial differential equation is elaborated to formulate the Spectral stiffness matrix. Effectively, the nonclassical end boundary conditions of the beam are the primordial task to calibrate the phenomenon of the Timoshenko beam-soil foundation interaction. Non-dimensional natural frequencies and shape modes are obtained by solving the partial differential equations, numerically. Upon solving the eigenvalue problem, non-dimensional frequencies are computed for the first three modes of vibration. Obtained results of this study are intended to describe multiple objects, such as: (1) the establishment of the modal analysis with and without elastic springs, (2) the quantification of the influence of the beam soil foundation interaction, (3) the influence of soil foundation stiffness’ on free vibration characteristics of Timoshenko beam. For this propose, the first three eigenvalues of Timoshenko beam are calculated and plotted for various stiffness of translational and rotational springs.

This paper presents a Galerkin Spectral element method for solving a fractional diffusion equation, considering initial and boundary conditions. We construct a discrete scheme for time, employing the Crank-Nicolson method to approximate the Caputo fractional derivative on a uniform mesh. Then we introduce a Galerkin variational formulation to establish the unconditional stability of the scheme. Moreover, we apply the Spectral element method based on Legendre polynomials in the space direction and obtain the fully discrete scheme. The error analysis of the fully discrete scheme is treated in $L_2$ sense. we present a computational analysis to deal with the Galerkin Spectral element method, to compute the corresponding bilinear form, on the implementation process. Finally, we prove the effectiveness of the method through numerical experiments and some simulations using MATLAB software.