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.      

one of the simplest numerical integration method which provides a large saving in computational efforts, is the well known one-point Gauss quadrature which is widely used for 4 nodes quadrilateral elements. On the other hand, the biggest disadvantage to one-point integration is the need to control the zero energy modes, called hourglassing modes, which arise. The efficiency of four different anti-hourglassing approaches, Flanagan (elastic approach), Dyna3d, Hansbo and Liu have been investigated. The first two approaches have been used in 2 and 3-D explicit codes and the latter have been employed in 2-D implicit codes. For 2-D explicit codes, the computational time was reduced by 55% and 60% for elastic and Dyna3d, respectively. However, for 3-D codes the reduction was dependent on the number of elements and was obtained between 50% and 70%. Also, the error due to the application of elastic methods was less than that for Dyna3d when the results were compared with those obtained from 2-points Gauss quadrature. Nevertheless, the convergence occurred more rapidly and the oscillations were damped out more quickly for Dyna3d approach. For implicit codes, the anti-hourglassing methods had no effect on the computations and therefore a 2-points Gauss quadrature is recommended for implicit codes as it provide the results more accurately

In this article, wavelet-based spectral finite element (WSFE) is formulated for time domain and wave domain dynamic analysis of Timoshenko beam subjected to a uniform axial tensile or compressive force (prestressed). Daubechies wavelet basis functions transform the time and space-dependent governing partial differential equations into a set of coupled space-dependent ordinary differential equations (ODE). The resulting ODEs are decoupled through an eigenvalue analysis and then solved exactly to obtain the shape functions and dynamic stiffness matrix. In the WSFE model, a beam can be divided into only a single element, but larger number of elements may be used in a finite element (FE) model. The accuracy of present WSFE model is validated by comparing its results with those of FE method. The results display advantages of WSFE model compared to FE one in reducing number of elements as well as increasing numerical accuracy. These advantages are more visible in higher frequency content excitations. In addition, the effects of axial tensile or compressive force on time domain analysis and system natural frequencies are investigated. Divergence instability of beam subjected to critical axial compressive force is investigated.

Uncertainty inherently exists in quantity of a system’s parameters (e.g., loading or elastic modulus of a structure), and thus its effects have always been considered as an important issue for engineers. Meanwhile, numerical methods play a significant role in stochastic computational mechanics, particularly for the problems without analytical solutions. In this article, spectral finite element method is utilized for stochastic spectral finite element analysis of 2D continua considering material uncertainties. Here, Lobatto family of higher order spectral elements is extended, and then influence of mesh configuration and order of interpolation functions are evaluated. Furthermore, Fredholm integral equation due to Karhunen Loeve expansion is numerically solved through spectral finite element method such that different meshes and interpolation functions’ orders are also chosen for comparison and assessment of numerical solutions are solved for this equation. This method needs fewer elements compared to the classic finite element method, and it is specifically useful in dynamic analysis as it supplies desirable accuracy by having diagonal mass matrix. Also, these spectral elements accelerate the computation process along with Karhunen Loeve and polynomial chaos expansions involving numerical solution of Fredholm integral equation. This research examines elastostatic and elastodynamic benchmark problems to demonstrate the effects of the undertaken parameters on accuracy of the stochastic analysis. Moreover, results demonstrate the effects of higher-order spectral elements on speed, accuracy and efficiency of static and dynamic analysis of continua.

In this article, a spectral finite element (SFE) formulation and its solution are described for free and force vibrations of cracked Euler-Bernoulli beam. The formulation based on SFE algorithm includes deriving partial differential equations of motion, spectral displacement field, dynamic shape functions, and dynamic stiffness matrix. Frequency-domain dynamic shape functions are derived from an exact solution of governing wave equations. The cracked beam with an open crack is modeled as two segments connected by a massless rotational spring at the crack position and frequency-domain dynamic stiffness matrix for cracked Euler-Bernoulli beam is extracted. By considering free vibration of the cracked beam, its natural frequencies are derived for different boundary conditions. In the SFE model, It is possible to represent the whole length of beam only by two spectral elements, while it may not be possible to do that in finite element (FE) model, for reaching the same order of accuracy. The accuracy of results obtained from SFE formulation is compared with that of either FE method or analytical formulations. The SFE results display remarkable superiority with respect to those of FE, for reducing the number of elements as well as increasing numerical accuracy.

In the present study, a spectral finite element method is developed for free and forced transverse vibration of Levy-type moderately thick rectangular orthotropic plates based on first-order shear deformation theory. Levy solution assumption was used to convert the two-dimensional problem into a one-dimensional problem. In the first step, the governing out-of-plane differential equations are transformed from time domain into frequency domain by discrete Fourier transform theory. Then, the spectral stiffness matrix is formulated, using frequency-dependent dynamic shape functions which are obtained from the exact solution of the governing differential equations. An efficient numerical algorithm, using drawing method is used to extract the natural frequencies. The frequency domain dynamic responses are obtained from solution of the spectral element equation. Also, the time domain dynamic responses are derived by using inverse discrete Fourier transform algorithm. The accuracy and excellent performance of the spectral finite element method is then compared with the results obtained from closed form solution methods in previous studies. Finally, comprehensive results for out-of-plane natural frequencies and transverse displacement of the moderately thick rectangular plates with six different combinations of boundary conditions are presented. These results can serve as a benchmark to compare the accuracy and precision of the numerical methods used.

The implementation of high-order (spectral) approximations associated with FEM is an approach to overcome the difficulties encountered in the numerical analysis of complex problems. This paper proposes the use of the spectral finite element method, originally developed for computational fluid dynamics problems, to achieve improved solutions for these types of problems. Here, the interpolation nodes are positioned in the zeros of orthogonal polynomials (Legendre, Lobatto, or Chebychev) or equally spaced nodal bases. A comparative study between the bases in the recovery of solutions to 1D and 2D elastostatic problems are performed. Examples are evaluated, and a significant improvement is observed when the SFEM, particularly the Lobatto approach, is used in comparison to the equidistant base interpolation.

There are different finite element models in place for predicting the bending behavior of shear deformable beams and plates. Mostly, the literature abounds with traditional equi-spaced Langrange based low order finite element approximations using displacement formulations. However, the finite element models of Timoshenko beams and Mindlin plates with linear interpolation of all generalized displacements have suffered from shear locking, which has been alleviated with the help of primarily reduced/selective integration techniques to obtain acceptable solutions [1-4]. These kinds of 'fixes' have come into existence because the element stiffness matrix becomes excessively stiff with low-order interpolation functions. In this study we propose an alternative spectrally accurate hp/spectral method to model the Timoshenko beam theory and first order shear deformation theory of plates (FSDT) to eliminate shear and membrane locking. Beams and isotropic and orthotropic plates with clamped and simply supported boundary conditions are analyzed to illustrate the accuracy and robustness of the developed elements. Full integration scheme is employed for all cases. The results are found to be in excellent agreement with those published in literature.

Changes of up to 80°C has been reported for oral cavity temperature. This could well effecti on the nature of restorations for example failure of bonding of adhesive restorations. It is advocated that using opaque layer in porcelane to restorations could reduce this problem. This experimental study was designed to evaluate the effect mentioned using finitelement analysis method.Results showed that cooling has a more destructive effect than warming process restorations with the presence of opaque having a finitelement analysis effect on restorations.