عنوان مقاله [English]
The present paper deals with transverse elastic wave propagation using spectral element method (SEM), based on the first order shear deformation plate theory. In this method, the element nodes are located at the zeros of an orthogonal function, such as the Legendre function, which causes the mass matrix to be diagonal. On the other hand, the differential method is implemented for solving the problem of limited intermediate. Dealing with a diagonal mass matrix is the main advantage of the present approach, because the analysis only requires the
inverse of a diagonal matrix. Therefore, the analysis time is much less than the one with the conventional Finite Element Method.
In this study, a 10 layered square composite plate with 1 m side and 10 mm thickness is investigated. A time-variable force is applied at the centre of the plate during 120 msec, with 25 kH frequency. It is assumed that the fibers are aligned parallel to the side of the plate at all layers. Elements with 9, 25 and 36 nodes are used for simulations. In each study case, the total number of elements of the plate is selected in such a way by increasing the number of elements and taking into account a 2% difference for two consecutive steps.
Therefore, the plate is divided into $200times200$ elements (40000 elements) with 9 node elements. The selection of $100times100$ and $60times60$ elements for 25 and 36-noded elements, respectively, result only in 2 percent reduction in convergence time.
Having solved the problem, three-dimensional deflection and two-dimensional deflection contour on paths passing through the center of the plate and parallel to the sides have been obtained. Then, using the two-dimensional
deflection results, group wave velocity values ??in the longitudinal (fiber direction) and transverse directions (perpendicular to fibers) of the plate have been obtained. Creating the element stiffness and mass matrices, and
assembling these matrices, makes the whole process analysis time. Therefore, for comparison between the elements with 9, 25 and 36 nodes, the time required for matrix assembly are registered. Besides, using theoretical formulations, the value of the group velocity for the desired frequency in this direction has
been obtained and compared with those obtained via finite element analysis. A three-dimensional deflection contour has shown that the wave front is elliptical and a large diameter of oval is located in the fiber direction. In
addition, the value of the maximum deflection of the plates has been decreased with increasing time. As shown by our findings, by increasing the number of element nodes, the time required for assembling the stiffness and mass matrices can be accelerated significantly. In addition, generally, by increasing the number of nodes per element, the differences between these results with those obtained from the analytical method are higher, and only in some cases lead to better correlations. Consequently, slightly better correlations incur very high
computational costs. It is noted that elements with fewer numbers of nodes have lower degree polynomials, and may not establish the continuity of stress and strain. For the 9-node element used, for example, with a function of order two, the curvature function is obtained by double differentiation. Since the curvature is constant, the adjacent elements cannot have the same curvature.
However, this must be the same for all elements. Therefore, from one element to the adjacent element, there will be a discontinuity in the stress and strain distribution. However, this is not the case for higher-order elements. In conclusion, if only the total amount of computational time and volume is required, a 9-node element is preferred. On the other hand, if the continuity condition of stress and strain distribution is preferred, then the advantage is in using higher-order elements.