Richardson extrapolation based integration scheme using FSDT over Quad elements: Application to thick FGM plates

In this work, our new quadrature schemes based on the Richardson extrapolation (RE) and centroid or midpoint rule are further applied to the quadrilateral Mindlin plate element for static and free vibration analysis. In the process of generating element matrices, to apply RE and midpoint quadrature rule, there are two approximations are particularly considered. As the first approximation, element matrices are calculated at centroid of whole plate element. For stabilizing function or second approximation, each quadrilateral element is divided into four sub-quadrilaterals, either centroid of the each sub-quadrilateral plate for Element midpoint method (EM-plate method) or midpoint of the element edges are used for Element edge method (EE-plate method) to compute the all the element matrices. Then, both the approximations are added with the RE based weighting functions. Generally, midpoint based quadrature avoids the shear-locking phenomenon in the plate elements and the RE enhances accuracy and rate of convergence of the final solution without hourglass or zero-energy issues. The numerical examples validated that the RE based EM-plate method and EE-plate method are free of shear-locking, ability to pass the patch test, better rate of convergence with less number of sampling points. Also, the new schemes confirms the three important properties such as (i) can produce high accurate solutions with better rate of convergence for problems for irregular geometries in the static analysis; (ii) can produce high accurate solutions without hourglass issues in free vibration analysis; and (iii) can generate the accurate values of high frequencies of plate elements over irregular meshes.

Automated summary

In this work, new quadrature schemes based on Richardson extrapolation (RE) and centroid or midpoint rule are applied to the quadrilateral Mindlin plate element for static and free vibration analysis. Two approximations are considered in generating element matrices, and the schemes show improved accuracy, convergence rate, and ability to handle irregular geometries and high frequencies over irregular meshes.