Gradient reproducing kernel based Hermite collocation method (GHCM) for eigenvalue analysis of functionally graded thin plates with in-plane material

Abstract

A meshfree Hermite collocation method based on gradient reproducing kernel approximations is proposed for the eigenvalue analysis of thin functionally graded plates with in-plane material inhomogeneity. Compared with direct collocation method (DCM), Hermite collocation method (HCM) can improve the accuracy of the solutions, especially on the boundaries. Moreover, HCM results in a determined system for the eigenproblems while DCM leads to an over-determined system where the solutions may not be the real results. Since derivative calculations of the reproducing kernel function are complex and time-consuming, gradient reproducing kernel approximations are introduced for solving the thin plate problems to avoid the high order direct differentiations which are required in the thin plate strong form solutions. Proper weights that should be imposed for the boundary conditions to balance the errors in the solutions are derived for both DCM and HCM. Convergence studies are also derived to evaluate the convergence of these two methods. Comparison studies with analytical solutions indicate the high accuracy and good performance of the proposed method. Numerical simulations demonstrate that material inhomogeneity has considerable effects on the natural frequencies and mode shapes.