Free vibration of variable-thickness plates via adaptive finite elements

New technologies, such as additive manufacturing combined with topology optimisation or bio-inspired design, can produce lightweight structures with better dynamical properties but more complex geometries. Analysing these components with the finite element method can become time-consuming because of fine 3D meshes. By exploiting the node-dependent kinematic approach of Carrera's unified formulation and using Lagrange expanding functions, this work presents the implementation of adaptive finite elements for the free-vibration analysis of plates with an arbitrary thickness variation through the in-plane domain. In other words, the kinematics of the 2D model on which the element is based can be adapted to the geometry of the plate. The formulation is mainly based on the 3D integration of the approximating functions and computation of a 3D Jacobian matrix inside the element for the derivation of stiffness and mass matrices; substantially, the resulting elements are 3D elements in which the order of expansion through the thickness can be different from that in the plane of the plate. The free vibration analysis of some plates with different thickness variations is performed. The results demonstrate that the present elements allow us to accurately study these innovative 2D structures by employing much fewer degrees of freedom with respect to classical 3D finite elements.