Finite beam elements based on Legendre polynomial expansions and node-dependent kinematics for the global-local analysis of composite structures

This work introduces an innovative type of FEM beam models with node-dependent kinematics. A variety of global-local approaches have been proposed to reduce the consumption of computational resources in FEM analysis, in which mostly the main idea is to couple the elements in the locally refined region (with either refined mesh or higher-order theories) and those in the less refined area. As a new method to build FEM models in a global-local analysis, node-dependent kinematics makes it possible to construct elements with different kinematic theories on different nodes and implement a kinematic variation conveniently within an element to bridge a local model to a global one. With the help of the introduced cross-section functions, CUF allows the definition of different kinematics on each node and their interpolation over the axial domain of the beam element. Without using any ad hoc coupling method, beam elements with node-dependent kinematics have very compact and coherent formulations through the Fundamental Nucleus (FN). Corresponding FEM governing equation is derived from the Principle of Virtual Displacements (PVD), and the expressions of FNs of the stiffness matrix and load vector are given. Both ESL (Equivalent Single-layer) and LW (Layer-wise) models are addressed. In fact, in this work, Legendre polynomials are used to construct refined beam models, obtaining cross- section functions (nodal kinematics) with Hierarchical Legendre Expansions (HLE) and, eventually, LW accuracy. In the numerical examples, refined models with HLE are employed in the local area with a higher stress gradient, and in the less critical regions ESL models are adopted; meanwhile, in the kinematic transition zone, a beam element with node-dependent kinematics are used to connect these two domains. By comparing the numerical results with those in literature and from 3D FEM modeling, it is demonstrated that when used in the analysis of composite beams with local effects to be considered, node-dependent kinematic beam elements can reduce the computational costs significantly without losing numerical accuracy.