The Cuf Based fem2.0 as the Second Generation Finite Element Methods and Software for Structural Mechanics

Current Finite Elements implementations, herein referred to as FEM1.0, including those available in commercial software, are characterized by a fixed/limited number of degrees of freedom (DOF) per node. Normally, these are ‘six’ for structural elements (beams, plates, shells) and ‘three’ for 3D (solids) ones. These constraints could lead to severe limitations in solving ‘localized’ stress fields, laminated composite and/or metallic structures, electromechanical problems, and structures subjected to multifield loadings. Such assumptions originate from the well-known pioneering hypotheses by Euler-Bernoulli/Timoshenko as well as by Kirchhoff and Reissner-Mindlin, for beams and plates/shells, respectively. In recent years, the authors have successfully introduced and extended the Carrera Unified Formulation (CUF), a hierarchical framework for developing structural theories for beams, plates, and shells, including laminated structures and multifield loadings. These have been extended and applied to various linear and nonlinear problems, achieving excellent and unique levels of accuracy. Finite Element (FE) applications based on CUF have been extensively developed and are herein referred to as Second-Generation Finite Element Methods, e.g., FEM2.0. This work describes some of the most interesting problems solved by FEM2.0: accurate stress and vibration response of laminated beams, plates, and shells; buckling and post-buckling of thin-walled structures; plasticity, progressive failure in laminates, and low-velocity impact. It shows that such accuracy could otherwise be achieved only through solid-3D Finite Elements in commercial software. Nevertheless, the computational costs of full 3D analysis could become prohibitive due to the well-known aspect-ratio constraints. FEM 2.0 would further lead to Finite Elements in which the number of degrees of freedom per node can vary from element to element. This is the Node Dependent Kinematic (NDK) version of the FEM. In other words, each node can refer to a different structural theory, and the FE matrices are weighted not only by classical shape functions but also with respect to structural theory. In particular, the possibility of applying NDK to global-local problems without the need to use transition elements and/or penalty procedures will be highlighted. The advantages of NDK-FEM2.0 over traditional FEM1.0, in terms of both accuracy and computational cost reduction, will be clearly demonstrated. The lecture further describes a later development: the X-DOF capabilities of the FEM 2.0 method to develop structural theories that can exhibit ‘any’ X-assumptions for each displacement component (i.e., DOF). Applications to beams, plates, and shells made of laminated composites are presented. It is concluded that X-DOF combined with NDK provides the most natural and effective way to describe deformation at each point/direction of the structure, leading to the most successful and effective method for building the ‘best’ computational model for a given problem related to composite structures.