Hierarchical theories of structures based on Legendre polynomial expansions with finite element applications

Abstract

This paper discusses classical and refined beam and plate theories based on the Carrera Unified Formulation (CUF). Attention is focussed on (but not limited to) a new refined beam element with enhanced kinematics based on Legendre polynomial expansions of the primary mechanical variables. By employing CUF, the governing equations and the related finite element arrays are written in a hierarchical, compact and general manner. Readily, these characteristics are used to arbitrarily tune the finite element model at the cross-sectional level, by locally enriching the theory kinematics up to the desired accuracy. The uncompromising accuracy of the present beam model is demonstrated by considering various numerical examples, including solid and thin-walled beams with open and close cross-sections as well as plate structures. The results are compared with those from classical and already established refined CUF models. Eventually, three-dimensional elasticity solutions by the commercial tool MSC Nastran are also given to underline the high accuracy of the present methodology. The numerical efficiency and the capabilities of the Legendre-based CUF beam models to deal with complex structures with no geometrical approximations result clear from the analyses conducted.

Cross-sectional mapping for refined beam elements with applications to shell-like structures

“…This class of CUF beam theories was devised in Carrera et al [42] by adopting hierarchical series of Legendre-type polynomials to expand the generalized displacement variables over the cross-section. Hierarchical Legendre Expansion (HLE) models combine the main features of the previous CUF beam models, i.e.…”

“…where φ p corresponds to the one-dimensional internal Legendre-type modes, see [24,42] and Eq. (31).…”

“…as defined in [24,42]. The set of polynomials is defined in a hierarchical manner and, as a consequence, the polynomial order of the element can be increased straightforwardly by adding internal shape functions of higher orders.…”

Micromechanics Modeling of Unit Cells Using CUF Beam Models and the Mechanics of Structure Genome

“…fiber reinforced composites, can be modeled by means of refined beam models in which the different phases are represented by non-local expansions of the cross-sectional coordinates. The properties of the Hierarchical Legendre Expansion (HLE) [9] are exploited to generate hierarchical models of the UC in which the exact geometry of the constituents is introduced in the model by a non-isoparametric mapping technique and the accuracy of the model is controlled by the polynomial order of the theory.…”

Component-Wise Method for Macro-Meso-Micro Modeling and Failure Analysis of Composite Structures

“…The classical finite element, 1D approach is used along one direction. On the other hand, the unknown variables are expanded over the remaining two local coordinates via Legendre polynomials, referred to as Hierarchical Legendre Expansions (HLE) [3]. Also, non-local expansion domains with curved boundaries are defined to capture the exact shape of the constituents independently of the refinement of the model.…”