Michele D’Ottavio

Hierarchical Beam Finite Elements Based Upon a Variables Separation Method

A family of hierarchical one-dimensional beam finite elements developed within a variables separation framework is presented. A Proper Generalized Decomposition (PGD) is used to divide the global three-dimensional problem into two coupled ones: one defined on the cross-section space (beam modeling kinematic approximation) and one belonging to the axis space (finite element solution). The displacements over the cross-section are approximated via a Unified Formulation (UF). A Lagrangian approximation is used along the beam axis. The resulting problems size is smaller than that of the classical equivalent finite element solution. The approach is, then, particularly attractive for higher-order beam models and refined axial meshes. The numerical investigations show that the proposed method yields accurate yet computationally affordable three-dimensional displacement and stress fields solutions.

The Ritz – Sublaminate Generalized Unified Formulation approach for piezoelectric composite plates

ABSTRACT This paper extends to composite plates including piezoelectric plies the variable kinematics plate modeling approach called Sublaminate Generalized Unified Formulation (SGUF). Two-dimensional plate equations are obtained upon defining a priori the through-thickness distribution of the displacement field and electric potential. According to SGUF, independent approximations can be adopted for the four components of these generalized displacements: an Equivalent Single Layer (ESL) or Layer-Wise (LW) description over an arbitrary group of plies constituting the composite plate (the sublaminate) and the polynomial order employed in each sublaminate. The solution of the two-dimensional equations is sought in weak form by means of a Ritz method. In this work, boundary functions are used in conjunction with the domain approximation expressed by an orthogonal basis spanned by Legendre polynomials. The proposed computational tool is capable to represent electroded surfaces with equipotentiality conditions. Free-vibration problems as well as static problems involving actuator and sensor configurations are addressed. Two case studies are presented, which demonstrate the high accuracy of the proposed Ritz-SGUF approach. A model assessment is proposed for showcasing to which extent the SGUF approach allows a reduction of the number of unknowns with a controlled impact on the accuracy of the result.

Robust Displacement and Mixed CUF-Based Four-Node and Eight-Node Quadrilateral Plate Elements

This paper presents two classes of new four-node and eight-node quadrilateral finite elements for composite plates. Variable kinematics plate models are formulated in the framework of Carrera’s Unified Formulation, which encompass Equivalent Single Layer as well as Layer-Wise models, with the variables that are defined by polynomials up to 4th order along the thickness direction z. The two classes refer to two variational formulations that are employed to derive the finite elements matrices, namely the Principle of Virtual Displacement (PVD) and Reissner’s Mixed Variational Theorem (RMVT). For the PVD based elements, the main novelty consists in the extension of two field compatible approximations for the transverse shear strain field, referred to as QC4 and CL8 interpolations, which eliminate the shear locking pathology by constraining only the ɀ–constant transverse shear strain terms, to all variable kinematics plate elements. Moreover, for the first time the QC4 and CL8 interpolations are introduced for the transverse shear stress field within RMVT based elements. Preliminary numerical studies are proposed on homogeneous isotropic plates that demonstrate the absence of spurious modes and of locking problems as well as the enhanced robustness with respect to distorted element shapes. The new QC4 and CL8 variable kinematics plate elements display excellent convergence rates and yield accurate responses for both, thick and thin plates.