Master's Dissertation
DOI
10.11606/D.18.2009.tde-21082009-101537
Document
Author
Full name
Maria do Socorro Martins Sampaio
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2009
Supervisor
Committee
Bravo Castillero, Julian
Coda, Humberto Breves
Title in Portuguese
O método de Galerkin descontínuo aplicado na investigação de um problema de elasticidade anisotrópica
Keywords in Portuguese
Método de Galerkin descontínuo
Método dos elementos finitos
Problema de equilíbrio
Abstract in Portuguese
Title in English
The discontinuous Galerkin method applied to the investigation of an anisotropic elasticity problem
Keywords in English
Anisotropic elasticity
Discontinuous Galerkin method
Equilibrium problem
Finite element method
Abstract in English
The equilibrium problem without body force of an anisotropic sphere under radial compression that is uniformly distributed on the sphere's boundary is investigated in the context of the classical linear elasticity theory. The solution of this problem predicts the unacceptable phenomenon of self-intersection in a vicinity of the center of the sphere for a given range of material parameters. This phenomenon can be eliminated in the context of a theory that minimizes the total potential energy of classical linear elasticity subjected to the restriction that the deformation field be injective. Two formulations of the Finite Element Method using Discontinuous Galerkin (MEFGD) are used to obtain approximate solutions for the unconstrained problem. The first formulation of the MEFGD approximates both the displacement and the strain fields. The consideration of the strain as an additional field in the formulation of the MEFGD increases the number of degrees of freedom associated to the finite elements and, therefore, the computational cost. With the objective of reducing the number of degrees of freedom, an alternative formulation of the MEFGD is introduced in this work. In this formulation, the strain field is not obtained directly from the inversion of the resulting linear system of equations, but from a post-processing calculation using the approximate displacement field. The approximate solutions obtained with both formulations of the MEFGD are compared with the exact solution of the problem without restriction and with approximate solutions obtained with the Finite Element Method using Classical Galerkin (MEFGC). Both formulations of the MEFGD yield better approximations for the exact solution than the approximations obtained with the MEFGC. The errors between the exact solution and the approximate solutions obtained with the alternative formulation of the MEFGD are slightly higher than the corresponding errors obtained with the original formulation of the MEFGD. These errors are compensated by the fact that the alternative formulation requires less computational effort than the computational effort required by the original formulation. This work serves as a basis for the study of problems with the injectivity restriction using the discontinuous Galerkin method.