Undergraduate Thesis of Walter Rudolph
|Career||Mathematical Civil Engineering, Universidad de Concepción|
|Thesis Title||A priori and a posteriori Error Analysis of Increased Double Seat Point Formulations for Nonlinear Elasticity Problems|
In this report new methods of augmented mixed finite elements are introduced and analyzed for a class of equations of nonlinear elasticity that arise in hyperelasticity. The original mixed method is based on the incorporation of the strain tensor as an auxiliary variable, so that, together with the usual variables of stress, displacement and rotation used in linear elasticity, a variational formulation is given in the form of a Nonlinear equation of double chair point operators. Firstly, known results on the existence, uniqueness and stability of the Galerkin scheme associated with PEERS of order k = 0 to the case k? 1. The Galerkin schemes, both partial and total, are obtained by adding the consistent terms arising from the constitutive equation, the equilibrium equation, and the relations that define the rotation as a function of the displacement, and the tensor of Deformations as an independent variable, all multiplied by stabilization constants chosen conveniently. We apply the classical results of the analysis of nonlinear double-saddle point schemes and of equations with strongly monotonous operators to prove that the continuous and discrete augmented schemes are well proposed. In particular, it is shown that the partially enlarged Galerkin scheme is well defined with any finite element subspace for the strain tensor, and with the PEERS space of order k? Ge 0 for all other unknowns, whereas any subspace of finite elements that approximates all unknowns can be used for the case of the completely increased scheme. Residual a posteriori error estimators are then deduced for each of the schemes, and it is proved that all of them are reliable and efficient. Finally, several numerical examples are provided which illustrate the good performance of the resulting mixed finite element methods, confirm the theoretical properties of the estimators, and show the behavior of the associated adaptive algorithms.
|Thesis Director(s)||Gabriel N. Gatica|
|Thesis Project Approval Date||2010, October 05|
|Thesis Defense Date||2012, August 24|
|Professional Monitoring||Professor Instructor,Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción.|
|PDF Tesis||Download Thesis PDF|
ISI Publications from the Thesis
Gabriel N. GATICA, Antonio MARQUEZ, Walter RUDOLPH: A priori and a posteriori error analyses of augmented twofold saddle point formulations for nonlinear elasticity problems. Computer Methods in Applied Mechanics and Engineering, vol. 264, 1, pp. 23-48, (2013).