Thesis Summary:

In this thesis we develop new mixed finite element methods for the modelling of a class of nonlinear problems in incompressible elasticity on Lipschitz domains in the plane. We consider the following model problems: • an exterior transmission problem, defined by the coupling of a certain nonlinear incompressible elastic material in a bounded domain with a linear incompressible elastic material in the unbounded exterior domain. • a boundary value problem with mixed boundary conditions, defined by a nonlinear incompressible elastic material on a bounded domain. For the analysis of the transmission problem we use the Dirichlet-to-Neumann method, which consists in transforming the exterior problem in a boundary value problem on a bounded domain, by introducing a suitable artificial boundary on which the Neumann data is defined in terms of the Dirichlet data. This function, named Dirichlet-to-Neumann (DtN) mapping, is an exact nonlocal artificial boundary condition that is expressed in terms of an infinite Fourier series. This approach allows us to define a mixed variational formulation in which the displacement and the hydrostatic pressure are the unknowns. For the second problem, the variational approach is based on the Hu-Washizu principle, which is characterized by the fact that, besides the displacement and the stress, it includes the deformation as a third unknown. In addition, the symmetry of the stress tensor is imposed weakly, and the trace of the displacement on the Neumann boundary is also incorporated as an additional unknow, which finally yields a two-fold saddle point operator equation as the resulting variational formulation. We prove that the associated Galerkin schemes of both mixed formulations are well-posed, provide the corresponding optimal rates of convergence, and derive reliable a-posteriori estimates for the adaptive computation of the respective discrete solutions. Finally, several numerical results illustrating the good performance of the adaptive algorithm for the two-fold saddle point Galerkin scheme are presented.

ISI Publications from the Thesis

Gabriel N. GATICA, Luis F. GATICA, Ernst P. STEPHAN: A dual-mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions. Computer Methods in Applied Mechanics and Engineering, vol. 196, pp. 3348–3369, (2007)

Gabriel N. GATICA, Luis F. GATICA: On the a-priori and a-posteriori error analysis of a two-fold saddle point approach for nonlinear incompressible elasticity. International Journal for Numerical Methods in Engineering, vol. 68, 8, pp. 861-892, (2006)

Gabriel N. GATICA, Luis F. GATICA, Ernst P. STEPHAN: A FEM-DtN formulation for a nonlinear exterior problem in incompressible elasticity. Mathematical Methods in the Applied Sciences, vol. 26, 2, pp. 151-170, (2003)

Other Publications (ISI)

Tomás BARRIOS, Gabriel N. GATICA, Luis F. GATICA: On the numerical analysis of a nonlinear elliptic problem via mixed-FEM and Lagrange multipliers. Applied Numerical Mathematics, vol. 48, 2, pp. 135-155, (2004)