Abstract:
This thesis is concerned with three-field mixed methods for elasticity (often referred to as Hu-Washizu formulations) in which the variables are, for small-strain problems, the displacement, stress and strain. For problems in nonlinear elasticity the corresponding variables are the displacement, first Piola-Kirchhoff stress, and deformation gradient. Of particular interest is the design and analysis of mixed formulations that are uniformly stable in the incompressible limit. The first part of the thesis deals with problems in linear elasticity. Lamichhane, Reddy and Wohlmuth (Numer. Math., 104 (2006)) have shown that the conditions for stability and uniform convergence include an ellipticity condition and, secondly, a condition that the displacement together with a discrete pressure, suitably defined, constitute a stable Stokes pair. The latter condition implies that the inf-sup condition for the three-field formulation is satisfied. In the thesis, families of new stable mixed elements are generated by the following approach. First, a stable Stokes pair is chosen. Then, the space of discrete stresses is defined such that the associated discrete pressure corresponds to that of the Stokes pressure. The space of strains is defined such that it forms a superset of the space of stresses. The final task is that of showing that the spaces chosen in this way satisfy the discrete ellipticity condition. A number of new families of mixed elements are designed and analyzed in this way, and numerical examples in two and three space dimensions are presented to illustrate the theory. The second part of the thesis comprises a short chapter in which the displacement-dilatation- pressure formulation of Taylor (Int. J. Numer. Meth. Engng, 47 (2000)) is shown to be a special case of the general three-field formulation, and is then shown to be uniformly convergent. The final part of the thesis is concerned with the extension of the earlier approach to problems of nonlinear elasticity. The problem considered is the incremental or linearized version, of the kind that forms part of a Newton-Raphson process in numerical implementations, with the unknown variables being the increments in displacement, first Piola-Kirchhoff stress, and deformation gradient. In the discrete formulation the elasticity tensor (that is, the second derivative of the strain energy with respect to deformation gradient) is approximated by its mean value on each element. Conditions are established for the resulting incremental formulation to be stable and uniformly convergent, assuming that the continuous problem is stable. The analysis is illustrated through selected numerical examples.
Reference:
Chama, A. 2013. Three-field mixed finite element approximations for problems in elasticity. University of Cape Town.
Includes bibliographical references.