Validated Continuation for a simple two-dimensional elastic solid

• Die Methode der ‚Validierten Pfadverfolgung‘ angewandt auf ein vereinfachtes Modell eines zweidimensionalen Festkörpers

Schulze, Friederike; Maier-Paape, Stanislaus (Thesis advisor); Healey, Timothy J. (Thesis advisor)

Aachen (2017)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2017

Abstract

In this thesis, we validate solutions of a fourth order boundary value problem arising from the study of microstructure in shape memory alloys using the computer-assisted ’Validated Continuation’ method described in Gameiro & Lessard (2010). The problem, as formulated in Healey&Miller (2007), considers a simplified two-dimensional model of an elastic solid undergoing forced anti-plane shear in the presence of interfacial energy.The resulting parameter-dependent differential equation can be subdivided into a fourth order linear part andtwo nonlinearities, one of which, for a particular loading parameter, vanishes. This poses an important special case for the equation.Using the reciprocal of the capillarity coefficient as continuation parameter, several paths of equilibrium solutions bifurcating from the trivial solution are followed. Each numerical solution is then validated by a constructive analytical proof yielding a so-called validation radius. Existence of this radius allows to conclude that in the vicinity of the approximated numerical zero of the finite dimensional approximated equation there has to be a truesolution of the partial differential equation. We achieve validation results for the special case for several numerical solutions at the beginning of different paths. However, due to computational limits, we can not pursue the paths to higher values of the continuation parameter, i.e. to solutions corresponding to a small capillarity coefficient and, thus, to small interfacial energy. We also analyze the difficulties of treating the general case with the same method and conclude with some ideas on different approaches for that problem.