Numerical applications of free discontinuity problems to folding and fracture

The aim of the thesis is the application of new numerical methods based on the theory of free discontinuities of E. De Giorgi to challenging fields of relevant practical interest in engineering, such as folding of thin walled tubes and propagation of fracture in brittle solids. Both the mathematical models to describe folding and fracture are based on the minimization of two competing energy forms: a volume and a surface energy. The variational formulation leads to the minimization (under displacement boundary conditions) of a functional F(K;u), where K is the set of discontinuous points of u. In order to perform the numerical search for the minimum of F, a powerful open code (developed by K. Brakke) named Surface Evolver has been adapted according to the purposes of the research at hand. Since the energy is non-convex the solution obtained through the algorithm is strongly dependent on the initial point. Starting from an initial faceted surface, the numerical code evolves the surface towards minimum energy through the nonlinear conjiugate gradient method.