My main area of interest is in the variational approach to fracture mechanics, and in particular, to its application in thin-film fracture. As an inherit requirement for very fine discretization, I have become increasingly interested in high performance computing.
Thin-film fracture represents some of the most challenging aspects of theoretical mechanics and applied mathematics. Since resolving the film in vertical direction (through thickness) is computationally prohibitive, reduced dimensional modeling becomes imperative. Theoretical mechanics aspires to offer a sound foundation to derive and develop such models. During my Ph.D. studies I have developed such a model through asymptotic analysis of 3-D elasticity. In this particular model, the out-of-plane and in-plane parts of the displacement field decouple.
Variational fracture mechanics, through its need for large scale simulations, requires state of the art computational techniques. In general, the condition numbers for matrices representing the systems for these fine discretizations worsen as fracture develops. Moreover, the non-linearity and in-convexity of the systems resulting from the variational approach to fracture mechanics necessitates many alternate minimization steps and inflates the computational cost. Furthermore, thin-film fracture introduces a 4th order PDE that not only is inherently more costly to compute, but is also more prone to instabilities and spurious modes as fracture develops and propagates in the film.