Local/Global model order reduction techniques for fracture

The idea of local/global model order reduction is to apply reduced basis techniques in the region of the domain where they are valid. Elsewhere, direct numerical simulation is performed.

  

In the case of quasi-brittle fracture, reduced spaces are unable to represent the solution in the process zone. The successive damage states need to be represented accurately, and are unfortunately not correlated to one another.

The process zones are extracted by an empirical procedure (a zone of predefined size around the points of the domain undergoing maximum damage rate). A reduced model of the complementary domain is built by a projection of the initial model in a reduced space spanned by a few snapshot-POD basis vectors (figure above). In the process zone itself, the reduced space is not completely discarded, as it contains valuable information about the solution. Instead, it is used as a coarse-grid preconditioner for an iterative algorithm to the direct solution. The iterative algorithm coupling the two solvers is illustrated in the figure below

We are currently investigating an algebraic extraction of the process zone, such that problem-dependence is avoided.