Extraction of low-dimensional subspaces for the representation of unknown fields in stochastic fracture mechanics.

Our previous research has shown that Model Order reduction could help alleviate the computational burden associated with the simulation of propagating damage in composite structures. However, the region where damage localises (i.e. crack tip) must be treated using a full order model (i.e. no reduction). This is not necessarily a huge limitation. Indeed, damaged regions are very localised in a complex engineering system (e.g. vicinity of joints, holes or other stress concentrators).

The key point of this research is to find the region that must be excluded from the reduction. While our previous investigations used empirically defined damaged region, basically spheres on the materials points exhibiting maximum energy dissipation), we propose here an algebraic interpretation of the process zone: this will be defined as a region of minimum measure that needs to be excluded from the reduction technique in order to obtain approximate numerical predictions up to a given level of accuracy. Typically, the process zone becomes the region where the mechanical state changes rapidly with respect to parameter variations (i.e. chaotic behaviour).

We developed an algorithm to extract the algebraic process zone, which we called progressive Proper orthogonal decomposition. In essence, this is a Greedy algorithm that automatically finds a domain of given size over which a Singular Value Decomposition converges optimally (i.e. maximum correlation in the data). The size of this domain, together with the order of truncation of the SVD, is then selected in order to maximise the computational speed-up.

The algorithm has been applied successfully to the prediction of propagating damage in random media.