Predicting failure using conditioning on damage history: demonstration on percolation and hierarchical fiber bundles

We formulate the problem of probabilistic predictions of global failure in the simplest possible model based on site percolation and on one of the simplest models of time-dependent rupture, a hierarchical fiber bundle model. We show that conditioning the predictions on the knowledge of the current degree of damage (occupancy density p or number and size of cracks) and on some information on the largest cluster improves significantly the prediction accuracy, in particular by allowing one to identify those realizations which have anomalously low or large clusters (cracks). We quantify the prediction gains using two measures, the relative specific information gain (which is the variation of entropy obtained by adding new information) and the root mean square of the prediction errors over a large ensemble of realizations. The bulk of our simulations have been obtained with the two-dimensional site percolation model on a lattice of size L x L=20 x 20 and hold true for other lattice sizes. For the hierarchical fiber bundle model, conditioning the measures of damage on the information of the location and size of the largest crack extends significantly the critical region and the prediction skills. These examples illustrate how ongoing damage can be used as a revelation of both the realization-dependent preexisting heterogeneity and the damage scenario undertaken by each specific sample.