Fracture Network Localization Preceding Catastrophic Failure in Triaxial Compression Experiments on Rocks

We quantify the spatial distribution of fracture networks throughout six in situ X-ray tomography triaxial compression experiments on crystalline rocks at confining stresses of 5–35 MPa in order to quantify how fracture development controls the final macroscopic failure of the rock, a process analogous to those that control geohazards such as earthquakes and landslides. Tracking the proportion of the cumulative volume of fractures with volumes >90th percentile to the total fracture volume, ∑v90/vtot indicates that the fracture networks tend to increase in localization toward these largest fractures for up to 80% of the applied differential stress. The evolution of this metric also matches the evolution of the Gini coefficient, which measures the deviation of a population from uniformity. These results are consistent with observations of localizing low magnitude seismicity before large earthquakes in southern California. In both this analysis and the present work, phases of delocalization interrupt the general increase in localization preceding catastrophic failure, indicating that delocalization does not necessarily indicate a reduction of seismic hazard. However, the proportion of the maximum fracture volume to the total fracture volume does not increase monotonically. Experiments with higher confining stress tend to experience greater localization. To further quantify localization, we compare the geometry of the largest fractures, with volumes >90th percentile, to the best fit plane through these fractures immediately preceding failure. The r2 scores and the mean distance of the fractures to the plane indicate greater localization in monzonite than in granite. The smaller mean mineral diameter and lower confining stress in the granite experiments may contribute to this result. Tracking these various metrics of localization reveals a close association between macroscopic yielding and the acceleration of fracture network localization. Near yielding, ∑v90/vtot and the Gini coefficient increase while the mean distance to the final failure plane decreases. Macroscopic yielding thus occurs when the rate of fracture network localization increases.

Distributed localization using Levenberg-Marquardt algorithm

In this paper, we propose a distributed algorithm for sensor network localization based on a maximum likelihood formulation. It relies on the Levenberg-Marquardt algorithm where the computations are distributed among different computational agents using message passing, or equivalently dynamic programming. The resulting algorithm provides a good localization accuracy, and it converges to the same solution as its centralized counterpart. Moreover, it requires fewer iterations and communications between computational agents as compared to first-order methods. The performance of the algorithm is demonstrated with extensive simulations in Julia in which it is shown that our method outperforms distributed methods that are based on approximate maximum likelihood formulations.