A Regularized Ensemble Kalman Filter for Stochastic Phase Field Models of Brittle Fracture

2026-03-10

AI Summary

This study addresses the significant impact of local parameter uncertainty—arising from material defects—on crack path prediction and residual strength assessment in phase-field models of brittle fracture. Moving beyond conventional Bayesian inversion that updates only model parameters, this work proposes a novel framework that directly infers the full system state, comprising both displacement and phase fields, by constructing a joint Bayesian prior over these fields. The ensemble Kalman filter (EnKF) is employed to assimilate sparse displacement observations for real-time state updating, augmented with a phase-field–based regularized proximal correction mechanism to enforce physical consistency with fracture mechanics principles. Numerical experiments in one and two dimensions demonstrate that the implicit crack evolution can be accurately reconstructed solely from displacement measurements, achieving high fidelity in both crack topology and propagation dynamics.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

The phase-field approach to brittle fracture provides a continuum framework for modeling crack initiation and propagation without explicit representation of discrete crack surfaces, provided the spatial discretization is fine enough to resolve the regularization length scale. However, uncertain local material parameters due to material defects can strongly influence simulation results, such as crack paths and remaining structural strength. At the same time, the ability to continuously monitor structures using sensors allows complementing modeling predictions with, e.g., displacement measurements. In this contribution, we connect these two complementary sources of information and present a Bayesian inference procedure that allows updating the current model state with incoming sensor data. We construct a Bayesian prior for the model state (both displacements and phase-field) and employ an ensemble Kalman filter (EnKF) to perform the update. In the EnKF, the update is computed by performing a Kalman shift on each ensemble member. Since the standard EnKF may produce assimilated states that violate common modeling assumptions, we present a phase field-based regularization technique as a proximal step correction toward model-consistent updates. 1D and 2D numerical examples demonstrate the performance and accuracy of the proposed method and show that the updated state matches the ground truth reasonably well. Unlike traditional Bayesian inversion techniques, which have already been applied to brittle fracture, we infer not the model parameters but the model state, i.e., the displacement field and the phase-field. Although only displacements are observed, the strong correlation between both fields also allows inference of the posterior phase-field.

Problem

• brittle fracture
• phase-field model
• uncertainty quantification
• state inference
• sensor data assimilation

Innovation

• ensemble Kalman filter
• phase-field fracture
• Bayesian state inference
• regularization
• data assimilation