Mechanical State Estimation with a Polynomial-Chaos-Based Statistical Finite Element Method

2024-12-06

arXiv.org

AI Summary

This work addresses the computational inefficiency of statistical finite element methods (statFEM) for Bayesian state estimation under non-conjugate, non-Gaussian priors. We propose a sampling-free, non-conjugate statFEM that integrates polynomial chaos expansion with Smolyak sparse grid projection. Crucially, we establish the first coupling of sparse grid projection with Gaussian–Markov–Kalman filtering, enabling unified treatment of prior non-Gaussianity, model misspecification, and measurement noise. Hyperparameters are jointly optimized via marginal likelihood maximization. Numerical validation on 1D and 2D linear elastostatic problems demonstrates high accuracy, rapid convergence, and superior computational efficiency compared to conventional sampling-based statFEM. The method provides a scalable uncertainty quantification framework for structural health monitoring and digital twin applications.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

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• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

The Statistical Finite Element Method (statFEM) offers a Bayesian framework for integrating computational models with observational data, thus providing improved predictions for structural health monitoring and digital twinning. This paper presents an efficient sampling-free statFEM tailored for non-conjugate, non-Gaussian prior probability densities. We assume that constitutive parameters, modeled as weakly stationary random fields, are the primary source of uncertainty and approximate them using Karhunen-Lo`eve (KL) expansion. The resulting stochastic solution field, i.e., the displacement field, is a non-stationary, non-Gaussian random field, which we approximate via Polynomial Chaos (PC) expansion. The PC coefficients are determined through projection using Smolyak sparse grids. Additionally, we model the measurement noise as a stationary Gaussian random field and the model misspecification as a mean-free, non-stationary Gaussian random field, which is also approximated using KL expansion. The coefficients of the KL expansion are treated as hyperparameters. The PC coefficients of the stochastic posterior displacement field are computed using the Gauss-Markov-K'alm'an filter, while the hyperparameters are determined by maximizing the marginal likelihood. We demonstrate the efficiency and convergence of the proposed method through one- and two-dimensional elastostatic problems.

Problem

• Estimates mechanical states using Bayesian statFEM for structural monitoring
• Handles non-Gaussian uncertainties via Polynomial Chaos expansion
• Optimizes hyperparameters via marginal likelihood maximization

Innovation

• Sampling-free statFEM for non-Gaussian priors
• Polynomial Chaos expansion for displacement fields
• Gauss-Markov-Kálmán filter for posterior estimation