The statistical finite element method (statFEM) for coherent synthesis of observation data and model predictions (1905.06391v3)

Abstract: The increased availability of observation data from engineering systems in operation poses the question of how to incorporate this data into finite element models. To this end, we propose a novel statistical construction of the finite element method that provides the means of synthesising measurement data and finite element models. The Bayesian statistical framework is adopted to treat all the uncertainties present in the data, the mathematical model and its finite element discretisation. From the outset, we postulate a data-generating model which additively decomposes data into a finite element, a model misspecification and a noise component. Each of the components may be uncertain and is considered as a random variable with a respective prior probability density. The prior of the finite element component is given by a conventional stochastic forward problem. The prior probabilities of the model misspecification and measurement noise, without loss of generality, are assumed to have zero-mean and known covariance structure. Our proposed statistical model is hierarchical in the sense that each of the three random components may depend on non-observable random hyperparameters. Because of the hierarchical structure of the statistical model, Bayes rule is applied on three different levels in turn to infer the posterior densities of the three random components and hyperparameters. On level one, we determine the posterior densities of the finite element component and the true system response using the prior finite element density given by the forward problem and the data likelihood. On the next level, we infer the hyperparameter posterior densities from their respective priors and the marginal likelihood of the first inference problem. Finally, on level three we use Bayes rule to choose the most suitable finite element model in light of the observed data by computing the model posteriors.

• The paper introduces statFEM, a Bayesian framework that integrates observation data with finite element models to address uncertainties in engineering systems.
• It details a hierarchical Bayesian approach that decomposes model predictions, mis-specification, and measurement noise, improving calibration and model comparison.
• Numerical experiments in 1D and 2D validate the method's convergence and efficiency, highlighting its potential for industrial applications.

Statistical Finite Element Method for Synthesis of Observation Data and Model Predictions

The paper proposes a novel approach, termed the statistical finite element method (statFEM), which integrates observation data with finite element models using Bayesian inference. This method provides a probabilistic framework, leveraging uncertainties present in the data, the mathematical model, and its discretization through finite elements. Herein, key aspects and practical implementations of this approach are discussed.

Overview of statFEM

Motivation and Background

Engineering systems often involve uncertainties owing to variations in material properties, geometry, loading conditions, and other parameters. Traditional deterministic models, despite the use of safety factors, may significantly differ from actual system responses. With the advent of sensor technologies, measurement data can be continually collected from systems, providing an opportunity to refine model predictions by integrating these observations.

Core Methodology

StatFEM begins by decomposing data into three components: the finite element model prediction, model misspecification, and measurement noise. Adopting a hierarchical Bayesian framework, each component is treated as a random variable with specified prior distributions.

1. Finite Element Component: Standard probabilistic methods solve stochastic PDEs, treating the finite element solution with a Gaussian prior, estimating mean and covariance.
2. Model Misspecification: Decomposes discrepancies between real and modeled systems using zero-mean Gaussian processes with predefined covariance structures.
3. Measurement Noise: Assumed to be white Gaussian noise, representing the observation errors.

Bayes' theorem iteratively updates the posterior distributions, incorporating observed data to refine model predictions.

Practical Implementation

Finite Element Discretization

The finite element method discretizes the domain into elements, solving weak forms of governing equations. This paper describes implementing stochastic PDEs under Gaussian assumptions, approximating random fields with perturbation methods.

Hierarchical Inference

Using Bayes rule at multiple hierarchical levels allows for the efficient computation of posterior densities for finite element solutions, true systems, and hyperparameters involved. Initially, predictions start from prior densities and subsequently iterate through levels to utilize observation data.

Numerical Examples

The authors demonstrate statFEM on various 1D and 2D problems. These examples validate the convergence of probabilistic predictions towards actual data with increasing observations and highlight the efficiency of Bayesian methods in calibration and model comparison.

Computational Considerations

Efficient algorithmic strategies, such as perturbation techniques and MCMC sampling, are employed to manage high-dimensional problems. The methods discussed circumvent the need for exhaustive sampling, optimizing computational resources while ensuring robust predictions.

Implications and Future Directions

Theoretical and Practical Significance

StatFEM contributes significantly to uncertainty quantification in engineering modeling. By coherently integrating observational data with deterministic models, it enhances predictive capabilities while accommodating inherent uncertainties.

Potential Enhancements

The research prompts further exploration into alternative covariance kernels for model misspecification. Moreover, extending statFEM to time-dependent or highly nonlinear systems could offer advanced predictive insights.

Industrial Applications

The proposed method is applicable across various domains such as structural health monitoring, materials engineering, and reliability analysis. As sensor technology and computation power evolve, the use of statFEM can be extended, offering real-time model updates and predictive analytics.

Conclusion

The statistical finite element method provides a comprehensive framework to merge data observations with mesh-based predictions under uncertainty. By adopting Bayesian perspectives, statFEM enhances predictive modeling fidelity, paving the way for integrated data-centric engineering applications.