Verification and Validation under Uncertainty Applied to Finite Element Models of Historic Masonry Monuments

The goal of this paper is to take a step towards illustrating the application of tools collectively referred to as Verification and Verification (V&V) to improve the predictive accuracy of large-scale Finite Element (FE) simulations of the National Cathedral, D.C. For this purpose, we first complete a series of code and solution verification studies. "Code verification" quantifies errors due to a potentially deficient implementation of mathematical models in a computer code. "Solution verification" quantifies the numerical error introduced by discretizing mathematical models on a computational mesh. Next, we integrate large amounts of experimental and computational information collected from the choir vaults of the Cathedral. Measurement uncertainty is assessed from the replicated experiments. In parallel, a design of computer experiments is analyzed by perturbing model parameters, which enables us to explore variability of the model parameter domain. Both from the physical measurements and computer experiments, comparative features are extracted probabilistically as mean and variance statistics. Using a Phenomenon Identification and Ranking Table (PIRT), the uncertain parameters that are candidates for calibration are ranked based on the sensitivity of test- analysis comparative features. Parameters whose influence on the output of interest is not significant are disqualified from being calibrated. Once the comparative features and calibration parameters are defined, Bayesian inference is used to compound our prior knowledge about the calibration parameters together with experimental observations collected from vibration testing. Prior probability distribution incorporates expert judgment while the variance of measured features account for the experimental uncertainty. Bayesian inference, then, results in updated knowledge of the calibration parameters in the form of a posterior probability distribution. The point of this exercise is not to obtain a calibrated FE model that identically matches the experimental data. It is, rather, to better understand where modeling uncertainty originates from and to obtain model predictions that are statistically consistent with the measurements and their uncertainty. The complete V&V "story" is told of how to establish confidence in the model of an existing structure.