Optimal physical experimental designs to calibrate functional Weibull models under cost constraints. Application to a structural reliability problem
Résumé
We focus on the specific issue of identifying (static) optimal physical experimental designs when the aim is to calibrate functional Weibull models in an informed context and under cost constraints. In this nonlinear parametric context, a major difficulty is that the exact evaluation of the expected utility function to maximize is intractable. Among the proposed approaches to overcome this difficulty, one can distinguish near optimal Bayesian experimental designs defined by maximizing either an analytical approximation or a numerical estimation of the expected utility. These latest approaches are based on quadratic techniques or simulations produced by stochastic algorithms. They remain computationally intensive, which makes them often limited in practice to very simple designs. In this work, we explore the potential of the Bayesian approach to account for legacy data as a prior informative sample when designing physical experiments. We propose generic design criteria based on a weighted combination of utility functions to balance the statistical gain yielded by a given design and the prospective experimental cost, in an understandable way for the engineer. We consider different utility functions to quantify the statistical gain yielded by a given design when fitting Bayesian functional Weibull models. Finally, we derive analytical asymptotic approximations of the associated expected utility functions that we maximize with a simulated annealing algorithm. Our method is applied to a structural reliability problem informed by a real dataset of U.S. ferric steels. The design problem consists in selecting an optimal number of temperature levels, optimal values of temperature levels and optimal numbers of Charpy destructive impact tests to allocate to each temperature level, in order to produce the most precise estimates of the parameters defining the functional Weibull distribution of steel fracture toughness for a French production park under cost constraints.