To content
Department of Mechanical Engineering
Operator norm theory as a tool to efficiently bound failure probabilities under imprecise information

DFG FA 2004/2-1

Since structures are nowadays designed up to their performance limits, uncertainty quantification (UQ) is being increasingly applied in computer aided engineering to account for the fact that not all parameters of the computer models can be exactly quantified. However, criticism exists with respect to the pure virtual optimisation of functional components due to often large discrepancy of numerical model predictions with experimental data. This discrepancy originates from epistemic (lack of knowledge) and aleatory (inherently variable) uncertainty in the numerical model. These sources of non-determinism are abundantly encountered in almost all engineering fields. Therefore, the modelling should account for the joint occurrence of these aleatory and epistemic uncertainties to ensure model credibility, as well as guarantee design robustness and reliability, even when only limited information on the model is available.

The framework of imprecise probabilities allows for explicitly modelling aleatory and epistemic uncertainty into a single coherent mathematical framework. As such, it allows for effectively incorporating all sources of uncertainty explicitly into the modelling procedure. In this context, especially probability boxes are interesting since they provide an understandable and objective, yet rigorous description of this kind of uncertainty. In essence, a p-box considers a set of probabilistic models that is consistent with the description of the governing epistemic uncertainty. Based on this description, for instance, the probability of failure of a structure or component can be bounded, and the sensitivity with respect to the uncertainty can be assessed. However, the main drawback lies in the required computational cost since these kinds of uncertainties require typically a double-loop problem to be solved. Preliminary work of the applicant introduced a proof of principle of an intrusive method based on Operator Norm theory that can effectively reduce this computational cost with three orders of magnitude for linear dynamical models subjected to Gaussian-distributed uncertainties.

This project aims at generalizing this approach to more realistic engineering practices such as non-linear dynamical models, black-box models and models subjected to non-Gaussian loads. Hereto, the Operator Norm framework will be extended by (1) integrating appropriate statistical linearization and Volterra series expansion methods, (2) exploring alternative Karhunen-Loeve series expansion formulations and other polynomial representations of non-Gaussian loads and (3) resorting to non-intrusive model order reduction and operator inference methods. All developments will be properly benchmarked against state-of-the-art solvers for these kinds of problems on case studies ranging from academic single-degree-of-freedom oscillators to realistic engineering finite element models.