A new methodology for efficient reduced basis reliability based design – ReBReD

To achieve this goal, we proposed a new method using numerical models whose fidelity is automatically adjusted according to the needs of the reliability analysis. This is achieved through reduced models built by projection onto a base constructed on the fly during simulation calls. This reduced base is enriched during simulations, thus adjusting its fidelity to the needs of the reliability analysis. Furthermore, we defined a new enrichment criterion for Kriging substitution models that will be able to replace numerical models during reliability analysis. This new criterion makes it possible to take into account both the uncertainty related to the substitution model and the uncertainty related to the limited number of samples to decide what the next enrichment point should be. These methods have enabled a very significant reduction in computational times. For example, on a reliability analysis problem on a rocket engine, a reduction of an order of magnitude could be obtained by using the developed methods.

To achieve this goal, a new method was proposed, using numerical models whose fidelity is automatically adjusted according to the needs of the reliability analysis. This is achieved through reduced models built by projection onto a base constructed on the fly during simulation calls. This reduced base is enriched during simulations, thus adjusting its fidelity to the needs of the reliability analysis. Furthermore, a new enrichment criterion was defined for Kriging substitution models that will be able to replace numerical models during reliability analysis. This new criterion makes it possible to take into account both the uncertainty related to the substitution model and the uncertainty related to the limited number of samples to decide what the next enrichment point should be. These methods have enabled a very significant reduction in computational times. For example, on a reliability analysis problem on a rocket engine, a reduction of an order of magnitude could be obtained by using the developed methods. The methodological developments carried out in this project also made it possible to initiate, after the end of the project, a collaboration with Airbus on the semi-probabilistic dimensioning of aeronautical structures with funding for a CIFRE thesis.

The coupling between model reduction and reliability optimization was investigated using the ReBRBDO method. This method is the one on which we were able to spend the least amount of time. Indeed, the development of the coupling criterion of the VbAGP method took longer than expected, so we were not able to devote as much time to the ReBRBDO method as initially planned. This method would therefore merit further development in order to reach its full potential.

Menz, M.; Dubreuil, S.; Morio, J.; Gogu, C.; Bartoli, N.; et al. Variance based sensitivity analysis for Monte Carlo and importance sampling reliability assessment with Gaussian processes. Structural Safety. 2021, 93, 102116.

Menz, M.; Gogu, C.; Dubreuil, S.; Bartoli, N.; Morio, J. Adaptive coupling of reduced basis modeling and Kriging based active learning methods for reliability analyses. 2020.

Fernandez-Godino, M. G.; Dubreuil, S.; Bartoli, N.; Gogu, C.; Balachandar, S.; et al. Linear regression-based multifidelity surrogate for disturbance amplification in multiphase explosion. Structural and Multidisciplinary Optimization. 2019, 1-16.

Workshop organized by the project leader in the form of the «Robust and Reliable Design Day« which took place on April 2, 2019 in Toulouse. This workshop brought together around sixty people from both academic and industrial backgrounds and gave rise to 13 scientific presentations.

Over the past decades progress in modeling, parallel computing and available computational resources made possible the analysis of complex, large scale models of increasingly detailed structures. On a different front, new methodological developments for reliability analysis and reliability based design allowed significant improvements in the efficiency of these approaches. In particular, recent developments in so called active learning reliability analysis approaches have proved to be very efficient for problems with a moderate number of random variables. Active learning (also known as adaptive sampling) approaches consist in constructing a kriging surrogate model (or metamodel) for calculating the reliability constraints and adaptively enriching the surrogate model based on the kriging uncertainty structure.
In spite of these recent developments, reliability analyses involving large scale structural models may still pose computational cost issues because the expensive simulations still need to be carried out many times. To address the computational cost issue of running one expensive simulation, reduced order modeling has been proposed in the past and has recently regained a lot of interest. We will consider here reduced order modeling by projection, also known as reduced basis modeling, which consists in solving the large scale system projected on an appropriately defined basis. Drastic reductions by many orders of magnitude are thus achieved in the size of the problem to be solved.
The objective of the proposed project is to develop a new methodology for efficient reliability analysis of large scale structures. The novel approach resides in defining an interaction between adaptive sampling and reduced order modeling by projection, by adaptively enriching the kriging metamodel using a reduced order model tuned to have at each step the appropriate fidelity based on the accuracy requirements of the reliability analysis. These accuracy requirements are derived from the kriging uncertainty structure, thus guiding the level of fidelity of the reduced basis model. Far from the limit state leading to failure a very low fidelity (but very cost efficient) reduced basis model may be sufficient. As one approaches the limit state, the fidelity of the reduced basis model will be automatically increased based on the accuracy requirements precisely at the current sampling point. Such a combined approach can thus be seen as a tunable fidelity approach since it tunes the fidelity of the reduced order model to the requirements of the current step of the adaptive reliability analysis. This new methodology is expected to lead to a reduction by several orders of magnitude in the computational time of reliability analyses for large scale structures compared to current state of the art methods. It thus has the potential to be transformative for industry practice, by allowing to undertake reliability based design where it would have not been practical before.
The project will first investigate the most appropriate coupling criterion between adaptive sampling and reduced basis modeling, then, based on this coupling, develop and implement new reliability analysis and reliability based design optimization methodologies. Finally, the developed approaches will be applied to two structural mechanics application problems within the aerospace domain. The first one concerns a classical aeronautical certification test on a composite open-hole laminate. The second problem involves a wingbox structural model, which has all the ingredients of large scale structural models and aims at demonstrating the computational savings potential of the proposed approach.