Advanced methods using stochastic modeling in high dimension for uncertainty modeling, quantification and propagation in computational mechanics of solids and fluids. – TYCHE

The objectives of all development efforts in computational mechanics of solids and fluids for 20 years by the international community, aim at improving predictions of numerical simulations to better reflect the physical reality and to make more robust predictions with respect to uncertainties and to the complexity of the systems modeled.

This research is part of the challenge of developing new methodologies and advanced tools for stochastic modeling, for the quantification and for the propagation of uncertainties in computational mechanics of solids and fluids for a single scale or for a multi-scale modeling.

Today, some of these issues are addressed and some less, and there are many results, formulations and methodologies available. However, these available techniques essentially apply to low dimensional stochastic modeling and they cannot easily be extended to the high stochastic dimension case. Moreover, there is no constructive method today to address stochastic modeling and its identification in high dimension.

This project addresses key questions, unresolved to this day, regarded by the international community as challenging problems which must be solved, for a single or for two coupled scales of stochastic modeling, and which are:

(1) The parametric stochastic modeling in high dimension of mechanical system parameters in solids and fluids mechanics, their quantification and their identification by solving inverse stochastic boundary value problems.
(2) The nonparametric stochastic modeling of model errors related to the construction of a reduced-order model of the computational mechanical model in presence of a parametric stochastic modeling in high dimension of the computational model parameters, their quantification and their intrinsic identification performed by solving inverse stochastic boundary value problems.
(3) Methods and formulations to analyze the propagation of high dimensional parametric stochastic models through linear and nonlinear computational models in fluid and solid mechanics, with or without considering a nonparametric modeling of model errors.
(4) Methodologies and formulations for stochastic multi-scale modeling corresponding to the coupling of two scales with stochastic models in high dimension for linear and nonlinear solid mechanics.

The application fields covered in this project are representative of the major challenges facing today in modeling and multi-scale simulation of complex systems in computational fluid and solid mechanics. The methods developed in this project will allow the handling of problems that are currently regarded as extremely difficult or even unreachable. The tools and methods developed in this project could be applicable to the numerical simulations conducted in the industry to optimize the design and performance (1) for automative vehicles of the future, (2) for aerospace and space technologies, (3) for complex systems of energy production, particularly concerning the seismic risk assessment on nuclear power plant, (4) for composites with complex microstructures, including living tissues, (5) for microsystems and even nanosystems, (6) for fluid dynamics, for solid mechanics and also for fluid-solid coupling system where multi-scale and multi-physics modeling are most often required.

The proposed project represents an important gap in the knowledge and methodologies in this domain and is not a simple incremental contributions from previous contributions. It corresponds to a true technological breakthrough with respect to the capabilities of current methods to address numerical simulations of complex mechanical systems in presence of uncertainties and random media. The project explores new directions and methodologies related to unresolved issues of high dimensional stochastic modeling in computational mechanics.