Section: Overall Objectives

Overall Objectives

ACUMES aims at developing a rigorous framework for numerical simulations and optimal control in engineering, with focus on multi-scale, heterogeneous, unsteady phenomena subject to uncertainty. Starting from established macroscopic Partial Differential Equation (PDE) models, we pursue a set of innovative approaches to include small-scale phenomena, which impact the whole system. Targeting real-life applications, we couple these models with robust control and optimization techniques accounting for errors and uncertainties.

Modern engineering sciences make an important use of mathematical models and numerical simulations at the conception stage. Effective models and efficient numerical tools allow for optimization before production and to avoid the construction of expensive prototypes or costly post-process adjustments. Most up-to-date modeling techniques aim at helping engineers to increase performances and safety and reduce costs and pollutant emissions of their products. For example, mathematical traffic flow models are used by civil engineers to test new management strategies in order to reduce congestion on the existing road networks and improve crowd evacuation from buildings or other confined spaces without constructing new infrastructures. Similar models are also used in mechanical engineering, in conjunction with concurrent optimization methods, to improve aerodynamic performance of aircrafts and cars, or to increase thermal and structural efficiency of buildings while, in both cases, reducing ecological cost.

Nevertheless, current models and numerical methods exhibit some limitations:

• Most simulation-based design procedures used in engineering rely on steady (time-averaged) state models. However, objectives for reduction of pollutant emissions for cars, or noise reduction for aircrafts at take-off, require finer models taking into account unsteady phenomena.

• The classical purely macroscopic approach, while offering a framework with a sound analytical basis, performing numerical techniques and good modeling features to some extent, is not able to reproduce some particular phenomena related to specific interactions occurring at lower (possibly micro) level. We refer for example to self-organizing phenomena observed in pedestrian flows, to vesicle trafficking impact on wound repair, or to the dynamics of turbulent flows for which large scale / small scale vortical structures interfere. These flow characteristics need to be taken into account to obtain a realistic model and reliable optimal solutions.

• Uncertainty related to operational conditions (e.g. inflow velocity in aerodynamics), or models (e.g. individual behavior in crowds) is still rarely considered in engineering analysis and design, yielding solutions of poor robustness.

Therefore, this project focuses on the analysis and optimal control of classical and non-classical evolutionary systems of Partial Differential Equations (PDEs) arising in a variety of applications, ranging from fluid-dynamics and structural mechanics to traffic flow and biology. The complexity of the involved dynamical systems is expressed by multi-scale, time-dependent phenomena subject to uncertainty, which can hardly be tackled using classical approaches, and require the development of unconventional techniques.