Reduced-Order Modeling for Interfaces in Fluid Dynamics Applications – ROMINA

Fluid flows involving interfaces between different materials, or between different phases of a same material, are of paramount importance in many fields of science and engineering, from vehicle dynamics to process engineering or biomedical applications. The current economical, regulatory and environmental context, it is of utmost importance for industrial companies to be able to understand, predict and control these flows. Numerical simulation is often the only way to reach this goal.

Many numerical methods have been devised for the simulation of interfacial flows in the last decades. Those that can handle large and complex interface deformations, for instance the merging and splitting of gas bubbles in a liquid, often do so at the expense of accuracy. The loss of accuracy has then to be compensated by local mesh refinement in the vicinity of the interface. Because the interface evolves in time, the mesh has to be periodically adapted to follow it. This results in simulations that are computationally extremely intensive.

On the other hand, an alternative known as Model Order Reduction (MOR) is gaining interest. It aims to approximate a large-scale system with
a reduced model that is much less computationally demanding. Even if the "offline" construction of the surrogate may be computationally very
costly, the reduced model can then be solved "online" very quickly as many times as desired. Applied to flow problems in industrial processes
and products, such a fast prediction capability opens new possibilities for engineers: the early design phases of industrial processes and
products, that typically require parametric studies and frequent design changes, could be greatly accelerated. Automatic optimal design
procedures would also benefit from such progress. Regarding production, real-time reduced model could even be employed in control devices to
optimise the operation of processes. Up to now, however, MOR procedures have not be adapted to the type of mesh-adaptive simulation that is
required for interfacial flows.

The goal of the proposed research is to help close the technological gap between mesh-adaptive, complex-physics simulation methods and MOR
techniques. To our knowledge, only few attempts at applying MOR to multi-phase flows have been reported, and none of them focused on the correct representation of the interfaces. The main challenge is to develop MOR methods that can deal with the mesh adaptivity, with the complex
physics of the interface and the materials involved, and with the high computational cost implied by the construction of the reduced model.

In order to tackle these ambitious challenges, we propose a progressive methodology that builds on proven methods originating in two research
laboratories of MINES ParisTech. The large-scale multi-phase problems will be solved through the fluid dynamics solvers of CEMEF. They include
stabilized Finite Element methods for the Navier-Stokes equations (possibly with complex constitutive laws) and for the level-set equation that models the evolution of fluid interfaces. These numerical schemes can be used in combination with anisotropic adaptive remeshing. Excellent parallel performance is obtained. Concerning the MOR procedures, they will benefit from the hyper-reduction techniques invented at by the MOR team at CDM. In a first step, we will develop a MOR procedure that is able to build reduced models with meshes that deform in time without changing connectivity. In a second step, we will upgrade the MOR procedure to work with h-adaptive meshes. In addition to academic verification test cases, the resulting methods will be assessed on three cases that reflect industrial applications.