Reduced Order Models for Fluid-Structure Interaction problems
Fluid — Structure Interaction (FSI) problems are a wide spread topic in the applied mathematics community. Despite their intrinsic complicated nature, they are frequently used in the simulation of a lot of situations: in naval engineering, they are used to study the interaction between the water and the hull of a ship; in biomedical applications FSI problems describe the interaction between the blood flow and the deformable walls of a vessel; in aeronautical engineering, FSI describes the way the air interacts with a plane or with (parts of) a shuttle.
All the aforementioned example of applications are modelled, from the mathematical point of view, through a system of Partial Differential Equations (PDEs): some equations, such as the Navier-Stokes equations, describe the motion of the fluid, some equations, such as the linear (or nonlinear) elasticity equations, describe the motion of the solid, and some others describe the actual interaction between the fluid and the structure.
As we can see from the two pictures on the side, FSI problems are able to describe very complicated phenomena. The coupled system of PDEs in the FSI problem is not only complicated because of its mathematical nature; indeed, it is even more complex, given the fact that we usually have to take into account a number of parameters: for example, the physical parameters indicating how soft/hard the material is; the physical parameter indicating how turbulent/laminar a fluid flow is, and many more others..
The Reduced Basis Method: reducing the computational costs.
Given the extreme complexity of coupled systems, and given the number of parameters that one usually has to take into consideration, it is easy to imagine that obtaining a simulation of the solution of such system, for given values of the parameters in the problem, is extremely costly, both in terms of computational time and computer memory. Now, say you were able to obtain these simulations, and say that you are interested in changing one (or more) of the values of the parameters in the problem. Do you have to compute everything from scratch? NO! For this reason, we rely on the well known Reduced Basis Method (RBM), which is a method to approximate solutions of PDEs (systems of PDEs), starting from a set of previously computed accurate solutions (snapshots).
Reduced Basis Method for Fluid-Structure Interaction problems.
We focused our study on two different approaches that can be adopted to solve numerically a FSI problem: monolithic approach and partitioned (segregated) approach.
- Monolithic approach: we solve simultaneously the fluid and the solid problem. We mostly focused on incorporating a preprocessing technique, during the offline phase of the method, in order to lower the complexity of the problem, due to a transportation phenomenon.
- Partitioned approach: we solve separately the fluid and the solid problem. We then couple the two physics by means of some iterative procedure. We designed a segregated model order reduction method, that, thanks to a particular coupling technique, is able to handle complex problems such as FSI problems formulated within an Arbitrary Lagrangian Eulerian (ALE) formalism, where the structure is thick and two dimensional.
Some examples of applications.
A very nice problem that we considered, arises from medical applications. Imagine we have a 2D-section of a vessel in the human body. We have a channel (the vessel), and two leaflets (in the vessel, made by some biological tissue).
Problem: study how the two leaflets bend and deform, under the influence of a fluid flowing in the rectangular cavity [3]. What happens if we change the lenght of the leaflets? What happens if we harden/soften the material?
1. Some physical parameters of the biological tissue make the leaflets more soft and hence more likeable to deform, under the influence of the same fluid.
2. Keeping the same physical parameters for the solid, we see that the longer the leaflets, the bigger the deformation under the influence of the same fluid.
Another interesting problem: the behaviour of a fluid that flows in a channel with deformable walls, with an initial pressure impulse at the entrance of the channel [1].
Conclusions.
It is clear that the RBM represents an extremely powerful tool, that finds thousands of applications within the world of coupled, multiphysics problems.
With our work, through the study of different test cases, we have seen how the RBM can be coupled with various discretization techniques and with various algorithms, in order to obtain reduction procedures that can be applied to several situations: coupled problems where we want to reduce the dimensionality of the system to be solved, advection dominated problems and problems with a significant change in the physical domain.
Future perspectives?
- Incorporate techniques that can handle efficiently the nonlinearities in the online phase of the method, such as the Empirical Interpolation Method.
- Study alternative decoupling techniques for the partitioned approaches (different ways to decouple the fluid and the solid problem).
- Combine the RBM with discretization techniques other than Finite Element Method (or CutFEM [2])?
References:
[1] M. Nonino, F. Ballarin, G. Rozza, Y. Maday. Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and Fluid-Structure Interaction problems. https://arxiv.org/abs/1911.06598 , 2019
[2] E. Karatzas, M. Nonino, F. Ballarin, G. Rozza. A Reduced Order Cut
Finite Element Method for geometrically parametrized steady and unsteady
Navier–Stokes. https://arxiv.org/abs/2010.04953, 2020
[3] M. Nonino, F. Ballarin, G. Rozza, Y. Maday. Projection based semi-implicit partitioned reduced order models for Fluid-Structure Interaction problems. Submitted, 2020.
[4] multiphenics. https://mathlab.sissa.it/multiphenics
[5] RBniCS. https://www.rbnicsproject.org/
