Collaborative Research: Time Accurate Fluid-Structure Interactions

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In realistic problems describing fluid flow, sometimes the dynamics are not known, or the variables are changing rapidly. Hence, to accurately compute the solution, one might need to use small temporal discretization parameters. For example, in simulations of blood flow, the pressure rapidly increases and then decreases during the systole, which lasts 3/8 of the cardiac cycle, followed by slower and smaller changes in the pressure during diastole, lasting 5/8 of the cardiac cycle. To accurately capture the peak systolic flow, a small time step has to be used in that interval. However, that same time step might be unnecessary small during diastole and could lead to longer computational times. Therefore, robust adaptive time-stepping is central to accurate and efficient long-term predictions of the solution. The adaptive time-stepping methods for partial differential equations describing flow problems are under-investigated and this project will make a major contribution in that field. The methods developed in this project will be used to model problems involving transport and fluid-elastic/poroelastic structure interaction, such as the transport of contaminants in hydrological systems where surface water percolates through rocks and sand, transport of nutrients and oxygen between capillaries and tissue, or spread of a disease across a border. This project will involve the training of graduate students. The focus of this project is the development of adaptive time-stepping methods for two classes of coupled flow problems: the fluid-porous medium coupled problems and the fluid-structure interaction problems. A monolithic and a partitioned method will be developed for the fluid-porous medium problem described using the Stokes-Darcy system. Partitioned numerical methods will be developed for the fluid-structure interaction problems with both thin and thick structures. The proposed methods will be semi-discretized in time based on the refactorized Cauchy’s one-legged theta-like method, which is B-stable when used with a variable time step. Furthermore, when theta is 0.5, the method is also second-order accurate and conserves all linear and quadratic Hamiltonians. However, the application of this method to coupled problems, especially when partitioned methods are designed, has to be carefully performed to allow the use of black-box and legacy codes. The proposed methods will be mathematically and computationally analyzed. Various adaptive strategies will be considered. The performance of each method will be investigated with respect to the parameters in the problem. In both classes of multi-physics problems, the underlying equations will be coupled with a transport equation. The proposed techniques will also be applied to the transport problem, with a particular attention to mass and energy conservation. Conservative properties of the transport problem will be investigated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.