Collaborative Research: Time Accurate Fluid-Structure Interactions

在描述流体流动的实际问题中，有时动力学未知，或者变量变化很快。因此，为了准确地计算解，可能需要使用小的时间离散化参数。例如，在血流模拟中，压力在收缩期间迅速上升，然后下降，持续心脏周期的3/8，然后是舒张期压力变化较慢和较小的变化，持续心脏周期的5/8。为了准确地捕获峰值收缩期血流，必须在该间隔中使用一个小的时间步长。然而，相同时间步长在舒张期可能不必要地小，且可能导致更长的计算时间。因此，稳健的自适应时间步长是解决方案的准确和有效的长期预测的核心。描述流动问题的偏微分方程组的自适应时间步长方法还没有得到充分的研究，本项目将在这一领域做出重大贡献。该项目中开发的方法将用于模拟涉及传输和流体-弹性/孔弹性结构相互作用的问题，例如地表水通过岩石和沙子渗入的水文系统中的污染物传输、毛细血管和组织之间的营养物质和氧气传输，或疾病跨越边界的传播。该项目将涉及研究生的培养。本项目的重点是发展两类耦合渗流问题的自适应时间步长方法：流体-多孔介质耦合问题和流固耦合问题。对于用Stokes-Darcy系统描述的流体-多孔介质问题，将发展一种整体和分区方法。对于薄结构和厚结构的流固耦合问题，将发展分段数值方法。所提出的方法将基于重构的Cauchy单腿类theta方法，在时间上是半离散的，该方法在变时间步长时是B-稳定的。此外，当theta为0.5时，该方法也是二阶精度的，并且守恒所有的线性和二次哈密顿量。然而，将这种方法应用于耦合问题，特别是在设计分区方法时，必须仔细执行，以允许使用黑盒和遗留代码。将对所提出的方法进行数学和计算分析。将考虑各种适应战略。将根据问题中的参数来研究每种方法的性能。在这两类多物理问题中，基本方程将与输运方程耦合。拟议的技术也将应用于运输问题，特别关注质量和能量守恒。交通问题的保守性质将被调查。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.