Collaborative Research: Time Accurate Fluid-Structure Interactions

在描述流体流动的实际问题中，有时动力学是未知的，或者变量变化很快。因此，为了精确地计算解，可能需要使用小的时间离散化参数。例如，在血流的模拟中，压力在心脏收缩期间快速增加，然后降低，其持续心动周期的3/8，随后在心脏收缩期间压力的变化更慢且更小，持续心动周期的5/8。为了准确地捕获收缩期峰值流量，必须在该间隔中使用小的时间步长。然而，在计算过程中，相同的时间步长可能不必要地小，并且可能导致更长的计算时间。因此，鲁棒的自适应时间步进是准确和有效的长期预测的解决方案的核心。自适应时间步方法描述流动问题的偏微分方程的研究下，这个项目将在该领域作出重大贡献。在这个项目中开发的方法将被用来模拟涉及运输和流体弹性/多孔弹性结构相互作用的问题，如在水文系统中的污染物的运输，其中地表水通过岩石和沙子，毛细血管和组织之间的营养物质和氧气的运输，或跨越边界的疾病传播。该项目将涉及研究生的培训。本计画的重点是发展两类耦合流动问题的自适应时步方法：流体-多孔介质耦合问题和流体-结构耦合问题。将开发一个整体和分区方法的流体多孔介质问题描述使用斯托克斯-达西系统。将发展分区数值方法来处理薄结构和厚结构的流固耦合问题。所提出的方法将在时间上半离散化的重构柯西的单腿θ类方法的基础上，这是B-稳定的，当使用一个可变的时间步长。此外，当θ为0.5时，该方法也是二阶精度的，并且保持所有线性和二次哈密顿量。然而，这种方法的应用程序耦合的问题，特别是当分区方法的设计，必须仔细执行，以允许使用黑盒和遗留代码。所提出的方法将进行数学和计算分析。将考虑各种适应性战略。每种方法的性能将被调查的问题中的参数。在这两类多物理问题中，基本方程将与输运方程耦合。所提出的技术也将被应用到运输问题，特别注意质量和能量守恒。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估的支持。

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.