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，或者变量是快速变化的。因此，为了准确地计算解，可能需要使用较小的时间离散化参数。例如，在血流模拟中，在心脏收缩期血压迅速上升，然后下降，持续3/8的心脏周期，随后在心脏舒张期血压变化缓慢和较小，持续5/8的心脏周期。为了准确地捕获峰值收缩流量，必须在该间隔内使用一个小的时间步长。然而，相同的时间步长在舒张期间可能太小，并且可能导致更长的计算时间。因此，鲁棒自适应时间步进是准确和有效的解决方案的长期预测的核心。对描述流动问题的偏微分方程的自适应时间步进方法进行了研究，本项目将在该领域做出重大贡献。本项目开发的方法将用于模拟涉及运输和流体-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透的水文系统中污染物的运输，毛细血管和组织之间营养物质和氧气的运输，或疾病跨越边界的传播。这个项目将涉及研究生的培训。本项目的重点是发展两类耦合流动问题的自适应时间步进方法：流体-多孔介质耦合问题和流体-结构相互作用问题。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割的方法。对于薄结构和厚结构的流固耦合问题，将发展分块数值方法。该方法基于重构柯西单足类方法在时间上进行半离散化，该方法在可变时间步长下是b稳定的。此外，当θ为0.5时，该方法也是二阶精度的，并保留所有线性和二次哈密顿量。然而，将这种方法应用于耦合问题时，特别是在设计分区方法时，必须小心执行，以允许使用黑盒和遗留代码。提出的方法将进行数学和计算分析。将考虑各种适应策略。每种方法的性能将根据问题中的参数进行研究。在这两类多物理场问题中，基础方程将与输运方程耦合。拟议的技术也将应用于运输问题，特别注意质量和能源的节约。我们将研究输运问题的保守性。该奖项反映了美国国家科学基金会的法定使命，并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，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.