Higher order methods for fluid structure interaction problems

多物理问题在现实世界和工程设计中普遍存在。流体-结构相互作用(FSI)是一类重要的多物理问题，它涉及由固体力学控制的物体与流体的相互作用(例如涡轮机与风的相互作用)。对于这些问题，拥有可靠和有效的计算机模型对于理解不同的自然物理系统和基本的工程问题至关重要。FSI问题用耦合偏微分方程组(PDE)进行数学建模。已发展出复杂的数值方法来逼近单个偏微分方程组的解。然而，发展最先进的耦合偏微分方程组的数值方法仍然是一个具有挑战性的任务，并且仍然是一个非常活跃和动态的研究领域。预计将对至少一名研究生进行有关该项目主题的培训。作为这项努力的一部分，将致力于为FSI问题开发可证明稳定的高阶分裂方法。分裂方法利用微调数值方法处理流体问题和固体问题。一个人单独解决每个问题，他们通过边界条件进行交流。挑战是使方法既稳定又高阶准确。事实上，大多数被证明是稳定的数值方法在时间上都是低阶精度的。PI将建立在使用Robin-Robin方法与合作者一起完成的工作的基础上。先前开发的Robin-Robin方法将被用作更复杂的纠正方法的基础方法。该奖项反映了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。

Multi-physics problems are ubiquitous in the real world and in engineering design. An important class of multi-physics problems are those of fluid-structure interactions (FSI) which involve an object governed by solid mechanics interacting with a fluid (e.g. a turbine interacting with the wind). Having reliable and efficient computer models for these problems are vital to understanding different natural physical systems and essential engineering problems. FSI problems are modeled mathematically by coupled partial differential equations (PDEs). Sophisticated numerical methods have been developed to approximate solutions of individual PDEs. However, the development of the state-of-the-art numerical methods for coupled PDEs remains a challenging task and continues to be a highly active and dynamic area of research. Training of at least one graduate student on the topics of the project is expected.As part of this effort, the development of provably stable higher-order splitting methods for FSI problems will be pursued. Splitting methods take advantage of fine-tuned numerical methods for fluid problems and solid problems. One solves each problem separately and they communicate via boundary conditions. The challenge is to make the methods both stable and higher-order accurate. Indeed, most of the numerical methods that are provably stable are low order accurate in time. The PI will build on the work done with collaborators using Robin-Robin methods. The Robin-Robin methods previously developed will be used as a base method for a more sophisticated correction method.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.