Mathematical and computational modeling of fluid-structure-control interactions with multidisciplinary applications in science and engineering

如何有效求解流体与结构的复杂非线性相互作用，一直是计算数学中的一个难题。此类应用通常涉及多个物理过程的复杂动态相互作用，这在表示所涉及的多物理场和处理由此产生的耦合行为方面都提出了重大挑战。如果将控制和设计系统的愿望添加到画面中，那么复杂性甚至会进一步增加。因此，拟议研究的重点将是系统地开发非一致性有限元方法，以适应高性能计算，应用于涉及流固控相互作用的几个计算挑战性的多学科应用。重点将是在数学和计算上研究各种非一致性有限元技术的稳定性、收敛性和控制性，并利用这些信息为流体结构控制应用开发一种有效和通用的解决方法。更具体地说，拟议的研究将通过调查(a)计算稳定性和全耦合算法的收敛性来探索这种方法的鲁棒性；(b)迭代耦合的计算稳定性和收敛性；(c)形状、边界和分布式控制应用的理论和计算研究。作为本研究的一部分开发的计算算法的性能将应用于两个现实的流体结构应用：(a)父动脉-动脉瘤多结构中的血流和(b)微型飞行器的计算气动弹性。本研究旨在为科学和工程应用中出现的流固控相互作用问题开发最佳计算算法。所提出的工作是高度多学科的，并且作为本研究的一部分开发的算法可以迅速采用到广泛的工程和医学应用中。例如，这项研究可以用来更好地了解动脉瘤的破裂，这是该国发病率和死亡率很高的原因。这项工作还可用于开发具有柔性飞机机翼的微型飞行器的增强和高效设计，可用于各种任务，如侦察和监视，瞄准，标记，生化传感等。与研究组成部分相结合的还有一项教育计划，该计划将鼓励跨学科研究，这将涉及课程开发中拟议研究的教学含义，并将有助于研究生、本科生、高中生和教师的科学发展。更具体地说，拟议的研究将用于开发学习模块，这些模块将用于培训学生和教师如何有效地使用计算数学来解决科学和工程中的多学科问题。这项研究还将极大地鼓励女性和未被充分代表的少数族裔从事计算数学方面的职业，特别是在连接生物、数学和计算科学的跨学科领域。

The efficient solution of modeling the complex nonlinear interaction of a fluid with a structure has remained a challenging problem in computational mathematics. Such applications often involve complex dynamic interactions of multiple physical processes which present a significant challenge, both in representing the multiphysics involved and in handling the resulting coupled behavior. If the desire to control and design the system is added to the picture, then the complexity increases even further. The focus of the proposed research will therefore be to sytematically develop non-conforming finite element methods tuned to high performance computing applied to several computatationally challenging multidisciplinary applications involving fluid-strucuture-control interaction. The thrust will be to mathematically and computationally investigate the stability, convergence and control of a variety of non-conforming finite element techniques and use this information to develop an efficient and general solution methodology for fluid-structure-control applications. More specifically, the proposed research will explore the robustness of this methodology by investigating (a) computational stability and convergence of fully-coupled algorithms; (b) computational stability and convergence for iterative coupling and; (c) theoretical and computational investigation for shape, boundary and distributed control applications. The performance of the compuational algorithms developed as a part of this research will be applied to two realistic fluid-structure applications: (a) Blood flow in a parent-artery-aneurysm multistructure and (b) Computational aeroelasticity of micro-air-vehicles. The proposed research aims to develop optimal computational algorithms for fluid-stucture-control interaction problems arising in science and engineering applications. The proposed work is highly multidisciplinary and the algorithms developed as a part of this research can be quickly adopted to a wide range of engineering and medical applications. For instance, this research may be used to better understand the rupture of aneurysms which are responsible for significant morbidity and mortality in the country. This work may also be used to develop enhanced and efficient design of micro air vehicles with flexible aircraft wings, that may be used for a variety of missions such as reconnaissance and surveillance, targeting, tagging, bio-chemical sensing and many more. Integrated with the research component is also an educational plan that will encourage interdisciplinary research, that will involve the pedagogical implications of the proposed research in curriculum development and that will contribute to the scientific development of graduate students, undergraduate students, high school students and teachers. More specifically, the proposed research will be used to develop learning modules that will be used to train students and teachers on the efficient use of compuatational mathematics to solve multidisciplinary problems in science and engineering. The research will also greatly encourage women and underrepresented minorities to pursue careers in computational mathematics, especially in interdisciplinary areas that bridge the biological, mathematical, and compuational sciences.