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

如何有效地模拟流体与结构之间复杂的非线性相互作用一直是计算数学中的一个具有挑战性的问题。这种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和处理所产生的耦合行为方面都是一个巨大的挑战。如果将控制和设计系统的愿望添加到画面中，那么复杂性将进一步增加。因此，建议的研究重点将是系统地开发适用于高性能计算的非协调有限元方法，用于涉及流体-结构-控制相互作用的几个具有计算挑战性的多学科应用。其主旨将是从数学和计算上研究各种非协调有限元技术的稳定性、收敛和控制，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求解方法。更具体地说，拟议的研究将通过调查(A)全耦合算法的计算稳定性和收敛；(B)迭代耦合的计算稳定性和收敛；(C)形状、边界和分布式控制应用的理论和计算研究来探索该方法的稳健性。作为这项研究的一部分，计算算法的性能将被应用于两个现实的流体结构应用：(A)母动脉-动脉瘤多结构中的血液流动和(B)微型飞行器的计算气动弹性。这项研究旨在为科学和工程应用中出现的流体-结构-控制相互作用问题开发最优计算算法。拟议的工作是高度多学科的，作为这项研究的一部分开发的算法可以迅速应用于广泛的工程和医学应用。例如，这项研究可能被用来更好地了解动脉瘤破裂，而动脉瘤破裂是该国显著的发病率和死亡率的原因。这项工作还可用于开发具有灵活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影响，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，用于培训学生和教师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妇女和代表性不足的少数群体从事计算数学领域的职业，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.