High-order Geometry, Mesh and Adaptivity for Fluid-Structure Interaction Simulations

在过去的几十年里，为了基于模拟的工程科学，在领域表示和离散化领域取得了非常显著的进展。这些发展大多基于区域边界的线性表示和使用低阶单元的空间离散化。最近，对于几种类型的偏微分方程，特别是结构问题，在p型有限元分析(FEA)、等距分析(IgA)和高阶(HO)模拟方面取得了非常重要的进展。在这些新的方法中，使用了HO单元和形状函数，从而增加了每个离散单元所支持的自由度数，并且边界的空间离散化由保持使用CAD系统建模的几何形状的精度的高次插值法来表示。在提高解精度的同时，可以显著减少单元数量和总体计算成本。在计算流体力学(CFD)领域也可以观察到类似的趋势，用于流体流动模拟的HO方法也得到了发展，特别是在大涡模拟和DNS类型的模拟中，使用了各种类型的有限元离散方法。这些单一学科进展的自然延伸包括开发多物理模拟方法来模拟例如流体和固体之间的相互作用(FSI)。然而，由于需要高质量和稳健的空间离散化方法来生成适应所模拟的物理现象的具体特征的曲线元素，这一进展受到了阻碍。这项研究计划旨在通过开发高效和健壮的HO网格生成和操作方法，并在代表真实世界应用的测试用例上进行验证，来促进HO方法在工程师和科学家的实践中的推广和渗透。

During the last decades, very significant advances have been made in the field of domain representation and discretization for the purpose of simulation-based engineering sciences. These developments have been mostly based on linear representations of domain boundaries and space discretization using low-order elements. Recently, very significant developments have been made towards p-version finite element analysis (FEA), isogeometric analysis (IgA) and high-order (HO) simulations, for several types of partial differential equations, and in particular for structural problems. In these new approaches, HO elements and shape functions are used, thereby increasing the number of degrees of freedom supported by each discrete element, and spatial discretization of boundaries are represented by high degree interpolants that preserve the accuracy of the geometric shapes modeled using CAD systems. Significant reduction in the number of elements and overall computational cost may be achieved, while increasing solution accuracy. A similar trend may be observed in the field of computational fluid dynamics (CFD), where HO methods for fluid flow simulations are also gaining momentum, especially in LES and DNS-type simulations, using various types of Finite Element discretization approaches. A natural extension to these single-discipline advancements involves the development of multi-physics simulation methodologies to model, for instance, the interaction between fluids and solids (FSI). This progress is however hampered by the need for high-quality and robust spatial discretization approaches to generate curved elements which are adapted to the specific features of the physical phenomena being simulated. This research program aims to contribute to the promotion and penetration of HO methods in the practice of engineers and scientists through the development of efficient and robust HO mesh generation and manipulation methods and their validation on test cases representative of real-world applications.