全封闭区域流固耦合问题的样条时空有限元方法
Spline-Based Space-Time Finite Element Approach for Fluid-Structure Interaction Problems With a Focus on Fully Enclosed Domains
摘要：
非均匀有理B样条（NURBS）曲面通常在计算机辅助设计（CAD）工具中用于表示几何对象。当使用等几何分析（IGA）时，可以直接将此类NURBS几何用于数值分析。然而，分析流体流动需要复杂的三维几何形状来表示流动域。使用NURBS定义这种体积域的参数化可能具有挑战性，并且仍然是IGA社区中正在进行的主题。近年来发展起来的NURBS-增强有限元法（NEFEM）将NURBS良好的几何特性应用于标准有限元法中。这是通过使用NURBS几何体本身增强接触边界的元素来实现的。在目前的工作中，一个新的变化的NEFEM介绍，这是适合于三维时空有限元公式。所提出的方法利用一个新的映射，导致在非笛卡尔制定适合于流体-结构相互作用（FSI）。这是证明了结合的方法与IGA制定在一个强耦合的分区框架解决FSI问题。该框架通过一个单一的NURBS产生一个完全基于样条曲线的流体-结构界面的表示。通过Robin-Neumann型耦合方案，在流体-结构界面处的耦合条件被强制执行。当考虑完全狄利克雷有界和弯曲问题中的不可压缩流体时，该方案特别有用，因为它满足耦合过程中每个步骤对流体的不可压缩性约束。介绍了基于样条的时空有限元方法的精度和性能，并在拟议的耦合FSI框架内使用证明使用一系列的二维和三维基准问题。
英文摘要：
Non-Uniform Rational B-Spline (NURBS) surfaces are commonly used within Computer-Aided Design (CAD) tools to represent geometric objects. When using isogeometric analysis (IGA), it is possible to use such NURBS geometries for numerical analysis directly. Analyzing fluid flows, however, requires complex three-dimensional geometries to represent flow domains. Defining a parametrization of such volumetric domains using NURBS can be challenging and is still an ongoing topic in the IGA community. With the recently developed NURBS-enhanced finite element method (NEFEM), the favorable geometric characteristics of NURBS are used within a standard finite element method. This is achieved by enhancing the elements touching the boundary by using the NURBS geometry itself. In the current work, a new variation of NEFEM is introduced, which is suitable for three-dimensional space-time finite element formulations. The proposed method makes use of a new mapping which results in a non-Cartesian formulation suitable for fluid-structure interaction (FSI). This is demonstrated by combining the method with an IGA formulation in a strongly-coupled partitioned framework for solving FSI problems. The framework yields a fully spline-based representation of the fluid-structure interface through a single NURBS. The coupling conditions at the fluid-structure interface are enforced through a Robin-Neumann type coupling scheme. This scheme is particularly useful when considering incompressible fluids in fully Dirichlet-bounded and curved problems, as it satisfies the incompressibility constraint on the fluid for each step within the coupling procedure. The accuracy and performance of the introduced spline-based space-time finite element approach and its use within the proposed coupled FSI framework are demonstrated using a series of two- and three-dimensional benchmark problems.
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