Spline-Based Space-Time Finite Element Approach for Fluid-Structure Interaction Problems With a Focus on Fully Enclosed Domains

Abstract

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 fluidstructure 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 frame∗Corresponing Author Email addresses: make@cats.rwth-aachen.de (Michel Make), spenke@cats.rwth-aachen.de (Thomas Spenke), hosters@cats.rwth-aachen.de (Norbert Hosters), behr@cats.rwth-aachen.de (Marek Behr) Preprint submitted to Computers and Mathematics with Applications. March 31, 2022 ar X iv :2 20 3. 16 15 2v 1 [ cs .C E ] 3 0 M ar 2 02 2 work are demonstrated using a series of twoand three-dimensional benchmark problems.