# Coupled finite element and boundary element analysis for fluid-structure interaction with deformable shells

• Gekoppelte Finite-Elemente- und Boundary-Elemente-Analyse für Fluid-Struktur-Interaktion mit verformbaren Schalen

Harmel, Maximilian; Sauer, Roger Andrew (Thesis advisor); Itskov, Mikhail (Thesis advisor)

Aachen : RWTH Aachen University (2022, 2023)
Dissertation / PhD Thesis

Dissertation, Rheinisch-Westfälische Technische Hochschule Aachen, 2022

Abstract

This thesis presents an efficient mathematical model for fluid-structure interactions (FSI) by coupling a finite element (FE) formulation for thin-walled deformable Kirchhoff-Love shells and a boundary element (BE) formulation for Stokes flow. This coupling allows to represent both the nonlinear surface effects of Kirchhoff-Love shells and the linear volume effects of Stokes flow, discretizing only the conformal surface - no volumetric mesh is required. Due to this dimensional reduction, the coupled FE-BE formulation is very efficient in terms of mesh generation as well as computational effort. The surface is discretized with isogeometric NURBS shape functions that provide the exact representation of surfaces and a higher continuity than classical Lagrangian shape functions. First, the nonlinear theory of Kirchhoff-Love shells and the corresponding FE formulation are presented. However, the main focus of this thesis is on the isogeometric BE formulation for three-dimensional Stokes flow and on the monolithic coupling of the two formulations. The theory underlying the boundary integral equation (BIE) for Stokes flow is treated for incompressible fluids in external, internal, and two-sided flow problems at open and closed surfaces. A single BIE applicable to all ow problems mentioned is derived and presented. The BIE is collocated, discretized and then assembled into a linear system of equations that can be solved for Dirichlet, Neumann and mixed boundary value problems. A novel hybrid quadrature approach for the approximation of strongly singular boundary integrals is presented. The hybrid quadrature efficiently combines classical Gaussian quadrature and a Duffy transformation based quadrature. The novel quadrature approach is investigated on plane NURBS sheets and on exact NURBS spheres of quadratic to quintic order. In particular, the hybrid quadrature approach is outstandingly accurate for coarse discretizations. The developed BE formulation is applied to various stationary problems considering spherical and ellipsoidal NURBS surfaces in Stokes flow, achieving high accuracy for rotating and translating bodies. The presented coupling condition models the interactions between surface and flow considering slip as well as bounded and unbounded flow. The viscous forces from the BE formulation are integrated into the discretized FE weak form by means of a discrete damping matrix. A generalized-α method is used for temporal discretization. The resulting expressions for the kinematic quantities are explicitly given in this thesis. The application of the presented FE-BE coupling to various FSI problems with balloons, bending resistant shells, red blood cells and beams in Stokes flow yields physically comprehensible results. In absence of analytical or experimental reference solutions, successful convergence studies with respect to a heavily refined numerical solution are presented.

Institutions

• Continuum Mechanics Teaching and Research Department [418220]