Simulation of Hydrodynamics in Flexible Fibre Arrays

SWANSEA UNIVERSITY

Simulation of Hydrodynamics in Flexible Fibre Arrays

by Rui Liang

A thesis submitted in partial fulfillment for the degree of Doctor of Philosophy

in the College of Engineering

September 2018

Declarations 1. This work has not previously been accepted in substance for any degree and is not being concurrently submitted in candidature for any degree.

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(candidate)

Date

2. This thesis is the result of my own work and investigation, except where otherwise stated. Other sources have been acknowledged by giving explicit references. A bibliography is appended.

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(candidate)

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3. I hereby give consent for my thesis, if accepted, to be available for photocopying and for inter-library loan, and for the title and summary to be made available to outside organisations.

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Acknowledgements I would like to express my profound gratitude to my supervisors Prof. Djordje Perić and Prof. Wulf Dettmer for their invaluable guidance and encouragement during all stages of this research work. The finicial support comes from Engineering and Physical Sciences Research Council (EPSRC), Unilever UK Ltd and College of Engineering is gratefully acknowledged. My special thanks go to Prof. Perić for making this possible. I am especially grateful to Prof. Yun tian Feng for his unconditional guidance and support during my study in Swansea University. My sincere thanks also go to my group member Dr. Chennakesava Kadapa for his continuous support during my whole PhD career, and especially for sharing with me his code without reservation. I am grateful to my colleagues in Zienkiewicz Centre for Computational Engineering, with whom I had many helpful and interesting discussions. I would like to thank all my friends who made my stay in Swansea enjoyable. Last my deep gratitude goes to my parents for their constant support.

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宇宙来自于平衡附近的星球来自于回声沼泽来自于地面的失眠褶皱来自于海

一定有人离开了会回来腾空的竹篮装满爱一定有某种破碎像泥土某个谷底像手一样摊开

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Abstract This work is concerned with modelling of interaction between incompressible fluid flow and flexible fibres in two- and three-dimensions. To this end, a fictitious domain/distributed Lagrange multiplier (FD/DLM) based strategy with hierarchical b-spline grids is developed to solve the fluid-structure interaction (FSI). A stabilised finite element method is used for solving fluid flow, in which the bodyfitted mesh is initially employed for the spatial discretisation of fluid-solid interfaces. A detailed analysis of standard Galerkin formulation and its deficiencies in modelling incompressible Navier-Stokes flow is performed, leading to the choice of streamline upwind Petrov-Galerkin/pressure-stabilising Petrov-Galerkin (SUPG/PSPG) formulation with velocity-pressure equal-order elements for the fluid solver. However, the main disadvantage of the body-fitted methodology is the computational cost associated with the underlying fluid mesh update or the need to use expensive re-meshing algorithms in FSI problems where the solid body undergoes large deformations. This is particularly important in the case of three-dimensional simulations. Therefore, an immersed type method, FD/DLM scheme, is introduced to solve fluid flow, in which a fixed fluid mesh is used over the entire simulation domain extending over the interior of the solid domain. As a result, a Cartesian grid can be used for the fluid domain. In addition, hierarchical b-splines are adopted for the local refinement of fluid mesh near the immersed solid. Two- and three-dimensional geometrically exact beam formulations are introduced to capture the extremely large deformations of slender fibres. The representation of 3D finite rotations is recognised as a main difficulty in beam analysis. In this work, the rotation tensor is represented by a vector-like parametrization through the Euler-Rodrigues formula. The mass, stiffness and residual of the beam element are then obtained in a simple and straightforward fashion. Several static and transient problems are investigated to demonstrate the excellent performance of the chosen formulations. A staggered scheme is employed to solve the FSI system in a partitioned way. This force-predictor based scheme is very efficient as only one iteration is required within each time step. Another main feature of the current scheme is that by tuning an averaging parameter, it is applicable to the case with small solid over fluid mass ratio. Finally, a number of 2D and 3D fluid-flexible fibre interaction examples are studied in detail, which demonstrate the robustness and the efficiency of the overall algorithm. Some conclusions during this work and some suggestions for the future research are given.

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Contents Declarations

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Acknowledgements

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Abstract

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List of Figures

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List of Tables

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1 Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . 1.2 FSI solution procedures . . . . . . . . . . . . . . . . . 1.2.1 Monolithic and partitioned approaches . . . . . 1.2.2 Conforming and non-conforming mesh methods 1.3 Immersed boundary method . . . . . . . . . . . . . . . 1.4 Aim of the thesis . . . . . . . . . . . . . . . . . . . . . 1.5 Layout of the thesis . . . . . . . . . . . . . . . . . . .

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2 Stabilised Finite Elements for Fluid Flow 2.1 Navier-Stokes equations . . . . . . . . . . 2.1.1 Governing equations . . . . . . . . 2.1.2 Weak form . . . . . . . . . . . . . 2.2 Galerkin finite element solution . . . . . . 2.2.1 Galerkin formulation . . . . . . . . 2.2.2 Velocity oscillation . . . . . . . . . 2.2.3 Pressure oscillation . . . . . . . . . 2.3 Velocity stabilisation . . . . . . . . . . . . 2.3.1 A Petrov-Galerkin formulation . . 2.3.2 Stabilised methods . . . . . . . . . 2.3.2.1 SUPG method . . . . . . 2.3.2.2 GLS method . . . . . . . 2.4 Pressure stabilisation . . . . . . . . . . . . 2.4.1 PSPG method . . . . . . . . . . . 2.4.2 GLS method . . . . . . . . . . . . 2.4.3 LBB-stable element . . . . . . . . 2.5 SUPG/PSPG formulation . . . . . . . . . 2.6 Discrete time integration schemes . . . . .

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3 Fictitious Domain/distributed Lagrangian Multiplier Method for Fluid Flow 3.1 Fictitious domain method . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Main features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Governing equations for fluid . . . . . . . . . . . . . . . . . . . . . 3.2.2 Governing equations for solid . . . . . . . . . . . . . . . . . . . . . 3.2.3 Governing equations at the interface . . . . . . . . . . . . . . . . . 3.2.4 Weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hierarchical b-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Bézier curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 B-spline basis functions . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 B-spline curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 B-splines in higher dimensions . . . . . . . . . . . . . . . . . . . . 3.3.5 Hierarchical refinement . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Conservation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Flow past a circular cylinder . . . . . . . . . . . . . . . . . . . . .

37 38 39 39 40 40 40 41 41 42 42 43 44 45 46 47 47 51

4 Geometrically Exact Beam Theory 4.1 Two-dimensional geometrically exact beam formulation 4.2 Parametrization of finite three-dimensional rotations . . 4.2.1 Euler-Rodrigues formula . . . . . . . . . . . . . . 4.2.2 Rotation tensor . . . . . . . . . . . . . . . . . . . 4.2.3 Rotation gradient . . . . . . . . . . . . . . . . . . 4.2.4 Second derivative of rotation tensor . . . . . . . 4.2.5 Second derivative of rotation gradient . . . . . . 4.3 Strain measures . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Bending strains . . . . . . . . . . . . . . . . . . . 4.3.2 Axial and shear strains . . . . . . . . . . . . . . 4.4 Stress measures . . . . . . . . . . . . . . . . . . . . . . . 4.5 Variational formulation . . . . . . . . . . . . . . . . . . 4.6 Residual vector . . . . . . . . . . . . . . . . . . . . . . . 4.7 Stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . ∂Ra 4.7.1 First block: ∂ua1 . . . . . . . . . . . . . . . . . .

55 56 59 59 60 61 62 65 65 65 66 66 67 69 69 70

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2.6.1 Backward Euler method . . . 2.6.2 Generalised midpoint rule . . 2.6.3 Generalised-α method . . . . Numerical examples . . . . . . . . . 2.7.1 Lid-driven cavity problem . . 2.7.2 Flow past a circular cylinder 2.7.3 Backward facing step flow . .

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∂Ra 1 ∂θ a a ∂R Third block: ∂ua2 . ∂Ra Fourth block: ∂θa2

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4.8 4.9

Mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 A curved cantilever under tip force . . . . . . . . . . . 4.9.2 A cantilever beam subject to a pure bending moment 4.9.3 A frame structure subject to a concentrated load . . . 4.9.4 A cantilever beam bent to a helical form . . . . . . . . 4.9.5 Dynamic response of a cantilever under tip force . . . 4.9.6 Right-angle cantilever beam . . . . . . . . . . . . . . .

5 Solution Strategies for Fluid-structure Interaction 5.1 Strongly coupled strategy . . . . . . . . . . . . . . . 5.1.1 Monolithic algorithm . . . . . . . . . . . . . . 5.1.2 Partitioned algorithm . . . . . . . . . . . . . 5.2 Weakly coupled strategies - a staggered scheme . . . 5.2.1 Governing equation . . . . . . . . . . . . . . 5.2.2 Solution algorithm . . . . . . . . . . . . . . . 5.2.3 Main features . . . . . . . . . . . . . . . . . .

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6 Fluid-flexible Fibre Interaction: 2D Numerical Examples 6.1 Flow-induced vibrations of a flexible beam . . . . . . . . . . 6.2 Flexible fibre in a soap film . . . . . . . . . . . . . . . . . . 6.3 Neutrally buoyant fibre in shear flow . . . . . . . . . . . . . 6.4 Two flapping leaves . . . . . . . . . . . . . . . . . . . . . . . 6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Fluid-flexible Fibre Interaction: 3D Numerical Examples 109 7.1 Two flapping fibres subject to sinusoidally varying velocity . . . . . . . . 109 7.2 Fibre array subject to constant velocity . . . . . . . . . . . . . . . . . . . 119 7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8 Conclusions 8.1 Achievements and conclusions . . . . . . . . 8.1.1 SUPG/PSPG and FD/DLM for fluid 8.1.2 Geometrically exact beam . . . . . . 8.1.3 2D FSI numerical examples . . . . . 8.1.4 3D FSI numerical examples . . . . . 8.1.5 Computer Implementation . . . . . . 8.2 Suggestions for future work . . . . . . . . .

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Appendix A Newton-Raphson Method

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Bibliography

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List of Figures 1.1 1.2

1.3 2.1 2.2 2.3 2.4

2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19

Some applications of fluid-structure interaction. . . . . . . . . . . . . . . . Schematic of the monolithic and partitioned approaches for FSI, where S f and S s denote the fluid and structure solutions, respectively, whereas tn and tn+1 represent, respectively, the time instant n and n + 1. . . . . . Examples of conforming and non-conforming meshes. . . . . . . . . . . . . Galerkin finite element solution of 1D, steady AD equation for different P e. (a) P e = 0.25; (b) P e = 1; (c) P e = 2; (d) P e = 5; . . . . . . . . . . Definition of the lid-driven cavity problem and a mesh of 400 quadrilateral elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity contour and streamlines obtained using GFEM with 400 quadrilateral elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure contour plots of lid-driven cavity problem with 400 quadrilateral elements. (a) 2d pressure, Galerkin; (b) 3d pressure, Galerkin; (c) 2d pressure, PSPG; (d) 3d pressure, PSPG . . . . . . . . . . . . . . . . . . shape functions of 1D linear elements . . . . . . . . . . . . . . . . . . . . weight functions of 1D linear elements in Petrov GFEM . . . . . . . . . Petrov GFEM solution of 1D, steady AD equation for different β, P e = 5. (a) β = 0; (b) β = 0.5; (c) β = 1; (d) β = 2; (e) β = 0.8001; . . . . . . . Comparison of different stabilisations for 1D AD equation. (a) P e = 0.25; (b) P e = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Triangular (P2 P1 ) and quadrilateral (Q2 Q1 ) Taylor Hood elements . . . MINI (P1+ P1 ) element . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lid-driven cavity: geometry and boundary conditions . . . . . . . . . . Lid-driven cavity: mesh with (a) 400 quadrilateral elements; (b) 2048 triangular elements; (c) 8192 triangular elements . . . . . . . . . . . . . Lid-driven cavity: velocity contour plot and streamlines obtained with 8192 elements, Re = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . Lid-driven cavity: velocity contour plot and streamlines obtained with 8192 elements,Re = 1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . Lid-driven cavity: velocity contour plot and streamlines obtained with 8192 elements,Re = 3200 . . . . . . . . . . . . . . . . . . . . . . . . . . . Lid-driven cavity: u and v velocity profiles along x = 0.5 and y = 0.5 for Re = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lid-driven cavity: u and v velocity profiles along x = 0.5 and y = 0.5 for Re = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lid-driven cavity: u and v velocity profiles along x = 0.5 and y = 0.5 for Re = 3200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow past a circular cylinder: geometry and boundary conditions . . . . viii

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2.20 Flow past a circular cylinder: mesh with 2969 triangular elements . . . 2.21 Flow past a circular cylinder: mesh with 24330 triangular elements and a zoom view around the cylinder . . . . . . . . . . . . . . . . . . . . . . 2.22 Flow past a circular cylinder: contour plots of X-velocity for steady Stokes flow and steady Navier-Stokes flow, Re = 0.1 . . . . . . . . . . . . . . . 2.23 Flow past a circular cylinder: contour plots of velocity and pressure at t = 25, Re = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.24 Flow past a circular cylinder: plot of velocity vector at t = 25, Re = 100 2.25 Flow past a circular cylinder: evolution of drag and lift coefficients, Re = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.26 Backward facing step flow: geometry and boundary conditions . . . . . 2.27 Backward facing step flow: mesh with 12601 triangular elements . . . . 2.28 Backward facing step flow: streamlines around the step for different Re. (a) Re = 0.01; (b) Re = 1; (c) Re = 50; (d) Re = 100; (e) Re = 400; (f) Re = 800; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.29 Backward facing step flow: length of the vortex behind the step normalized by the step height s . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.30 Backward facing step flow: contour plots of velocity at different time instants for Re = 800, time step ∆t = 0.02 . . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 4.1 4.2 4.3 4.4 4.5 4.6

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Fictitious domain method: a schematic description . . . . . . . . . . . . B-spline basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Support of a equal-order b-spline basis function in 2D . . . . . . . . . . Two-scale relation of the b-spline functions . . . . . . . . . . . . . . . . Hierarchical refinement near the solid body . . . . . . . . . . . . . . . . Conservation test: geometry of the problem . . . . . . . . . . . . . . . . Conservation test: mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservation test: pressure contours . . . . . . . . . . . . . . . . . . . . Conservation test: X-velocity contours . . . . . . . . . . . . . . . . . . . Conservation test: X-velocity profiles along opening AC in Figure 3.6 . . Flow past a circular cylinder: mesh and 3 levels of hierarchical refinements. Number of DOFs: 27938 . . . . . . . . . . . . . . . . . . . . . . Flow past a circular cylinder: velocity contour plots. . . . . . . . . . . . Flow past a circular cylinder: pressure contour plots. . . . . . . . . . . . Flow past a circular cylinder: history of drag (left) and lift(right) coefficients, Re=100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow past a circular cylinder: history of drag (left) and lift(right) coefficients, Re=200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow past a circular cylinder: drag coefficient (left) and Strouhal number(right) v.s Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Kinematics of the beam . . . . . . . . . . . . . . . . . . Finite rotation of a vector [1] . . . . . . . . . . . . . . . Frequencies of a undamped cantilever beam in (a) 1st mode; (c) 3rd mode. . . . . . . . . . . . . . . . . . . . . A curved cantilever under free-end force . . . . . . . . . A cantilever beam subject to a pure bending moment . Deformed shape of the pure bending cantilever beam . .

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4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19

Deflection curve of the cantilever beam. (a) X-displacement; (b) Ydisplacement; (c) rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation of the cantilever beam. (a) Euler-Bernoulli beam; (b) geometrically exact beam 2D; (c) geometrically exact beam 3D. . . . . . . . Deformation of the cantilever beam from a circular configuration. . . . . A frame structure subject to a concentrated load, 20 elements. . . . . . Deformation of the frame. (a) Euler-Bernoulli beam; (b) 2D geometrically exact beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflection curve of the frame . . . . . . . . . . . . . . . . . . . . . . . . A cantilever beam subjected to end moment and out-of-plane force. . . Deformed shape of a beam bent in helical form at different load (M ) steps: 10π, 30π, 50π, 70π, 90π, 110π, 130π, 150π, 170π, 190π . . . . . . Deformed shape of a beam bent in helical form at different load (M ) steps: 20π, 40π, 60π, 80π, 100π, 120π, 140π, 160π, 180π, 200π . . . . . A cantilever beam subject to a time varying force . . . . . . . . . . . . . Evolution of cantilever tip displacement . . . . . . . . . . . . . . . . . . Right-angle cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of out-of-plane displacement of right-angle cantilever beam. (a) elbow; (b) tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Classification of solution algorithms [2]. . . . . . . . . . . . . . . . . . . . 88

6.1

Flow-induced vibrations of a flexible beam: geometry and boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow-induced vibrations of a flexible beam: hierarchical b-spline mesh with two levels of refinements. . . . . . . . . . . . . . . . . . . . . . . . . . Flow-induced vibrations of a flexible beam: vertical displacement of the tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow-induced vibrations of a flexible beam: contour plots of velocity at different time instants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flexible fibre in soap film: geometry and boundary conditions . . . . . . . Flexible fibre in soap film: mesh with hierarchical refinement . . . . . . . Flexible fibre in soap film: tip displacement and drag history for different inflow velocities, EI = 4.17e − 8. . . . . . . . . . . . . . . . . . . . . . . . Flexible fibre in soap film: tip displacement and drag history for different inflow velocities, EI = 1.25e − 7. . . . . . . . . . . . . . . . . . . . . . . . Flexible fibre in soap film: tip displacement and drag history for different inflow velocities, EI = 1.25e − 6. . . . . . . . . . . . . . . . . . . . . . . . Flexible fibre in soap film: deformed shapes at t = 0.085 with different inflow velocities and bending stiffnesses EI = 1.25e − 6 (left), EI = 1.25e − 7 (middle), EI = 4.17e − 8 (right) . . . . . . . . . . . . . . . . . . Flexible fibre in soap film: drag forces versus inflow velocity and its scaling analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flexible fibre in soap film: contour plots of vorticity at time t = 0.085 with different bending stiffnesses. The inflow velocity is u = 2.5. . . . . . Neutrally buoyant fibre in shear flow: geometry and boundary conditions Neutrally buoyant fibre in shear flow: mesh with hierarchical refinement . Neutrally buoyant fibre in shear flow: orbit classes [3] . . . . . . . . . . .

6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

6.11 6.12 6.13 6.14 6.15

x

94 94 95 96 97 97 98 98 99

99 100 101 102 102 103

6.16 Neutrally buoyant fibre in shear flow: fibre deformation at different time instants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Two flapping leaves: geometry and boundary conditions. . . . . . . . . . 6.18 Two flapping leaves: hierarchical refinements near the immersed leaves . 6.19 Two flapping leaves: tip displacements for Level-0 mesh with different time steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.20 Two flapping leaves: tip displacements for Level-2 mesh with different time steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.21 Two flapping leaves: tip displacements for different mesh levels with ∆t = 0.0025. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.22 Two flapping leaves: X-velocity contour plots at t = 0.5, ∆t = 0.0025. . 6.23 Two flapping leaves: pressure contour plots at t = 0.5, ∆t = 0.0025. . . 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18

Two flapping fibres in 3D: geometry and boundary conditions . . . . . . Two flapping fibres in 3D: level-0 mesh (9375 elements and 27556 DOFs). 7 beam elements for each fibre. . . . . . . . . . . . . . . . . . . . . . . . Two flapping fibres in 3D: level-1 mesh (13575 elements and 35586 DOFs). 14 beam elements for each fibre. . . . . . . . . . . . . . . . . . . . . . . Two flapping fibres in 3D: level-2 mesh (29575 elements and 70006 DOFs). 28 beam elements for each fibre. . . . . . . . . . . . . . . . . . . . . . . Two flapping fibres in 3D: X-displacements of the fibre tip with different time steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two flapping fibres in 3D: Y-displacements of the fibre tip with different time steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two flapping fibres in 3D: Z-displacements of the fibre tip with different time steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two flapping fibres in 3D: tip displacements for different levels of mesh with ∆t = 0.0025. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two flapping fibres in 3D: X-velocity contour plots at Z = 1 and t = 2.5 with ∆t = 0.0025. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two flapping fibres in 3D: X-velocity contour plots at Y = 1 and t = 2.5 with ∆t = 0.0025. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two flapping fibres in 3D: pressure contour plots at Z = 1 and t = 2.5 with ∆t = 0.0025. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two flapping fibres in 3D: pressure contour plots at Y = 1 and t = 2.5 with ∆t = 0.0025. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fibre array: geometry and boundary conditions. . . . . . . . . . . . . . Fibre array: level-1 mesh (16935 elements and 46664 DOFs). 20 beam elements for each fibre. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fibre array: velocity and pressure contour plots at Z=1.0 and t=0.4. . . Fibre array: velocity and pressure contour plots at Y=1.0 and t=0.4. . . Fibre array: Deformation of fibre array at different time instants. . . . . Fibre array: streamlines plot at t = 0.4 . . . . . . . . . . . . . . . . . . .

xi

. 104 . 104 . 105 . 107 . 107 . 107 . 108 . 108 . 110 . 111 . 112 . 112 . 113 . 113 . 113 . 114 . 115 . 116 . 117 . 118 . 119 . . . . .

120 121 122 123 124

List of Tables 3.1 3.2

Conservation test: flow rate through BC for b-spline elements. The theoretical value is 0.75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Flow past a circular cylinder: different cases . . . . . . . . . . . . . . . . . 52

4.1 4.2 4.3

Cantilever bend free-end displacement components . . . . . . . . . . . . . 74 Residual norm after the 2nd and 6th load steps . . . . . . . . . . . . . . . 75 Tip displacement components under M = 2.5π . . . . . . . . . . . . . . . 76

6.1

Flow-induced vibrations of a flexible beam: parameters used in the simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow-induced vibrations of a flexible beam: amplitude and frequency of tip displacement oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . Flexible fibre in soap film: parameters used in the simulations. . . . . . . Neutrally buoyant fibre in shear flow: parameters used in the simulations. Neutrally buoyant fibre in shear flow: parameters used in the simulations.

6.2 6.3 6.4 6.5

xii

94 95 97 103 105

Chapter 1

Introduction 1.1

Motivation

Fluid-structure interaction (FSI) plays a prominent role in many engineering applications, such as fluttering and buffeting of bridges [4], aeroelastic response of airplanes [5], opening and drag of parachutes [6], particle sedimentation [7], blood flow through arteries and heart valves [8], flying and swimming [9, 10], fluid-particle interaction in geomaterials [11–14]. See Figure 1.1 for some of these applications. Due to the strong nonlinearity and multidisciplinary nature of most FSI problems, analytical solutions are impossible to obtain, whereas laboratory experiments are limited in scope. Thus, numerical methods serve as a crucial way to study the FSI phenomena. In the past few decades, a variety of numerical methods have been developed to gain a deep insight into these problems, and we refer [15] for a detailed review of some numerical methods. Among different FSI applications, the interaction between incompressible fluid and flexible fibres (see its application in pulp and paper manufacture, polymer melts, and fibrereinforced composite materials in [16, 17] ) has its own difficulties due to the small solid over fluid mass ratio and the large deformation of the fibres. The first factor results in a large added mass which normally requires the FSI system to be solved in monolithic solution strategies [18, 19], whereas the second feature usually leads to a large distortion of the fluid mesh and hence re-meshing procedure is necessary [20]. Also, a proper choice of the model for the slender fibre with extremely large aspect ratio especially in three dimensional space is a challenging task [21]. The reliable prediction of the inter-fibre hydrodynamics and thus active transport, is dependent on robust modelling procedures for strong coupling of the fluid dynamics and

1

fibre array mechanics. The principal challenge is to define a robust and efficient computational framework for fluid-structure interaction incorporating large fibre deformations under different fluid flow scenarios and the effect this may have on fluid flow.

(a) Collapse of the Tacoma bridge

(b) Aerospace engineering

(c) Opening of parachutes

(d) Blood flow in vessels

Figure 1.1: Some applications of fluid-structure interaction.

1.2 1.2.1

FSI solution procedures Monolithic and partitioned approaches

The numerical strategies to solve FSI problems can roughly be classified into two approaches: the monolithic approach and the partitioned approach, in which the subsystems are solved simultaneously or sequentially, respectively, as shown in Figure 1.2. The monolithic approach [22–24] formulates a single equation system for the entire coupled problem, and solve the fluid and structure dynamics simultaneously by a unified algorithm. The interaction of the fluid and the structure at the interface is implicitly included in the solution procedure. This approach is often chosen to ensure the accuracy, stability and convergence of the coupled solution, but it requires more expertise to develop and maintain such a specialized code, and the computational cost is high. 2

In addition, the formulation of a single equation system may lead to ill-conditioned matrices.

S f (tn )

S f (tn+1 )

S s (tn )

S s (tn+1 ) (a) Monolithic approach

S f (tn )

S f (tn+1 )

Interface

Interface

S s (tn )

S s (tn+1 ) (b) Partitioned approach

Figure 1.2: Schematic of the monolithic and partitioned approaches for FSI, where S f and S s denote the fluid and structure solutions, respectively, whereas tn and tn+1 represent, respectively, the time instant n and n + 1.

On the contrary, in partitioned approaches [25–27], the fluid and structure are treated as two computational fields and solved separately. The interface conditions are enforced explicitly to transfer the information between the fluid and structure sub-systems. The sequential process allows for existing sub-solvers and the software modularity is preserved. However, the conservation properties of the continuum fluid-structure system are lost due to the time lag between the time integration of fluid and structure. Therefore, the partitioned algorithms are usually only conditionally stable and hence a small time step is necessary. Moreover, the accuracy of the solution may also be affected.

1.2.2

Conforming and non-conforming mesh methods

Based on the treatment of meshes, the FSI solution strategies can be classified into conforming mesh methods and non-conforming mesh methods. The conforming mesh methods, usually referred to as Arbitrary Lagrangian-Eulerian (ALE) methods, treat the fluid-structure interface location as part of the solution, and adopt body-fitted mesh to follow the movement of the interface. Owing to the deformation of the solid structure, mesh-update or re-meshing is required, which makes this method cumbersome in FSI problems with large structural deformations. A detailed description of ALE methods can be found in Chapter 14 in [28] and references therein. On the other hand, in non-conforming mesh methods, the interface location and the corresponding boundary conditions are treated as constraints, and thus non-conforming meshes can be employed. The immersed boundary method and its variation may be the most well-known non-conforming mesh methods. The distinguishing feature of this kind 3

of methods is the fact that the simulation is performed on a fixed Eulerian Cartesian grid, which does not conform to the current geometry of the deformed immersed structure [29]. The difference between the conforming and non-conforming meshes can be clearly observed in Figure 1.3.

(a) Initial configuration

(b) Conforming mesh (with mesh-update) method

(c) Conforming mesh (with remeshing)

(d) Non-conforming mesh

Figure 1.3: Examples of conforming and non-conforming meshes.

1.3

Immersed boundary method

The immersed boundary method (IBM) was developed by Peskin and his collaborators [30–32] for the simulation of the interaction of incompressible viscous flow and immersed elastic bodies. In IBM and its variation [33–36] the kinematic constraint at fluid-solid interface is weakly enforced via an elastic force which is obtained from the solid configuration. A discrete Dirac delta function is utilised to transfer the force and velocity between the Eulerian and Lagrangian reference frames. However, this method restricts the time steps to small values for both implicit or explicit fluid solvers [37]. Motivated by IBM, the immersed interface method (IIM) was introduced by [38–41] to deal with interface problems with complicated boundaries and obtain a higher order accuracy. In this approach, use is made of the jump conditions across the interface, which makes the method applicable to only fluid-structure interaction (FSI) problems with bulky solids. In most of the research work carried out with IBM and IIM the fluid is discretised by 4

finite difference or finite volume methods, which leads to the lack of local refinement capability. Finite element based IBM and IIM are investigated in [42, 43]. However, the amount of research in such methods is limited, and these methods still suffer from the disadvantages of IBM such as small time step limitation.

1.4

Aim of the thesis

This thesis aims to develop a robust and efficient computational framework to solve the interaction between fluid flow and flexible fibres. The focus is restricted on laminar incompressible Newtonian fluid flow and three-dimensional fibres with large deformations. The solution strategy to be developed is based on fictitious domain/distributed Lagrange multiplier method and hierarchical b-splines. Galerkin based finite element method is applied for both fluid and solid discretisation, and a staggered scheme is adopted for solving the coupled system. Furthermore, a 3D geometrically exact beam formulation is discussed in detail and implemented into the FSI framework. The objective is to investigate the 3D fluid-flexible fibres interaction and the performance of the presented numerical strategy.

1.5

Layout of the thesis

Chapter 2. The incompressible Navier-Stokes equations and their weak form are introduced. Two numerical issues that in standard Galerkin solutions are illustrated by employing two representative model problems, i.e. the one-dimensional advection-diffusion problem and the steady Stokes flow. Several techniques which overcome the numerical deficiencies are then presented and compared. Finally, the stabilised SUPG/PSPG finite element method is introduced. After giving a brief introduction of different time integration schemes, three numerical examples are presented. Chapter 3. To provide a background for solving fluid-structure interaction, fictitious domain/distributed Lagrange multiplier (FD/DLM) method is presented to solve the incompressible fluid flow. In contrast to body-fitted mesh in the previous chapter, in FD/DLM a structured mesh is employed to discretise the fluid. First the fictitious domain method is introduced, including its concept, history, main features and the formulations in an FSI framework. To produce the hierarchical refinement of the fluid mesh, b-spline function is then presented. A conservation test is carried out to assess whether the use of FD/DLM with b-splines can circumvent the numerical oscillations

5

provided by standard Galerkin method, as seen in Chapter 2.The flow past a circular cylinder is studied again to validate the performance of the presented methodology. Chapter 4. The 2D and 3D geometrically exact beam formulations are presented. In 3D case, the representation of finite rotations is a main difficulty for beam element. In this work a vector-like parametrization is used for the rotation tensor, but the stiffness matrix and residual vector are obtained in a simpler and more straightforward way. For simplicity, the mass matrix is chosen by properly modifying the mass matrix of EulerBernoulli beam element. Finally, several of static and transient analyses are performed. Chapter 5. As both the fluid solver and solid solver have been established, the solution strategy for fluid-structure interaction is presented in this chapter. Several coupled algorithms are briefly reviewed, including strongly coupled and weakly coupled procedures. A force predictor-based staggered scheme is then discussed, and it is the one adopted in the current work. Chapter 6. Four numerical examples involving two-dimensional fluid-flexible fibre interaction are studied in detail to demonstrate the proposed computational framework for fluid-flexible fibre interaction. Chapter 7. Numerical examples in three dimensional space are presented in this chapter. Since there are few 3D fluid-flexible fibre interaction examples that can be found in literature, the author has created two problems associated with the interaction between fluid and flexible fibre and fibre arrays. The investigation on these two examples helps to assess the convergence of the numerical strategy in this work. Chapter 8. Some important conclusions and observations are drawn. The contributions made by this work are summarised. Some suggestions for future research are also given.

6

Chapter 2

Stabilised Finite Elements for Fluid Flow In this chapter the focus is on the solution of incompressible Navier-Stokes equations by employing finite element method. The fact that in history the finite element method was less popular than the finite difference and finite volume in the Computational Fluid Dynamics (CFD) community is mainly due to two reasons: the first reason is the lack of upwind techniques without which oscillatory velocity solutions occurs in high Reynolds number flows, whereas the second is that traditional velocity-pressure equal order element fails to satisfy the so-called Ladyzhenskaya-Babuska-Brezzi (LBB) condition, which, as a result, produces spurious pressure solutions. In addition, the absence of pressure in the continuity equation induces zeros diagonal entries in the matrix of linear systems of equations, and this gives difficulties to many iterative solvers. However, during the past three decades some stabilisation methods have been developed to overcome the issues addressed above, among which the streamline upwind PetrovGalerkin (SUPG) and pressure stabilising Petrov-Galerkin (PSPG) methods are best know. This chapter presents a SUPG/PSPG stabilised finite element method for incompressible fluid flow. First the Navier-Stokes equations and their weak form are introduced in Section 2.1. The employment of traditional Galerkin finite element method to solve N-S equations is demonstrated in Section 2.2, in which the mentioned oscillation problems are clearly illustrated. Sections 2.3 and 2.4, respectively, present some techniques to stabilise the velocity and pressure solutions, followed by a formulation based on SUPG and PSPG given in Section 2.5. In Section 2.6 some discrete time integration schemes are presented for solving the unsteady flows. At last a couple of numerical examples are shown in Section 2.7. 7

2.1 2.1.1

Navier-Stokes equations Governing equations

The Navier-Stokes equations for incompressible fluid flow, consisting of momentum equation and continuity equation, are given as ρ

∂u + (u · ∇) u = ∇ · σ + f ∂t ∇·u=0

in Ω

(2.1)

in Ω

(2.2)

where ρ denotes the fluid density, u the velocity vector, t the time, σ and f the Cauchy stress tensor and body force per unit mass, respectively. Equation (2.1) shows the conservation of momentum, and can be derived from the second Newton’s law. For an incompressible and isotropic fluid the Cauchy stress term σ can be expressed as σ = −pI + 2µ∇S u

(2.3)

where p, I and µ are the pressure, identity tensor and dynamic viscosity, respectively. The operator ∇S (·) denotes the symmetric part of the gradient, e.g. ∇S = 12 ∇ + ∇T . Substitute (2.3) into (2.1) by making use of the continuity equation (2.2) the momentum equation can be rewritten as ρ

∂u + (u · ∇) u − µ∇2 u + ∇p = f ∂t

in Ω

(2.4)

Since the unknown variables in (2.2) and (2.4) are velocity and pressure, this formulation is also known as mixed formulation of the Navier-Stokes equations. To form a well-posed initial boundary value problem, the Navier-Stokes equations need to be supported by proper initial and boundary conditions. A divergence free velocity distribution should be given as the initial condition, i.e. u = u0 ,

∇ · u0 = 0

in Ω

(2.5)

whereas the boundary conditions are typically of two types, specifying velocity and traction as Velocity(Dirichlet) BC:

u = uD

Traction(Neumann) BC: σ · n = tN 8

on ΓD

(2.6)

on ΓN

(2.7)

where n is the unit outward normal vector of the boundary Γ, t is the traction force. The Navier-Stokes equations (2.2) and (2.4), with the initial condition (2.5) and boundary condition (2.6) − (2.7) together govern the incompressible fluid flows, and act as the start of the finite element discretisation in the following sections.

2.1.2

Weak form

The first step towards the finite element discretisation of N -S equations is to construct their corresponding weak form. To do so, two classes of functions need to be defined: the test or weighting functions and the trial solutions. The space of test functions, denoted by W, consists of functions that are square integrable, have square integrable first derivatives over the domain Ω and vanish the Dirichlet boundary ΓD . It is defined as W = {w ∈ H1 (Ω)

nsd

|wΓD = 0}

(2.8)

where nsd = 2 or nsd = 3 is the dimension of space. The trial solutions have similar properties to the test functions expect they are required to satisfy the Dirichlet conditions on ΓD . The space of these trial solutions is denoted by U and is defined as U = {u ∈ H1 (Ω)

nsd

|uΓD = uD }

(2.9)

By projecting the Equations (2.2) , (2.4) and (2.5) − (2.7) onto the space of weighting

functions w ∈ W for the momentum equation and q ∈ P for the continuity equation, and by employing integration by parts and the divergence theorem, the weak form of Navier-Stokes equations is obtained, which can be stated as: given u0 , uD , f and t, find u ∈ U and p ∈ P, such that for all w ∈ W and q ∈ P Z ∂u + (u · ∇) u + µ∇w : ∇u − (∇ · w) p dΩ− w·ρ ∂t Ω Z Z Z w · f dΩ − w · tdΓ + q · (∇ · u) dΩ = 0

(2.10)

P = {p ∈ L2 (Ω)}

(2.11)

Ω

Γ

Ω

where

Note that there are no explicit boundary conditions on pressure, thus the space P can serve as the space for both trial solution and weighting function. In addition, since 9

spatial derivatives of pressure do not appear in (2.10), functions in P are required to be square-integrable only [44].

2.2

Galerkin finite element solution

2.2.1

Galerkin formulation

The Galerkin formulation can be obtained from (2.10) simply by replacing the space U , W and P with the finite element spaces. The end result is the following problem: find uh ∈ U h and ph ∈ P h such that for all wh ∈ W h and q h ∈ P h h Z ∂u h h h h h h + u · ∇ u + µ∇w : ∇u − ∇ · w ph dΩ− w ·ρ ∂t Ω Z Z Z h h h q h · ∇ · uh dΩ = 0 w · f dΩ − w · t dΓ + Ω

Γ

Ω

(2.12)

with n U h = {uh ∈ H 1 (Ω) sd |uhΩe ∈ (Pk (Ωe ))nsd = uD , uhΓD = uD } n W h = {wh ∈ H 1 (Ω) sd |whΩe ∈ (Pk (Ωe ))nsd = wD , whΓD = 0}

(2.13)

P h = {ph ∈ H 1 (Ω) |phΩe ∈ Pk (Ωe )} where Ωe is the element domain and Ω =

Snel

e=1 Ω

e,

and the space Pk (Ωe ) consists of all

the k-th order polynomials on Ωe . Note that the above standard Galerkin formulation leads to central approximations of the convective term uh · ∇ uh and thus the velocity undergoes spurious oscillations in convection dominated case, i.e. high Reynolds number flows. In addition, using velocitypressure equal order element results in the pressure oscillations. Both issues are shown in the following.

2.2.2

Velocity oscillation

To better illustrate the velocity oscillation, consider the one-dimensional, steady advectiondiffusion equation u

dT d2 T −ν 2 =f dx dx T (0) = 0 T (L) = 0 10

in [0, L] (2.14)

where T (x) is the scalar unknown, f is the force function, u and ν are the known constant velocity and diffusivity, respectively. The standard Galerkin finite element formulation yields the following elemental stiffness e and elemental matrix, written as the summation of elemental convection matrix Cij e. diffusion matrix Dij

e Kij

=

e Cij

+

e Dij

dNj uNi dx + = dx Ωe Z

Z ν Ωe

dNi dNj dx dx dx

(2.15)

where N is the shape function of finite element, i and j are the element nodes. Using e and D e matrices can be calculated as continuous piecewise linear elements, Cij ij

Z

e

C =u

" 1 N1 dN dx 1 N2 dN dx

Ωe

De = ν

Z Ωe

"

dN1 dx dN2 dx

" # u −1 1 dx = 2 2 −1 1 N2 dN dx " # # dN1 dN2 1 −1 ν dx dx dx = e dN2 dN2 h −1 1 2 N1 dN dx

dN1 dx dN1 dx

dx

#

(2.16)

(2.17)

dx

where he is the element length. It is clearly seen that the diffusion term is symmetric, whereas the advection term is not. Assuming the force function f to be constant gives the elemental force vector as Fe = f

N1

Z Ωe

!

N2

f he dx = 2

! 1

(2.18)

1

By assembling the elemental stiffness matrix and force vector to the global system, the discretised equation is obtained as Kg · T g = F g

(2.19)

where K g , T g and F g are, respectively, the global stiffness matrix, nodal unknowns and force vector. The algebraic equation for an interior node n can be written as u ν (Tn+1 − Tn−1 ) − e (Tn+1 − 2Tn + Tn1 ) = f he 2 h

(2.20)

Dividing the equation by he gives u

Tn+1 − Tn−1 2he

−ν

Tn+1 − 2Tn + Tn−1 (he )2

=f

which is the same as the equation obtained by central difference scheme.

11

(2.21)

To study advection-diffusion equation, the use of elemental Peclet number P e =

uhe 2ν

helps to distinguish diffusion dominated (P e < 1) and convection dominated (P e > 1) problems, since this number is ratio of convection and diffusion. We solve the above one-dimensional advection-diffusion problem with u = 1, f = 1 and L = 1 on a mesh of 10 equi-length elements, which gives element length he = 0.1. Figure 2.1 shows the results obtained by Galerkin finite element method and the exact solution[44] Texact

1 = u

1 − eux/ν x− 1 − eu/ν

! (2.22)

for different elemental Peclet numbers.

(a)

(b)

(c)

(d)

Figure 2.1: Galerkin finite element solution of 1D, steady AD equation for different P e. (a) P e = 0.25; (b) P e = 1; (c) P e = 2; (d) P e = 5;

As seen, the exact solution develops a boundary layer at the outflow of the boundary as P e increases. For high P e cases the numerical results show strong node to node oscillations. This oscillatory behaviour is clearly observed for P e > 1.

12

2.2.3

Pressure oscillation

Consider the steady Stokes problem −µ∇2 u + ∇p = f

in Ω

(2.23)

∇·u=0

in Ω

(2.24)

on ΓD

(2.25)

Velocity(Dirichlet) BC: u = uD

which can be directly obtained by neglecting the acceleration term and convective term of the Navier-Stokes equations in (2.4), (2.2) and (2.6). For simplicity, pure Dirichlet boundary conditions are taken into account. The Galerkin formulation of the Stokes equations is obtained in a similar way to (2.12): finduh ∈ U h and ph ∈ P h such that for all wh ∈ W h and q h ∈ P h Z h Ω

h

h

µ∇w : ∇u − ∇ · w

h

p

h

i

dΩ −

Z

h

Z

w · f dΩ + Ω

Ω

q h · ∇ · uh dΩ = 0 (2.26)

If U h , P h and W h are chosen the same as in (2.12), then the formulation (2.26) gives spurious oscillatory pressure filed. The lid-driven cavity problem is investigated with equal order linear velocity-pressure elements to demonstrate the pressure oscillation issue. Figure 2.2 depicts the geometry, boundary conditions and the mesh, in which 400 quadrilateral elements are used. Both width and height of cavity are 1, the horizontal velocity at the lid u0 = 1, and the fluid viscosity µ = 0.01.

Figure 2.2: Definition of the lid-driven cavity problem and a mesh of 400 quadrilateral elements

Figure 2.3 shows the velocity contour and streamlines obtained by using the standard Galerkin formulation (2.26) and bilinear quadrilateral elements. The solutions seem to be reasonable. However, the pressure distribution rendered with the same formulation, shown in Figure 2.4, has large un-physical oscillations, and they do not disappear as the 13

mesh is refined. The solutions obtained by a stabilised formulation are also given here, and this formulation will be discussed in Section 2.4. Note that the solution still has oscillations in the regions close to the top corners, which results from the sharp change of the horizontal velocity at the two top corner nodes.

Figure 2.3: Velocity contour and streamlines obtained using GFEM with 400 quadrilateral elements

(a)

(b)

(c)

(d)

Figure 2.4: Pressure contour plots of lid-driven cavity problem with 400 quadrilateral elements. (a) 2d pressure, Galerkin; (b) 3d pressure, Galerkin; (c) 2d pressure, PSPG; (d) 3d pressure, PSPG

14

2.3 2.3.1

Velocity stabilisation A Petrov-Galerkin formulation

The spurious oscillations seen in Section 2.2.2 stem from the truncation error in the diffusion term given by the standard Galerkin formulation. Precisely, the Galerkin formulation (and central difference scheme as well) provides a solution with less diffusivity than the actual one. Therefore the idea is to use an increased artificial diffusivity in the equation. In the framework of the finite difference method this can be achieved by using first-order upwind differencing for the convective term rather than the central differencing. In a finite element framework, however, the upwind effect can be provided by a Petrov-Galerkin formulation in which the weight functions are different from the element shape functions.

1 N1e+1

N2e e+1

e n−1

n

n+1

Figure 2.5: shape functions of 1D linear elements

Consider a one dimensional mesh with equal length linear elements. As shown in Figure 2.5, the shape functions that contribute to node n are N2e and N1e+1 , which are piece wise linear 1 (1 + ξ) 2 1 = (1 − ξ) 2

N2e = N1e+1

(2.27)

where ξ is the coordinate in master element. In Petrov-Galerkin method, rather than choosing weight functions the same as shape functions, weight functions are distorted to the upwind side as shown below. This kind of distorted test functions give more weight to the upwind element e than to the downwind element e + 1, and can be obtained by adding bubble functions to the

15

1 w1e+1

w2e e n−1

e+1 n

n+1

Figure 2.6: weight functions of 1D linear elements in Petrov GFEM

original linear shape functions, for example 1 (1 + ξ) + 2 1 = (1 − ξ) − 2

w2e = w1e+1

3 β 1 − ξ2 4 3 β 1 − ξ2 4

(2.28)

where β is a parameter that controls the amount of upwinding. Apparently β = 0 gives traditional Galerkin formulation. By following the procedure in Section 2.2.2, the elemental stiffness matrix now becomes e Kij

=

e Cij

+

e Dij

" # " # ν + uβhe /2 −1 1 u −1 1 + = 2 −1 1 he −1 1

(2.29)

It is clearly seen that the modified weight functions lead to an artificial diffusion, whose amplitude is ν¯ = uβhe /2. There is an optimum value of the upwinding parameter β = coth (P e) − 1/P e [44]. The previous advection-diffusion example for P e = 5 is

studied again by using Petrov-GFEM, and the results obtained with different β values

are shown in Figure 2.7. Stable solutions can be obtained by tuning β, and when the value is larger than its optimum (0.8001 in this example) the results are excessively diffusive.

16

(a)

(b)

(c)

(d)

(e)

Figure 2.7: Petrov GFEM solution of 1D, steady AD equation for different β, P e = 5. (a) β = 0; (b) β = 0.5; (c) β = 1; (d) β = 2; (e) β = 0.8001;

17

2.3.2

Stabilised methods

Proposed by Hughes and co-workers [45] [46], several stabilisation techniques are popular in convection dominated problem because they stabilise the convective term in a consistent manner. These methods introduce an extra term to the Galerkin weak form over the element domain, and this term is a function of the residual of the differential equation which ensures the consistency. Hence this class of stabilised formulations is called consistent Petrov Galerkin formulation. Consider a two dimensional AD equation u · ∇T − ∇ · (ν∇T ) = f

(2.30)

R = L (T ) − f

(2.31)

L ( ) = u · ∇ ( ) − ∇ · [ν∇ ( )]

(2.32)

whose residual is given as

where L ( ) is a differential operator

Consistent Petrov GFEM stabilisations can be generalised as the following weak form Z Ωe

[wu · ∇T + ∇w · (ν∇T )] dΩ +

Z Ωe

P (w) τ R (T ) dΩ =

Z

Z f wdΩ +

Ωe

whΩ (2.33) Γe

where h is the normal flux on the boundary, and the second term is the stabilisation term in which τ is a stabilisation parameter, and different choices of P operator results in different stabilised methods.

2.3.2.1

SUPG method

One of the most popular stabilised method is known as Streamline Upwind Petrov Galerkin (SUPG) which is defined by taking P = u · ∇w and the parameter τ is selected as τ=

ν¯ |u|2

(2.34)

(2.35)

where ν¯ = uβhe /2 as previously defined. Some alternative definitions of τ can be found in [44].

18

Apply SUPG to the one dimensional AD equation in previous sections with constant u and ν we have P=u

dw , dx

τ=

ν¯ βhe = , u2 2u

R (T ) = u

dT d2 T −ν 2 −f dx dx

(2.36)

Use these in 2.33, and note that no integration by parts is applied to the higher order derivatives in the stabilisation integral, we obtain the following element stiffness matrix and force vector βhe dNi dNj dNi dNj βhe dNi d2 Sj K = u Ni + dΩ +ν +ν 2 dx dx dx dx 2 dx dx2 Ωe Z βhe dNi e f Ni + dΩ F = 2 dx Ωe e

Z

(2.37) (2.38)

If linear elements are adopted, the term involving second derivative of the unknown vanishes and the element stiffness reduces to βhe dNi dNj dNi dNj K = u Ni + +ν dΩ 2 dx dx dx dx Ωe e

Z

(2.39)

which actually leads to the same stiffness as the one obtained using Petrov-Galerkin formulation. However, note that this is only valid for the employment of linear elements, and even with linear elements SUPG will give better results than Petrov-Galerkin method since the force term has been modified.

2.3.2.2

GLS method

An alternative stabilised technique is Galerkin Least Squares (GLS) formulation in which P = L = u · ∇w − ∇ · (ν∇w)

(2.40)

In this formulation, the least squares term of the residual of the original equation is added to the standard Galerkin formulation. As seen, the first term in P operator is

the same as the one in SUPG, and the second term has second order derivatives which vanishes for linear element interpolations. Therefore SUPG and GLS provide the same

results for AD equation solved with linear elements, but this is not the case for other differential equations.

19

To make a comparison of different stabilisation techniques, consider the following 1D AD problem u

d2 T dT − ν 2 = 10e−5x − 4e−x , dx dx T (0) = 0

in [0, 1] (2.41)

T (1) = 1 A non-constant force is used to better observe the difference between different formulations. The mesh refinement and U value are taken the same as in the previous 1D AD problem, and by changing ν, two different Peclet numbers, 0.25 and 5, are selected. Figure 2.8 shows the exact solutions and the results obtained by Galerkin, Petrov-Galerkin and SUPG formulations. Note that the optimum value of β = coth (P e) − 1/P e is used for Petrov-Galerkin and SUPG.

(a)

(b)

Figure 2.8: Comparison of different stabilisations for 1D AD equation. (a) P e = 0.25; (b) P e = 5

As seen in the figure, for low P e all three formulations provide nice results, however, to obtain a reasonable solution stabilisation is necessary for high P e case. As mentioned above, although linear elements produce same stiffness matrix for Petrov-Galerkin and SUPG, the force vector in SUPG is modified to give a more accurate result. Also note that, as mentioned, GLS provide the same results as SUPG in this problem.

20

2.4 2.4.1

Pressure stabilisation PSPG method

As mentioned in Section 2.2.3, the failure of the Galerkin formulation with a velocitypressure equal order element is due to the violation of the Ladyzhenskaya-Babuska-Brezzi (LBB) condition. One way to circumvent this condition is the pressure-stabilising/PetrovGalerkin (PSPG) method [47], which adds an extra term to the formulation (2.26) as Z h Z Z i h h h h h µ∇w : ∇u − ∇ · w p dΩ − w · f dΩ + q h · ∇ · uh dΩ+ Ω

Ω

nel Z X e=1

Ω

e

Ωe

h

h

2 h

(2.42)

τ (h , µ) ∇q · ∇p − µ∇ u − f = 0

where τ is a parameter which tends to zero as the mesh is refined, and this property guarantee the consistency with the strong form of the problem. Similar to the SUPG for AD equation, no integration by parts is used in the stabilisation term, and the diffusion part disappears if linear elements are employed. The performance of the method is already shown in the lid-driven cavity problem in Section 2.2.3, where a smooth and reasonable pressure filed is obtained by using piecewise linear, velocity-pressure equal order elements. The stabilisation parameter was selected as [2] τ (he , µ) =

(he )2 12µ

(2.43)

where µ is the dynamic viscosity of the fluid, he is element size defined by r he =

4Ae π

(2.44)

in which Ae is the element area.

2.4.2

GLS method

The Galerkin Least Squares (GLS) method can be obtained by simply replacing ∇q h with ∇q h + µ∇2 wh in (2.42), and similar to the stabilisation methods for AD equation, GLS scheme is reduced to PSPG for Stokes problem when linear element is employed.

21

2.4.3

LBB-stable element

Rather than to circumvent the LBB condition in PSPG and GLS, some special elements with higher order interpolations for velocity than pressure can be adopted to satisfy this compatibility condition, and therefore provide non-oscillatory pressure solutions. Among these LBB-stable elements, Taylor Hood elements are typical one, as shown in Figure 2.9

Figure 2.9: Triangular (P2 P1 ) and quadrilateral (Q2 Q1 ) Taylor Hood elements

Note that in the figure the solid circles represent velocity nodes whereas the others denote pressure nodes. As clearly seen, in this type of elements, the velocity is interpolated in a quadratic or bi-quadratic way, while the pressure is in a linear or bilinear way, and both are continuous. Another type of LBB-stable element, known as the MINI element, can be obtained by adding a bubble functions to the velocity space of the piecewise linear element, which can be seen in the figure below. The pressure is continuous and piecewise linear. Interestingly, the MINI element formulation can be proved to be equivalent to the PSPG method [2].

Figure 2.10: MINI (P1+ P1 ) element

There are some other LBB-stable elements such as Crouzeix-Raviart family, and more details can be found in references [44] and [48]. However. these elements are either more complex or provide a discontinuous pressure interpolation, and hence are less popular than the Taylor Hood and MINI elements.

22

2.5

SUPG/PSPG formulation

In Section 2.3, the SUPG formulation was introduced to solve the advection-diffusion problem. It suppressed the unphysical oscillation of the solution in convection dominated situation, and provided reliable results. In addition, in Section 2.4 the use of PSPG method in Stokes equation successfully circumvented the LBB condition and gave a non-oscillatory pressure distribution. Therefore, a stabilised formulation consists of the SUPG and PSPG strategies that are employed to model the unsteady Navier-Stokes equations. This formulation, often referred to as the SUPG/PSPG formulation [49], allows the use of piecewise linear, velocity-pressure equal order element, and can be stated as: find uh ∈ U h and ph ∈ P h such that for all wh ∈ W h and q h ∈ P h Z

h

w ·ρ Ω

∂uh h h h h h + u · ∇ u + µ∇w : ∇u − ∇ · w ph dΩ− ∂t Z Z Z h h h w · f dΩ − w · t dΓ + q h · ∇ · uh dΩ+ Ω

nel Z X e=1

Ωe

h

Γ

Ω

i ∂uh h h h h h h τu ρ u · ∇ w + τp ∇q · ρ + u · ∇ u + ∇p − f dΩ = 0 ∂t

(2.45) with U h = {uh ∈ H 1 (Ω)

nsd

|uhΩe ∈ (P1 (Ωe ))nsd = uD , uhΓD = uD }

n W h = {wh ∈ H 1 (Ω) sd |whΩe ∈ (P1 (Ωe ))nsd = wD , whΓD = 0}

(2.46)

P h = {ph ∈ H 1 (Ω) |phΩe ∈ P1 (Ωe )} The SUPG/PSPG formulation (2.45) consists of the Galerkin formulation (2.12) and a stabilisation term, in which the diffusion term involving second derivative of the velocity is neglected due to the employment of piecewise linear element. The use of two different stabilisation parameters, τu and τp , enables us to control the velocity and pressure stabilisations independently. Different choices of these two parameters are mentioned and discussed in references [44] and [2]. For simplicity, in this context both τu and τp are chosen to be the same as the one in (2.43), that is τu = τp =

23

(he )2 12µ

(2.47)

2.6

Discrete time integration schemes

For unsteady problem, a proper time integration scheme is usually required. In this work the semi-discrete procedure is adopted, which indicates the combination of the discrete time integration schemes and the spatial finite element discretisations. Three implicit schemes, backward Euler method, generalised midpoint rule and generalised-α method are introduced in the following context.

2.6.1

Backward Euler method

In backward Euler method, the governing equations are formulated at time instant n+1, and the time derivative of the unknown variable reads u˙ n+1 =

un+1 − un ∆t

(2.48)

where ∆t is time increment, un+1 and u˙ n+1 are, respectively, the velocity and acceleration in time instant n + 1, whereas un is the velocity at time instant n. Backward Euler method is simple and straightforward, but gives a result with only first order accuracy.

2.6.2

Generalised midpoint rule

Within the generalised midpoint rule all the unknown variables and its time derivative and the right hand side are approximated at time instant tn+γ as un+γ = γun+1 + (1 − γ) un

(2.49)

pn+γ = γpn+1 + (1 − γ) pn

(2.50)

un+1 − un ∆t

(2.51)

tn+γ = γtn+1 + (1 − γ) tn

(2.52)

u˙ n+γ =

where 0 ≤ γ ≤ 1. The unconditional stability requires γ ≥ 0.5. For γ = 0.5 the integration scheme becomes

second order accurate, otherwise it is first order accurate. Note that when the spectral radius ρ∞ =

(1−γ) γ

= 0, the backward Euler method is obtained, whereas ρ∞ = 1 yields

the trapezoidal rule.

24

2.6.3

Generalised-α method

For generalised-α method the velocity, pressure and traction force are approximated at time instant tn+αf , while the acceleration is evaluated at time instant tn+αm as un+αf = (1 − αf ) un + αun+1

(2.53)

pn+αf = (1 − αf ) pn + αpn+1

(2.54)

tn+αf = (1 − αf ) tn + αtn+1

(2.55)

u˙ n+αm = (1 − αm ) u˙ n + αm u˙ n+1

(2.56)

un+1 = un + ∆t ((1 − γ) u˙ n + γ u˙ n+1 )

(2.57)

with relation

These expressions give u˙ n+αm

αm αm αm = un+1 − un + 1 − u˙ n γ∆t γ∆t γ

u˙ n+1 =

1 1 1−γ un+1 − un − u˙ n γ∆t γ∆t γ

(2.58) (2.59)

The condition for second order accuracy reads γ=

1 + αm − αf 2

(2.60)

For the method to be unconditionally stable and second order accurate, αm and αf are usually chosen as αm =

1 3 − ρ∞ , 2 1 + ρ∞

αf =

1 1 + ρ∞

(2.61)

where 0 ≤ ρ∞ ≤ 1. Similar to the generalised midpoint rule, the generalised-α method

becomes identical to the trapezoidal rule if ρ∞ = 1.

By employing any one of these implicit time integration schemes, all unknown variables in Equation (2.45) are to be solved at time instant tn+1 . However, before applying the finite element method for spatial discretisation, this non-linear system need to be linearised. Usually Newton-Raphson method (see Appendix A for an introduction) is adopted for the linearisation.

25

2.7 2.7.1

Numerical examples Lid-driven cavity problem

The classical lid-driven cavity benchmark problem is displayed in Figure 2.11. The flow is inside a square cavity with a side length of L = 1. On the left, right and bottom boundaries the velocity is fixed to be zero, while on the top wall (lid) the horizontal velocity is prescribed as u0 = 1. Besides, pressure at the lower left corner is set to zero. By selecting a constant fluid density ρ = 1 and three different dynamic viscosities µ = 0.01, µ = 0.001 and µ = 0.0003125, the problem is studied at three Reynolds numbers Re = ρu0 L/µ = 100, Re = 1000 and Re = 3200, respectively.

u = u0 , v = 0

u = 0, v = 0

u = 0, v = 0

u = 0, v = 0 p=0 Figure 2.11: Lid-driven cavity: geometry and boundary conditions

Three different mesh refinements with bilinear quadrilateral elements and linear triangular elements are employed in this study, as shown in Figure 2.12.

(a)

(b)

(c)

Figure 2.12: Lid-driven cavity: mesh with (a) 400 quadrilateral elements; (b) 2048 triangular elements; (c) 8192 triangular elements

26

As discussed in Section 2.2, the standard Galerkin formulation fails in producing a nonoscillatory pressure filed in Stokes problem, and it has the same behaviour in unsteady N-S flows. In fact, the Galerkin formulation diverges in the case with Re = 1000 and Re = 3200, which verifies the velocity oscillation problem as discussed previously. The velocity contours and streamlines for different Reynolds number with 8192 elements are illustrated in Figures 2.13-2.15. It is seen that for Re = 100, two small vortices are observed at the bottom corners of the cavity. As the Reynolds number increases the core of centreline vortex moves towards the centre of cavity, and the vortices at bottom corners grow larger. The velocity components profile along the horizontal and vertical lines through the centre of cavity are also investigated. These diagrams, along with the reference results from Ghia et al [50], are displayed in Figures 2.16-2.18. For Re = 100 the results provided by different meshes match reference solution well, but only a dense mesh gives an acceptable accuracy when Re > 1000.

27

Figure 2.13: Lid-driven cavity: velocity contour plot and streamlines obtained with 8192 elements, Re = 100

Figure 2.14: Lid-driven cavity: velocity contour plot and streamlines obtained with 8192 elements,Re = 1000

Figure 2.15: Lid-driven cavity: velocity contour plot and streamlines obtained with 8192 elements,Re = 3200

28

Figure 2.16: Lid-driven cavity: u and v velocity profiles along x = 0.5 and y = 0.5 for Re = 100.



29

2.7.2

Flow past a circular cylinder

In this example the flow around a circular cylinder is considered. This problem has been often used as a benchmark to test the numerical solution algorithms. The geometry and boundary conditions of this problem are shown in Figure 2.19, in which U , V , p, d denote the velocity components in x and y directions, the pressure and diameter of the cylinder, respectively. The analysis domain has a size of 30 × 30, while the cylinder has a diameter of 1 and its origin is at (10,15).

v=0

15

u = uin v=0

d=1

p=0

15

v=0 10

20

Figure 2.19: Flow past a circular cylinder: geometry and boundary conditions

Uniform flow of magnitude U is prescribed at the left inlet. Both top and bottom walls are specified to be sliding with the inlet velocity. The cylinder is fixed and a no-slip condition is provided on its surface by fixing both velocity components to zero. Finally the pressure is fixed to be zero at the centre point of the outlet. The simulations are performed to obtain the velocity and pressure field around the cylinder for two different Reynolds numbers Re = U dρ/µ = 0.1 and 100, where ρ and µ are density and dynamic viscosity of the fluid. These different Reynolds numbers are achieved by setting U = 0.005 and 5.0 with ρ = 1.0 and µ = 0.05. Two different meshes are employed, shown in Figures 2.20-2.21, in which the dense one with 24330 elements is for the case Re = 100 while the coarse one with 2969 elements for the low Reynolds number case Re=0.1. In the dense mesh, refinement is performed around the cylinder and along the outflow direction to capture the von Karman vortex street.

30

Figure 2.20: Flow past a circular cylinder: mesh with 2969 triangular elements

Figure 2.21: Flow past a circular cylinder: mesh with 24330 triangular elements and a zoom view around the cylinder

For Re = 0.1, different solutions are obtained for steady Stokes flow and steady NavierStokes flow with standard Galerkin formulation, and figure 2.22 depicts the velocity fields for the two cases. It is clearly seen that for sufficiently low Reynolds number Stokes equations provide a highly similar solution to that obtained by Navier-Stokes equations. 31

(a) steady Stokes

(b) steady Navier-Stokes

Figure 2.22: Flow past a circular cylinder: contour plots of X-velocity for steady Stokes flow and steady Navier-Stokes flow, Re = 0.1

As the Reynolds number increases, solutions obtained from Stokes and Navier-Stokes equations become completely different. Besides, for relatively high Reynolds numbers flow becomes unsteady. Therefore, the unsteady SUPG/PSPG formulation (2.45) is employed in the case Re = 100. Generalised-α method is applied for temporal discretisation with a time step = 0.01. At the inflow boundary, the U velocity is increased from zero to 5.0 gradually in the time interval [0,1]. As mentioned before, the dense mesh with 24330 elements is adopted. Figure 2.23 depicts the contour plots of velocity and pressure. A plot of velocity vectors is given in Figure 2.24. The von Karman vortex street in the wake of the cylinder is clearly seen in these pictures. This vortex shedding phenomenon is induced by the oscillating drag and lift forces Fx and Fy parallel and perpendicular to the direction of the flow, respectively. Three dimensionless numbers, drag and lift coefficients CD and CL and Strouhal number St can be expressed as follows

CD =

2Fx , ρU 2 d

CF =

2Fy , ρU 2 d

St =

fd U

(2.62)

where f is the frequency of the lift force. The evolution of drag and lift coefficients shown in Figure 2.25 are consistent with the results in [2]. The Strouhal number calculated from Equation 2.62 is 0.164, which also agrees with that obtained in [51].

32

(a) Velocity contour

(b) Pressure contour

Figure 2.23: Flow past a circular cylinder: contour plots of velocity and pressure at t = 25, Re = 100

Figure 2.24: Flow past a circular cylinder: plot of velocity vector at t = 25, Re = 100

Figure 2.25: Flow past a circular cylinder: evolution of drag and lift coefficients, Re = 100

33

2.7.3

Backward facing step flow

The behaviour of flow over a channel with a backward facing step is investigated in this example. The long term solution is mainly dependent on the expansion ratio and Reynolds number. The expansion ratio is defined by the ratio of downstream height H = h + s to upstream height h, H/h = 1 + s/h, where s denotes the step height, as shown in Figure 2.18. The Reynolds number is given as Re = ρua D/µ, where ρ and µ are density and dynamic viscosity, respectively. Here ua = 2/3uin is the average velocity of the parabolic inlet flow with a maximum value uin , D the characteristic length. We note, however, that different definitions of D used in the literature result in various definitions of Reynolds number. In the present work D is chosen as the hydraulic diameter of the inlet channel, which equals to twice its height, D = 2h. The geometry and boundary conditions are sketched in Figure 2.26. The heights of step and inlet are s = 0.9423 and h = 1, which leads to a expansion ratio 1.9423 that is consistent with the value used by Biswas et al [52]. The lengths of upstream and downstream are l = 1 and L = 34, respectively. No-slip boundary conditions are prescribed at both top and bottom walls. Similar to the previous example, the pressure is set to zero at the centre of the outlet. The density and viscosity are set to ρ = 1 and µ = 0.01. By choosing different maximum values for uin , the simulation is performed for a Reynolds number ranging from 0.01 to 800. In this example a mesh with 12601 elements and 6631 nodes is adopted, and the region around the step is refined, as shown in Figure 2.27.

u = uin v=0

h s

u = 0, v = 0 p=0 u = 0, v = 0

l

L

Figure 2.26: Backward facing step flow: geometry and boundary conditions

Figure 2.27: Backward facing step flow: mesh with 12601 triangular elements

34

Figure 2.28 gives the streamlines of unsteady flow field for a Reynolds number range 0.01 ≤ Re ≤ 800. In the first two cases, given by Re = 0.01 and Re = 1, the flow

follows the upper convex corner without a flow separation. Besides, a small vortex can be seen in the concave corner behind the step. Note that the size of the vortex is almost constant in this small Reynolds number range. For Re > 1 the size of the vortex strongly increases. In the last case Re = 800, a secondary vortex is found near the top wall.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2.28: Backward facing step flow: streamlines around the step for different Re. (a) Re = 0.01; (b) Re = 1; (c) Re = 50; (d) Re = 100; (e) Re = 400; (f) Re = 800;

35

The growth of the first vortex behind the step as Reynolds number increases can be seen more clearly in Figure 2.29, in which the length of the vortex x normalized by the step height s is illustrated. The result agrees well with that from Biswas et al [52], which is depicted by the squares. As mentioned above, the size of the first vortex is nearly the same for Re ≤ 1. Figure 2.30 shows the velocity distribution at different time instant for Re = 800.

Figure 2.29: Backward facing step flow: length of the vortex behind the step normalized by the step height s

(a) t=5

(b) t=10

(c) t=20

(d) t=60

Figure 2.30: Backward facing step flow: contour plots of velocity at different time instants for Re = 800, time step ∆t = 0.02

36

Chapter 3

Fictitious Domain/distributed Lagrangian Multiplier Method for Fluid Flow In the previous chapter a stabilised finite element formulation has been introduced to solve the incompressible fluid flow. In that formulation the fluid domain is described in an Eulerian frame of reference, i.e. the mesh is fixed in space and the fluid moves with respect to the grid. In structural mechanics, on the other hand, the solid is generally formulated in a Lagrangian description, in which each mesh point is attached to the material and follows the associated material particle during motion. Therefore, in modelling fluid-structure interaction(FSI), the well-known Arbitrary Lagrangian Eulerian (ALE) method which combines the features of these two approaches was naturally proposed [53] and is extensively studied (see Chapter 14 in [28] and references therein). However, for large deformations of the solid the fluid elements tend to become ill-shaped, and need to be re-meshed, which is a complex and time-consuming task. Besides, each re-meshing process introduces artificial diffusion and loss of accuracy during the data transfer from the degenerated mesh to the new mesh [20]. Therefore to overcome these difficulties some solution schemes based on fixed Cartesian grids are developed, among which the fictitious domain method is a powerful and easily implemented one. In this chapter the fictitious domain/distributed Lagrange multiplier (FD/DLM) method is introduced to solve the incompressible Navier-Stokes equations and the fluid-flexible structure interaction. In contrast to body-fitted mesh, in FD/DLM a structured mesh is used to discretise the fluid. B-splines are also employed in the fluid discretisation to produce the hierarchical refinement of the mesh. This chapter presents some basic concepts and features of the FD/DLM and b-splines and their application in solving 37

incompressible fluid flow. In the next few chapters this methodology will be applied to fluid-flexible structure interaction in 2D and 3D. This chapter is organised as follows. First, a brief history and main features of FD/DLM is introduced in Section 3.1. The formulation for fluid-structure interaction is then given in Section 3.2. Section 3.3 gives a overview of b-splines and their hierarchical refinement, followed by two numerical examples presented in Section 3.4.

3.1

Fictitious domain method

The fictitious domain methods (FDM) are a class of domain embedding methods used for solving partial differential equations on a complicated domain. In FDM, the fluid domain is extended into the interior of the solid domain, and the use of non-body-fitted mesh is allowed on a simple shape auxiliary domain (also called fictitious domain) [54]. As seen in seen in Figure 3.1, the fluid Ωf , with its boundary Γf is modelled in an Eulerian reference whereas the immersed solid Ωs is discretised in a Lagrangian frame, and the fluid grid does not need to match the solid boundaries Γs . Therefore, FDM are simpler, easy to implement and computationally more efficient than the methods based on body-fitted meshes (seen in Chapter 2) for problems where the immersed solids experience large deformation and/or topological changes [37].

Ωf

Ωs

Γf

Γs

Figure 3.1: Fictitious domain method: a schematic description

38

3.1.1

A brief history

The fictitious domain method was first introduced by Hyman [55] in 1952, and was discussed by Saul’ev (gave the name ”fictitious domain”) [56] in early 1960s. Since then, several authors have employed this method for the solution of elliptic partial differential equations [57–61]. Pioneered by Glowinski [62–68], a cluster of fictitious domain methods showed their power in modelling particulate flows and FSI problems. In these methods, Lagrange multipliers are employed to enforce the kinematic constraint at the fluid-solid interface. Compare to the IBM [32] and IIM [38], FDM offers two main advantages. First, while the kinematic constraint at the fluid-solid interface is enforced weakly (via a force), it is applied strongly in FDM by using Lagrange multipliers. Second, the Lagrange multipliers are tractions on the boundary of the immersed body which can be used directly for FSI problems [37]. However, in the majority of literatures, the fluid is discretised with the element in which the velocity has a higher order interpolation than the pressure, such as Taylor-Hood or Crouziex-Raviart family elements [69]. Recently, Kadapa [37] proposed a fictitious domain formulation for FSI problems based on hierarchical b-spline Cartesian grid. In that work, b-splines are employed in the fluid discretisation, and the fluid grid near the immersed solids can be refined locally using hierarchical b-splines, which leads to significant savings in computational cost.

3.1.2

Main features

By following Kadapa’s work [37], a formulation based on fictitious domain/distributed Lagrange multiplier is adopted in the present work. This approach provides some advantages over the body-fitted method and can be summarised as: • the fluid is discretised in an Eulerian frame, therefore there is no need for computationexpensive body-fitted mesh, which makes discretisation easy and efficient.

• complicated re-meshing procedure due to large solid deformation can be circumvented.

• in this scheme data-mapping and its error can be avoided, which is a difficulty in body-fitted ALE method.

• properties of structured grids can be exploited in developing parallel solvers.

39

3.2

Formulation

In this section the main equations of the FD/DLM method are summarised.

3.2.1

Governing equations for fluid

The governing equations for incompressible fluid flow has been introduced in Chapter 2. Here we present them again: ρ

f

∂uf f + u · ∇ uf ∂t

− µf ∇2 uf + ∇p = f f

in Ωf

(3.1)

in Ωf

(3.2)

uf = ufD

on ΓfD

(3.3)

σ f · nf = tfN

on ΓfN

(3.4)

uf (·, 0) = uf0

in Ωf

(3.5)

∇ · uf = 0

where uf and p are the fluid velocity and pressure, respectively, in the domain Ωf , ρf is the fluid density, µf is the dynamic viscosity, f f is the fluid body force, nf is the unit outward normal vector of the boundary Γf , tf is the traction vector, and σ f is the fluid Cauchy stress. The superscript f is used here to denote fluid quantities. More details about the governing equations for fluid can be found in Chapter 2.

3.2.2

Governing equations for solid

The governing equations for an initial-boundary value problem of elasticity, in the current configuration, is stated as ρs

∂ 2 ds + ∇ · σs = f s ∂t2

in Ωs

(3.6)

ds = dsD

on ΓsD

(3.7)

σ s · ns = tsN

on ΓsN

(3.8)

ds (·, 0) = ds0

in Ωs

(3.9)

s s d˙ (·, 0) = d˙ 0

in Ωs

(3.10)

s where ds and d˙ are, respectively, the displacement and velocity of the solid, ρs is the

solid density, f s is the solid body force, σ s is the solid Cauchy stress, and ns is the unit 40

outward normal vector of the boundary Γs which consists of two parts: ΓsD is the part of boundary of the domain Ωs where Dirichlet boundary condition dsD is applied and ΓsN is the part of the boundary where Neumann boundary condition tsN is applied. Here, s Γs = ΓsD ∪ ΓsN and ΓsD ∩ ΓsN = ∅, while ds0 and d˙ 0 are the initial displacement and initial

velocity of the solid, respectively.

3.2.3

Governing equations at the interface

Two conditions need to be satisfied at the fluid-structure interface, denoted as Γf −s . The first one is known as no-slip condition, which enforces that the fluid at the interface moves at the same velocity as the solid boundary. The second condition states the equilibrium of stresses along the interface. These two conditions can be mathematically expressed as uf = us tf + ts = 0

on Γf −s

(3.11)

on Γf −s

(3.12)

where us is the solid velocity, tf and ts are, respectively, the tractions exerted by the fluid and solid on the interface.

3.2.4

Weak form

The variational equations for fluid-structure interaction problems can be stated as: find the fluid velocity uf ∈ U f and pressure p ∈ P, the structural velocity us ∈ U s and the Lagrange multiplier λ ∈ λ such that for all weight functions wf ∈ W f , q ∈ P, ws ∈ W s and φ ∈ W λ B

f

Z f f {w , q}, {u , p} − F {w , q} + f

f

wf · λdΓ = 0

(3.13)

ws · λdΓ = 0

(3.14)

φ · uf − us dΓ = 0

(3.15)

Γf −s

s

s

s

s

s

B (w , u ) − F (w ) − Z Γf −s

41

Z Γf −s

where, f

B ({w, q}, {u, p}) =

Z f

ZΩ

Ωf

F f ({w, q}) =

Z Ωf

Z

s

B (w, u) = Γs

F s (w) =

∂uf w·ρ + (u · ∇) u dΩf + ∂t Z q∇ · udΩf (w) : σ f (u, p) dΩf + f

Z Ωs

(3.16)

Ωf

w · f f dΩf + w·ρ

s ∂u

∂t

Z ΓfN

w · tfN dΓfN

Z

s

(w) : σ (u) dΩs

dΩ +

w · f s dΩs +

(3.17) (3.18)

Γs

Z ΓsN

w · tsN dΓsN

(3.19)

Here U f and P are, respectively, the spaces of fluid velocity and pressure solutions, which are defined in Equations (2.9) and (2.11). Similarly, U s is the space of structural velocity, defined as n U s = {us ∈ H1 (Ωs ) sd |usΓD = usD }

(3.20)

W f , defined in (2.8), is the space of weight functions for fluid velocity, whereas W s is the space of weight functions for solid velocity, which has a similar form to (2.8) but defined in solid domain Ωs . λ and W λ are the spaces of solution and weight functions for Lagrange multiplier, respectively. Note that the variational formulation given by Equation (3.15) enforce the no-slip condition, while (3.13) and (3.14) give the Euler-Lagrange conditions on the fluid-structure interface Γf −s as λ = −σ f nf = σ s ns

(3.21)

where nf = −ns .

3.3

Hierarchical b-splines

In this section a brief overview of the B´ ezier and b-spline curves are presented, followed by a introduction of b-splines in higher dimensions and hierarchical refinement that will be adopted in the fluid discretisation.

3.3.1

B´ ezier curves

A Bézier curve is a parametric curve frequently used in computer graphics and geometric modelling. Given n + 1 control points P0 , P1 , P2 , ... and Pn in space, the Bézier curve 42

defined by these control points is C (ξ) =

n X

Bi,n (ξ) Pi

i=0

for 0 ≤ ξ ≤ 1

(3.22)

where the coefficients, often known as B´ ezier basis functions or Bernstein polynomials, are given by Bi,n (ξ) =

n! ξ i (1 − ξ)n−i i! (n − i)!

(3.23)

Hence, the point corresponds to ξ on the B´ ezier curve is the weighted average of all control points, where the weights are Bi,n (ξ). Note that the degree of the curve is n.

3.3.2

B-spline basis functions

B-spline curves are generalizations of B´ ezier curves and have higher degree of freedom for curve design. A b-spline basis function consists of a number of piecewise polynomials joined with the desired continuity. To define a b-spline basis function, a knot vector and the degree of the polynomial are required. For a given m + 1 non-decreasing numbers, ξ0 ≤ ξ1 ≤ ξ2 ≤ ... ≤ ξm , the set Ξ of these numbers is called knot vector, and ξi ’s are knots. The i-th b-spline basis function of degree p, written as Ni,p (ξ), is evaluated from the following recurrence relations,  1, if ξi ≤ ξ ≤ ξi+1 . Ni,0 (ξ) = 0, otherwise. Ni,p (ξ) =

ξi+p+1 − ξ ξ − ξi Ni,p−1 (ξ) + Ni+1,p−1 (ξ) ξi+p − ξi ξi+p+1 − ξi+1

(3.24)

(3.25)

The above is usually referred to as the Cox-de Boor recursion formula. A set of b-spline basis functions Ni,p of different degrees p span in the knot vector Ξ = [ξi , ξi+1 , ...ξi+p+1 ] are shown in Figure 3.2. In this thesis, only uniform b-splines are considered, i.e. ∆ξ = ξi+1 −ξi is constant throughout the domain when no local refinement is performed. Some

of the important properties of b-spline basis functions can be concluded from the above formula and figure: • Ni,p (ξ) is a degree p polynomial in ξ. • Non-negativity: for all i, p and ξ, Ni,p (ξ) is non-negative. This property is important in dynamic analysis cause the coefficients of mass matrix computed from B-Spline basis functions are non-negative. 43

• Local support: Ni,p (ξ) is a non-zero polynomial on [ξi , ξi+p+1 ). • On any span [ξi , ξi+1 ), at most p + 1 degree p basis functions are non-zero, namely: Ni−p,p (ξ), Ni−p+1,p (ξ), ..., Ni,p (ξ).

• Partition of Unity: for any given value ξ, the sum of all degree p basis functions

on span [ξi , ξi+1 ) are equal to unity. This states that the sum of all p + 1 non-zero basis functions in the previous property is 1.

• At a knot of multiplicity k, basis function Ni,p (ξ) is C p−k continuous. • For a knot vector with (m + 1) knots and degree of the polynomial p, the number of basis functions is n + 1, where n = m − p − 1.

Figure 3.2: B-spline basis functions

3.3.3

B-spline curves

For a given n + 1 control points P 0 , P 1 , ..., P n and a knot vector Ξ = {ξ0 , ξ1 , ..., ξm }, the b-spline curve of degree p defined by these control points and knot vector Ξ is C (ξ) =

n X

Ni,p (ξ) P i

(3.26)

i=0

where Ni,p (ξ) are b-spline basis functions of degree p. The point on the curve that corresponds to a knot ξi , C (ξi ), is known as a knot point. These knot points divide a b-spline curve into curve segments which are all B´ ezier curves of degree p. Some of the important properties of b-spline curves are: • B-spline curve C (ξ) is a piecewise curve with each component a curve of degree p.

44

• End-point interpolation: b-spline curve C (ξ) passes through the two end control point P 0 and P n , or mathematically, C (0) = P 0 and C (1) = P n .

• Local modification scheme: moving a control point P i only changes the curve C (ξ) on interval [ξi , ξi+p+1 ).

• B-spline curve C (ξ) is C p−k continuous at a knot of multiplicity k. • Affine invariance: an affine transformation of a b-spline curve is obtained by applying transformation to its control points.

• B´ ezier curves are special cases of b-spline curves. If n = p and there are 2 (p + 1)

knots with p + 1 of them clamped at each end, then this b-spline curve reduces to a B´ ezier curve.

3.3.4

B-splines in higher dimensions

One of the significant advantages of b-splines is that they can easily be extended to higher dimensions. This is achieved by using the tensor products property of b-splines. For univariate b-spline functions N ξ , N η , N ζ in, respectively, ξ, η, ζ parametric directions, the multivariate b-spline basis functions in two- and three-dimensions are expressed as N (ξ, η) = N ξ ⊗ N η N (ξ, η, ζ) = N ξ ⊗ N η ⊗ N ζ

in 2D

(3.27)

in 3D

(3.28)

The support of a 2D b-spline basis function which has same polynomial order in both parametric directions is shown in Figure 3.3.

Constant Linear Quadratic

η

Cubic

ξ Figure 3.3: Support of a equal-order b-spline basis function in 2D

45

3.3.5

Hierarchical refinement

The local refinement strategy of b-splines is based on their subdivision property, also known widely as two-scale relation and illustrated in Figure 3.4, given as Np (ξ) =

p+1 X i=0

αi Np (2ξ − i)

(3.29)

where αi are functions of binomial coefficients which are expressed as 1 αi = p 2

p+1

! (3.30)

i

This property enables any b-spline function of order p and defined on a knot vector with knot span ∆ξ to be represented as a linear combination of p + 2 b-splines of the same order and defined on a knot vector with knot span ∆ξ/2. Therefore, the local refinement can be achieved by replacing b-spline basis functions N k (ξ) at level k by a set of functions N k+1 (ξ) at level k + 1 as N k (ξ) = SN k+1 (ξ)

(3.31)

where S is the subdivision matrix contains the coefficients α in Equation (3.29). By applying hierarchical refinement, the b-spline grid near the immersed bodies is locally refined to different levels, as shown in Figure 3.5, which leads to an excellent computational efficiency. 1.00

1.00

Original Refined

0.75 0.50

0.50

0.25

0.25

0.000

1

2

3

4

Original Refined

0.75

0.000

5

1

(a) Linear(Q1)

0.75

0.50

0.25

0.25 3

5

4

Original Refined

0.75

0.50

2

4

1.00

Original Refined

1

3

(b) Quadratic(Q2)

1.00

0.000

2

0.000

5

(c) Cubic(Q3)

1

2

3

(d) Quartic(Q4)

Figure 3.4: Two-scale relation of the b-spline functions

46

4

5

Ωf

Ωs

Γf

Γs

Figure 3.5: Hierarchical refinement near the solid body

3.4 3.4.1

Numerical examples Conservation test

Studied by [70] and [37], flow over a cylinder in a narrow channel was used as an example to investigate the conservation properties of different numerical schemes. Here this example is employed to demonstrate the conservation properties of the presented fictitious domain/distributed Lagrange multiplier (FD/DLM) method. The geometry of the problem is shown in Figure 3.6. The boundary conditions are chosen as ux = 1.0, uy = 0.0 on the entire outer boundary and no-slip condition is applied on the surface of the cylinder. The density and dynamic viscosity of the fluid are ρf = 1.0 and µf = 0.1, respectively. The whole domain is discretised with a uniform mesh of 30, 000 quadrilateral elements, while the cylinder is represented by 200 equally spaced Lagrange points, as illustrated in Figure 3.7.

C B A D E

D = 1.0

L = 4.5

Figure 3.6: Conservation test: geometry of the problem

47

H = 1.5

Use is made of formulations (3.13) - (3.15) in the simulation of the problem. Since the cylinder is fixed and there is no fluid-structure interaction in this example, only the fluid need to be solved. As discussed in previous chapter, for velocity-pressure equalorder elements pressure stabilisation is necessary due to LBB condition. Hence in this example two sets of simulations are performed to assess the conservation, one without any stabilisation and the other with PSPG stabilisation. In both cases, linear (Q1), quadratic (Q2) and cubic (Q3) b-splines are employed for the element interpolation.

Figure 3.7: Conservation test: mesh

As depicted in Figure 3.8, the pressure contour plots show that without pressure stabilisation, the velocity-pressure equal-order elements lead to unphysical oscillations in the pressure field for linear and cubic b-splines. On the contrary, use of PSPG circumvents the LBB condition and generates a smooth pressure field as expected, which was already seen in the previous chapter. However, the X-velocity contour plots presented in Figure 3.9 indicate that pressure stabilisation reduces the amount of flow across the openings BC and DE (seen in Figure 3.6). The flow reduction through the openings is due to the flow leaking into the cylinder. To assess this influence quantitatively, the X-velocity profiles along the opening AC are plotted in Figure 3.10, in which the variation of the peak of velocity is clearly seen. Besides, flow rates through the opening BC are calculated and compared with the theoretical value, as illustrated in Table 3.1.

Table 3.1: Conservation test: flow rate through BC for b-spline elements. The theoretical value is 0.75.

degree of b-spline Q1 Q2 Q3

With PSPG Flow rate % error 0.6271 16.38 0.6934 7.55 0.7174 4.35

Without PSPG Flow rate % error 0.7385 1.54 0.7373 1.69 0.7388 1.49

Table 3.1 clearly indicates that the use of FD/DLM with pressure stabilisation affects mass conservation significantly, and the results agree well with those from [37]. It is observed that the combination of equal-order elements with quadratic b-spline(Q2) and 48

no-stabilisation gives small error in flow rates and more importantly, a smooth pressure filed, although the combination of velocity-pressure is LBB unstable. In fact the same phenomenon has also been observed in [71], in which the stabilisation parameter is reduced in the vicinity of the immersed solid bodies. However, the amount of reduction and the area of the fluid domain where the stabilisation parameter has to be lowered is difficult to choose in order to prove the accuracy. Therefore in this thesis the author follows [37] and choose to employ velocity-pressure equal-order elements with quadratic b-splines and without any stabilisation in all the following examples.

49

(a) Q1 without stabilisation

(b) Q1 with stabilisation

(c) Q2 without stabilisation

(d) Q2 with stabilisation

(e) Q3 without stabilisation

(f) Q3 with stabilisation

Figure 3.8: Conservation test: pressure contours

(a) Q1 without stabilisation

(b) Q1 with stabilisaton

(c) Q2 without stabilisation

(d) Q2 with stabilisation

(e) Q3 without stabilisation

(f) Q3 with stabilisation

Figure 3.9: Conservation test: X-velocity contours

50

Figure 3.10: Conservation test: X-velocity profiles along opening AC in Figure 3.6

3.4.2

Flow past a circular cylinder

The example flow past a circular cylinder is studied again by using FD/DLM presented in this chapter. The geometry and boundary condition of the problem are chosen to be the same as those in Section 2.7.2 (seen in Figure 2.19), while the mesh is generated with 3 different levels of hierarchical b-spline refinements, as shown in Figure 3.11.

Figure 3.11: Flow past a circular cylinder: mesh and 3 levels of hierarchical refinements. Number of DOFs: 27938

The fluid density and viscosity are also chosen the same as those in previous chapter, i.e. ρ = 1.0 and µ = 0.05. By tuning the inflow velocity Uin , a number of simulations are performed to observe the variation in flow patterns corresponding to different Reynolds 51

number. Table 3.2 illustrates the inflow velocity, Reynolds number and time step for each cases. Generalised-α method is adopted for temporal discretisation.

Table 3.2: Flow past a circular cylinder: different cases

CASE CASE CASE CASE CASE CASE CASE

1 2 3 4 5 6 7

Re 0.1 1.0 10 40 100 150 200

Uin 0.005 0.05 0.5 2.0 5.0 7.5 10.0

∆t steady steady steady 0.5 0.01 0.005 0.002

The contour plots of velocity and pressure are, respectively, presented in Figure 3.12 and Figure 3.13, which clearly show that as Re increases, the steady and symmetric flow pattern becomes unsteady and asymmetric, and von Karman vortex begins to occur. The history of drag and lift coefficients for CASE 5 and CASE 7 are, respectively, given in Figures 3.14 and 3.15. The frequency and amplitude of the curves for Re=100 agree with those obtained by using body fitted mesh and SUPG/PSPG linear elements previously. However, note that it takes a shorter time to observe the oscillations of the drag and lift coefficients in the body-fitted case because the asymmetric body-fitted mesh gives a natural imperfection for the oscillations. CD =

8π Re (2.002 − lnRe)

20.1 St = 0.195 × 1 − Re

21.1 St = 0.212 × 1 − Re

52

(3.32) (3.33) (3.34)

(a) Re=0.1

(b) Re=10

(c) Re=40

(d) Re=100

Figure 3.12: Flow past a circular cylinder: velocity contour plots.

(a) Re=0.1

(b) Re=10

(c) Re=40

(d) Re=100

Figure 3.13: Flow past a circular cylinder: pressure contour plots.

The drag coefficients calculated in each cases, drag coefficient obtained from the experiments [72] and the result obtained using Oseen approximation in Equation (3.32) [51] are shown together in Figure 3.16. In addition, the Strouhal numbers calculated in CASE 5-7, along with those obtained from approximated formulas established by Rayleigh (Equation 3.33) and Roshko (Equation 3.34) [51] are given in the same figure.

53

Figure 3.14: Flow past a circular cylinder: history of drag (left) and lift(right) coefficients, Re=100.

Figure 3.15: Flow past a circular cylinder: history of drag (left) and lift(right) coefficients, Re=200.

Figure 3.16: Flow past a circular cylinder: drag coefficient (left) and Strouhal number(right) v.s Re.

54

Chapter 4

Geometrically Exact Beam Theory To describe the behaviour of beam with large deformation, the beam is modelled with geometrically exact formulation by following Zienkiewicz and Taylor [73] for 2D beam element and Ibrahimbegović [1] for 3D beam element. In geometrically exact beam model, large displacement is taken into account, and the beam cross-section does not necessarily remain normal to the beam axis during the deformation. However, the small strain assumption still holds. In 2D beam formulation, the second Piola-Kirchhoff stress and Green-Lagrange strain are used as the stress and strain measures, while in 3D case, Cauchy stress and strain are employed. Note that in 3D space, the representation of finite rotations is the main difficulty. In this work a vector-like parametrization is used for the rotation tensor as in [1], but the stiffness matrix and residual vector are obtained in a simpler and more straightforward way. This chapter consists of two parts.

Section 4.1 introduces a two-dimensional geo-

metrically exact beam formulation, and the remaining sections contains the discussion of three-dimensional geometrically exact beam. Section 4.2 presents a vector-like parametrization to describe the rotation tensor of the three-dimensional beam. The strain and stress measures are then, respectively, introduced in Sections 4.3 and 4.4. The variational formulation is discussed in Section 4.5, and the explicit forms of the residual vector and stiffness matrix are presented in Sections 4.6 and 4.7. After the discussion of mass matrix in Section 4.8, several numerical examples are studied in Section 4.9.

55

4.1

Two-dimensional geometrically exact beam formulation

To model the 2D geometrically nonlinear beam, the current work employs the geometrically exact formulation by following Zienkiewicz and Taylor[73]. In this model the large displacement is taken into account, and the beam cross section does not necessarily remain normal to the beam axis during the deformation. However, the small strain assumption still holds. Assuming that the beam is aligned with the X-axis and deforms in X-Y plane, as depicted in Figure 4.1, the deformed configuration can be described by

β

Y,y v u X,x Figure 4.1: Kinematics of the beam

x = X + u + Y sin β (4.1)

y = v + Y cos β z=Z

where u and v are the X- and Y- displacements, while β is the rotation of cross section. The deformation gradient for this displacement then becomes   F = 

∂u ∂X + ∂v ∂X − Y

1+



∂β Y cos β ∂X

sin β

∂β sin β ∂X

 cos β 0   0 1

0

0

(4.2)

By ignoring the quadratic terms in Y, the two non-zero components of the GreenLagrange strain tensor E = 21 (F T F − I) are EXX = E 0 + Y K b EXY = 12 Γ

56

(4.3)

where the quantities E 0 , Γ and K b refer to the average axial strain, the shear strain and the bending strain caused by the beam curvature, respectively 2 2 ! ∂u ∂v 1 ∂u + + E0 = ∂X 2 ∂X ∂X ∂u ∂v Γ= 1+ sin β + cos β ∂X ∂X ∂v ∂u ∂β cos β − Kb = 1+ sin β ∂X ∂X ∂X

(4.4)

(4.5) (4.6)

The first variation of total potential energy of the beam can be written, in terms of second Piola-Kirchhoff stresses, as Z δΠ = Ω

(δEXX SXX + 2δEXY SXY ) dV − δΠext

(4.7)

where δΠext denotes the potential energy due to external forces. By introducing the axial force, shear force and bending moment acting on the cross section as Z

Z

T =

SXX dA, S = A

Z SXY dA, M =

A

SXX ZdA

(4.8)

A

where A is the cross section area. The variational form now can be rewritten as

δΠ =

Z L

δE 0 T + δΓS + δK b M b dX − δΠext

(4.9)

Note that δ indicates the variation of strains with respect to u, v and β. From Equations (4.4) - (4.6) we can obtain the variation of these strain quantities as 0

δE =

∂u 1+ ∂X

δΓ = sin β δK b = Λ where Λ = 1 +

∂u ∂X

∂δu ∂v ∂δv + ∂X ∂X ∂X

(4.10)

∂u ∂δv + cos β + Λδβ ∂X ∂X

(4.11)

∂δβ ∂β ∂δu ∂β ∂δv ∂β − Γδβ + cos β − sin β ∂X ∂X ∂X ∂X ∂X ∂X

cos β −

∂v ∂X

(4.12)

sin β.

The finite element discretisation for the displacements and rotation can be employed:         u u     a     = Na v va        β    β   a 57

(4.13)

where Na is the shape function and [ua , va , βa ]T the nodal variables. Substitute the finite element interpolations into the variation of total potential energy, a non-linear equation is obtained:

    T     T Ba Φ=f− =0 S   L   Mb   Z

(4.14)

Under the small strain assumption addressed above, a linear elastic relation between the second Piola-Kirchhoff stresses and the Green-Lagrange strains can be established, which leads to T = EAE 0 , S = κµAΓ, M = EIK b

(4.15)

in which E, µ, I and κ = 5/6 represent the Young’s modulus, the shear modulus, the second moment of area and the shear correction factor, respectively. Using these relations and performing linearisation by Newton-Raphson method on Equation (4.14) gives the tangent stiffness Z (K T )ab =

L

B Ta DB b dX + (K G )ab

(4.16)

where D is the constitutive matrix 



EA

 D= 

  

κGA EI

(4.17)

B associates the strain and displacement vectors   Ba =  

∂u ∂Na ∂X ∂X a sin β ∂N ∂X ∂β ∂Na ∂X cos β ∂X

1+

∂v ∂Na ∂X ∂X a cos β ∂N ∂X ∂β a − ∂X sin β ∂N ∂X



0 ΛNa

a Λ ∂N ∂X

−

∂β Γ ∂X Na

  

(4.18)

and K G is the geometric stiffness matrix coming from the linearisation of B matrix  Z (K G )ab =

L



 ∂Na     ∂X  

+

∂Na   ∂X 

T

0

0

T

M b cos β





0 0

0



  ∂Nb    −M b sin β   ∂X + Na  0 0 0  Nb 0 0 G3 M b cos β −M b sin β 0     0 0 G1 0 0 0  ∂Nb      0  N + N 0 0 G2  0 0 a b    ∂X  dX G1 G2 −M b Γ 0 0 −M b Γ (4.19)

58

in which ∂β G1 = S cos β − M b ∂X sin β

∂β G2 = −S sin β − M b ∂X cos β

G3 = −SΓ −

4.2

(4.20)

∂β M b ∂X Λ

Parametrization of finite three-dimensional rotations

Finite three-dimensional rotations can be represented by an orthogonal tensor Λ, an element of the SO(3) rotation group [1], i.e. ΛT Λ = I. Hence the nine components of the rotation matrix can be expressed by only three independent parameters. However, it has long been established that a unique global representation based on only three parametres does not exist. Instead, 5-parameter and 4-parameter representations of finite rotations are established in early works [74] [75]. In contrast, in the following a finite rotation representation with only three parameters is introduced.

4.2.1

Euler-Rodrigues formula

Figure 4.2: Finite rotation of a vector [1]

According to Euler’s theorem, any three-dimensional rotation can be interpreted as a two-dimensional rotation which takes place in a plane orthogonal to a direction identified by the unit rotation vector n, and the two quantities (n, θ) as shown in Figure 4.2, are sometimes called the principal axis of rotation and the principal angle of rotation, respectively. They completely define the rotational displacement represented by the rotation tensor Λ, and by this way the notion of a rotation vector θ = θn can be employed to parametrize the three-dimensional rotation, which leads to the Euler-Rodrigues formula [1]

59

Λ = cos θI +

sin θ ˆ 1 − cos θ θ⊗θ Θ+ θ θ2

(4.21)

where Λ and θ are the rotation tensor and its corresponding vector, respectively, and θ = p ˆ represents θ12 + θ22 + θ32 is the magnitude of the rotation vector. The capital letter Θ a skew-symmetric tensor for which θ is the axial vector, i.e.  ˆ = θ × b, Θb

∀b ∈ R3 ;

0

 ˆ =  θ3 Θ 

−θ3 0

−θ2

θ1



θ2

 −θ1  , 0

  θ1    θ = θ2   θ3

(4.22)

To derive the strain and stress measures in the following context, the first and second derivatives of rotation tensor and rotation gradient are now calculated.

4.2.2

Rotation tensor

Note that the derivative of rotation tensor with respect to rotation vector

∂Λ ∂θ

is a 3rd-

order tensor, also note that ∂θ 2θi θi = p 2 = 2 2 ∂θi θ 2 θ1 + θ2 + θ 3

(4.23)

∂ cos θ ∂ cos θ ∂θ − sin θ θi = = ∂θi ∂θ ∂θi θ

(4.24)

∂ sinθ θ cos θ · θ − sin θ θi θ cos θ − sin θ = · = θi 2 ∂θi θ θ θ3

(4.25)

θ ∂ 1−cos sin θ · θ2 − (1 − cos θ) · 2θ θi θ sin θ + 2 cos θ − 2 θ2 = · = θi 4 ∂θi θ θ θ4

(4.26)

and   0 0 0 ˆ   ˆ ,1 = ∂ Θ = 0 0 −1 , Θ   ∂θ1 0 1 0



0

0 1



ˆ   ˆ ,2 = ∂ Θ =  0 0 0 , Θ   ∂θ2 −1 0 0

  0 −1 0 ˆ   ˆ ,3 = ∂ Θ = 1 0 0 Θ   ∂θ3 0 0 0 (4.27)

The tensor product of two rotation vectors and its derivatives read 

θ12

θ1 θ2 θ 1 θ3



  2  θ⊗θ = θ θ θ θ θ 2 1 2 3 2   θ3 θ1 θ3 θ2 θ32

60

(4.28)

      2θ1 θ2 θ3 0 θ1 0 0 0 θ1        , (θ⊗θ),2 = θ1 2θ2 θ3  , (θ⊗θ),3 =  0 0 θ2  (4.29) (θ⊗θ),1 =  θ 0 0 2       θ1 θ2 2θ3 θ3 0 0 0 θ3 0 Then, the derivative of rotation tensor is obtained as ∂Λ ˆ + f3 · Θ ˆ ,i + f 4 · θi θ ⊗ θ + f 5 · (θ ⊗ θ),i = f 1 · θi I + f 2 · θ i Θ ∂θi

(4.30)

f1 =

− sin θ θ

(4.31)

f2 =

θ cos θ − sin θ θ3

(4.32)

where

f3 = − f1

(4.33)

f4 =

θ sin θ + 2 cos θ − 2 θ4

(4.34)

f5 =

1 − cos θ θ2

(4.35)

Since in finite element method θi = N a θia + N b θib , where N a and N b are the shape functions at node a and b, respectively, the derivative of rotation tensor at node a and b, respectively, are represented by ∂Λ ∂Λ = Na , a ∂θi ∂θi

4.2.3

∂Λ ∂Λ = Nb b ∂θi ∂θi

(4.36)

Rotation gradient

The rotation gradient is the derivative of rotation tensor with respect to the coordinates of initial beam axis, and can be obtained as [73] Λ,S =

dΛ ˆ ,S Λ =Θ dS

(4.37)

ˆ ,S is the skew-symmetric matrix where S is the arc length coordinate of the beam, and Θ of its axial vector θ ,S which comes from

θ ,S

    θ1a θb  dN b  1  dN a a dN b b dN a  θa  + θb  = θ + θ = dS dS dS  2  dS  2  θ3a θ3b 61

(4.38)

The derivative of the skew-symmetric tensor gradient is ˆ ,S ∂Θ ∂θ ,S = skew[ ] ∂θi ∂θi

(4.39)

where   ∂θ ,S = a  ∂θ1

dN a /dS



0

 , 

0







0

  ∂θ ,S = dN a /dS  a  , ∂θ2 0

0



  ∂θ ,S  = 0 a   ∂θ3 a dN /dS

(4.40)

and hence their skew-symmetric tensors can be obtained as ˆ ,S ∂Θ ∂θ1a

ˆ ,S ∂Θ ∂θ2a

ˆ ,S ∂Θ ∂θ3a

Similarly,

ˆ ,S ∂Θ ∂θib

 0 0  =  0 0 a 0 dN /dS  0   =  0 −dN a /dS  0  a =  −dN /dS 0

0



 −dN a /dS  

(4.41)

0

0 dN a /dS



0

0

0

0

  

−dN a /dS 0 0

0

(4.42)



 0 

(4.43)

0

can be obtained. With these results in hand, we are able to calculate

the derivative of rotation gradient at each node of an element

4.2.4

ˆ ∂Λ,S ˆ ,S ∂Λ + ∂ Θ,S Λ =Θ a a ∂θi ∂θi ∂θia

(4.44)

ˆ ∂Λ,S ˆ ,S ∂Λ + ∂ Θ,S Λ = Θ ∂θib ∂θib ∂θib

(4.45)

Second derivative of rotation tensor

To calculate the stiffness matrix, we need to obtain the second derivative of rotation tensor. From Equations (4.30)-(4.36) we have ∂Λkl = c1 + c2 + c3 + c4 + c5 ∂θia

62

(4.46)

where − sin θ θ cos θ − sin θ ˆ kl , c3 = N a sin θ · Θ ˆ kl,i , · θi δkl , c2 = N a · θi Θ 3 θ θ θ (4.47) a θ sin θ + 2 cos θ − 2 a 1 − cos θ c4 = N · θi (θ ⊗ θ)kl , c5 = N · (θ ⊗ θ)kl,i θ4 θ2

c1 = N a

Now the derivative of rotation tensor with respect to θjb is performed for each term in the right hand side of Equation (4.46) ∂(− sin θ/θ) − sin θ ∂θi ∂c1 = Na · θi δkl + N a · b δkl b b θ ∂θj ∂θj ∂θj cos θ sin θ − sin θ = N (− 2 + 3 )θj N b θi δkl + N a · δij N b δkl θ θ θ

(4.48)

a

3 ˆ ∂c2 a ∂((θ cos θ − sin θ)/θ ) ˆ kl + N a θ cos θ − sin θ · ∂θi Θ ˆ kl + N a θ cos θ − sin θ · θi ∂ Θkl Θ = N · θ i θ3 θ3 ∂θjb ∂θjb θjb θjb

sin θ cos θ sin θ ˆ kl + N a θ cos θ − sin θ · δij N b Θ ˆ kl − 3 4 + 3 5 )θi θj N b Θ 3 θ θ θ θ3 ˆ kl θ cos θ − sin θ ∂Θ + Na · θi N b 3 θ θj

= N a (−

(4.49) ˆ kl,i ∂c3 ∂(sin θ/θ) ˆ sin θ ∂ Θ · = Na · Θkl,i + N a b b θ ∂θj ∂θj ∂θjb

(4.50)

cos θ sin θ ˆ kl,i + N a sin θ · 0 = N ( 2 − 3 )θj N b Θ θ θ θ a

4 ∂c4 θ sin θ + 2 cos θ − 2 ∂θi a ∂((θ sin θ + 2 cos θ − 2)/θ ) = N · θi (θ ⊗ θ)kl + N a · b (θ ⊗ θ)kl b b θ4 ∂θj ∂θj θj

+ Na

θ sin θ + 2 cos θ − 2 ∂(θ ⊗ θ)kl · θi θ4 θjb

sin θ cos θ cos θ 8 + 4 − 8 6 + 6 )θj N b θi (θ ⊗ θ)kl θ5 θ θ θ θ sin θ + 2 cos θ − 2 θ sin θ + 2 cos θ − 2 + Na · δij N b (θ ⊗ θ)kl + N a · θi (θ ⊗ θ)kl,j N b 4 θ θ4 (4.51)

= N a (−5

2 ∂c5 1 − cos θ ∂(θ ⊗ θ)kl,i a ∂((1 − cos θ)/θ ) = N · (θ ⊗ θ)kl,i + N a · b b θ2 ∂θj ∂θj θjb

sin θ cos θ 2 1 − cos θ = N ( 3 + 2 4 − 4 )θj N b (θ ⊗ θ)kl,i + N a · (θ ⊗ θ)kl,ij N b θ θ θ θ2 a

63

(4.52)

where (θ ⊗ θ)kl,ij is a 4th-order tensor whose component form can be obtained from

expression (4.29) as

(θ ⊗ θ)kl,11

(θ ⊗ θ)kl,21

(θ ⊗ θ)kl,31

 2  = 0 0  0   = 1 0  0   = 0 1

0 0



 0 0 , 0 0  1 0  0 0 , 0 0  0 1  0 0 , 0 0

(θ ⊗ θ)kl,12

(θ ⊗ θ)kl,22

(θ ⊗ θ)kl,32

 0  = 1 0  0   = 0 0  0   = 0 0

 1 0  0 0 , 0 0  0 0  2 0 , 0 0  0 0  0 1 , 1 0

(θ ⊗ θ)kl,13

(θ ⊗ θ)kl,23

(θ ⊗ θ)kl,33

 0  = 0 1  0   = 0 0  0   = 0 0

0 1



 0 0 , 0 0  0 0  0 1 , 1 0  0 0  0 0  0 2 (4.53)

Therefore the second derivative of rotation tensor can be obtained as ∂c1 ∂c2 ∂c3 ∂c4 ∂c5 ∂ 2 Λkl = b + b + b + b + b a b ∂θi ∂θj ∂θj ∂θj ∂θj ∂θj ∂θj ˆ kl d3 = N a · N b · (θi θj δkl d1 + δij δkl d2 + θi θj Θ

(4.54)

ˆ kl d4 + θi Θ ˆ kl,j d4 + θj Θ ˆ kl,i d4 + θi θj (θ ⊗ θ)kl d5 + δij Θ + δij (θ ⊗ θ)kl d6 + θi (θ ⊗ θ)kl,j d6 + θj (θ ⊗ θ)kl,i d6 + (θ ⊗ θ)kl,ij d7)

where cos θ sin θ + 3 θ2 θ sin θ d2 = − θ sin θ cos θ sin θ d3 = − 3 − 3 4 + 3 5 θ θ θ cos θ sin θ d4 = 2 − 3 θ θ sin θ cos θ cos θ − 1 d5 = −5 5 + 4 − 8 θ θ θ6 sin θ cos θ − 1 d6 = 3 + 2 θ θ4 1 − cos θ d7 = θ2

d1 = −

64

(4.55)

4.2.5

Second derivative of rotation gradient

The second derivative of rotation gradient is obtained from Equations (4.44) and (4.45) as 2 ˆ ,S )km ∂Λml ∂ 2 (Λ,S )kl ∂(Θ ˆ ,S )km ∂ Λml + ( Θ = ∂θia ∂θia ∂θjb ∂θjb ∂θia ∂θjb

(4.56)

ˆ ,S )km ∂Λml ˆ ,S )km ∂(Θ ∂ 2 (Θ Λml + + a b ∂θia ∂θi ∂θj ∂θjb Equations (4.41)-(4.43) clearly indicate that the term

ˆ ,S )km ∂ 2 (Θ ∂θia ∂θjb

vanishes, and the other

quantities can be easily obtained in the previous sections.

4.3 4.3.1

Strain measures Bending strains

A skew-symmetric tensor for the bending strains is [1] ˆ = ΛT Λ,S K

(4.57)

ˆ at node a is The derivative of K ˆ ˆ a = ∂K = dK ∂θia

∂Λ ∂θia

T

T

Λ,S + Λ

∂Λ,S ∂θia

(4.58)

and its notation form â = dK jki

ˆ ∂K ∂θa

!

=

jki

ˆ Then we have the axial vector of K 

∂Λlj ∂θia

(Λ,S )lk + Λlj

∂ (Λ,S )lk ∂θia

 ˆ 2) K(3,       ˆ κ= κ2  = K(1, 3) ˆ 1) K(2, κ3 κ1



(4.59)



and the axial matrix of its derivative  â h i dK (3, 2, 1) â dκa = dκa1 dκa2 dκa3 =  dK (1, 3, 1) ˆ a (2, 1, 1) dK 65

 ˆ a (3, 2, 2) dK ˆ a (3, 2, 3) dK  ˆ a (1, 3, 2) dK ˆ a (1, 3, 3) dK  ˆ a (2, 1, 2) dK ˆ a (2, 1, 3) dK

(4.60)

(4.61)

of which each column represents the axial vector of

4.3.2

ˆ ∂K ∂θia

Axial and shear strains

The finite strain measures for the axial and shear components reads [1] = ΛT φ,S − g 1

(4.62)

where φ is the current configuration vector, whereas g 1 is the unit vector along the beam axis in the reference configuration. If the reference beam axis is assumed to be straight and set parallel to the X axis in global coordinate system, one has h

g1 = 1 0 0

iT

(4.63)

The configuration gradient can be obtained as

φ,S

    1 1   du   dN a a dN b b d    (S + u) =  = 0 + dS = 0 + dS u + dS u dS 0 0

(4.64)

where u is the displacement. The derivative of axial and shear strains with respect to rotation and displacement at node a are, respectively, ∂ = ∂θ a

4.4

∂Λ ∂θ a

T φ,S ,

a ∂ T ∂(du/dS) T dN = Λ = Λ ∂ua ∂ua dS

(4.65)

Stress measures

The stress resultants and couples are determined through the constitutive equations n = C n ,

m = C mκ

(4.66)

For a linear elastic beam, the constitutive matrices are  EA  Cn =  GA2 



GA3

 , 

Cm

 GJ  = 

 EI2 EI3

  

(4.67)

where E is the Young’s modulus, G is the shear modulus, I2 , I3 and J are the moment of inertia along local y−, z− and x− axis, respectively, whereas A2 and A3 refer to the cross-sectional geometric areas including shear factors.

66

4.5

Variational formulation

With the variables calculated above, the total potential energy of a beam element can be obtained as

Z Π (u, θ) = L

(T n + κT m) dL − Πext

(4.68)

where and κ are the axial and shear strains and curvature, respectively, whereas n and m are, respectively, stress resultants and couples, L is the length of the beam, and Πext is the potential energy due to external forces acting on the beam, given as Πext

Z h = uT

θT

i

L

!

f

dL

M

(4.69)

Therefore, Z Π (u, θ) = L

(T n + κT m) dL −

Z h

uT

θT

f

i

M

L

! dL

(4.70)

For the system to be in equilibrium the potential energy has to be stationary and hence the first variation of total potential energy vanishes. Z δΠ = L

Z h (δ n + δκ m) dL − δuT T

T

δθ T

L

i

f M

! dL = 0

(4.71)

Note that δ indicates the variation of strains with respect to the displacement vector u and rotation vector θ, which can be approximated by using the same shape functions at element nodes, ¯, u = Nu

¯ θ = Nθ

(4.72)

¯ δθ = N δ θ

(4.73)

and hence ¯, δu = N δ u

¯ are the nodal displacement and rotations, ¯ and θ where N are the shape functions, u respectively. By substituting equation (4.72) into (4.68) the variation of total potential

67

energy can be written as Z δΠ = L

(δT n + δκT m) dL −

Z =

=

!T

n

δuT

δθ T

f

i

L

! dL −

!T δu

Z

f

M !

! dL

dL m M L δθ     ! !T Z ∂ ∂ T Z n ¯ f δu  ∂ u¯ ∂ θ¯    T N dL    dL −  ¯ M δθ L L ∂κ ∂κ m ¯ ¯ ∂u θ  ∂ ! !T Z Z n ¯ f δu   NT dL = 0 B T   dL − ¯ M δθ L L m

L

=

δ

Z h

δκ

(4.74)

where the strain-displacement matrix B is evaluated at element nodes h i a b B= B B in which





∂ a  ∂u

Ba = 



∂ ∂θ a 

∂κ ∂ua

∂κ ∂θ a

(4.75)



∂ ∂θ b 

∂  ∂ub

Bb = 

,

∂κ ∂ub

∂κ ∂θ b



(4.76)

where ua and ub are, respectively, the displacements at nodes a and b, whereas θ a and θ b are the corresponding nodal rotations. Note that

∂κ ¯ ∂u

= 0 since κ is independent of

displacement u. Substituting Equations (4.61) and (4.65) into (4.76) leads to a

B =

" a ΛT dN dS

∂Λ ∂θ a

T

φ,S

# (4.77)

dκa

0

 b ΛT dN dS Bb =  0

∂Λ ∂θ b

T

φ,S

dκb

 

(4.78)

!T ¯ δu is arbitrary, the sum of other terms must ¯ δθ

Since in equation (4.74) the variation

vanish, which results in a discrete system, Φ = P − f ext = 0 where



Z P = L



n   B   dL m T

68

(4.79)

(4.80)

Z

NT

f ext = L

4.6

f

! dL

M

(4.81)

Residual vector

From Euler-Rodrigues formula it is easily known that the strain is a nonlinear function of displacement and rotation, and hence the system in (4.79) is nonlinear. Applying Newton-Raphson method to linearise this system gives a matrix system of equations in an incremental form

! d¯ u = −R ¯ dθ

K

(4.82)

¯ are respectively, the increment of displacement and rotation, and the where d¯ u and dθ residual denotes   R=

4.7

R

a

Rb



 Z

n



  dL − f ext L m   Z h iT n   = B a B b   dL − f ext L m

 =

T

 B 

(4.83)

Stiffness matrix

The stiffness matrix is obtained by performing derivative of the residual vector with respect to displacement and rotation variables

K=



∂Ra a ∂U 

# " K 11 K 12

∂R = = ∂U K 21 K 22 ∂Rb

∂U a



∂Ra ∂U b  ∂Rb ∂U b



(4.84)

h iT where U = u θ , therefore each block of the stiffness can also be written in a matrix form with four sub-blocks. For example, the first block K 11 can be expressed as ∂Ra 1 a  ∂u

 K 11 = 

where Ra1

Z = L

∂ ∂ua

∂Ra 2 ∂ua

T

n+

69

∂Ra 1 ∂θ a 



∂Ra 2 ∂θ a

∂κ ∂ua

(4.85)



T

m dL − f ext

(4.86)

Ra2

Z = L

∂ ∂θ a

T

n+

∂κ ∂θ a

T

m dL − f ext

(4.87)

By employing the results obtained in previous sections, each block of K 11 can be calculated explicitly.

4.7.1

First block:

∂Ra 1 ∂ua

# ∂κ T n+ m ∂ua " # ∂ ∂ T ∂n ∂ T = n+ ∂ua ∂ua ∂ua ∂ua

∂Ra1 ∂ = a ∂u ∂ua

"

From Equation (4.65) we know that ∂Ra1 = ∂ua

4.7.2

Second block:

∂ ∂ua

∂ ∂ua

∂ ∂ua

T

∂ ∂ua

T Cn

T

= 0, and hence

a dN a ∂ T dN = ΛC Λ n ∂ua dS dS

"

# ∂κ T n+ m ∂ua " # ∂ T ∂n ∂ ∂ T n+ = ∂θ a ∂ua ∂ua ∂θ a dN a ∂Λ dN a ∂Λ T = C + ΛC φ,S n n dS ∂θ a dS ∂θ a

Third block:

∂ ∂ua

T

(4.90)

∂Ra 2 ∂ua

# ∂ T ∂κ T n+ m ∂θ a ∂θ a (" #T ) " # ∂ ∂Λ T ∂ ∂κ T = φ,S C n + C mκ ∂ua ∂θ a ∂ua ∂θ a " #T " #T ∂Λ T dN a ∂Λ T ∂ = C n + φ,S C n a ∂θ a dS ∂θ a ∂u " #T a ∂Λ dN a ∂Λ T T dN = C + φ C Λ n n ,S ∂θ a dS ∂θ a dS

∂Ra2 ∂ = ∂ua ∂ua

(4.89)

∂Ra 1 ∂θ a

∂ ∂Ra1 a = ∂θ ∂θ a

4.7.3

(4.88)

"

70

(4.91)

4.7.4

Fourth block:

∂Ra 2 ∂θ a

# ∂κ T n+ m ∂θ a " # ∂ T ∂ T ∂ ∂ n+ Cn a = a a a ∂θ ∂θ ∂θ ∂θ " T # T ∂ ∂κ ∂κ ∂κ + m+ Cm a a a a ∂θ ∂θ ∂θ ∂θ

∂Ra2 ∂ a = ∂θ ∂θ a

"

∂ ∂θ a

T

(4.92)

where ∂ ∂θ a

"

∂ ∂θ a

T #

∂ n= ∂θ a

"

∂Λ ∂θ a

and

∂ ∂θ a

T

∂ Cn a = ∂θ

#T

T

"

n=

φ,S

∂Λ ∂θ a

"

∂2Λ ∂θ a ∂θ a

T

T

#T

T φ,S

Cn

∂Λ ∂θ a

#T φ,S

φ,S

C n

(4.93)

(4.94)

Once K 11 is obtained, the other blocks in stiffness matrix K 12 , K 21 , and K 22 can be calculated in the same way, and hence the system of Equations (4.6) can be solved directly.

4.8

Mass matrix

For dynamic problems, the mass matrix of a Euler-Bernoulli beam element is employed with all the rotational terms being neglected in this work. The reason is given as follows. On one hand, since nonlinear effects come from the rotation, translational terms in the mass matrix of geometrically exact beam could be chosen the same as in Euler-Bernoulli beam. On the other hand, all the rotational terms are of second and even higher order of element length. Hence, as the number of elements increases, the element length decreases, which in turn leads to a higher rate decrease of the rotational terms that could be neglected when the mesh is fine enough. To verify the above assumption, the author investigates free vibration of a undamped 2D cantilever by using Euler-Bernoulli beam element, whose consistent mass matrix (in which axial displacement is neglected) reads [76] 

156

22l

54

−13l



  2 2 22l 4l 13l −3l ρAl    me =    420  54 13l 156 −22l   −13l −3l2 −22l 4l2 71

(4.95)

where ρ, A, l are the density, area of beam cross section and element length, respectively. By removing all the rotational terms (Note that a very small non-zero value need to be placed in the diagonal to circumvent the numerical difficulty in this case study, but unnecessary in the following numerical examples), the simplified mass matrix then becomes

  156 0 54 0     0 1e − 8 0 0 ρAl   m0e =    420  54 0 156 0   0 0 0 1e − 8

(4.96)

For a cantilever beam with bending stiffness EI = 0.8333, length L = 4 and density per unit length ρA = 1, the frequencies of first three modes with different mesh refinements are computed, as shown in Figure 4.3. The plots clearly indicate that the frequencies generated by consistent mass matrix and simplified mass matrix tend to the same value as the mesh is refined, especially for the low frequency mode. Therefore in this thesis the mass matrix of 2D geometrically exact beam is taken to be the mass matrix of 2D Euler-Bernoulli beam element in [76] by neglecting all the rotational terms:

m2D e

  140 0 0 70 0 0    0 156 0 0 54 0      0 0 0 0 0 ρAl  0  =  420  0 0 140 0 0  70    0 54 0 0 156 0   0 0 0 0 0 0

(4.97)

In the same way the mass matrix of 3D geometrically exact beam is obtained: 

m3D e

            ρAl   = 420             

140

0

0

156

0

0

0

0

0

0

0

0

70

0

0

54

0

0

0

0

0

0

0

0

0

0

0 0

70

0

0

0

0 0



 0 0    156 0 0 0 0 0 54 0 0 0   0 140 0 0 0 0 0 70 0 0    0 0 0 0 0 0 0 0 0 0    0 0 0 0 0 0 0 0 0 0    0 0 0 0 140 0 0 0 0 0   0 0 0 0 0 156 0 0 0 0    54 0 0 0 0 0 156 0 0 0    0 70 0 0 0 0 0 140 0 0   0 0 0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0 0 0 0

0

0 0

72

0

54

0

0

(4.98)

Note that, to avoid shear locking only one Gauss point is used for the integration over the element length for both stiffness and mass matrix.

(a)

(b)

(c)

Figure 4.3: Frequencies of a undamped cantilever beam in (a) 1st mode; (b) 2nd mode; (c) 3rd mode.

73

4.9 4.9.1

Numerical examples A curved cantilever under tip force

This example is taken from the work of Bathe and Bolourchi [77]. As shown in Figure 4.4, the initial configuration (dashed line) of the curved beam is defined as a 45◦ circular segment with radius R = 100 in X-Y plane, and subject to a force F = 600 at its tip in Z direction. The beam has a square cross section of unit area, and the physical properties are chosen as: EA = 107 , EI = 8.333×105 , GA = 5×106 , GJ = 7.03×105 , where E and G are the Young’s modulus and shear modulus, respectively, A and I are, respectively, the area and second moment of inertia, and J is the torsional constant.

Y Z

X

F F

Figure 4.4: A curved cantilever under free-end force

Eight beam elements are employed for the spatial discretisation, and six equal load steps are used to apply the total load of F . The final configuration and tip displacement components are shown in Figure 4.4 (solid line) and Table 4.1, respectively. Solutions from other references are also presented in the table, from which we see that the present geometrically exact beam formulation gives a very similar results. Table 4.1: Cantilever bend free-end displacement components

Model Present Ibrahimbegović1 Bathe and Bolourchi Simo and Vu-Quoc Cardona and Geradin Crisfield Ibrahimbegović2

ux 13.659 13.668 13.4 13.50 13.73 13.63 13.601

uy -23.686 -23.697 -23.5 -23.48 -23.67 -23.87 -23.746

uz 53.449 53.498 53.4 53.37 53.50 53.71 53.407

This example is also selected to investigate the quadratic convergence rate of the Newton solution procedure in a three-dimensional setting. The residual norm after each 74

iteration at the second and sixth load steps are shown in Table 4.2. The results from Ibrahimbegović [1] are also given for comparison. Fewer iterations are needed in the final steps, and the quadratic convergence rates of the Newton method are clearly displayed. Table 4.2: Residual norm after the 2nd and 6th load steps

Iter 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6

4.9.2

Present Ibrahimbegović1 2nd load step 1.000 × 102 1.000 × 102 5 9.273 × 10 9.267 × 105 4 3.013 × 10 3.012 × 104 6.772 × 104 6.591 × 104 2 1.320 × 10 1.274 × 102 3.263 × 104 3.124 × 104 1 2.294 × 10 2.125 × 101 0 1.071 × 10 8.520 × 10−1 5.169 × 10−6 3.420 × 10−6 9.533 × 10−9 1.425 × 10−8 6th load step 1.000 × 102 1.000 × 102 4 7.531 × 10 7.194 × 104 1.913 × 102 1.755 × 102 2 7.933 × 10 7.723 × 102 3.582 × 10−1 3.036 × 10−1 6.109 × 10−3 4.897 × 10−3 1.626 × 10−8 2.223 × 10−8

A cantilever beam subject to a pure bending moment

A cantilever beam subject to a pure bending moment at its end, as shown in Figure 4.5, is usually used as the first example to test the accuracy of presented element formulation under extreme bending and large deformation. This problem has been studied by a number of researchers [77] [78].

M

L Figure 4.5: A cantilever beam subject to a pure bending moment

Assuming that bending deformation is constant along the beam means the deformed shape must be a part of circle. The analytical solution is not difficult to obtain by 75

analysing the structure during deformation, as depicted in Figure 4.6. uX = −(L − R sin θ) = −(L −

uY = R − R cos θ = θ

=

ML EI

L θ (1

L θ

sin θ) (4.99)

− cos θ)

where R is the radius of the deformed curve and follows that L = Rθ, E and I are the Young’s modulus and second moment of inertia, respectively.

θ R sin θ R uY

Y

X uX Figure 4.6: Deformed shape of the pure bending cantilever beam

In the present study the beam is discretised with 10 elements. The length of the cantilever is L = 10, the Young’s modulus E and second moment of inertia I are 79577 and 0.0013, respectively, its cross section area A = 0.1257, Poisson’s ratio µ = 0. The tip displacement components provided by the analytical solution under the value of the bending moment M = 2.5π are given in Table 4.3. The results obtained from some other formulations are also presented. Table 4.3: Tip displacement components under M = 2.5π

Model Present Ibrahimbegović1 Zienkiewicz and Taylor2 Euler-Bernoulli beam3 Analytical

uX -0.994522 -0.994522 -0.995316 0 -0.996837

uY 3.73019 3.73019 3.73037 3.92699 3.72923

uZ 0 0 0 0 0

It is clearly seen that the displacement obtained in the present work is the same as that from Ibrahimbegović [1], both of which, as well as that from Zienkiewicz and Taylor [73], are very close to the analytical solution. Note that since the Euler-Bernoulli beam is linear, it generates a displacement with no component in X direction. 76

To observe an extremely large rotation, the bending moment is then increased from 0 to 20π with an constant increment of π and a mesh of 20 beam elements. The loaddisplacement curves at the tip of the cantilever obtained from the analytical solution, Euler-Bernoulli beam, 2D and 3D geometrically exact beam formulations are shown in Figure 4.7. The curves produced by analytical solution and the two geometrically exact beam models agree very well during the whole deformation process. The Euler-Bernoulli beam element, on the other hand, only gives a liner relation between the solution and bending moment as expected. Specifically, it produces a acceptable result for M − θ curve since the rotation is linearly dependent on the bending moment.

The deformed shapes from the different beam formulations are given in Figure 4.8, which clearly shows that the Euler-Bernoulli beam formulation generates no X-displacement. For geometrically exact beam models, a fully closed circle is obtained by both 2D and 3D formulations when the bending moment reaches 20π. Note that from these plots it is clearly seen that no visible difference exists in the results produced by two geometrically exact beam formulations, therefore in the following context the 2D and 3D geometrically exact beam formulations will be employed to model beam which experiences large deformations in 2D and 3D spaces, respectively. Finally the fully deformed beam in Figure 4.8(b) is chosen to be the initial configuration and subject to a clockwise bending moment with the same increment. Figure 4.9 shows how it deforms back to its initial configuration as a straight line and then continues to bend in the opposite direction and finally form another closed circle. The expected symmetry is clearly depicted.

77

(a)

(b)

(c)

Figure 4.7: Deflection curve of the cantilever beam. (a) X-displacement; (b) Ydisplacement; (c) rotation.

78

(a)

(b)

(c)

Figure 4.8: Deformation of the cantilever beam. (a) Euler-Bernoulli beam; (b) geometrically exact beam 2D; (c) geometrically exact beam 3D.

79

Figure 4.9: Deformation of the cantilever beam from a circular configuration.

4.9.3

A frame structure subject to a concentrated load

A frame structure subject to a concentrated force [79], as shown in Figure 4.10, is studied. The frame is discretised with 20 elements, and both Euler-Bernoulli and 2D geometrically exact beam elements are employed to investigate the load deflection behaviour of the structure.

F

Material properties: EA = 106 EI = 2 × 105 GA = 105

30

300

270

Figure 4.10: A frame structure subject to a concentrated load, 20 elements.

The deformed configurations under a load F = 45, obtained from the two formulations are depicted in Figure 4.11. It is easy to see that the geometrically exact beam formulation produces a larger deformation than Euler-Bernoulli beam model. The load deflection curves obtained by an increasing load are shown in Figure 4.12, which gives a very similar path rendered by the two beam formulations when the force is less than 10, while only geometrically exact beam model captures the non-linear behaviour of the 80

structure when the force keeps increasing. Note that in this case Newton method is applied for the linearisation, and therefore the post-critical behaviour of the frame cannot be properly captured. To do so some other approaches such as arc-length method need to be employed.

(a)

(b)

Figure 4.11: Deformation of the frame. (a) Euler-Bernoulli beam; (b) 2D geometrically exact beam.

Figure 4.12: Deflection curve of the frame

81

4.9.4

A cantilever beam bent to a helical form

This example investigates a cantilever beam which is subject to an increased bending moment M = 200π and an out-of-plane force F = 50, as shown in Figure 4.13. As already known in the previous cantilever example, a bending moment M = 20π at tip deforms the cantilever beam to a circular shape. However, the force will drag the circle out of the plane. As a result, 10 circles in a helical form along the direction of the force would be expected. The physical properties of the beam is taken to be the same as in the pure bending cantilever example, while 100 beam elements are employed in order to capture the sufficient accuracy of the large deformation.

Figure 4.13: A cantilever beam subjected to end moment and out-of-plane force.

The sequence of the deformed shapes at different load steps are shown in Figures 4.14 and 4.15, from which the 3D geometrically exact beam formulation presented in this work produces a very nice result. More importantly, when the rotation reaches the value equal to 2π and its multiples, the author do not come across any divergent issue which emerges when using Ibrahimbegović’s formulation [21].

82

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 4.14: Deformed shape of a beam bent in helical form at different load (M ) steps: 10π, 30π, 50π, 70π, 90π, 110π, 130π, 150π, 170π, 190π

83

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 4.15: Deformed shape of a beam bent in helical form at different load (M ) steps: 20π, 40π, 60π, 80π, 100π, 120π, 140π, 160π, 180π, 200π

84

4.9.5

Dynamic response of a cantilever under tip force

This example present a dynamic response of a undamped cantilever beam under a tip force which varies in time, as depicted in Figure 4.16. Material properties of the cantilever are given as EI = GJ = 3.194e − 6, EA = GA = 1e6, ρA = 1, while the force

varies sinusoidally as F = 3.194 sin(πt) on 0 ≤ t ≤ 1, and after that the cantilever undergoes free vibration. Newmark-β method [80] with time step ∆t = 0.0125 is adopted for the time integration.

F

F 3.194

L=1

1

t

Figure 4.16: A cantilever beam subject to a time varying force

The evolution of the vertical tip displacement of the cantilever beam is computed by using different types of beam formulations and different number of elements. All the results, along with an analytical solution [80] are plotted in Figure 4.17. First we see that as the number of elements increases, the solution obtained by geometrically exact beam formulation converges to the analytical solution (no visible difference of the solution with 10 and 100 elements can be seen in the plot). However, one can clearly observe that the Euler-Bernoulli beam still provides a better result even than the geometrically exact beam with 100 elements. The reason, the author believes, is that the analytical solution is obtained under an Euler-Bernoulli beam assumption so that an extremely accurate solution can be provided by Euler-Bernoulli beam formulation even with only one element.

Figure 4.17: Evolution of cantilever tip displacement

85

4.9.6

Right-angle cantilever beam

The last example is adopted from [81]. An L shape cantilever beam subjected to an out-of plane force at its elbow is shown in Figure 4.18. The force increases from 0 to 50 linearly on 0 ≤ t ≤ 1, and then decreases back to 0 on 1 ≤ t ≤ 2 and keep unchanged afterwards. Material properties are listed as: EA = GA = 1e6, EI = GJ = 1e3, ρA = 1.

For comparison, different mesh refinements are established with 4, 20, and 100 elements. Time step is ∆t = 0.025, and also Newmark-β method is employed for time integration.

F

F 50

Material properties: EA = GA = 106 EI = GJ = 103 ρA = 1

L t 1

2

L

Figure 4.18: Right-angle cantilever beam

The out-of-plane displacements at the elbow and tip are computed and shown in Figure 4.19. Note that the amplitude of vibration has the same order of magnitude as the structure dimensions. We also see that the spatial convergence is achieved as the mesh is refined.

(a)

(b)

Figure 4.19: Evolution of out-of-plane displacement of right-angle cantilever beam. (a) elbow; (b) tip.

86

Chapter 5

Solution Strategies for Fluid-structure Interaction At this stage, a complete description of modelling the fluid flow and geometrically nonlinear beam problems and the formulation of fluid-structure interaction has been given. Thus, a solution algorithm need to be employed to solve the coupled system of equations. In this work, the overall problem is decomposed into three subdomains: the fluid, the solid and the interface domains, and a weakly coupled solution strategy is adopted to solve the system of equations in a partitioned procedure. In Chapter 1 a very brief introduction has been given to the solution approaches, where classification was made into monolithic and partitioned strategies according to wether the subsystems of FSI are solved simultaneously or sequentially. In this chapter, on the other hand, the solution algorithms are classified according to whether both kinematic consistency and traction equilibrium at the fluid-structure interface are satisfied in each time step, resulting in strongly coupled and weakly coupled strategies, respectively. The relationship of these terminologies are illustrated in Figure 5.1. In the following context of this chapter, Section 5.1 gives a brief overview of strongly coupled strategies, including monolithic and partitioned algorithms. Section 5.2 introduces weakly coupled algorithms and a force-predictor based staggered scheme which will be employed in the FSI simulations in this work.

87

Figure 5.1: Classification of solution algorithms [2].

5.1

Strongly coupled strategy

To solve the coupled systems of equations exactly, strongly coupled strategies are usually employed, either monolithic or partitioned. Monolithic schemes treat the systems of equations simultaneously, and all the variables are solved at the same time. On the contrary, partitioned methodologies decompose the coupled system into subsystems and solve them separately. The monolithic approaches require a code developed for this particular combination of physical problems, whereas in partitioned approaches the software modularity is preserved. Hence, partitioned schemes have a higher flexibility, but iterative calls of the subsystem solvers are necessary in each time step in order to obtain the exact solution. A brief review of some strongly coupled methods is presented in the following subsections.

5.1.1

Monolithic algorithm

In monolithic schemes Newton-Raphson technique is usually applied to linearise the whole system of equations. Thus a matrix system is obtained which contains all subsolvers. In the context of fictitious domain/distributed Lagrange multiplier method, the matrix system can be expressed as [37]  K f f  u u K f  pu   K λuf  0

K uf p K uf λ 0

0

0

0

0

K us λ

  Ruf      R   d¯ p 0   p     = −  ¯  Rλ     s K λu   dλ    Rus d¯ us K us us 0



d¯ uf



(5.1)

where uf , p, us and λ are fluid velocity, fluid pressure, solid velocity and Lagrange multipliers, respectively. K and R are stiffness matrices and residual vectors associated

88

¯ and d¯ with their corresponding variables. d¯ uf , d¯ p, dλ us are the incremental nodal solution variables to be computed. Note that all the stiffness matrices in (5.1) are obtained by performing the derivative of residual vectors with respect to solution variables, and the matrix system of equations are solved at once. Thus, the monolithic method does not allow the employment of any existing sub-solver, and its computational cost is expensive. But this strategy has a good accuracy and robustness.

5.1.2

Partitioned algorithm

Enabling use of existing sub-solvers, partitioned strategies are more popular in resolving the fluid-structure problems [2, 82, 83, 83–88]. Both Newton-Raphson and fixed-point methods (also known as block Gauss-Seidel method) can be used to solve the coupled system of equations.

• Block Newton methods. These methods solve the coupled systems by using NewtonRaphson method in a partitioned procedure. The Jacobian of the fluid equation

with respect to the interface’s position, known as the shape derivative, is not easy to obtain [89]. Instead approximation of Jacobian is therefore made and hence this kind of strategies is called inexact block Newton methods [90]. Dettmer [2] developed an arbitrary Lagrangian-Eulerian (ALE) based FSI framework, in which the overall problem is regarded as a coupled three-field problem, involving the fluid flow, the motion of the fluid mesh and the structural dynamics. The adopted solution strategy represents the decomposition of a monolithic Newton-Raphson procedure into several sub-solvers, and it can be classified as an exact block Newton method. • Block Gauss-Seidel method. The coupled system can be solved by using block

Gauss-Seidel iterations [91], in which the fluid and solid subsystems are solved successively until the convergence criterion is achieved. However, the convergence deteriorates as the incompressibility of the fluid or the ratio of fluid over solid mass increases [18, 92]. If only one fixed-point iteration is performed in each time step, then the so-called staggered scheme is achieved, which will be introduced in the following section.

89

5.2

Weakly coupled strategies - a staggered scheme

As presented above, a strongly coupled solution strategy refers to a scheme in which both kinematic consistency and equilibrium of the traction forces at the interface are satisfied in each time step. On the contrary, in a weakly coupled strategy usually only the kinematic consistency is required, which is referred to as staggered scheme. Staggered schemes naturally belong to the class of partitioned strategies. Thus, such methodologies allow the use of existing computer programs specifically designed for fluid and/or solid subsystems. Also, the high computational efficiency make them applicable to large scale problems, which may not be tackled with strongly coupled methods. On the other hand, the loss of accuracy and restriction to very small time steps contribute to the disadvantages of staggered schemes. More importantly, the stability of the weakly coupled method is an important issue. To solve these drawbacks, significant research has been carried out, see a review of the progress in weakly coupled schemes in [93]. In the present work a force-predictor based staggered scheme proposed by [94] is employed as the solution strategy to solve the fluid-structure system. This method is proved to provide good stability, robustness and accuracy.

5.2.1

Governing equation

After spatial and temporal discretisation, the fluid-structure interaction problem in Equations (3.13) - (3.15) can be expressed as r f (uf , ui ) = 0 tf (uf , ui ) + ts (ds , ui ) = 0 r s (ds , ui ) = 0

(fluid)

(5.2)

(interface)

(5.3)

(solid)

(5.4)

where uf and ds denote, respectively, the degrees of freedom inside the fluid and solid domains. In the context of incompressible fluid-flexible fibre interaction, uf is the nodal fluid velocity and pressure values, whereas ds is the displacements and rotations of the beam element. ui denotes the interface degrees of freedom, and it is usually obtained by interpolation of the degrees of freedom on the interface boundary of the fluid and solid meshes (not including the fluid pressure). The vectors r f and r s denote the residual forces in the fluid and solid domain, while tf and ts represent the traction forces on the interface given by the fluid and the solid, respectively. In a strongly coupled strategy, Equations (5.2)-(5.4) are satisfied exactly in each time instant. In the staggered schemes, however, by introducing the interface traction force 90

ti (5.2)-(5.4) are rewritten as  r f (uf , ui ) = 0  tf (uf , ui ) = ti 

 r s (ds , ui ) = 0  ts (ds , ui ) = −ti 

fluid

(5.5)

solid

(5.6)

In this way the overall problem is decoupled to fluid and solid subsystems, which interact with each other via interface degrees of freedom and traction force. In staggered schemes the main difference is that the traction force equilibrium is not required, and only one iteration is needed within each time step, and that is why staggered schemes are known as weakly coupled.

5.2.2

Solution algorithm

Let the subscripts n − 1, n and n + 1 denote the time instants, the staggered scheme for

the solution of system (5.5)-(5.6) at time instant tn+1 can be summarised as follows [94], i i 1. perform traction force predictor tiP n+1 = 2tn − tn−1

2. load solid with tiP n+1 and solve equations (5.6) for solid displacement and interface degrees of freedom, respectively, dsn+1 and uin+1 3. substitute uin+1 into fluid subsystem (5.5) and obtain the interface traction force ti∗ n+1 iP 4. average the traction force tin+1 = βti∗ n+1 + (1 − β)tn+1

5. go to next time step In each time step the structure is solved under tiP n+1 , whereas the traction force calculated iP i∗ from the fluid is ti∗ n+1 . Hence, the difference tn+1 − tn+1 naturally reflects the amount

of violation of the system of equations (5.5) and (5.6). Note that the traction force tin+1

iP is computed from a weighted average of ti∗ n+1 and tn+1 . The parameter β is dependent

on the solid over fluid mass ratio. To achieve unconditional stability β may need to be a very small value [94]. A usual chosen value is β = 0.5. If β is set to zero, as can be seen in Step 4, the fluid exerts no impact on the solid.

91

5.2.3

Main features

Compared to other weakly coupled strategies, the present staggered scheme provides some advantages which can be summarised as follows [94]: • this scheme gives second-order accuracy in time. When generalised-α method is

adopted for both fluid and solid solver, the accuracy of overall solution strategy can be maintained to be second order.

• the strategy is unconditionally stable, and allows for the simulation of problems of fluid-thin-structure interaction, which has been believed to require a strongly

coupled scheme. This feature makes it possible to apply the staggered scheme in the present work. • the use of the averaging parameter β allows the scheme to be applicable in the case with small solid over fluid mass ratio. However, small solid over fluid mass

ratio leads to large truncation errors, and small values of β results in reduction of high frequency damping of the overall system. • the current staggered scheme provides some applicability very similar to the GaussSeidel strategy which is a strongly coupled strategy, but has a remarkable computational efficiency.

92

Chapter 6

Fluid-flexible Fibre Interaction: 2D Numerical Examples All the preliminary work has been made to solve fluid-structure interaction problems in previous chapters. In the following context, a number of problems involving two- and three-dimensional fluid-flexible fibre interactions will be presented to demonstrate the accuracy and robustness of the current computational framework. Note that in all the problems studied, backward-Euler method is adopted for the temporal discretisation for both fluid and solid, whereas the staggered scheme is employed to solve the coupled system of equations except for one problem which is treated with monolithic procedure. The direct solver PARDISO [95, 96] is employed as fluid and solid solver. In this chapter, four numerical examples are investigated. Section 6.1 presents a flexible beam attached to an immersed fixed square rigid body, which is usually used as a benchmark problem to assess the performance of numerical strategies for fluid-flexible body interaction. Section 6.2 gives a detailed analysis of a flexible fibre whose centre is fixed in a soap film, in which extreme deflection is observed. A buoyant fibre immersed in shear flow is studied in Section 6.3, where we can follow its complex movement. The final example, a leaflet in cross flow, is shown in Section 6.4, where periodic and extreme deformation is captured. A brief discussion in Section 6.5 concludes this chapter.

6.1

Flow-induced vibrations of a flexible beam

This problem was first studied by [97] and then used as a benchmark to evaluate the accuracy and robustness of different numerical schemes for fluid-flexible body interaction [2, 22, 71, 82, 94]. The geometry and boundary conditions of the problem are given in 93

Figure 6.1. A flexible beam is attached to a fixed square rigid body which is immersed in incompressible fluid flow with a uniform inflow velocity. In the initial configuration, the beam is aligned with the far field flow, and later the lift forces generated by vortices excite the vibrations of the flexible beam. v=0 6 u = uin v=0

p=0

1 1

4

6 v=0 20

Figure 6.1: Flow-induced vibrations of a flexible beam: geometry and boundary conditions.

Figure 6.2: Flow-induced vibrations of a flexible beam: hierarchical b-spline mesh with two levels of refinements.

The fluid density and viscosity are taken as ρf = 1.18 × 10−3 and µf = 1.82 × 10−4 ,

respectively. The density of the beam is ρs = 0.1, its Young’s modulus is E = 2.5 × 106

and Poison’s ratio is νs = 0.35. The area and second moment of area of cross section of the beam are, respectively, A = 0.06 and I = 1.8 × 10−5 . These physical properties

are listed in Table 6.1. The inflow velocity is Uin = 51.3, and the resulting Reynolds number is Re = ρf DUin /µf ≈ 333, where D = 1.0 is the diameter of the square body.

A time step ∆t = 0.005 is chosen for the simulation.

Table 6.1: Flow-induced vibrations of a flexible beam: parameters used in the simulations.

ρf 1.18 × 10−3

µf 1.82 × 10−4

ρs 0.1

E 2.5 × 106 94

νs 0.35

A 0.06

I 1.8 × 10−5

The simulation is performed with two level of hierarchical refinement, as shown in Figure 6.2. The beam is modelled with 40 geometrically exact beam elements as presented in previous Section, and the total degrees of freedom (DOF) is 19018. The time history of vertical displacement of the beam-tip is shown in Figure 6.3, which indicates that periodic oscillations of the flexible beam are obtained. The amplitude and the frequency of the oscillations are compared with those from literature in Table 6.2. The contour plots of velocity at different time instants during oscillations are given in Figure 6.4.

Figure 6.3: Flow-induced vibrations of a flexible beam: vertical displacement of the tip.

Table 6.2: Flow-induced vibrations of a flexible beam: amplitude and frequency of tip displacement oscillations

Wall [97] Dettmer and Perić [2] Kadapa, el at. [37] Present

Amplitude 1.12-1.32 1.1-1.4 1.27 1.21

95

Frequency (Hz) 2.78-3.22 2.96-3.31 3.41 3.55

(a) t = 4.154

(b) t = 4.236

(c) t = 4.306

(d) t = 4.358

(e) t = 4.384

(f) t = 4.456

Figure 6.4: Flow-induced vibrations of a flexible beam: contour plots of velocity at different time instants.

96

6.2

Flexible fibre in a soap film

A flexible fibre immersed in a soap film is used in experimental and numerical studies [98–100] to investigate the drag of a flexible fibre in viscous flows. The geometry and boundary conditions are depicted in Figure 6.5. A hierarchical b-spline refinement is performed near the immersed fibre, as shown in Figure 6.6. Note that the weight of the film and the air resistance are neglected here, and hence the soap film is placed in a horizontal direction. Fixed at its centre, the fibre is subject to a uniform inflow velocity. No-slip conditions are applied on both top and bottom sides of the domain. no-slip 0.045 u = uin v=0

p=0

L

0.045 no-slip 0.04

0.14

Figure 6.5: Flexible fibre in soap film: geometry and boundary conditions

Figure 6.6: Flexible fibre in soap film: mesh with hierarchical refinement

The parameters used in the simulations, summarised in Table 6.3, are chosen the same as in [100]. The axial stiffness of the fibre EA is relatively high, meaning that the fibre is assumed to be inextensible. Different bending stiffnesses EI and inflow velocities Uin are adopted in the simulations. The Reynolds number, computed from Uin and L, varies from Re = 50 to 312.5. Time step ∆t = 0.001 is adopted for all the cases.

Table 6.3: Flexible fibre in soap film: parameters used in the simulations.

ρf 0.003

µ 1.2e-6

u 1.0-2.5

ρs A 7e-3

EA ∞

EI (4.17 − 125) × 10−8 , 1.25 × 10−4 97

L 0.05

Shown in Figure 6.8, the history of horizontal displacement and drag force of the fibre are plotted for different inflow velocities and an unchanged bending stiffness EI = 1.25 × 10−7 . It is clearly seen that the fibre tip begins to oscillate with a diminishing

amplitude until reaching a steady state. A larger inflow velocity results in a larger amplitude, a higher frequency of the oscillation and a shorter time the steady state is reached. The drag force reduces to a minimum each time when the fibre bends to its maximum, and in turn, reaches a maximum when the fibre swings back.

(a) horizontal tip displacement

(b) drag force

Figure 6.7: Flexible fibre in soap film: tip displacement and drag history for different inflow velocities, EI = 4.17e − 8.


(b) drag force


The deformed fibre shapes at time t = 0.085 with different inflow velocities and bending stiffnesses are depicted in Figure 6.10. As expected, the deflection of the fibre is proportional to the inflow velocity and inversely proportional to its bending stiffness. The contour plots of vorticity with u = 2.5 are given in Figure 6.12. Note that the fibre tips with smallest stiffness is closing at time instant t = 0.085, and this extreme deflection

98


(b) drag force


Figure 6.10: Flexible fibre in soap film: deformed shapes at t = 0.085 with different inflow velocities and bending stiffnesses EI = 1.25e − 6 (left), EI = 1.25e − 7 (middle), EI = 4.17e − 8 (right)

gives a huge difficulty to the body-fitted mesh without re-meshing, while in the present methodology the fixed Eulerian grid shows its remarkable advantage. Figure 6.11 gives a quantitative analysis for the drag force, in which figure (a) plots the variation of drag force with the inflow velocity. The drag force is taken as the one in steady state, say the drag at time t = 2. Clearly seen in the figure, for softer fibres, the drag grows more slowly as the velocity increases. In [100], the drag of the most rigid fibre (EI = 1.25e − 4) scales quadratically with the velocity, while in the plots here that

drag grows even more quickly. This is due to the neglect of the weight of the soap film and the air resistance, as mentioned earlier. Following references [98, 100], the relation between the drag divided by the stiffness p F/(EI) and the parameter η = u/ (EI) is plotted in Figure 6.11 (b), which shows how F/(EI) varies with η. For different stiffnesses, all the functional dependences fall in O(η 2 ) and O(η 4/3 ). This result matches with the experimental [98] and numerical [100] 99

observations. Note that in [98], the fluid density and fibre thickness and length are also included in η. Since these parameters are constant (see Table 6.3), they are unnecessary in the scaling analysis.

(a) drag v.s inflow velocity

(b) scaling analysis of drag v.s inflow velocity

Figure 6.11: Flexible fibre in soap film: drag forces versus inflow velocity and its scaling analysis.

100

(a) EI = 1.25e − 6

(b) EI = 1.25e − 7

(c) EI = 4.17e − 8

Figure 6.12: Flexible fibre in soap film: contour plots of vorticity at time t = 0.085 with different bending stiffnesses. The inflow velocity is u = 2.5.

101

6.3

Neutrally buoyant fibre in shear flow

This problem was studied by [3, 101] to test different numerical schemes in the framework of immersed boundary method. In the example the behaviour of a neutrally buoyant fibre immersed in a shear fluid flow is investigated. The geometry and boundary conditions are shown in Figure 6.13. A constant velocity uin is prescribed at top and bottom walls in opposite directions. At the inlet, a shear profile u (x, 0) = (G (y − H/2) , 0) is given,

which leads to a shear rate G = 2U/H. The fibre of length L is placed horizontally at the half height of the channel. However, to excite its motion, a slight curvature is given as an imperfection. A mesh with three levels of hierarchical refinement near the fibre is depicted in Figure 6.14. u = uin , v = 0 u = uin v=0

H = 0.5

L

u = −uin , v = 0 0.3

1.7

Figure 6.13: Neutrally buoyant fibre in shear flow: geometry and boundary conditions

Figure 6.14: Neutrally buoyant fibre in shear flow: mesh with hierarchical refinement

The density and viscosity of the fluid are 10 g/cm3 and 7.8 g/cm · s, respectively. The

velocity at top and bottom walls is U = 1.0cm/s. The axial stiffness of the fibre EA is set to infinitely large, while the bending stiffness is EI = 6e − 4g · cm3 /s2 . Note

that in this problem the density of the fibre ρs is set to be zero, meaning that only the static response of the fibre is considered. The length of the fibre is L = 0.1cm. All the parameters used in the simulation are listed in Table 6.4. ∆t = 0.001 is chosen as the time step. Note that to keep consistency with [3, 101], dynamics of the fibre is neglected, and the fibre density is set to zero. Because of the extremely small (zero) solid over fluid mass ratio the staggered scheme diverges. Hence, the monolithic strategy is adopted in this problem.

102

Table 6.4: Neutrally buoyant fibre in shear flow: parameters used in the simulations.

ρf 10

µf 7.8

U 1.0

ρs A 0

EA ∞

EI 6 × 10−4

L 0.1

To describe the fibre flexibility during the motion, a dimensionless parameter is defined as [102] χ=

µf GL3 EI

(6.1)

which can be interpreted as the ratio of fibre deflection to fibre length. This parameter gives a useful measure of fibre flexibility and characterises four fibre orbit classes in [3, 101], as shown in Figure 6.15. Note that [3] extends this simulation to three dimensional, and the fibre diameter is included in the χ definition.

Figure 6.15: Neutrally buoyant fibre in shear flow: orbit classes [3]

According to the parameters in Table 6.4, the flexibility dimensionless parameter in the present study is χ = 52, and its corresponding fibre deformation captured at different time instants are shown in Figure 6.16. Clearly, this orbit falls into the Snake turn class. Note that in Figure 6.15 the initial configuration of the fibre in each orbit class is a straight line, while in this study the fibre is, as mentioned above, placed with a curvature initially.

103

(a) t = 0.15

(b) t = 0.25

(c) t = 0.35

(d) t = 0.5

(e) t = 0.8

(f) t = 1.0

Figure 6.16: Neutrally buoyant fibre in shear flow: fibre deformation at different time instants.

6.4

Two flapping leaves

Proposed by [103] and investigated by [37, 71, 104–106], the two flapping leaves example is used to model the behaviour of an idealised two-leaflet valve. The geometry and boundary conditions of the problem are shown in Figure 6.17. The problem consists of two cantilevered leaf valves of equal length immersed in a 2D channel. No-slip boundary conditions are applied on the top and bottom sides of the channel, the outlet is set to be traction-free, and the inlet has a time-dependent velocity profile, given by uin = 5y (1.61 − y) (1.1 + sin (2πt)) .

(6.2)

The fluid and solid have an equal density of ρf = ρs = 100. The fluid viscosity is µf = 10. Young’s modulus and Poisson’s ratio of the valve are E = 5.6 × 107 and

νs = 0.4, respectively. The thickness of the valve is h = 0.0212, as listed in Table 6.5.

u = uin v=0

1.61 0.7

no-slip

2

6

Figure 6.17: Two flapping leaves: geometry and boundary conditions.

104

p=0

no-slip

0.7

Table 6.5: Neutrally buoyant fibre in shear flow: parameters used in the simulations.

ρf 100

µf 10

ρs 100

E 5.6 × 107

νs 0.4

h 0.0212

To assess the convergence of spatial discretisation, this problem is studied with three different levels of hierarchical b-spline refinement near the immersed leaves, as shown in Figure 6.18. In this example, each leaf is modelled with 10, 20, 40 and 80 elements in Level-0, Level-1, Level-2 and Level-3 meshes, respectively.

(a) Level-0 mesh.

(b) Level-1 mesh.

(c) Level-2 mesh.

(d) Level-3 mesh.

Figure 6.18: Two flapping leaves: hierarchical refinements near the immersed leaves

For each mesh level, different time steps are used in the simulation in order to investigate the temporal convergence. Figures 6.19 and 6.20 depict the X- and Y-displacements of free end of a leaf for Level-0 and Level-2 meshes obtained with ∆t = 0.01,s ∆t = 0.005 and ∆t = 0.0025. These plots show that the differences between numerical results obtained with these three time steps are obvious at the beginning, where some spurious oscillations are observed. As the time progresses, the results tend to be smooth, and their differences become very small. Compared to [37] in which the same problem is studied with monolithic coupled scheme, the oscillations in current work may result from the use of staggered scheme. Figure 6.21 shows the evolution of X- and Y-displacements of the leaf tip for different mesh levels with ∆t = 0.0025. It is apparent that the solution converges as the mesh is refined. Contour plots of X-velocity and pressure at time instant t = 0.5 are given, 105

respectively, in Figures 6.22-6.23. As seen in these plots, the solution improves as the mesh is refined.

6.5

Discussion

Based on the computational framework introduced previously, this chapter presents several two dimensional numerical problems involving flexible fibre-fluid interactions. As already seen in these examples, very nice results are obtained both qualitatively and quantitatively. The use of hierarchical b-splines provides a very efficient way of fluid mesh refinement. On the other hand, the geometrically exact beam element successfully captures the extremely large deformation of the fibres. Results generated by the staggered scheme are in overall excellent, apart from small oscillations observed at the beginning of simulations. However, for cases with very small solid over fluid mass ratio, the convergence of staggered scheme cannot be guaranteed.

106

Figure 6.19: Two flapping leaves: tip displacements for Level-0 mesh with different time steps

Figure 6.20: Two flapping leaves: tip displacements for Level-2 mesh with different time steps

Figure 6.21: Two flapping leaves: tip displacements for different mesh levels with ∆t = 0.0025.

107

(a) Level-0 mesh.

(b) Level-1 mesh.

(c) Level-2 mesh.

(d) Level-3 mesh.

Figure 6.22: Two flapping leaves: X-velocity contour plots at t = 0.5, ∆t = 0.0025.

(a) Level-0 mesh.

(b) Level-1 mesh.

(c) Level-2 mesh.

(d) Level-3 mesh.

Figure 6.23: Two flapping leaves: pressure contour plots at t = 0.5, ∆t = 0.0025.

108

Chapter 7

Fluid-flexible Fibre Interaction: 3D Numerical Examples In this chapter, two numerical examples are presented to demonstrate the proposed computational scheme for fluid-flexible fibre interaction in three dimensional space. Since no 3D fluid-flexible fibre interaction examples can be found in literature, in Section 7.1 a problem inspired by a 2D benchmark is investigated to assess the convergence of the numerical strategy. Section 7.2 gives a study on fibre array immersed in the fluid flow. A discussion is presented in Section 7.3.

7.1

Two flapping fibres subject to sinusoidally varying velocity

This problem is inspired by the 2D heart-valve benchmark studied in the previous chapter [37, 71, 103–106]. Here the geometrically exact beam element is employed to model the solid as circular fibres rather than shells. The geometry and boundary conditions of this problem are shown in Figure 7.1.

109

0.805

no-slip 0.7

u = Uin v=0 w=0

0.805

y z

1.61

0.7 no-slip

2

1.61

6

x

Figure 7.1: Two flapping fibres in 3D: geometry and boundary conditions

This problem consists of two elastic fibres of equal length l = 0.7 fixed to the boundaries of a 3D channel with a dimension of L × W × H = 8 × 1.61 × 1.61. The fluid and fibres

have equal densities of ρf = ρs = 100. The viscosity of fluid is µf = 10. The diametre of fibre is h = 0.1. The Young’s modulus and Poisson’s ratio of the fibre are, respectively, E = 5.6 × 107 and ν = 0.4. No-slip boundary conditions are applied on the top and

bottom faces of the channel, whereas on the front and back faces the Z-velocity w is set

to be zero. The outlet is chosen to be traction-free, and the inlet boundary condition is satisfied with a sinusoidally varying horizontal velocity profile Uin = 10y(1.61 − y)(1.1 + sin(2πt)).

(7.1)

This problem is studied with three different time steps and three different levels of hierarchical b-spline refinement to investigate the convergence with respect to temporal and spatial discretisations, respectively. The refinements are shown in Figures 7.2 to 7.4. Each fibre in Level-k mesh is modelled with 7 × 2k beam elements. Since fibres are

located inside the fluid domain, the 3D view of all mesh levels are the same. Hence only X-Z and X-Y section views are presented for Level-1 and Level-2 meshes.

110

(a) 3D view

(b) X-Z section

(c) X-Y section

Figure 7.2: Two flapping fibres in 3D: level-0 mesh (9375 elements and 27556 DOFs). 7 beam elements for each fibre.

111

(a) X-Z section

(b) X-Y section


(a) X-Z section

(b) X-Y section


To assess temporal convergence of the numerical method, the evolution of X-, Y- and Z-displacements of the fibre tip for Level-0 and Level-1 meshes with three different time steps ∆t = 0.005 and ∆t = 0.0025 are investigated, as depicted in Figures 7.5-7.7. These graphs show the periodical variation of both X- and Y-displacements of the tip of the fibres, which indicate that the difference of the displacements obtained from adjacent mesh levels decreases. In other words, the temporal convergence has been achieved. Note that the geometry and boundary conditions are symmetric about X-Y plane and no inflow velocity in Z-direction is prescribed. Thus, the fibres are expected to deform in X-Y plane only, and Z-displacements of the tip are negligibly small, as seen in Figure 7.7.

112

(a) Level-0 mesh

(b) Level-1 mesh

Figure 7.5: Two flapping fibres in 3D: X-displacements of the fibre tip with different time steps.

(a) Level-0 mesh

(b) Level-1 mesh

Figure 7.6: Two flapping fibres in 3D: Y-displacements of the fibre tip with different time steps.

(a) Level-0 mesh

(b) Level-1 mesh

Figure 7.7: Two flapping fibres in 3D: Z-displacements of the fibre tip with different time steps.

113

Figure 7.8 shows the evolution of X- and Y-displacements of the fibre tip for all the three levels of mesh with time step ∆t = 0.0025. Clearly, the amplitude of tip displacements decrease as the mesh is refined, but no convergence can be observed. The author believes that reason for this behaviour is due to the ”size effect” of beam element in 3D space. Since a beam element is formulated as one dimensional in space, no thickness is given in its geometry. We notice that in X-Z section view the beam is a dot rather than a rectangle in a fluid element. Thus, the thickness of the beam cannot be detected by the fluid, and the beam is always infinitely thin no matter how the fluid element is refined. This singularity property results in the lack of convergence of the beam displacements. In addition, the Y-displacement is much smaller than the X-displacement in comparision to the 2D case. In other words, the ”size effect” makes the beam stiffer in Y-direction.

(a) X-displacement

(b) Y-displacement

Figure 7.8: Two flapping fibres in 3D: tip displacements for different levels of mesh with ∆t = 0.0025.

The X-velocity and pressure contours of different levels of refinement at time instant t = 2.5 are presented in Figures 7.9-7.12. These plots also shows that the refined mesh gives an improved solution.

114

(a) Level-0

(b) Level-1

(c) Level-2

Figure 7.9: Two flapping fibres in 3D: X-velocity contour plots at Z = 1 and t = 2.5 with ∆t = 0.0025.

115

(a) Level-0

(b) Level-1

(c) Level-2

Figure 7.10: Two flapping fibres in 3D: X-velocity contour plots at Y = 1 and t = 2.5 with ∆t = 0.0025.

116

(a) Level-0

(b) Level-1

(c) Level-2

Figure 7.11: Two flapping fibres in 3D: pressure contour plots at Z = 1 and t = 2.5 with ∆t = 0.0025.

117

(a) Level-0

(b) Level-1

(c) Level-2

Figure 7.12: Two flapping fibres in 3D: pressure contour plots at Y = 1 and t = 2.5 with ∆t = 0.0025.

118

7.2

Fibre array subject to constant velocity

In this example a 4 × 4 fibre array that are subjected to a time independent velocity are investigated, as shown in Figure 7.13. The dimension of fluid domain is similar to the

previous example, but differs in width and height, that is W = H = 2. The fibres have a length l = 1, and are placed in a uniformly distributed array. Boundary conditions on the top, bottom, front, back faces and the outlet are the same as in first example, while the inlet velocity is defined by the following formula, Uin = 10y(2 − y).

(7.2)

no-slip 2 u = Uin v=0 w=0

no-slip

0.4

y z

1

0.4

x

2.8

0.4 0.4 0.4

2

4

Figure 7.13: Fibre array: geometry and boundary conditions.

The material properties of fluid are the same as the previous example, that is, ρf = 100, µf = 10. The fibre density is also ρs = 100, the Young’s modulus and shear modulus of the fibre are, respectively, E = 3.0 × 106 and G = 1.0714 × 106 . The cross section of

each fibre is a 0.1 × 0.1 square, thus, the sectional area is A = 0.01, the second moments

of area are Iy = Iz = 8.3333 × 10−6 , the polar moment of inertia is Ixx = 1.4083 × 10−5 .

Level-1 hierarchical b-spline refinement is employed in this study, as shown in Figure 7.14. Time step is chosen as ∆t = 0.0025. Figures 7.15-7.16 depict the contour plots of velocity and pressure, respectively, in X-Y and X-Z sections at time instant t = 0.4. From these plots we clearly see that the fibres in front rows have larger deformations than the fibres in back rows.

119

(a) 3D view

(b) X-Z section (bottom view)

(c) X-Y section

Figure 7.14: Fibre array: level-1 mesh (16935 elements and 46664 DOFs). 20 beam elements for each fibre.

120

(a) X-velocity

(b) Y-velocity

(c) Pressure

Figure 7.15: Fibre array: velocity and pressure contour plots at Z=1.0 and t=0.4.

121

(a) X-velocity

(b) Y-velocity

(c) Pressure

Figure 7.16: Fibre array: velocity and pressure contour plots at Y=1.0 and t=0.4.

122

(a) t=0

(b) t=0.2

(c) t=0.3

(d) t=0.4

(e) t=0.5

(f) t=0.5, top view

Figure 7.17: Fibre array: Deformation of fibre array at different time instants.

Figure 7.17 shows the configuration of fibre array at five different time instants. Clearly, the fibre array undergoes large deformation, and at t = 0.5, some fibres start to deform in transverse direction. A typical plot of streamlines is shown in Figure 7.18.

123

Figure 7.18: Fibre array: streamlines plot at t = 0.4

7.3

Conclusion

In this chapter two 3D FSI problems have been created and investigated. Again, as clearly seen, the hierarchical b-splines enables the fluid mesh and the local refinement to be generated very easily and efficiently in 3D. Reasonable fluid velocity and pressure fields have been obtained even for a relatively large time step, and the large deformation of fibres can be captured. Temporal convergence has been achieved without any extra effort. However, the spatial convergence has not been fully achieved. It should also be noted that in the first example, the X-displacements of the beam are much smaller than Ydisplacements, which means the beam response has stiffened in 3D. The reason for these two problems, believed by the author, is the ”size effect” of beam element in 3D space, on which requires a further detailed study.

124

Chapter 8

Conclusions This thesis has presented a robust and efficient computational framework for interactions of incompressible Newtonian fluid flow and flexible fibre/fibre arrays by employing a finite element based immersed methods. Use has been made of hierarchical b-spline Cartesian grids and fictitious domain/distributed Lagrange multipliers, which significantly improves the computational efficiency in comparison to the traditional arbitrary Lagrangian-Eulerian method. The large displacements and rotations of the slender fibres have been captured accurately by using geometrically exact beam formulations. A staggered scheme has been adopted in solving the coupled system. A number of 2D and 3D numerical examples have been presented in detail to demonstrate the performance of the overall strategy.

8.1 8.1.1

Achievements and conclusions SUPG/PSPG and FD/DLM for fluid flow

Both stabilised finite element method and fictitious domain/distributed Lagrange multiplier scheme for the modelling of incompressible fluid flow have been elucidated. The deficiencies of standard Galerkin formulation have been illustrated, and it has been shown that these problems can been addressed by either SUPG/PSPG formulation or quadratic b-spline based FD/DLM strategy. In FD/DLM, the fluid is discretised in a structured Eulerian frame, and by applying hierarchical refinement, the b-spline grid near the immersed bodies is locally refined to different levels, therefore the computationally expensive body-fitted mesh is avoided. These properties show a great deal of potential in modelling the fluid-structure interaction where the solid undergoes large

125

deformations. Hence, the FD/DLM method has been chosen for FSI simulation in this work.

8.1.2

Geometrically exact beam

Different geometrically exact beam formulations have been presented for modelling twoand three-dimensional slender bodies. Following the classical approaches, the representation of 3D finite rotations has been achieved by applying Euler-Rodrigues formula. However, in this thesis, the stiffness matrix and residual vector of the beam element are obtained in a new simple and straightforward fashion. The mass matrix has also been obtained based upon a physically based simplification. Several static and transient problems have been studied to show that the present geometrically exact beam formulation provides results that are consistent with the reference values.

8.1.3

2D FSI numerical examples

A partitioned staggered scheme has been introduced to solve the FSI system in a robust and efficient way even for the cases when the added-mass effect is significant. Several 2D benchmark problems have been investigated. The obtained results show a very good agreement with the reference values, thus demonstrating the correctness of the adopted fluid solver (FD/DLM method), solid solver (2D geometrically exact beam) and staggered scheme. In addition, the ease of generating Cartesian grid for the fluid and the local refinement around immersed bodies by employing hierarchical b-splines are clearly illustrated. Moreover, the quadratic b-spline basis functions enable the use of standard Galerkin formulation without any stabilisation term to provide a smooth pressure field even when the LBB condition is not satisfied.

8.1.4

3D FSI numerical examples

Since very few 3D fluid-flexible fibre interaction examples can be found in literature, two problems have been created and studied. To the author’s knowledge, this is the first time that 3D nonlinear beam formulation is implemented in FSI problems. The results illustrated reasonable fluid velocity and pressure fields even in a relative large time step, with the large deformation of fibres captured by the geometrically exact beam element. Good convergence in time has also been achieved. These again indicate the correctness of the presented methodology. Note that the hierarchical b-spline refinement gives significant savings in computational time for 3D problems. However, the convergence in

126

spatial discretisation has not been fully achieved. The reason is believed to be related to the ”size effect” of beam element in 3D space.

8.1.5

Computer Implementation

In this work, the complete solution algorithm is based on an existing in-house C++ software [37]. The 3D geometrically exact beam formulation has been implemented in the code. Besides, the stabilised finite element fluid solver, the FD/DLM fluid solver and the geometrically exact beam solvers in both 2D and 3D have also been implemented in MATLAB.

8.2

Suggestions for future work

Some directions for future research work are given as follows: • The first issue need to work on is obtaining the spatial convergence in 3D FSI

problems. The ”size effect” may be dealt with by using Dirac delta function which is extensively used in many immersed boundary methods.

• In the fibre arrays example, fibre-fibre contact has been neglected, which could be

taken into account in the future work. In addition, particles could be involved in the flow to better simulate many engineering problems, and hence fibre-particle contact and particle-particle contact can also be considered.

• Although the present staggered scheme can deal with FSI problems, limitations

have been seen in the case where solid over fluid mass ratio are extremely small. Even staggered scheme works in those cases the chosen value of time step and averaging parameter β are limited. Some experimentations performed by the author have shown that the applicability of the staggered scheme can be extended significantly by making use of Nitsche’s method.

• In this work a direct solver PARDISO [95, 96] has been used to solve the global

matrix system of equations, which results in a high computational cost in 3D problems. Therefore, the work on efficient iterative solvers is considered essential for very large scale simulations.

127

Appendix A

Newton-Raphson Method Newton-Raphson method is a numerical scheme to successively find the roots of a nonlinear system. Consider a system of nonlinear equations: R (x) = 0

(A.1)

which for a simple case with two equations and two unknowns can be written as R (x) =

# " R1 (x1 , x2 ) R2 (x1 , x2 )

,

x=

" # x1 x2

(A.2)

In Newton-Raphson method, given an estimate solution xk at iteration k, a new solution xk+1 = xk +u is obtained in terms of an increment u by taking linearised approximation of equation (A.2) R (xk+1 ) = R (xk ) + DR (xk ) [u]

(A.3)

where DR (xk ) [u] indicates the directional derivative of R at xk in the direction of an increment u, which is evaluated as [107] d DR (xk ) [u] = R (xk + u) d # " d R1 (x1 + u1 , x2 + u2 ) = d R2 (x1 + u1 , x2 + u2 )

(A.4)

= Ku where K, known as tangent matrix, is given as h

i K (xk ) = Kij (xk ) ,

128

∂Ri Kij (xk ) = ∂xj xk

(A.5)

By substituting equation (A.4) into (A.3) a Newton-Raphson iteration can be obtained as K (xk ) u = −R (xk ) ;

xk+1 = x + u

(A.6)

This iteration process stops once a convergence criteria is satisfied. In general, the L2 norm of residual R (xk ) is compared against a tolerance value, say, specified by the user. In the finite element methods the residual function R is usually a set of equilibrium equations given as the sum of internal forces, F int , and external load, F ext , as R = F int − F ext

(A.7)

In practice, the external load F ext is applied in a series of increments, and the NewtonRaphson algorithm is given in Box A.

Box A: Newton−Raphson algorithm • Input initial geometry X, material properties and solution tolerance .

• Initialise x = X, F = 0, R = 0. • Loop over load increments – Find ∆F . – Set F = F + ∆F . – Set R = R − ∆F . – Do while ||R|| > ∗ Solve Ku = R ∗ Update x = x + u ∗ Find F int and K. ∗ Find R = F int − F ext – End Do • End Loop

129

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Simulation of Hydrodynamics in Flexible Fibre Arrays

Simulation of Hydrodynamics in Flexible Fibre Arrays

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