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... coupling strength The coupling of the fluid and the structure comes from the continuities of velocities and normal stresses along the FS interface According to this we can classify the coupling into... By combining the governing equations for fluid and incompressible structure, and their coupling conditions at the interface, we have the full FSI problem in the strong form, i.e Fluid structure. .. representation of the computational domain 28 ix Chapter Introduction 1.1 An overview Fluid- structure interaction (FSI) is the interaction of some movable or deformable structure with an internal
ANALYSIS OF DIRICHLET-NEUMANN AND NEUMANN-DIRICHLET PARTITIONED PROCEDURES IN FLUID-STRUCTURE INTERACTION PROBLEMS XUE HANSONG (B.Sc.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2011 Acknowledgements First of all, I would like to express my uttermost gratitude to my supervisor, Dr Liu Jie, for offering me the opportunity to work under him, putting in great efforts and spending precious time to help, guide and encourage me, not only in this entire research work, but also in non-academic related aspects Within the past two years he has broadened my view and knowledge in the area of computational fluid dynamics and fluid-structure interaction I have benefited greatly from his enlightenment and inspiration in this area Everything I have learned from him is of endless benefit to my whole life It has been really a great pleasure to work under Dr Liu Jie and I really appreciate his patience I would like to thank especially Simon and Yvonne for their care and encouragement from Karlsruhe, Germany during my graduate studies although we are so far apart I am very grateful to Dr and Mrs Ong for their support and good care during the past twelve years in Singapore ii Acknowledgements iii A thousand thanks to Cai Ruilun, Gaobing, Gaorui, Gongzheng, Jiang Kaifeng, Li Xudong, Ma Jiajun, and Wangkang, for helping me in my graduate course studies I would like to thank my beloved parents in China and other who have helped me in one way or the other Last but not least, to dedicate this work to my beloved wife, Zhihan, for her endless support, meticulous care and magnanimous understanding she has given to me Xue Hansong October 2011 Contents Acknowledgements ii Summary vii List of Figures ix Introduction 1.1 An overview 1.2 The organization of the thesis Problem Setting 2.1 Structure domain 2.2 Fluid domain 2.3 The Arbitrary Lagrangian Eulerian(ALE) formulation of the Navier- 2.4 Stokes equation 10 Coupling conditions 14 iv Contents v Time discrete system and domain decomposition method 3.1 3.2 Full FSI problem and time discrete system 15 3.1.1 Semi-implicit scheme 17 3.1.2 Implicit scheme 18 Domain decomposition method 19 Convergence analysis of simplified problems 4.1 4.2 4.3 15 23 Heat-wave(HW) 1D model 23 4.1.1 ND partitioned procedure 24 4.1.2 DN partitioned procedure 27 Stokes-algebraic generalized string(SAGS) 1D model 27 4.2.1 The structural problem 28 4.2.2 The fluid problem 29 4.2.3 Fluid-structure interaction 29 4.2.4 ND partitioned procedure 30 4.2.5 DN partitioned procedure 36 Stokes-linear elasticity(SLE) 2D model 37 4.3.1 ND partitioned procedure 39 4.3.2 DN partitioned procedure 45 Geometric convergence of domain decomposition method 47 5.1 Geometric convergence for ND partitioned procedure 49 5.2 Geometric convergence for DN partitioned procedure 58 5.3 Parameter estimation and improved convergence rate for the case of heat-wave equations coupling 61 vi Contents Conclusion 68 Bibliography 71 Summary In recent decades, the development and application of respective modeling and simulation approaches for fluid-structure interaction (FSI) problems have grasped much attention While solving FSI problems, partitioned scheme shows its efficiency by using a modular algorithm in which the equations of fluid and structure are solved separately in an iterative manner through the exchange of suitable transmission conditions at the FS interface The goal of this work is to verify in terms of the convergence behavior that using structure normal stress as the boundary condition along the FS interface in the fluid solver and hence prescribing displacement boundary condition for the structure is actually better than the opposite approach In fact, the opposite approach has great numerical instabilities, especially when the iteration time step is small, but our proposed approach can reduced this instabilities and hence has a better convergence behavior Based on three different simplified models of the fluid and the structure, i.e Heatwave 1D model, Stokes-algebraic generalized string 1D mode and Stokes-linear vii viii Summary elasticity 2D model, we present a detailed analysis of the convergence behavior to substantiate our claim by deriving a reduction factor at each iteration of the partitioned algorithm In particular, these model problems are used to highlighted some aspects that probably will arise in the context of applying partitioned scheme to FSI problems Furthermore, if we ignore the fluid domain deformation and also the convection term in fluid equation, we can prove the geometric convergence of the iteration that enforces the continuities of velocities and normal stresses along the FS interface An example of heat-wave equations coupling is also given to show an improved convergence rate and estimate the parameters in its geometric convergence List of Figures 2.1 Example of the computational fluid domain Ωft 2.2 Example of the computational solid domain Ωst 2.3 A longitudinal section of the fluid domain Ωft 2.4 Comparison between the Lagrangian and the ALE approach The 11 reference computational domain Ωf0 is mapped by (a) the Lagrangian 4.1 mapping Lt and by (b) the ALE mapping At 12 Schematic representation of the computational domain 28 ix Chapter Introduction 1.1 An overview Fluid-structure interaction (FSI) is the interaction of some movable or deformable structure with an internal or surrounding fluid flow It can also be considered as a coupled problem, consisting of two or more domains which interact at common boundaries This kind of mutual influence of a flexible structure with a flowing fluid in which it is submersed or by which it is surrounded gives rise to a rich variety of frequently occurring physical phenomenon with applications in many fields of engineering as well as in applied sciences Furthermore, it is a crucial consideration in the design of many engineering systems[33] In recent years, it has received much attention and has become one of the major research activities due to its growing importance Some of the examples are: aeroelasticity[14, 44, 29]; helicopters[37, 24]; the vibration of turbine and compressor blades; the sloshing in tanks[34]; the response of bridges and skyscrapers to winds; membranous structures[51]; the description of the mechanical behavior of cells[13] or, more generally, the organic fluid mechanics; acoustic problems; hemodynamics[43, 6, 22] 66 Chapter Geometric convergence of domain decomposition method Proof s H∆ (wk ) s s = H∆t (E∆t2 γΣ ek ) f f ≤ c−1 H∆t2 (E∆t2 γΣ ek ) f f −2 ≤ c−1 H∆t (E∆t γΣ ek )2 ∆t f −2 = c−1 H∆t (ek )2 ∆t −2 2 ≤ c−2 H∆t (dk−1 ) ∆t where we have used (5.63) in the second step, (5.65) in the third step and (5.34) in the last step Next, we will prove the geometric convergence for the ND partitioned procedure s of the heat-wave equations coupling case Firstly by applying E∆t to both sides of the equation (5.8) to have s s s E∆t (γΣ dk ) = θ∆tE∆t2 (γΣ ek ) + (1 − θ)E∆t2 (γΣ dk−1 ) s Let wk = E∆t (γΣ ek ), then dk = θ∆twk + (1 − θ)dk−1 Since similar to (5.33), we have σ s (dk , lk ), ∇dk s = σ s (dk , 0), ∇dk s Then s H∆t (dk ) = ∆t−2 ρs dk , dk s + µs ∇dk + ∇dk , ∇dk + ∇dk s = ∆t−2 ρs θ∆twk + (1 − θ)dk−1 , θ∆twk + (1 − θ)dk−1 s + µ2 s ∇(θ∆twk + (1 − θ)dk−1 ) + ∇(θ∆twk + (1 − θ)dk−1 ) , ∇(θ∆twk + (1 − θ)dk−1 ) + ∇(θ∆twk + (1 − θ)dk−1 ) = ∆t−2 ρs (θ2 ∆t2 wk µs s + (1 − θ)2 dk−1 + (θ2 ∆t2 ∇wk + ∇wk s s s + 2θ(1 − θ)∆t wk , dk−1 s ) + (1 − θ)2 ∇dk−1 + ∇dk−1 +2θ(1 − θ)∆t ∇wk + ∇wk , ∇dk−1 + ∇dk−1 s ) s 5.3 Parameter estimation and improved convergence rate for the case of heat-wave equations coupling = θ2 ∆t2 (∆t−2 ρs wk s + µs +(1 − θ)2 (∆t−2 ρs dk−1 s ∇wk + ∇wk + µs +2θ(1 − θ)∆t(∆t−2 ρs wk , dk s s) ∇dk−1 + ∇dk−1 2s ) + µs ∇dk−1 + ∇dk−1 , ∇wk + ∇wk s ) s 2 s = θ2 ∆t2 H∆t (wk ) + (1 − θ) H∆t2 (dk−1 ) +2θ(1 − θ)∆t(∆t−2 ρs wk , dk s + µs ∇dk−1 + ∇dk−1 , ∇wk + ∇wk + ∇wk − ∇wk s ) s 2 s = θ2 ∆t2 H∆t (wk ) + (1 − θ) H∆t2 (dk−1 ) +2θ(1 − θ)∆t(∆t−2 ρs wk , dk s + µs (∇dk−1 + ∇dk−1 ), ∇wk s ) 2 s s = θ2 ∆t2 H∆t (wk ) + (1 − θ) H∆t2 (dk−1 ) +2θ(1 − θ)∆t(∆t−2 ρs wk , dk s + σ s (dk−1 , 0), ∇wk s ) 2 s s = θ2 ∆t2 H∆t (wk ) + (1 − θ) H∆t2 (dk−1 ) +2θ(1 − θ)∆t(∆t−2 ρs wk , dk s + σ s (dk−1 , lk−1 ), ∇wk s ) f s 2 s 2 = θ2 ∆t2 H∆t (wk ) + (1 − θ) H∆t2 (dk−1 ) − 2θ(1 − θ)∆tH∆t (ek ) −1 s 2 ≤ (θ2 c−2 ∆t + (1 − θ) − 2θ(1 − θ)∆tc2 )H∆t2 (dk−1 ) where we have used (5.40) in the second last step, (5.66) and left half of (5.34) in the last step Hence, −1 s s 2 −2 H∆t (dk ) ≤ (θ c1 ∆t + (1 − θ) − 2θ(1 − θ)∆tc2 )H∆t2 (dk−1 ) f When ∆t < ( νµs )4 , we can take θ = 1, this gives s s H∆t (dk ) ≤ H∆t2 (dk−1 ) and we have verified our claim 67 Chapter Conclusion In this work we have presented a mathematical contribution to explain that the numerical instabilities encountered using time-partitioned procedures in the simulation of FSI problems to a certain extend is due to the choice of boundary conditions at the interface for the fluid and structure solver The convergence analysis for the ND and DN partitioned procedures of respective above-mentioned reduced models has been done by using a defined reduction factor at the FS interface The dependence of the partitioned schemes especially on the iteration time step has also been highlighted We can see that in all proposed models, when time step is small, the reduction factor for the DN partitioned procedure tends to blow up, but for the ND partitioned procedure it tends to zero Therefore, these models substantiate our claim that using structure normal stress as the boundary condition along the FS interface in the fluid solver and hence prescribing displacement boundary condition for the structure is actually better than the opposite approach We have only theoretically proved that the ND partitioned procedure exhibits enhanced convergence behavior with respect to the DN partitioned procedure In next phase of our work, we are going to focus on the numerical tests and enforce our claim using numerical experimentation Most current existing partitioned FSI 68 69 solvers use the classical DN partitioned procedure, but it is negatively affected by the add-mass effect Therefore, for FSI applications where this effect is important, DN needs a strong relaxation and its convergence is very slow We are also going to include the added-mass effect in our models and further test the efficiency and convergence behavior for the ND partitioned procedure Our results here are derived based on implicit scheme We not expect similar conclusions to hold for other schemes such as the explicit scheme This is because for implicit scheme, the iteration will only go to next step when the interface conditions between the fluid and the structure are enforced In other words, it guarantee the fluid and the structure to reach same interface position for each iteration step However, the explicit approach not hold this property and the iteration will probably become unstable In our geometric convergence analysis, ideally we would like to show that C., can be chosen independent of time step ∆t, but only depend on other physical parameters Therefore, when the time step is small, the ND partitioned procedure converges faster than the DN one However, for some technical reason reasons, we are not able to that, only for a simplified heat-wave equations coupling model Further work will be put in to explore this part Every design of a numerical scheme for FSI problems hopes to achieve converges without any relaxation and moreover, the convergence is insensitive to the addedmass effect There are many aspects to be considered in the modeling and simulation of FSI problems However, in this work we focus on the choice of the boundary condition for the fluid and the structure solvers We believe that the discussion in this work also applies to FSI problems encountered in other applications as well and can provide some insights for the modeling of and 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ANALYSIS OF DIRICHLET-NEUMANN AND NEUMANN-DIRICHLET PARTITIONED PROCEDURES IN FLUID-STRUCTURE INTERACTION PROBLEMS XUE HANSONG NATIONAL UNIVERSITY OF SINGAPORE 2011 partitioned procedures in fluid-structure interaction problems Analysis of Dirichlet-Neumann and Neumann-Dirichlet Xue Hansong 2011