Masters thesis of engineering performance of fluid structure interaction based on analytical and computational techniques

Performance of Fluid-Structure Interaction based on Analytical and Computational Techniques A thesis submitted in fulfilment of the requirements for for the degree of Master of Engineering Pongpat Thavornpattanapong BEng Supervisor: Prof Jiyuan Tu Associate Supervisor: Dr Sherman C.P Cheung School of Aerospace, Mechanical and Manufacturing Engineering RMIT University March, 2011 c Copyright by Pongpat Thavornpattanapong 2011 to my MOTHER and FATHER with love iii Contents List of Tables ix List of Figures x Abstract xiii Acknowledgments xvi Publications xviii Nomenclature xix Introduction 1.1 Background 1.2 Outline of thesis 1.3 Summary of contributions Literature review 11 2.1 Frame of reference 11 2.1.1 Lagrangian frame 11 2.1.2 Eulerian frame 12 2.1.3 Arbitrary Lagrangian-Eulerian frame 13 2.2 Fluid-structure interaction 14 2.2.1 Monolithic approach 15 2.2.2 Partitioned approach 15 iv 2.3 Added-mass instability in partitioned approach 20 2.4 Existing Techniques for FSI Closely Coupling 21 2.4.1 Under-relaxation 21 2.4.2 Reduced order models 23 2.4.3 Artificial compressibility 23 2.5 Methods for computatonal mesh in FSI 25 2.5.1 Fixed grid algorithm 25 2.5.2 Moving mesh 28 2.5.3 Automatic remeshing 29 2.6 Summary and concluding remarks 29 Methodology 31 3.1 Governing equations 31 3.1.1 Fluid governing equation 31 3.1.2 Solid governing equation 33 3.2 Discretization for governing equations 33 3.2.1 Discretization for fluid governing equations 33 3.2.2 Discretization for solid governing equation 40 3.3 Fundamental conditions for FSI coupling 42 3.4 Implementation of artificial compressibility 43 3.5 Procedure of calculation 46 3.5.1 Under-relaxation 47 3.5.2 Aritificial compressibility 49 3.6 Convergence of interface loads 52 3.7 General procedure of FSI simulation setup 53 3.8 Construction and setup for mesh in FSI 54 3.9 Summary and concluding remarks 59 Theoretical analysis of stability in FSI 61 4.1 Introduction 61 4.2 Simplified problem for stability analysis 63 4.3 Theoretical analysis: under-relaxation 65 v 4.3.1 Stability condition: minimum numerical damping is applied 65 4.3.2 Stability condition: maximum numerical damping is applied 68 4.3.3 Generalization of stability condition 70 4.4 Summary and concluding remarks 72 Validation and numerical results 75 5.1 Validation 75 5.1.1 Geometric and material descriptions 76 5.1.2 Computational domain 76 5.1.3 Analytical solutions 76 5.1.4 Comparison between numerical and analytical solutions 78 5.2 Numerical results 80 5.2.1 Influence of structural time integration scheme 80 5.2.2 Influence of fluid time integration scheme 83 5.2.3 Influence of time step size 84 5.2.4 Influence of density ratio 85 5.2.5 Influence of vessel radius 85 5.3 Implicit solution using artificial compressibility 87 5.4 Performance comparison 87 5.5 Summary and concluding remarks 88 Fluid-structure interaction in carotid bifurcation 93 6.1 Abstract 93 6.2 Introduction 94 6.2.1 General background of atherosclerosis 94 6.2.2 Hypothesis on the developement of artherosclerosis 94 6.2.3 Types of atherosclerosis 95 6.2.4 Clinical implications of blood pressure and heart rate 96 6.2.5 Aim and scope 97 6.3 Methodology 97 6.3.1 Partitioned approach for fluid-structure interaction 98 6.3.2 Idealistic geometry of carotid bifurcation 98 vi 6.3.3 Computational domain 101 6.3.4 Boundary conditions and material properties 101 6.3.5 Flow properties to be examined 105 6.4 Results 105 6.4.1 Healthy carotid bifurcation 105 6.4.2 Diseased carotid bifurcation – parameters affected based on different degrees of stenosis 106 6.5 Discussion 119 6.5.1 Blood flow pattern and wall shear stress 119 6.5.2 Deformation 119 6.5.3 Principal stress 120 6.5.4 Relationship of pulse pressure, MD and minimum value of MPS121 6.6 Summary and concluding remarks 122 Concluding Remarks 123 7.1 Achievements and their significances 123 7.1.1 Numerical and theoretical analysis on influence of structural time integration schemes on added mass instability in FSI 123 7.1.2 Performance comparision between under-relaxation technique and artificial compressibility 124 7.1.3 Application of artificial compressibility to artherosclerosis in carotid bifurcation 125 7.2 Future Work 125 Appendix A Detail derivation 127 A.1 Discretization of structural acceleration 127 A.1.1 When minimum numerical damping is applied 127 A.1.2 When maximum numerical damping is applied 130 A.2 Discretization of fluid acceleration 131 A.2.1 First order accurate backward Euler 132 A.2.2 Second order accurate backward Euler 132 A.2.3 Trapezoidal rule 132 vii Appendix B Analysis on added mass instability 133 B.1 Analysis 133 B.1.1 Leap-Frog and Euler scheme with under-relaxation 134 B.1.2 Leap-Frog and Euler scheme with artificial compressibility 137 B.2 Conclusion 140 References 141 viii List of Tables 3.1 Table of mesh motion analysis 55 5.1 Influence of amplitude decay factor on the critical relaxation factor when time step size is 0.001s 83 5.2 The critical relaxation factors when time step size is 0.001s and first order accurate backward Euler scheme is used for fluid acceleration 83 5.3 Influence of amplitude decay factor on required density ratio when time step size is 0.0001s 84 5.4 Influence of amplitude decay factor on required density ratio when time step size is 0.00001s 84 5.5 The critical value of relaxation factor when using time step size is 0.001s and radius is increased to 0.7 cm 85 5.6 Comparison between the consumed computation time when using artificial compressibility and under-relaxation technique 89 6.1 FSI parameters, material properties and dimensions of carotid bifurcation used in this work 102 ix List of Figures 1.1 The collapse of the Tocoma narrow bridge 1.2 Fluid-structure interaction of human blood vessel 1.3 Fluid-structure interaction of oscillating rear windshield of convertible car 1.4 Introduction to fluid-structure interaction 2.1 Classification of analysis types in fluid-structure interaction 14 2.2 Diagram of partitioned approach 16 2.3 Procedure of reduced order models for fluid-structure interaction 24 3.1 Interpolation using upwind differencing scheme when Fc is greater or equal to 36 3.2 Interpolation using upwind differencing scheme when Fc is less than 36 3.3 Interpolation using central differencing scheme 37 3.4 Diagram of FSI coupling 44 3.5 Flowchart of standard procedure of partitioned approach when using under-relaxation technique 47 3.6 Flowchart of standard procedure of partitioned approach when using artificial compressibility 50 3.7 Mesh motion of case 56 3.8 Mesh motion of case 57 3.9 Mesh motion of case 58 4.1 Graphical representation of flexible vessel 64 x 128 APPENDIX A DETAIL DERIVATION dn dn1 dn2 = dn1 + dăn1 t + n2 = d + dăn2 t + n3 = d + dăn3 t + ăn d t, ăn1 d t, ăn2 d t, (A.3) (A.4) (A.5) and, ¨n d = = d¨n−1 = = d¨n−2 = =   △t n n−1 n−1 n−1 d d td dă t2 4 n n1 n1 ăn1 d d d d △t2 △t2 △t   △t n−2 n−1 n2 n2 dă , d d td t2 4 n2 n2 ăn2 n1 d d − d −d 2 △t △t △t   t2 ăn3 n2 n3 n3 d d −d − △td − △t2 4 n−2 n−3 n3 ăn3 d d d d 2 △t △t △t (A.6) A.1 DISCRETIZATION OF STRUCTURAL ACCELERATION dăn+1, 129 = dăn+1   t2 n+1 n n = n ă d d td + t2 4 n1 d 2dăn1 3dăn = [4dn+1 4dn ] t t ˙n−1 12 n = [4dn+1 − 4dn ] d d 2dăn1 t t t2 12 12 n1 + dn1 + d + 3dăn1 △t △t ˙n−1 n+1 n = [4d − 4d ] d 2dăn1 t2 t 12 12 n1 12 n1 dn + d + 3dăn1 d + t t2 t n1 ăn1 = [4dn+1 − 16dn + 12dn−1] + d +d △t t n2 d + 4dn2 + 4dăn1 + dăn1 [4dn+1 16dn + 12dn1] + = t △t ˙n−2 ˙n−2 = d + d + 5dăn1 [4dn+1 16dn + 12dn1] + t △t △t ˙n−2 = d˙n−2 + d + 5dăn1 [4dn+1 16dn + 12dn1] + t △t △t ˙n−2 20 n−1 ˙n−2 = d + d + [4dn+1 − 16dn + 12dn−1] + d △t △t △t △t2 20 20 ˙n−2 d 5dăn2 dn2 t t 12 n2 ¨n−2 d −d [4dn+1 − 16dn + 32dn−1 − 20dn−2] − = △t △t 12 ˙n−3 = d 6dăn3 7dăn2 [4dn+1 16dn + 32dn1 − 20dn−2] − △t △t 12 ˙n−3 = d 6dăn3 [4dn+1 16dn + 32dn1 20dn2] − △t △t 28 28 n−3 28 ˙n−3 − dn2 + d + d + 7dăn3 t △t △t [4dn+1 − 16dn + 32dn−1 − 48dn−2 + 28dn3 ] = t2 16 dn3 + dăn3 (A.7) △t 130 APPENDIX A DETAIL DERIVATION A.1.2 When maximum numerical damping is applied Amplitude decay factor is set to when maximum numerical damping is applied which yields αm = −1 , αf = , α = and δ = 32 Therefore, we get = dn dăn t + dăn+1 t, 2  △t2 n+1 n n ˙ = d d td + n ă , t2 dn+1 dăn+1 (A.8) (A.9) therefore, dn dn1 dn2 = dn1 dăn1 t + = dn2 dăn2 t + = dn3 dăn3 t + ăn d t, ăn1 d t, ăn2 d t, (A.10) (A.11) (A.12) and, dăn dăn1 dăn2 = t2 = t2 = t2  t2 ăn1 n1 d , d −d − △td +   △t n2 n1 n2 n2 dă , d d td +   t2 ăn3 dn2 dn3 △td˙n−3 + d  n n−1 (A.13) (A.14) (A.15) A.2 DISCRETIZATION OF FLUID ACCELERATION dăn+1, A.2 131 = 2dăn+1 dăn  t2 n+1 n n = d d td + n ă dn △t2 2 n+1 n n+1 = d − d − d 2 △t △t △t n+1 n n1 ăn1 = d +d 3dăn d d 2 t t t n+1 n ˙n−1 n−1 = d − d − [d − dn−2 − △td˙n−2 d − 2 t t t t t2 ăn1 t2 ăn2 + d − 3dn + 3dn−1 + 3d˙n−1 − d ] 2 n n−1 n−2 ˙n−1 = d [2dn+1 − d − d + d ]− 2 △t △t △t △t △t t ăn2 3t ăn1 d + d ) 3d˙n−1 ) − ((d˙n−2 − △t 2 n n−1 n−2 ˙n−1 n+1 = d [2d − d − d + d ] − △t2 △t2 △t △t △t − (d˙n−1 − 3d˙n−1 ) △t n n−1 n−2 [2dn+1 − d − d + d ] (A.16) = 2 △t △t △t △t Discretization of fluid acceleration In this section, the detail derivation of discretized fluid acceleration is presented The zero order accurate structural predictor is used n dn+1 Γ,P = dΓ (A.17) The calculation for fluid velocity at FSI interface can be written as un+1 Γ n dn+1 Γ,P − dΓ,P = △t (A.18) 132 APPENDIX A DETAIL DERIVATION A.2.1 First order accurate backward Euler n un+1 Γ,P − uΓ,P = △t = (dnΓ − dΓn−1 − dΓn−1 + dΓn−2 ) △t = (dn − 2dΓn−1 + dΓn−2 ) △t2 Γ u˙ n+1 Γ A.2.2 Second order accurate backward Euler u˙ n+1 Γ A.2.3 (A.19) n+1 [ uΓ − 2unΓ + uΓn−1 ] △t 2 n n−1 1 = [ dΓ − dΓ − 2dΓn−1 + 2dΓn−2 + dΓn−2 − dΓn−3 ] △t 2 2 n n−1 n−2 n−3 = [ d − d + dΓ − dΓ ] (A.20) △t2 Γ Γ 2 = Trapezoidal rule u˙ n+1 Γ n+1 − unΓ ) + u˙ Γn−1 (u △t Γ (dnΓ − dΓn−1 − dΓn−1 + dΓn−2 ) + u˙ Γn−1 = △t (dn − 2dΓn−1 + dΓn−2 ) + u˙ Γn−1 = △t2 Γ = (A.21) Appendix B Analysis on added mass instability In the solution procedure of fluid-structure interaction (FSI), the so-call artificial added mass effect plays an important role in determining the stability of the computation We propose a derivation using Von Neumann stability analysis, which shows its significance as a tool for studying this numerical instability Our derivation demonstrates that FSI solution is severely unstable when density ratio is high, solid structure is thin and flexible It also shows that this instability can be eliminated by introducing aritificial compressibility B.1 Analysis In this section, the instability of fluid-structure interaction using partitioned approach is discussed This section is divided into two sub-sections The first one is the derivation of instability which occurs when using Leap-Frog and forward Euler schemes without artificial compressibility The second one is the derivation of instability which occurs when using Leap-Frog and forward Euler schemes with artificial compressibility 133 134 APPENDIX B ANALYSIS ON ADDED MASS INSTABILITY B.1.1 Leap-Frog and Euler scheme with under-relaxation We refer to the governing equation of deformation of a flexible cylindrical tube in (Causin et al., 2005), which can be written as ρs hs where a0 = Ehs r (1−ν ) ∂ dfΓ ∂ dsΓ ∂ dsΓ s + ρ M + a d − b = Pext,Γ , f a Γ ∂t2 ∂t2 ∂x2 (B.1) and b = KT Bhs The Leap-Frog scheme for structural acceleration is written as (dn+1 2dn + dn1 ) dăn+1,s = ∆t2 Γ (B.2) The Euler scheme for FSI interface acceleration for fluid domain is written as dăn+1,f = (d˙n ∆t Γ − d˙Γn−1) = (dnΓ ∆t2 − 2dΓn−1 + dΓn−2 ) (B.3) It is noted that superscript m represents pseudo time step or stagger iteration, while n represents physical time step Also, x and t are space and time respectively Due to the nature of sequential approach, the information used for calculation of FSI interface acceleration for fluid domain is always one step behind solid From (B.1), since we are interested in added-mass instability, where the mass term is dominated the stiffness term, we will neglect some non-linearity Therefore, (B.1) is reduced to equation (B.4) such that ρs hs ∂ dfΓ ∂ dsΓ + ρ M + a0 dsΓ = Pext,Γ f a ∂t2 ∂t2 (B.4) Discretization of (B.4) is achieved by substituting (B.2) and (B.3) to give ρs hs n+1 ρf Ma n n (dΓ − 2dnΓ + dΓn−1 ) + (d − 2dΓn−1 + dΓn−2 ) + a0 dnΓ = Pext,Γ (B.5) △t △t2 Γ 135 B.1 ANALYSIS To simplify it, let’s look at only one position on the interface Notice that the added-mass operator MA can be represented by the ith eigenvalues of MA , µi This make (B.5) reduced further to (B.6) as given by ρf µi n ρs hs n+1 n (di − 2dni + din−1 ) + (d − 2din−1 + din−2) + a0 dni = Pext,i △t △t2 i (B.6) Next we will use the idea of Von Neumann stability analysis to prove stability of the system of discrete equation (Strang, 2007) This technique based on understanding in fourier analysis that any function can be represented by a superposition of basis functions like sine, cosine and exponential functions The idea is to represent the solution in terms of initial value eiat eikx , which depends only on spatial variation x In other words, we are trying to separate variables x and t in our solution This will allow us to monitor how the solution is growing with time Initial value of solution can be written as d(x, t) ≈ or Z d(x, t) ≈ ∞ eiat eikx dk −∞ ∞ X eiat eikx (B.7) −∞ Note that a is a temporal wave frequency, which is also a function of spatial wave frequency, k For simplicity, we drop the summation sign in equation (B.7) since our aim is to prove the instability, which can be done by monitoring only one mode of the solution Therefore (B.7) becomes d(x, t) ≈ eiat eikx (B.8) 136 APPENDIX B ANALYSIS ON ADDED MASS INSTABILITY Now we have separated the time and space variation in our solution From (B.8), the initial solution is in the form of multiplication of complex exponential terms and our initial value eiat eikx The next step is to represent the other terms (dni , din−1 ) in term of the initial value of the system (din−2 ) This is set to be the initial value because in the discrete equation (B.6), din−2 is the oldest information available Let din−2 ≈ eiat eikx , which is our initial value Therefore, displacement at time step n − and n can be represented as din−1 = eia(t+∆t) eikx and dni = eia(t+2∆t) eikx By substituting (B.9) into (B.6), we get ρs hs n+1 ρf µi ia(t+2∆t) (di − eia(t+2∆t) eikx + eia(t+∆t) eikx ) + (e △t △t2 n −2eia(t+∆t) eikx + eikx ) + a0 eia(t+2∆t) eikx = Pext,i (B.9) The growth factor is determined by rearranging above equation as dn+1 i = = = =  n  
ρf µi ia2∆t △t2 Pext,i ia2∆t ia∆t ia2∆t ia∆t − a0 e − e − 2e + + 2e −e eiat eikx ρs hs eiat eikx △t2  
ρf µi ia2∆t ia∆t ia2∆t ia∆t e − 2e + + 2e −e eiat eikx ∵ △t2 → − ρs hs   ρf µi ia2∆t ρf µi ia∆t ρf µi ia2∆t ia∆t − e +2 e − + 2e −e eiat eikx ρh ρs hs ρs hs  s s     ρf µi ρf µi ρf µi ia2∆t ia∆t (B.10) − +2 e + −1 e − e|iat{z eikx} ρs hs ρs hs ρs hs Initial value | {z }  Growth Factor 137 B.1 ANALYSIS The solution is unstable if the absolute value of growth factor is greater than Therefore, instability occurs if (B.11) that is given as     ρf µi ρ µ ρ µ f i f i ia2∆t ia∆t > ... equations consist of the following equations: • Continuity equation: conservation of mass; • Momentum equaton: conservation of momentum; • Energy equation: conservation of energy Continuity equation... value of MPS 118 B.1 ρf µi ρs hs > 0, R > (Unconditionally unstable) 138 xii Performance of Fluid- Structure Interaction based on Analytical and Computational Techniques Pongpat... xiv Performance of Fluid- Structure Interaction based on Analytical and Computational Techniques Declaration I certify that except where due acknowledgement has been made, the work is that of the