Add force and/or change underlying projection method to improve accuracy of Explicit Robin-Neumann and fully decoupled schemes for the coupling of incompressible fluid with thin-walled structure
AAdd force and/or change underlying pro jection methodto improve accuracy of Explicit Robin-Neumann andfully decoupled schemes for the coupling ofincompressible fluid with thin-walled structure
Yiyi HUANG ([email protected], Dept. of Math, HKUST)August 18, 2020 a r X i v : . [ phy s i c s . c o m p - ph ] A ug ontents Table of Contents 2List of Figures 3List of Tables 4Abstract 51 Introduction 12 The simplified model problem 23 Notations 44 Fernandez’s Explicit Robin-Neumann and fully decoupled schemes 45 Two schemes with added force 66 A fully decoupled scheme based on Van Kan’s projection method andwith added force 87 Numerical experiments 108 Conclusions from numerical results 20
Algorithm 1 from Table 1 . . . . . . . . . . . . . . . . . . 208.2 Conclusions for
Algorithm 2 from Table 2 . . . . . . . . . . . . . . . . . . 218.3 Conclusions for
Algorithm 3 from Table 3 . . . . . . . . . . . . . . . . . . 21 β, θ and ξ for practical applications 2211 Discussions and future work 2612 Acknowledgments 27 ist of Figures β = 43 at final time . . . . . . . . . . 263 Structure displacement of Fernandez fully decoupled scheme (Fern FD) andAlgorithm 2 (Algo 2) with θ = 25 at final time . . . . . . . . . . . . . . . . . 274 Structure displacement of Fernandez fully decoupled scheme (Fern FD) andAlgorithm 3 (Algo 3) with ξ = 31 at final time . . . . . . . . . . . . . . . . . 285 Structure displacement of Fernandez fully decoupled scheme (Fern FD) andAlgorithm 3 (Algo 3) with ξ = 32 at final time . . . . . . . . . . . . . . . . . 296 Structure displacement of Fernandez fully decoupled scheme (Fern FD) andAlgorithm 3 (Algo 3) with ξ = 33 at final time . . . . . . . . . . . . . . . . . 303 ist of Tables β . . . . . . . . . . . . . . . . . . . . . 235 Numerical results of Fernandez fully decoupled scheme (Fern FD) and Algo-rithm 2 (Algo 2) at selected θ . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Numerical results of Fernandez fully decoupled scheme (Fern FD) and Algo-rithm 3 (Algo 3) at selected ξ . . . . . . . . . . . . . . . . . . . . . . . . . . 254 bstract Enlightened by added-mass effect and viscosity of fluid, in Explicit Robin-Neumannand fully decoupled schemes for the coupling of incompressible fluid with thin-walledstructure, the force between fluid and structure corresponding to viscosity is increased.Numerical experiments demonstrate improvement of accuracy under such modification.To further improve accuracy of fully decoupled schemes, the underlying projectionmethod is replaced. Introduction
The coupling of incompressible fluid with thin-walled structure typically arises in subjectslike bio-mechanics of blood flow in vessels [10], for which the blood is governed by Navier-Stokes equations, while the structure assumed to be deformable. Generally, like the numericalmethods to other fluid-structure interaction problems, there are two types of numerical ap-proaches, namely the monolithic (also named fully implicit ) and partitioned (also named decoupled ) [14]. The monolithic is accurate but less efficient, while the partitioned is muchmore efficient but not that accurate.Among various partitioned algorithms on this topic [10, 14, 9, 6, 8, 7, 12, 3] , Fernandez’s
Explicit Robin-Neumann [9] and fully decoupled schemes [6] [8] are exceptional, dueto their high efficiency, theoretical and/or numerical stability and applicability to the topicwith a vast variety of structure models. The fully decoupled scheme is more efficient thanExplicit Robin-Neumann scheme, because the velocity and pressure of fluid are decoupled.The infamous added-mass effect is known to cause instability to tremendous partitionedalgorithms for fluid-structure interaction problems with large fluid/structure density ratio(for many problems the density of fluid is lower than that of structure, so this implies thedensity of fluid is large enough to be close to that of structure) and long thin geometry [18].Intuitively, due to viscosity of fluid, when the fluid or structure moves, some amount offluid is attached to it near the interface, resulting in ”added” mass. By Newton’s SecondLaw, the mass together with acceleration results in extra force on the structure. By Newton’sThird Law, there is an equal and opposite force on the fluid. Thus, it is reasonable to expect,for a partitioned algorithm, increasing the force due to viscosity of fluid between the fluidand structure can make it more realistic, namely more close to the actual behaviour of thecoupling system, which might lead to better accuracy. The amount of extra force resultingfrom ”added” mass is difficult (if not impossible) to compute; however, it can be approached.Up to certain values, increasing the force gradually should improve accuracy gradually.The idea is applied to Fernandez’s two schemes mentioned above. Note that, however,this does not mean the two schemes are unstable; in fact, their stabilities have proved bytheoretical analysis and/or numerical experiments (availability of theoretical analysis depends1n extrapolation orders) to be free of added-mass effect. In what follows, the two schemesare presented. Afterwards, coefficients of the terms corresponding to force from viscosityare increased, generating
Algorithm 1 and
Algorithm 2 . Fernandez’s fully decoupledscheme is based on Chorin-Temam projection method. To further improve accuracy, theunderlying projection method is replaced with Van Kan’s, leading to
Algorithm 3 .Numerical results are reported later. The results indicate improvement of accuracy asthe force from viscosity increases. Appropriate values for such increment are recommendedfor practical applications.
Remark 1
Based on the intuition mentioned above, it might not be difficult to understandthe reasons why added-mass effect causes instability to tremendous partitioned algorithmsfor fluid-structure interaction problems with large fluid/structure density ratio and long thingeometry. As the density of fluid increases, the part of fluid attached to the structure dueto viscosity influences more seriously on the movement of structure. If the geometry is longand thin, such a part of fluid takes large portion of the whole fluid, which influences seriouslyon the movement of fluid. If it decouples the fluid and structure without paying sufficientattention to such effect, a partitioned algorithm might be far from the actual movement ofthe coupling system and thus might fail to finish the simulation, which constitutes stabilityissues.
For sake of clarity and simplicity, the simplified test-case used in [5] is adopted. This workis expected to be also applicable to the a bit more general model described in [9] [6] [8]. Thefluid dominated by Stokes equations is defined on Ω = [0 , L ] × [0 , R ], where L = 6 , R = 0 . ∂ Ω = Γ ∪ Γ ∪ Σ ∪ Γ (see Figure 1 ) . Thedomain is extracted as upper half of the rectangle [0 , L ] × [ − R, R ] which simulates a tube intwo-dimensional space with horizontal centerline Γ and top boundary Σ. As a result, Γ isimposed symmetric boundary condition. The structure is assumed to be a generalized stringdefined on Σ with the two end points ( x = 0 , L ) fixed. When the fluid flows from the leftto the right, structure deforms vertically. Equations read as follows.2igure 1: Geometrial configurationFind the fluid velocity u : Ω × R + → R , the fluid pressure p : Ω × R + → R , the structurevertical displacement η : Σ × R + → R and the structure vertical velocity ˙ η : Σ × R + → R such that ρ f ∂ t u − div σ ( u , p ) = in Ω , div u = 0 in Ω , u · n = 0 , σ ( u , p ) n · t = 0 on Γ , σ ( u , p ) n = − P ( t ) n on Γ , σ ( u , p ) n = on Γ , (1) u · n = ˙ η, u · t =0 on Σ ,ρ s (cid:15)∂ t ˙ η − c ∂ xx η + c η = − σ ( u , p ) n · n on Σ , ˙ η = ∂ t η on Σ ,η =0 on ∂ Σ , (2)with initial conditions u (0) = , η (0) = 0 , ˙ η (0) = 0 , where normal vector is denoted by n , tangent vector is t , fluid Cauthy stress tensor σ ( u , p ) def = − p I + 2 µ ε ( u ) , ε ( u ) def = ( ∇ u + ∇ u T ) , fluid dynamic viscosity µ = 0 . ρ f = 1 .
0, pressure P ( t ) = P max (1 − cos (2 tπ/T (cid:63) )) / P max = 2 ∗ when 0 ≤ t ≤ T (cid:63) and P max = 0 when t > T (cid:63) , T (cid:63) = 5 ∗ − , structure density ρ s = 1 . , c def = E(cid:15) ν ) , c def = E(cid:15)R (1 − ν ) , (cid:15) = 0 .
1, Young’s modulus E = 0 . ∗ , Poisson’s ratio ν = 0 . Notations
For all the algorithms mentioned in this work, τ denotes time step, while h stands for spacediscretization parameter.Given arbitrary variable x , the notation x n,(cid:63) def = r = 0 ,x n − if r = 1 , x n − − x n − if r = 2 (3)is used for interface extrapolations of order r . The time semi-discrete form of
Explicit Robin-Neumann scheme (Fernandez [9]) is citedhere. (Fernandez) Explicit Robin-Neumann scheme (time semi-discrete)
For n ≥ r + 1, find u n : Ω → R , p n : Ω → R , η n : Σ → R and ˙ η n : Σ → R such that1. Fluid step (interface Robin condition) ρ f u n − u n − τ u n − div σ ( u n , p n ) = in Ω , div u n = 0 in Ω , u n · n = 0 , σ ( u n , p n ) n · t = 0 on Γ , σ ( u n , p n ) n = − P ( t ) n on Γ , σ ( u n , p n ) n = on Γ , σ ( u n , p n ) n · n + ρ s (cid:15)τ u n · n = ρ s (cid:15)τ ( ˙ η n − + τ ∂ τ ˙ η n,(cid:63) )+( − p n,(cid:63) I + 2 µ ε ( u n,(cid:63) ) ) n · n on Σ , u n · t = 0 on Σ . (4)4. Solid step (Neumann condition) ρ s (cid:15) ˙ η n − ˙ η n − τ − c ∂ xx η n + c η n = − ( − p n I + 2 µ ε ( u n ) ) n · n on Σ , ˙ η n = ∂ τ η n on Σ ,η n =0 on ∂ Σ , (5)The fully decoupled scheme is proposed in Fernandez [6], [8]. There are non-incrementaland incremental forms, of which both deliver close numerical results on accuracy. Here onlypresents the non-incremental form. (Fernandez) fully decoupled scheme (time semi-discrete) For n ≥ r + 1,(1) Fluid viscous sub-step: find ˜u n : Ω → R such that ρ f ˜u n − u n − τ − µ div ε ( ˜u n ) = 0 in Ω , ˜u n · n = 0 , µ ε ( ˜u n ) n · τ = 0 on Γ , µ ε ( ˜u n ) n · τ = 0 on Γ , µ ε ( ˜u n ) n · τ = 0 on Γ , ˜ u n = 0 , µ ε ( ˜u n ) n · n + ρ s (cid:15)τ ˜u n · n = ρ s (cid:15)τ ˙ η n − on Σ , (6)(2) Fluid projection sub-step: find φ n : Ω → R such that − τρ f ∆ φ n = −∇ · ˜u n in Ω ,∂φ n ∂n = 0 on Γ ,φ n = P ( t n ) on Γ ,φ n = 0 on Γ ,τρ f ∇ φ n · n + τρ s (cid:15) φ n = τρ s (cid:15) φ n,(cid:63) + ˜u n,(cid:63) · n − ˙ η n,(cid:63) on Σ , (7)Thereafter set p n = φ n , u n = ˜u n − τρ f ∇ φ n in Ω .
53) Solid sub-step: find η n : Σ → R such that ρ s (cid:15) ˙ η n − ˙ η n − τ − c ∂ xx η n + c η n = − µ ε ( ˜u n ) n · n + p n on Σ , ˙ η n = η n − η n − τ on Σ ,η n = 0 on ∂ Σ , (8) Remark 2
Substituting u n = ˜u n − τρ f ∇ φ n into (6) leads to a more compact style, with u n eliminated (see [6]). Replacing the coefficient 2 in the term 2 µ ε ( u n,(cid:63) ) at the right hand side of (4) and theterm 2 µ ε ( u n ) at the right hand side of (5) with a real number named β larger than 2generates Algorithm 1 as follows. Analogously, substituting the coefficient 2 in the term − µ ε ( ˜u n ) n · n at the right hand side of (8) with a real number named θ larger than 2 leadsto Algorithm 2 . Remark 3
There is no such a term in (6) like ( − p n,(cid:63) I + 2 µ ε ( u n,(cid:63) ) ) n · n of (4) , so thereis no such a term in (11) like ( − p n,(cid:63) I + βµ ε ( u n,(cid:63) ) ) n · n of (9) Algorithm 1
For real number β > , n ≥ r + 1, find u n : Ω → R , p n : Ω → R , η n : Σ → R and ˙ η n : Σ → R such that 6. Fluid step (interface Robin condition) ρ f u n − u n − τ − div σ ( u n , p n ) = in Ω , div u n = 0 in Ω , u n · n = 0 , σ ( u n , p n ) n · t = 0 on Γ , σ ( u n , p n ) n = − P ( t ) n on Γ , σ ( u n , p n ) n = on Γ , σ ( u n , p n ) n · n + ρ s (cid:15)τ u n · n = ρ s (cid:15)τ ( ˙ η n − + τ ∂ τ ˙ η n,(cid:63) )+( − p n,(cid:63) I + βµ ε ( u n,(cid:63) ) ) n · n on Σ , u n · t = 0 on Σ . (9)2. Solid step (Neumann condition) ρ s (cid:15) ˙ η n − ˙ η n − τ − c ∂ xx η n + c η n = − ( − p n I + βµ ε ( u n ) ) n · n on Σ , ˙ η n = ∂ τ η n on Σ ,η n =0 on ∂ Σ , (10) Algorithm 2
For real number θ > n ≥ r + 1,(1) Fluid viscous sub-step: find ˜u n : Ω → R such that ρ f ˜u n − u n − τ − µ div ε ( ˜u n ) = 0 in Ω , ˜u n · n = 0 , µ ε ( ˜u n ) n · τ = 0 on Γ , µ ε ( ˜u n ) n · τ = 0 on Γ , µ ε ( ˜u n ) n · τ = 0 on Γ , ˜ u n = 0 , µ ε ( ˜u n ) n · n + ρ s (cid:15)τ ˜u n · n = ρ s (cid:15)τ ˙ η n − on Σ , (11)72) Fluid projection sub-step: find φ n : Ω → R such that − τρ f ∆ φ n = −∇ · ˜u n in Ω ,∂φ n ∂n = 0 on Γ ,φ n = P ( t n ) on Γ ,φ n = 0 on Γ ,τρ f ∇ φ n · n + τρ s (cid:15) φ n = τρ s (cid:15) φ n,(cid:63) + ˜u n,(cid:63) · n − ˙ η n,(cid:63) on Σ , (12)Thereafter set p n = φ n , u n = ˜u n − τρ f ∇ φ n in Ω . (3) Solid sub-step: find η n : Σ → R such that ρ s (cid:15) ˙ η n − ˙ η n − τ − c ∂ xx η n + c η n = − θµ ε ( ˜u n ) n · n + p n on Σ , ˙ η n = η n − η n − τ on Σ ,η n = 0 on ∂ Σ , (13) Fernandez’s fully decoupled scheme is based on Chorin-Temam projection method, whoseaccuracy is of first order in time (see e.g. [16] [13] ). It is expected, if the underlying projectionmethod is replaced with Van Kan’s projection method, which is of second order in time (seee.g. [16]), such schemes could be more accurate. This idea produces
Algorithms 3 . Algorithm 3
For real number ξ ≥ n ≥ r + 1, 81) Fluid viscous sub-step: find ˜u n : Ω → R such that ρ f ˜u n − u n − τ = −∇ p n − + 12 (2 µ div ε ( ˜u n ) + 2 µ div ε ( u n − )) in Ω , ˜u n · n = 0 , µ ε ( ˜u n ) n · τ = 0 on Γ , µ ε ( ˜u n ) n · τ = 0 on Γ , µ ε ( ˜u n ) n · τ = 0 on Γ , ˜ u n = 0 , µ ε ( ˜u n ) n · n + ρ s (cid:15)τ ˜u n · n = ρ s (cid:15)τ ˙ η n − on Σ , (14)(2) Fluid projection sub-step: find φ n : Ω → R such that − τρ f ∆ φ n = −∇ · ˜u n in Ω ,∂φ n ∂n = 0 on Γ ,φ n = P ( t n ) − P ( t n − )2 on Γ ,φ n = 0 on Γ ,τρ f ∇ φ n · n + τρ s (cid:15) φ n = τρ s (cid:15) p n,(cid:63) − p n − ,(cid:63) ˜u n,(cid:63) − ˜u n − ,(cid:63) · n − ˙ η n,(cid:63) − ˙ η n − ,(cid:63) , (15)Thereafter set p n = p n − + 2 φ n , u n = ˜u n − τρ f ∇ φ n in Ω . (3) Solid sub-step: find η n : Σ → R such that ρ s (cid:15) ˙ η n − ˙ η n − τ − c ∂ xx η n + c η n = − ξµ ε ( ˜u n ) n · n + p n on Σ , ˙ η n = η n − η n − τ on Σ ,η n = 0 on ∂ Σ , (16) Remark 4
Boundary conditions for the fluid projection sub-step of
Algorithm 3 are de-duced from that of
Fernandez’s fully decoupled scheme by noting that φ n = p n − p n − for Algorithm 3 and that p n = φ n for Fernandez’s fully decoupled scheme . For example,on Γ , (7) indicates p n = P ( t n ) p n − = P ( t n − )9aking the difference and didvided by 2 yields (15) φ n = p n − p n − P ( t n ) − P ( t n − )2The same procedure applies to the deduction of (15) , , Fernandez’s two algorithms and
Algorithms 1-3 are all discretized with Galerkin finiteelement method in space and implemented with FreeFem++ [15] using Lagrange P elementfor both the fluid and structure with symmetric pressure stabilization method [2]. In orderto observe the order of convergence, the time and space are refined at the same rate ,( τ, h ) = (5 ∗ − , . rate , rate = 0 , , , , , , .... The reference solution is generated using monolithic scheme at high time-space grid res-olution τ = 10 − , h = 3 . × − . All algorithms run from initial time t = 0 to final time t = 0 . rates of space and time refinement.Computation of relative errors and preparation of data for writing are completed withPerl [20] as well as an amount of Perl modules [17] and Bash [11]. Graphs are drew usinggnuplot [21]. All codes run on x86 64 Linux 5.6.0 [19] with one Intel R (cid:13) Xeon R (cid:13) E-2186M CPU@ 2.90GHz.Tables 1, 2 and 3 report relative errors of Fernandez’s two algorithms and
Algorithms1-3 with β, θ and ξ ranging from integers 10 to 45 respectively, at refinement rate = 2 , , , rate = 0 and 1 are of no interest and not presented, since all of Fernandez’stwo algorithms and Algorithms 1-3 perform poorly in accuracy at such low rates . Numericalresults of
Algorithms 1-3 with β, θ and ξ ranging from 2 to 10 are not presented, becausethey do not yield obvious improvement of accuracy at these intervals.Both of Fernandez’s two algorithms achieve both highest accuracy and optimal first-order convergence rate in time with first-order extrapolation, so Tables 1, 2 and 3 include10heir results at first-order extrapolation only. For purpose of comparison, Algorithm 1 and are also computed with first-order extrapolation. However, Algorithm 3 reacheshighest accuracy at zeroth-order extrapolation, so its results at zeroth-order extrapolationare presented. 11able 1: Numerical results of Fernandez Explicit Robin-Neumann scheme (Fern ERN) andAlgorithm 1 (Algo 1)rate Fern ERN Algo 1 Algo 1 Algo 1 Algo 1 Algo 1 β = 10 β = 11 β = 12 β = 13 β = 142 0.435176 0.423118 0.421689 0.420281 0.418895 0.4175323 0.241766 0.233158 0.232109 0.231066 0.230030 0.2290014 0.128616 0.123319 0.122668 0.122021 0.121377 0.1207355 0.064847 0.061810 0.061437 0.061066 0.060696 0.060328rate Algo 1 Algo 1 Algo 1 Algo 1 Algo 1 Algo 1 β = 15 β = 16 β = 17 β = 18 β = 19 β = 202 0.416193 0.414879 0.413592 0.412334 0.411104 0.4099063 0.227979 0.226965 0.225959 0.224962 0.223972 0.2229924 0.120097 0.119462 0.118831 0.118202 0.117578 0.1169575 0.059961 0.059597 0.059234 0.058873 0.058514 0.058156rate Algo 1 Algo 1 Algo 1 Algo 1 Algo 1 Algo 1 β = 21 β = 22 β = 23 β = 24 β = 25 β = 262 0.408739 0.407606 0.406510 0.405836 unstable unstable3 0.222021 0.221059 0.220107 0.219165 0.218234 0.2173134 0.116340 0.115727 0.115117 0.114512 0.113911 0.1133145 0.057802 0.057449 0.057097 0.056748 0.056402 0.05605712ate Algo 1 Algo 1 Algo 1 Algo 1 Algo 1 Algo 1 β = 27 β = 28 β = 29 β = 30 β = 31 β = 322 unstable unstable unstable unstable unstable unstable3 0.216403 0.215505 0.214619 0.213744 0.212883 0.2120344 0.112722 0.112134 0.111551 0.110973 0.110399 0.1098315 0.055715 0.055375 0.055038 0.054703 0.054371 0.054041rate Algo 1 Algo 1 Algo 1 Algo 1 Algo 1 Algo 1 β = 33 β = 34 β = 35 β = 36 β = 37 β = 382 unstable unstable unstable unstable unstable unstable3 0.211198 0.210376 0.209568 0.208775 0.207996 0.2072334 0.109268 0.108710 0.108157 0.107610 0.107069 0.1065365 0.053714 0.053389 0.053068 0.052749 0.052433 0.052121rate Algo 1 Algo 1 Algo 1 Algo 1 Algo 1 Algo 1 β = 39 β = 40 β = 41 β = 42 β = 43 β = 442 unstable unstable unstable unstable unstable unstable3 0.206485 0.205754 0.205038 0.204341 0.203850 unstable4 0.106004 0.105481 0.104966 0.104455 0.103949 0.1034525 0.051811 0.051504 0.051201 0.050901 0.050604 0.05031013ate Algo 1 β = 452 unstable3 unstable4 0.1029615 0.050021 14able 2: Numerical results of Fernandez fully decoupled scheme (Fern FD) and Algorithm 2(Algo 2) rate Fern FD Algo 2 Algo 2 Algo 2 Algo 2 Algo 2 θ = 10 θ = 11 θ = 12 θ = 13 θ = 142 0.437713 0.420421 0.418264 0.416110 0.413961 0.4118173 0.243562 0.231346 0.229813 0.228279 0.226744 0.2252084 0.129731 0.123637 0.122873 0.122109 0.121345 0.1205805 0.065497 0.063052 0.062746 0.062441 0.062136 0.061830rate Algo 2 Algo 2 Algo 2 Algo 2 Algo 2 Algo 2 θ = 15 θ = 16 θ = 17 θ = 18 θ = 19 θ = 202 0.409654 unstable unstable unstable unstable unstable3 0.223672 0.222135 0.220599 0.219063 0.217527 0.2159924 0.119816 0.119050 0.118285 0.117519 0.116754 0.1159885 0.061525 0.061220 0.060915 0.060610 0.060305 0.060000rate Algo 2 Algo 2 Algo 2 Algo 2 Algo 2 Algo 2 θ = 21 θ = 22 θ = 23 θ = 24 θ = 25 θ = 262 unstable unstable unstable unstable unstable unstable3 0.214458 0.212924 0.211393 0.209862 0.208879 unstable4 0.115222 0.114456 0.113690 0.112924 0.112158 0.1113925 0.059695 0.059391 0.059086 0.058782 0.058477 0.05817315ate Algo 2 Algo 2 Algo 2 Algo 2 Algo 2 Algo 2 θ = 27 θ = 28 θ = 29 θ = 30 θ = 31 θ = 322 unstable unstable unstable unstable unstable unstable3 unstable unstable unstable unstable unstable unstable4 0.110627 0.109861 0.109096 0.108330 0.107566 0.1068015 0.057869 0.057565 0.057261 0.056958 0.056654 0.056351rate Algo 2 Algo 2 Algo 2 Algo 2 Algo 2 Algo 2 θ = 33 θ = 34 θ = 35 θ = 36 θ = 37 θ = 382 unstable unstable unstable unstable unstable unstable3 unstable unstable unstable unstable unstable unstable4 0.106036 0.105272 0.104509 0.103746 0.102984 0.1022225 0.056048 0.055745 0.055442 0.055139 0.054837 0.054534rate Algo 2 Algo 2 Algo 2 Algo 2 Algo 2 Algo 2 θ = 39 θ = 40 θ = 41 θ = 42 θ = 43 θ = 442 unstable unstable unstable unstable unstable unstable3 unstable unstable unstable unstable unstable unstable4 0.101463 0.100700 0.099940 0.099186 unstable unstable5 0.054232 0.053931 0.053629 0.053327 0.053027 0.05272616ate Algo 2 θ = 452 unstable3 unstable4 unstable5 0.052425 17able 3: Numerical results of Fernandez fully decoupled scheme (Fern FD) and Algorithm 3(Algo 3) rate Fern FD Algo 3 Algo 3 Algo 3 Algo 3 Algo 3 ξ = 10 ξ = 11 ξ = 12 ξ = 13 ξ = 142 0.437713 0.413443 0.408376 0.403425 0.398606 0.3939393 0.243562 0.248517 0.243651 0.238858 0.234148 0.2295354 0.129731 0.136471 0.132807 0.129173 0.125574 0.1220165 0.065497 0.069889 0.067621 0.065367 0.063132 0.060917rate Algo 3 Algo 3 Algo 3 Algo 3 Algo 3 Algo 3 ξ = 15 ξ = 16 ξ = 17 ξ = 18 ξ = 19 ξ = 202 0.389442 0.385136 0.381043 0.377186 0.373591 0.3702813 0.225031 0.220650 0.216408 0.212321 0.208408 0.2046864 0.118508 0.115057 0.111672 0.108363 0.105142 0.1020215 0.058728 0.056566 0.054439 0.052351 0.050309 0.048320rate Algo 3 Algo 3 Algo 3 Algo 3 Algo 3 Algo 3 ξ = 21 ξ = 22 ξ = 23 ξ = 24 ξ = 25 ξ = 262 0.367286 0.364632 0.362348 0.360462 0.359005 0.3580053 0.201178 0.197904 0.194886 0.192147 0.189712 0.1876044 0.099015 0.096139 0.093410 0.090847 0.088472 0.0862995 0.046394 0.044540 0.042769 0.041095 0.039531 0.03809518ate Algo 3 Algo 3 Algo 3 Algo 3 Algo 3 Algo 3 ξ = 27 ξ = 28 ξ = 29 ξ = 30 ξ = 31 ξ = 322 0.357492 0.357517 0.358058 0.359165 0.360862 0.3631693 0.185846 0.184461 0.183471 0.182897 0.182757 0.1830654 0.084358 0.082669 0.081254 0.080136 0.079329 0.0788575 0.036802 0.035672 0.034723 0.033973 0.033437 0.033131rate Algo 3 Algo 3 Algo 3 Algo 3 Algo 3 Algo 3 ξ = 33 ξ = 34 ξ = 35 ξ = 36 ξ = 37 ξ = 382 0.366107 0.369690 0.373933 0.378845 0.384418 0.3921123 0.183835 0.185076 0.186796 0.188994 0.191672 0.1948264 0.078730 0.078957 0.079543 0.080489 0.081789 0.0834345 0.033061 0.033234 0.033648 0.034298 0.035172 0.036259rate Algo 3 Algo 3 Algo 3 Algo 3 Algo 3 Algo 3 ξ = 39 ξ = 40 ξ = 41 ξ = 42 ξ = 43 ξ = 442 unstable unstable unstable unstable unstable unstable3 0.198449 0.202535 0.207070 0.212044 0.217443 0.2232534 0.085413 0.087711 0.090310 0.093194 0.096343 0.0997415 0.037542 0.039004 0.040630 0.042402 0.044305 0.04632619ate Algo 3 ξ = 452 unstable3 0.2294594 0.1033705 0.048453 Conclusions can be drawn from Tables 1, 2 and 3 respectively as follows.
Relative errors of
Algorithm 1 decrease in a regular manner as β or refinement rate increase.All the relative errors are less than that of Fernandez Explicit Robin-Neumann scheme except for unstable ones. All algorithms roughly achieve the same convergence order in time,namely O ( t ).Stability is conditional. For a specific value of β , the algorithm is stable at high refinement rates ; for a specific refinement rate , it is stable at small values of β . At rate = 2 and 3, Algorithm 1 is stable up to β = 24 and 43 respectively. At rate = 4 and 5, it is stable forall tested values of β .That relative errors keep decreasing as β increases up to 45 implies that the amount offorce added to the algorithm keeps approaching the actual amount of force resulting fromadded-mass effect in the coupling system. Based on the intuition mentioned in Section 1 ,it is guessed continuing increasing β up to certain value larger than 45 might decrease therelative errors further. However, since the algorithm performs worse in stability at larger β and the stability is already frustrating at β = 45, it is not worth doing so.20 .2 Conclusions for Algorithm 2 from Table 2 Relative errors of
Algorithm 2 decrease in a regular manner as θ or refinement rate increase.All the relative errors are less than that of Fernandez fully decoupled scheme except forunstable ones. All algorithms roughly achieve the same convergence order in time, namely O ( t ).Stability is conditional. For a specific value of θ , the algorithm is stable at high refinement rates ; for a specific refinement rate , it is stable at small values of θ . At rate = 2 , Algorithm 2 is stable up to θ = 15 , 25 and 42 respectively. At rate = 5 , it is stable for alltested values of θ .That relative errors keep decreasing as θ increases up to 45 implies that the amount offorce added to the algorithm keeps approaching the actual amount of force resulting fromadded-mass effect in the coupling system. Based on the intuition mentioned in Section 1 ,it is guessed continuing increasing θ up to certain value larger than 45 might decrease therelative errors further. However, since the algorithm performs worse in stability at larger θ and the stability is already frustrating at θ = 45, it is not worth doing so. Relative errors of
Algorithm 3 decrease in a regular manner as ξ or refinement rate increaseup to ξ = 27 at rate = 2, ξ = 31 at rate = 3, ξ = 33 at rate = 4 and ξ = 33 at rate = 5.For larger ξ at that refinement rate , relative errors augment. All the relative errors are lessthan that of Fernandez fully decoupled scheme except for unstable ones. All algorithmsroughly achieve the same convergence order in time, namely O ( t ).Stability is conditional. For a specific value of ξ , the algorithm is stable at high refinement rates ; for a specific refinement rate , it is stable at small values of ξ . At rate = 2, Algorithm3 is stable up to ξ = 38 . At rate = 3 , ξ . Comparedwith Algorithm 2 , Algorithm 3 possesses better stability and accuracy.That relative errors keep decreasing as ξ increases up to 27 and increasing as ξ increasesfrom 33 complies with the intuition mentioned in Section 1 . It is guessed the amount offorce added to the algorithm by taking ξ between 27 and 33 is close to the actual amount of21orce resulting from added-mass effect in the coupling system. Theoretical analysis is not available yet. Here states a possible and non-rigorous explanation.It remains unknown whether such an explanation is correct.All algorithms mentioned lead to linear equations after space discretizations at each timestep. Compared with Fernandez’s two algorithms,
Algorithms 1-3 modify the right handside of those linear equations generated, causing perturbations to the solutions. Since Fernan-dez’s two algorithms are stable, it is expected solutions still exist and do not change obviouslyunder small perturbations. However, as β, θ or ξ increases, such perturbations become moresignificant and affect the existence and values of solutions more seriously. As time stepsgo on, perturbations accumulate and at some time steps cause the solutions to the linearequations generated at that step non-existent. This perhaps explains why Algorithms 1-3 become unstable at large β, θ or ξ for a specific refinement rate .For a specific value of β, θ or ξ , as the refinement rate increases, the number of nodesof mesh enlarges. Let an integer m denote the number of nodes on Σ, m = L/h + 1. Thenumber of nodes on Ω is approximately m . Note that the added force is only imposed onthe interface. Thus, only m nodes are affected. The ratio of affected and non-affected nodesis approximately m/ ( m − m ) = 1 / ( m − m increases. As refinement rate increases, m increases and therefore the forces added to Algorithms 1-3 disturbs thesystem less, which yields better stability.
10 Selection of β, θ and ξ for practical applications Practical applications should take into account both efficiency and accuracy. At refinement rate = 2, time step τ = 0 . h = 0 . Algorithms1-3 regardless of values of β, θ and ξ . It is quite fast. However, all of them are far from22ccurate. Therefore, practical applications are not expected to run at such low rate ofrefinement; it suffices to consider rate = 3 , rate = 3 , β = 43 , θ = 25 , ≤ ξ ≤
33 are recommended for
Algorithm 1-3 respectively.Tables 4, 5 and 6 report their values and percents of decrement of relative errors comparedwith Fernandez’s algorithms respectively. The values of decrement of relative errors equalto relative errors of Fernandez’s algorithms minus that of
Algorithm 1-3 , while percentsequal to values divided by relative errors of Fernandez’s algorithms times 100. Structuredisplacements are displayed in Figures 2, 3, 4, 5 and 6.Table 4: Numerical results of Fernandez Explicit Robin-Neumann scheme (Fern ERN) andAlgorithm 1 (Algo 1) at selected β rate Fern ERN Algo 1 decrement of errors β = 43 values percents(%)2 0.435176 unstable N/A N/A3 0.241766 0.203850 0.037916 15.68294 0.128616 0.103949 0.024667 19.17885 0.064847 0.050604 0.014243 21.9640Table 5: Numerical results of Fernandez fully decoupled scheme (Fern FD) and Algorithm 2(Algo 2) at selected θ rate Fern FD Algo 2 decrement of errors θ = 25 values percents(%)2 0.437713 unstable N/A N/A3 0.243562 0.208879 0.034683 14.23994 0.129731 0.112158 0.017573 13.54575 0.065497 0.058477 0.00702 10.7180234able 6: Numerical results of Fernandez fully decoupled scheme (Fern FD) and Algorithm 3(Algo 3) at selected ξ rate Fern FD Algo 3 decrement of errors ξ = 31 values percents(%)2 0.437713 0.360862 0.076851 17.55743 0.243562 0.182757 0.060805 24.96494 0.129731 0.079329 0.050402 38.85125 0.065497 0.033437 0.03206 48.9488rate Fern FD Algo 3 decrement of errors ξ = 32 values percents(%)2 0.437713 0.363169 0.074544 17.03033 0.243562 0.183065 0.060497 24.83844 0.129731 0.078857 0.050874 39.21505 0.065497 0.033131 0.032366 49.4160rate Fern FD Algo 3 decrement of errors ξ = 33 values percents(%)2 0.437713 0.366107 0.071606 16.35913 0.243562 0.183835 0.059727 24.52234 0.129731 0.078730 0.051001 39.31295 0.065497 0.033061 0.032436 49.522925 a) rate = 2 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 1, β = 43 Fern ERN (b) rate = 3 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 1, β = 43 Fern ERN (c) rate = 4 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 1, β = 43 Fern ERN (d) rate = 5
Figure 2: Structure displacement of Fernandez Explicit Robin-Neumann scheme (Fern ERN)and Algorithm 1 (Algo 1) with β = 43 at final time
11 Discussions and future work
The numerical results validate the ideas that adding force corresponding to viscosity andreplacing underlying projection method can improve accuracy; particularly, Table 6 indicatesas large improvement as up to 49 . Algorithm 3 with ξ = 33 compared with Fernandez fully decoupled scheme at refinement rate = 5. It is expected, for otherfluid-structure interaction problems, if the fluid is also viscous, adding force might also helpwith accuracy.As a direction of future work, it is worth trying investigating how adding force improveaccuracy theoretically. Reading works [4] [3] on added-mass effect might benefit such analysis.This work deals with accuracy. On the other hand, it is possible to improve efficiency by26 a) rate = 2 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 2, θ = 25 Fern FD (b) rate = 3 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 2, θ = 25 Fern FD (c) rate = 4 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 2, θ = 25 Fern FD (d) rate = 5
Figure 3: Structure displacement of Fernandez fully decoupled scheme (Fern FD) and Algo-rithm 2 (Algo 2) with θ = 25 at final timeparallelism. A choice is to take advantage of extrapolation (1 st order might be better than0 th ). To implement such ideas, MPI [1] might work.
12 Acknowledgments
This research did not receive any specific grant from funding agencies in the public, commer-cial, or not-for-profit sectors.The proposal of ideas as well as algorithms, numerical experiments, conclusions, explana-tions and draft of this article are completed entirely independently, without any assistanceor guide from anyone else.This article serves as the author’s PhD thesis. Compared with the dradt written beforeThesis Exam held on August 10 2020, this is a revised version. The Thesis ExaminationCommittee (consisting of Prof. Mo MU, Prof. Jian-Feng CAI, Prof. Kun XU, Prof. Eric27 ξ = 31 Fern FD (a) rate = 2 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 3, ξ = 31 Fern FD (b) rate = 3 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 3, ξ = 31 Fern FD (c) rate = 4 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 3, ξ = 31 Fern FD (d) rate = 5
Figure 4: Structure displacement of Fernandez fully decoupled scheme (Fern FD) and Algo-rithm 3 (Algo 3) with ξ = 31 at final timeTsz Shun CHUNG, Prof. Jidong ZHAO and Prof. Jensen Tsan Hang LI), especially Prof.Mo MU, advised on the writing and provides lots of very useful suggestions, precisely on thewriting of Sections 1, 8 and 9 after the Thesis Exam. The author greatly appreciates theiradvices. 28 ξ = 32 Fern FD (a) rate = 2 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 3, ξ = 32 Fern FD (b) rate = 3 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 3, ξ = 32 Fern FD (c) rate = 4 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 3, ξ = 32 Fern FD (d) rate = 5
Figure 5: Structure displacement of Fernandez fully decoupled scheme (Fern FD) and Algo-rithm 3 (Algo 3) with ξ = 32 at final time 29 ξ = 33 Fern FD (a) rate = 2 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 3, ξ = 33 Fern FD (b) rate = 3 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 3, ξ = 33 Fern FD (c) rate = 4 -0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 1 2 3 4 5 6referenceAlgo 3, ξ = 33 Fern FD (d) rate = 5
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