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A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows

We extend [Shravan K. Veerapaneni, Denis Gueyffier, Denis Zorin, George Biros, A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D, Journal of Computational Physics 228(7) (2009)
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  A numerical method for simulating the dynamics of 3D axisymmetricvesicles suspended in viscous flows Shravan K. Veerapaneni a, * , Denis Gueyffier b , George Biros c , Denis Zorin a a Courant Institute of Mathematical Sciences, New York University, NY 10012, United States b NASA Goddard Institute for Space Studies, New York, NY 10025, United States c College of Computing, Georgia Institute of Technology, Atlanta, GA 30332, United States a r t i c l e i n f o  Article history: Received 18 March 2009Received in revised form 7 June 2009Accepted 19 June 2009Available online 24 June 2009 Keywords: Particulate flowsIntegral equationsAxisymmetric flowsNumerical methodsFluid membranesInextensible vesiclesMoving boundaries a b s t r a c t Weextend[ShravanK.Veerapaneni,DenisGueyffier,DenisZorin,GeorgeBiros,Aboundaryintegral method for simulating the dynamics of inextensible vesicles suspended in a vis-cous fluid in 2D, Journal of Computational Physics 228(7) (2009) 2334–2353] to the caseof three-dimensional axisymmetric vesicles of spherical or toroidal topology immersedin viscous flows. Although the main components of the algorithm are similar in spirit tothe 2D case—spectral approximation in space, semi-implicit time-stepping scheme—themain differences are that the bending and viscous force require new analysis, the lineari-zation for the semi-implicit schemes must be rederived, a fully implicit scheme must beused for the toroidal topology to eliminate a CFL-type restriction and a novel numericalscheme for the evaluation of the 3D Stokes single layer potential on an axisymmetric sur-face is necessary to speed up the calculations. By introducing these novel components, weobtain a time-scheme that experimentally is unconditionally stable, has low cost per timestep, and is third-order accurate in time. We present numerical results to analyze the costand convergence rates of the scheme. To verify the solver, we compare it to a constrainedvariational approach to compute equilibrium shapes that does not involve interactionswith a viscous fluid. To illustrate the applicability of method, we consider a few vesicle-flow interaction problems: the sedimentation of a vesicle, interactions of one and threevesicles with a background Poiseuille flow.   2009 Elsevier Inc. All rights reserved. 1. Introduction Vesicles areclosedlipidmembranessuspendedina viscoussolution. Theyarecommoninbiological systemsandplayanimportant role in intracellular and intercellular transport; artificial vesicles are used in a variety of drug-delivery systemsand to study the properties of biomembranes. The vesicle evolution dynamics are characterized by a competition betweenmembraneelasticenergy,nonlinearity,surfaceinextensibilityandnon-localinteractionsduetothehydrodynamiccoupling.The designof efficient computational methodsfor suchflows has receivedlimitedattentioncomparedto other types of par-ticulate flows. In [25], we introduced an algorithmfor vesicle simulations in two dimensions. In this paper, we take the firststep towards efficient high-order three-dimensional simulations by considering axisymmetric vesicle flows for the casewhere there is no viscosity contrast across the vesicle membrane. The equations that govern the motion of a single vesiclein three dimensions are 0021-9991/$ - see front matter   2009 Elsevier Inc. All rights reserved.doi:10.1016/ *  Corresponding author. Tel.: +1 2129983510. E-mail addresses:, (S.K. Veerapaneni), (D. Gueyffier), (G. Biros), (D. Zorin). Journal of Computational Physics 228 (2009) 7233–7249 Contents lists available at ScienceDirect  Journal of Computational Physics journal homepage:  @   x  @  t   ¼  v  1  þ S  ½ f  b  þ  f  r  ð vesicle position evolution Þ ; div c @   x  @  t   ¼  0  ð surface inextensibility Þ ; ð 1 Þ where c  is the vesicle membrane, div c  is the surface divergence operator,  x   is a Lagrangian point on c ; f  r  is a force (tension)duetosurfaceinextensibility, f  b  isaforceduetobending,  v  1  isthefar-fieldvelocityofthebulkfluidand S   isthesinglelayerStokes operator, defined in Section 2. The first equation describes the motion of the vesicle boundary; the second equationexpresses the local inextensibility of   c .Our main goal is to extend the ideas presented in [25] to the axisymmetric case of vesicles with spherical or toroidaltopology. The extension is non-trivial because in three dimensions the bending energy has a much more complicated formandcannotbereducedtoalinearexpressioninarc-lengthderivativesasinthetwo-dimensionalcase.Furthermore,thequal-itativenumericalbehaviorofbendingforcesisalsodifferent:anunconditionallystablesemi-implicitlinearizedschemewithnoCFL-typerestrictiononthetimestep, similartothetwo-dimensionalcase, couldonlybefoundforthesphericaltopology.For vesicles with toroidal topology (admittedly less common, but observed in nature [19]), eliminating CFL-type time-steprestrictionsrequiresafullyimplicittime-marchingscheme.Ourmaincontributionisthedevelopmentofefficientnumericalschemes for (1) for spherical and toroidal topologies with the following components:   Theyrequireasinglelinearsolvepertimestepforsphericaltopologyandasmallnumberofnonlineariterationsfortoroi-dal topology;   They include an efficient preconditioner to enforce the surface-incompressibility constraint;   They are spectrally accurate in space and third-order accurate in time.Anotherimportantpartofthealgorithmisanovelnumericalschemeforevaluationofthe3DStokessinglelayerpotentialon an axisymmetric surface, needed to achieve an optimal complexity of the algorithm. For verification, we compare equi-libriumshapesobtainedusingtheproposedmethodwithshapesobtainedusingavariationalapproachthatdoesnotinvolvecomputing viscous forces. Finally, to illustrate the capabilities of the method, we consider sedimentation of vesicles undergravity and interactions of multiple vesicles with a background Poiseuille flow. 1.1. Limitations In the current form, our scheme is not adaptive. Incorporating adaptivity in space and time requires suitable error esti-mators. In addition, while  p -type spatial adaptivity can be incorporated with less effort, more fundamental changes to thecurrent scheme are required for  h -type adaptivity because a non-uniform discretization would require a different approachto compute high-order derivatives accurately.Our scheme has a mild time-steprestriction in the case of shear flows: the stable time-step size is inversely proportionalto the shear rate. While one would hope that a fully implicit scheme would eliminate or reduce this restriction, our exper-iments indicate that even a fully implicit Newton scheme (Section 4.3) does not yield noticeable speed-ups. This is becausethe Newtoniterations donot converge for large time-stepsizes. The time-steps for whichtheydo convergeare very close tothe time-steps for which the semi-implicit scheme is stable.An additional limitation of the overall scheme is that we do not consider topology changes or vesicles flows with a vis-cosity contrast across the membrane, which would require solution of an additional boundary integral equation. 1.2. Related work There has been a lot of work on modeling 3D axisymmetric particulate flows. In [25], we discussed vesicle-related algo-rithms. An excellent review of such methods can be found in [16] (Table 1, p. 289; for vesicles see the ‘‘liquid capsules”entry).Several groupshavefocusedondeterminingstationaryshapes ofthree-dimensional vesiclesusingsemi-analytic[20,3,5],ornumericalmethodslikethephase-field[8,7]andmembranefiniteelementmethods[9,13].Theseapproachesarebasedon a constrained variational approach (i.e., minimizing the bending energy subject to area and volume constraints) and cannotbe used for interactions of multiple vesicles in shear flows.Afullthree-dimensional simulationofasinglevesicleincorporatingthehydrodynamiccoupling,local inextensibilityandthe bending forces has been reported in [10,21]. A closely related work is also that of  [17], in which, a nearly inextensible interface was considered for the axisymmetric motion of red blood cells inside a cylindrical tube.In all, however, there has been little work in developing fast algorithms for axisymmetric vesicle flows. 1.3. Contents InSection2,weformulatetheintegro-differentialEq.(1)thatgovernvesicledynamics.Thespatialandtemporaldiscretiza- tionsaredescribedinSections3and4,respectively.InSection5,wepresentnumericalresultsforanumberofproblemsinvolv- 7234  S.K. Veerapaneni et al./Journal of Computational Physics 228 (2009) 7233–7249  ingsingleandmultiplevesiclessuspendedinaviscousfluid.Weconductnumericalexperimentstoinvestigatethestabilityandconvergenceorderofdifferenttime-steppingschemes.Theverificationofthesolverandseveralimportantdetails(semi-ana-lyticsolutionsforthequiescentcase,expressionsfortheforceandStokesconvolutionsintheaxisymmetriccaseandananalysisoftheapproximationerrorforhigh-orderderivativesandwaystoimproveaccuracy)arepresentedintheAppendix. 2. Problem formulation For simplicity, we first discuss the formulation for a single vesicle suspended in an unbounded viscous fluid. Let  p ð  x  Þ  and  v  ð  x  Þ  denotethefluidpressureandvelocityfieldsandlet c denotethevesiclemembrane. Themotionof thebackgroundfluidis described by the Stokes equations,  l D  v   þ  r  p  ¼  0 and div  v   ¼  0 in  R 3 n c ;  ð 2 Þ where l istheviscosityof thefluid. Theno-slipboundaryconditionon c andthefree-spaceboundaryconditionrequirethat  v   ¼  _  x   on  c ;  lim  x  !1  v  ð  x  Þ   v  1 ð  x  Þ ¼  0 ;  ð 3 Þ where  _  x   is the total derivative of the motion of material point on the vesicle surface (i.e., its velocity) and  v  1  is the far-fieldvelocityof thebackgroundfluid. Themembraneforcesof magnitude f   arebalancedbyatractionjumpacross theinterface c .That is, if   R  denotes the stress tensor, then ½½ R n  c  ¼  f  ;  ð 4 Þ where  n  is the normal to the interface. To derive an expression for  f   we have to consider the constitutive properties of thevesicle membrane. The standard assumptions for vesicles [19] consider a surface elastic energy that consists of two terms: E  ð H  ; r Þ ¼ Z  c 12 j B H  2 þ r d c ;  ð 5 Þ where j B  is the bending modulus and  H   is the mean curvature. The first term is the bending energy and the second term isrequired to enforce the local inextensibility constraint of the surface. In other words, the tension r  is a Lagrange multiplierthat enforces the constraint. The interfacial force can be derived from the surface energy by taking its  L 2 -gradient f   ¼  D E  D  x  : In order to derive a formula for  f   in terms of the curvature and the parameterization of the surface, we need to introduce afew quantities. Let  x  ð u ; v  Þ  :  U   !  c  be a parametrization of the surface. The corresponding fundamental form coefficients are[12], E   ¼  x  u    x  u ;  F   ¼  x  u    x  v  ;  G  ¼  x  v     x  v   ð first fundamental form Þ ;  ð 6 Þ L  ¼  x  uu    n ;  M   ¼  x  u v     n ;  N   ¼  x  vv     n  ð second fundamental form Þ :  ð 7 Þ The normal to the surface and the area element are defined by n  ¼ ð  x  u    x  v  Þ =  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG   F  2 p   ;  dA  ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG    F  2 p   dud v   ¼  Wdud v  :  ð 8 Þ We can now define the mean and Gaussian curvatures as H   ¼  12 EN    2 FM   þ  GLW  2  ;  K   ¼  LN     M  2 W  2  :  ð 9 Þ Then, following [26], the gradient or first variation of  (5) is given by D E   ¼ Z  c ð D S  H   þ 2 H  ð H  2  K  ÞÞ n    D  x    ð r D S   x   þ  r S  r Þ   D  x  d c ;  ð 10 Þ where  D S   is the Laplace-Beltrami operator defined by D S  /  ¼  1 W E  / v     F  / u W    v  þ  G / u    F  / v  W    u   ;  for some scalar function  / :  ð 11 Þ From (10), we define the bending and tension forces as f  b  ¼ ð D S  H   þ 2 H  ð H  2   K  ÞÞ n ;  f  r  ¼  r D S   x   þ  r S  r :  ð 12 Þ These two forces constitute the interfacial force, that is,  f   ¼  f  b  þ f  r . Using classical potential theory [15], the solution of (2)–(4), combined with the local inextensibility constraint of the membrane can be written as _  x   ¼  v  1 ð  x  Þ þ S  ½ f  b  þ  f  r ð  x  Þ  and div c ð S  ½ f  r Þ ¼   div c ð  v  1  þ S  ½ f  b Þ :  ð 13 Þ S.K. Veerapaneni et al./Journal of Computational Physics 228 (2009) 7233–7249  7235  This is a systemof two integro-differential equations for two unknowns: the position of the membrane  x   and the tension r .The single layer potential operator is defined by S  ½ f  ð  x  Þ ¼ R  c  G ð  x  ;  y  Þ f  ð  y  Þ d c ð  y  Þ , where  G  is the free-space Green’s function forthe Stokes operator and is given by G ð  x  ;  y  Þ ¼  18 pl 1 j r  j I  þ  r     r  j r  j 3  ! ;  r   ¼  x     y  :  ð 14 Þ Next, we present the reduction of these equations to one spatial variable in the axisymmetric case.  2.1. Axisymmetric formulation Assuming symmetry in the ‘ v ’ direction, the positions and the interfacial forces take the following form  x   ¼  x 1 ð u Þ cos v   x 1 ð u Þ sin v   x 2 ð u Þ 264375 ;  f   ¼  f  1 ð u Þ cos v   f  1 ð u Þ sin v   f  2 ð u Þ 264375 :  ð 15 Þ The parametric domain  f u ; v  g 2  U   is  ½ 0 ; 2 p   ½ 0 ; 2 p   for toroidal topologies; representing all variables in thetrigonometric basis guarantees that the resulting functions, are well-defined on the toroidal domain ½ð R  þ cos u Þ cos v  ; ð R  þ cos u Þ sin v  ; sin u  , with  R  ¼  ffiffiffi 2 p   . A sphere can be regarded as a degenerate torus with  R  ¼  0, with eachpointof thespherecorrespondingtotwopointsonthetorus. Tomakethismappingone-to-oneweconsideronlyone halfof the parametric domain  ½ 0 ; p   ½ 0 ; 2 p  . For  x   to be a smooth function on the sphere, it is necessary and sufficient that  x 1  is anodd and  x 2  is an even periodic function of   u ; in other words, a trigonometric series for  x 1  and  x 2  have only nonzero coeffi-cients for sines and cosines, respectively. Similarly, any scalar function defined on the surface needs to be even in  u .We can now write the bending and tension forces in terms of   u . Let  s  be the arc-length parameter, that is, s ð u Þ ¼ R  u 0  jj  x  ð u 0 Þjj du 0 . In the Appendix B, we derive the expressions for the forces in terms of the principal curvatures  j ; b given by  j  ¼  x 1 s  x 2 ss    x 2 s  x 1 ss  and  b  ¼  x 2 s  x 1 ; here, we just state the result: f  b  ¼  12  D S  ð j þ  b Þ þ ð j þ  b Þð j   b Þ 2 2  ! n ;  f  r  ¼ ð r  x  s Þ s   r b n ;  ð 16 Þ and at the poles ;  we have lim  x 1 ! 0 f  b  ¼  j ss n ;  lim  x 1 ! 0 f  r  ¼  r s  x  s   2 r j n :  ð 17 Þ Next,wederivetheaxisymmetricformofthesinglelayerpotential.Withoutlossofgenerality,weassumethatthetargetsonthe surface are located at the cross-section  v   ¼  0. Then, the target and source points have the form  x   ¼ ½  x 1 ;  0 ;  x 2  T  and  y  ð u ; v  Þ ¼ ½  y 1 cos v  ;  y 1 sin v  ;  y 2  T  , respectively (for notational clarity, we drop the explicit dependence of   x i  and  y i ; i  ¼  1 ; 2,on  u ). The single layer potential can be written as S  ½ f   ¼ F  1 0 F  2 264375 ¼ Z   2 p 0 d v  Z   p 0 du  1 j r  j I  þ  r     r  j r  j 3  !  f  1 cos v   f  1 sin v   f  2 264375  y 1 j  y  u j ; where  r   ¼  y 1 cos v     x 1  y 1 sin v   y 2    x 2 264375 ;  j r  j ¼  x 21  þ  y 21   2  x 1  y 1 cos v   þ ð  x 2    y 2 Þ 2 h i 1 = 2 : After further simplification, we get S  ½ f   ¼ F  1 F  2    ¼ Z   2 p 0 d v  Z   p 0 du cos v  j r  j  þ  ð  y 1  cos v    x 1 Þð  y 1   x 1  cos v  Þj r  j 3 ð  y 1  cos v    x 1 Þð  y 2   x 2 Þj r  j 3 ð  y 1   x 1  cos v  Þð  y 2   x 2 Þjj r  j 3 1 j r  j  þ  ð  y 2   x 2 Þ 2 j r  j 3 2435  f  1  f  2    y 1 j  y  u j :  ð 18 Þ All theintegrals withrespect to‘ v ’ arecomputedanalyticallyusingEqs. (56)–(60). Insummary, theaxisymmetric formof the 3D Stokes operator is given by S  ½ f  ð  x  Þ ¼ Z   p 0 K  ð  x  ; u Þ f  ð u Þ  y 1 ð u Þj  y  u j du :  ð 19 Þ The kernel  K   is composed of   elliptic integrals  of first and second kind.  2.1.1. Gravitational force If there is a density difference across the membrane of a vesicle, then the vesicle experiences an additional force due togravity given by f   g   ¼ ð q in  q out Þð g     x  Þ n :  ð 20 Þ 7236  S.K. Veerapaneni et al./Journal of Computational Physics 228 (2009) 7233–7249  Then, the governing equations that include gravitational forces are _  x   ¼  v  1  þ S  ½ f  b  þ  f  r  þ  f   g   ;  div c ð S  ½ f  r Þ ¼   div c ð  v  1  þ S  ½ f  b  þ  f   g  Þ :  ð 21 Þ  2.1.2. Scaling  Following [10], we set the length and time scales as  R 0  ¼  ffiffiffiffi  A 4 p q   and s  ¼  l R 30 j B , respectively, where  A  is the surface area of thevesicle. In the absence of external flows and gravity, it is known that the vesicle dynamics are characterized by a singleparameter [10], the reduced volume  m  ¼  6  ffiffiffi p p   V  A 3 = 2  .Since we are dealing with an axisymmetric problem, we must consider an axisymmetric  v  1 , for example a velocity fieldwith parabolic profile that smoothly decays to zero away from the axis of symmetry, to resemble the profile of a Poiseuilleflow. Typically, we consider velocity profiles of the form  v  1  ¼  c  ð w 2   x 21 ð u ÞÞ , where  c   and  w  are constants. Notice that thecurvature of this velocity profile is  @  2  v  1 @   x 21 ¼  2 c   and the corresponding shear rate is  @   v  1 @   x 1 ¼  2 cx 1 ð u Þ . We introduce the nondi-mensional entity  ^ c   ¼  c  l R 40 j B that parametrizes such external flows.In the presence of gravity, an additional parameter that governs the vesicle dynamics is the nondimensional gravityparameter, given by  ^  g   ¼  ð q in  q out Þ  gR 40 j B .  2.1.3. Multiple vesicles The governing equations for the  j th-vesicle, in a suspension of   N  v   vesicles, are given by _  x   j  ¼  v  1 ð  x   j Þ þ S   j ½ f  b  þ  f  r ð  x   j Þ þ X N  v  k ¼ 1 k –  j S  k ½ f  b  þ  f  r ð  x   j Þ ;  ð 22 Þ div c  j ð S   j ½ f  r Þ ¼  div c  j  v  1 ð  x   j Þ þ S   j ½ f  b ð  x   j Þ þ X N  v  k ¼ 1 k –  j S  k ½ f  b  þ  f  r ð  x   j Þ 0BB@1CCA :  ð 23 Þ where we separate the terms accounting for the interactions with other vesicles.To summarize, (22) and (23) give the update and the incompressibility constraint, the forces  f  b  and  f  r  are given by (16)and (17) and the single layer is given by (18). These equations form a closed system of equations for the positions and tensions. 3. Spatial discretization scheme We have chosen the spatial discretization scheme to enable efficient and high-order computationof derivatives for com-puting bending and tension forces  f  b  and  f  r  and accurate computation of integrals (18) involving singular kernels. We usethe trigonometric polynomial bases to represent the position of the interface and functions defined on it. The coordinatefunctions  x 1  and  x 2  are given by the coefficients  ^  x 1 ð k Þ  and  ^  x 2 ð k Þ : 1  x 1 ð u Þ ¼ X M k ¼ 1 ^  x 1 ð k Þ  sin ð ku Þ ;  x 2 ð u Þ ¼ X M k ¼ 0 ^  x 2 ð k Þ  cos ð ku Þ ;  u  2 ½ 0 ; p  :  ð 24 Þ (Recall that for smoothness  x 1 ð u Þ  is required to be odd and  x 2 ð u Þ  even). Similarly,  r ð u Þ ¼  P M k ¼ 0 ^ r ð k Þ  cos ð ku Þ . The spatial tospectral transform and vice-versa are computed efficiently using the forward and inverse fast sine- and cosine-transforms.This representation allows for an efficient derivative computation:  x 1 u ð u Þ ¼ X M k ¼ 1 k ^  x 1 ð k Þ  cos ð ku Þ ;  x 2 u ð u Þ ¼  X M k ¼ 1 k ^  x 2 ð k Þ  sin ð ku Þ :  ð 25 Þ Assuming that the shape of the vesicle is smooth, this derivative approximation is spectrally accurate. We make a few moreremarks on the derivative accuracy and the effects of round-off error in the Appendix D.  3.1. Quadrature scheme The kernels in (19) have a logarithmic singularity, which can be verified by examining their asymptotic expansions. Let  z   2 ð 0 ; 1 Þ , then we have the following expansions around  z   ¼  0,EllipticK ð 1   z  Þ ¼  c  0    12  ln  z   þ  c  1    14  ln  z     z   þ  c  2    532  ln  z     z  2 þ O  ð  z  3 Þ  (  first kind ),EllipticE ð 1   z  Þ ¼  d 0  þ  d 1    12  ln  z     z   þ  d 2    18  ln  z     z  2 þ O  ð  z  3 Þ  ( second kind ), 1 In the case of torus, we use Fourier basis,  x  ð u Þ ¼ P M  = 2  1 k ¼ M  = 2 ^  x  ð k Þ e  iku : S.K. Veerapaneni et al./Journal of Computational Physics 228 (2009) 7233–7249  7237
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