AN EFFICIENT AND ROBUST METHOD FOR SIMULATING TWO-PHASE GEL DYNAMICS

Transcription

1 AN EFFICIENT AND ROBUST METHOD FOR SIMULATING TWO-PHASE GEL DYNAMICS GRADY B. WRIGHT, ROBERT D. GUY, AND AARON L. FOGELSON Abstract. We develop a computational metod for simulating models of gel dynamics were te gel is described by two-pases, a networked polymer and a fluid solvent. Te models consist of transport equations for te two pases, two coupled momentum equations, and a volume-averaged incompressibility constraint, wic we discretize wit finite differences/volumes. Te momentum and incompressibility equations present te greatest numerical callenges since i) tey involve partial derivatives wit variable coefficients tat can vary quite significantly trougout te domain (wen te pases separate), and ii) teir approximate solution requires te inversion of a large linear system of equations. For solving tis system, we propose a box-type multigrid metod to be used as a preconditioner for te generalized minimum residual (GMRES) metod. Troug numerical experiments of a model problem, wic exibits pase separation, we sow tat te computational cost of te metod scales nearly linearly wit te number of unknowns, and performs consistently well over a wide range of parameters. For solving te transport equation, we use a conservative finite volume metod for wic we derive stability bounds. Key words. multipase flow, mixture teory, multigrid, Vanka relaxation, coupled relaxation, box relaxation, Krylov, GMRES, preconditioning AMS subject classifications. 65M55, 65F1, 65M6, 92E2, 76T99 1. Introduction. Polymer gels are used in a variety of industrial applications and appear naturally in many biological systems. Gels are made of two materials: a polymer network and a solvent. By weigt and volume gels are mostly solvent, but te reology of te gel can range from a very viscous fluid to an elastic solid. In addition to viscoelastic stresses, gels exibit cemical stresses wic result in swelling and deswelling beavior. A commonly used model for te mecanics of gels is te two-pase flow (or twofluid) model [3, 9, 11, 12, 13, 17, 29]. Eac pase, network and solvent, is treated as a continuum and moves according to its own velocity field. Eac region in space is composed of a mixture of te two pases tat is described by te volume fractions of te pases. Te equations of motion for te gel consist of two coupled momentum equations and two continuity equations. Tese models of gels are based on mixture teory [14, 15], wic as been used for many applications and is becoming increasingly popular in models of biological materials suc as tissue [19], tumors [7, 21], cytoplasm [3, 11, 12, 17], and biofilms [2, 8, 9]. Wile muc work as gone in to advancing tese models, te same is not true for te development of efficient numerical metods for simulating tem. Since repeated simulation of models is crucial for validating teir usefulness, te need for fast and robust computational metods is essential. In tis study, an efficient and robust computational metodology for simulating gel dynamics is proposed for a model problem tat sares several caracteristics of many oter models in te literature [3, 9, 11, 12, 17]. Bot te network and solvent are modeled as viscous fluids. Te viscosity of te network is muc larger tan tat June 3, 27,Tis work was supported by NSF-DMS grants and Department of Matematics, Boise State University, Boise, ID , USA (wrigt@boisestate.edu). Department of Matematics, University of California, Davis, CA 95616, USA (guy@mat.ucdavis.edu). Departments of Matematics and Bioengineering, University of Uta, Salt Lake City, UT , USA (fogelson@mat.uta.edu). 1

2 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 2 of te solvent. Te gel feels a cemical pressure tat makes te system bistable, tat is, it drives te network volume fraction to one of two preferred states. Under appropriate conditions te cemical pressure induces pase separation of te mixture and is a good test for te robustness of any procedure. Numerical tests of te proposed metod are carried out for te model in one and two dimensions, wit te primary investigations of efficiency and robustness done for te one dimensional case. In many models of gel dynamics, including our model problem, viscous terms are assumed to dominate so tat inertial terms are negligible. Tis assumption leads to an elliptic system for te coupled momentum and incompressibility equations tat resembles Stokes equations. However, tere are many key differences wic increase te complexity of te problem. First, since tere are two pases, tere are two sets of momentum equations. Second, tere are off-diagonal terms coupling te two velocity fields (and te components of te two fields) togeter. Tird, te Laplacians involve time-dependent variable coefficients related to te volume fractions of te two pases. Finally, incompressibility is replaced wit a volume averaged incompressibility over te two fluids. We use second-order finite-differences on a marker-and-cell (MAC) grid [16] to discretize tis system. As wit Stokes (and linearized Navier-Stokes) equations, tis gives rise to a large, sparse linear system of saddle point type. To treat tis system we propose an extension of a multigrid metod initially proposed for Stokes and Navier-Stokes equations by Vanka [33], and wic as seen considerable development in te past several years (see, for example, [1, 22, 28, 3, 31, 36]). Te metod is caracterized by te smooter used and is referred to in literature as box [31, pp ], coupled [36], or Vanka [22] relaxation. Te extension of tis metod to te gel system is straigtforward since te basic idea of te smooter is to compute updates to te solution by collectively solving for te velocity and pressure in te discrete momentum equations locally, computational cell (or box) by cell. For te gel system, tis means solving a (2 d+1 + 1)-by-(2 d+1 + 1) saddle point system for eac cell, were d is te number of spatial dimensions. Depending on ow te cells are processed (Jacobi or Gauss-Seidel like), te metod can also be viewed as an additive or multiplicative Scwarz domain decomposition metod, were te subdomains consist of a single computational cell [22]. We use te smooter wit standard prolongation and restriction operators and do a direct discretization of te equations on te coarse grids, wic makes te implementation relatively simple. We compare bot te V- and F-cycle tecniques for cycling troug te grids. Wen te network and solvent are well mixed, te multigrid box-relaxation metod performs quite well as a solver for te model problem considered ere. However, as pase separation occurs, te solver performance degrades quite dramatically, and it, in fact, fails in some cases. One particularly effective way of improving te robustness of a non-optimal (or even non-converging) multigrid metod is to combine it wit a Krylov subspace metod (see, for example, [31, 7.8] or [25]). In tis case, multigrid is viewed as preconditioner for te Krylov metod. We use exactly tis idea for te gel system, combining te box multigrid metod wit te generalized minimum residual (GMRES) metod [27]. Tis combination appears to result in an efficient and robust metod for solving te momentum and incompressibility equations. It scales nearly linearly wit te number of unknowns and sows little dependence on te mixture of te two pases and on te parameters. In addition to te momentum and incompressibility equations, models for gel dynamics include equations for te transport of te network and solvent. For te

3 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 3 transport equations we use a conservative finite-volume discretization in wic te advection term is treated explicitly wit first-order upwinding and te diffusion term is andled implicitly wit backward Euler. Since te coupled system of transport and momentum equations is nonlinear, additional constraints on te time-step beyond te Courant-Friedrics-Lewy (CFL) condition are needed. Using linear stability analysis, we derive stability constraints tat also appear to old in te full nonlinear problem. Te rest of te paper is organized as follows. In Section 2, te model problem for describing and testing te new computational metod is introduced. Section 3 describes te discretization (in space and time) for te 1-D model. Tis is followed by a derivation of te stability restriction on te time step in Section 4. In Section 5, te multigrid box-relaxation sceme is introduced followed by a description of ow it is used as a preconditioner for GMRES. Numerical results for te 1-D model problem are presented in Section 6. We compare te performance of te multigrid tecnique as a solver and as a preconditioner, and investigate te spectra of te operators to understand te convergence of bot metods. In Section 7, extensions of te metod to te 2-D model problem are discussed and numerical results presented. Te paper concludes wit some remarks on parallelization and oter Krylov tecniques for solving te momentum equations. 2. Model Problem. Te model problem we consider is for a gel composed of two immiscible materials, a polymer network and a fluid solvent. We assume tat te total amount of gel remains constant and tat te transport of te network and solvent is governed by te equations: θ n t + (θn u n ) = (2.1) θ s t + (θs u s ) =, (2.2) were θ n, and θ s = 1 θ n are te respective volume fractions of te network and solvent, < θ n < 1, and u n and u s are te respective transport velocities. Adding tese two equations, and using θ n +θ s = 1 gives te incompressibility-type constraint (θ n u n + θ s u s ) =. (2.3) Te transport velocities are determined by conservation of momentum. We assume te network acts as a constant density viscous material and te solvent acts as a Newtonian fluid of muc less viscosity. Viscous terms are assumed to dominate so tat inertial terms are negligible (similar to zero-reynolds number flow), i.e. te system responds instantaneously to applied forces. Te network and solvent are eac subject to a number of intrapase stresses. We assume tat te viscous stress tensors σ n and σ s for te network and solvent are proportional to te respective gradient of te network and solvent velocities, i.e. σ n = µ n ( u n + u nt) + λ n δ ij u n, (2.4) σ s = µ s ( u s + u st) + λ s δ ij u s, (2.5) were µ n,s are sear viscosities and λ n,s + 2µ n,s /d are te bulk viscosities of te network and solvent (d is te dimension). Te network and solvent are also subject to a frictional drag since te motion of te solvent influences te network. We model tis by βθ n θ s, were β > is te drag coefficient. Te tird force on eac pase is

4 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 4 due to ydrostatic pressure. Since te polymer is assumed to be cemically active witin te gel and te solvent is assumed to be cemically neutral, te final force is generated by a cemical pressure, Ψ(θ n ), and acts only on te network. Te form of Ψ(θ n ) used in tis paper is described below. Balancing te above forces on te network and solvent yields te equations: (θ n σ n ) βθ n θ s (u n u s ) θ n p = Ψ(θ n ), (2.6) (θ s σ s ) βθ n θ s (u s u n ) θ s p =, (2.7) were p is te ydrostatic pressure. Tese two equations, combined wit (2.1) and (2.3), and subject to suitable boundary conditions, govern te gel dynamics. For our test problem we assume no-slip boundary conditions for te network and solvent velocities, and no-flux boundary conditions for θ n and θ s. It is pysically reasonable to assume tere is a small amount of diffusion between te network and solvent in te gel. It also matematically and computationally useful since it means te transition between te network and solvent will be smoot even in areas were te two pases ave separated. Tis additional assumption canges (2.1) and (2.2) to θ n t + (θn u n ) = κ 2 θ n, (2.8) θ s t + (θs u s ) = κ 2 θ s, (2.9) were κ is te diffusion constant. Note tat tis modification does not cange te volume conservation constraint (2.3). Te cemical pressure includes osmotic pressure, but it may also include active, contractile stresses suc as in te actomyosin gels of cytoplasm. In tis paper we are not concerned wit te origins of te cemical stress, and refer to it simply as te osmotic pressure. We assume te osmotic pressure is of te form Ψ(θ n ) = γθ n (θ n θ n )(θ n θ n ), (2.1) were γ >, < θ n < θ n < 1; see Figure 2.1 for a plot of Ψ(θ n ) wit parameters used in te numerical experiments tat follow. Tis functional form is cosen since it can produce pase separation, or canneling. In regions of space were Ψ (θ n ) > te mixture is stable, but were Ψ (θ n ) < te mixture tends to pase separate; see Cogan and Keener [9] for an analysis. Tis form of Ψ does not allow te pases to separate completely. Pase separation is generally observed in gels [29], and is vital for locomotion and for transporting nutrients in some amoeboid cells [1, 18] One dimension. We describe and do te primary numerical investigations of our proposed computational metodology for te model problem in one dimension. Letting te spatial domain be x 1, te transport equation is θ n t + (θn u n ) x = κθ n xx, (2.11) wile te momentum equations and incompressibility constraint are α n (θ n u n x) x βθ n θ s (u n v n ) θ n p x = Ψ (θ n )θ n x, (2.12) α s (θ s u s x ) x + βθ n θ s (u n v n ) θ s p x =, (2.13) (θ n u n + θ s u s ) x =, (2.14)

5 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 5 Ψ(θ) θ (a) Ψ(θ) x θ (b) Fig (a) Osmotic pressure function (2.1) from te model problem wit parameters γ = 5, θ n =.1 and θn =.15 used in te numerical experiments. (b) A detailed plot of tis function near te origin. were α n,s = 2µ n,s + λ n,s. We can alternatively express te system in matrix-vector form as α n x (θ n x ) βθ n θ s βθ n θ s βθ n θ s α s x (θ s x ) βθ n θ s θ n x θ s x } x θ n {{ x θ s } A un v n = p Ψ x θ n. (2.15) Te negative sign in te volume-averaged incompressibility constraint is included to impose a useful restriction on te eigenvalues of te discrete approximation to te operator A as discussed in Section 3. No-slip boundary conditions are used for (2.15), wile no-flux conditions are used for (2.11). Finally, some initial network volume fraction θ n is provided. Note tat te above system could be reduced quite significantly by analytically integrating (2.14) to obtain a relationsip for u n in terms of u s. Indeed, tis is exactly wat is done in [3, 9]. However, since we are ultimately interested in developing computational tecniques for solving te model in two (and tree) dimensions for wic tis analytical reduction would not be possible (see Section 7.1), we focus on te full system (2.15) An example. Before going into detail on te computational tecniques for solving (2.11) (2.14), we first give an example illustrating te interesting pattern formations tat can occur wit tis model. Tis also allows us to illuminate some potential computational issues tat arise. Note tat te simulation results tat follow are generated using our proposed metod. We start wit an initial distribution of network θ n tat is perturbed about te unstable region of te osmotic pressure term Ψ (cf. Figure 2.1): θ n (x) =.5(1 +.1 cos(2πcx)), (2.16) were c = 3. We set te network viscosity α n =.1, te solvent viscosity α s =.1, te frictional coupling parameter β = 1, and te diffusion coefficient κ = 1 7. Figure 2.2 displays snapsots of te network volume fraction and velocity u n at various times

6 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 6 corresponding to different stages of pase separation. Clearly as time increases, te network and solvent pases separate and tree cannels wit sarp edges form. Te system of equations for computing te network/solvent velocity and te pressure contains variable-coefficient Laplacians (see (2.12) and 2.13)), as well as a variable-coefficient divergence equation (see (2.14)). Te coefficients in tese cases are te volume fractions θ n and θ s, and as we can see from Figure 2.2, tese can vary strongly in some regions. Tus, any algoritm for solving tis system must be able to andle variable coefficients effectively. 3. Discretization of te 1-D model. Te basic strategy we propose for simulating te 1-D gel model (2.11) (2.14) is as follows: 1. For a given θ n at time t, solve a discrete analog of (2.15) for u n, u s, and p at time t. 2. Solve for θ n (and tus θ s ) at time t + t using a discrete analog of (2.11) wit te value of u n at time t. 3. Repeat step 1, wit te θ n at t + t Spatial discretization. We use a MAC grid [16] for te unknowns. Values of te network and solvent velocities u n, u s reside at te cell edges {x i+ 1 }, wile values 2 for te network and solvent volume fractions θ n, θ s, and te pressure p reside at cell centers {x i }; see Figure 3.1. Here x i+ 1 = i, i =,...,N, and x i = (i ), i = 1,...,N, were N = 1. All equations in (2.15) are discretized using secondorder, centered finite differences. Te MAC grid is used for several reasons it allows for a natural approximation to te variable-coefficient Laplacians appearing in (2.12) and (2.13); it requires no artificial boundary condition on te pressure; it is favorable for volume-conserving time integration. However, since θ n and θ s appear in te frictional coupling term, te gradient of te pressure, and te incompressibility-type constraint in (2.12) (2.14), we need values of θ n and θ s at cell edges. To obtain tese we use simple two-point aritmetic averaging. Te respective approximations for (2.12) and (2.13) at x = x i+ 1, i = 1, 2,...N 2 1, are given by were α n 2 [θn i+1(u n i+ u n 3 2 i+ ) θ 1 i n (u n 2 i+ u n 1 2 i )] βθ n 1 i+ 1θs 2 2 i+ 1[un 2 i+ u s 1 2 i+ ] 1 2 θ n i+ 1 2 [p i+1 p i ] = Ψ(θn i+1 ) Ψ(θn i ), (3.1) α s 2 [θs i+1(u s i+ u s 3 2 i+ ) θi(u s s 1 2 i+ u s 1 2 i )]+βθ n 1 i+ 1θs 2 2 i+ 1[un 2 i+ u s 1 2 i+ ] 1 2 θ s i+ 1 2 [p i+1 p i ] =, (3.2) θ n i+ 1 2 = θn i + θn i+1 2 and θ s i+ 1 2 = 1 θn i Note tat te no slip boundary conditions give u n 1/2 = un N+1/2 = us 1/2 = us N+1/2 =. Similarly, (2.14) is approximated at x i, i = 1,...,N, by θn i+ 1 2 un i+ + θn i un i θs i us i+ + θs i us i 1 2 =. (3.3)

7 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS Network volume fraction θ, time=. 2 x 1 4 Network velocity u n, time= x.15 Network volume fraction θ, time=13.78 Stage x 3 x 1 3 Network velocity u n, time= x.15 Network volume fraction θ, time=16.84 Stage x.1 Network velocity u n, time= x.15 Network volume fraction θ, time=22.92 Stage x.1 Network velocity u n, time= x.15 Network volume fraction θ, time=79.37 Stage x 1 x 1 3 Network velocity u n, time= x Stage x Fig Network volume fraction (left column) and network velocity (rigt column) from a numerical simulation of te 1-D model problem (2.11) and (2.15) at various stages of gel pase separation. Note te fixed scale on te θ n plots and te variable scale on te u n plots.

8 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 8 u n i 1/2 u s i 1/2 θ i u n i+1/2 u s i+1/2 p i+1 p i θ i+1 u n i+3/2 u s i+3/2 Fig Location of te unknowns in te MAC grid for te 1-D gel model. θ ere represents bot te network and solvent volume fractions. We can represent te (3N 2)-by-(3N 2) system of equations (3.1) (3.3) in matrix-vector form as L n C C G n C L s C Gs un u s = Ψ x θ n, (3.4) Gn T Gs T p }{{} A wic forms our discrete approximation of (2.15); see Figure 3.2 for an illustration of te structure of A. Since Gn and G s anniilate constant vectors, A as at least one zero eigenvalue. More can be said about te spectrum of tis saddle point system by rewriting A as [ ] A A B = B T. (3.5) Since we assume < θ n < 1, A is symmetric and negative definite and B as rank N 1. It follows, for example, from [4, Teorem 3.6] tat te eigenvalues of A ave nonpositive real part (A is negative semistable), i.e. Re(λ) for all λ σ(a ). Tis can be advantageous for (preconditioned) Krylov subspace metods [4], wic are te basis for te metod we develop to solve (3.4) nz = 236 Fig Structure of te A matrix in (3.4) for te case of N = 2 4 grid cells; solid dots mark non-zero entries in te matrix, wile solid orizontal and vertical lines igligt te structure sown in (3.4) Temporal discretization. To advance te system in time according to (2.11), we use explicit first-order upwinding for te flux (θ n u n ) x and treat te diffusion implicitly wit backward Euler. Te solvent volume fraction is updated according to te relation sip θ s = 1 θ n. We use adaptive time-stepping for efficiency and stability as discussed next.

9 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 9 4. Stable time stepping. We postpone te discussion of a metod for solving (3.4) for u n, v n, and p until te next section, and focus ere on analyzing te stability of te upwind time stepping sceme discussed in te previous section. For a given u n and κ =, (2.11) looks like a linear, variable-coefficient advection equation. Te CFL condition for upwinding tis equation would tus be t max 1 i N 1 un i+1/2. However, because (2.11) depends nonlinearly on u n, tis is not a sufficient restriction on t for stability. To get a better idea of ow t must scale for stability we do a linear stability analysis Linear stability analysis. A spatially uniform network volume fraction and zero network and solvent velocities is a solution to (3.1) (3.3). We perturb about tis state and assume θ n (x, t) = θ n, + ǫθ n,1 (x, t) + O(ǫ 2 ), u n (x, t) = + ǫu n,1 (x, t) + O(ǫ 2 ), u s (x, t) = + ǫu s,1 (x, t) + O(ǫ 2 ), p(x, t) = p + ǫp 1 (x, t) + O(ǫ 2 ). were < θ n, < 1 and x 1; also recall θ s = 1 θ n. Since p is unique up to a constant, we set p =. Plugging te above expansions into (2.12) (2.14) and looking at te O(ǫ) terms gives te linearized equations α n θ n, xx (u n,1 ) βθ n, θ s, (u n,1 u s,1 ) θ n, x (p 1 ) = Ψ (θ n, ) x (θ n,1 ), (4.1) α s θ s, xx (u s,1 ) + βθ n, θ s, (u n,1 u s,1 ) θ s, x (p 1 ) =, (4.2) θ n, x (u n,1 ) θ s, x (u s,1 ) =. (4.3) We discretize tese equations using finite-differences on a MAC grid as discussed in te previous section and let x and xx denote te second-order accurate, centered approximation to x and xx. We assume an O(ǫ) perturbation to te θ n initial condition of form θ n,1 = cos(ωπx), were ω is an integer, and assume Ψ (θ n, ) >, so tat te gel is stable. To solve te discrete form of te equations (4.1) (4.3) for tis θ n,1, we first exploit (4.3) to get te following analytical expression relating u s,1 to u n,1 : u s,1 = θn, θ s, un,1. (4.4) Tis allows us to use te discrete forms of te two equations (4.1) and (4.2) to determine u n,1 and p 1. Second, we plug θ n,1 in te rigt and side of te discrete approximation to (4.1) to obtain ( ωπ Ψ (θ n, ) x ( θ n,1 ) x=x i+1/2 = 2Ψ (θ n, ) sin Tird, we note tat te discrete operator xx satisfies [ xx (sin(ωπx)) ] x=x i+1/2 = 4 2 sin2 ( ωπ 2 2 ) [sin(ωπx)] x=xi+1/2. ) [sin(ωπx)] x=xi+1/2.

10 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 1 Tese tree results suggest solutions of te form u n,1 = B 1 sin(ωπx) and p 1 = B 2 cos(ωπx), for some constants B 1 and B 2. To determine tese constants, and tus u n,1 and p 1, we plug te suggested solutions for u n,1 and p 1 into te discrete forms of (4.1) and (4.2) and solve for B 1 and B 2. We only present te final result for u n,1 since tis is all we need to know for te stability estimate: u n,1 = 2 sin( ) ωπ 2 Ψ (θ n, ) sin(ωπx), (4.5) ζ were [ ζ = θn, θ s, β + 4 ( ) ] ωπ 2 sin2 (α n θ s, + α s θ n, ). 2 Now tat we ave computed u n,1, we can determine ow θ n is updated in time. Since tere are no O(1) terms in te expansions for u n, u s, and p, te O(ǫ) term in te expansion for θ n is updated according to te linearized equation t (θ n,1 ) + θ n, x (u n,1 ) = κ xx (θ n,1 ). Discretizing tis equation according to te metod discussed in Section 4 yields θ n,1 (x i, t + t l ) θ n,1 (x i, t) + θ n, [ t x(u n,1 ) ] x=x l i = κ [ xx(θ n,1 (x i, t + t l )) ] x=x i. (4.6) Using te O(ǫ) perturbation θ n,1 = cos(ωπx) as te initial condition for θ n and te corresponding O(ǫ) solution u n,1 from (4.5) in (4.6), we obtain te tridiagonal system κ t ( l 2 θn,1 (x i 1, t + t l ) κ t ) l 2 θ n,1 (x i, t + t l ) κ t l 2 θn,1 (x i+1, t + t l ) = [ 4 sin 2 ( ωπ 2 Ψ (θ n, )θ n, 1 t l 2 ζ ) ] cos(ωπx i ), for wic te solution is θ n,1 1 Ψ (x i, t + t l ) = 1 (θ n, )θ s, 1 + 4κ t t l l cos(ωπx α 2 n θ s, + α s θ n, + 2 β i ) 4sin 2 ( ωπ 2 ) = A ω cos(ωπx i ). Te linearized sceme is stable if A ω < 1. To bound t l so tat tis restriction on A ω is satisfied, we assume ω 1 (i.e. te frequency of te perturbation is ig) so tat sin 2 ( ) ωπ 2 1. Also, since we are interested in very small diffusion coefficients, we set κ =. Tese assumptions lead to te time-step bound [ ] 2 α n θ s, + α s θ n, + 2 β 4 < t l < Ψ (θ n, )θ s,, wic is sufficient for te stability of te linearized sceme.

11 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS Extension to te nonlinear equations. For te full nonlinear system of equations (2.11) (2.14), we assume te perturbation is about some initial state at eac discretization point and tat te above linear analysis applies locally. Tis gives an adaptive time stepping restriction as follows: [ ] < t l < min 2 α n θi s + α sθi n + 2 β 4 1 i N Ψ (θi n, (4.7) ) θs i were θi n and θs i indicate te respective approximate values of θn and θ s at te i t cell center and current time (i.e. θi n θn (x i, t j ), 1 i N, j ). For te nonlinear case, te restriction (4.7) may not be sufficient since it may still allow a time step large enoug to violate te CFL condition. To prevent tis problem, we coose te time step according to te rule [ ] t = min η l min 2 α n θi s + α sθi n + 2 β [ ] 4,η 1 i N Ψ (θi n c min ) θs i 1 i N 1 u n i+1/2, (4.8) were u n i+1/2 is te approximate network velocity at te rigt edge of te it cell center and < η l, η c < 1. Wile te linear stability analysis allows for η l = η c = 1, for te full nonlinear case we find in practice tat coosing tese values sligtly less tan 1 results in a robust adaptive sceme. For example, in all experiments tat follow, as well as te one from Figure 2.2, we use η l = η c = Solving te coupled momentum and incompressibility equations. As mentioned in te introduction, te coupled momentum and incompressibility equations (2.15) are similar to Stokes equations. However, te coupling of te two fluids (given by te βθ n θ s term), te time-dependent, strongly varying variable-coefficient Laplacian terms, and te volume-averaged incompressibility equation make numerically solving te equations more complex. In tis section, we introduce a particularly effective metod for solving te discrete form of tis system (3.4) tat is based, in part, on a multigrid tecnique first proposed by Vanka [33] for Stokes and Navier-Stokes. We first discuss te extension of tis tecnique (wic we call box relaxation [31, p.32]) to te gel system. We follow tis wit a discussion of ow to make it more efficient and robust by using it as a preconditioner for te GMRES Multigrid box relaxation. Any geometric multigrid metod is caracterized by te following five components: te grid coarsening, te transfer operators (restriction and prolongation) for te unknowns, te coarse grid discretization, te smooter, and te grid cycling. For our proposed metod, te smooter component is te only one tat is not standard, so we discuss tis in detail and only briefly discuss te oter components. For a more detailed description of multigrid in general, we refer te reader to te many excellent books on te subject (e.g. [6, 31, 35]). Grid coarsening and transfer operators. For a given value of N = 2 k, a sequence of coarser grids Ω N, Ω N 1,..., Ω M (N > M) is defined were eac grid is a factor of two coarser tan te previous; see Figure 5.1 for an illustration. Since we are using a staggered-discretization, te transfer operators for te defects of te two velocity components u n, u s will be different from tat for te defect for te pressure p. For te velocity components we use tree-point, full-weigting for restriction, wile for te pressure we use two-point aritmetic averaging [31, p.69]. For prolongation, we

12 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 12 use linear interpolation for te two velocities and te pressure [31, p.7]. Because θ n and θ s are variable coefficients, it is also necessary to transfer tese components from te fine to te coarse grid, but not te reverse since tese are not unknowns of te system. For tis transfer we use two point aritmetic averaging. Level : Level 2: Fig Example of coarsening an N = 2 3 staggered grid to an M = 2 2 grid. Network and solvent velocities u n, u s reside at cell edges, wile pressure p and volume fractions θ n, θ s reside at te cell centers. Coarse grid discretization. We use te simple DCGA (discretization coarse grid approximation) strategy, were te equations are directly discretized on te coarser grids. Smooter. Since tere is a zero diagonal block in te (3, 3) entry of te matrix in (3.4), standard decoupled relaxation scemes (e.g. Jacobi or Gauss-Siedel) cannot be directly applied. Instead, following te work of Vanka [33] we use a collective (box) relaxation sceme. Te basic idea of box relaxation is to compute te network velocity u n, solvent velocity u s, and pressure p (or more accurately corrections to tese values) by collectively solving te discrete gel equations (3.4) locally computational cell (or box) by cell (see Figure 3.1). Tus for eac box, a 5-by-5 system of equations must be solved. Letting te superscript k denote te iteration of te relaxation sceme, te linear system for solving for te five unknowns in box i (2 i N 1 ) at te k+1 iteration is given by: L n C C G n (u n C L s C G i )k+1 b u n s (u s i i )k+1 = b u s T T p k+1 i, (5.1) i b pi G n G s were [ ] [ ] (u (u n i ) k+1 n = i 1/2 ) k+1 (u, and (u s i) k+1 s = i 1/2 ) k+1 (u n i+1/2 )k+1 (u s i+1/2 )k+1 are te unknown network and solvent velocities at te edges of box i and L n = α [ n (θ n i + θi 1 n ) ] θn i 2 θi n (θi+1 n + θn i ), L s = α [ s (θ s i + θi 1 s ) ] θs i 2 θi s (θi+1 s + θs i ), C = β 2 [ (θ n i θi s + θn i 1 θs i 1 ) ] (θi+1 n θs i+1 + θn i θs i ), G n = 1 [ ] θ n i + θi 1 n 2 (θi+1 n + θn i ), G s = 1 [ ] θ s i + θi 1 s 2 (θi+1 s + θs i ),

13 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 13 b u n i = [ Ψ(θ n i ) Ψ(θi 1 n ) αn Ψ(θi+1 n ) Ψ(θn i ) 2 p k+1 i 1 θ n 2 i 1 (un i 3/2 )k+1 θn i +θn i 1 αn θ n 2 i+1 (un i+3/2 )k + θn i+1 +θn i 2 p k i+1 [ ] α s θ s b u s i = 2 i 1 (us i 3/2 )k+1 θs i +θs i 1 2 p k+1 i 1, b αs θ s 2 i+1 (us i+3/2 )k + θs i+1 +θs i 2 p k pi =. i+1 Note, for i = 1 and i = N, only tree equations need to be solved because of te no-slip boundary conditions. Te above procedure updates te unknowns box-by-box in a Gauss-Seidel-type manner. For all te numerical results, we process te blocks using red-black ordering. In te context of Stokes and Navier-Stokes equations, it is sometimes necessary to combine te box sceme wit underrelaxation to acieve better smooting properties (cf. [31, p.321]). We also find tis to be te case for te gel system, for wic te relaxation sceme takes te form (u n i )k+1 (u s i )k+1 p k+1 i = (1 ω) (un i )k L n C C G n (u s i )k + ω C L s C G s p k T T i G n G s ], 1 b u n i b u s i b pi, (5.2) were < ω < 1. Multigrid cycles. We use two different strategies for cycling troug te grids: te V- and F-cycles, see Figure 5.2 for an illustration. Te V-cycle is ceaper and simpler, wile te F-cycle is more expensive, but also more robust [23]. Letting ν 1 and ν 2 be te number of pre- and post-smooting operations for eac grid, we denote tese respective metods as V (ν 1, ν 2 ) and F(ν 1, ν 2 ). In all of our numerical results, we coarsen te grid until tere are two cell centers, i.e. N = 2, at wic point we solve te system (3.4) directly. In tis case, te total computational work for one V-cycle is W V = 4CN 8C, wile for te F-cycle it is W F = 8CN 8C(log 2 (N)+1), for some constant C independent of N. Tus, te computational complexity of bot cycles is O(N), wit te cost of an F-cycle about twice tat of a V-cycle in 1-D. finest coarsest V-cycle F-cycle Fig Illustration of te grid cycling strategies used in te multigrid metod in te case of five-grids. Te above multigrid procedure as te benefit of being relatively straigtforward to implement. However, as sown in te numerical experiments in te next section,

14 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 14 it may converge slowly or even fail to converge wen using it as a standalone solver for te gel system (3.4), especially as pase separation increases, i.e. as te variation in te variable coefficient increases. More sopisticated multigrid components like operator-dependent interpolation/restriction [31, p.272], Galerkin coarse-grid correction [31, p.273], and distributive relaxation [5] could potentially be developed to improve te convergence and obtain an O(N) solver. However, a muc simpler idea is to use te multigrid metod above as a preconditioner for a Krylov subspace metod. Te idea of combining multigrid wit Krylov metods as been successfully used for complex systems of linear and non-linear equations (e.g., Trottenberg et al. [31, 7.8], [23, 24, 25, 34]). Wile -independence convergence may not be acieved, robustness in te range of systems and parameters is typically gained, and te metods are easier to implement Multigrid box-relaxation as a preconditioner for GMRES. We propose using te multigrid procedure described above as a rigt-preconditioner for GMRES(m) [27] for solving te system (3.4). Tis Krylov metod is applicable to non-symmetric matrices and non-positive definite preconditioners, bot of wic are found for te gel system. Te parameter m represents te maximum dimension te Krylov subspace can reac before te algoritm is restarted. All of our results are for m = 2. Following te description of GMRES(m) in [26], tis metod requires one matrix-vector product wit A in (3.4) and one preconditioning step per iteration. Upon termination of te algoritm, or wen a restart occurs, one extra preconditioning step is required. If we let y T = [(u n ) T (u s ) T p T ] and f be te rigt and side of (3.4), te preconditioned system of equations to solve wit GMRES(m) is given by A M z = f, (5.3) were z = (M ) 1 y, and M represents an application of te multigrid box relaxation sceme. Letting y () be some initial guess to te solution of (3.4), r (l) = f A M z (l), and defining ( K l = span r (), A M r (),..., ( A M ) ) l 1 r (), l m, GMRES(m) finds a z (l) z () + K l suc tat r (l) 2 is minimal. Tis minimization can alternatively be expressed as { r (l) ql ( 2 = min A M ) } r () 2 ql is a poly. of degree l and q l () = 1. (5.4) Since y (l) = M z (l), minimizing r (l) 2 is equivalent to minimizing f A y (l) 2 (i.e., te standard residual); see, for example, [26, p. 272]. Te spectrum of te preconditioned matrix is key to understanding te convergence beavior of tis Krylov metod (see, for example, [23]). To calculate te spectrum we need a way to construct te operator M. Tis can be done using te procedure outlined in Trottenberg et al. [31, p. 28]. Briefly, te idea is to set f = wit zero boundary conditions and perform one multigrid cycle (V or F, in our case) for eac unit vector as an initial approximation to te system (3.4). Te resulting solution from te first standard basis vector initial guess becomes te first column of te iteration matrix, te solution for te second standard basis vector becomes te second column, and so fort.

15 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 15 It turns out tat knowing te spectrum of A M gives us information about te spectrum of te multigrid procedure as a standalone metod. As sown in [26], one iteration of te multigrid procedure can be written in te form y (m+1) = (I (B ) 1 A )y (m) + (B ) 1 f, (5.5) were B is determined by te multigrid components defined above. Applying tis once wit zero initial guess gives y (1) = (B ) 1 f. Tis is exactly wat we do in applying multigrid once as a preconditioner, so M = (B ) 1. It follows tat te multigrid iteration matrix (I (B ) 1 A ) can also be written (I M A ). Since M A and A M ave te same spectra, σ(i M A ) = 1 σ(a M ). For well-beaved multigrid problems, te spectrum of te multigrid iteration matrix will be clustered around zero and te spectral radius will be small making te solver very efficient [34]. For more difficult problems, it as been often found tat most of te eigenvalues of te multigrid iteration matrix are clustered around zero, wit a few outliers wic can slow convergence or even cause te metod to diverge [23]. For tis situation, combining multigrid wit GMRES can be very effective. A full analysis on ow GMRES(m) eliminates tese outliers and te resulting bounds on te residual reduction are given by Oosterlee and Wasio [23, 34]. To understand te key idea of teir analysis, suppose tat A M as isolated eigenvalues λ 1,...,λ j, j < l far from 1 wit te remaining eigenvalues contained in a small disk centered at 1. Consider te polynomial q l (x) = (1 x) l j j i=1 λ i x λ i. wic is in te class of polynomials defined in (5.4). Now applying q l (A M ) to an eigenvector of A M corresponding to one of te eigenvalues λ k for k > j, we see tat te factor (I A M ) l j in q l (A M ) substantially reduces te magnitude of tis eigenvector. On te and, applying ( λ i A M ) /λ i to an eigenvector tat corresponds to an eigenvalue λ k wit k j, causes a similar reduction. Since q l bounds te action of te true minimizing polynomial selected by GMRES, tis suggests tat tis is an effective metod for operators wit spectra of tis form. As noted in [23, Remark 2.1.3], te asymptotic reduction of te residual of te multigrid preconditioned GMRES(m) for a reasonable initial guess is at least as fast as for te multigrid solver wit an initial guess for wic te residual does not contain any components corresponding to te outlying eigenvalues. Te spectral analysis of A M for our model problem is included in te next section. 6. Numerical results for te 1-D model. In te first two tests, we compare te multigrid box relaxation metod as a standalone solver wit GMRES(2), preconditioned by tis multigrid metod, for solving te momentum and incompressibility equations at various stages of gel pase separation. Te upwind sceme discussed in Section 4 for te transport equation (2.11) is used for all tests. We use te same setup for te model problem as given in Section 2.2 and illustrated in Figure 2.2. We

16 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 16 consider te system (3.4) at time t solved wen te residual of te l t iterate of te respective metod satisfies r (l) 2 f 2 1 6, (6.1) were f equals te rigt and side of (3.4). Unless oterwise specified, for all time steps but te initial, we use te previous timestep s values for te network and solvent velocities and pressure as initial guesses for te iterative metods. All metods were implemented in Matlab, and run on an AMD Opteron GHz processor under Matlab version 7.1 (R14). Te first test is to see ow well te momentum and incompressibility equations solvers scale as te grid is refined from = 2 7 to = 2 11 (recall N = 1/ is te number of cell-centers). Table 6.1 displays te results for te standalone multigrid V(1,1)-cycle (MGV) and F(1,1)-cycle (MGF) metods and te rigt-preconditioned versions of tese metods for GMRES(2) (GMRES-MGV and GMRES-MGF, respectively). Integer values listed in te table are te number of multigrid cycles (V(1,1) or F(1,1)) required to satisfy (6.1) using values from te previous time-step as te initial guess (non-bracket), and using all zeros as te initial guess (bracket). Decimal values in parentesis are te corresponding wall-clock times (in seconds) for te former initial guess metod. Since te computational cost of one preconditioning step of GMRES(2) by a V or F-cycle dwarfs te oter costs of te metod, we ave cosen to list te number of tese iterations for te GMRES(2) metod. Looking first at te MGV results, we see tat te computational cost is increasing faster tan linearly. However, wat is more problematic is tat te metod fails to converge for Stage 5 pase separation, and even for Stage 4 wen is fine enoug. Te MGF scaling results are muc better, sowing a linear growt in te computational cost. However, we again see tat, for fine enoug, te metod failed to converge for Stage 5 pase separation. Te results for GMRES-MGV and GMRES-MGF are muc more encouraging. Bot of tese metods converged for all stages of pase separation and teir computational cost scales very well. Te cost of GMRES-MGV appears to be increasing at a rate sligtly larger tan linear wit a decreasing slope as te pases become increasingly separated, wile te cost of GMRES-MGF is increasing linearly for all stages of separation. However, in terms of wall-clock time our implementation of GMRES-MGV is more efficient tan GMRES-MGF for almost all and stages of pase separation. Wile te teory from Section 5 predicts te F-cycle to be twice te cost of a V-cycle, we see tat in actual wall-clock time it is about tree times te cost in our implementation. Finally, we note tat te F-cycle, weter used as a standalone solver or a preconditioner, is muc less sensitive to te initial guess. Using te values for te pressure and two velocities from te previous time-step as te initial guess instead of all zeros drops te iterations for te V-cycle, but as little effect for te F-cycle. Figure 6.1 sows details on te convergence istory of te solvers at te various stages of pase separation sown in Figure 2.2 for te case of = 2 1. Comparing te MGV and GMRES-MGV results for Stages 1-3, we see tat te slope of te relative residual is muc steeper for te latter. Te MGV metod diverged for Stages 4-5, so no results are presented. For te MGF and GMRES-MGF metods, te slopes are nearly identical for Stages 1-3 and differ very sligtly for Stage 4. No MGF results are presented for Stage 5 since te metod diverged in tis case.

17 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 17 Stage = 2 7 = 2 8 = 2 9 = 2 1 = 2 11 MGV 1 7 [7] (.74) 9 [9] (.17) 11 [11] (.388) 14 [14] (.945) 18 [18] (2.379) 2 7 [7] (.74) 8 [9] (.152) 9 [11] (.318) 1 [14] (.678) 12 [18] (1.59) 3 7 [7] (.74) 7 [9] (.134) 8 [11] (.286) 9 [14] (.69) 11 [18] (1.45) 4 7 [8] (.74) 1 [13] (.189) 25 [34] (.88) DIV DIV 5 DIV DIV DIV DIV DIV MGF 1 3 [3] (.89) 3 [3] (.158) 3 [3] (.289) 2 [2] (.36) 2 [2] (.687) 2 3 [3] (.89) 3 [3] (.158) 3 [3] (.289) 3 [3] (.54) 3 [3] (1.31) 3 3 [3] (.89) 3 [3] (.158) 3 [3] (.289) 3 [3] (.54) 3 [3] (1.31) 4 3 [3] (.89) 3 [3] (.158) 3 [3] (.289) 3 [3] (.54) 2 [3] (.687) 5 3 [5] (.89) 5 [8] (.262) DIV DIV DIV GMRES-MGV 1 7 [7] (.75) 8 [8] (.153) 9 [9] (.321) 1 [1] (.688) 11 [11] (1.477) 2 7 [7] (.75) 7 [8] (.135) 7 [9] (.25) 8 [1] (.551) 8 [11] (1.65) 3 7 [7] (.75) 7 [8] (.135) 7 [9] (.25) 7 [1] (.483) 8 [11] (1.65) 4 6 [8] (.65) 7 [8] (.135) 7 [9] (.25) 8 [1] (.551) 8 [11] (1.65) 5 6 [8] (.65) 6 [8] (.114) 7 [9] (.25) 7 [9] (.483) 8 [1] (1.65) GMRES-MGF 1 4 [4] (.119) 4 [4] (.212) 4 [4] (.386) 3 [3] (.542) 3 [3] (1.37) 2 4 [4] (.119) 4 [4] (.212) 4 [4] (.386) 4 [4] (.724) 4 [4] (1.384) 3 4 [4] (.119) 4 [4] (.212) 4 [4] (.386) 4 [4] (.724) 4 [4] (1.384) 4 4 [4] (.119) 4 [4] (.212) 4 [4] (.386) 4 [4] (.724) 3 [4] (1.37) 5 4 [5] (.119) 4 [6] (.212) 5 [5] (.479) 6 [6] (1.82) 6 [6] (2.65) Table 6.1 Comparison of te multigrid V-cycle metod (MGV), multigrid F-cycle (MGF), GMRES(2) preconditioned wit MGV, and GMRES(2) preconditioned wit MGF for solving te system (3.4) at te various stages of pase separation sown in Figure 2.2. Integer values are te number of V(1,1)- or F(1,1)-cycles required to satisfy (6.1) wen using values from te previous time-step as te initial guess (non-bracket) and using all zeros as te initial guess (bracket). Decimal values in parentesis are te wall clock times (in seconds) for te metods using former of te initial guess metods. DIV means tat te metod diverged. Te underrelaxation was set to ω =.675 for all results Number of cannels. In te next set of experiments, we test ow well te metod scales as te number of cannels tat form in te gel increases. Tis is accomplised by canging c to te number of desired cannels in te initial perturbation to te gel given by (2.16). Te results for = 2 1 are presented in Table 6.2. Te stages of pase separation for different c are similar to wat is sown in Figure 2.2, wic is for c = 3. Te table clearly sows tat te MGV standalone solver depends quite significantly on te number of cannels, wile te MGF solver as less of a dependence for te lower stages of pase separation and numbers of cannels. However, we see tat MGF fails to converge at Stage 4 for 6 or more cannels. Te GMRES-MGV and GMRES-MGF metods are muc less sensitive to te number of cannels for Stages 1-4 of pase separation and ave a very tolerable increase for Stage 5. It is also interesting tat te number of cycles for te GMRES-MGV and GMRES-MGF seem to approac one anoter for Stage 5 as te number of cannels increases Spectrum analysis of te preconditioned operator. To get a better understanding of te convergence of te multigrid solver and preconditioner, we re-

18 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS Stage 1 MGV MGF GMRES MGV GMRES MGF Stage 2 MGV MGF GMRES MGV GMRES MGF r (l) 2/ r () r (l) 2/ r () iterations (l) iterations (l) Stage 3 MGV MGF GMRES MGV GMRES MGF Stage 4 MGF GMRES MGV GMRES MGF r (l) 2/ r () r (l) 2/ r () iterations (l) iterations (l) Stage 5 GMRES MGV GMRES MGF r (l) 2/ r () iterations (l) Fig Convergence istory of te MGV and MGF standalone solvers and te preconditioned GMRES-MGV and GMRES-MGF solvers for (3.4) at te various stages of pase separation sown in Figure 2.2. Missing results for MGV and MGF are because tese metods diverged for tat particular stage. Te mes spacing is given by = 2 1 and te underrelaxation was set to ω =.675 for all results. turn to te tree cannel example in Figure 2.2 and analyze te spectrum of te preconditioned operator A M in (5.3). Te eigenvalues of A M for te multigrid V- and F-cycle for te various stages of pase separation are sown in Figure 6.2 (a) and (b), respectively. As discussed in Section 5.2, te spectrum of te multigrid iteration matrix in (5.5) is simply 1 σ ( A M ), so we can also easily obtain tis spectrum from Figure 6.2. Te majority of te eigenvalues of A M are clustered

19 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 19.4 Stage 1.15 Stage Im(λ) Im(λ) Re(λ) Re(λ).4 Stage 2.15 Stage Im(λ) Im(λ) Re(λ) Re(λ).4 Stage 3.15 Stage Im(λ) Im(λ) Re(λ) Re(λ).4 Stage 4.15 Stage Im(λ) Im(λ) Re(λ) Re(λ).4 Stage 5.15 Stage Im(λ) Im(λ) Re(λ) (a) Re(λ) (b) Fig Eigenvalues of multigrid box-relaxation preconditioned matrix A M at te five stages of pase separation sown in Figure 2.2 using te (a) V(1,1) and (b) F(1,1) cycles. Note te different scales on te plots in te (a) and (b) column as well as te different scale for Stage 5. All results are for a mes-spacing of = 2 1 and an underrelaxation of ω =.675.

20 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 2 Stage c = 1 c = 2 c = 3 c = 4 c = 5 c = 6 c = 7 MGV DIV DIV DIV DIV DIV 5 39 DIV DIV DIV DIV DIV DIV MGF DIV DIV 5 3 DIV DIV DIV DIV DIV DIV GMRES-MGV GMRES-MGF Table 6.2 Comparison of te four solution metods as te number of cannels, c, tat form in te gel is increased. Te stages of pase separation are similar to tose in Figure 2.2. Integer values are te number of V(1,1)- or F(1,1)-cycles required to satisfy (6.1). DIV means te metod did not converge. All results are for = 2 1 and an underrelaxation of ω =.675. very close to λ = 1 for Stages 1-3 wit te V-cycle preconditioner wit outliers tat are still close to λ = 1. Te corresponding spectrum of te multigrid V-cycle iteration matrix is tus similarly clustered around λ =. Tese results explain te convergence rates in Table 6.1 and Figure 6.1 for te V-cycle solver and preconditioner. At Stages 4 and 5, we see tat te outlying eigenvalues become increasingly far from λ = 1. For te V-cycle solver, tis means te spectral radius is increasing and for Stages 4 and 5 we see tat it increases beyond 1, wic explains te failure to converge sown in Table 6.1. However, for te V-cycle preconditioner, we see tat tese outliers ave little effect on te convergence of te preconditioned GMRES(2) metod since te majority of te eigenvalues remain clustered at λ = 1. Te spectrum for te F-cycle in Figure 6.2 (b) is even more clustered near λ = 1 and canges very little for Stages 1-4. Tis explains te very rapid convergence of tis metod as bot a solver and preconditioner for tese stages as sown in Table 6.1 and Figure 6.1. However, we see tat at Stage 5 two outlying eigenvalues ave appeared wic are a distance more tan 1 from λ = 1; tis explains te failure of te F-cycle solver for tis stage. Again, owever, tis as little effect on te preconditioned GMRES(2) metod as discussed in Section 5.2. Te results displayed so far ave sown tat for te V-cycle, it is almost always more efficient to use it as a preconditioner tan as a solver, regardless of gel separation.

21 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 21 Te same is not true of te F-cycle since it appears to perform very efficiently as a standalone solver wen te gel mixture is still relatively omogeneous, and te number of cannels is not large. For tese cases, using it as a preconditioner for GMRES(2) typically takes one more F-cycle iteration due to te overead of te GMRES metod (see Section 5.2). However, since we want a robust metod for simulation, te remainder of te tests will only include results for te preconditioned V- and F-cycles wic ave yet to fail in our experiments. In actual practice, te entire simulation could be made efficient and robust by using a ybrid metod, were te V- or F-cycle preconditioners are only used wen needed Model parameters. In tis next experiment, we test te robustness of te preconditioned solvers as te parameters of te model are canged. We vary te network and solvent viscosity, α n and α s, respectively, wile olding te frictional coupling parameter β constant at 1. Te solvers are tested over five stages of pase separation similar to wat is sown in Figure 2.2 (for wic β = 1, α n =.1, and α s =.1). Te number of cannels is set to tree and = 2 1. Table 6.3 displays te number of V- and F-cycle iterations for te different parameters. Te results indicate te metod is quite robust over a wide range of parameters. Te only moderate increase in te number of iterations occurs in te last column for α n = 1 3 and α s = 1 6. It is less pronounced in te F-cycle tan te V-cycle. α n = 1 1 α n = 1 2 α n = 1 3 α s = α s = α s = α s = α s = α s = α s = α s = α s = Stage GMRES-MGV, β = GMRES-MGF, β = Table 6.3 Comparison of GMRES-MGV and GMRES-MGF metods for solving (3.4) for different model parameter values and different stages of pase separation. Integer values are te number of V(1,1)- or F(1,1)-cycles required to satisfy (6.1). Te stages of separation are similar to tose in Figure 2.2. All results are for = 2 1 and an underrelaxation of ω = Underrelaxation parameter. For all previous experiments, te underrelaxation parameter as been fixed at ω =.675. So in tis final experiment, we test ow te number of multigrid cycles for te GMRES-MGV and GMRES-MGF depends on ω. Te results are displayed in Figure 6.3 for te example sown in Figure 2.2. We see from te figure tat for bot te GMRES-MGV and GMRES-MGF metods, te optimal ω is nearly te same for all five stages of pase separation. Furtermore, te range of acceptable coices for ω is quite large and is similar between te V and F-cycles.

22 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 22 # MG iterations Stage 1 Stage 2 Stage 3 Stage 4 Stage ω Fig Comparison of te number of multigrid cycles need for te GMRES-MGV (black) and GMRES-MGF (gray) to solve (3.4) as te underrelaxation parameter ω is canged. Symbols represent te different stages of pase separation sown in Figure Extension to two dimensions. In two and iger dimensions te complexity of te gel dynamics model from Section 2 increases; owever, te extension of te computational metod is relatively straigtforward. We present ere an overview of te two-dimensional model problem, discuss te extension of te computational procedure, and present some numerical results Model problem. Let u n = (u n, v n ) T and u s = (u s, v s ) T, were u n, v n and u s, v s are te respective network and solvent velocity components in te x and y directions, and let te spatial domain be Ω = {(x, y) x a, y b}. Ten te momentum equations (2.6) (2.7) and volume-averaged incompressibility (2.3) are given in matrix-vector form as L n C C G n C L s C G s un u s = Ψ θ n, (7.1) Dn T Ds T p were L n,s = [ ] αn,s x (θ n,s x ) + µ n,s y (θ n,s y ) µ n,s y (θ n,s x ) + λ n,s x (θ n,s y ) µ n,s x (θ n,s y ) + λ n,s y (θ n,s x ) α n,s y (θ n,s y ) + µ n,s x (θ n,s, x ) C = [ ] [ ] [ ] βθ n θ s θ βθ n θ s, G n,s = n,s x x θ θ n,s, D n,s = n,s y y θ n,s, and α n,s = (2µ n,s + λ n,s ). Since θ s = 1 θ n, we need only an equation for te transport of θ n, wic is again given by (2.8). As in 1-D, we use no-slip boundary conditions for (7.1) and no-flux for (2.8). Finally, te cubic function (2.1) is again used for modeling te osmotic pressure Discretization. Te same computational strategy as outlined at te beginning of Section 3 is used for te 2-D model. For te spatial discretization we again use a MAC grid were te position of te unknowns are indicated in Figure 7.1. Te mes-spacing in te x and y direction is set equal and is given by.

23 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 23 j + 1 j j 1 i 1 i i + 1 Fig Location of te unknowns in te MAC grid for te 2-D gel model: =network/solvent orizontal velocity, =network/solvent vertical velocity, =pressure, and =network/solvent volume fractions. All equations in (7.1) are discretized using second-order, centered finite differences, wic leads to te following approximation of te first row of (7.1) at te interior point (x i+ 1,j, y 2 i+ 1,j): 2 α [ n 2 θi+1,j n (un i+ 3 2,j un i+ 1,j) θn i,j (un 2 i+ 1 2,j un i ]+ 1,j) 2 µ [ n 2 θ n i+ 1 2,j+ 1(un 2 i+ 1 2,j+1 un i+ 1,j) θn i ,j 1(un 2 i+ 1 2,j un i+ ]+ 1,j 1) 2 µ n v n i,j+ ) θ n 1 i ,j 1 (vn 2 i+1,j 1 2 [θ n 2 i+ 1 2,j+ 1(vn 2 i+1,j+ 1 2 λ n [θ n 2 i+1,j(v n i+1,j+ v n 1 2 i+1,j ) θi,j(v n n 1 2 i,j+ v n 1 2 i,j ) 1 2 βθ n i+ 1 2,jθs i+ 1 2,j(un i+ 1 2,j us i+ 1 2,j) θn i+ 1 2,j p i+1,j p i,j ] v n i,j + (7.2) 2) 1 ] = Ψ(θn i+1,j ) Ψ(θn i,j ), wile te approximation to te second row at te interior point (x i,j+ 1, y 2 i,j+ 1) is 2 given by: µ [ n 2 θ n i+ 1 2,j+ 1 (un 2 i+ 1 2,j+1 un i+ 1,j) θn i 1 2 2,j+ 1 (un 2 i 1 2,j+1 un i ]+ 1,j) 2 λ [ ] n 2 θi,j+1(u n n i+ 1 2,j+1 un i 1,j+1) θn i,j(u n 2 i+ 1 2,j un i 1,j) + 2 α n v n i,j+ ) θ 1 i,j n (vn 2 i,j+ v n 1 2 i,j 2) 1 ] + (7.3) ] [θ n 2 i,j+1 (vn i,j+ 3 2 µ n [θ n 2 i+ 1 2,j+ 1 (vn 2 i+1,j+ v n 1 2 i,j+ ) θ n 1 i 1 2 2,j+ 1(vn 2 i,j+ v n 1 2 i 1,j+ 2) 1 βθ n i,j+ 1 2 θs i,j+ 1 2 (vn i,j+ 1 2 v s i,j+ ) θ n 1 i,j p i,j+1 p i,j Bars over θ n and θ s represent aritmetic averages of te values of tese variables at nearest neigbor cells wit two-point averages wen tere is a mix of integer and alf-integer indices, and four-point averages wen tere are two alf-integer indices. Te discretization of te tird and fourt row of (7.1) are te same, but wit variables =.

24 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 24 for te network replaced accordingly by te variables for te solvent. Finally, te last row of (7.1) is approximated at (x i,j, y i,j ) by θ n i+ 1 2,jun i+ 1 2,j + θn i 1 2,jun i 1 2,j θ n i,j+ 1 vn + θ n 2 i,j+ + 1 i,j 1vn 2 2 i,j θ s i+ 1 2,j u s i+ 1 2,j + θs i 1 2,j u s i 1 2,j θ s i,j+ 1 vs + θ s 2 i,j+ + 1 i,j 1vs 2 2 i,j 1 2 =. (7.4) Were necessary, we use second-order extrapolation to account for te no-slip boundary conditions. For a grid wit N cell-centers in te x direction and M cell-centers in te y direction, te above approximations can be collected in a (5NM 2(N +M))-by-(5NM 2(N + M)) linear system, wic we denote by L n C C G n C L s C Gs un u s = Ψ θ n. (7.5) Gn T Gs T p }{{} A Tis forms te discrete approximation to (7.1). Te properties tat old for te 1-D system (3.4) mentioned at te end of Section 3.1, also old for te 2-D system. Te structure of (7.5) is illustrated in Figure 7.2 for te case of a square domain nz = 294 Fig Structure of te A matrix in (7.5) for te case of a square domain wit N = M = 2 3 grid cells; solid dots mark non-zero entries in te matrix, wile solid orizontal and vertical lines igligt te structure sown in (7.5). For te temporal discretization of (2.8), we again use explicit, first-order upwinding for te flux (θ n u n ) (wit LeVeque s transverse propagation correction [2]) and treat te diffusion implicitly wit backward Euler. We use te same adaptive time-stepping strategy as discussed in Section Solving te coupled momentum and continuity equations. To solve te discrete system (7.5), we extend te multigrid box-relaxation sceme introduced in Section 5 to te 2-D system and use it as a rigt-preconditioner for GMRES(m). We briefly discuss te extension of te multigrid components for te 2-D system. Transfer operators. As mentioned above, te mes-spacing in bot te x and y directions is given by, wit 1/ a power of 2. We define a sequence of coarser

25 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 25 grids, were eac grid is a factor of two coarser tan te previous in bot te x and y direction. For restriction and prolongation operators between te grids, we use standard full-weigting and bilinear interpolation defined on a MAC grid [31, p.69-7]. Coarse grid discretization. Te simple DCGA strategy for discretizing te gel system of equations on te coarser grids is again employed. Smooter. We extend te box-relaxation metod described in Section 5.1. For te 2-D system, tis involves solving te discrete equations (7.5) locally in eac computational cell (or box). For eac interior box, tis requires solving a 9-by-9 linear system (4 equations for te network velocity, 4 equations for te solvent velocity, and 1 equation for te pressure) for corrections to te unknowns u n, u s, and p. Boxes in te corners of te domain require solving 5-by-5 linear systems, wile boxes on te edges require solving 7-by-7 systems. Te exact form of all tese systems can be worked out from (7.2)-(7.4). We update corrections to te unknowns in a Gauss- Seidel-type manner and combine tis wit underrelaxation similar to (5.2). Te boxes are processed using red-black ordering. Multigrid cycles We again compare te V- and F-cycle tecniques illustrated in Figure 5.2 for cycling troug te coarse grids. In all of our numerical results, we coarsen te grid until tere are two cell-centers on te smaller side of te rectangular domain. At tis point we solve te system (3.4) directly using Matlab s sparse solver library. For a square domain in 2-D wit N = 2 k cell-centers in eiter direction (N 2 total points), te computational complexity of one V-cycle is W V = 8 3 C(N2 4), wile for te F-cycle it is W F = 32 9 C(N2 (3 log 2 (N)+1)), for some C independent of N. For general rectangular domains wit equal mes spacing in bot directions and standard coarsening, te constants in front of tese work units will cange, owever, te ratio will remain te same wit te F-cycle being 4/3 te cost of a V-cycle. Compare tis to 1-D were te F-cycle is twice te cost of a V-cycle. We use one iteration of te above multigrid procedure wit an initial guess of zero as a rigt-preconditioner for GMRES(2). Since te GMRES metod is black-box, it requires no special modifications for te 2-D system Numerical results. To test te metod in 2-D, we let te domain be te unit square and start wit an initial distribution of network θ n tat is perturbed about te unstable region of te osmotic pressure Ψ (cf. Figure 2.1): θ n = (cos(6πx) + cos(4πy)). Te model parameters are set to µ n =.1, λ n =.3, µ s =.25, λ s =, β = 1, and κ = 1 7. Te first four of tese values make α n =.5 and α s =.5. As in 1-D, we define five stages of pase separation for te gel as illustrated in Figure 7.3, and test te performance of te standalone MGV and MGF metods and preconditioned GMRES-MGV and GMRES-MGF metods at tese stages. Table 7.1 displays te scaling results for tese solvers as te grid is refined from = 2 7 to = 2 9. Integer values listed in te table are te number of multigrid cycles (V(1,1) or F(1,1)) required to satisfy (6.1) using values from te previous timestep as te initial guess (non-bracket), and using all zeros as te initial guess (bracket). Decimal values in parentesis are te corresponding wall-clock times (in seconds) for te former metod of generating initial guesses. Te results are similar to tose in 1-D. Te MGV solver does not scale well and cannot solve te system for all stages. Te MGF metod scales linearly for all but Stage 5, for wic te metod actually diverges for = 2 9. Te GMRES-MGV metod performs very well for all stages of pase separation and sows a near linear scaling as is refined. Finally, as wit te

26 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 26 MGF metod, te GMRES-MGF metod scales linearly for all but Stage 5, were we see a jump in te number of iterations at = 2 9. However, unlike te MGF metod, te GMRES-MGF metod does not diverge, but converges in a reasonable number of iterations. Basing te initial guesses on te previous time-steps typically reduces te number of iterations for all te metods compared wit using all zeros. Tis decrease is more pronounced in te MGV and GMRES-MGV metods. Comparing te wallclock times, we see tat te MGF metod as a standalone solver or preconditioner is almost always more efficient tan te MGV metods; tis was not te case in 1-D (cf. Table 7.1). Stage = 2 7 = 2 8 = 2 9 MGV 1 11 [11] (18.5) 17 [17] (123) 27 [27] (798) 2 11 [11] (18.5) 14 [17] (11) 21 [26] (614) 3 1 [12] (16.9) 14 [16] (11) 22 [26] (648) 4 11 [13] (18.5) 27 [33] (197) DIV 5 DIV DIV DIV MGF 1 5 [5] (15.2) 5 [5] (65.1) 5 [5] (266) 2 5 [5] (15.2) 4 [5] (52.2) 3 [5] (161) 3 5 [5] (15.2) 4 [5] (52.2) 3 [5] (161) 4 5 [5] (15.2) 4 [5] (52.2) 5 [5] (266) 5 5 [7] (15.2) 4 [6] (52.2) DIV GMRES-MGV 1 1 [1] (17.4) 11 [11] (87.5) 12 [12] (389) 2 1 [1] (17.4) 1 [11] (79.5) 1 [12] (336) 3 1 [1] (17.4) 1 [11] (79.5) 1 [12] (336) 4 1 [11] (17.4) 1 [12] (79.5) 11 [14] (37) 5 9 [12] (15.9) 9 [15] (71.4) 1 [17] (336) GMRES-MGF 1 6 [6] (18.9) 6 [6] (79.3) 6 [6] (359) 2 6 [6] (18.9) 5 [6] (66.4) 4 [6] (223) 3 5 [6] (15.7) 5 [6] (66.4) 4 [6] (223) 4 5 [6] (15.7) 5 [6] (66.4) 5 [6] (288) 5 5 [7] (15.7) 5 [7] (66.4) 1 [13] (542) Table 7.1 Comparison of te multigrid V-cycle metod (MGV), multigrid F-cycle (MGF), GMRES(2) preconditioned wit MGV, and GMRES(2) preconditioned wit MGF for solving te system (7.5) at te various stages of pase separation sown in Figure 7.3. Integer values are te number of V(1,1)- or F(1,1)-cycles required to satisfy (6.1) wen using values from te previous time-step as te initial guess (non-bracket) and using all zeros as te initial guess (bracket). Decimal values in parentesis are te wall clock times (in seconds) for te metods using former of te initial guess metods. DIV means tat te metod diverged. Te underrelaxation was set to ω =.675 for all results. To better understand te convergence beavior of te four metods for te finer = 2 9 mes, we plot in Figure 7.4 teir convergence istory at te five stages of pase separation. Te results for Stages 1-3 are similar for te four metods, wit te GMRES-MGF metod aving te steepest slope and te MGV metod te flattest. Te individual slopes do not cange significantly over tese stages. At Stage 4, all but te MGV metods converged, and we see tat te slopes of te remaining tree metods are similar to tose in Stages 1-3. Te big difference comes wit Stage 5.

27 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 27 Te MGF metod diverges, wile te decrease in te residual for te GMRES-MGV and GMRES-MGF metods is not very consistent until about ten iterations, at wic te slopes of te lines again appear to resemble tose from te previous stages. As discussed at te end of Section 5.2 and observed in te 1-D experiments, tis is most likely due to te GMRES metod eliminating te problem eigenvalues of te MGF or MGV iteration matrices. Once tese ave been eliminated, te rapid convergence of tese metods is again observed. 8. Concluding remarks. We ave presented a computational metodology for simulating models of two-pase gel dynamics. Te main computational callenge of tese models is in solving te momentum and incompressibility equations wic involve variable-coefficient differential terms and terms coupling te two fluids. Wen discretized, tese equations lead to a large, sparse linear system of saddle point type. Our metod of solving tis system by using multigrid wit a box-type relaxation procedure as a preconditioner for GMRES(m) appears to be very effective. Numerical results from model problems in one and two dimensions indicate te metod is bot robust and efficient, wit a near linear scaling in te computational cost. Furtermore, te metod is straigtforward to implement since it uses standard transfer operators and direct coarse-grid discretization. For 2-D, using an F-cycle in te multigrid preconditioner appears to be more efficient tan te classical V-cycle. In our numerical experiments, storage was not an issue since te size of te Krylov subspaces was not required to grow excessively large in order to solve te systems. If tis appens to be te case and te number of preconditioned GMRES iterations is muc larger tan te restart value, one migt consider using te BiCGSTAB metod of van der Vorst [32]. Tis Krylov metod only requires storing six intermediate vectors per iteration. However, BiCGSTAB would require two applications of te box-type multigrid preconditioner per iteration, wereas GMRES(m) only requires one. Te obvious way to improve te efficiency of te momentum and incompressibility equations solver is troug parallelization. Tis will be especially important for 3-D applications. Parallel implementations of GMRES(m) are relatively straigtforward and readily available [26, C. 11]. For Stokes and Navier-Stokes, an efficient parallel implementation of multigrid box-relaxation is presented in [1]. Te extension of tis parallel sceme to te gel system will be pursued in a future study. 9. Acknowledgements. Te autors wis to tank Dr. Oren E. Livne for discussing various multigrid metods for problems arising in fluid mecanics wit tem. REFERENCES [1] R. D. Allen and R. R. Cowden, Syneresis in ameboid movement: Its localization by interference microscopy and its significance, J. Cell Biol., 12 (1962), pp [2] E. Alpkvist and I.Klapper, A multidimensional multispecies continuum model for eterogenous biofilm development, Bull. Mat. Biol., 69 (27), pp [3] W. Alt and M. Dembo, Cytoplasm dynamics and cell motion: Two-pase flow models, Mat. Biosci., 156 (1999), pp [4] M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numerica, 14 (25), pp [5] A. Brandt and N. Dinar, Multigrid solutions to flow problems, in Numerical metods for partial differential equations, S. Parter, ed., Academic Press, New York, 1979, pp [6] W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, SIAM Books, Piladelpia, 2. Second edition.

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29 AN EFFICIENT AND ROBUST METHOD FOR GEL DYNAMICS 29 Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Fig Network volume fraction (left column) and magnitude of te network velocity (rigt column) from a numerical simulation of te 2-D model problem at various stages of gel pase separation. Note te fixed scale on te volume fraction plots and te variable scale on te velocity plots.