An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure

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1 An operator splitting approac or te interaction between a luid and a multilayered poroelastic structure M. Bukač I. Yotov P. Zunino Abstract We develop a loosely coupled luid-structure interaction inite element solver based on te Lie operator splitting sceme. Te sceme is applied to te interaction between an incompressible, viscous, Newtonian luid and a multilayered structure, wic consists o a tin elastic layer and a tick poroelastic material. Te tin layer is modeled using te linearly elastic Koiter membrane model, wile te tick poroelastic layer is modeled as a Biot system. We prove a conditional stability o te sceme and derive error estimates. Teoretical results are supported wit numerical examples. 1 Introduction Many natural materials including soil, wood and some biological tissues ave a multilayered structure consisting o two or more constituent materials. Multilayered structures can ave distinct properties rom teir constituent materials. Tis caracteristic is oten used in engineering to produce a new material wic is stier or ligter wen compared to traditional materials. In many cases, suc structures are surrounded by a luid. In tis setting, we are interested in permeable structures. Examples o multilayered permeable structure tat are in contact wit a luid can be ound in groundwater low modeling, reservoir engineering, and modeling o blood low troug te major blood vessels. Tus, in order to detect te damage in a reservoir or certain patologies o blood vessels, it is important to understand te interaction between a luid and a multilayered permeable structure. We study te interaction between an incompressible viscous, Newtonian luid and a multilayered poroelastic structure. Tis model eatures two dierent kinds o coupling, eac widely studied in te literature: te low-porous media coupling [16, 3, 4, 34, 45, 46, 53, 54, 64, 68] and te luid-structure coupling [5,7,11,6,8,37,49,55,60,6]. Main callenges in te low-porous media interaction problems arise rom te coupling o two domains, a luid region and a porous media region, along wit te two pysical processes occurring in eac region. Introducing te poroelastic media, our domain becomes time-dependent, and tus we must resolve diiculties related to a moving domain. Furtermore, classical partitioned solvers or te luid-structure interaction problems are known to ave stability issues wen te density o te structure is comparable to te density o te luid [17]. Tis diiculty will be taken into account ere, since in tis work we are interested on applications in emodynamics, among oters, and te density o blood is almost equal to te density o blood vessels. Department o Matematics, University o Pittsburg, Pittsburg, PA 1560, martinab@pitt.edu Department o Matematics, University o Pittsburg, Pittsburg, PA 1560, yotov@mat.pitt.edu Department o Mecanical Engineering & Materials Science, University o Pittsburg, Pittsburg, PA 1561, paz13@pitt.edu 1

2 Te material properties o arteries ave been widely studied [1, 9, 15, 33, 41, 65, 73]. Pseudoelastic [33, 74], viscoelastic [1, 9, 15] and nonlinear material models represent well known examples. To our knowledge, only a ew o tem ave been deeply analyzed in te time dependent domain, namely wen coupled wit te pulsation induced by eartbeat. Tese considerations also apply to poroelasticity, wic is addressed ere. Poroelasticity becomes particularly interesting wen looking at te coupling o low wit mass transport. Tis is a signiicant potential application o our model, since mass transport provides nourisment, remove wastes, aects patologies and allows to deliver drugs to arteries [61]. Poroelastic penomena are interesting in dierent applications were sot biological tissues are involved. We mention or example cerebro-spinal low [48], wic also involves luid-structure interaction FSI, te study o ysteresis eects observed in te myocardial tissue [38, 39], as well as te modeling o lungs as a continuum material [63]. Besides biological applications, tis model can also be used in numerous oter applications: geomecanics, ground-surace water low, reservoir compaction and surace subsidence, seabed-wave interaction problem, etc. Wile tere exist many complex and detailed models or mutilayered structures in dierent applications, te interaction between te luid and a multilayered structure remains an area o active researc. To our knowledge, te only teoretical result was presented in [55], were te autors proved existence o a solution to a luid-two-layered-structure interaction problem, in wic one layer is modeled as a tin viscoelastic sell and te oter layer as a linearly elastic structure. Several studies ocused on numerical simulations. An interaction between te luid and a two-layer anisotropic elastic structure was used in [69] to model te uman rigt and let ventricles. Sligtly dierent models were used in [43] to model ully coupled luid-structure-soil interaction or cylindrical liquid-contained structures subjected to orizontal ground excitation. Te work in [10] ocused on studying velocity o acoustic waves excited in multilayered structures in contact wit luids. A luid-multilayered structure interaction problem coupled wit transport was studied in [18], wit te purpose o investigating low-density lipoprotein transport witin a multilayered arterial wall. However, none o tese studies present a numerical sceme supported wit stability and error analysis. In tis work, we propose a model tat captures interaction between a luid and a multilayered structure, wic consists o a tin elastic layer and a tick poroelastic layer. In te context o cardiovascular applications, we assume tat te tin layer represents a omogenized combination o te endotelium, tunica intima, and internal elastic lamina, and tat te tick layer represents tunica media. Te tin elastic layer is modeled using te linearly elastic Koiter membrane model, wile te poroelastic medium is modeled using te Biot equations. Te Biot system consists o an elastic skeleton and connecting pores illed wit luid. We assume tat te elastic skeleton is omogeneous and isotropic, wile te luid in te pores is modeled using te Darcy equations. Te Biot system is coupled to te luid and te elastic membrane via te kinematic no-slip and conservation o mass and dynamic conservation o momentum boundary conditions. More precisely, we assume tat te elastic membrane cannot store luid, but allows te low troug it in te normal direction. In te tangential direction, we prescribe te no-slip boundary condition. Tis assumption is reasonable in blood low modeling, since it as been sown in [47] tat predominant direction o intimal transport is te radial direction normal to te endotelial surace, or all ranges o relative intimal tickness. Te coupling between a luid and a single layer poroelastic structure as been previously studied in [7, 44, 57, 67, 71]. In particular, te work in [7] is based on te modeling and a numerical solution o te interaction between an incompressible, Newtonian luid, described using te Navier Stokes equations, and a poroelastic structure modeled as a Biot system. Te problem was solved using bot a monolitic and a partitioned approac. Te partitioned approac was based on te domain decomposition procedure, wit te purpose o solving te Navier-Stokes equations separately

3 rom te Biot system. However, sub-iterations were needed between te two problems due to te instabilities associated wit te added mass eect. Namely, in luid-structure interaction problems, te classical loosely-coupled metods ave been sown to be unconditionally unstable i te density o te structure is comparable to te density o te luid [17, 3], wic is te case in emodynamics applications. To resolve tis problem, several dierent splitting strategies ave been proposed [,3,5,6,11,13,0,1,7,8,9,30,36,58,60,6]. More precisely, in [5], te autors present a strongly-coupled partitioned sceme based on Robin-type coupling conditions. In addition to te classical Diriclet-Neumann and Neumann-Diriclet scemes, tey also propose a Robin-Neumann and a Robin-Robin sceme, tat converge witout relaxation, and need a smaller number o sub-iteration between te luid and te structure in eac time step tan classical strongly-coupled scemes. In [13, 14], Burman and Fernández propose an explicit sceme were te coupling between te luid and a tick structure is enorced in a weak sense using Nitsce s approac [37]. Te ormulation in [13] still suers rom stability issues, wic were corrected by adding a weakly consistent penalty term tat includes pressure variations at te interace. Te added term, owever, lowers te temporal accuracy o te sceme, wic was ten corrected by proposing a ew deect-correction sub-iterations to acieve optimal accuracy. A novel loosely coupled partitioned sceme, called te kinematically coupled sceme, was introduced by Guidoboni, Glowinski et al. in [36], and applied to FSI problems wit tin structures. Te sceme is based on embedding te no-slip kinematic condition into te tin structure equations. Using te Lie operator splitting approac [35], te structure equations are split so tat te structure inertia is treated togeter wit te luid as a Robin boundary condition, wile te structure elastodynamics is treated separately. Tis metod as been sown to be unconditionally stable, and tereore independent o te luid and structure densities. Stability is acieved by combining te structure inertia wit te luid sub-problem to mimic te energy balance o te continuous problem. Additionally, Mua and Čanić sowed tat te sceme converges to a weak solution o te ully nonlinear FSI problem [56]. Te main eatures o te kinematically coupled sceme are simple implementation, modularity, no need or sub-iterations between te luid and structure sub-problems, and very good perormance in terms o stability, accuracy, and computational cost. Hence, modiications o tis sceme ave been used by several autors to study dierent multi-pysics problems involving FSI. A modiication o te sceme was proposed by Lukáčová et al. to study FSI involving non-newtonian luids [4, 49]. An extension o te kinematically coupled sceme was proposed in [11] were a parameter β was introduced to increase te accuracy. It was sown in [11] tat te accuracy o te kinematicallycoupled β-sceme wit β = 1 is comparable to tat o monolitic sceme by Badia, Quaini, and Quarteroni in [6] wen applied to a nonlinear bencmark FSI problem in emodynamics. A modiied kinematically-coupled sceme, called te incremental displacement-correction sceme, tat treats te structure displacement explicitly in te luid sub-step and ten corrects it in te structure sub-step was recently proposed by Fernández et al. [7, 8, 30].. Inspired by te kinematically coupled sceme, in tis manuscript we propose a loosely-coupled inite element sceme or te luid-membrane-poroelastic structure interaction problem based on te Lie operator splitting metod. We use te operator splitting to separate te luid problem Navier-Stokes equations rom te Biot problem. Te no-slip kinematic condition in te tangential direction is embedded into te membrane equations. We recall tat tis coupled problem is particularly callenging, because it combined te ree luid-porous media low and te luid-structure coupling mecanisms. Tis work sows tat te kinematically coupled Lie splitting metod can be successully applied also in tis case. In particular, te operator splitting is preormed so tat te tangential component o 3

4 te structure inertia is treated togeter wit te luid as a Robin boundary condition. Assuming te pressure ormulation or te Darcy equations, te continuity o te normal lux and te balance o normal components o stress between te Navier Stokes luid and te luid in te pores is treated in a similar way as in te partitioned algoritms or te Stokes-Darcy coupled problems [45, 66]. Te membrane elastodynamics is embedded into te Biot system as a Robin boundary condition. In contrast wit domain decomposition metods proposed in [7], te operator splitting approac does not require sub-iterations between te luid and te Biot problem, making our sceme more computationally eicient. We prove a conditional stability o te proposed sceme, were te stability condition does not depend on te luid and structure densities, but it is related to te decoupling o te Stokes-Darcy interaction problem. Furtermore, we derive error estimates and prove te convergence o te sceme. Te rates o convergence and te stability condition are validated numerically on a classical bencmark problem typically used to test te results o luid-structure interaction algoritms. In te second numerical example, we investigate te eects o porosity on te structure displacement. Namely, we consider a ig storativity and a ig permeability case in te Darcy equations, and compare tem to te results obtained using a purely elastic model. Depending on te regime, we observe a signiicantly dierent beavior o te coupled system. Tis conclusion is also supported by te sensitivity analysis, based on bot teoretical and numerical approac, addressed by te autors in [1]. Te main contributions o tis work can be summarized as ollows. We propose a novel model to study interaction between a luid and a composite poroelastic structure, and a novel, loosely-coupled numerical sceme. Te sceme is based on existing work [36, 46], wic was combined and modiied to resolve bot issues due te luid-structure coupling, and te luid-porous medium coupling. We present te stability and convergence analysis o te proposed sceme, completed wit te numerical examples. Te rest o te paper is organized as ollows. In te ollowing section we introduce te model equations and te coupling conditions. In Section 3 we propose a loosely-coupled sceme based on te operator-splitting approac. Te weak ormulation and stability o te sceme is presented in Section 4. In Section 5 we derive te error analysis o te sceme. Finally, te numerical results are presented in Section 6. Description o te problem Consider a bounded, deormable, two-dimensional domain Ωt = Ω t Ω p t o reerence lengt L, wic consists o two regions, Ω t and Ω p t, see Figure 1. We assume tat te region Ω t as reerence widt R, and is illed by an incompressible, viscous luid. We denote te widt o te second region Ω p t by r p, and assume tat Ω p t is occupied by a ully-saturated poroelastic matrix. Te two regions are separated by a common interace t. We assume tat t as a mass, and represent a tin, elastic structure. Namely, we assume tat te tickness o te interace r m is small wit respect to te radius o te luid domain, r m << R. Tus, te volume o te interace is negligible, so it acts as a membrane tat can not store luid, but allows te low troug it in te normal direction. We are interested in simulating a pressure-driven low troug te deormable cannel wit a two-way coupling between te luid, tin elastic interace, and poroleastic structure. Witout loss o generality, we restrict te model to a two-dimensional D geometrical model representing a deormable cannel. We consider only te upper al o te luid domain supplemented by a symmetry condition at te axis o symmetry. Tus, te reerence luid and structure domains in our problem 4

5 in Ωt p t r m Ωt out Figure 1: Deormed domains Ω t Ω p t. sowed by dased lines in Figure 1 are given, respectively, by ˆΩ := {x, y 0 < x < L, 0 < y < R}, ˆΩ p := {x, y 0 < x < L, R < y < R + r p }, and te reerence lateral boundary by ˆ = {x, R 0 < x < L}. Te inlet and outlet luid boundaries are deined, respectively, as in = {0, y 0 < y < R} and out = {L, y 0 < y < R}. We model te low using te Navier-Stokes equations or a viscous, incompressible, Newtonian luid: v ρ t + v v = σ + g in Ω t 0, T,.1 v = 0 in Ω t 0, T,. were v = v x, v y is te luid velocity, σ = p I + µ Dv is te luid stress tensor, g is a body orce, p is te luid pressure, ρ is te luid density, µ is te luid viscosity and Dv = v + v T / is te rate-o-strain tensor. Denote te inlet and outlet luid boundaries by in and out, respectively. At te inlet and outlet boundary we prescribe te normal stress: σ n in = p in tn in on in 0, T,.3 σ n out = 0 on out 0, T,.4 were n in /n out are te outward normals to te inlet/outlet luid boundaries, respectively. Tese boundary conditions are common in blood low modeling [4, 5, 59] even toug tey are not pysiologically optimal since te low distribution and pressure ield in te modeled domain are oten unknown [7]. Along te middle line o te cannel 0 = {x, 0 0 < x < L} we impose te symmetry conditions: v x y = 0, v y = 0 on 0 0, T..5 Te lateral boundary represents a deormable, tin elastic wall, wose dynamics is modeled by te linearly elastic Koiter membrane model, given in te irst order Lagrangian ormulation by: ˆξx ρ m r m t C ˆη y ˆx C ˆη x 1 ˆx = ˆ x on ˆ 0, T,.6 ˆξy ρ m r m t + C ˆη x 0ˆη y + C ˆx = ˆ y on ˆ 0, T,.7 ˆη t = ˆξ on ˆ 0, T,.8 5

6 were ˆηˆx, t = ˆη x ˆx, t, ˆη y ˆx, t denotes te axial and radial displacement, ˆξˆx, t = ˆξ x ˆx, t, ˆξ y ˆx, t denotes te axial and radial structure velocity, ˆ = ˆ x, ˆ y is a vector o surace density o te orce applied to te membrane, ρ m denotes te membrane density and C 0 = rm R µm λ m λ m+µ m + µ m, C1 = r m µm λ m λ m+µ m + µ m, C = rm R µ m λ m λ m+µ m..9 Te coeicients µ m and λ m are te Lamé coeicients or te membrane. Note tat we can write te system.6-.7 more compactly as ˆξ ρ m r m t + ˆLˆη = ˆ, C1 ˆL := ˆxˆx C ˆx C ˆx C Te luid domain is bounded by a deormable porous matrix consisting o a skeleton and connecting pores illed wit luid, wose dynamics is described by te Biot model, wic in te irst order, primal, Eulerian ormulation reads as ollows: DV ρ p Dt σp = in Ω p t 0, T,.11 DU Dt = V in Ωp t 0, T,.1 D Dt s 0p p + α U κ p p = s in Ω p t 0, T,.13 were D Dt denotes te classical concept o material derivative. Te stress tensor o te poroelastic medium is given by σ p = σ E αp p I, were σ E denotes te elasticity stress tensor. Wit te assumption tat te displacement U = U x, U y o te skeleton is connected to stress tensor σ E via te Saint Venant-Kirco elastic model, we ave σ E U = µ p DU + λ p trdui, were λ p and µ p denote te Lamé coeicients or te skeleton, and, wit te ypotesis o small deormations, DU = U + U T /. Te displacement velocity is denoted by V = V x, V y, is a body orce, and s is a source or sink. System consists o te momentum equation or te balance o total orces.11, and te storage equation.13 or te luid mass conservation in te pores o te matrix, were p p is te luid pressure. Te density o saturated porous medium is denoted by ρ p, and κ denotes te uniormly positive deinite ydraulic conductivity tensor. For simplicity o te presentation we assume tat κ is a scalar constant. Te coeicient c 0 > 0 is te storage coeicient, and te Biot-Willis constant α is te pressure-storage coupling coeicient. Te relative velocity o te luid witin te porous structure q can be reconstructed via Darcy s law q = κ p p in Ω p t 0, T. Denote te inlet and outlet poroelastic structure boundaries, respectively, by p in = {0, y R < y < R + r p } and p out = {L, y R < y < R + r p}, and te reerence exterior boundary by ˆ p ext = {x, R + r p 0 < x < L}. We assume tat te poroelastic structure is ixed at te inlet and outlet boundaries: U = 0 on p in p out 0, T,.14 tat te external structure boundary p ext t is exposed to external ambient pressure n ext σ E n ext = p e on p ext t 0, T,.15 6

7 were n ext is te outward unit normal vector on p ext t, and tat te tangential displacement o te exterior boundary is zero: U x = 0 on p ext t 0, T..16 On te luid pressure in te porous medium, we impose drained boundary conditions []: p p = 0 on p ext t p in p out 0, T..17 Initially, te luid, elastic membrane and te poroelastic structure are assumed to be at rest, wit zero displacement rom te reerence coniguration v = 0, U = 0, DU Dt = 0, ˆη = 0, ˆη t = 0, q = 0, p p = To deal wit te motion o te luid domain we adopt te Arbitrary Lagrangian-Eulerian ALE approac [5, 40, 59]. In te context o inite element metod approximation o moving-boundary problems, ALE metod deals eiciently wit te deormation o te mes, especially at te boundary and near te interace between te luid and te structure, and wit te issues related to te approximation o te time-derivatives v/ t vt vt n / wic, due to te act tat Ω t depends on time, is not well deined since te values vt and vt n correspond to te values o v deined at two dierent domains. Following te ALE approac, we introduce two amilies o arbitrary, invertible, smoot mappings A t and S t, deined on reerence domains ˆΩ and ˆΩ p, respectively, wic track te domain in time: A t : ˆΩ Ω t R, x = A t ˆx Ω t, or ˆx ˆΩ,.19 S t : ˆΩ p Ω p t R, x = S t ˆx Ω p t, or ˆx ˆΩ p..0 Note tat te luid domain is determined by te displacement o te membrane ˆη, wile te porous medium domain is determined by its displacement Û, were Û is te displacement o te porous medium evaluated at te reerence coniguration.. However, because o condition.4, we can deine a omeomorpism over Ω t Ω p t by setting mappings A t and S t equal on t. For te structure, we adopt te material mapping S t ˆx = ˆx + Ûˆx, t, ˆx ˆΩ p..1 Since te mapping A t is arbitrary, wit te only requirement tat it matces S t on t, we can deine A t as A t ˆx = ˆx + Extˆηˆx, t = ˆx + ExtÛˆx, t ˆ, ˆx ˆΩ...1 Te coupling conditions In order to prescribe te coupling conditions on te pysical luid-structure interace t, let η := ˆη At 1 t and ξ := ˆξ A 1 t t were A t is deined in.19. Note tat ξ = Dη Dt. Wile te lumen and te poroelastic medium contain luid, we assume tat te elastic membrane does not contain luid, but allows te low troug it in te normal direction. Tis is a reasonable assumption because te elastic membrane represents tunica intima. It as been sown by experimental studies tat te normal transport in tunica intima is signiicantly greater tan tangential transport [47]. Denote by n te outward normal to te luid domain and by τ te tangential unit vector. Tus, te luid, elastic membrane and poroelastic structure are coupled via te ollowing boundary conditions: 7

8 Mass conservation: since te tin lamina allows te low troug it, te continuity o normal lux is DU v n = Dt κ p p n on t..3 Since we do not allow iltration in te tangential direction, we prescribe no-slip boundary conditions between te luid in te lumen and te elastic membrane, and between te elastic membrane and poroelastic medium: Balance o normal components o te stress in te luid pase: v τ = ξ τ, η = U on t..4 n σ n = p p on t..5 Te conservation o momentum describes balance o contact orces. Precisely, it says tat te sum o contact orces at te luid-porous medium interace is equal to zero: σ n σ p n + J 1 = 0 on t,.6 were := ˆ A 1 t t, and J denotes te Jacobian o te transormation rom t to ˆ given by J = 1 + ˆη x ˆηy +..7 ˆx ˆx. Weak ormulation o te monolitic problem For a domain Ω, we denote by H k Ω te norm in te Sobolev space H k Ω. Te norm in L Ω is denoted by L Ω, and te L Ω inner product by, Ω. We introduce te ollowing bilinear orms a v, φ = µ Dv : Dφ dx, b p, φ = Ω t a e U, φ p = µ p a p p p, ψ p = Ω p t b ep p p, φ p = α Ω t p φ dx, Ω p t Ω p t a m ˆη, ˆζ = r m L c p p p, φ = c ep p p, φ p = t t 0 DU : Dφ p dx + λ p κ p p ψ p dx, p p φ p dx, ˆη x 4µ ˆζ x m ˆx ˆx + 1 R ˆη y ˆζ y p p φ ndx, p p φ p ndx, 8 Ω p t U φ p dx, dˆx + r m L 0 4µ m λ m ˆηx λ m + µ m ˆx + 1 R ˆη ˆζ x y ˆx + 1 R ˆζ y dˆx

9 and te trilinear orm d v, u, φ = ρ Ω t v u φdx. For more details on te derivation o te bilinear orm a m, or te elastic part o te Koiter membrane.6-.8, see [15]. To ind a weak orm o te Navier-Stokes equation, introduce te ollowing test unction spaces: V t = {φ : Ω t R φ = ˆφ A t 1, ˆφ H 1 ˆΩ, φ y = 0 on 0 },.8 Q t = {ψ : Ω t R ψ = ˆψ A t 1, ˆψ L ˆΩ },.9 or all t [0, T. Te variational ormulation o te Navier-Stokes equations now reads: given t 0, T ind v, p V t Q t suc tat or all φ, ψ V t Q t ρ Ω t v t φ dx + d v, v, φ + a v, φ b p, φ + b ψ, v = t σ n φ dx + g φ dx + Ω t p in tφ xdy. in.30 In order to write te weak orm o te linearly elastic Koiter membrane, let ˆV m = H 1 0 0, L. Ten te weak ormulation reads as ollows: given t 0, T ind ˆη, ˆξ ˆV m ˆV m suc tat or all ˆζ, ˆχ ˆV m ˆV m L ρ m r m Finally, let us introduce 0 ˆξ ˆη t ˆχdˆx + ρ m r m L 0 ˆξ L t ˆζdˆx + a m ˆη, ˆζ = ˆ ˆζdˆx V p t = {φ : Ω p t R φ = ˆφ S t 1, ˆφ H 1 ˆΩ p, φ = 0 on p in p out, φ x = 0 on p ext t}, Q p t = {ψ : Ω p t R ψ = ˆψ S t 1, ˆψ H 1 ˆΩ p, ψ Ω p t\t = 0}. Now te weak orm o te Biot system reads as ollows: given t 0, T ind U, V, p p V p t V p t Q p t suc tat or all φ p, ϕ p, ψ p V p t V p t Q p t ρ p V DU ϕ p DV dx + ρ p Ω p t Dt Ω p t Dt φp dx + a e U, φ p b ep p p, φ p Dp p + s 0 Ω p t Dt ψp dx +b ep ψ p, DU Dt +a pp p, ψ p = t σ p n φ p dx t κ p p p e φ p ydx+ φ p dx+ p ext Ω p t Ω p t sψ p dx..3 To write a weak ormulation o te coupled Navier-Stokes/Koiter/Biot system, deine a space o admissible solutions W t = {φ, ˆζ, ˆχ, φ p, ϕ p V t ˆV m ˆV m V p t V p t ζ = φ p t, χ = ϕ p t, φ t τ = ζ τ },.33 9

10 were ζ := ˆζ A 1 t t, χ := ˆχ A 1 t t, and add togeter equations.30,.31 and.3: v ρ Ω t t φ dx + d v, v, φ + a v, φ b p, φ + b ψ, v L +ρ m r m ˆξ ˆη L ˆξ ˆχdˆx + ρ m r m 0 t 0 t ˆζdˆx + a m ˆη, ˆζ + ρ p V DU ϕ p dx Ω p t Dt DV +ρ p Ω p t Dt φp dx + a e U, φ p b ep p p, φ p Dp p + s 0 Ω p t Dt ψp dx + b ep ψ p, DU Dt + a pp p, ψ p L = σ n φ dx σ p n φ p dx κ p p nψ p dx + ˆ ˆζdx t + g φ dx + Ω t t p in tφ xdy in Denote by I t te interace integral I t = t p ext t p e φ p ydx + Ω p t φ p dx + 0 Ω p t σ n φ σ p n φ p κ p p nψ p + J 1 ζdx. sψ p dx..34 Decomposing te stress terms and tin sell orcing term into teir normal and tangential components and employing conditions.3 and.5 we ave I t = p p φ n n σ p nφ p n + J 1 nζ n + v nψ p DU t Dt nψp +τ σ nφ τ τ σ p nφ p τ + J 1 τ ζ τ dx..35 For eac triple o test unctions φ, ˆζ, φ p W t, due to te condition.6 we ave I t = p p φ n n σ p nφ p n + J 1 nφ p n + v nψ p DU Dt nψp dx. t Finally, employing conditions.5 and.6 we ave I t = t p p φ n + p p φ p n + v nψ p DU Dt nψp dx. Tus, te weak ormulation o te coupled Navier-Stokes/Koiter/Biot system reads as ollows: given t 0, T ind X = v, ˆη, ˆξ, U, V, p, p p V t ˆV m ˆV m V p t V p t Q t Q p t, wit η = U t, ξ = V t, and v τ t = ξ τ, suc tat or all Y = φ, ˆζ, ˆχ, φ p, ϕ p, ψ, ψ p W t Q t Q p t PX, Y + d v, v, φ = FY,.36 10

11 were L +ρ m r m +ρ p Ω p t and FY = 0 PX, Y = ρ ˆξ ˆη t Ω t ˆχdˆx + ρ m r m L DV Dt φp dx + a e U, φ p b ep p p, φ p + Ω t v t φ dx + a v, φ b p, φ + b ψ, v 0 ˆξ t ˆζdˆx + a m ˆη, ˆζ + ρ p Ω p t Ω p t V DU ϕ p dx Dt s 0 Dp p Dt ψp dx + b ep ψ p, DU Dt + a pp p, ψ p +c p p p, φ c ep p p, φ p c p ψ p, v + c ep ψ p, DU,.37 Dt g φ dx + p in tφ xdy p e φ p ydx + φ p dx + sψ p dx..38 in p ext Ω p t Ω p t Note tat te interace terms are contained in bilinear orms c p, and c ep,. In te error analysis, or simplicity, we will ocus on te time dependent Stokes problem, in wic case term d v, v, φ will be dropped..3 Energy equality To derive te energy o te coupled problem let Y = v, ˆη t, ˆξ t, DU Dt, DV Dt, p, p p in equation.36. Ten, te energy equality or coupled system is given as ollows: { 1 d ρ v dt L Ω t + ρ ˆη x mr m ˆη y } t + ρ m r m L 0,L t L 0,L [ ˆη y +r m 4µ m ˆη x R + 4µ m L 0,L ˆx + 4µ mλ m ˆη x L 0,L λ m + µ m ˆx + ˆη ] y R L 0,L DU } +ρ p Dt + µ p DU L Ω p t + λ p U L Ω p t + s 0 p p L Ω p t L Ω p t +µ Dv L Ω t + κ p p L Ω p t = g vdx + p in tv x dy Ω t in DU y p e p Dt dx + sp p dx + ext Ω p t Ω p t DU Dt dx. 3 A loosely-coupled sceme based on te operator-splitting approac To approximate te luid-multilayer structure interaction problem described in Section 1, we propose a loosely coupled sceme based on a time-splitting approac known as te Lie splitting [35]. Details o te Lie splitting are described below. 11

12 3.1 Te Lie sceme Let A be an operator rom a Hilbert space H into itsel, and suppose ϕ 0 H. Consider te ollowing initial value problem: ϕ P t + Aϕ = 0, in 0, T, were A = A i, ϕ0 = ϕ Te Lie sceme consists o splitting te ull problem into P sub-problems, eac deined by te operator A i, i = 1,..., P. Te original problem is discretized in time wit te time step > 0, so tat t n = n. Te Lie splitting sceme consist o solving a series o problems ϕ i t + A i ϕ i = 0, or i = 1,..., P, eac deined over te entire time interval t n, t, but wit te initial data or te i t problem given by te solution o te i 1 st problem at t. More precisely, set ϕ 0 = ϕ 0. Ten, or n 0 compute ϕ by solving i=1 ϕ i t + A iϕ i = 0 in t n, t, ϕ i t n = ϕ n+i 1/P, 3. and ten set ϕ n+i/p = ϕ i t, or i = 1,....P. Tis metod is irst-order accurate in time. To increase te accuracy in time to second-order, a symmetrization o te sceme can be perormed [35]. 3. Te irst-order system in te ALE ramework To deal wit te motion o te luid domain, we consider te Navier-Stokes equation in te ALE ormulation. To write te Navier-Stokes equations in te ALE orm, we notice tat or a unction z = zx, t deined on Ω t 0, T te corresponding unction ẑ := z A t deined on ˆΩ 0, T is given by ẑˆx, t = za t ˆx, t. Dierentiation wit respect to time, ater using te cain rule, gives z = z t ˆx t + w z, wx, t = A tˆx, 3.3 t were w denotes te domain velocity. We will apply tis rule to write te time-derivative o te velocity in Navier-Stokes equations on te reerence domain. Note tat we do not ave to apply te same rule to te time-derivatives in te Biot system and in Koiter membrane equations since te material time-derivative is suitable or te time discretization, due to Dz Dt = z t, and te membrane ˆx equations are given on te reerence coniguration. Wit tese assumptions, our problem now reads: Given t 0, T, ind v = v x, v y, p, ˆη = ˆη x, ˆη y, ˆξ = ˆξ x, ˆξ y, U = U x, U y, V = V x, V y and p p, 1

13 wit ηx, t = ˆηA 1 t x, t, or x t, suc tat ρ v t + v w v = σ + g in Ω t 0, T, 3.4a ˆx v = 0 in Ω t 0, T, 3.4b ˆξ ρ m r m t + ˆLˆη = ˆ on ˆ 0, T, 3.4c ρ m r m ˆξ ˆη = 0 on t ˆ 0, T, 3.4d DV ρ p Dt = σp + in Ω p t 0, T, 3.4e D s 0 Dt p p + α DU Dt κ p p = s in Ω p t 0, T, 3.4 ρ p V DU = 0 in Ω p t 0, T, 3.4g Dt wit te kinematic coupling conditions on t: dynamic coupling conditions on t: and te continuity o normal lux on t: ξ τ = v τ, η = U, 3.5 σ n σ p n + J 1 = 0, 3.6 v n = n σ n = p p, 3.7 DU Dt κ p p n, 3.8 wit te boundary and initial conditions given in Section 1. Denote by L te inverse Piola transormation o ˆL, namely L = J 1 ˆLF T, were F = x A t. Ten, composing te Koiter membrane equations.10 wit A 1 t, and employing te irst condition in 3.5, and condition 3.7 we can write te tangential and normal component o condition 3.6 as ollows: v ρ m r m t τ + τ Lη + Jτ σ n Jτ σ p n = 0, on t 3.9a Dξ ρ m r m Dt n + n Lη Jp p Jn σ p n = 0, on t. 3.9b We will use condition 3.6 written te orm 3.9a-3.9b wen perorming te operator splitting. 3.3 Details o te loosely-coupled sceme In tis section we will apply te Lie splitting sceme to problem 3.4, were te discretization in time will be done using te backward Euler sceme. We will denote te discrete time derivatives by d t φ = φ φ n and d tt φ = d tφ d t φ n, 13

14 were all quantities are evaluated on te reerence domain. In our case, using te notation rom Section 3.1, ϕ tat appears in equation 3.1 is a vector ϕ = v, v t τ, ξ n, η, V, p p, U T. We will split te irst-order system into two main sub-problems, separating te problem deined on te luid domain Ω t rom te problem deined on te poroelastic medium domain Ω p t. In tat case, we will split te sum o all operators tat appear in te system 3.4 into two parts, as A 1 + A, were A i = A v i, Aξτ i, A ξn i, A η i, AV i, App i, A U i T, or i = 1,. For eac o te equations, tis will be done in te ollowing way: Equation 3.4a will be split so tat A v 1 = ρ v w v σ and A v = 0, Equation 3.4c will be used in orm 3.9, were equation 3.9a will be split so and equation 3.9b will be split so A ξ τ 1 = Jτ σ n and A ξ τ = τ Lη Jτ σp n, A ξ n 1 = 0 and A ξ n = n Lη Jp p Jn σ p n. Equation 3.4d will be split so tat A η 1 = 0, and Aη 1 = ξ, Equation 3.4e will be split so tat A V 1 = 0 and AV = σp, Equation 3.4 will be split so tat A pp 1 Equation 3.4g will be split so tat A U 1 = 0 and AU = V. = 0 and App = α V κ p p, and inally, Using tis approac, our system is decoupled into a luid problem and te Biot problem. Furtermore, we not only split te coupled problem into two dierent domains, but we also treat dierent pysical penomena separately. Details o te loosely coupled sceme are given as ollows. Step 0. Step 0 is a geometry problem wic involves computation o a luid domain and ALE velocity w n : A t nˆx = ˆx + Extˆη n, Ω t n = A t nˆω, w n = d t x n, 3.10 were ˆx ˆΩ, x n Ωt n, and x n 1 Ωt n 1. Step 1. Step 1 involves solving te Navier-Stokes equations, and equation ρ m r m v t τ + Jτ σ n = 0 on t n t n, t, 3.11 wile time-derivatives o all te oter unctions are equal to zero. Tereore, equation 3.11 can be seen as a Robin-type boundary condition or luid velocity. Now, in te time-discrete ramework, Step 1 reads as ollows: Find v and p, wit V n and p n p obtained at te previous time step, suc tat ρ d t v + v w n v = σ v, p + g in Ω t n, 3.1a v = 0 in Ω t n, 3.1b τ σ v, p v τ V n τ n + ρ m r m = 0 on t n, 3.1c n σ v, p n = p n p on t n. 3.1d 14

15 wit te ollowing boundary conditions on in out 0 : v x y = v y = 0 on 0, v 0, R, t = v L, R, t = 0, σ v, p n = p in tn on in, σ v, p n = 0 on out. Step : Step involves solving te Biot problem togeter wit equations v ρ m r m t τ + τ Lη Jτ σp n = 0 on t n t n, t, 3.13a Dξ ρ m r m Dt n + n Lη Jp p Jn σ p n = 0 on t n t n, t, 3.13b and conditions Dη Dt n = v n + κ p p n on t n t n, t, 3.14 η = U, v τ = ξ τ on t n t n, t, 3.15 wile te luid velocity in Ω t n does not cange in tis step. Since η = U t n, ξ = V t n, and v τ = ξ τ, we can rewrite conditions 3.13a, 3.13b, and 3.14 in te ollowing way: ρ m r m DV Dt τ + τ LU Jτ σp n = 0 on t n t n, t, 3.16 ρ m r m DV Dt n + n LU Jp p Jn σ p n = 0 on t n t n, t, 3.17 κ p p n = DU Dt n v n on tn t n, t Finally, using te Backward Euler sceme as in e.g. [8] or te Biot equations, Step reads as ollows: Find U, V, and p p, wit v computed in Step 1 and U n, V n, and p n p computed in te previous time-step, suc tat ρ p d t V = σ p U, p p + t in Ω p t n, 3.19a s 0 d t p p + α d t U κ p p = st in Ω p t n, 3.19b ρ p V d t U = 0 in Ω p t n, 3.19c Jτ σ p U, pp V τ v τ n = ρ m r m + τ LU on t n, 3.19d Jn σ p U, p p n = ρ m r m d t V n + n LU Jp p on t n, 3.19e κ p p n = d t U n v n on t n, 3.19 wit boundary conditions: p p = 0 on Ω p t n \t n, U = 0 on p in p out, n ext σ E U n ext = p e on p ext, U x = 0 on p ext. 15

16 Remark 1. Once U and V are computed, we can ind te membrane displacement η and velocity ξ via η = U t n, ξ = V t n. Remark. In practice, te structure is usually andled in Lagrangian ramework. Togeter wit te ypotesis o small deormations we assume tat te structure is linearly elastic, in wic case we can easily recast Step in te reerence domain, were te boundary conditions 3.19d-3.19e simpliy as ollows: e 1 ˆσ p Û ˆV, ˆp x ˆv x p ˆn = ρ m r m e ˆσ p Û, ˆp p ˆn = ρ m r m d t ˆV y + C 0 Ûy / were e 1 and e are te Cartesian unit vectors. Û y / C ˆx Û x / + C ˆx Ûx / C 1 ˆx on ˆ t n, t, ˆp p on ˆ t n, t, Te proposed sceme is an explicit loosely-coupled sceme were te irst step consists o a luid Navier-Stokes problem, and te second step consists o a poroelastic problem. Bot subproblems are solved wit a Robin-type boundary conditions, wic take into account tin-sell inertia and kinematic conditions implicitly. We note tat te original monolitic problem becomes ully decoupled, and tere are no sub-iterations needed between te two sub-problems. Remark 3. One can apply additional splitting to Step 1 and Step o te algoritm described above. Namely, te luid problem described in Step 1 can be split into its viscous part te Stokes equations or an incompressible luid and te pure advection part incorporating te luid and ALE advection simultaneously. Te Biot system described in Step can be split so te elastodynamics is treated separately rom te pressure. For te details o possible Biot splitting strategies see [51] and te reerences terein. 4 Weak ormulation and stability In tis section we write te variational ormulation and prove te conditional stability o te loosely coupled sceme proposed in Section. For simplicity, we work out te analysis assuming tat te displacement o te boundary is small enoug and can be neglected. Under tese assumptions, domains Ω t and Ω p t are ixed: Ω t = ˆΩ, Ω p t = ˆΩ p, t 0, T. Altoug simpliied, tis problem still retains te main diiculties associated wit te added-mass eect and te diiculties tat partitioned scemes encounter wen modeling luid-porous medium coupling. Since rom now on all te variable are deined on te ixed domain, we will drop te at notation to avoid cumbersome expressions. Let t n := n or n = 1,..., N, were T = N is te inal time. Let te test unction spaces V, Q, V p and Q p be deined as in.8,.9,.3, and.3 respectively. To discretize te problem in space, we use te inite element metod. Tus, we deine te inite element spaces V V, Q Q, V p V p, Q p Qp, and V m := V p. Te deinition o tese discrete spaces will 16

17 be made precise at te beginning o Section 5. We assume tat all te inite element initial conditions are equal to zero: v 0 = 0, U 0 = 0, V 0 = 0, η0 = 0, ξ0 = 0, p0 p, = 0. Finally, te ully discrete numerical sceme is given as ollows: Step 1. Given t 0, T ], n = 0,..., N 1, ind v V and p, Q, wit V n and p n p, obtained at te previous time step, suc tat or all φ, ψ V Q : ρ d t v φ dx+d v, v, φ +a v, φ +ρ mr m Ω b p,, φ +b ψ, v +c p p n p,, φ = Ω gt φ dx+ Step. Given v computed in Step 1, ind U V p, V tat or all φ p, ϕp, ψp V p V p Qp : ρ p +ρ m r m Ω p V v τ V n τ φ τ dx p in t φ x,dy. 4.1 in V p, and p p, Qp, suc d t U ϕ p dx + ρ p d t V φ p dx + a eu, φ p + s 0 d t p p, ψp dx Ω p Ω p +a p p p,, ψp b epp p,, φp + b epψ p, d tu + ρ m r m d t V nφ p ndx 4.1 Stability analysis V τ v τ φ p τ dx + a mu, φ p c ep p p,, φp + c epψ p, d tu c p ψ p, v = p e φ p y, Ω dx + t φ p dx + st ψ p dx. 4. p Ω p p ext To present our results in a more compact manner, in te analysis we study te Stokes equations instead o te Navier-Stokes equations. Handling te convective term in te Navier-Stokes equations can be done using classical approaces, see or example [55, 70]. Let us introduce te ollowing seminorms and Furtermore, we deine te time discrete norms: φ l 0,T ;S = φ p E := a e φ p, φp 1/ φ p V p, 4.3 ζ M := a m ζ, ζ 1/ ζ V m / φ S, φ l 0,T ;S = max 0 n N φn S, were S {H k Ω, H k Ω p, H k 0, L, E, M}. 17

18 Let E n denote te discrete energy o te luid problem, En p denote te discrete energy o te Biot problem, and E n m denote te discrete energy o te Koiter membrane at time level n, deined respectively by E n = ρ vn L Ω, 4.5 E n p = ρ p V n L Ω p + 1 U n E + s 0 pn p, L Ω p, 4.6 Em n = ρ mr m ξ n L 0,L + 1 ηn M. 4.7 Beore proceeding, let us address te ollowing property tat will serve as auxiliary result or te stability and error analysis. Lemma 1. Suppose U, V, p p, is a solution to 4.. Ten V = d t U. 4.8 Proo. Let φ p, ϕp, ψp = 0, V d t U, 0 in 4.. Ten, we ave V d t U L Ω p = 0, and te assertion ollows. Te stability o te loosely-coupled sceme is stated in te ollowing result. constants tat appear in 4.10 are deined in te Appendix. Te Teorem 1. Assume tat te luid-poroelastic system is isolated, i.e. p in = 0, p e = 0, g = 0, = 0 and s = 0. Let {v n, pn p,, V n, U n, ξn, ηn, pn p, } 0 n N be te solution o Ten, under te condition µ C K C T ICT C P F µ s 0 γ > 0 i.e. < s 0 CK C T ICT C, 4.9 P F te ollowing estimate olds: E N +EN p +E N m + 4 ρ d t v l 0,T ;L Ω + 4 ρ mr m + ρ mr m V τ v τ L v τ V n τ L + γ Dv l 0,T ;L Ω + ρ mr m d t ξ n l 0,T ;L + ρ p d t V l 0,T ;L Ω p + d tu l 0,T ;E + d tη l 0,T ;M + δ p, l 0,T ;L Ω + 4 s 0 d t p p, l 0,T ;L Ω p were δ is given in te proo. + 1 κ p p, l 0,T ;L Ω p E0 + E0 p + E 0 m,

19 Proo. To prove te energy estimate, we test te problem 4.1 wit φ, ψ = v, p,, and problem 4. wit φ p, ϕp, ψp = d tu, d t V, p p,. Note tat, due to 4.8, we ave V = d t U. Adding te equations togeter and multiplying by we get ρ v L Ω vn L Ω + v v n L Ω +µ Dv L Ω +ρ mr m v τ L V n τ L + v + 1 τ V n τ L + ρ p U E U n E + U U n s 0 E + + κ p p, L Ω p + ρ mr m + 1 U Canceling v + ρ mr m V L Ω p V n L Ω p + V V n L Ω p p p, L Ω p pn p, L Ω p + p p, pn p, L Ω p V n L V n n L + V n V n n L V τ L v τ L + V τ v τ L M U n M + U U n M cp p p,, v c p p n p,, v. τ L and using te discrete energy deined by , we ave E + Ep + Em + ρ d t v L Ω + µ Dv + ρ mr m V τ v τ L + ρ mr m d t V n L + ρ p + d tu E + d tu M + s 0 c p p p,, v L Ω + ρ mr m v τ V n τ L d t V L Ω p d t p p, L Ω p + κ p p, L Ω p c p p n p,, v + E n + En p + E n m Te term c p p p, pn p,, v arises in classical partitioned scemes or Navier Stokes/Stokes- Darcy coupling, and as been previously addressed in [45]. Following te similar approac as in [45], we can estimate te interace term using Caucy-Scwarz inequality A.5, Young s inequality A.3 or ϵ 1 > 0, and te local trace-inverse inequality A.4 in te ollowing way: c p p p, pn p,, v = p ndx p, ϵ 1 p p, pn p, L + v ϵ 1 pn p, v L ϵ 1C T I p p, pn p, L Ω + v ϵ L. 1 Finally, using trace inequality A.7, Poincaré-Friedrics inequality A.6, and Korn s inequality A.8, we ave c p p p, pn p,, v ϵ 1C T I p p, pn p, L Ω + C T C K C P F Dv ϵ L Ω

20 In order to recover control on te pressure in te luid domain, we exploit te in-sup stability o te approximation spaces V and Q. Namely, spaces V and Q are in-sup stable provided in p sup, Q φ V b p,, φ φ H 1 Ω p, L Ω = β > Combining te in-sup condition 4.13 wit 4.1 tested wit ψ = 0 we obtain β p, k=1, L Ω sup T kφ φ H 1 Ω φ V 4.14 were β > 0 is a constant independent o te mes caracteristic size and T k φ is a sortand notation or te ollowing terms T 1 φ := ρ d t v φ dx + ρ mr m Ω T φ := a v, φ + c pp n p,, φ. v τ V n τ φ τ dx, Exploiting te Caucy-Scwarz A.5, trace A.7, and Poincaré - Friedrics A.6 inequalities, we obtain te ollowing upper bounds T 1 φ sup φ C T C P F ρ d t v v τ V n L Ω + ρ m r m τ, H 1 Ω φ V L sup φ V T φ φ H 1 Ω µ Dv L Ω + C T C P F κ 1/ κ p n p, L Ω p. Let us now multiply te square o 4.14 as well as te bounds or T k φ by ϵ and combine te resulting inequality wit 4.11 and 4.1 to get, E + Ep + Em + +µ 1 C T C K C P F + ρ mr m 1 4ϵ CT CP F ρ m r m v ϵ µ 4µ ϵ 1 τ V n τ L Ω ρ 1 4ϵ CT CP F ρ d t v Dv L Ω L + ρ mr m V τ v τ + ρ mr m d t ξ n L + ρ p d t V L Ω p + d tu E + d tη + s 0 ϵ 1C T I d t p p, L Ω p + ϵ β p, L Ω + κ p p, L Ω p ϵ C T C P F κ M L κ p n p, L Ω p En + En p + E n m

21 Note tat ere we used equalities η = U and ξ = V. Ater summing up wit respect to te time index n we observe tat [ κ p p, L Ω p ϵ C T C P F ] κ p n p, κ L Ω p By setting ϵ 1 = = s 0 C T I, and 1 ϵ C T C P F κ p n p, κ L Ω p + κ p N p, L Ω p. ϵ = 1 min ρ CT, C P F ρ m r m CT, C T C K C P F C T I C P F µ µ s, 0 we prove te desired estimate wit δ = ϵ β. 5 Error Analysis κ C T C P F In tis section we analyze te convergence rate o te proposed metod. For te spatial approximation we apply Lagrangian inite elements o polynomial degree k 1 or all te variables, except or te luid pressure, or wic we use elements o degree s < k. We assume tat te regularity assumptions reported in Lemma 1 o te Appendix are satisied and tat our FEM spaces satisy te usual approximation properties, as well as te luid velocity-pressure spaces satisy te discrete in-sup condition Let P be te Lagrangian interpolation operator onto V p, and let Π/p be te L -ortogonal projection onto Q /p, satisying p r Π r p r, ψ = 0, ψ Q r, r {, p}. 5.1 Ten I := P is a Lagrangian interpolation operator onto V m operator S, R : V V Q, deined or all v V by. Introduce a Stokes-like projection S v, R v V Q, 5. S v = I v, 5.3 a S v, φ br v, φ = a v, φ, φ V suc tat φ = 0, 5.4 bψ, S v = 0, ψ Q. 5.5 Te inite element teory or Lagrangian interpolants and L projections [19] gives te classical approximation properties reported in Lemma 3. Since P is te Lagrangian interpolant, so is its trace on. Tereore, we inerit optimal approximation properties also on tis subset. We reer to Lemma 3 or a precise statement o tese properties. Since te test unctions or te partitioned sceme do not satisy te kinematic coupling condition between te luid and te membrane, we start by deriving te monolitic variational ormulation wit te test unctions belonging to te test spaces required by te partitioned sceme. Assuming 1

22 η = U t, ξ = V t, and φ p = ζ, ϕ p = χ, te weak ormulation is given as ollows: Find X = v, U, V, p, p p V V p V p Q Qp suc tat or all Y = φ, φp, ϕp, ψ, ψp V V p V p Q Qp we ave PX, Y = FY + τ σ nφ τ φp τ dx. 5.6 To analyze te error o our numerical sceme, denote e = v v, e p = p p,, e u = U U, e v = V V, and e p = p p p p,. We start by subtracting rom 5.6, giving rise to te ollowing error equations: ρ d t e φ dx+a e Ω +ρ p v d t e u ϕ p dx + ρ p +ρ m r m Ω p e, φ b e p +a p e p, ψ p b epe p, φ p + b epψ p, d te u, φ +b ψ, e e τ e n v τ +ρ m r m φ τ dx d t e v φ p dx + a ee u, φ p + s 0 d t e p ψ p dx Ω p Ω p + ρ m r m d t e nφ p ndx e v τ e τ φ p τ dx + a me u, φ p c ep e p, φ p + c epψ p, d te u c p ψ p, e + c p e n p, φ = R φ + R s v φ p + R v ϕ p + R p ψ p +R os φ φp, Y V V p V p Q Qp, were te consistency errors are given by R φ = ρ d t v t v φ dx + c pp n p p p, φ Ω R s φ p = ρ p d t V t V φ p dx + ρ mr m d t V t V φ p dx, Ω p R v ϕ p = ρ p d t U t U ϕ p dx, Ω p R p ψ p = s 0 d t p p t p p ψ p dx + b epψ p, d tu t U Ω p R os φ φp = +c ep ψ p, d tu t U, τ σ v τ V n τ n + ρ m r m φ τ φp τ dx. Let us split te error o te metod into te approximation error θ r and te truncation error δ r, wit r =, p, u, v, p, as ollows: e = v v = v S v + S v v =: θ + δ, Π p + Π p p, =: θ p + δ p, e u = U U = U P U + P U U =: θu + δu, e p = p p, = p e v = V V = V P V + P V V =: θv + δv, e p = p p p p, = p p Π p p p + Π p p p p =: θ p + δp. p,

23 Our plan is to rearrange te terms in te error equations so tat we ave te truncation errors on te let and side, and te consistency and interpolation errors on te rigt and side. Ater tat, we will coose φ = δ, ψ = δ p, φp = d tδu, ϕ p = d tδv, and ψ p = δ p, and use te stability estimate or te truncation errors. Finally, we will bound te remaining terms, and use te triangle inequality to get te error estimates or e, e u, e v, and e p. Rearranging te terms in te error equations, and using property 5.1 o te projection operator, we ave Π p ρ d t δ φ dx + a δ, φ + c pδp n, φ + ρ mr m Ω δ τ δ n v τ φ τ dx b δ p, φ +b ψ, δ + c p δp n, φ + ρ p δv d t δu ϕ p dx + ρ p d t δv φ p dx + a eδu, φ p Ω p Ω p + s 0 d t δp ψ p dx + a pδp, ψ p b epδp, φ p + b epψ p, d tδu + ρ m r m d t δv nφ p ndx Ω p +ρ m r m ρ δv τ δ τ φ p τ dx+a mδu, φ p c ep δ = R φ + R s Ω d t θ φ dx a θ φ p + R v, φ + b θ p, φ p +c epψ p, d tδu ϕ p + R p ψ p + R os φ φp θ τ θ n v τ ρ m r m φ τ dx ρ p θv ϕ p dx+ρ p Ω p +b ep θp, φ p b epψ p, d tθu ρ m r m d t θv nφ p ndx ρ mr m p c p ψ p, δ v, φ b ψ, θ + ρ p d t θu ϕ p dx Ω p d t θv φ p dx a eθu, φ p a pθp, ψ p Ω p θv τ θ τ p φ τ dx a m θu, φ p + c ep θp, φ p c epψ p, d tθu + c p ψ p, θ + c p θp n, φ, 5.7 or all Y V V p V p Q Qp. Let Eδ n be deined as Eδ n = ρ δn L Ω + ρ p δn v L Ω p + 1 δn u E + s 0 δn p L Ω p + ρ mr m δv n L + 1 δ u n M. Note tat Eδ n corresponds to te total discrete energy o te sceme tat appears in Teorem 1 in terms o te truncation error. Teorem. Consider te solution v, p p,, V, U, ξ, η, p p, o Assume tat te time step condition 4.9 olds, and tat te true solution v, p p, V, U, ξ, η, p p satisies A.9. Ten, te ollowing estimate olds: v v l 0,T ;L Ω + γ v v l 0,T ;H 1 Ω + V V l 0,T ;L Ω p + p p p p, l 0,T ;L Ω p + p p p p, l 0,T ;H 1 Ω p + ξ ξ l 0,T ;L 0,L + U U l 0,T ;H 1 Ω p + η η l 0,T ;H 1 0,L + p p, l 0,T ;L Ω Ck B 1 v, U, η, p p + C k+ B U, V, η, ξ, p p 3

24 were +C B 3 v, U, V, η, ξ, p p + C s+ p l 0,T ;H s+1 Ω, B 1 v, U, η, p p = v l 0,T ;H k+1 Ω + v l 0,T ;H k+1 Ω + p p l 0,T ;H k+1 Ω p + tv l 0,T ;H k+1 Ω + t U l 0,T ;H k+1 Ω p + tp p l 0,T ;H k+1 Ω p + tη l 0,T ;H k+1 0,L + p p l 0,T ;H k+1 Ω p + U l 0,T ;H k+1 Ω p + η l 0,T ;H k+1 0,L, B U, V, η, ξ, p p = t p p l 0,T ;H k+1 Ω p + ttu l 0,T ;H k+1 Ω p + ttv l 0,T ;H k+1 Ω p + tt ξ l 0,T ;H k+1 0,L + tv l 0,T ;H k+1 Ω p + tξ l 0,T ;H k+1 0,L + p p l 0,T ;H k+1 Ω p + V l 0,T ;H k+1 Ω p + tu l 0,T ;H k+1 Ω p + tv l 0,T ;H k+1 Ω p + tξ l 0,T ;H k+1 0,L, B 3 v, U, V, η, ξ, p p = tt v L 0,T ;L Ω + tp p L 0,T ;H 1 Ω p + ttp p L 0,T ;L Ω p + tt U L 0,T ;H 1 Ω p + tttu L 0,T ;L Ω p + tttη L 0,T ;L 0,L + tttv L 0,T ;L Ω p + tt U l 0,T ;L Ω p + ttη l 0,T ;L 0,L + ttv l 0,T ;L Ω p + η l 0,T ;H 0,L + tt v L 0,T ;L. Proo. Wit te purpose o presenting te proo in a clear manner, we will separate te proo into our main steps. Step 1: Application o te stability result 4.10 to te truncation error equation. Coose φ = δ, ψ = δ p, φp = d tδu, ϕ p = d tδv, and ψ p = δ p in equation 5.7. Tanks to 5.5, te pressure terms simpliy as ollows Furtermore, we ave ρ m r m = ρ m r m θ since, due to.4, b δ p, θ = b δ p, S v = τ θv n τ δ τ dx ρ m r m d t θv τ δ τ dx ρ m r m = ρ m r m θ θ v τ θ τ d t δu τ dx v τ θ τ d t δu τ δ τ dx d t θv τ δ τ dx 5.9 θ v τ θ τ = V τ I V τ v τ + I v τ = 0 on. Ten, multiplying equation 5.7 by, summing over 0 n N 1, and using te stability estimate 4.10 or te truncation error, we get E N δ + ρ d t δ L Ω + ρ mr m 4 δ τ δv n τ L + γ Dδ L Ω

25 + ρ mr m + Eδ 0 + ρ ρ m r m δv τ δ τ d t δu E + L + ρ mr m d t δv n L + ρ p d t δv L Ω p d t δu M + 4 d t δ p L Ω p + κ δ p L Ω p R δ + R s d t δ u + R v d t δ v + R p δ p + R os δ d t δ u Ω d t θ δ dx d t θv τ δ τ dx ρ p a e θu, d t δu ρ m r m d t θv nd t δu c ep δp, d t θu a θ, δ a p θp, δp + Ω p θ + ndx + b θ p v d t δv dx+ρ p b ep θp, d t δu a m θu, d t δu c p δp, θ, δ +ρ p d t θu d t δv dx Ω p d t θv d t δu dx Ω p + b ep δ p, d t θ u c ep θ p, d t δ u c p θ n p, δ Te rigt and side o 5.10 consists o consistency error terms R, R s, R v, R p, te operator splitting error R os, and mixed truncation and interpolation error terms. We will proceed by bounding te consistency error terms. Step : Te consistency error estimate. In tis step we will use te ollowing bound o te consistency error terms proved in Lemma 5 in te Appendix: R δ + R s d t δu + R v d t δv + R p δp + R os δ d t δu C tt v L 0,T ;L Ω + tp p L 0,T ;H 1 Ω p + ttp p L 0,T ;L Ω p + ttu L 0,T ;H 1 Ω p + ttt U L 0,T ;L Ω p + tttη L 0,T ;L 0,L + tttv L 0,T ;L Ω p + ttu l 0,T ;L Ω p + tt η l 0,T ;L 0,L + ttv l 0,T ;L Ω p + η l 0,T ;H 0,L + ttv L 0,T ;L +Aδ, δ p, δ v, δ u, 5

26 were Aδ, δ p, δ v, δ u = γ Dδ L Ω + ρ mr m 4 δ τ δv τ L κ δ p L Ω p + ϵ δ N v L Ω p + δn v L + DδN u L Ω p +C δv n L Ω p + δn v L + Dδn u L Ω p. Reerring to te terms collected into te expression Aδ, δ p, δ v, δ u, we observe tat te discrete Gronwall Lemma is required to obtain an upper bound. Step 3: Te mixed truncation and interpolation error terms estimate. In tis step we estimate te remaining terms o 5.10, wic are terms tat contain bot truncation and interpolation error. Using Caucy-Scwartz A.5, Young s A.3, Poincaré - Friedrics A.6, and Korn s A.8 inequalities, we ave te ollowing: ρ d t θ δ dx C d t θ Ω L Ω + γ Dδ 8 L Ω. Furtermore, using Young s A.3, Korn s A.8, and trace A.7 inequalities we can estimate a θ, δ +a p θp, δp c p δp, θ +c p θp n, δ C + γ 8 Dδ In a similar way, we bound ρ m r m d t θv τ δ L Ω + C τ dx C Te next two terms can be controlled as ollows b ep δp, d t θu +c ep δp, d t θu We bound te pressure term as ollows b θ p, δ C θp L Ω p d t θv L + γ 6 κ δ p θ p L Ω + γ 8 κ δ p L Ω p. L Ω p +C Dθ L Ω Dδ L Ω. Dδ L Ω. d t θ u L Ω p. To estimate te remaining terms, we use discrete integration by parts in time. Using equation A., we ave b ep θp, d t δu = α θp N δu N dx α d t θp δudx n C ϵ θp N L Ω p Ω p Ω p 6

27 and +ϵ Dδu N L Ω p + C c ep θp, d t δu = α θp N δu N d t θ p ndx α L Ω p + C Dδ n u L Ω p, d t θ p δ n u ndx C ϵ θ N p L Ω p Also, ρ p +ϵ Dδu N L Ω p + C d t θ p L Ω p + C Ω p d t θ u d t δ v dx = ρ p Ω p d t θ N u δ N v dx ρ p Dδ n u L Ω p. d tt θu Ω p δ n v dx and C ϵ d t θu N L Ω p + ϵ δn v L Ω p + C ρ p d tt θ u θv d t δv dx = ρ p θv Ω Ω N δu N dx + ρ p p p C ϵ θv N L Ω p + ϵ DδN u L Ω p + C In a similar way, and ρ p d t θ v Ω p d t θ v d t δ u dx = ρ p Ω p d t θ N v δ N u dx ρ p C ϵ d t θv N L Ω p + ϵ DδN u L Ω p + C Furtermore, ρ m r m +ρ m r m d tt θ v L Ω p + C L Ω p + C d t θv Ω p δ n v L Ω p, δ n v dx Dδ n u L Ω p. d tt θv Ω p L Ω p + C δ n udx Dδ n u L Ω p. d t θv nd t δu ndx = ρ m r m d t θv N nδu N ndx a e θu, d t δu d tt θv nδu n ndx C ϵ d t θv N L + ϵ δn u M +C d tt θv L + C δu n M. = a e θu N, δu N + 7 a e d t θ u, δ n u C ϵ Dθ N u L Ω p

28 Lastly, +ϵ Dδu N L Ω p + C a m θu, d t δu Dd t θ u L + C = a m θu N, δu N + Dδ n u L Ω p. a m d t θ u, δ n u C ϵ θu N M + ϵ δu N M + C d t θu M + C δ u M. Using te estimates rom Steps 1-3, we ave T E N ϵ + G + S + T, 5.11 were T E denotes te terms on te let-and side o te stability estimate or te truncation error + γ + T E = E N δ + ρ Dδ L Ω + ρ mr m 4 d t δu E + d t δ L Ω + ρ mr m δv τ δ τ d t δu M + 4 δ L τ δv n τ + ρ p d t δp L Ω p + L d t δ v L Ω p κ δ p L Ω p, N ϵ denotes te terms at time N, multiplied by ϵ, tat can be incorporated in te let-and side N ϵ = ϵ Dδu N L Ω p + δn v L Ω p + δn v L + δn u M, G denotes te terms or wic we ave to apply te discrete Gronwall lemma G = C S denotes te approximation error terms S = C Dθ Dδu n L Ω p + δn v L Ω p + δn v L + δn u M, L Ω + θ p L Ω + θ p L Ω p + d tθ L Ω + d tθu L Ω p + d t θp L Ω p + d tθp L Ω p + d ttθu L Ω p + d ttθv L Ω p + d ttθv + d t θv L Ω p + d tθv L + Dd tθu L Ω p + d tθu M +C max θp n 0 n N L Ω p + θn p L Ω p + θn v L Ω p + Dθn u L Ω p + θu n M + d t θu n L Ω p + d tθv n L Ω p + d tθv n L. L 8

29 Using Lemma 6, we bound te approximation error terms as ollows S C k v l 0,T ;H k+1 Ω + v l 0,T ;H k+1 Ω + p p l 0,T ;H k+1 Ω p + tv l 0,T ;H k+1 Ω + t U l 0,T ;H k+1 Ω p + tp p l 0,T ;H k+1 Ω p + tη l 0,T ;H k+1 0,L + p p l 0,T ;H k+1 Ω p + U l 0,T ;H k+1 Ω p + η l 0,T ;H k+1 0,L +C k+ t p p l 0,T ;H k+1 Ω p + ttu l 0,T ;H k+1 Ω p + ttv l 0,T ;H k+1 Ω p + ttξ l 0,T ;H k+1 0,L + t V l 0,T ;H k+1 Ω p + tξ l 0,T ;H k+1 0,L + p p l 0,T ;H k+1 Ω p + V l 0,T ;H k+1 Ω p + t U l 0,T ;H k+1 Ω p + tv l 0,T ;H k+1 Ω p + tξ l 0,T ;H k+1 0,L. 5.1 In te statement o te teorem, terms multiplying k are denoted by B 1 v, U, η, p p, and terms multiplying k+ are denoted by B U, V, η, ξ, p p. Lastly, T denotes te bound on te consistency error terms T = C tt v L 0,T ;L Ω + tp p L 0,T ;H 1 Ω p + ttp p L 0,T ;L Ω p + ttu L 0,T ;H 1 Ω p + ttt U L 0,T ;L Ω p + tttη L 0,T ;L 0,L + tttv L 0,T ;L Ω p + ttu l 0,T ;L Ω p + tt η l 0,T ;L 0,L + tt V l 0,T ;L Ω p + η l 0,T ;H 0,L + ttv L 0,T ;L. Te terms multiplying C in T give rise to terms denoted by B 3 v, U, V, η, ξ, p p in te statement o te teorem. Finally, using estimates 5.1, approximation properties A.10-A.15, triangle inequality, and te discrete Gronwall inequality, we prove te desired estimate, except or te pressure error in te luid domain. Step 4: analysis o te luid pressure error. To control tis part o te error we proceed as or te stability estimate. More precisely, we start by taking ψ = 0, φp = 0, ϕp = 0, and ψp = 0 in 5 and rearranging it as ollows b δ p, φ = ρ d t e φ dx + a e, φ + ρ e τ e n v τ mr m φ Ω τ dx + c p e n p, φ b θ p, φ R φ R os φ For simplicity o notation, let us group te terms on te rigt and side o te previous equation, T φ := ρ d t e φ dx + a e, φ Ω e τ e n v τ + ρ m r m φ τ dx + c pe n p, φ + b θ p, φ Owing to te in-sup condition 4.13 between spaces V independent o te mes caracteristic size suc tat, and Q tere exists a positive constant β β δ p L Ω sup φ V T φ R φ R os φ 9 φ H 1 Ω. 5.14

30 Moving along te lines o te stability estimate, te ollowing upper bounds or te rigt and side o 5.14 old true, wit a generic constant C wic depends on te trace A.7, Korn A.8 and Poincaré-Friedrics A.6 inequalities, as well as on te parameters o te problem, sup φ V T φ φ H 1 Ω + d t e L Ω + C e H 1 Ω + δp n H 1 Ω p + θp n H 1 Ω p + θ p L Ω δ τ δv n τ L Using te bounds detailed in Lemma 5 o te Appendix, we get + d t θ τ L. sup φ V R φ R os φ τ C d t v t v L Ω + p p p n p L Ω p φ H 1 Ω + τ v τ V n τ σ n + ρ m r m L. Finally, we replace te previous estimates into 5.14, square all terms, sum up wit respect to n and multiply by. Tere exists a positive constant c small enoug suc tat c δ p L Ω d t e L Ω + δ τ δv n τ + e H 1 Ω + δn p H 1 Ω p + θn p H 1 Ω p + θ + p p L + d t θv L p L Ω + d tv t v L Ω. p n p L Ω p + τ σ n + ρ m r m v τ V n τ To conclude, combining te triangle inequality wit te approximation properties o te discrete pressure space and bounding te rigt and side using Lemma 4 and equation 5.11, we obtain p p, l 0,T ;L Ω C C k B 1 v, U, η, p p + C k+ B v, U, V, η, ξ, p p + C B 3 v, U, V, η, ξ, p p + C s+ p l 0,T ;H s+1 Ω. L 6 Numerical results Te ocus o tis section is on veriication o te results presented in tis work and exploration o poroelastic eects in te model. We test te sceme on a classical bencmark problem used or convergence studies o luid-structure iteration problems [4,8,11,13,31]. In Example 1, we present te convergence o our sceme in space and time. Furtermore, we validate te necessity o te stability condition 4.9. In Example we analyze te role o poroelastic eects in blood low. In particular, we compare our results to te ones obtained using a purely elastic model in Example. We distinguis a ig permeability and a ig storativity case, and present a comparison between te two cases and te purely elastic model. 30

31 6.1 Example 1. We consider te classical test problem used in several works [4, 11, 13, 31] as a bencmark problem or testing te results o luid-structure interaction algoritms or blood low. In our case, te low is driven by te time-dependent pressure data: p in t = { pmax 1 cos πt T max i t T max 0 i t > T max, 6.1 were p max = dyne/cm and T max = s. For te elastic skeleton, we consider te ollowing equation o linear elasticity: ρ p D U Dt + βu σp = 0. Te additional term βu comes rom te axially symmetric ormulation, accounting or te recoil due to te circumerential strain. Namely, it acts like a spring term, keeping te top and bottom structure displacements connected in D, see, e.g., [6, 8, 50]. Te values o te parameters used in tis example are given in Table 1. Parameters Values Parameters Values Radius R cm 0.5 Lengt L cm 6 Membrane tickness r m cm 0.0 Poroelastic wall tickness r p cm 0.1 Membrane density ρ m g/cm Poroelastic wall density ρ p g/cm Fluid density ρ g/cm 3 1 Dyn. viscosity µ g/cm s Lamé coe. µ m dyne/cm Lamé coe. λ m dyne/cm Lamé coe. µ p dyne/cm Lamé coe. λ p dyne/cm Hydraulic conductivity κcm 3 s/g Mass storativity coe. s 0 cm /dyne Biot-Willis constant α 1 Spring coe. βdyne/cm Table 1: Geometry, luid and structure parameters tat are used in Example 1. Parameters given in Table 1 are witin te range o pysiological values or blood low. Te problem was solved over te time interval [0, 0.006] s. In order to veriy te convergence estimates rom Teorem, let te errors between te computed and te reerence solution be deined as e = v v re, e p = p p,re, e v = V V re, e p = p p p p,re, and e u = U U re. We start by computing te rates o convergence in time. In order to do so, ix x = and deine te reerence solution to be te one obtained wit t = Table sows te error between te reerence solution and solutions obtained wit t = 10 6, , 10 5, and or te luid velocity v, luid pressure p, pressure in te pores p p, displacement U and its velocity V, respectively. To study te convergence in space, we take = and deine te reerence solution to be te one obtained wit x = r p /14 = Table 3 sows errors between te reerence solution and te solutions obtain using x = 0.01, 0.015, and To veriy te necessity o te time-step condition 4.9, we compute te total energy E N o te system using dierent time steps. Te time at wic E N is computed is eiter te time wen E N becomes greater tan 10 50, or te inal time t N = 6 ms. Figure sows te relation o te energy o 31

32 t e l L rate e l H 1 rate e p l L rate e v l L rate e 1-8.8e - 8.9e 1-3.0e e e e e e e e e e e e e t e p l L rate e p l H 1 rate e u l H 1 rate e 1-4.1e 1-7.7e e e e e e e e e e Table : Convergence in time. x e l L rate e l H 1 rate e p l L rate e v l L rate r p /4.4e 1-6.8e 1-3.1e 1 -.5e - r p /5 1.9e e e e 1.46 r p /6 1.5e e e e 1.37 r p /7 1.e e e e 1.6 x e p l L rate e p l H 1 rate e u l H 1 rate r p /4 1.e e 1-3.1e 1 - r p /5 0.9e e e r p /6 0.8e e e r p /7 0.6e e e Table 3: Convergence in space. E N x = r p /6 x = r p /4 x = r p / t 14 x t x Figure : Veriication o te time-step condition 4.9. Let: Relation between te total energy o te system and te time step. Rigt: Relation between x and te critical. te system and te time step let, and te relation between x and te critical rigt. Indeed, we observe a linear relation between x and te critical value o, wit te proportionality constant.4e 3. Tis is less restrictive tan te prediction 4.9 rom te teory, were te proportionality 3

33 constant or te parameters in Table 1 can be estimated as 3.5e 7, indicating tat te sceme enjoys better stability properties ten prescribed by Example. In tis example we compare our numerical results to te ones obtained using a purely elastic model or te outer layer o te arterial wall. More precisely, wile te luid and te membrane are modeled as beore, we assume tere is no luid contained witin te wall, and we model te tick wall using D linear elasticity ρ p D U Dt + βu σe = 0 in Ω p t or t 0, T. Te problem is solved using an operator-splitting approac perormed in te same spirit as in tis manuscript. For te purpose o understanding te poroelastic eects to te structure displacement, we distinguis two cases: te ig storativity case s 0 >> κ, and te ig permeability case κ >> s 0. We give a comparison o te results obtained using te elastic model or te outer wall, and poroelastic model using two dierent values or s 0, and two dierent values or κ. Te irst test case or te poroelastic wall will correspond to te parameters s 0 and κ rom Example 1 s 0 = , κ = , te second test case will correspond to te increased value o s 0 s 0 = 10 5, κ = , and te tird example to te increased value o κ s 0 = , κ = Figure 3 sows te pressure pulse colormap and velocity streamlines obtained wit te two models. Te velocity magnitude is sown in Figure 4. To quantiy te dierences, we compute average quantities on eac vertical line Si r o te computational mes Ω r, corresponding to te position x i = i x, were x = and r {, m}. Te quantities o interest are membrane displacement, te mean pressure, and te lowrate in te lumen: p x i = 1 S i S i p, ds, p p x i = 1 S p i S p i p p, ds, Qx i = S i v e x ds. Figure 5 sows a comparison between te lowrate in te lumen, membrane displacement, and te mean pressure in te lumen and in te wall, obtained using a poroelastic model and an elastic model. In te ig permeability regime te structure displacement is te smallest, wile in tis case we observe te largest mean pressure in te wall. In te ig storativity regime, we observe a delay in te pressure wave propagation speed, and qualitatively dierent displacement. 7 Conclusion Te ocus o tis paper is on modeling and implementation o a luid-poroelastic structure interaction problem. In particular, we study te interaction between te luid a multilayered wall, were te wall consists o a tin membrane, and a tick poroelastic medium. We proposed an explicit numerical algoritm based on te Lie operator splitting sceme. An alternative discrete problem ormulation based on Nitsce s metod or te enorcement o te interace conditions is under study. Tis new metod can accommodate a mixed ormulation or te Darcy s equations. We prove te conditional stability o te algoritm, and derive error estimates. Stability and convergence results are validated by te numerical simulations. Te drawback o te sceme is tat it requires pressure ormulation or te Darcy equation. Concerning te application o te sceme to 33

34 Figure 3: Pressure in te lumen, velocity streamlines, and pressure in te wall at times t = 1.5 ms, t = 3.5 ms and t = 5.5 ms. Te outer layer o te arterial wall is model using a elastic model top, poroelastic model wit s 0 = , κ = middle top, poroelastic model wit s 0 = 10 5, κ = middle bottom, and poroelastic model wit s 0 = , κ = 10 4 bottom. blood low in arteries, we test numerically te porous eects in te wall, comparing results obtained wit dierent coeicients to te ones obtained using a purely elastic model. We observe dierent beavior depending on te storativity or permeability dominant regime. 34

35 Figure 4: Velocity magnitude at times t = 1.5 ms, t = 3.5 ms and t = 5.5 ms. Te outer layer o te arterial wall is model using a elastic model top, poroelastic model wit s 0 = , κ = middle top, poroelastic model wit s 0 = 10 5, κ = middle bottom, and poroelastic model wit s 0 = , κ = 10 4 bottom. 35