MOX-Report No. 61/2015

Transcription

1 MOX-Report No. 61/2015 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for te treatment of eart valves as mixed time varying boundary conditions Tagliabue, A.; Dedè, L.; Quarteroni, A. MOX, Dipartimento di Matematica Politecnico di Milano, Via Bonardi Milano (Italy) mox-dmat@polimi.it ttp://mox.polimi.it

2 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for te treatment of eart valves as mixed time varying boundary conditions Anna Tagliabue 1,, Luca Dedè 2, and Alfio Quarteroni 1,2, 1 MOX Modeling and Scientific Computing Matematics Department F. Briosci Politecnico di Milano via Bonardi 9, Milano, 20133, Italy 2 CMCS Cair of Modeling and Scientific Computing MATHICSE Matematics Institute of Computational Science and Engineering EPFL École Polytecnique Fédérale de Lausanne Station 8, Lausanne, CH 1015, Switzerland Abstract In tis work, we study te blood flow dynamics in idealized left ventricles (LV) of te uman eart modelled by te Navier-Stokes equations wit mixed time varying boundary conditions (BCs). Te latter are introduced for simulating te functioning of te aortic and mitral valves. Based on te extended Nitsce s metod firstly presented in [Juntunen and Stenberg, Matematics of Computation, 2009], we propose a formulation allowing an efficient and straigtforward numerical treatment of te opening and closing pases of te eart valves wic are associated to different kind of BCs, namely natural and essential. Moreover, our formulation includes terms preventing te numerical instabilities associated to backflow divergence, i.e. nonpysical reinflow at te valves. We present and discuss numerical results for te LV obtained by means of Isogeometric Analysis for te spatial approximation wit te aim of bot analysing te formulation and sowing te effectiveness of te approac. In particular, we sow tat te formulation allows to reproduce meaningful results even in idealized LV. Key words. extended Nitsce s metod; mixed time varying boundary conditions; eart fluid dynamics; left ventricle. Corresponding autor. anna.tagliabue@polimi.it. Pone: , Fax: Currently on leave from Politecnico di Milano. 1

3 2 A. Tagliabue, L. Dedè, A. Quarteroni 1 Introduction In te last years, several efforts ave been dedicated to describing, studying, and understanding te blood flow dynamics inside te uman eart, especially focusing on te uman Left Ventricle (LV) [13, 42, 65]. Te latter is te eart camber wic distributes te oxygenated blood to te systemic circulation and plays a primal role in te cardiac activity. In particular, due to te recent tecnological advances in medical imaging tecniques [58], a preliminary understanding of te main features caracterizing te cardiac emodynamics as been performed by using e.g. pase-contrast magnetic resonance imaging [41, 38] and by ecocardiograpy tecniques [32, 40]. Neverteless, bot of tese metods present some deficiencies [13], among wic we recall te filtering and reconstruction related to te image processing, te presence of aliasing artefacts, te dependence on te temporal and spatial resolution available, and te use of pase-averaged flows description, wic may neglect small-scale instabilities and eart beat to beat variations. More recently, pysical and computational models ave become of paramount importance in providing more detailed informations on te blood flow features, especially for predicting eart diseases [2, 46, 65]. Indeed, te potentiality of Scientific Computing in understanding te emodynamics bot in pysiological and patological conditions can bring a significant complement to medical imaging or in vitro studies [26], altoug te complexity of te penomena at and remains a callenging task from bot te matematical and numerical points of view. In fact, even if one only focuses on te LV fluid dynamics, te problem sould be addressed by accounting for te complex sape of te eart camber wic expands, contracts, and experiences large displacements and, at te same time, wose movement is governed by complex electrical-fluid-structure interactions. Moreover, te blood flow is igly influenced by te presence of te valves tat yield an additional fluid-structure interaction problem, wit a regime varying during te eart beat from laminar to transitional, and eventually turbulent [13]. Because of tese aspects and te additional difficulty in obtaining accurate clinical data describing, e.g., te mecanical properties of te wall or te blood flow profile troug te mitral valve, simplified models ave been introduced and studied, starting from te earliest works based on te immersed boundary metod [51, 52, 53, 47] up to more recent studies, e.g. [1, 2, 20, 21, 22, 67]. In tis work, we perform a preliminary study of te emodynamics inside te LV by considering an idealized LV represented as a truncate prolate ellipsoid, wose motion is prescribed by a wall law. By assuming a Newtonian reology for te blood flow, we model it by te incompressible Navier-Stokes equations in te Arbitrary Lagrangian Eulerian (ALE) formulation [23, 24, 25, 27, 28], wic we spatially approximate by using Isogeometric Analysis (IGA) [3, 5, 16]. Indeed, in te last years, IGA as been largely used for te numerical approximation of a wide range of problems providing accurate and efficient solutions, also in te context of cardiovascular system modelling [5, 62]. Specifically, we refer to NURBS-based IGA [16] in te framework of te Galerkin metod, bot for te matematical properties of te basis functions, e.g. NURBS basis functions wic can be globally C k -continuous in te computational domain for some 0 k p 1, wit p te polynomial degree, and for te possibility to exactly represent conic sections, as it is te idealized LV. Indeed, te configuration of a truncate prolate speroid used as idealized LV is considered in literature [1, 2] as a sufficiently accurate geometry to represent te average endocardial sape of different uman subjects; in our model, te motion of te LV is completely defined troug time variations of te upper diameter and of te major semi-axis of te ellipsoid. Suc time dependent functions sould, in principle, be computed by solving an electrical-fluid-structure interaction problem; owever, in te present work we instead prescribe te ventricle wall displacement derived from a simple elastic

4 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for eart valves 3 model and a flow-discarge function following [1, 2]. In describing te complex fluid dynamics in te LV, wic includes asymmetric vortex structures allowing an efficient filling and a natural redirection towards te aorta, one of te most difficult aspects consists in accounting for te valves in a realistic and pysiologically meaningful manner. In tis respect, tis work focuses on te formulation of mixed time varying (MTV) boundary conditions (BCs) for te Navier-Stokes equations wic allow a simplified but realistic treatment of te mitral and aortic valves for te study of te LV fluid dynamics. Specifically, we propose a weak treatment of suc BCs by embedding te valves opening and closing pases into te variational formulation of te problem. Indeed, we remark tat, from bot te matematical and numerical points of view, accounting for te presence of te valves troug commonly used BCs is a callenging task due to te different nature of te BCs associated to te opening and closing pases, wic namely switc during te eart beat from essential to natural and viceversa. In tis respect, even te well-posedness analysis of a simpler equation, as a parabolic equation, requires a careful treatment [57]. Te formulation of tese new kind of MTV BCs for te Navier-Stokes equations in ALE formulation is done in te framework of te extended Nitsce s metod (ENM) firstly proposed by Juntunen and Stenberg [36]. Specifically, by using a penalty tecnique we allow te switcing in time between te imposition of Diriclet and defective BCs of natural type [5, 31, 63]. On te one and, te Diriclet BCs are imposed weakly rater tan strongly in te test functions space and it as been sown in [7, 8, 33] tat tis leads to a better resolution of boundary layers wit respect to te strong imposition, eventually avoiding computationally expensive fine meses. On te oter and, wen imposing defective BCs of te natural type, we consider additional terms wit respect to te standard imposition of te natural BCs in weakly formulated problems; tese additional terms control te backflow velocity troug te outflow. In particular, suc terms prevent te numerical instabilities associated to backflow divergence [49] witout neiter perturbing pysically te problem nor introducing additional unknowns, as e.g. for te Lagrange multiplier metod [39]; in addition, te consistency of te metod is also preserved. Controlling backflow divergence is very important for te LV; indeed, te LV fluid dynamics is potentially affected by suc beaviour correspondingly to te outlet LV boundary representing te open aortic valve, possibly leading to numerical instabilities due to partial or total flow reversal, as well as local flow recirculation. Because of te similar beaviour of te mitral and aortic valves, bot of wic undergo opening and closing processes, we use te MTV BCs for bot of te valves modelled as orifices of infinitesimal tickness located at te upper equatorial diameter of te idealized LV. Neverteless, we furter improve te formulation of te BCs on te mitral valve by adding a regularizing ig order boundary term in a penalty fasion wic induces someow realistic inlet profiles. Indeed, altoug tere is a general lack of accurate clinical data describing te inflow profile troug te mitral valve, it as been igligted from visualizations [9, 55] tat te mitral valve as a strong influence on te intra-ventricular vortex structures. As a matter of fact, te study of te fluid dynamics of te LV, even in an idealized setting, cannot neglect an accurate and pysically meaningful modelling of te aortic and mitral valves. Terefore, in tis paper, we propose a general framework for modelling te valves functioning by means of mixed type BCs of te Navier-Stokes equations tat are treated numerically by means of te ENM. Te outline of tis work is as follows. In Sec. 2, we define te problem of modelling te blood flow in an idealized two-dimensional LV for wic we consider te incompressible Navier-Stokes equations in ALE formulation; we describe te LV geometry and te governing law for te imposed LV motion. In Sec. 3 we recall te Galerkin metod in te framework of NURBS-based IGA wit

5 4 A. Tagliabue, L. Dedè, A. Quarteroni VMS-LES formulation [4, 12, 62] and te generalized-α metod [14, 35, 66] for te discretization in space and time, respectively. In Sec. 4, we provide te formulation of te MTV BCs for te Navier-Stokes equations describing te valves; te equations are approximated by using te ENM. Firstly, we focus on modelling te aortic valve, ten we extend it to te mitral valve by introducing a suitable regularizing term yielding realistic inflow velocity profiles. Finally, in Sec. 5, we present and discuss some numerical results regarding te fluid dynamics of te LV and sow te effectiveness of te metod compared to results available in literature. Conclusions follow. 2 Modeling of blood flows in te idealized left ventricle We describe te model for te blood flow in an idealized LV. In Sec. 2.1, by assuming a Newtonian reology for te fluid in an expanding and contracting cavity undergoing large displacements, we consider te incompressible Navier-Stokes equations in ALE formulation [5, 23, 24, 25, 27, 28]. In Sec. 2.2, we describe a two-dimensional idealized LV geometry, in first approximation alf of an ellipse [1, 64, 67]. We focus on a model wit prescribed wall movement, for wic te governing law associated to te mecanical LV displacement following [1, 2]. Finally, in Sec. 2.4, we describe te matematical formulation of a set of BCs able to describe a pysiologically compatible two-dimensional LV model. 2.1 Navier-Stokes equations in ALE formulation Let Ω (t) R d, wit d = 2 or 3, be a time dependent spatial domain wit boundary Ω (t) representing te configuration at te current time t (0, T ) of a reference domain Ω R d. Specifically, te reference domain is mapped to te current configuration troug te ALE mapping φ (t) : Ω Ω (t) ; we indicate wit x and X te spatial coordinates of te current and reference configurations, respectively. Te dimensionless Navier-Stokes equations for an incompressible fluid in ALE (convective) formulation read: for all t (0, T ), find u : Ω (t) R d and p : Ω (t) R suc tat: u t + ((u û) ) u σ(u, p) = f in Ω (t), (2.1a) X u = 0 in Ω (t), (2.1b) complemented wit suitable initial and boundary conditions, were te partial time derivative is taken wit respect to te (fixed) reference domain Ω, wile te partial spatial derivatives are evaluated in te current configuration. Te vector field f : Ω (t) R d for all t (0, T ) indicates te external body forces and û is te (unknown) domain velocity. Moreover, te Caucy stress tensor σ(u, p) is defined as σ(u, p) = pi+ 2 Re D(u), wit Re te Reynolds number, D(u) := ( u + ut ) 2 is te strain rate tensor, and I is te identity tensor. Te initial condition of te fluid velocity is a divergence-free velocity field u 0 : Ω (0) R d. Moreover, we consider eiter essential or natural BCs [5, 6], te latter associated to te total momentum flux: Φ(u û; u, p) = u (u û) + σ(u, p), (2.2)

6 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for eart valves 5 as we will detail in Sec Specifically, if te Diriclet and natural BCs are expressed in terms of te vector fields g and G, tese read for any t (0, T ): u = g on Γ (t) D, Φ(u û; u, p)n = G on Γ (t) N, (2.3a) (2.3b) respectively, were Γ (t) D and Γ(t) N are subsets of Ω(t) at any time t (0, T ) and n indicates te outward directed unit vector normal to Γ (t) N. Te weak formulation of Eq. (2.1) complemented wit te BCs (2.3) considers te following time dependent trial and weigting spaces for te velocity field at t (0, T ): V (t) g := {u [H 1 (Ω (t) )] d : u = g on Γ (t) D }, W (t) V (t) 0 := {u [H1 (Ω (t) )] d : u = 0 on Γ (t) D }, (2.4a) (2.4b) respectively, wile for te pressure we use Q (t) := L 2 (Ω (t) ). Ten, te variational formulation of problem (2.1) reads: for all t (0, T ), find (u, p) V g (t) Q (t) : B((ϕ, q), (u, p); û) = F(ϕ, q) (ϕ, q) W (t) Q (t), (2.5) were: B((ϕ, q), (u, p); û) := ϕ u Ω (t) t dω + 2 D(ϕ) : D(u) dω + Re Ω (t) p ϕ dω + q u dω, Ω (t) Ω (t) F(ϕ, q) := Ω (t) ϕ f dω + Γ (t) N ϕ G dγ. Ω (t) ϕ ((u û) ) u dω (2.6a) (2.6b) 2.2 Left ventricle geometry and wall displacements We consider now te two-dimensional case, for wic d = 2. We geometrically represent te LV as alf of an ellipse wit moving walls caracterized by te time dependent functions D : (0, T ) R and H : (0, T ) R representing te upper diameter and te major semiaxis of te ellipsoid, respectively; see Fig. 1a. Te geometry at te beginning of te diastolic filling pase is taken as te reference configuration Ω; te diameter of te reference configuration is D(0) D 0 = 3.4 cm wit x coordinate between 1.7 cm and 1.7 cm, moreover D 0 is cosen as te reference lengtscale (L 0 ). Similarly, we set te eigt H(0) = H 0 = 6.8 cm, for wic H 0 /D 0 = 2 as e.g. in [21]; te base of te ellipsoid is fixed, for wic te apex is moving during te eart beat. Te inlet and outlet subsets of te boundary representing te mitral and aortic valves are located on te upper part of te LV. In te reference configuration, te mitral valve position is fixed between te x coordinates cm and cm, wile te aortic valve is fixed between te x coordinates cm and cm. Teir positions and sizes are fixed in time, despite te LV base moves accordingly wit te governing wall movement law, described in Sec. 2.3.

7 6 A. Tagliabue, L. Dedè, A. Quarteroni (a) Idealized 2D LV geometry. (b) Volumetric flow rate Q V (t) vs. time t for a 3D LV. Figure 1: Geometry representing te idealized uman LV (1a) wit te aortic valve in red and mitral valve in blue; flow rates in a 3D LV (1b). 2.3 Prescribed wall displacement of te left ventricle Te large displacements experienced by te LV wall and te strong fluid-structure interactions (FSI) play a fundamental rule in te LV flow. In te present work, we do not analyse te FSI problem governed by te coupling of te fluid and wall dynamics; rater we refer te interested reader to e.g. [43] for a state of te art in FSI problem for te LV. In tis paper, we are more interested in te fluid dynamics aspects by prescribing a priori te LV wall displacement based on a simple elastic model and a given flow-discarge profile, following [1, 2]. We remark tat for patient-specific LV geometries, a time dependent ventricle model derived from in vivo images data (MRI, Ecocardiograpy) could be used; see e.g. [37, 46]. We prescribe te LV wall motion as a dilatation map by controlling te time evolution of te equatorial diameter D(t) and te LV eigt H(t). We recall tat, in literature [13, 21, 45, 67], for te idealized uman LV te ratio between H and D is typically fixed at H(0)/D(0) During te cardiac cycle, tis ratio basically varies in tis range, as sown in Fig. 2b and it is determined by te governing law used to prescribe te LV wall movement. Following [1], we consider D(t) and H(t) for a tree-dimensional LV as te solution of te system of ordinary differential equations: dd dt = 6Q 8H 2 D 2 π 20DH 3 2HD 3 in (0, T ), (2.7a) dh find D : (0, T ) R and H : (0, T ) R : dt = H dd 4H 2 D dt 8H 2 D 2 in (0, T ), (2.7b) D(0) = D 0, (2.7c) H(0) = H 0. (2.7d) Te system (2.7) as been derived in [1] by considering a simplified elastic membrane model and enforcing te equality of te volumetric flow rate and LV volume variation, i.e. Q V (t) := dv dt (t) for all t (0, T ), were te tree-dimensional LV volume is V (t) = π 6 D2 (t)h(t). Te prescribed inlet/outlet volumetric discarge Q V, reported in Fig. 1b and used in [2], as been adapted from clinical data and it is caracterized by two maxima, te first corresponding to te early LV filling (E-wave) and te secondary filling mainly associated to te atrial contraction (A-wave). Negative values of Q V, instead, represent te systolic pase in wic te ventricle contracts. Once H(t)

8 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for eart valves 7 (a) Area flow rate Q(t) vs. time t for a 2D LV. (b) Ratio between te eigt (H(t)) and te upper diameter (D(t)) vs. time. (c) Area A(t) vs. time t for a 2D LV. Figure 2: Flow rates for a 2D LV (2a); evolution of te ratio H(t)/D(t) vs. time [s] for one eart beat (2b) and evolution of te area A(t) for 2D LV vs. time [s] in an eart beat (2c). and D(t) are obtained by solving problem (2.7), for example by means of te explicit Runge-Kutta metod [11], we deduce, for te two-dimensional LV model, bot te area and te area flow rate, defined as te time variation of te area and wose profile is reported in Figure 2a. Specifically, tese are given by A(t) := π da D(t)H(t) and Q(t) = 4 dt (t) = π ( 1 4 D(t)H(t) dd D(t) dt (t) + 1 ) dh H(t) dt (t), for all t (0, T ], respectively. In Fig. 2c, we report te area variation of te LV vs. te time for an eart beat, i.e. A(t) vs t. Moreover, we obtain te functions λ : (0, T ) R and µ : (0, T ) R given by λ(t) := D(t) and µ(t) := H(t), respectively, wic define te affine transformation D 0 H 0 governing te LV ( camber dilatation) and wose derivatives provide te velocity at te boundary, d as v W (x, t) := dt λ(t) x, d dt µ(t) y. 2.4 Boundary conditions for pysiological flows in te left ventricle We are interested in properly accounting for te valves effects in te idealized LV by prescribing suitable, pysiologically compatible BCs. In tis respect, on te subset of te boundary Ω (t) corresponding to te muscle LV walls, say Γ W,(t) for any t (0, T ), we impose te so called no-slip condition, a Diriclet BC, wic reads for any t (0, T ): u = v W on Γ W,(t), (2.8) dd were v W denotes te velocity of te LV wall, determined by te expressions of and dh dt dt in Eq. (2.7). Te remaining part of te boundary is constituted by te subsets Γ AO and Γ MT wic correspond to te aortic and mitral valves, respectively, modelled as orifices of infinitesimal tickness; in terms of Eq. (2.3), Γ W,(t) Γ (t) D and g = vw on Γ W,(t). A common practice [1, 2, 20, 21, 22, 46], motivated by pysical considerations, consists in prescribing a velocity profile on Γ MT, i.e. a Diriclet condition on Γ MT. In te first instance, we follow tis approac focusing on te aortic valve to account troug suitable BCs te opening and closing stages; te latter are associated to different types of BCs, namely, essential and natural, wic switc during te eart beat. A similar treatment can be applied to te mitral valve.

9 8 A. Tagliabue, L. Dedè, A. Quarteroni Prescribed inflow velocity at te mitral valve: pulsatile flow Te filling of te LV camber occurs troug te mitral valve in two steps, te Early-filling wave (E-wave) and te A-wave. In pysiological conditions, te blood enters te LV as an impulsive jet wic is redirected toward one portion of te moving wall and interacts wit it, due to te asymmetric position of te mitral valve wit respect to te central axis of te LV. Tere is a strong dependence of te intraventricular fluid dynamics on te inflow velocity [1], and specifically, on te intensity of te E- and A- filling waves, te velocity profile, and te size and sape of te mitral valve orifice. First, we consider a prescribed pulsatile periodic flow wit eiter parabolic or flat velocity profiles compatible wit te flow rate Q(t) of Fig. 2a. In tis respect, by denoting wit ɛ MT te eccentric position of te mitral orifice wit respect to te vertical axis of te LV and wit r MT te radius of te mitral orifice, respectively, we assume tat te function g MT defining te Diriclet BC of Eq. (2.3a) on Γ MT reads: g MT : Γ MT (0, T ) R 2 : g MT (x, t) := v MT (x)q(t), (2.9) wit v MT : Γ MT R 2 and q : (0, T ) R a vector and a scalar valued functions, respectively. Specifically, by considering dimensionless ( quantities { for Q, ɛ MT and r MT, ( ) }) 1 x 8 we define q(t) := Q(t)χ {Q(t)>0} and v MT := vx W ɛmt, exp or v MT := c NI r MT ( vx W, 6 ) 8 (x2 2ɛ MT x + ɛ 2 MT r 2 MT ) for flat or parabolic velocity inlet profiles, respectively; c NI { rmt ( ) } 1 x 8 ɛmt is suc tat exp = 1 in analogy wit [20]. r MT c NI r MT Boundary conditions for te treatment of te aortic valve For te modeling of te aortic valve, we consider essential BCs as no-slip BCs wen te valve is closed, wile natural BCs wen open. In particular, for tis latter stage, in order to ensure realistic simulations and to account for te downstream circulation in te aorta, a geometrical multiscale approac can be reproduced in a simplified manner by considering resistance or defective BCs at Γ AO [5, 6, 63]. Specifically, we prescribe at te open aortic valve orifice a resistance BCs of te natural type, similarly to [5, 6]. In tis case, on Γ (t) N := { if Q(t) 0, Γ AO, we consider te if Q(t) < 0, BC given in Eq. (2.3b) by decomposing te prescribed stress on Γ (t) N in its normal and tangential components as G := G n + G t, wit t te unit tangent vector to Γ (t) N. Specifically, we set G = 0 and G = (C out Q AO out + P V ). Te coefficient C out represents a resistance constant, wose corresponding dimensional counterpart as dimensions in te units dyn s/cm 5 and wose values can be obtained by in vivo measurements [56]. Te term P V sets a pysiologically realistic pressure. Finally, Q AO out refers to te flowrate troug Γ AO, wic we indicate as Q AO out := π 2 r AO u n dγ, Γ wit r AO AO te radius of te aortic orifice. We remark tat numerical instabilities may arise wit natural BCs on Γ AO in te presence of back flow reversal [49]; to overcome tis drawback in [6] te defective BC is augmented by a term acting only in presence of reversal flow. Instead, we refer to te form proposed in [5] wic does

10 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for eart valves 9 not add any add oc term to control back flow reversal, and allows a straigtforward treatment in te framework of te ENM. 3 Numerical approximation of te Navier-Stokes equations To numerically approximate te incompressible Navier-Stokes equations in ALE formulation (2.5), we consider IGA in te framework of te Galerkin metod [3, 5, 16, 18] for its spatial approximation, wile te time discretization wit te generalized-α metod [14, 35, 66]. Specifically, in order to provide a stable formulation in te sense of te inf-sup condition [10, 54], to control te numerical instabilities due to te advective regime occurring at ig Reynolds numbers, and to numerically model turbulence under te Large Eddy Simulation paradigm, we consider te Variational Multiscale Metod wit terms accounting for LES modeling (VMS-LES) [4]. In Sec. 3.2, we provide te semi-discrete formulation in space based on te Galerkin metod in te framework of NURBS-based IGA wit VMS-LES formulation, tat we later fully discretize by te generalized-α metod. 3.1 Spatial approximation: IGA In order to semi-discretize te problem (2.5) for any t (0, T ), we firstly represent te reference domain Ω as a NURBS geometry and we observe tat te idealized two-dimensional LV can be exactly represented by quadratic NURBS; see e.g. [16, 17, 18]. We remark tat, wit te Finite Element metod only an approximation of Ω wit a polygonal saped domain Ω can be obtained; tis induces a geometrical error wic potentially perturbs te formulation of te ENM. Let { P A } N b A=1 and { N A } N b A=1 be te control points and te set of NURBS basis functions defining N b te NURBS geometry as X = P A NA (X). Ten, te discrete ALE mapping is defined as: A=1 N b φ (t) (X) := ( P A + d A (t)) N A (X) (3.1) A=1 were {d A (t)} N b A=1 are te displacement vectors of te control points associated to te prescribed displacement of te LV. As consequence, we obtain a representation of te current configuration Ω (t), wose NURBS mes is denoted by T (t), wit (t) T := diam T (t) te diameter of a general element T (t) T (t) and (t) := max (t) T te caracteristic mes size. Moreover, we define T (t) T (t) te space of NURBS in Ω (t) as te pus-forward of te space N := span { N A } N b (t) A=1, i.e. N := span { N A φ (t) 1 } N b (t) A=1 = span {N A }N b A=1. Te problem (2.5) is formulated in terms of (u, p) in te current configuration. Wen considering NURBS-based IGA in te framework of te Galerkin metod, we look for an approximate solution as an element of te NURBS space N (t), i.e. u (x, t) := and p (x, t) := N b A=1 p A (t)n (t) A (x), were {u A(t)} N b A=1 and {p A(t)} N b A=1 N b A=1 u A (t)n (t) A (x) are te control variables

11 10 A. Tagliabue, L. Dedè, A. Quarteroni associated to te velocity and te pressure at t (0, T ), respectively. Specifically, te semi-discrete formulation of problem (2.5), reads: for all t (0, T ), find (u, p ) V (t) Q(t) : B((ϕ, q ), (u, p ); û ) = F(ϕ, q ), (ϕ, q ) W (t) Q(t), (3.2) were V (t) := V (t) N (t), W(t) := W (t) N (t), and Q(t) := Q (t) N (t) ; te form B((ϕ, q ), (u, p ); û ) and te functional F(ϕ, q ) read as in Eqs. (2.6a) and (2.6b), respectively. 3.2 VMS-LES modeling of te Navier-Stokes equations in ALE formulation Due to te fact tat we are considering a couple of spaces wic does not satisfy te Babuška-Brezzi condition [10, 54] as te same basis functions are used for bot te velocity and te pressure spaces, we need to consider a stabilized Galerkin formulation of te Navier-Stokes equations. Altoug several stabilized metods can be used to overcome tis issue, see e.g. [10], we consider te Variational Multiscale formulation for Large Eddy simulation (VMS-LES) [4]. Te metod yields a stable formulation in te sense of inf-sup problem, controls numerical instabilities due to transport dominated regime, and provide LES modeling of turbulence. Te latter metod as been firstly proposed in [4], ten furter developed in [16, 17, 18] and extended to te ALE case in [5, 62]. Problem (3.2) in VMS-LES formulation reads: for all t (0, T ), find (u, p ) V (t) Q(t) : R VMS ((ϕ, q ), (u, p ); û ) = 0 were R VMS ((ϕ, q ), (u, p ); û ) is defined as: (ϕ, q ) W (t) Q(t), (3.3) R VMS ((ϕ, q ), (u, p ); û ) := B((ϕ, q ), (u, p ); û ) F(ϕ, q ) (3.4) + T T [ ((u û ) ϕ, τ M Res M (u, p )) L 2 (T ) + ( ϕ, τ C Res C (s )) L 2 (T ) + ( (u û ) ϕ T, τ MRes M (u, p ) ) L 2 (T ) ( ϕ, τ M Res M (u, p ) τ M Res M (u, p )) L 2 (T ) ] + ( q, τ M Res M (u, p )) L 2 (T ), wit B((ϕ, q ), (u, p ); û ) defined in (2.6a) and F(ϕ, q ) in Eq. (2.6b); te residuals (in strong form) Res M (u, p ) and Res C (u, p ) and te parameters τ M and τ C are given by: Res M (u, p ) := u + ((u û ) )u + p 2 t Re (D(u )) f, (3.5a) Res C (u, p ) := u, (3.5b) [ ] 4 τ M := t 2 + (u 1 1 û ) G(u û ) + C I Re 2 G : G 2, (3.5c) 1 τ C := τ M g g, (3.5d)

12 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for eart valves 11 respectively, wit G defined componentwise as G i,j := d k=1 η k x i η k x j for i, j = 1,..., d, te covariant d η k metric tensor related to te geometrical mapping wit η = (η 1,..., η d ), wile g i :=, for x i k=1 i = 1,..., d. Te constant C I > 0 is independent of te mes size, but depends on te degree p of te basis functions, and, following an element-wise inverse estimate, we set it as C I := 60 2 p 2 according to [66]; t refers to te time step size, even if we ave not formally introduced te time discretization yet. For te time discretization of te above problem (3.3), we use te generalized-α metod [35, 66]. Specifically, te full discrete problem, wic is implicit, is solved by a predictor-multicorrector algoritm at eac time step; see e.g. [19]. 4 Extended Nitsce s metod (ENM) for mixed time varying (MTV) BCs In tis section, we introduce te ENM for te formulation of te Navier-Stokes equations wit tat account for valves functioning. Specifically, in Sec. 4.1, we focus on te aortic valve. In tis respect, we introduce a proper set of BCs embedded in te weak formulation in te framework of te ENM. Suc BCs are able bot to describe te opening and closing of te aortic valve and to control te occurrence of te numerical instabilities due to te partial or total flow reversal associated to defective BCs of Sec Ten, in Sec. 4.2, we propose a similar treatment for te BCs mimicking te mitral valve by using natural BCs [29] for te inflow; in tis case, we add a suitable regularizing term to te BCs to model realistic inflow profiles. 4.1 MTV BCs: te extended Nitsce s metod (ENM) In order to introduce te ENM for te treatment of te aortic valve as MTV BCs of te Navier-Stokes equations in weak formulation, we follow te framework of [36] proposed for elliptic PDEs and ten [61] for parabolic PDEs; for te original Nitsce s metod, we refer te reader to [50]. We complement Eq. (2.1) wit generalized Robin BCs on Γ AO in te form: Φ(u û; u, p)n + γ AO (x, t)u(x, t) = G(x, t) + γ AO (x, t)g(x, t) on Γ AO, (4.1) were γ AO : Γ AO (0, T ) (0, + ), G is a function defining te defective BC described in Sec used for te open valve, and g defines te no-slip BC mimicking te closed valve (ideally, in a fixed ventricle, we ave g = 0). We notice tat in te limit γ 0, Eq. (4.1) tends to te natural BC (2.3b), wile, in te limit γ, we recover te Diriclet BC (2.3a). Finally, we recall tat on Γ W,(t) for any t (0, T ) we impose te no-slip condition of Sec. 2.4 and on Γ MT we prescribed a pulsatile periodic flow as described in Sec , i.e. u = v W Te ENM: application to te aortic valve We consider te BC of Eq. (4.1) on Γ AO in te semi-discrete spatial approximation of te Navier-Stokes equations (3.3) introduced in Sec. 3. Wereas, on Ω (t) \ Γ AO Γ W,(t) Γ MT we

13 12 A. Tagliabue, L. Dedè, A. Quarteroni strongly impose te essential BCs by defining te finite dimensional test and trial function spaces V (t) and W (t) as V (t) := V(t) g N (t), W(t) := W(t) N (t), respectively, were, for t (0, T ): V (t) g := {u [H 1 (Ω (t) )] d : u = g on Γ W,(t) Γ MT } (4.2) and W (t) := {u [H 1 (Ω (t) )] d : u = 0 on Γ W,(t) Γ MT }, (4.3) { vw on Γ W,(t), wit g =. In tis manner, it is sufficient to look for a solution u of Eq. (3.3) in v MT on Γ MT [H 1 (Ω (t) )] d satisfying te BCs u = g on Γ W,(t) Γ MT and for t (0, T ) by considering velocity test functions in [H 1 (Ω (t) )] d vanising on Γ W,(t) Γ MT. Before introducing te weak imposition of te MTV BCs of Eq. (4.1) defined on Γ AO by te ENM, we recall te notation introduced in Sec Let us consider te pysical mes T (t) associated to te pysical domain in te current configuration Ω (t) R d ; moreover, let te boundary Γ (t) be split into N eb parts corresponding to te edges of te elements T b T (t) suc tat T b Γ AO, for b = 1,..., N eb. We define Γ b := T b Γ AO and, for d = 2, we introduce te size b of Γ b, for b = 1,..., N eb, by using te covariant element metric tensor G as: b := 2(t T Gt) 1/2, (4.4) were t is te unit vector tangent to te boundary element T b. By considering a positive bounded time dependent real parameter ξ > 0, acting as a penalty parameter and associated to te stability properties of te metod in case of flow reversal, te problem (3.3) wit te weak imposition of te MTV BCs by means of te ENM reads: find, for all t (0, T ), u (t) V and p (t) Q : were: R ((ϕ, q ), (u, p ); û) = 0 (ϕ, q ) W Q, (4.5) R ((ϕ, q), (u, p), û) := R VMS ((ϕ, q), (u, p), û) N eb [ ( + γ b b=1 Γ b ξ + γ b ( + γ b Γ b ξ + γ b ( ξγ + Γ b Re(ξ + γ b ) ( + Re b Γ b ξ + γ b ( ξ ξ + γ b Γ b ) ϕ (Φ(u û; u, p)n) dγ ) (Φ in(u û; ϕ)n) (u g) dγ ) ϕ (u g) dγ ) Φ in(u û; ϕ)n (Φ(u û; u, p)n G) dγ ) ϕ G dγ wit R VMS ((ϕ, q), (u, p); û) defined in Eq. (3.4); Φ in is te adjoint inflow total flux, wic, witin te ALE formulation, reads Φ in(u û; ϕ) := (ϕ (u û))χ {(u û) n<0} + 2 D(ϕ), for wic Re Φ in(u û; ϕ)n = {(u û) n} ϕ+ 2 Re D(ϕ)n, were {(u û) n} (u û) n (u û) n =. 2 ], (4.6)

14 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for eart valves 13 Te adjoint inflow total flux Φ in as been introduced in analogy wit te weak enforcement of te Diriclet BC for te advection-diffusion equation of [7] to andle te BCs for te inflow and outflow subsets of te boundary. In tis manner, if Γ b lays on an outflow boundary (for wic u n > 0), only te diffusive part of te total flux operator Φ is considered for te weigting function ϕ, wile, if Γ b is on an inflow boundary, we take into account bot te diffusive and advective parts for te weigting function ϕ. We igligt tat, by considering te ENM of Eq. (4.5) in te limit for γ, we recover te original weak imposition of te Diriclet BCs for te Navier-Stokes equations (2.1) presented in [7] in te ALE case. Contrarily, in te limit γ 0 in Eq. (4.5), we are weakly imposing natural BCs, even if we get an additional term wit respect to te standard formulation; tis latter term does not affect te consistency of te metod, but rater assumes a stabilization role preventing te occurrence of te numerical instabilities caused by te partial or total flow reversal troug Γ AO. Tis term can be compared to te artificial traction term weakly added for te metod proposed in [6] and furter developed in [49] to prevent backflow divergence. Remark 4.1. We observe tat for γ, i.e. for te weak imposition of te essential BCs on Γ AO, te larger te value of ξ, te stronger te penalization associated to suc BCs. Converserly, for γ 0, tus considering te imposition of te natural BCs on Γ AO, te smaller te value of ξ, te larger te contribute of te stabilization term preventing backflow divergence. Terefore, similarly to γ, we also consider ξ as a time dependent real positive function ξ : (0, T ) R : t ξ(t). Specifically, we suppose ξ to be a measurable bounded function for wic tere exist two positive constants suc tat 0 < ξ 0 < ξ(x, t) < + and 0 < ξ(x, t) < ξ < + for all (x, t) N b b=1 Γ b (0, T ). We observe tat, te constant ξ 0 can not assume arbitrary positive values, but, in order to ensure stability at te discrete level, it must be larger tan some positive constant dependent on te data of te problem (i.e. te Reynolds number Re and te sape of te domain), on local boundary inverse estimates, and on te order of interpolation used in te finite dimensional space; see [7, 15]. 4.2 MTV BCs for inflow boundaries: te mitral valve Boundaries wic are mainly associated wit inflows can be treated in a manner wic is analogous to te case described in Sec Tis is te case of te mitral valve, wic for an idealized LV is treated as a MTV BC, similarly to Eq. (4.1). Indeed, wen te mitral valve is open, one can set natural BCs (e.g. resistance BC), wile, wen it is closed, Diriclet BCs. Using Diriclet BCs for te full eart beat corresponds to assume for te mitral valve a given inflow velocity profile, wic, as a matter of fact, sould be an unknown of te problem. Te use of te MTV BCs prevents te need to make suc coice a priori, since a natural BC can be weakly set wit an open valve. Neverteless, suc natural BC may still lead to unrealistic velocity profiles at te mitral valve because te standard resistance BC is too simple to account for te inflow, especially in a moving domain. In order to prevent unrealistic inflow profiles, we add to te MTV BC of Eq. (4.1) a regularization term; tis as te role of weakly penalizing te inflow velocity profiles wic are too dissimilar from te parabolic one. Terefore, for te mitral valve, we propose on Γ MT a generalized Robin BC wit penalization based on te second-order Laplace-Beltrami operator, reading: Φ(u û; u, p)n + γ MT (x, t)u(x, t) α MT Γ u = G(x, t) + γ MT (x, t)g(x, t) on Γ MT, (4.7)

15 14 A. Tagliabue, L. Dedè, A. Quarteroni were α MT is a positive real parameter constant in time. At te discrete level, te BC (4.7) can be treated by means of te ENM as in Sec , wit te addition of te regularization term. Terefore, we consider for Eq. (3.3) te time dependent trial solution and weigting spaces for t (0, T ): and V (t) := {u (H 1 (Ω (t) ) : u = v W on Γ W,(t) } (4.8) W (t) := {u (H 1 (Ω (t) ) : u = 0 on Γ W,(t) }, (4.9) respectively. In tis case, we account for Γ V := Γ AO Γ MT as a subset in wic natural BCs are prescribed and we consider te N eb elements suc tat T b Γ V for b = 1,..., N eb. Ten, te problem reads as in Eq. (4.5), were te residual of Eq. (4.6) is augmented by a regularizing term as: R ((ϕ, q), (u, p), û) := R N eb ] ((ϕ, q), (u, p), û) + [ α MT χ Γ MT (D(u) : D(ϕ)) dγ, (4.10) Γ b were χ Γ MT(x, t) := 5 Numerical tests { 1 if x Γ MT, 0 oterwise. b=1 We present te numerical study of te fluid dynamics in te idealized LV wit prescribed wall motion. In Sec. 5.1 we assess te MTV BCs used to treat te aortic and mitral valve and we analyse te blood flow inside te LV. Ten, we compare te flow patterns obtained wit prescribed inflow profiles, parabolic or flat, at te mitral valve wit tose obtained wit MTV BCs on bot te valves, see Sec In Sec. 5.2, we study te role of te parameters involved in te MTV BCs and ENM. Specifically, we consider a prescribed parabolic pulsatile inlet profile at te mitral valve and te MTV BCs only for modelling te aortic valve. We perform a sensitivity analysis on te function γ AO of Eq. (4.1) and compare te additional term preventing te numerical instabilities caused by flow reversal of Eq. (4.6) to te metod proposed in [49] by sowing te effectiveness of our formulation. Ten, by using te MTV BCs to treat te mitral valve, we perform a sensitivity analysis on te parameter α MT introduced in Eq. (4.10) and we sow tat te typical inlet jet profile is correctly identified. We consider te time dependent domain Ω (t) of Sec. 2.2, wose time dependent equatorial diameter D(t) and eigt H(t) are te solutions of te systems of ODEs (2.7). Specifically, by setting te initial configuration at te beginning of te diastolic pase, te initial conditions are set to D(0) = D 0 = 3.4 cm and H(0) = H 0 = 2D 0, yielding D = 5.5 cm and H/D = 1.58 at te end of te diastolic pase, te latter being te mean values of uman LV ealty-normal subjects [2, 44]. Te eart beat period is set as T HB = s and te initial condition of te fluid velocity is set to te zero function, i.e. u(x, 0) = u 0 = 0 in Ω (0). Moreover, te blood is considered a Newtonian fluid wit constant density ρ = 1.06 g/cm 3 and dynamic viscosity µ = g/(cm s); te resistance constant is set to C out = 0. For our numerical study, we simulate up to ten eart beats and we disregard te initial tree beats to remove non-pysiological solutions from our analysis. Te domain Ω (t) is exactly represented by means of globally C 0 -continuous NURBS basis functions of degree p = 2 wit a mes wit N el = 2,096 elements, for a total of 9,576 Dofs for te velocity

16 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for eart valves 15 (a) (b) Figure 3: (a) Times at wic te numerical results are visualized during te eart beat for te diastolic ( ) and systolic ( ) pases. (b) Evolution of te pressure during te sixt eart beat in tree points near te valves and LV base. and pressure variables. Following [60], we define te time step by referring to te caracteristic mes size and velocity, corresponding to a dimensional time step equal to t = s. 5.1 Blood flows in te LV We describe te intraventricular blood flow pattern obtained by solving te Navier-Stokes equations endowed wit te MTV BCs for te treatment of bot te aortic and mitral valves. By referring to Eq. (4.10), we set ξ AO (t) = 10 4 χ [0, 0.68) (t) χ [0.68, 1.068) (t), ξ MT (t) = 10 2 χ [0, 0.68) (t) χ [0.68, 1.068) (t), γ AO (t) = χ [0, 0.68) (t) χ [0.68, 1.068) (t), γ MT (t) = 10 7 χ [0, 0.68) (t) χ [0.68, 1.068) (t), and α MT = 100. Ten cardiac cycles are simulated and we analyse bot instantaneous quantities of interests as te velocity or pressure, for example at te sixt eart beat, and some pase averaged quantities over te last N av = 7 eart beats. Te numerical results are sown at te caracteristic times reported in Fig. 3a. In Fig. 3b we report te evolution of te pressure field vs. te time in te sict eart beat. Te results are in agreement wit te typical Wiggers diagram [48]. By using te same definitions of [13], we firstly consider te pase average velocity (reported in Fig. 4) wic is defined as ū(x, t) := 1 N av N av 1 j=0 u(x, t + jt HB ). In order to igligt te vortex structures we consider te Q-criterion wic detects regions of positive values of te second invariant of te velocity gradient tensor were Q = 1 ( ) X S 2 2 > 0, wit X = 1 ( u u T ) and S = 1 ( u + u T ) ; see 2 2 e.g. [34, 59]. In Fig. 5 we report te results of te pase-averaged Q-criterion, say Q-criterion. Te transitional nature of te fluid from laminar to nearly turbulent over eac eart beat and te cycle-to-cycle variations are quantitatively analysed by considering te fluctuating kinetic energy (E FKE ) (in Fig. 6) wic is defined as E FKE := 1 2 u rms(x, t) 2 2, were u rms(x, t) = u 2 (x, t) ū 2 (x, t) is te root mean square velocity and 2 stands for te usual Euclidean norm. Te above results provide an insigt of te blood flow patterns inside te LV. During te diastolic pase, te LV dilates and te cavity is filled wit blood coming from te left atrium troug te

17 16 A. Tagliabue, L. Dedè, A. Quarteroni mitral valve wic is open. As sowed in Fig. 3a, te mitral flow rate is caracterized by a first peak (E-wave) corresponding to a rapid filling at t = 0.16 s and a second one (A-wave) in wic te flow enters te camber wit a smaller velocity, due to te atrial contraction at t = 0.58 s; te two contro-rotating vortexes developing close to te mitral valve are clearly igligted wit te Q-criterion. Te strong jet entering te cavity forms a central vortex structure wic dominates te entire flow field. Suc vortex, wic interacts wit te jet entering troug te mitral valve, induces a secondary one near te wall. Tis latter vortex is soon dissipated, wile te primary vortex grows in intensity and dimensions moving toward te apex of te LV cavity. We notice also te presence of smaller vortices inside te LV cavity wit an intense one at te apex. Te pase averaged velocity u also igligts te recirculations forming inside te LV. At te peak E-wave, as expected, a ig velocity is observed in te proximity of te mitral valve. At te end of te diastolic pase te E FKE reaces its largest values indicating large cycle-to-cycle variations and possible transitional beaviour of te flows from laminar to nearly turbulent. Tis results is in agreement wit te numerical simulations and considerations of [13] for nearly realistic tree-dimensional LV. At te systolic pase te aortic valve opens, wile te mitral valve closes. Te flow is naturally redirected towards te aortic valve and ejected from te LV. Moreover, a smaller region of recirculation under te mitral valve can be observed at te beginning of te ejection pase (Fig. 5). At tis pase te vortexes are expelled from te LV and te flow regularized as igligted bot by te Q-criterion and te E FKE wic presents smaller values.

18 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for eart valves (a) t = 0.03 s. (b) t = 0.10 s. (c) t = 0.16 s. (d) t = 0.21 s. (e) t = 0.29 s. (f) t = 0.36 s. (g) t = 0.45 s. () t = 0.50 s. (i) t = 0.58 s. (j) t = 0.68 s. (k) t = 0.74 s. (l) t = 0.77 s. (m) t = 0.83 s. (n) t = 0.90 s. (o) t = 0.98 s. (p) t = 1.03 s. 17 Figure 4: Pase averaged velocity magnitude u (cm/s) at different instants of te eart beat (Fig. 3a).

19 18 A. Tagliabue, L. Dede, A. Quarteroni (a) t = 0.03 s. (b) t = 0.10 s. (c) t = 0.16 s. (d) t = 0.21 s. (e) t = 0.29 s. (f) t = 0.36 s. (g) t = 0.45 s. () t = 0.50 s. (i) t = 0.58 s. (j) t = 0.68 s. (k) t = 0.74 s. (l) t = 0.77 s. (m) t = 0.83 s. (n) t = 0.90 s. (o) t = 0.98 s. (p) t = 1.03 s. Figure 5: Pase averaged Q-criterion (Hz) at different instants of te eart beat (Fig. 3a).

20 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for eart valves (a) t = 0.03 s. (b) t = 0.10 s. (c) t = 0.16 s. (d) t = 0.21 s. (e) t = 0.29 s. (f) t = 0.36 s. (g) t = 0.45 s. () t = 0.50 s. (i) t = 0.58 s. (j) t = 0.68 s. (k) t = 0.74 s. (l) t = 0.77 s. (m) t = 0.83 s. (n) t = 0.90 s. (o) t = 0.98 s. (p) t = 1.03 s. 19 Figure 6: Fluctuating kinetic energy EFKE (cm2 /s2 ) at different instants of te eart beat (Fig. 3a).

21 20 A. Tagliabue, L. Dedè, A. Quarteroni (a) Parabolic velocity profile (b) Flat velocity profile. Figure 7: Imposed inflow velocity profiles troug te mitral valve Comparison of results wit different treatments of te BCs at te mitral valve. We analyse te dependency of te intraventricular fluid dynamics on te inlet condition, especially for vortex structures as igligted by flow visualizations [9, 55]. Specifically, we consider eiter te MTV BCs for modelling bot te aortic and mitral valves or te strong imposition of inflow velocity at te mitral valve by considering eiter parabolic or flat profiles as sown in Fig. 7. In Figs. 8 10, we report te velocity magnitudes obtained in te tree cases under consideration at te times reported in Fig. 3a and at te sixt eart beat. As expected, te velocity field and vortexes are significantly different during te diastolic pase, wile more uniform at te systolic pase, i.e. wen te flow is regularized during te ejection pase. Even if te inflow velocity profile of te MTV BCs is similar to te imposed parabolic one, te flow pattern are sensibly different, for wic we igligt tat te treatment of te mitral valve as a crucial importance in te study of te blood flows in te LV.

22 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for eart valves 21 MTV BCs Parabolic inlet Flat inlet t = 0.03 s t = 0.16 s t = 0.29 s Figure 8: Magnitude of te velocity field u (cm/s) at different instants of te #6 -t eart beat for MTV BCs at te mitral valve (left), imposed parabolic (center), and flat (rigt) velocity profiles.

23 22 A. Tagliabue, L. Dedè, A. Quarteroni MTV BCs Parabolic inlet Flat inlet t = 0.45 s t = 0.58 s t = 0.68 s Figure 9: Magnitude of te velocity field u (cm/s) at different instants of te #6 -t eart beat for MTV BCs at te mitral valve (left), imposed parabolic (center), and flat (rigt) velocity profiles.

24 Fluid dynamics of an idealized left ventricle: te extended Nitsce s metod for eart valves 23 MTV BCs Parabolic inlet Flat inlet t = 0.74 s t = 0.84 s t = 0.98 s Figure 10: Magnitude of te velocity field u (cm/s) at different instants of te #6 -t eart beat for MTV BCs at te mitral valve (left), imposed parabolic (center), and flat (rigt) velocity profiles.

25 24 A. Tagliabue, L. Dedè, A. Quarteroni 5.2 Sensitivity to parameters We analyse te dependency of our model on te time dependent function ξ AO of Eq. (4.5), wic bot controls te penalization on te Diriclet BC (for te closed valve) and manages te rule of te stabilizing term preventing numerical instabilities due to backflow divergence (for te open valve). We sow tat suc terms effectively act as stabilizing terms preventing numerical instabilities due to backflow divergence and we compare te solution to te case of a do-noting approac and to te metod proposed by Bazilevs et al. in [6, 49]. In bot te cases we consider te MTV BCs for modelling only te aortic valve and we impose an inflow parabolic profile at te mitral valve. Finally, we study te dependency on te parameter α MT in Eq. (4.10) during te diastolic pase by considering MTV BCs for modelling te mitral valve Dependency on time dependent function ξ AO For te study of te function ξ AO in te MTV BCs (4.5), we focus on te velocity profiles on te LV diameter corresponding to te aortic valve section. As observed in Remark 4.1, we consider a time dependent function to weakly enforce essential BCs or te resistance BC on Γ AO. As mentioned in Remark 4.1, we deduce tat during te diastolic pase, for γ AO (e.g. γ AO = ), te larger te value of ξ AO, te larger is te penalization on te Diriclet, and ence te better is te approximation of te Diriclet data. Te velocity profiles at te aortic valve reported in Fig. 11 numerically confirm te expected beaviour of te function ξ AO, i.e. te larger is ξ AO, te stronger is te enforcement of te Diriclet BCs at te aortic valve during te diastolic pase. Neverteless, we can notice in Fig. 12 tat te amplitude of te oscillations caused by te coice of a small value of ξ AO is negligible wit respect to te values tat te velocity takes at te time t = 0.16 s corresponding to te peak E-wave of te diastolic pase. On te oter and, wen considering te weak imposition of te resistance BCs for γ 0 (e.g. γ AO = 10 7 ) in Eq. (4.6), te smaller te value of ξ AO, te larger is te contribute of te stabilizing term. Tis is igligted in Fig. 13, were we observe tat te velocity profiles are qualitatively similar for different values of γ AO bot during te early systole (t = 0.74 s) and at te peak systolic pase (t = 0.84 s), even if tese may differ in te presence of backflow penomena troug Γ AO at te late systolic pase. Specifically, te smaller te values of ξ AO, te larger is te contribute of te term to prevent te insurgence of numerical instabilities associated to backflow divergence during te decelerating pase of te flow. We quantify te difference between solutions sown in Fig. 13d by computing te flowrate troug te subsets of te boundary were u n < 0 wic is equal to 3.76, 4.62, and 4.71 cm 2 /s, respectively. We conclude tat γ AO sould be large during te diastolic pase and small during te systolic pase.