UNIVERSITÀ DEGLI STUDI DI BERGAMO DIPARTIMENTO DI INGEGNERIA DELL INFORMAZIONE E METODI MATEMATICI QUADERNI DEL DIPARTIMENTO Department of Information Tecnology and Matematical Metods Working Paper Series Matematics and Statistics n. 10/MS 2011 Numerical treatment of boundary conditions to replace lateral brances in aemodynamics by A. Porpora, P. Zunino, C. Vergara, M. Piccinelli Viale Marconi. 5, I 24044 Dalmine (BG), ITALY, Tel. +39-035-2052339; Fax. +39-035-562779
COMITATO DI REDAZIONE Series Information Tecnology (IT): Stefano Parabosci Series Matematics and Statistics (MS): Luca Brandolini, Ilia Negri L accesso alle Series è approvato dal Comitato di Redazione. I Working Papers della Collana dei Quaderni del Dipartimento di Ingegneria dell Informazione e Metodi Matematici costituiscono un servizio atto a fornire la tempestiva divulgazione dei risultati dell attività di ricerca, siano essi in forma provvisoria o definitiva.
Numerical treatment of boundary conditions to replace lateral brances in aemodynamics Azzurra Porpora, Paolo Zunino, Cristian Vergara, Marina Piccinelli Abstract In tis paper, we discuss a tecnique for weakly enforcing flow rate conditions in computational emodynamics. In particular, we study te effectiveness of cutting lateral brances from te computational domain and replacing tem wit non perturbing boundary conditions, in order to simplify te geometrical reconstruction and te numerical simulation. All tese features are investigated bot in te case of a rigid and of a compliant wall. Several numerical results are presented in order to discuss te reliability of te proposed metod. 1 Introduction and motivations Te prescription of suitable boundary conditions on artificial sections is a major issue in computational aemodynamics [27]. An artificial section is a part of te domain boundary wic does not correspond to any pysical wall, but it is just introduced by te truncation of te computational domain, in order to separate a vascular district selected for a computational fluid-dynamics (CFD) analysis from te proximal and distal parts of te arterial tree [15]. In te last years, te development in te biomedical acquisition tecniques, suc as pase contrast-magnetic resonance imaging (PC-MRI), allowed to obtain satisfying pointwise information about te velocity field at a section. However, tese procedures are nowadays still very onerous bot from computational (3-4 minutes are needed for eac acquisition, wic often as to be repeated), and economic point of view. For tis reason, a great attention in computational aemodynamics is still paid to averaged data, wic can be more easily obtained, for example wit a Doppler-ultrasound tecnique, wic unlike MRI is very fast and inexpensive. Tis work as been partially supported by te ERC Advanced Grant N.227058 MATH- CARD. MOX, Dipartimento di Matematica Francesco Briosci, Politecnico di Milano, Italy MOX, Dipartimento di Matematica Francesco Briosci, Politecnico di Milano, Italy Dipartimento di Ingegneria dell Informazione e Metodi Matematici, Università di Bergamo, Italy Department of Matematics & Computer Science, Emory University, Atlanta, GA, USA 1
In particular, wit tis tecnique it is possible to obtain at eac time te flow rate Q = u n ds, Γ were u is te fluid velocity, Γ te section at and, and n te unit outward normal. Wen suc data are prescribed as boundary conditions at te artificial sections, tey are usually referred as defective boundary conditions, because some additional requirement is needed to close te system of governing equations. Anoter situation were averaged data are needed is wen one couples te 3D model wit reduced models (one-dimensional, 1D, or zero-dimensional) to take into account te complementary part of te arterial tree. In tis case, due to te reduced dimensionality of suc models, just averaged quantities, suc as te flow rate, can be excanged wit te 3D model. We refer te interested reader to [11] for a review of tese topics. Several strategies ave been introduced for te management of defective flow rate conditions. Te first ad oc treatment is found in [15], were te autors proposed a suitable variational formulation, wic owever requires to build up null-flow rate functional spaces, wic are ardly discretized by means of te finite element metod. For tis reason, oter tecniques ave been developed in te last decade. We cite te Lagrange multipliers approac [10, 30, 31], an optimal control-based metod [12], and te coupled momentum metod [33, 34]. Recently, in [37] te autors proposed to manage te flow rate condition troug Nitsce s metod [23]. Tis metod does not introduce additional unknowns suc as Lagrange multipliers and it does not need any iterative scemes to solve te discrete problem as in te optimal control-based approac (for a numerical comparison wit te Lagrange multipliers tecnique, see [32]). Wen applied to flow rate conditions, Nitsce s tecnique weakly perturbs te flow profile naturally determined by te continuity and momentum equations inside te computational domain. However, a weak boundary control can lead to lack of stability of te numerical sceme. On tis basis, te analysis of [5, 8, 20] is devoted to develop and compare numerical treatments of outflow boundary conditions to avoid backflow effects tat could destabilize te numerical sceme. Suc obstacle will be overridden by means of a suitable stabilization tecnique. In tis work, we consider Nitsce s metod for te prescription of defective conditions applied to aemodynamics. More precisely, te main goal of tis work is to study its reliability wen lateral brances are neglected in te geometry reconstruction from biomedical images. We are interested in tis situation for many reasons. Firstly, te geometry reconstruction and te related mes formation to obtain te patient-specific computational domain may be simplified if lateral brances are not considered. Indeed, focusing just on te main vessel can in some cases facilitate te reconstruction of te surface since often lateral brances are small and difficult to recognize. For wat concerns te mes set up, avoiding small 2
angles and vessels wit small radius facilitates te formation of regular triangulations. Secondly, te presence of lateral brances poses some difficulties in te set up of fluid-structure interaction (FSI) models. Artificial sections almost ortogonal to te vessel centerline are usually kept fixed in FSI models. In te case of vascular districts wit several secondary brances, te previous approac could introduce an artificial stiffening of te arterial district (see for example [5] for an application to te ascending aorta, and [36] for te toracic aorta). Furtermore, te tree-dimensional arterial wall geometry and mes are obtained usually from extrusion of te fluid domain, since te vessel walls are not easily detected from biomedical images. Ten, te presence of small bifurcation angles could lead to a failure in te structure mes generation. Lastly, wen considering a membrane model for te deformation of te arterial walls, te main and Gaussian curvatures of te vessel surface are needed in order to feed te governing equations, see [25]. Obviously, removing lateral brances from te main vessel makes bot curvatures almost constant, simplifying te application of suc FSI models. As an example among many oter cases, te approac tat we propose could be particularly effective to account for te presence of intercostal arteries brancing from te descending toracic aorta. Indeed, several studies ave recently analyzed te effect of suc arteries on wall sear stresses, by means of te classical approac were lateral brances are fully accounted in te geometrical model, see [35, 18, 19]. We will sow on a simple test case tat te metod proposed ere could provide comparable results wit a simplified geometrical setting. Te outline of te work is as follows. In Sect. 2 we review te problem of te geometry reconstruction of patient specific models in aemodynamics. In Sect. 3 we describe Nitsce s metod for te prescription of defective conditions, discussing te application to problems were lateral brances are neglected from te geometrical model and we address some numerical experiments to validate te metod at and. In Sect. 4 we discuss te reliability of te metod wen a fluid-structure interaction model wit a membrane structure is considered, complementing te description of te sceme wit extensive numerical results. Finally, in Sect. 5 we address some conclusions of te work. 2 Set up of patient specific geometrical models in aemodynamics Image segmentation is te operation of partitioning an image into different specific objects. In te context of patient-specific computational fluid dynamics it provides te surface identifying te interface between blood and vessel wall and consequently te volumetric domain for te simulation itself. Crucial features of te images, typically dimensionality, spatial resolution and, given te recent tecnological advances in te radiological field, temporal resolution, determine 3
te segmentation tecnique to be applied and te accuracy of te results. In an ideal situation all te vessels in a specific vascular district sould be reconstructed and included in te model to properly represent te in-vivo condition. In many real situations some collateral or side brances may be difficult to model due to teir small caliper compared to te image resolution or to suboptimal quality of te available images. In tese cases te proposed approac could be an important improvement to te simplistic solution of omitting unavailable side brances. Te aortic arc is a callenging bencmark test for aemodynamics, because of ig speed flow and corresponding complex flow patterns as well as for te study of non perturbing boundary conditions for inflow and outflow artificial sections, see for instance [5, 8, 20, 21, 22, 36]. For tis reason, we exploit a patient specific aortic arc model to test te validity of te present metod. Te package VMTK (ttp://www.vmtk.org) was used for te preparation of te 3D model. An MRI scansion of te aortic arc was obtained at Ospedale Borgo Trento in Verona, Italy. Te images were acquired wit a 1.5 Tesla macine (Magnetom Simpony, Siemens Medical Systems, Erlangen, Germany) after injection of contrast agent and saved in DICOM format for subsequent processing. Te following parameters were used: TE=1.6 ms, flip angle=65 o, slice tickness=6 mm, field of view=400 mm, acquisition matrix =256 256. Te images were segmented by means of te segmentation tool available witin VMTK based on a gradient-driven level-set approac [1]; te complete model of te aortic arc was reconstructed from te aortic root to te first portion of te descending aorta including te tree main brances tat originate from it: te braciocepalic, te left common carotid and te left subclavian arteries. We depict in Figure 1 (top, left) te complete model of te aortic arc. To provide a model of te aortic arc were side brances ave been removed, two sets of te VMTK tools for editing 3D triangulated surfaces were employed: first, te brances were manually clipped from te arc; secondly te resulting oles on te arc surface were re-mesed by progressively and smootly, i.e. following te local surface curvature, fill tem in wit concentric triangulated circular strips. Te triangles tat eventually capped te aortic arc surface were coerently marked in order to keep trace of were te removed arteries were originally positioned, see Figure 1 (rigt column) for a description of te model after te removal of sovra-aortic brances. In bot cases, valve leaflets opening and closing mecanisms were not taken into account, so tat numerical simulations ave been performed in te systolic configuration. In particular, an analytical model of te aortic valve orifice was defined as a circle of area 3.0 cm 2 (see Figure 1, bottom, left). Moreover, te leaflets were not drawn, since teir modelization is outside of te goals of tis work. Te solid model was successively turned into a volumetric mes of linear tetraedra in order for computational fluid-dynamics simulations to be carried 4
Figure 1: Patient specific model of te aortic arc including te braciocepalic, te left common carotid and te left subclavian arteries (top, left), compared wit te analogous model witout side brances (top, rigt). On te bottom, a detail of te valve orifice at systole (bottom, left) and of te outflow sections reconstructed on te surface of te aortic wall in correspondence of te removed lateral brances (bottom, rigt). 5
out. We ave 227858 tetraedra for te mes wit lateral brances and 198292 tetraedra for te mes witout lateral brances. Tese dimensions were reaced after successive mes refinements, wit te aim of obtaining a mes-independent numerical solution. 3 Outflow boundary conditions for lateral brances. 3.1 Steady Navier-Stokes model wit rigid walls. We assume tat blood beaves as a omogeneous and Newtonian fluid [11], and tat te arterial walls could be considered as rigid boundaries. Reminding tat u denotes te blood flow velocity, p te pressure and µ te blood dynamic viscosity, we define te strain rate and te Caucy stress tensor as follows D(u) := 1 2( u + u ), T(u, p) := 2µD(u) pi. Let Ω be a domain representative of bot full (Ω f ) and reduced (Ω r ) flow domains, see Figure 1. We assume tat suc domains feature N outflow sections (distal artificial sections wit respect to te eart) and a single inflow section (proximal to te eart). We denote by Γ i suc artificial sections, were Γ 0 corresponds to te inflow and Γ k wit k = 1,..., N represent te outflow sections. We denote wit Γ := N k=0 Γ k te union of all artificial sections, ten Ω \ Γ corresponds to te arterial wall. As previously observed, one of te main issues in computational aemodynamics consists in providing boundary conditions for te artificial sections Γ k tat separate te considered vascular district Ω from te remaining part of te arterial tree. In absence of accurate data on inflow and outflow velocity profiles, boundary conditions tat minimally perturb blood flow sould be applied. In particular, mean flow rate or mean pressure conditions (or equivalently mean normal stress) are often applied to computational aemodymamics studies [15, 10, 30, 31, 12, 34, 37]. A more general model consists in enforcing a linear combination of te two previous conditions [33, 5, 32]. Denoting by ρ f te constant fluid density, by Q k a reference flow rate, by P k te mean stress and by R k a constant coefficient, suc condition prescribes tat ( ) ( ) ρ f R k u n ds Q k + n T(u, p)n ds + P k = 0. (1) Γ k Γ k Previous conditions can be interpreted as mean resistance conditions, tat prescribe a constitutive law between te blood flow rate troug an artificial section Γ k and te resistance to discarge, in te spirit of Poiseuille law for a laminar flow in a straigt tube. In tis case, te parameter R k assumes te role of resistance to flow induced by te arterial tree distal to te artificial section. We observe tat boundary conditions (1) only constrain te mean value of normal velocity components and stresses. Terefore, tey are not sufficient to 6
ensure tat, combined wit Navier-Stokes equations for conservation of mass and momentum, te problem is uniquely solvable. For tis reason suc type of conditions is often classified as being defective. To override tis drawback, additional conditions on stresses at te artificial sections must be prescribed. More precisely, we require tat te normal stress vector is constant along eac artificial section Γ k, wit unknown modulus along te normal direction to te considered section, tat is T(u, p)n = c k n on Γ k or equivalently n T(u, p)n = c k, n T(u, p)n = 0 on Γ k, for suitable constants c k. Ten, given Q k, P k and R k wit k = 0,..., N, and te forcing term f, our reference problem consists to find velocity and pressure fields u, p and constants c k suc tat µ ( u + u ) ( + ρ f u )u + p = f in Ω, u = 0 in Ω, u = 0 ( on Ω \ Γ, ) ( ) ρ f R k Γ k u n ds Q k + Γk n T(u, p)n ds + P k = 0 on Γ k, T(u, p)n = c k n on Γ k. Most often parameters R k, P k sould be provided by suitable matematical models for te distal arterial tree, see for instance [26]. Altoug te problem setting is kept general trougout tis section, for te numerical investigation we restrict to flow rate boundary conditions, wic correspond to (1) wit R k. In particular, we assume tat te entire flow division is provided in our case. More precisely, for eac outflow section Γ k wit k = 1,..., N, te flow rate Q k is known, wile te inflow flow rate is provided by enforcement of mass conservation constraint, i.e. Q 0 = N k=1 Q k. Remark 1 For te sake of completeness, we notice tat it is possible to generalize defective boundary conditions to account for te angle of incidence between te main cannel and te outgoing vessels. Tis gives rise to te following condition, ( ) ( ) ρ f R k u ds Q k + T(u, p)n ds + P k = 0, Γ k Γ k were Q k R 3, P k R 3. By tis way, te mean value of te tangential components of te velocity and of te normal stress vector are also taken into account. We refer to tis condition as vector valued defective condition. 3.2 Numerical treatment of resistance boundary conditions. Let T be a family of admissible, sape regular and quasi-uniform triangulations of Ω wit caracteristic mes size. We denote by V and Q respectively, te (2) 7
selected finite element spaces for velocity and pressure approximation, wic will be specified later on. For te fortcoming metod te coice of suc spaces is independent of te definition of boundary conditions on artificial sections. Ten, te main difficulty for te approximation of problem (2) in te framework of te finite element metod consists in embedding resistance boundary conditions into te sceme. Our approac is based on te reinterpretation of (1) as an averaged Robin-type (or mixed type) boundary condition, wic consists in te following rearrangement, n T(u, p)n ds = G k µr k u n ds (3) Γ k Γ k wit R k := ρ f R k µ, G k := µr k Q k P k. We aim to develop a general approximation sceme capable to andle te entire range of admissible Robin coefficients R k, from te case of mean stress conditions, corresponding to te limit R k 0, to te flow rate conditions obtained wen R k. Furtermore, te selected approximation sceme sould be robust, namely te stability of te sceme must not depend on R k. In particular, by multiplying te momentum equation by a test function v and apply Green s formula to te viscous term and to te pressure, and by exploiting te fact tat T(u, p)n is constant and aligned wit te normal direction on eac section Γ k, we obtain, ( 2µD(u), D(v) ) Ω + ( ρ f (u )u, v ) Ω ( p, v ) Ω ( T(u, p)n, v ) Ω\Γ n T(u, p)n, v n = ( f, v ) Γ k Ω (4) k were (, )Ω is te L2 inner product and, Σ denotes te following symmetric positive semidefinite bilinear form [37], u, v Σ := 1 u ds v ds. Σ Σ Σ We observe tat equation (3) could now be naturally enforced into (4), by substitution into te last term on te left and side, namely n T(u, p)n, v n Γk. However, to improve te generality and te robustness of te resulting sceme, we opt for a tecnique recently proposed in [17] to weakly enforce Robin conditions for te Laplacian, ten extended in [32, 37] to te defective-stokes case. Tis approac is based on suitable linear combinations of (3) and (4), and contrarily to te approac based on simple substitution, it as te advantage to be applicable to te entire range of parameters R k, including R k tat corresponds to te pure flow rate boundary conditions. Given a penalty parameter γ to be specified later on, by following [32] we obtain te governing equations for te desired finite element approximation of 8
problem (2), wic consists to find u V and p Q suc tat a R, (u ; u, v ) + b R, (p, v ) = F R, (v ) v V (5) b R, (q, u ) + c R, (p, q ) = G R, (q ) q Q were a R, (, ), b R, (, ), c R, (, ) are defined as follows, a R, (w; u, v) := ( 2µD(u), D(v) ) Ω + ( ρ f (w )u, v ) Ω + k k k µr k 1 + γ R k u n, v n Γ k γ R k [ 2µn D(u)n, v n Γk 1 + γ R k + 2µn D(v)n, u n Γk] γ 2µn D(u)n, 2µn D(v)n 1 + γ R k µ Γ k b R, (q, v) := ( p, v ) Ω + γ R k 1 + γ R k q, v n Γ k k + γ q, 2µn D(v)n 1 + γ R k µ Γ k k c R, (p, q) := k γ 1 + γ R k µ p, q Γ k were, to simplify te description of te sceme, we ave assumed tat te finite element space conforms wit no-slip boundary conditions on Ω \ Γ, i.e. V H 1 Ω\Γ (Ω) so tat ( T(u, p)n, v ) = 0. However, in te fortcoming Ω\Γ numerical experiments we will apply Nitsce s type approximation of no-slip conditions at te interface wit rigid walls. Te rigt and side terms in (5) are, F R, (v) := ( f, v ) Ω 1 1 1 + γr k Γ k G k, v n Γk k k G R, (q) := γ 1 1 + γr k µ Γ k G k, q Γk. k γ 1 Gk, 2µn D(v)n, 1 + γr k µ Γ k Γ k In previous definitions of bilinear and linear forms, te suffix R stands for te collection of values [R 0,..., R N ], igligting te dependence of te forms on te value of te resistances. It as been proved in [32] tat te linearized version of problem (5) (tat is te Oseen problem) is well posed and robust provided tat te stability parameter γ is smaller tan a given tresold. 3.3 Stabilization tecnique for inflow sections and flow reversal. In te seminal work [15] Heywood, Rannacer and Turek observed tat, in contrast to Diriclet boundary conditions, te application of mean pressure or 9
mean flow rate conditions results in a difficulty in estimating te energy balance across artificial sections. A similar issue consists on te simulation divergence due to flow reversal at outflow boundaries as addressed in [5] and extensively analyzed in [8], were different stabilization approaces are compared. In our view, te teory developed in [15] already seds ligt on an efficient and sound tecnique to override tis drawback. In particular, owing to te identity 1 2 u 2 = u( u) T, te convection term ( ρ f (u )u, v ) can be replaced Ω wit its symmetric form ( ρf (u )u, v ) Ω ( ρ f (v )u, u ) Ω + 1 2 ( ρf u 2, v ) Ω, (6) were te kinetic energy 1 2 ρ f u 2 can be combined wit te ydrostatic pressure to give te total pressure P := p + 1 2 ρ f u 2. Ten, reformulating te resistance conditions in terms of total pressure and resorting to te symmetrization of te convective term could restore te control on kinetic energy balance. However, it as been observed tat suc tecnique provides unsatisfactory results of artificial outflow boundaries, see for instance Figure 8 in [15], wic is te most significant case for us. In alternative, we consider te introduction of a stabilization term to restore control on te kinetic energy at inflow and outflow sections. Suc tecnique, originally proposed in [5], can be reinterpreted in te present framework. Setting v = u into (6) and applying integration by parts we easily conclude tat ( ρf (u )u, u ) Ω = ρ f 2 Ω u n u 2 ds. It is straigtforward to see tat neiter resistance nor flow rate conditions allow to control suc residual boundary terms. Tis task is acieved by te introduction into te momentum bilinear form a R, (w; u, v) of te following stabilization term s k (w; u, v) := ρ f 2 Γ k ( w n w n)u v ds, (7) complemented by te rigt and side s k (w; U k, v), being U k a given velocity profile. In particular, for outflow sections we consider U k = 0 wit k = 1, 2, 3,..., wile for te inflow section Γ 0 we propose to apply a flat velocity profile U 0 = (Q 0 / Γ 0 )n. We immediately verify tat, if no-slip conditions on te arterial wall are embedded into te solution searc space, te stabilization term restores te positivity of te convective terms at te boundaries, ( ρf (u )u, u ) Ω + k s k (u; u, u) = ρ f 2 Γ u n u 2 ds 0. Unfortunately, te introduction of tis new term violates te strong consistency of te sceme (5) wit te governing equations (2). Ten, a natural question to address concerns te caracterization of te new governing equations after te 10
introduction of te stabilization term, wit special attention to te boundary conditions. In particular, te term s k (w; u U k, v) can be interpreted in multiple ways. On te one and, it can be seen as a penalization term corresponding to te weak enforcement of te pointwise conditions u = U k on Γ k. In tis case, te penalty parameter directly depends on te negative part in te inflow / outflow convective velocity, namely 1 2 ( w n w n). By tis way, te stabilization is active only for te part of te boundary were te flow is entering te domain. On te oter and, as previously remarked in [5, 8], te introduction of s k (w; u, v) can be compared wit te aforementioned application of total pressure to resistance boundary conditions. Indeed, te stabilization tecnique corresponds to replacing te constraint T(u, p)n = c k n wit T(u, p)n + 1 2 ( u n u n)u = c k n. Ten, equation (3) becomes ( ) T(u, p)n ds = G k µr k u n n ds 1 ( u n u n)u ds on Γ k. (8) Γ k Γ k 2 Γ k Ten, replacing (8) into te weak form of te Navier-Stokes momentum equation, te additional term k 1 ( ) 2 ( u n u n)u, v appears, wic is exactly k s k(u; u, v). Tis sows tat te proposed stabilization tecnique cor- Γ k responds to modify te traction force at te artificial sections were flow reversal takes place. 3.4 Numerical solver Problem (5) does not yet correspond to an uniquely solvable discrete linear problem and several steps are necessary to cast it into suc a framework. As previously remarked, formulation (5) does not set any specific constraints for te definition of te finite element spaces V, Q. To obtain an easily implementable sceme, we opt for te equal order affine approximation for velocity and pressure fields. It is well known tat suc coice violates te inf-sup stability condition for mixed problems suc as (5). To restore stability of te discrete problem we resort to te so called Brezzi-Pitkaranta stabilized formulation (we refer te interested reader to [29] and references terein), wic is based on te relaxation of te incompressibility constraint by means of te introduction of te following additional term d (p, q ) := γ p 2 p q µ into te continuity equation of (5), wic becomes b R, (q, u ) + c stab R, (p, q ) = G R, (q ), q Q, were we ave set c stab R, (, ) := c R,(, ) + d (, ). Te parameter γ p is cosen to guarantee stability and appropriate conditioning of te discrete system of equations. 11 Ω
Anoter source of instability comes from te fact tat te momentum equation for blood flow in large or medium sized arteries is usually convection dominated. To treat te lack of stability of te finite element metod in tese conditions, we apply te classical streamline upwind metod wic consist to modify te bilinear form a R, (, ) as follows, a stab R, (w; u, v) := a R,(w; u, v) + γ vρ f 2 ((w ) u) ((w ) v), µ were γ v is a parameter to be suitably cosen. We finally observe tat for te fortcoming numerical investigations we will consider te particular case obtained by taking R k, corresponding to mean flow rate conditions on eac artificial section Γ k. Owing to te generality and te robustness of te metod, te limit cases R k 0 or R k can be straigtforwardly obtained by restriction of te bilinear forms, indeed all te scaling expressions depending on R k remain bounded for any value of te parameter. For te sake of clarity, we report ere te problem wit flow rate conditions, wic takes te form (we omit te suffix R to identify tis case, tat is R k, k = 0,..., N) a stab (u ; u, v ) + b (p, v ) = F (v ) v V, (9) b (q, u ) + c stab (p, q ) = G (q ) q Q, wit a stab (w; u, v) := ( 2µD(u), D(v) ) Ω + ( ρ f (w )u, v ) Ω + γvρ f 2 µ ((w )u, (w )v) Ω + µ k γ u n, v n Γ k ] k [ 2µn D(u)n, v n Γk + 2µn D(v)n, u n Γk, b (q, v) := ( p, v ) Ω + k q, v n Γ k, Ω c stab (p, q) := γp2 µ ( p, q ) Ω, F (v) := ( f, v ) Ω ρ f Q k, v n Γk ρ f Q k, 2µn D(v)n Γk, G (q) := ρ f Q k, q Γk. (10) We observe tat te momentum equation of (9) is nonlinear because of te convective term. For simplicity, we apply dumped Picard (or fixed point) iterations to linearize it. More precisely, we replace (9) wit te following sequence of problems. Given u 0, for k = 1, 2,... aim to find uk V and p k Q suc tat σ ( u k ) uk 1, v (u k 1 ; u k, v ) + b (p k, v ) = F (v ) v V, Ω + astab b (q, u k ) + cstab (p k, q ) = G (q ) q Q, 12
were σ > 0 is te damping parameter to be suitably cosen for eac test case. Te corresponding algebraic problem is solved by means of a pressure matrix metod, wic consists in te elimination of te velocity vector unknowns and te solution of te pressure Scur complement matrix. Since suc system is usually ill conditioned (see for instance [29]) especially wen te velocity mass matrix appears in te discrete momentum equation, te Caouet-Cabard preconditioner as been applied to speed up GMRES iterations. We ave applied suc solver, implemented in te finite element library Freefem++ (see ttp://www.freefem.org/ff++/) for te approximation of te test cases addressed in te fortcoming sections. 3.5 Numerical results Te aim of tis section is to validate (9) for te approximation of outflow conditions at te intersection wit lateral brances. We consider two test cases, an idealized model addressed in [18] and a realistic aortic arc. Since we focus on aemodynamics applications, pressure, flow profiles and wall sear stresses (WSS) will be analyzed. In all te numerical tests of tis section and of Section 4 we ave used te following values for parameters: γ = 10 4, ρ f = 1.0 g/cm 3, µ = 0.035 P oise, R k = R k =, k, γ p = 5 10 4, γ v = 5 10 2. 3.5.1 Application to an idealized model of lateral branc Te present test case as been proposed in [18] to study te effect of small lateral side brances, suc as intercostal arteries, on te sear stress patterns at te wall of large arteries, suc as te descending toracic aorta. Following [18] we consider te domain depicted in Figure 2 (top), were te main box measures 1 26 0.5 cm, being (x, y, z) te lateral, longitudinal and vertical directions, respectively, and te side branc is modeled as a cylindrical segment 0.1 cm wide. Te visualized computational mes consists on 31778 tetraedral elements. Te bulk flow is oriented form left to rigt, driven by a steady semi-parabolic inflow profile u x = 0, u y (z) = 3 2 U(1 (z/0.5)2 ), u z = 0. Te upper surface of te main cannel corresponds to a no-slip boundary, were te velocity is fixed to zero, wile on te lateral sides of te box we set omogeneous Neumann conditions. Te inflow flow rate Q 0 is computed according to te inflow profile, wile te lateral branc flow rate is set to Q 1 = 0.0079 Q 0. Te flow rate at te main outflow section on te rigt and side of te box, denoted wit Q 2, is set to satisfy mass conservation. Te profile modulus U is cosen to make sure tat te flow Reynolds number is equal to 250, wic corresponds to test case (a) of Figure 2 in [18], in order to allow for a visual comparison wit present results. As depicted in Figure 2 (middle row) we consider two configurations. A full model were te lateral branc is accounted (on te left) and a simplified model 13
were te flow rate Q 1 is weakly enforced at te intersection of te lateral branc wit te main branc (on te rigt). As a preliminary validation, we observe te flow profiles at te bifurcation of te side branc are rater similar. A more significant validation is acieved in Figure 2 (bottom row) by comparing te WSS patterns on te surface of te main vessel, in te neigborood of te side branc. In particular, we report ere te modulus of te WSS vector, normalized wit respect to te WSS at te inflow of te main cannel. Again, te agreement between te full model and te incomplete one is satisfactory. We observe tat te WSS pattern is also significantly similar to te one reported in Figure 2(a) of [18], for te full geometrical model and equivalent flow conditions. 3.5.2 Application to an aortic arc model In tis test case we consider te full aortic arc model described in Section 2, togeter wit te variant were te ascending brances ave been omitted. For simplicity, we consider steady flow conditions, were te inflow rate Q 0 is set to 80 cm 3 s 1, corresponding to te mean value over a eart beat. Concerning te outflow, te flow division between te braciocepalic, left common carotid and left subclavian arteries is kept constant to te values Q 1 = 5%, Q 2 = 5%, Q 3 = 7% of te inflow Q 0, respectively, wic corresponds to realistic aemodynamic conditions. In te full model suc flow rates are weakly enforced on te distal section of eac branc, wile for te reduced model equivalent boundary conditions are set on te outflow sections Γ k, wit k = 1, 2, 3, laying on te reconstructed surface of te main branc, as depicted in Figure 1 (bottom, rigt). According to tese conditions, te inflow and outflow velocity profiles were not a priori enforced, but determined by te flow governing equations to satisfy te constraints of given flow rate and constant stresses. In Figure 3 we compare te pressure fields (top), velocity profiles (middle) and WSS distributions (bottom) for te full and te incomplete model (left and rigt columns respectively). Again, te results for te two models remarkably agree. Tis confirms te effectiveness of te defective flow rate conditions to correctly capture te outflow profile, even toug te geometrical features of te lateral brances are omitted. 4 Fluid structure interaction wit outflow boundary conditions on lateral brances 4.1 Preliminaries We notice tat now te domain moves in time, so tat we distinguis between te current configuration Ω t, wic canges in time, and te reference configuration Ω 0, wic corresponds to te initial fixed geometry. Wen te fluid domain is moving, a classical Eulerian approac is not suitable for writing te fluid equations since one would like to follow te movement of te fluid-structure 14
Figure 2: Details of te idealized model of lateral branc (as in [18]). Geometry (top), lateral outflow velocity profiles (middle) and corresponding WSS (bottom) are depicted. For te last two rows, we report te case were te lateral side branc is accounted (left) as well as te one witout it (rigt). 15
Figure 3: From top to bottom we depict te pressure fields, velocity profiles and WSS distributions for te full and te incomplete aortic arc models, corresponding to left and rigt columns respectively. 16
(FS) interface. On te oter and, a pure Lagrangian framework would deform te fluid domain also at te artificial sections. Terefore, te fluid equations are written in a ybrid configuration, namely te Arbitrary Lagrangian-Eulerian (ALE) framework (see e.g. [16, 7]). To tis aim we consider te ALE map A. A classical coice in aemodynamic applications to define te ALE map is to consider a armonic extension of te structure displacement at te FS interface Σ t := Ω t \ Γ t in te reference domain (see, e.g., [6]), and to fix te artificial section (at least) in te normal direction. For any function v living in te current configuration, we denote by v := v A its counterpart in te reference configuration. As pointed out previously, wit current imaging devices displacement is mainly retrieved on te interface between fluid and structure, so tat extrusion of fluid mes is needed to obtain structure meses. However, te tickness of te vessel is not acquired by tese devices, so tat a priori assumptions on it are mandatory. Furtermore, te vascular wall is usually tinner tan te lumen. For tese reasons, in order to reduce te computational time, we consider ere te vascular wall as a tin membrane. In particular, te membrane law considered in tis work is te simple interial-algebraic law [25] ρ s s 2 η t 2 + β s η = f s in Σ 0, (11) were ρ s is te structure density, β = E β wit β = (4ρ 2 1 ν 2 1 2(1 ν)ρ 2), were ρ 1 and ρ 2 are te mean and te Gaussian curvature, s is te tickness of te membrane, E te Young modulus and ν te Poisson ratio. Here, η is te structure displacement in te normal direction wit respect to Σ 0, and f s te external forces. We observe tat we ave written te structure equation as usual in te Lagrangian framework. As it emerges from te membrane law (11), te knowledge of te curvature of te FS interface is mandatory in order to write te structure problem, and consequently also te FS coupled problem, as it will be clear from te next subsection. Te cut of lateral brances provides a computational domain wit almost constant curvatures. As an example, in Figure 4 we depict te value of β (wic depends on curvatures) for te computational domains considered in tis work. We observe tat by cutting te lateral brances te value of β is almost constant, wile viceversa in te original domain it features values in te range [0, 100]. Since te case wit uniform parameters leads to a better conditioned system, te case witout brances is preferable for te numerical simulations. Tis provides anoter motivation for cutting lateral brances in te computational setting of cardiovascular applications. 4.2 Coupled formulation Te numerical solution of te FSI problem features two major difficulties: te treatment of te interface position, wic is an unknown of te problem, and te 17
Figure 4: Magnitude of β = (4ρ 2 1 2(1 ν)ρ 2), ρ 1, ρ 2 being te mean and Gaussian curvature respectively, in te case wit (left) and witout (rigt) lateral brances. prescription of te continuity interface conditions, namely continuity of velocities and normal stresses, wic reads as follows [11] { u n = η on Σ t, T(u, p)n n = f s on Σ t (12). In te tangential direction, tere is no coupling, so tat we ave to prescribe furter conditions, namely eiter u n = 0 or T n n = 0. For te prescription of (12) two strategies ave been proposed and widely studied in te literature, namely te partitioned and te monolitic approaces. In te first case, one solves te fluid and structure subproblems in an iterative framework, until fulfillment of te interface continuity conditions (see, e.g., [28, 6, 2]. Here, owever, we consider te second strategy, based on building te wole FSI matrix, and ten by solving it wit an efficient metod [14, 4, 13]. In tis way te interface continuity conditions are automatically satisfied. Te drawback of tis approac is te non-modularity, in te sense tat an ad oc code as to be implemented, witout any possibility to exploit existing fluid and structure solvers. However, ere we consider te monolitic strategy introduced in [25] in te case of a membrane structure, wic requires just te solution of te fluid subproblem. In particular, te structure subproblem is embedded into te fluid one leading to a Robin boundary condition for te fluid at te FS interface. Concerning te treatment of te interface position, we consider ere an explicit treatment as proposed in [9, 3], based on a suitable extrapolation of te interface position from previous time steps. Recently, in [24] it as been igligted te effectiveness and accuracy of tis treatment for aemodynamics applications. Moreover, in order to obtain pysical solutions, one as to deal wit anoter difficulty, namely te formation of spurious pressure reflections at te outlets, 18
were non ad-oc boundary conditions are prescribed. To tis aim, we consider te following absorbing boundary condition at a generic outlet, introduced in [25] and obtained by a 1D reduced model, ( ( ) ) 2 Qout T(u, p)n n = β + (A 0 4χA out) 1/4 (A 0 out) 1/2, out were χ := β 2ρ f and Q out (t) and A out (t) (wit A 0 out = A out (t = 0)) are te flow rate and te area at te outlet at and. Tis condition gives an implicit relation among te normal stress, te flow rate and te area at te outlet. In te numerical simulations, we will use an explicit expression of flow rate and area, by using a suitable extrapolation of previous time steps, relying to a Neumann boundary condition. In view of writing te time discrete problem, given a quantity z, we denote wit z n its approximation at time t n = t 0 + n t, t n [t 0, T ], were t is te time discretization parameter. Moreover, we indicate wit w te finite element approximation of te fluid domain velocity. Ten, after time discretization (implicit Euler for te fluid wit a semi-implicit treatment of te convective term, and Backward Differentiation Formulae (BDF) of first order for te structure), te FSI problem wit flow rate conditions on te artificial sections reads at eac time t n+1 as follows [25] 1. Given u n, ηn, ηn 1, solve te fluid problem tat is to find u n, pn suc tat for all v V and q Q te following equations old true ( 1 t (un+1 u n, v ) Ω n + a stab,n (u n wn ; un+1, v ) + β s t + ρ ) s s u n+1 Σ n t v ds ( +b n (pn+1, v )= F n (v ) + β s η n + ρ s s ( η n Σ n t 2 η n 1 ) ) n v ds ( ( ) ) Q n 2 + β out Γ n 4χA n + (A 0 out) 1/4 (A 0 out) 1/2 n v ds, out out b n (q, u n+1 ) + c stab,n (p n+1, q ) = G n (q ). Superscript n over bilinear forms a stab (, ), b (, ) and c stab (, ) and over functionals F ( ) and G ( ) denotes tat suc quantities are evaluated over Ω n. Te number of artificial sections were a flow rate is imposed are N, namely Γ k, k = 0,..., N 1, and we ave denoted te main outflow section wit Γ out. Te tird term at te left and side and te second term at te rigt and side of te momentum equation are related to te monolitic FSI formulation and represent a Robin boundary condition at te FS interface for te fluid due to te coupling wit te membrane [25]. Te tird term in te rigt and side of te momentum equation is due to te absorbing Neumann boundary condition at te outlet Γ out. 19
2. Compute te structure displacement η n+1 = η n + tun+1 n; 3. Compute te fluid domain displacement by solving for all ẑ Z H 1 (Ω 0 ) te following armonic extension problem ( ŵ n+1, ẑ ) + 1 (ŵn+1 Ω 0 γ, ẑ ) ( ŵ n+1 Σ 0 n, ẑ ) ( ẑ n, ŵ n+1 ) Σ 0 Σ 0 + 1 (ŵn+1 γ n, ẑ n ) ( n ŵ n+1 Γ 0 0 Γ0 n, ẑ n ) ( n ẑ n, ŵ n+1 out Γ 0 0 Γ0 n ) out Γ 0 0 Γ0 out = 1 (ûn+1 γ, ẑ ) ( ẑ n, û n+1 ) Σ 0 Σ 0 and ten update te fluid domain by x n+1 = x n + t ŵ n+1, were x are te coordinates of te fluid domain wit respect to te reference configuration. We notice Nitsce s treatment of te Diriclet boundary condition at te FS interface for te armonic extension. Tis coice allows more versatility in imposing different kind of boundary condition for te tangential and te normal components, as usually appens in tis problem. In particular, we observe tat we fixed te inlet Γ 0 and te outlet Γ out in te normal direction by imposing a omogeneous Diriclet condition, wilst we let tem free to move in te tangential direction by imposing a omogeneous Neumann condition. Finally, we observe tat at sections Γ k, k = 1,..., N 1, we ave imposed omogeneous Neumann conditions for all te components. We observe tat due to te explicit treatment of te interface position, te solution of te FSI problem involves just one fluid problem per time step. Terefore, tis metod is very interesting from te point of view of te computational time. 4.3 Numerical results We ave considered te computational domain witout lateral brances addressed in te rigid case (see Figure 1, rigt column). We ave used te same parameters and flow division of te rigid case as described in Section 3.5, as well as te same solver and code as described in Section 3.4, apart from te convective term wic as been treated ere explicitly. Moreover, we ave set t = 2 10 3 s, ρ s = 1.1 g/cm 3, and we ave imposed te pysiological flow rate depicted in Figure 5. In favor to te stability of te numerical sceme, te values of te flow rate ave been set to four time less tan pysiological ones. 20
Figure 5: Flow rate prescribed at te inlet. Terefore, te numerical results obtained by tese simulations are not pysiological and ave to be tougt as an intermediate result in view of realistic patient-specific simulations. We ave considered two sets of simulations: in te first case we ave added te term (7) to te bilinear form a stab (, ) (10) 1, wilst in te second case we ave dropped it. In tis way, we could evaluate te effect of (7) on te solution in terms of stabilization of flow reversals. In Figure 6 we report te vectors of te velocity field for bot cases (wit and witout (7)) at tree different times. In particular, we consider early, peak and late systole. In te same picture te background color refers to te pressure distribution. First of all we notice te effectiveness of Nitsce s metod for te prescription of te flow rate at te inlet. We observe indeed tat te peak velocity at tis section is moved towards te inner wall, as expected for te aortic arc. Moreover, we observe tat in bot cases te cut of lateral brances allows to recover a satisfying solution also in te case of a deformable vessel. In particular, te ALE map is stable and displacements of te artificial sections accounting for lateral brances are coerent to te ones of te neigboring arterial wall. Tis suggests tat te coice of prescribing omogeneous Neumann conditions for te armonic extension at tese sections is valid in terms of stability and accuracy. Finally, we remark te effectiveness of te stabilization induced by (7). At peak systole flow reversal does not occur at outflow sections. In tis case (Figure 6, middle row) te stabilization tecnique does not perturb te flow. Conversely, early and late systole are caracterized by transient flow patterns. During early systole flow acceleration takes place. In tis case it seems unlikely to develop flow reversal, wic owever appears on artificial sections corresponding to lateral 21
Figure 6: Velocity vectors on inlet and outlet sections and pressure distribution in te background for te case wit (left) and witout (rigt) stabilization term (7) - Time t = 0.05 s (top), t = 0.11 s (middle), and t = 0.22 s (bottom). 22
brances (Figure 6, top row). We believe tat tis is a spurious flow mode due to lack of pointwise control at tese outflow sections. Te stabilization tecnique is able to filter suc flow conditions, restoring outward velocity field at lateral sections. Te beavior at late systole is different (Figure 6, bottom row). In tis case, igly decelerating flow takes place, wic promotes pysiological flow reversal. Indeed, it occurs at all outflow sections and we notice tat te stabilization tecnique does not completely remove suc effects, but reduces teir magnitude in order to ensure stability of te discrete sceme. 5 Conclusions In tis work we ave considered te problem of te prescription of average data at artificial sections in computational aemodynamics, wen lateral brances are cut from te original computational domain. Te interest in cutting tese brances is multiple: for example, we mention a possible simplification for te geometry reconstruction and mes generation, as well as for te treatment of te FSI problem, in particular wen a membrane structure is considered (as usually done in aemodynamics). In particular, in tis work we ave studied te effectiveness of Nitsce s metod to prescribe flow rate conditions in tis context. Firstly, we ave compared te results obtained wit and witout lateral brances by performing steady Navier-Stokes simulation bot in an idealized and in a real computational domain. In te first case, te numerical results igligted te good accuracy of te solution obtained witout lateral brances wit respect to te full model also in proximity of te artificial sections. In te second case, we considered a real aortic arc as computational domain and again te results sowed a good agreement among te solutions obtained in te two cases. Secondly, we ave studied te effectiveness of Nitsce s metod for te prescription of te flow rate wen lateral brances are cut, in te context of te FSI problem wen a membrane structure is considered. Here, first of all we ave noticed te significant simplification in te model due to te cut of lateral brances, since te curvature of te simplified domain is almost constant. Moreover, we ave observed te effectiveness of Nitsce s metod wic allows to recover pysiological velocity profiles witout coosing tem a priori. Finally, we ave compared te case wit and witout a stabilization at te artificial sections to avoid flow reversal. Tese results igligted te effectiveness of tis tecnique, wic guarantees a stable distal flow profile witout affecting te solution far from te artificial sections. 23
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