Extended Finite Elements method for fluid-structure interaction with an immersed thick non-linear structure

Transcription

1 MOX-Report No. 26/2018 Extended Finite Elements metod for fluid-structure interaction wit an immersed tick non-linear structure Vergara, C.; Zonca, S. MOX, Dipartimento di Matematica Politecnico di Milano, Via Bonardi Milano (Italy) ttp://mox.polimi.it

2 Extended Finite Elements metod for fluid-structure interaction wit an immersed tick non-linear structure C. Vergara, S. Zonca April 15, 2018 MOX Modellistica e Calcolo Scientifico Dipartimento di Matematica Politecnico di Milano via Bonardi 9, Milano, Italy <cristian.vergara,stefano.zonca>@mate.polimi.it Keywords: Fluid-structure interaction, unfitted meses, valve dynamics AMS Subject Classification: 65M60, 74F10 Abstract We consider an Extended Finite Element metod to solve fluid-structure interaction problems in te case of an immersed tick structure described by non-linear finite elasticity. Tis metod, tat belongs to te family of te Cut Finite Element metods, allows to consider unfitted meses for te fluid and solid domains by maintaining te fluid mes fixed in time as te solid moves. We review te state of te art about te numerical metods for fluid-structure interaction problems and we present an overview of te Cut Finite Element metods. We describe te numerical discretization proposed ere to andle te case of a tick immersed structure wit size comparable or smaller tan te fluid mes element size in te case of non-linear finite elasticity. Finally, we present some tree-dimensional numerical results of te proposed metod. 1 Introduction Te interaction between a fluid and an immersed structure may be significant in many applications, for example in aeronautic engineering to study te response of te air on te aircraft [11, 32, 33, 61], in civil engineering to understand te effect of wind on bridges [20, 74, 92], towers [54], and suspended cable [13, 81], in energy engineering to study te modeling of wind-turbines, eat excangers and ydro-turbines [9, 80, 85], in sport engineering to investigate te impact of te waves over a rowing boat [34, 35] or te flow around a sailing yatc [78], in biomedical application, for instance in 1

3 emodynamics to study te stresses exerted by blood flow to te leaflets of an eart valve [67, 89], or to study te blood pressure exerted to te retinal vessels walls [3, 5]. In some cases, it may be interesting to consider a full tree-dimensional (3D) model for te structure, even toug te tickness of te structure is small wit respect to te caracteristic size of te domain. For example, in te simulation of eart valves, one sould consider te interaction between te blood and te valve leaflets. Often, for clinical purposes, tere is te need to accurately evaluate te internal structural stresses, wic can be computed only by means of a full 3D geometric model. Te numerical simulation of suc a fluid-structure interaction problem is very callenging. First of all, te structure undergoes large displacements, tus its movement cannot be ignored from te geometric point of view. Second, te immersed structure is very tin, often smaller tan te caracteristic mes size of te fluid problem, and tis leads to numerical and computational difficulties due to te cut of some fluid mes elements. In tis work, we propose an Extended Finite Elements discretization for FSI problem wit an immersed 3D non-linear elastic structure in te regime of large displacements. In Sect. 2 we briefly review te most important numerical strategies introduced so far to andle tis problem, wereas in Sect. 3 we specifically focus on te family of Cut Finite Elements, to wic our metod belongs. Ten, in Sect. 4 we introduce te matematical problem and te proposed numerical approximation, wereas in Sect. 5 we give some detail on te algoritm for te solution of te non-linear system arising after discretization. Finally, in Sect. 6 we present some numerical results. 2 State of te art Several numerical metods ave been developed so far to solve te FSI problem wit an immersed structure. We subdivide tem depending on te treatment of te computational meses. Accordingly, we arrange tem into two main categories: body-fitted mes metods and fixed/unfitted mes metods. In te first category, we place all te metods tat use a conforming and fitted mes at te fluid-structure interface. Among tem, we cite te Arbitrary Lagrangian Eulerian approac introduced in [29,55,58]. In presence of very large displacements, tis metod may fail due to te ig distortion of te fluid mes, so tat a remesing procedure is required. Moreover, tis procedure as te disadvantage tat may introduce an artificial diffusivity due to te need of interpolating from one mes to te new one. Noneteless, te ALE metod as been used by some autors to deal wit immersed structures. For example, in [64] an ALE approac wit remesing is proposed to simulate eart valve closure on a 2D simplified geometry, in [75] a syntetic 3D model was employed to study te valve opening, in [63] a 3D model is used to study te influence of te sinus of Valsalva in te aortic valve, in [91] a 2D simulation of te aortic valve is performed on a plane of symmetry along te center of a leaflet for an entire cardiac cycle. A similar approac based on local adaptation is presented in [8]. Tis Extended ALE metod allows te structure mes to move independently of te fluid one tat is kept fixed, 2

4 resulting in a pair of meses tat are not fitted at te interface. Ten, te fitting between te two meses is obtained via local remesing or local canges in te connectivity. Anoter fitted metod is given by Space-Time Finite Elements [30,57,59,76]. Te basic idea is to divide te time domain into slabs and ten to use te Finite Element (FE) basis functions on eac slab for bot te spatial and temporal discretizations. In te second family of numerical metods, we place te metods based on a fixed background mes and on an overlapped unfitted mes for te fluid and te solid, respectively. Tese approaces were developed to avoid te movement, or te remesing, of te fluid mes. In particular, tey ave been specifically designed for treating te case of large deformations. Te first fixed/unfitted mes metod proposed in te literature is te Immersed Boundary (IB) metod, introduced in [79] in te context of Finite Differences for an immersed membrane and specifically realized for studying te fluid-dynamics in te eart. In tis framework, te fluid is represented in Eulerian coordinates, wile te structure in Lagrangian coordinates by means of a forcing term for te fluid problem tat acts on te fluid-structure interface. Te extension of te IB approac to te FE metod is presented in [15, 16]. Te FE formulation is extendable to te case of tick structures and allows to easily manage te forcing term given by te action of te structure on te fluid, see, e.g., [17, 90, 94]. As regards te applications, te Finite Difference IB metod as been used in [47] to simulate te blood dynamics in a realistic domain of te eart and in [46, 48] to simulate te dynamics of te eart valves. Te Finite Element IB metod as been employed in [93] to study an immersed structure interacting wit a viscous fluid and in [66] for several biological applications regarding valve dynamics, vessel stents, red blood cells interaction and cells migrations. See also [56] for an application to bioprostetic eart valves. Anoter IB approac widely used in te context of eart valves is te Curvilinear Immersed Boundary (CURVIB) metod [36] wic is particularly suited for te 3D case. Te CURVIB metod was successfully employed for FSI problems for simulating te dynamics of prostetic eart valves, see [18, 19, 36, 65]. A different approac in te category of te fixed/unfitted mes metods is te Fictitious Domain (FD) metod. Tis metod was introduced in [43] for solving te viscous-plastic flow equations inside complex domains, ten in [14, 44] it was used for solving te Navier-Stokes equations around immersed objects, and in [45] it was extended to treat te case of moving rigid bodies inside incompressible viscous flows wit applications to particle flows, see also [41, 42]. Several works based on te FD metod ave been produced for te solution of FSI problems wit immersed structures: in [86, 87] an application to FSI for eart valves including contact wit rigid bodies is presented; in [7] a procedure for dealing wit te interaction of an incompressible fluid and different structures is proposed, allowing te contact among te deformable bodies; in [83] an application for eart valves is compared wit experimental data; in [62] an application to bioprostetic eart valves is considered; in [88] a comparison of some FD approaces wit te ALE one is presented for several FSI problems. A variant to te FD approac for dealing wit valve dynamics is proposed in [26, 27]. In te latter works, te FD approac is combined wit te ALE one in order to exploit teir advantages: te ALE approac is used to describe te movement of te root of te valve, tat 3

5 undergoes a limited displacement, so tat no remesing is necessary, allowing to build a fitted mes at te fluid-structure interface; te FD approac is instead used to describe te leaflets of te valve tat, on te contrary, undergo large displacements. 3 Cut Finite Element metods Te metods presented above feature two main limitations: te body-fitted mes metods require te remesing of te fluid domain due to te igly distorted fluid elements tat appear wen te displacement of te immersed structure is too large; te fixed/unfitted mes metods require te implementation of ad-oc strategies to sarply capture te interface. Indeed, in te case of diffuse interface metods (suc as te IB metod), te interface conditions are imposed troug forcing terms wic spread te effect of suc conditions on a cluster of neigbouring cells. Tis results in a lack of sarpness in capturing te boundaries and te inability of enforcing boundary conditions for strongly fluctuating quantities, suc as in turbulent flows, see, e.g., [72]. Here, we consider a specific class of metods belonging to te fixed/unfitted family tat tries to overcome tese limitations. Te advantages of tis class, referred to as Cut Finite Element (CFE) metods, are tat tey maintain te accuracy of classical FEM and can be developed by extending te features of FEM, see, e.g., [22] for a review of tis class of metods. Let us consider a finite set of domains Ω i R d, wit i = 1,...,N, and d = 3. We indicate wit background domain, a domain Ω suc tat Ω N i=1 Ω i, i.e. a domain tat covers all te domains Ω i ; foreground domain, eac of te domains Ω i, i = 1,...,N, tat overlaps te background domain Ω; interface, a curve Σ i, i = 1,...,N, of co-dimension one tat separates te background domain Ω to te foreground domain Ω i. Moreover, we distinguis te foreground domains into tree categories, depending on teir tickness: zero-tickness domain, a foreground domain of co-dimension one (contained in te background domain Ω) tat divides Ω into two parts (Σ in Figure 1); tin domain, a foreground domain suc tat its tickness is smaller tan te caracteristic size of te background mes (Ω 1 in Figure 1); tick domain, a foreground domain Ω i suc tat te tickness of te domain is comparable wit its caracteristic size (Ω 2 in Figure 1). 4

6 Ω 1 Ω Σ Σ 2 Ω 2 Σ 1 Figure 1: Te background domain Ω (wite) contains te zero-tickness domain Σ, te tin foreground domain Ω 1 (grey) and te tick foreground domain Ω 2 (grey). Te interfaces Σ 1 and Σ 2 delimit Ω 1 and Ω 2, respectively. Wit te final aim of describing ow te pysical processes are related to te domains and te meses, in wat follows we consider two processes, P B and P F, tat represent te background and foreground processes, respectively. Process P B occurs in te background process domain Ω BP, wile process P F occurs in te foreground domain Ω F, see Figure 2 (top). Te union of te domains Ω BP and Ω F generates te background domain Ω, see Figure 2 (bottom). For eac of tese tree domains we generate te corresponding mes. In particular, for te background domain Ω, we generate te background mes T, and for te foreground domain Ω F, we generate te foreground mes T F, see Figure 3 (top). Te background process mes T BP is instead generated by considering only te portions of te elements of te background mes T tat belong to te background process domain Ω BP, see Figure 3 (bottom), i.e. T BP = {K : K = K Ω BP, K T }. (1) We ave indicated by > 0 te mes size. Notice tat te background process mes contains elements of arbitrary sape, in particular polygons. We also define by T BP te smallest mes contained in T tat covers te background process domain Ω BP as T BP = {K T : K Ω BP /0}, (2) i.e. T BP is composed by te elements K tat belong (also partially) to te domain Ω BP, see Figure 4 (left). Finally, we define by Ω BP te domain associated wit te mes T BP, see Figure 4 (rigt), i.e. Ω BP = int K T BP K. (3) 5

7 Ω BP Ω F Ω Figure 2: Top, left: te background process domain Ω BP associated wit te process P B. Top, rigt: te foreground domain Ω F associated wit te process P F. Bottom: te background domain Ω is te union of te background process domain Ω BP and te foreground domain Ω F. 6

8 T T F T BP Figure 3: Top, left: te background mes T related to te background domain Ω. Top, rigt: te foreground mes T F related to te foreground domain Ω F. Bottom: te background process mes T BP (grey) associated wit te background process domain P B. T BP Ω BP Figure 4: Left: te mes T BP (grey). Rigt: te domain Ω BP (grey). 7

9 Wit te above definitions, we can provide te following statement: Te common idea of Cut Finite Element metods is i) to take a fixed background mes overlapped by foreground meses, ii) to cut te elements of te background mes wit te zero-tickness mes or wit te interfaces of te tin and tick foreground meses, generating elements of arbitrary sape (polytopes), and iii) to write a suitable weak formulation witin te background process and foreground domains. We stress tat te nature of te foreground domain may be eiter geometric or pysical. In particular: for a geometric foreground domain, te zero-tickness domain describes a discontinuity in te properties of te background domain, but it is not subject to a process; instead, te tin and tick domains may represent i) eiter a portion of te space wit different pysical properties wit respect to te background one, owever being described by te same process, ii) or a ole in te background geometry tat is not subject to any process (empty process); for a pysical foreground domain, te zero-tickness, tin and tick domains represent a different pysical process wit respect to te background one, represented by a different partial differential equation, as it appens in te FSI problem. Different formulations ave been proposed depending on te tickness and nature of te foreground domains, and eac of tese leads to a different CFE metod. A grapical representation of all te possible combinations in te case of two processes is sown in Table 1. In te literature, te geometric cases are considered in te following works: te zero-tickness case is treated in [4, 12, 49, 73] for elliptic problems, in [10, 50, 77] for solid mecanics and in [52] for te Stokes problem; te tin case is treated in [82] for solving te Navier-Stokes equations; te tick case is considered in [51, 68] for elliptic problems and in [69] for te Stokes problem. Te pysical cases ave been studied in te following works for FSI problems: te zero-tickness case in [2]; te tin case in [37 39] in a two-dimensional framework and in [95] for 3D problems; te tick case in [23] in two-dimensions, and in [37, 40, 70, 71] in tree-dimensions. A particular pysical case is considered in [53, 60] for solving PDEs only on te zero-tickness foreground domain, i.e. on immersed surfaces. In particular, te tickness of te foreground domain as a strong effect on te approac employed to tackle te problem in te background process domain. In fact, wen a foreground domain crosses te elements of te background mes, tree different configurations may appear, see Figure 5. In te case of a zero-tickness domain Σ, see Figure 5 (left), te background element K is split into two parts and a numerical approximation is required on eac portion of 8

10 Geometric Pysical zerotickness P1 P1 P1 P1 empty P2 tin P1 P1 P1 P1 P1 or empty P2 tick P1 P1 or empty P1 P2 Table 1: Scematic representation of possible scenarios wen considering overlapping background (wite) and foreground (grey) domains. On eac resulting domain, we ave indicated te underlying process (P1, P2 or empty). K. In te case of a tin domain Ω 1, see Figure 5 (center), te background element K is partially overlapped and is divided into tree parts: only te two parts tat belong to te background process domain require a numerical approximation for te background process. Finally, in te case of tick domain Ω 2, see Figure 5 (rigt), te background element K is divided into two portions and te numerical solution for te background process is required only on te background process mes. In any case, conforming grids at te interface between te background and foreground domains are difficult to generate, since te immersed foreground domain provides a severe constraint for te mes generation. For tis reason, unfitted overlapping meses are considered in order to avoid te computational issue tat may arise wen generating fitted meses. Te conditions to couple te problem at te interface are usually imposed via te Discontinuous Galerkin (DG) metod (or, as some autors refer to, te Nitsce s metod). Lagrange multipliers are also considered, see, e.g., [37 40]. 9

11 K Σ Ω 1 K Ω 2 K Figure 5: Configurations of te background element K depending on te tickness of te foreground domain. Left: a foreground zero-tickness domain Σ intersects te background element K. Center: a tin foreground domain Ω 1 overlaps te background element K. Rigt: a tick foreground domain Ω 2 overlaps te background element K. In wite, te background process element. Te majority of te metods presented above allow to consider a background domain overlapped by a tick foreground domain or crossed by a zero-tickness foreground domain. However, tey do not deal wit te case of a tin foreground domain immersed in te background one and, in fact, te extension of tese metods to te tin case is not straigtforward, since te tickness of te foreground domain is smaller tan te caracteristic size of te background mes elements. Some works in tis direction as been proposed in te literature: in [38 40], te autors consider two and tree-dimensional approaces tat are able to combine te feature of te Extended Finite Element metod (XFEM) wit overlapping meses in te case of tin and tick foreground domains, by employing Lagrange multipliers for imposing te interface conditions; in [82] a metod as been proposed to solve te Navier-Stokes equation solely wit immersed fixed obstacles in a tree-dimensional framework, were te interface conditions are imposed via te Nitsce s metod. Recently, in [95], an XFEM metod to andle te case of a tin foreground domain for 3D computations as been proposed for a linear structure. Te goal of tis work is to describe a metod for solving tree-dimensional FSI problems wit a non-linear immersed tin structure tat combine te features of te approaces presented above, i.e. i) te possibility of considering a numerical solution in eac portion of te fluid background elements split by te interface and ii) te employment of composite grids to represent two domains, see Figure 6. 4 Te continuous problem and te Extended Finite Elements approximation We consider a fluid background process domain Ω BP = Ω f and a structure foreground domain Ω F = Ω s suc tat Ω = Ω f Ω s R d, d = 2,3, is te background domain and Σ = Ω f Ω s is te fluid-structure interface, see Figure 7. We denote by Ω f and Ω s te boundaries of te fluid and solid domains, respectively, and we define Γ f = Ω f \Σ 10

12 Ω s Σ Σ Ω f,1 Ω f,2 T T s Figure 6: Left: te background domain is overlapped by a tin foreground domain Ω s. Te interface Σ separates te fluid background process domains (Ω f,1 and Ω f,2 ) to te solid foreground domain. Rigt: Te solid foreground mes T s overlaps te background mes T tat covers te entire domain. Te tickness of T s is smaller tan te size of te background mes elements. and Γ s = Ω s \ Σ. Finally, we indicate by n f and n s te outward unit normal to te domain Ω f and Ω s, respectively. On te interface Σ we ave n f = n s = n. Ω f Σ Ω s Γ f n s n f Γ s Figure 7: Sketc of te fluid and structure domain Ω f and Ω s wit te fluid-structure interface Σ. Te continuous fluid-structure interaction problem reads as follows: Find for any t (0,T ], te fluid velocity u(t) : Ω f (t) R d, te fluid pressure p(t) : Ω f (t) R, te solid displacement d(t) : Ω s R d, and te fluid domain displacement d f (t) : Σ R d, suc tat 11

13 Fluid problem: ρ f t u + ρ f u u T f (u, p) = 0 in Ω f (d f ), u = 0 in Ω f (d f ), u = g on Γ f D (d f ), T f (u, p)n f = on Γ f N (d f ); (4) Solid problem: ρ s tt d T s ( d) = 0 in Ω s, d = 0 on Γ s ; Coupling conditions: u = t d on Σ(d f ), T f (u, p)n f = T s (d)n s on Σ(d f ); (5) (6) Geometric condition: d f = d on Σ(d f ), (7) were (4) are te Navier-Stokes equations, (5) are te equations of elasto-dynamics, (6) are te pysical coupling conditions (kinematic and dynamic, respectively), and (7) is te condition tat guarantees te geometric aderence between te fluid and solid domains. We ave igligted te dependence of te fluid domain and of its boundaries on te interface displacement d f, wic in fact couples geometrically te fluid and te structure sub-problems. We point out tat te fluid problem (4) is written in an Eulerian framework, i.e. in te deformed configuration, wile te solid problem (5) is written in te Lagrangian framework, i.e. in te reference configuration. We ave indicated wit te superscript te quantities evaluated in te reference configuration. Te quantities witout are referred to te current instant t. We ave Γ f = Γ f D Γ f N and we ave considered a Diriclet boundary condition on Γ f D and a Neumann condition on Γ f N, wit g and suitable functions wit te required regularity.. Moreover, ρ f and ρ s are te fluid and structure densities, T f (u, p) = pi + 2µ f D(u) is te fluid Caucy stress tensor, wit µ f te fluid viscosity and D(w) = w + T w, T s ( d) is 2 te first Piola-Kircoff solid stress tensor. Moreover, T s = JT s F T is te formula to pass from te solid Caucy stress tensor T s to te Piola-Kircoff tensor, wit F = x te deformation gradient, i.e. te gradient of te coordinates in te current position wit respect to te reference space coordinates, and J = det(f) is its determinant. For 12

14 te sake of simplicity we ave considered omogeneous Diriclet conditions on Γ s. Moreover, te FSI problem given by equations (4)-(7) as to be completed wit initial conditions for te fluid and solid velocity and displacement. Te mecanical beaviour is described by a second order exponential model defined by te following te strain energy function: W (I 1 ) = κ 4 ( ) (J 1) 2 + ln 2 (J) + α ( ) e γ(i 1 3) 2 1, (8) 2γ were I 1 = tr (C), C = F T F is te rigt Caucy-Green tensor, α is te sear modulus tat represents te mecanical stiffness of te material, κ is te bulk modulus and γ is a positive parameter tat represents te level of non-linearity of te mecanical response of te body. Te corresponding first Piola-Kircoff solid stress tensor reads as follows: T s (F) = κ 2 ( J 2 J ln(j) ) F T ( ( ) + 2α J 2/3 I 1 3 e γ(j 2/3 I 1 3) 2 J 2/3 F 1 3 I 1F ). T In wat follows, wit te aim of writing te Extended Finite Elements/Discontinuous Galerkin (XFEM/DG) discrete formulation, we follow [95] and we introduce te meses and te numerical spaces. To ease te presentation, we assume tat Ω f, Ω s and Σ are polyedral. Referring to te notation of Sect. 3, we denote by T F = T s te solid foreground mes tat covers te domain Ω s and is fitted to Ω s, and by T te background mes tat covers te wole domain Ω and is fitted to Γ f, but in general not to Σ and Γ s. We indicate by > 0 te space discretization parameter, wic is a function tat may vary among te elements K of te meses and between te background and foreground meses. As a result, te solid foreground mes T s overlaps te background mes T, see Figure 8 (left). Ten, accordingly to definition (1), we denote by T BP = T f te fluid background process mes, i.e. te mes generated by considering te restriction of T to Ω f, defined as T f = {K : K = K Ω f, K T }. In te case of a tin foreground structure, te elements of te background mes T could be cut by te foreground mes and divided into several disconnected polyedra, wit a portion of te background elements overlapped by te foreground mes, see Figure 8. We refer to tese elements as split elements. We introduce te following mes G = {K : K T, K Σ /0, K Ω f is a non-connected set}, tat consists of all te elements K in T cut by te interface Σ wic are split elements, see Figure 9 (left). Tis means tat eac element K G is split into N K 2 fluid subparts, wic in general are polyedra. We denote by P K i, i = 1,...,N K, te polyedra of 13

15 T f T s Figure 8: Left: te structure foreground mes T s overlaps te background mes T. Rigt: representation of a background element split into tree disconnected polyedra (in blue/dark) by te solid foreground mes (in grey/ligt). a split element K. We define by G P te union of all suc polyedra PK i, for i = 1,...,N K and for eac K G, see Figure 9 (rigt) were N K = 2. More precisely P G P K G s.t. P K Ω f is a connected set G G P Figure 9: Left: representation of te mes G. Notice tat G contains also te portion of te elements overlapped by te structure. Rigt: representation of te non-connected mes G P. Te set G P in now partitioned into its N f = max K N K connected subsets Ω f,i. For example, by considering te same configuration in Figure 10 (left), we ave N f = 2 connected subregions (Ω f,1 and Ω f,2 ). Moving from tese definitions, we set Notice tat Ω f = i=0,...,n f Ω f,i Ω f,0 = Ω f \ K G K. f,i and tat Ω Ωf,j = /0, i j. We denote by T f,0 te, i.e. smallest mes composed of te elements K T tat covers te set Ω f,0 K T f,0 K Ω f,0 /0. 14

16 Ω f,0 T s T f T f, Ω f,1 Ω f,2 T f,1 T f, Figure 10: Left: sketc of te background (T ) and foreground (T s te sets Ω f,0 (blue), Ω f,1 (pink) and Ω f,2 represent te meses T f,0 (top) and T f,i ) meses (top) and (yellow) (bottom). Rigt: te saded regions (bottom). Finally, we denote by T f,i, for i = 1,...,N f, te smallest mes tat consists of all te elements of G tat covers te set Ω f,i, i.e. K T f,i K Ω f,i /0, i = 1,...,N f. Tus, eac element K G belongs to N K different meses T f,i, and tis will allow us to duplicate te dofs of K N K times. Te idea is to build te classical FEM approximation in T f,0, i.e. by using te classical dofs and sape functions, and to employ te XFEM strategy in T f,i, i = 1,...,N f, so tat te dofs associated wit te elements in T f,i, i = 1,...,N f, are duplicated: a set of dofs is used to compute te solution over eac mes T f,i. Te unfitted nature of te fluid and solid meses requires a specific treatment of te coupling conditions between te corresponding fluid and solid problems at te interface Σ. A possibility, considered ere, is to employ a Discontinuous Galerkin (DG) mortaring, see, e.g., [23, 25], in order to weakly impose te continuity of te fluid solution between te elements of te meses T f,i, i = 1,...,N f, see below. On te contrary, in T f,0 it is possible to use eiter a non-conforming or a conforming discretization. For te sake of simplicity, we consider a conforming discretization, tus we impose a strong continuity in T f,0. We also notice tat some operators of te discrete formulation will act on te domains Ω f,i Ω f, wile oter operators, suc as te stabilization terms, will act on te meses T f,i, as we explain later on. 15

17 We observe tat te set covered by T f,i is larger tan te one covered by te corresponding Ω f,i, see Figure 10 (rigt) for te case N f = 2. More complex configurations may appen for realistic tree-dimensional domains. Remark 4.1 We point out tat te elements of te background mes crossed by te interface Σ may be arbitrarily small due to te overlapping of te foreground domain. Tis may generate instabilities in te numerical solution and lead to an ill-conditioned matrix. For tese reasons, to prevent instabilities and to maintain te robustness of te metod, a possible strategy consists in te introduction of te gost-penalty stabilization, see below and [21]. We classify te faces involved in te discrete formulation as follows: te faces belonging to te fluid-structure interface Σ, were we impose weakly te continuity of te velocity and stresses by means of te DG formulation, see, e.g., [23, 25]; F f,i f,i,p, te faces in T, i = 1,...,N f, tat belong to te background process domain Ω f, were we impose weakly te continuity of te fluid velocity and stresses by means of te DG formulation, see below and, e.g., [6, 28]; F f,i f,i,σ, te faces of T, i = 1,...,N f, cut by te interface Σ, were te gost penalty stabilization term (10) is applied, see below and [21]. For a representation of tese faces, we refer to Figure 11. Σ F f,1,p F f,2,p F f,1,σ F f,2,σ Figure 11: Representation of te sets of faces involved in te discrete formulation (igligted in red): (left) faces of te interface Σ; (center) faces F f,1 f,2,p and F,p for te mortaring in te background process domain; (rigt) faces F f,1 f,2,σ and F,Σ for te gost penalty stabilization. After a suitable time discretization of te FSI problem (4)-(5)- (6)-(7), we denote by Ω f,n te approximation of Ω f at time t n. Te discrete spaces for te fluid velocity and pressure read as follows: V n = {v [X f,n ] d : v Γ f = 0}, Q n = {q X f,n }, W = {w [X s]d : w Γ s = 0}, 16

18 were and X f,n = {v L 2 (Ω f,n ) : v C 0 (Ω f,0,n ),v K P 1 (K), K T f,i,n for i = 0,...,N f }, X s = {v C 0 ( Ω s ) : v K P 1 (K), K T s }. To write te discrete formulation, we introduce te trace operators defined over an interface I tat separates a generic domain Ω 1,2 into Ω 1 and Ω 2. For a (scalar or vectorial) function q, we denote by I te jump and by {{ }} I,ε te ε-weigted mean across te interface I, defined as q I = q 1 q 2, {{q}} I,ε = εq 1 + (1 ε)q 2, (9) were q 1 and q 2 are te traces of q at te two sides of te interface and ε [0,1]. If te subscript ε is not indicated, we assume tat ε = 1 2. We consider a DG mortaring on Σ to impose te coupling conditions (6) and on te faces F f,i,p to impose te continuity of te background fluid solution, by mimicking te (symmetric) interior penalty metod, introduced for example in [6, 31] for te Poisson problem. Moreover, a gost penalty term, see [21], is applied on F f,i,σ to guarantee robustness of te metod wit respect to te elements crossed by te interface Σ, defined as N f g (u,v ) = γ g µ f F u F n v F n, (10) i=1 F F f,i,σ wit γ g > 0; We also introduce a stabilizing term s to andle spurious pressure and velocity instabilities due to equal order Finite Elements and to dominating convection regimes, respectively. In tis work we considered te interior penalty (IP) stabilization, see [24], as done in [82]. We now introduce te following forms. - Fluid form collecting te classical Navier-Stokes terms and te gost and IP stabilizations: F - Structure form: A f (z,u, p;v,q) r = ρ f t (u,v) Ω f,r + 2µ f (D(u),D(v)) Ω f,r (p, v) Ω f,r + (q, u) Ω f,r + ρ f (z u,v) Ω f,r + s (u, p;v,q) r + g (u,v) r ; ( d;ŵ ) A s = ρs + ( T s t ( d), ŵ) Ωs; 2( d,ŵ) Ωs 17

19 - Form related to te DG terms involving only te fluid unknowns and test functions: D f f (u, p;v,q) r = + N f i=1 N f i=1 N f i=1 F F f,i,r,p F F f,i,r,p F F f,i,r,p ({{ T f (u, p) }} F n, v F) ( u F, {{ T f (v, q) }} F n) F γ p µ f F ( u F, v F ) F, ( εt f (u, p)n,v ) Σ r ( u,εt f (v, q)n ) Σ r + γ Σµ f F (u,v) Σ r ; (11) - Form related to te DG terms involving only te structure unknowns and test functions: ) ss D ( d;ŵ ( = ( + γ Σµ f ) (1 ε) T s ( d)n, ŵ ) d t,(1 ε) T s (ŵ)n ( ) d t, ŵ ; Σ Σ Σ (12) - Form related to te DG terms involving mixed (fluid and structure) unknowns and test functions: D f s (u, p,d;v,q,w) r = ( εt f (u, p)n, w ) ((1 ε)t s (d)n,v) Σ r Σ r ( ) (u,(1 ε)t s (w)n) Σ r + γ Σµ f d t,εt f (v, q)n (u, w) Σ r + γ Σµ ( f d t ),v Σ r ; Σ r (13) - Rigt and side given by terms coming from time discretization and forcing terms: 18

20 F (u m, p m,d m ;v,q,w) r = ρ f t (um,v) Ω f,r + 2ρs t 2( d m ρs,ŵ) Ωs t 2( d m 1,ŵ) Ωs ( ) d m + t,εt f (v, q)n + (1 ε)t s (w)n Σ r γ ( ) Σµ f d m t,v w + (f f,m+1 s,m+1,v) Ω f,r + ( f,ŵ) Ωs. Σ r Notice tat te DG mortaring terms introduce two penalty parameters, γ p > 0 and γ Σ > 0. Te first parameter appears in te term D f f and it is related to te mortaring on te faces in F f,i,r,p, wile te latter appears in terms D f f,d f s,d ss and it is related to te mortaring on te fluid-structure interface Σ. Tus, te XFEM/DG approximation of te monolitic FSI problem (4)-(5)- (6)-(7) reads: For eac n, find (u n+1, p n+1, d n+1 ) V n+1 Q n+1 W suc tat A f ( u n+1,u n+1, p n+1 ) ) n+1 ;v,q + A s ( d n+1 ;ŵ +D f f ( u n+1, p n+1 ) ) n+1 ;v,q + D ss ( d n+1 ;ŵ + D f s ( u n+1, p n+1,d n+1 ) n+1 ;v,q,w = F ( u n, pn,dn ;v,q,ŵ ) n (14) for all (v,q,ŵ ) V n+1 Q n+1 W. In compact form we write ( ) n+1 H u n+1, p n+1, d n+1 ;v,q,ŵ = 0 for all (v,q,ŵ ) V n+1 Q n+1 W Remark 4.2 Notice tat in te previous formulation we could consider also a correction in te trilinear form to maintain te condition tat te latter vanises for z = u at te discrete level [28, 84] and a term to maintain te consistency of te formulation [95]. Tis is wat we did in te numerical experiments. However, to simplify te notation and focus on te XFEM/DG discretization, we omitted tese terms in (14). 5 An inexact-newton metod for te solution of te FSI problem For te solution of te FSI problem (14), we introduce in wat follows an inexact Newton-Krylov metod, used in combination wit a block Gauss-Seidel preconditioner. To tis aim, we indicate by δy (k) = y (k) y (k 1) te increment of a quantity y and we consider te following linearized forms: 19

21 - A ( ) s δ d (k) ;ŵ = ρs ( ) Ωs ( ) t 2 δ d (k),ŵ + D F T s ( d (k 1) ) : δ d (k), ŵ, Ω s were D F indicated te Gateaux derivative wit respect to F; - D ( ) ss δ d (k) ;ŵ ) = (((1 ε)d F T s ( d (k 1) ) : δ d (k) ( + γ Σµ f δ d (k), t ( ) n, ŵ ) ( ) (1 ε)d F T s ( d (k 1) ) : ŵ n δ d (k), ŵ t ) Σ ; Σ Σ - D f s ( u (k), p (k),δd (k) ;v,q,w ) r = ( εt f (u (k), p (k) )n, w ) Σ r (( (1 ε)d F T s ) (d (k 1) ) : δd (k) n,v )Σ ( ) ) r u (k), ((1 ε)d F T s (d m (k 1) ) : w n Σ ( ) r + γ Σµ f δd (k),εt f (v, q)n t Σ r ( u(k), w ) + γ ( Σµ f Σ r δd (k),v t ) Σ r. Moreover, we consider te approximated form D f s instead of D f s obtained by considering te following approximation T s F = F (J 1 T s F T ) J 1 T s F FT. Tis, togeter wit te fixed point iteration strategy (instead of te full Newton metod) used for te fluid problem (see (15)) leads to an inexact Newton metod. Finally, we point out tat at eac iteration of te inexact Newton metod, we ave to update te fluid mes obtained by te intersections generated by te moving structure mes onto te background fixed one (see Figure 4), and, accordingly, te velocity and pressure spaces. In particular, we introduce te compact notation updateddomainsandspaces() tat at time t n+1, iteration k, performs te following steps: 1. given te displacement at te previous iteration d n+1,(k 1), computation of te new position of te solid mes (T s)n+1 (k 1) ; 20

22 2. computation of te new fluid mes (T f )n+1 (k 1). Tis is done by intersecting te background mes T and te solid mes (T s)n+1 (k 1) ; 3. definition of te new discrete spaces V n+1,(k 1) and Qn+1,(k 1). Te FSI problem (14) is solved by means of te following algoritm: Algoritm 1 Inexact Newton metod for te FSI problem (14) At time t n+1, given an initial solution u n+1 for k = 1 : k max do 1. updatedomainsandspaces();,(0), pn+1,(0) :,(0), d n+1 2. Find (u n+1,(k), pn+1,(k) n+1,δ d,(k) ) Vn+1,(k 1) Qn+1,(k 1) W suc tat ( ) n+1 A f u n+1,(k 1),un+1,(k), pn+1,(k) ;v,q + ( A s δ d n+1 (k 1) ( ) n+1 +D f f u n+1,(k), pn+1,(k) ;v,q + D ( ) ss δ d n+1 (k 1),(k) ;ŵ + D f s ( ) n+1 u n+1,(k), pn+1,(k),δdn+1,(k) ;v,q,w (k 1) ( ) n+1 = H u n+1,(k 1), pn+1,(k 1),dn+1,(k 1) ;v,q,ŵ, (k 1),(k) ;ŵ ) (15) for all (v,q,ŵ ) V n+1,(k 1) Qn+1,(k 1) W ; 3. d n+1,(k) = d n+1 end for,(k 1) + δ d n+1,(k). Remark 5.1 Notice tat in tis case, due to te fixed nature of te background mes, we do not ave any geometric problem, tus no sape derivatives appear in te exact Jacobian. 6 Numerical results In tis section, we present some numerical results for te FSI problem given by equations (4)-(7). We consider Algoritm 1 for its numerical solution. We present te following test cases: - Blocked cannel: A time-dependent FSI problem wit an immersed non-linear elastic structure tat completely blocks a cannel; - Non-linear elastic slab: A time-dependent FSI problem wit an immersed non-linear elastic slab wit a ig Reynolds number; - Ideal aortic valve: A time-dependent FSI problem in te case of tree immersed linear elastic leaflets. 21

23 Te proposed examples are simulated in a tree-dimensional (3D) framework, and for te structure we use te non-linear strain energy function (8) for te blocked cannel and non-linear elastic slab tests, wereas te Hooke law for te ideal aortic valve. Te mortaring parameter ε for te fluid-solid coupling in te forms D f f, D f s, D ss and in functional F is set equal to 1, see [23]. Moreover, we point out tat at iteration k of Algoritm 1, te fluid velocity u n at te previous time step appearing in te term coming from time discretization (wic is defined in Ω f,n ) and te fluid velocity u n+1,(k 1) at te previous iteration used in te convective term (wic is defined in Ω f,n+1 (k 2)) are not defined in te current domain Ω f,n+1 (k 1) (remember tat at iteration k te fluid problem is solved in Ω f,n+1 (k 1)). Tus, tese terms sould be properly defined in te new computational domain Ω f,n+1 (k 1) in order to be used in te discrete formulation. In particular, issues may occur wen te uncovered portion of a fluid element cange between time n and n + 1 and/or between iteration k 1 and k. For te numerical treatment of tese cases, we employ te procedure proposed in [95]. Te simulations ave been performed wit te Finite Element library LifeV [1]. 6.1 Blocked cannel In tis experiment we consider a tick membrane placed in te middle of a cannel so tat te structure completely blocks te flow in te cannel, see Figure 12. Te aim of tis example is to assess te validity of te proposed metod. We consider a background domain Ω = 0.4cm 0.2cm cm and a structure domain Ω s = 0.01cm 0.2cm cm. Te resulting fluid domain is Ω f = Ω\Ω s. Notice tat, to ease te computational cost, we reduce te size of te domain along te z-axis. We impose T f n = ( 1000,0,0)dyne/cm 2 at te inlet Γ in, T f n = 0 at te outlet Γ out, u = 0 on Γ f wall, and u k = 0, (T f n) l = 0, l = {i,j}, on te remaining portions of te fluid boundary, i.e. for z = {0, }. Notice tat te latter coice allows te fluid to move in te xy-plane at te extreme surfaces z = {0, }. Te solid is fixed at Γ s wall, i.e. d = 0, and, like te fluid, is allowed to move in te xy-plane on te remaining portions of te boundary, i.e. d k = 0, (T s n) l = 0, l = {i,j}, for z = {0, }. As initial conditions, we set u(x, 0) = d(x, 0) = ḋ(x, 0) = 0. We also use te following values for te parameters: ρ f = 1g/cm 3, ρ s = 1.2g/cm 3, µ f = poise, α = dyne/cm 2, κ = dyne/cm 2, γ = 1, and T = 0.02s. We employ a background mes T composed of tetraedra ( ave = cm) and a solid mes T s composed of tetraedra ( ave = cm). Notice tat, te tickness of te structure domain (0.01cm) is iger tan te average size of te fluid elements, so tat we are in te tick case. Te time step t is s. We coose γ Σ = 10 4 (see equations (11)-(13)), γ p = 10 3 (see equation (11)) and γ g = 1 (see equation (10)). In Figure 13 and 14, we sow te numerical solution at different time steps. In particular, we plot te velocity and te pressure fields in te fluid domain and te structure displacement in te solid domain. A quantitative plot of te displacement of te center of mass of te structure is reported in Figure

24 Γ s wall Γ f wall Γ f wall Ω s Γ in Ω f Σ Σ Ω f Γ out z y x Γ f wall Γ s wall Γ f wall Figure 12: Top view of te fluid Ω f and structure Ω s domains. Blocking cannel test. Finally, in Figure 16 (left), we plot te beaviour in time of te total amount of fluid (in cm 3 ) tat goes troug Γ in (indicated by V in ) and Γ out (indicated by V out ), te variation of te structure volume (in cm 3 ) wit respect to te initial time (indicated by V s ), and te sum of tese tree quantities (indicated by V balance ) wic represents te error wit respect to te balance of volume. In Figure 16 (rigt), we plot relative error of te balance of volume, i.e. te ratio r = V balance /V f e f f, were V f e f f is te effective volume available for te fluid. We see tat te error committed by te metod is very small compared to te total amount of volume. 6.2 Non-linear elastic slab We consider a background domain Ω = (0,0.5) 3 cm and a structure domain Ω s = (0.025,0.425)cm (0.15,0.35)cm (0.10,0.13)cm, so tat te fluid domain is Ω f = Ω \ Ω s, see Figure 17. We impose u = (0,0,100)cm/s at te inlet Γ in, T f n = 0 at te outlet Γ out, u î = 0, (T f n) l = 0, l = {j,k}, at Γ symm, and u = 0 on te remaining portions of te fluid boundary. Te structure is fixed at x = 0.025cm, i.e. d = 0 at Γ s wall. Te fluid-structure interface is given by Σ = Ω s \ Γ s wall. As initial conditions, we set u(x,0) = d(x,0) = ḋ(x,0) = 0. We also use te following values for te parameters: ρ f = 1g/cm 3, ρ s = 1.2g/cm 3, µ f = poise, α = dyne/cm 2, κ = dyne/cm 2, γ = 1, and T = s. Te Reynolds number is equal to Re = We employ a background mes T composed of tetraedra ( ave = 0.025cm) and a solid mes T s composed of tetraedra ( ave = cm). Notice tat, te tickness of te structure domain (0.03cm) is comparable to te average size of te 23

25 Figure 13: Plot of te fluid velocity magnitude (in cm/s) and structure displacement magnitude (in cm) at different time steps. Top: t = s. Center: t = s. Bottom: t = s. Blocking cannel test. 24

26 Figure 14: Plot of te fluid pressure filed (in dyne/cm2 ) and structure displacement magnitude (in cm) at different time steps. Top: t = s. Center: t = s. Bottom: t = s. Blocking cannel test. 25

27 0.02 x-displacement [cm] t [s] Figure 15: Plot of te x-displacement (in cm) at te center of mass of te structure. Blocking cannel test. V [cm 3 ] V s V in V out V balance r t [s] t [s] Figure 16: Left: Plot of volumes (in cm 3 ) V s, V in, V out and V balance in time. Rigt: Plot of te relative error of te volume in time. Blocking cannel test. 26

28 Ω f Γ out Ω s Σ Γ symm Γ s wall y z x Γ in Figure 17: Sketc of te fluid Ω f and structure Ω s domains. Non-linear elastic slab test. fluid elements. Te time step t is 10 3 s. We coose γ Σ = 10 2 (see equations (11)- (13)), γ p = 10 3 (see equation (11)) and γ g = 1 (see equation (10)). In Figure 18, we sow te fluid velocity (in cm/s) and te structure displacement (in cm) at four different time steps. Te maximum velocity is about 160cm/s and te maximum displacement reaced by te structure is 0.35cm. We see tat te metod is able to deal wit ig Reynolds number and large displacement. In Figure 19, we sow te z-displacement (in cm) of te tip of te structure, i.e. at x tip = (0.425,0.25,0.115)cm, in time. In Figure 20, we plot te velocity field in te fluid domain and we represent te moving structure accordingly to te computed displacement at different time-steps. We see tat te fluid elements crossed by te structure may cange in time. We point out tat te refinement appearing near te structure is made only for a visualization purpose, in fact te background fluid mes never canges. 6.3 Ideal aortic valve In tis example, we consider te domain Ω defined by a cylinder of radius 0.5cm and eigt 1 cm and tree linear immersed structures tat are an ideal representation of te leaflets of an aortic valve, see Figure 21. Te tickness of te leaflets is 0.02cm. We impose te velocity profile u = (0,0,50sin( π 8 t))cm/s at te inlet Γ in, we set T f n = 0 at te outlet Γ out, and u = 0 on te remaining portions of te fluid boundary. At te fluid-structure interface, we impose te kinematic and dynamic coupling conditions, except on Γ wall were te leaflets are fixed, i.e. d = 0. As initial conditions, we set u(x,0) = d(x,0) = ḋ(x,0) = 0. We use te following values for te pysical parameters: 27

29 y z x Figure 18: Solution at two different time steps. We plot te fluid velocity (in cm/s) and te solid displacement (in cm). Top, left: t = Top, rigt: t = Bottom, left: t = Bottom, rigt: t = Non-linear elastic slab test z-displacement [cm] t [s] Figure 19: Plot of te z-displacement (in cm) at te tip of te structure in time. Nonlinear elastic slab test. 28

30 Figure 20: Velocity magnitude on te slice y = 0.25cm at time t = s (top) and at time t = s (bottom). Te element igligted in red at time t = s is partially overlapped by te interface, wile at time t = s is not crossed by te structure. Non-linear elastic slab test. 29

31 ρ f = 1g/cm 3, ρ s = 1.2g/cm 3, µ f = poise, α = dyne/cm 2, κ = dyne/cm 2 and γ = 1. We simulate only te initial pase of te movement of te leaflets, i.e. T = 0.45s. Te Reynolds number is equal to Re = Γ out Γ wall Ω f l z x y Ω s Ω s Γ in Ω s Γ wall A Γ wall Figure 21: Sketc of te fluid background domain Ω f and te tree foreground domains Ω s. To te rigt, we report te top view of te domains. Notice in black te region Γ wall were te leaflets are clamped. Ideal aortic valve test. For te numerical simulation, we employ a background mes T of elements wit te average size of te mes elements ave = 0.035cm, wile eac structure mes is composed of elements wit ave = 0.011cm. We set t = 0.05s. We coose γ Σ = 10 2 (see equations (11)-(13)), γ p = 10 3 (see equation (11)) and γ g = 1 (see equation (10)). In wat follows we report preliminary results for tis test. At te instant were te fluid flow reverses, numerical instabilities occurs. For tis reason, we ave reported te numerical results until te solution features a stable beaviour. Te study of suc oscillations is under investigation. A qualitative representation of te solution at time t = 0.4s is sown in Figure 22. More specifically, in Figure 23 (left), we plot te z- displacement at te tip (point A in Figure 21) of te tree leaflets in time. We observe tat te tree leaflets beave very similarly during time. In Figure 23 (rigt), we plot te fluid pressure along line l : x = 0cm, y = 0.25cm, 0 z 1cm at two different time steps, namely, t = 0.20s and t = 0.45s. From tis result, we see te different value of te fluid pressure upstream and downstream te leaflet. Notice tat, te position of te leaflet (dased lines) as canged in time. Te maximum z-displacement reaced by te leaflets is 0.24 cm and te maximum value of fluid velocity is 17.5cm/s. In Figure 24, we sow te pressure field (in dyne/cm 2 ) and te structure displacement (in cm) on te slice y = 0.5cm at time t = 0.20s and t = 0.45s. In particular, we observe te different position of te leaflets wit respect to teir initial position outlined 30

32 Figure 22: Solution at time t = 0.4s. We plot te fluid velocity (in cm/s) and te structure displacement (in cm). Ideal aortic valve test. 350 Leaflet 1 Leaflet 2 Leaflet 3 ] t = 0.20 s t = 0.45 s 300 Pressure [dyne/cm z-displacement [cm] t [s] z [cm] Figure 23: Left: evolution of te displacement (in cm) at te tip of te tree leaflets. Rigt: fluid pressure (in dyne/cm2 ) along te line l at two different time steps. Te position of te leaflet is denoted by te dased lines. Ideal aortic valve test. 31

33 in black. Again, it is possible to see te different values of te fluid pressure upstream and downstream te leaflets. Figure 24: Fluid pressure (in dyne/cm 2 ) and structure displacement (in cm) at slice y = 0.5cm. Te initial position of te leaflets is denoted by te black lines. Left: time t = 0.2s. Rigt: time t = 0.45s. Ideal aortic valve test. Acknowledgement Te autors gratefully acknowledge te financial support of te Italian MIUR by te grant PRIN12, number A4LX, Matematical and numerical models of te cardiovascular system, and teir clinical applications. C. Vergara as been partially supported by te H2020-MSCA-ITN-2017, EU project ROMSOC - Reduced Order Modelling, Simulation and Optimization of Coupled systems. References [1] LifeV. ttp:// Te parallel finite element library for te solution of PDEs. [2] F. Alauzet, B. Fabrèges, M. A. Fernández, and M. Landajuela. Nitsce-XFEM for te coupling of an incompressible fluid wit immersed tin-walled structures. Computer Metods in Applied Mecanics and Engineering, 301: , [3] M. Aletti, J.-F. Gerbeau, and D. Lombardi. Modeling autoregulation in treedimensional simulations of retinal emodynamics. Journal for Modeling in Optalmology, 1:88 115, [4] C. Annavarapu, M. Hautefeuille, and J. E. Dolbow. A robust Nitsce s formulation for interface problems. Computer Metods in Applied Mecanics and Engineering, :44 54,