Extended ALE Method for fluid-structure interaction problems with large structural displacements

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1 Extended ALE Metod for fluid-structure interaction problems wit large structural displacements Steffen Basting a,, Annalisa Quaini b, Sunčica Čanićb, Roland Glowinski b a Institute of Applied Matematics (LS III), TU Dortmund, Vogelpotsweg 8, Dortmund, Germany b Department of Matematics, University of Houston, 4800 Caloun Rd, Houston TX 77204, USA Abstract Standard Arbitrary Lagrangian-Eulerian (ALE) metods for te simulation of fluid-structure interaction (FSI) problems fail due to excessive mes deformations wen te structural displacement is large. We propose a metod tat successfully deals wit tis problem, keeping te same mes connectivity wile enforcing mes alignment wit te structure. Te proposed Extended ALE Metod relies on a variational mes optimization tecnique, were mes alignment wit te structure is acieved via a constraint. Tis gives rise to a constrained optimization problem for mes optimization, wic is solved wenever te mes quality deteriorates. Te performance of te proposed Extended ALE Metod is demonstrated on a series of numerical examples involving 2D FSI problems wit large displacements. Two way coupling between te fluid and structure is considered in all te examples. Te FSI problems are solved using eiter a Diriclet- Neumann algoritm, or a Robin-Neumann algoritm. Te Diriclet-Neumann algoritm is enanced by an adaptive relaxation procedure based on Aitken s acceleration. We sow tat te proposed metod as excellent performance in problems wit large displacements, and tat it agrees well wit a standard ALE metod in problems wit mild displacement. Keywords: Mes optimization, Arbitrary Lagrangian-Eulerian formulation, Fluid-structure interaction, Domain decomposition metods. 1. Introduction Te focus of tis work is on te numerical simulation of te motion of an elastic body immersed in an incompressible, viscous fluid, wit te structure undergoing large displacements. A motivating example is te interaction between eart valves and blood flow. Te difficulties associated wit accurate numerical simulation of tis class of fluid-structure interaction problems are: (1) large canges in te fluid domain tat occur due to te large structural displacements; (2) accurate approximation of te ydrodynamic force at te fluid-structure interface; and (3) added mass effect, wic is known to cause various numerical difficulties wen te fluid and structure ave comparable densities [76, 22]. Several numerical approaces ave been proposed in te literature for tis class of problems, eac dealing wit te above-mentioned difficulties in a different way. To deal wit te fluid domain motion associated wit structural displacements, numerical metods can be classified into te metods wit fixed meses and te metods wit moving meses. Te fixed mes metods include te Immersed Boundary Metod (IBM) [64, 66, 65], te Fictitious Domain Metod [45, 44, 1], te level set metod [27, 26, 42], te so called Eulerian FSI metods [31, 71, 81], and te deforming composite grids [6, 7, 57]. Tese metods rely on a fixed fluid mes used in a fluid solver, wile te presence of te structure is implemented in different ways. See also [72]. For example, in te Immersed Boundary Metod Corresponding autor addresses: steffen.basting@mat.tu-dortmund.de (Steffen Basting), quaini@mat.u.edu (Annalisa Quaini), canic@mat.u.edu (Sunčica Čanić), roland@mat.u.edu (Roland Glowinski) Preprint submitted to Elsevier November 4, 2016

2 te fluid feels te structure troug external forces acting on te fluid, were te coupling between te (fixed) fluid mes and a (Lagrangian) structure mes is performed via Dirac Delta functions. To get around te difficulties associated wit te discretization of te Dirac Delta, and te low accuracy it causes in te calculation of te ydrodynamic force, modifications of te IBM were introduced. Tey include te extended Immersed Boundary Metod [79], and te Immersed Finite Element Metod [82]. On te oter and, in te Fictitious Domain Metod, te coupling between te fluid and structure is enforced via Lagrange multipliers (imposing continuity of velocity, or te no-slip condition). Tis approac was applied first to problems wit rigid particles and later to problems wit flexible structures, were Lagrange multipliers were located along te structure surface [3, 50, 77, 78]. In all te cases discussed above, adaptive mes refinement typically needs to be used to obtain reasonable accuracy in te calculation of te ydrodynamic force acting on te structure. In te work presented in [57] te metod of deforming composite grids was used to capture te motion of a beam modeled using a generalized Euler-Bernoulli beam model. An underlying fixed Cartesian mes was used for te bulk of te fluid domain, wit an overlapping grid consisting of a set of structured component grids to resolve te structure and boundary-fitted curvilinear grids at te structure boundary. Te moving mes metods are typically based on Arbitrary Lagrangian-Eulerian (ALE) approaces, introduced in [53, 28] for FSI problems discretized wit te Finite Element metod. Earlier works [63, 51] introduced ALE metods for te Navier-Stokes equations in moving domains discretized wit te Finite Difference metod. Instead of being fixed, te fluid mes follows te motion of te elastic body via a mapping, called te ALE mapping, wic is calculated based on te current location of te structure (e.g., as a armonic extension of te current interface position onto te fluid domain). ALE metods were proved to be accurate and robust for emodynamics applications involving small mes displacements (see, e.g., [35]). Altoug tese metods offer many advantages provided by te explicit representation of te fluidstructure interface [52, 76, 9], problems arise wenever strong deformations or even topological canges of te interface lead to a degeneration of te computational mes. To deal wit large structural displacements, remesing was introduced in [30, 59, 58]. By remesing it is normally meant tat a new mes wit different connectivity is generated from scratc wen te quality of te given mes is poor. Mixed ALE and fictitious domain formulations ave also been proposed [49, 29]. Tese approaces also require adaptive mes refinement for an accurate calculation of te viscous sear stresses on te solid boundary. For completeness, we also mention a mes-free Lattice-Boltzmann metod [54, 34, 32, 21], wic as also been used for te simulation of FSI problems wit large structural displacements. In te present work we propose a metod wic is a variant of an ALE approac, but it elegantly captures large structural displacements witout canging mes connectivity. Te metod retains te positive features of te ALE approaces, suc as te accurate approximation of te fluid-structure interface and accurate representation of te ydrodynamic forces, witout te need for adaptive mes refinement. Te metod is based on a fixed base mes tat is adapted to approximate te interface via an ALE-type mapping, wile maintaining mes connectivity (nodes or elements are neiter inserted nor removed). Te fundamental building block is a variational mes optimization approac tat does not rely on any combinatorial considerations. Alignment of te optimized mes wit te structure interface is stated as a constraint of a mes optimization problem tanks to a level set description of te geometry. Our metod can be regarded as an extension of te tecniques introduced in [80, 12, 13] for two-pase flows and one-way coupled FSI problems (i.e., te structure moves wit a prescribed law). In contrast wit te strategy proposed in [80, 12, 13], in our approac te variational alignment procedure is only performed wen te mes becomes too distorted, making our metodology closer to a standard ALE approac, wic is wy we call it Extended ALE Metod. Te main features of te proposed Extended ALE approac are: - Non-degenerate meses of provably optimal quality; - Te alignment of te mes wit te interface, wic allows for a simple definition and efficient implementation of problem-specific finite element spaces, suc as te elements allowing discontinuities across te interface; - Fixed mes connectivity, wic makes te metod easy to implement in an existing standard ALE 2

3 code. Our approac is similar in spirit to te metods proposed in [15, 25, 5] (te fixed-mes ALE approac), and also to te metodology based on universal meses (see, e.g. [70]) recently proposed in [41, 40], wic all aim at providing discretization scemes defined on aligned meses wit fixed connectivity. In contrast to our optimization strategy, te mes adaptation presented in [15] relies on explicit combinatorial considerations to create interface aligned meses. Similarly, te metodology based on universal meses presented in [41, 40] makes use of a closest point projection to create a discrete interface consisting of certain edges tat ave to be cosen based on combinatorial considerations. In [70] it is discussed under wic conditions tis leads to suitable meses. On te oter and, te fixed mes ALE approac virtually evolves te fluid mes according to a standard ALE procedure, and ten projects back te results on a fixed background mes. However, cut elements ave to be considered in order to enable an accurate imposition of boundary conditions. Te main benefits of our variational approac are: no explicit combinatorial testing is needed and te resulting meses are guaranteed to be non-degenerate. To sow tat our approac as te desired features, we consider te interaction of an elastic beam wit a 2D incompressible fluid. Altoug dealing wit a simplified model, te problem under consideration retains important pysical features common to more complex models: large displacement and added mass effect. We sow tat te Extended ALE Metod allows to easily capture te pressure discontinuity across te interface, wic coincides wit te 1D elastic structure. Metods based on non-aligned fixed meses cannot capture suc a discontinuity, unless furter tecniques are used, suc as, e.g., te enricment of te function spaces as in X-FEM, see [39]. Moreover, tanks to te mes alignment wit te interface, te kinematic coupling condition is easily enforced. Once a mes as been obtained from te above mentioned constrained optimization problem, te FSI problem is solved wit classical Domain Decomposition algoritms (see, e.g., [69]): eiter te Diriclet- Neumann metod, wic is combined wit an Aitken s acceleration tecnique [55], or te Robin-Neumann metod. It was sown in [4] tat wen te structure lies on part of te fluid domain boundary, te Robin- Neumann metod features excellent convergence properties: it always converges witout any relaxation and its convergence is almost insensitive to te added-mass effect. Oter scemes for FSI problems tat use Robin interface conditions are presented in [62, 48, 20, 19, 8, 57]. In particular, we mention [57] were a second-order accurate loosely-coupled sceme incorporating a Robin-type interface condition was used to study fluid-structure interaction between an incompressible, viscous fluid and a generalized Euler-Bernoulli beam. To te best of our knowledge, te Robin-Neumann metod as never been applied to FSI problems involving an immersed structure. Te outline of te paper is as follows. In Section 2 we state te problem. Te constrained optimization approac, wic is at te core of our Extended ALE Metod, is explained in Section 3. We describe te Domain Decomposition algoritms in Section 4, and summarize te numerical metods tat we use for te time and space discretization of te fluid and structure problems in Section 5. In Section 6, we present numerical results obtained on a series of numerical tests carefully cosen to igligt te main features of te metod. Conclusions are drawn in Section Problem definition Consider a domain Ω R 2 containing an elastic beam forming a 1D manifold Γ(t) Ω wose location depends on time. Te beam is surrounded by an incompressible, viscous fluid occupying domain Ω, defining te time dependent fluid domain Ω f (t) := Ω \ Γ(t). Te beam can be periodic (closed curve, see Fig. 1(a)) or non-periodic (open curve, see Fig. 1(b)) Te fluid problem Te fluid flow is governed by te Navier-Stokes equations for an incompressible, viscous fluid: ( ) u ρ f + (u )u σ = 0 in Ω f (t), (1) t u = 0 in Ω f (t), (2) 3

4 (a) Periodic beam (b) Immersed beam Figure 1: Computational domain for (a) te periodic beam case and (b) te immersed beam case. for t (0, T ], were ρ f is te fluid density, u is te fluid velocity, and σ te Caucy stress tensor. For Newtonian fluids σ as te following expression σ(u, p) = pi + 2µɛ(u), were p is te pressure, µ is te fluid dynamic viscosity, and ɛ(u) = ( u + ( u) T )/2 is te strain rate tensor. Equations (1)-(2) need to be supplemented wit initial and boundary conditions. In order to describe te evolution of te fluid domain, we begin by adopting an Arbitrary Lagrangian- Eulerian (ALE) approac [53]. More precisely, let ˆΩ f R 2 be a fixed reference domain. We consider a smoot ALE mapping A : [0, T ] ˆΩ f R 2, A(t, ˆΩ f ) = Ω f (t), t [0, T ]. For eac time instant t [0, T ], A is assumed to be a omeomorpism. Te domain velocity w is defined as w(t, ) = t A(t, A(t, ) 1 ). For any sufficiently smoot function F : [0, T ] R 2 R, we may define te ALE time derivative of F as F = F F (t, A(t, ˆx)) = (t, x) + w(t, x) F (t, x), t ˆx t t were x = A(t, ˆx), ˆx ˆΩ. Wit tese definitions, we can write te incompressible Navier-Stokes equations in ALE formulation as follows: u ρ f + ρ f (u w) u σ = 0 t ˆx in Ω f (t), (3) u = 0 in Ω f (t), (4) for t (0, T ] Te structure problem For te structure problem, we consider two structural models, bot based on a linearly elastic beam equation: - a periodic beam (PB) described by a linear beam equation, wic results in balloon-type FSI problems; - an inextensible beam (IB) giving rise to a non-linear problem, were te nonlinearity comes from te inextensibility condition. 4

5 In bot cases, we assume negligible torsional effects for te beam. Let us denote by ρ s te linear density (i.e. mass per unit lengt), by L te lengt, and by EI te flexural stiffness of te beam. We will use te following notation, wit s denoting arc lengt and t time: y = y s, ẏ = y t, y = 2 y s 2, ÿ = 2 y t A periodic beam (PB) model Consider a periodic beam (closed curve). equations: Its dynamic beavior is governed by te Euler-Lagrange ρ s ẍ + EIx = f in (0, L), t (0, T ], (5) were x = x(t, s) is te parametric curve defining te beam position and f denotes te force acting on te beam. In our case, f is te ydrodynamic force, wic will be specified in Sec Problems similar to (5) were considered, for instance, in [23, 24] and te references terein. Equation (5) as to be supplemented wit initial and boundary conditions. We enforce periodic boundary conditions: x(0) = x(l), x (0) = x (L). (6) An inextensible beam (IB) model In te second case, we consider a non-periodic (open curve), inextensible beam [46, 43, 29]. Te structural model is based on equation (5) wit an additional constraint of inextensibility, i.e. te beam cannot srink or stretc during its interaction wit te fluid. Te resulting structure problem is non-linear, and te numerical treatment of tis problem is muc more callenging tan te problem described in Sec Using te virtual work principle, te beam motion for t (0, T ] is modeled by te following. Find x(t) K suc tat: wit L ρ s ẍ yds + L EI x y ds = L f yds, y dk(x), (7) and boundary conditions K = { y (H 2 (0, L)) 2, y = 1, y(0) = a, y (0) = b }, (8) dk(x) = { y (H 2 (0, L)) 2, x y = 0, y(0) = 0, y (0) = 0 }, (9) x(0) = a, x (0) = b, x (L) = x (L) = 0. (10) Te conditions at s = 0 are called te essential boundary conditions, describing a clamped beam, wile te conditions at s = L are called te natural boundary conditions. Te non-linear inextensibility condition for te beam, x = 1, is embedded into te set K Te coupled fluid-structure interaction problem We consider two-way coupling between te fluid and structure: te motion of te beam is driven by te contact force exerted by te fluid, wile at te same time te motion of te beam influences te fluid motion. Te coupling conditions are described by te following. Let us denote te interface by Γ(t) = {x(t, s), s [0, L]}. Let n 1 be te unit normal vector pointing to te left (left wit respect to te parameterization of x) and n 2 = n 1 is te unit normal pointing to te rigt, see Figure 1. Notice tat te fluid-structure interface 5

6 coincides wit te structure domain. Te ydrodynamic force acting on te structure (beam) is given by te jump in te normal fluid stress across te interface Γ(t): f Γ = σ 1 n 1 σ 2 n 2, (11) were σ i (x) = lim ɛ 0 σ(x + ɛn i ), x Γ, i = 1, 2. Using tis notation, we can now state te coupling conditions. For t (0, T ], te fluid problem (3),(4) and te structure problems (PB) or (IB) are coupled by te following two conditions: 1. kinematic coupling condition (continuity of velocity, i.e., te no-slip condition) 2. dynamic coupling condition (balance of contact forces) u = ẋ on Γ(t); (12) f Γ = f on Γ(t), (13) were f is given by Eq. (5) for problem (PB), or by Eq. (7) for problem (IB). Here, notation u = ẋ in (12) is used to express te relation u(t, x(t, s)) = ẋ(t, s), s [0, L] (analogously for f Γ and f in (13)). Since x denotes te location of structure points and not te structure displacement, bot te structure and fluid are given in Eulerian coordinates. For te purposes tat will be clear below wen we introduce te Robin-Neumann sceme, we remark ere tat te coupling conditions (12)-(13) can be written in an equivalent form by introducing te constants α f > 0 and α s > 0 (α f α s ), and writing: α f u f Γ = α f ẋ f on Γ(t), (14) α s u + f = α s ẋ + f Γ on Γ(t). 3. Numerical Representation of te Geometry Te main feature of our Extended ALE metod is a variational mes optimization tecnique combined wit an additional constraint to enforce te alignment of te structure interface wit te edges of te resulting triangulation. Te mes optimization, explained in Sec. 3.1 togeter wit te alignment constraint described in Sec. 3.2 corresponds to a reparametrization of te ALE mapping Optimal triangulations Let T be an initial triangulation of te domain Ω (not necessarily approximating te structure interface at tis stage). Following a variational mes optimization tecnique introduced by M. Rumpf in [73], we aim at finding an optimal triangulation T opt resulting from an optimal mes deformation χ opt of T, i.e. T opt = χ opt (T ). Deformation χ opt belongs to te set D of piecewise affine, orientation preserving, and globally continuous deformations: D = { χ ( C 0 (Ω) ) 2 : χ T GL(2), det( χ T ) > 0, T T }, (15) wit GL(2) = {A R 2 2 : det(a) 0}. Deformation χ opt D is optimal in te sense tat it is te argument for wic a certain functional F attains its minimum value: F(χ opt ) = min F(χ). (16) χ D We assume tat te functional in (16) can be represented by a sum of weigted, element-wise contributions F T : F(χ) = T T µ T F T (χ), 6

7 were µ T > 0 denotes a positive weigt wit T µ T = 1. For eac element T T, let R T denote te linear reference mapping from a prescribed reference element T opt (an equilateral simplex wit customizable edge lengt ) to T. We also define R T (χ) := χ R T for a given deformation χ. Te idea is to find mappings χ tat would map te element T onto te reference simplex T opt, i.e. suc tat te optimal deformation would fulfill χ opt = R 1 T. Under te assumptions of translational invariance, isotropy and frame indifference of te functionals, it can be sown (see [73]) tat in two dimensions F T may be expressed as a function of te invariants R T (χ) 2 and det( R T (χ)). For example, given a function F T : R R R we can write: F T (χ) = F T (a, d) := F T ( R T (χ) 2, det( R T (χ))). Here, denotes te Frobenius norm. Note tat te quantity R T (χ) 2 measures te cange of edge lengts wit respect to te reference element, and det( R T (χ)) measures te cange in area. In order to rule out deformations wit vanising determinant, we need lim F T (χ) =. det( R T (χ)) 0 Wit te additional assumption tat te local functional F T (χ) = F T (a, d) is polyconvex (i.e. FT (a, d) is convex wit respect to eac argument), it can be proven tat an optimal deformation exists and is globally injective [73]. A classical example of suc a function F T is given by F T (χ) = ( R T (χ) 2 2) 2 + det( R T (χ)) + Notice tat te optimally deformed simplex is obtained if χ opt T = R 1, i.e. if 1 det( R T (χ)). (17) T F T (χ opt ) = F T ( I 2, det(i)) = (2 2) = 2. We would also like to point out tat te mes optimization procedures proposed by Freitag and Knupp in e.g. [37, 38] wic are based on minimizing te condition number of te mapping R T (χ) are in te same spirit as te approac pursued ere. Te variational mes smooting approac described above as several advantages: - Minimization problem (16) yields triangulations wic are provably optimal in te sense of te local measure (17). - Tese triangulations can be sown to be non-degenerate, i.e. no self-intersection of elements occurs. Tis is te main property needed by our optimization approac. - Te element-wise representation of F provides built-in, local mes quality control. Te price to pay for tose advantages is tat functional F in (16) is igly non-linear, non-convex, and global minimizers may not be unique Interface aligned mes We are now interested in aving a triangulation tat is non-degenerate, optimal (as explained in te previous subsection) and aligned wit te beam position Γ(t), i.e. we want te optimal triangulation edges to approximate Γ(t). For tis purpose, we introduce te following auxiliary tools: - a tubular box around te structure of widt δ, denoted by Ω δ f (t) Ω f (t), witin wic mes optimization wit alignment will be performed, see Fig. 2(a); 7

8 φ(x e,2 ) > 0 Ω δ f (t) Γ δ (t) n 2 e n 1 φ(x e,1 ) < 0 Γ(t) Ω δ,1 f (t) Ωδ,2 f (t) (a) Tubular box Ω δ f (t) (b) Fluid mes intersected by Γ(t) Figure 2: (a) Tubular box Ω δ f (t) around te structure position x, zoomed in view of Fig. 1(b), and (b) Γ(t) intersecting elements of te fluid mes. - a continuous level set function φ : [0, T ] Ω δ f (t) R wose zero level set includes te structure position x: { } Ω δ,1 f (t) = y Ω δ f (t) : φ(t, y) > 0, { } Ω δ,2 f (t) = y Ω δ f (t) : φ(t, y) < 0, (18) } Γ δ (t) = {y Ω δ f (t) : φ(t, y) = 0, were Γ δ (t) denotes a natural extension of Γ(t) to te boundary of Ω δ f (t), and Ωδ,1 f (t) and Ωδ,2 f (t) denote te fluid sub-domains located on te left and rigt side of Γ δ (t), respectively. See Fig. 2(a). Notice tat n 1 (resp., n 2 ) is te outward unit normal on Γ δ (t) of Ω δ,1 (t) (resp., Ωδ,2 (t)). f f Witin Ω δ f (t) we perform te following procedure. Let e be an arbitrary edge of te triangulation T intersected by Γ δ (t) as sown in Fig. 2(b), and let x e,1 and x e,2 be its endpoints. Due to continuity of φ and assumption (18), we observe tat φ(x e,1 )φ(x e,2 ) < 0 if and only if e is intersected by Γ δ (t), provided tat te mes size is sufficiently small to resolve te sape of Γ δ (t). We terefore define te triangulation to be linearly aligned wit Γ(t) if φ(x e,1 )φ(x e,2 ) 0 for all e T. We introduce a scalar constraint c : D R + 0, defined on D given by (15), suc tat: c(χ) = e χ(t ) H(φ(x e,1 )φ(x e,2 )) were H(z) = { > 0 if z < 0, = 0 oterwise. Only deformations χ for wic c(χ) = 0 will give aligned triangulations. Tus, a linearly aligned triangulation of optimal quality is obtained from te following constrained minimization problem: min F(χ) suc tat c(χ) = 0. (19) χ D 8

9 Given an aligned triangulation T, we may define a linear approximation of te interface as Γ = {edges e T : φ(x e,i ) = 0 and x e,i Γ for i = 1, 2}. In order to obtain a more accurate representation of te structure, we also consider piecewise quadratic approximations of Γ(t). We make use of quadratic isoparametric finite elements equipped wit additional degrees of freedom located at te edges Details on te implementation In tis subsection we provide additional details on our numerical realization of te minimization problem (19). One of te difficulties is ow to andle te condition det( χ T ) > 0 for admissible deformations χ D. In practice, it is convenient to work wit a standard finite element space suc as X := {χ ( C 0 (Ω) ) 2 : χ T ( P 1 (T ) ) } 2 T T, wic consists of vector-valued, piecewise linear functions. Notice tat D X. In order to guarantee tat te condition det( χ T ) > 0 olds true for χ X, we replace te local functional F T (χ) in Eq. (17) by F T (χ) = (a 2) 2 + d + C(ε) ε + d + d, were again a = R T (χ) 2, d = det( R T (χ)), and 0 < ε 1 denotes a regularization parameter to avoid division by zero. For te numerical results in Sec. 6 we set ɛ = Te constant C(ε) = 1 (2 + ε)2 C(ε) guarantees tat d + ε+d+ d assumes its minimum value for d = 1. We remark tat F T blows up for d < 0 but coincides wit te original contribution (17) for d > 0 (up to regularization). Furter details on te numerical implementation togeter wit an evaluation of te mes, approximation quality, and computational costs can be found in [12, 10, 80]. In te following, for given a structure position Γ(t), we will denote te optimal interface aligned triangulation obtained from te strategy outlined above by T opt (Γ(t)). Te resulting computational domain is given by: Ω f,opt (t) = K. K T opt(γ(t)) 2 4. Partitioned metods for te fluid-structure interaction problem Te FSI problems described in Sec. 2 will be solved using two different partitioned strategies based on Domain Decomposition metods [69]: te Diriclet-Neumann (DN) and te Robin-Neumann (RN) algoritms. Partitioned metod are appealing for solving multi-pysics problems suc as tose discussed in tis manuscript, because tey allow te reuse of existing fluid and structure solvers wit minimal modifications. Because of te modularity of DN and RN algoritms, eac pysics sub-problem is solved separately, wit te coupling conditions enforced in an iterative fasion. In te DN algoritm te coupling boundary condition (12) is imposed at te interface as a Diriclet boundary condition for te fluid sub-problem, wereas in te RN algoritm te fluid sub-problem is endowed wit Robin interface condition (14). In bot algoritms, te structure sub-problem is supplemented wit te Neumann boundary condition (13). Eq. (13) is a proper Neumann boundary condition wen te structure is tick; for tin structures Eq. (13) prescribes a load on te structure. To describe te DN and RN algoritms, we introduce te time-discretization step t > 0 and set t n = n t, for n = 1,..., N, wit N = T/ t. At every time t n, te DN and RN algoritms iterate over te fluid and structure sub-problems until convergence. Tese are Ricardson (also called fixed point) iterations for te position of Γ(t n ). Let k be te index for tese iterations. 9

10 4.1. Te Diriclet-Neumann metod At time t n+1, iteration k + 1, assuming tat Ω n f, u k, p k, and x k performed: are known, te following steps are - Step 1: Solve te fluid sub-problem for te flow variables u k+1, p k+1 defined on Ω n f, wit Diriclet boundary condition u k+1 = ẋ k on Γ n. (20) - Step 2: Solve te structure sub-problem for te structure position x k+1, driven by te just calculated ydrodynamic force f Γ,k+1, i.e., f k+1 = f Γ,k+1 on Γ n. - Step 3: Ceck te stopping criterion, e.g. x k+1 x k x k < ɛ, (21) were ɛ is a given stopping tolerance. If violated, repeat Steps 1 3. If satisfied, set x n+1 = x k+1 and p n+1 = p k+1. - Step 4: Ceck te mes quality of Ω n f : - If good: Accept and set ũ n+1 = u k+1 and Ω n+1 f = Ω n f. - If bad: Apply mes optimization to get Ω n n+1 f,opt, set Ω f i.e. = Ω n f,opt. Project data onto new mes, ũ n+1 = I Ω n f Ω n+1 (u k+1 ). (22) f A mes is considered to be bad if te maximum angle of te elements exceeds a certain value, for instance 130 degrees. - Step 5: Standard ALE update: From te new structure position x n+1 obtain Γ n+1, and from te intermediate fluid domain obtain: Ω n+1 f Ω n+1 f = E(Γ n+1 n+1, Ω f ) using an extension operator E (see comment below). Set u n+1 = ũ n+1 and move to te next time step. Some remarks on te sceme outlined above are in order: 1. For our computations, we do not use a standard extension operator E (suc as armonic extension, or operators stemming from linear elasticity), but use te variational approac based on (16), (17). In our experience, tis approac is superior to linear extension operators in terms of mes quality. 2. Te inner loop, in wic te index k canges, corresponds to Steps 1 3 in te above iteration algoritm. In tose steps, te fluid domain is frozen, wic allows for important saving of computational time. 3. Te angle-based criterion used to detect bad meses in Step 4 is purely euristic and may be replaced by oter meaningful mes quality criteria. Te criterion sould be sufficiently mild in order to prevent te reparametrization at every time step. Notice tat te mes optimization procedure presented in Subsec. 3.1 aims at generating triangulations made of equilateral triangles, terefore te angles after optimization are usually bounded away from, e.g., 130 degrees. 10

11 4. A crucial point in te above algoritm is te coice of te mes transfer operator I Ωf,n+1 Ω n+1 f,opt appearing in Eq. (22) at Step 4, needed wenever reparameterization is performed. In our case, tis operator is te Lagrange interpolation operator wic was also proposed in [40]. However, it is known tat dynamically canging meses may lead to spurious oscillations of te pressure for small time step sizes [16, 18]. Indeed, we will observe tose oscillations in our numerical results, as sown in Section Transferring te solution from one mes to anoter witout introducing tese errors seems to be an open question. It is well known tat te convergence properties of te DN algoritm depend eavily on te added-mass effect [22]. In fact, wen te structure constitutes a part of te fluid domain boundary, te number of DN iterations required to satisfy te stopping criterion (21) increases as te structure density approaces te fluid density. Moreover, below a certain density ratio ρ s /ρ f, wic depends on te domain geometry, relaxation is needed for te DN algoritm to converge [60, 61, 22]. Tis is wy we adopt a simple Aitken s acceleration tecnique, wic is based on a relaxation approac, and is known to reduce te number of DN iterations. Tis strategy, introduced in [55], was proposed for a setting similar to ours in [2]. Te results in [2] indicate tat only a few accelerated DN sub-iterations are to be expected for FSI problems wit an immersed structure and large added-mass effect. Diriclet-Neumann algoritms, owever, ave been sown in [4] to fail for FSI problems wit ballon-type structures suc as te periodic beam case (PB). Tis is because in te DN algoritm te coupling conditions are satisfied asyncronously. As a result, te fluid sub-problem uses Diriclet boundary condition u k+1 = ẋ k wic is based on te velocity of te structure ẋ k calculated from te previous sub-iteration. Tis ẋ k may not be consistent wit te incompressibility condition (2), wic requires tat 0 = u k+1 n dγ n = ẋ k n dγ n. (23) Γ n Γ n Since te last integral is not necessarily equal to zero for all ẋ k, te fluid sub-problem in Step 1 is solved wit a velocity prescribed at te boundary tat does not satisfy te integral equality above, giving rise to an ill-posed problem at te semi-descrete level. Because of tis limitation, in te next subsection we consider a Robin-Neumann algoritm for te solution of te FSI problem involving te periodic beam case (PB) Te Robin-Neumann metod At time t n+1, iteration k + 1, te following steps are performed: - Step 1: Solve te fluid sub-problem for te flow variables u k+1, p k+1 defined on Ω n f, wit Robin boundary condition - Step 2, 3, 4 and 5 as in Sec. 4.1 α f u k+1 f Γ,k+1 = α f ẋ k f k on Γ n. (24) Recall tat f Γ denotes te jump in te normal stress across te structure, as defined in (11), and f stands for te rigt and-side of te structure equation (5). Notice tat te DN algoritm can be interpreted as a particular case of te RN algoritm for α f. It was sown in [4] tat wen te structure constitutes a part of te fluid domain boundary for a suitable coice of parameter α f te RN metod features excellent convergence properties: it always converges witout any relaxation and its convergence is insensitive to te added-mass effect. To our knowledge, te RN metod as never been applied to FSI problems involving an immersed structure and a complex structure model like (PB) or (IB). Our goals are: to ceck weter tese improved convergence properties still old wen te structure is immersed in te fluid domain, and to sow tat te RN algoritm can andle balloon-type FSI problems. 11

12 In [4], α f is estimated by considering a simplified structure model. Here, we follow te same approac. Under te ypotesis tat te term EIx (as well as te inextensibility condition wen problem IB is considered) is negligible, te structure model (5) reduces to a simple inertial model: ρ s ẍ = f Γ, in (0, L), t (0, T ]. By discretizing tis equation in time wit te implicit Euler sceme (see, e.g., [68]) and by using te kinematic coupling condition (12), at te time t n+1 we obtain: ρ s t un+1 f n+1 Γ = ρ s t 2 (xn x n 1 ), (25) wic is a Robin boundary condition for te fluid problem. Tus, wen a simple inertial model for te structure is adopted, te structure problem is not solved separately, but it is embedded in te fluid problem in te form of te Robin boundary condition (25). Tis simplified problem motivates te coice for te constant α f in (24): α f = ρ s t for general structure models. By plugging (26) into Robin condition (24) and taking into account Eq. (5) at iteration k, we get: ρ s t u k+1 f Γ,k+1 = ρ s (ρ s ẍ k + EIx k ), tẋk }{{} f k Tis equation is discretized in time wit te implicit Euler sceme to obtain: ρ s t u k+1 f Γ,k+1 = ρ s ẋ k ẋ n ρ s EIx k. tẋk t Te two terms containing ẋ k cancel out, and a furter discretization of ẋ n leads to: (26) ρ s t u k+1 f Γ,k+1 = ρ s t 2 (xn x n 1 ) EI(x k ). (27) Tis is a semi-discretized Robin condition (14) wit te coice of α f given by (26). Notice tat Eq. (27) is compatible wit Eq. (25), te only difference being te presence of te flexural stiffness term tat was neglected in te simplified model. 5. Te fully discrete problem We present te fully discrete problem for te case of te fluid problem (3),(4) wit Robin boundary condition (24), and recall tat a similar approac can be taken for te DN algoritm, since it is a particular case of te RN algoritm. We will state te problem in weak form by including only te boundary condition on Γ(t), since tose on Ω f (t)\γ(t) are understood and do not affect te presented metododology Te discrete fluid sub-problem For any given t [0, T ), we define te following spaces: { V (t) = v : Ω f (t) R 2, v = ˆv (A) 1, ˆv (H 1 (ˆΩ f )) 2}, { } Q(t) = q : Ω f (t) R, q = ˆq (A) 1, ˆq L 2 (ˆΩ f ). In te following we will use te notation V n := V (t n ) and Q n := Q(t n ) to denote te finite element spaces at te time instant t n. 12

13 We introduce te following linear forms: m(ω; u, v) = (u v) dω, a(ω; u, v) = µ (ɛ(u) : ɛ(v)) dω, Ω Ω c(ω; w; u, v) = ((w ) u v) dω, b(ω; p, v) = p v dω. Ω Te variational formulation of te fluid problem (3),(4) wit boundary condition (14) reads: given t (0, T ], find (u, p) V (t) Q(t) suc tat (v, q) V (t) Q(t) te following olds: ( ρ f m Ω f (t); u ), v + ρ f c(ω f (t); u w; u, v) + a(ω f (t); u, v) + b(ω f (t); p, v) t ˆx +m(γ(t); α f u, v) = m(γ(t); α f ẋ f, v), Ω b(ω f (t); q, u) = 0. Time and space discretization. For simplicity, te implicit Euler sceme is used to discretize te above weak formulation in time. Te convective term is linearized by a first order extrapolation formula. Notice tat iger order discretization scemes and extrapolation formulas are also possible. At time t n+1, and at te (k + 1)-st RN sub-iteration, te time discrete linearized fluid sub-problem reads as follows: Find (u k+1, p k+1 ) V n Q n suc tat ρ f m (Ω n f ; u k+1 u n ), v + ρ f c(ω n f ; u k w n ; u k+1, v) + a(ω n f ; u k+1, v) + b(ω n f ; p k+1, v) t +m(γ n ; α f u k+1, v) = m(γ n ; α f ẋ k f k, v), (28) b(ω n f ; q, u k+1 ) = 0, (29) for all (v, q) V n Q n. For te space discretization of problems (28)-(29), we coose te inf-sup stable Taylor-Hood finite element pair P 2 P 1. However, wile te velocity field is continuous at Γ n, te pressure space sould be able to capture discontinuities across Γ n, wic are needed also for te correct evaluation of te ydrodynamic force (11). In order to deal wit pressure discontinuities tat occur at Γ k, we use node doubling at te interface and apply a Subspace Projection Metod to enforce te continuity of te velocity; see also [14, 13, 67] for a description of tese tecniques. Te linear system resulting from linearization and discretization is solved wit te direct solver PARDISO [56, 74, 75] Te discrete structure problem For te time discretization of problem (5) or (7), we will consider a generalized Crank-Nicolson sceme [46]. At time t n+1, Diriclet-Neumann iteration k + 1, te time discrete structure problem (7) is as follows: Find x k+1 K suc tat: L x k+1 2x n + x n 1 ρ s t 2 yds + EI 0 L = 0 L 0 (αx k+1 + (1 2α)x n + αx n 1 ) y ds (αf k+1 + (1 2α)f n + αf n 1 ) yds, y dk(x k+1 ), (30) were K and dk are defined in (8) and (9), respectively. Time discretization of problem (5) is similar. Tis sceme is known to be second order accurate for linear problems. For te numerical results in Sec. 6, we will set α = 1/4 since in linear cases tis coice leads to an unconditionally stable sceme, wic possesses a very small numerical dissipation compared to oter scemes, e.g., te Houbolt metod [17, 11]. For te space discretization of problem (30), we use a tird order Hermite finite element metod [17]. To treat te inextensibility condition y = 1 in problem (IB), we use an augmented Lagrangian Metod [17, 36, 43, 46, 29]. We refer to [11] and te references terein, for details. 13

14 Γ s,n Γ f,n Figure 3: Fluid triangulation (black) aligned wit te structure mes Γ s,n (blue). Te fluid nodes are marked wit dots, wile te structure nodes are marked wit squares. Γ f,n (red) is te approximation of te interface given by an edge of te fluid mes Enforcement of te coupling conditions To describe te enforcement of te coupling conditions reported in Sec. 2.3 we first recall tat at every time step te fluid mes is aligned wit te structure position. However, in general te fluid and structure meses do not coincide, since tey are made up of different elements: cubic Hermite elements on te structure side, and quadratic isoparametric edges on te fluid side. Due to te alignment, te fluid nodes tat approximate te interface are always located on te structure mes, as sown in Fig. 3. In Fig. 3, we denote by Γ f,n te approximation of te location of Γ n given by te fluid mes, and by Γ s,n te approximation of Γ n by te structure mes. Enforcement of te kinematic coupling condition, i.e., te Diriclet condition (20). Denote by U Γ,k and Ẋk te arrays of te nodal values of te corresponding fluid and structure velocities at te interface. Let Bfs n be te interpolation matrix of te structure mes at te fluid interface nodes. To impose Diriclet condition (20), we set U Γ,k+1 = B n fsẋk. (31) Enforcement of te dynamic coupling condition. Te fluid load onto te structure is given by te ydrodynamic force (11). Te computation of te ydrodynamic force (11) is crucial for te numerical stability and accuracy of Domain Decomposition FSI solvers [33]. In te setting considered in tis paper (an immersed beam), te quality of approximation of te pressure jump across te beam is of great importance, as demonstrated by te results in Sections 6.1 and 6.3. Te load exerted by te fluid onto te structure f Γ can be computed as te variational residual R of te momentum conservation equation for te fluid, tested wit test functions v tat are different from zero on Γ(t): Γ(t) f Γ v dγ = Γ(t) σ 1 n 1 v dγ Γ(t) σ 2 n 2 v dγ ( = ρ f m Ω f (t); u ), v ρ f c(ω f (t); u w; u, v) a(ω f (t); u, v) b(ω f (t); p, v) t ˆx = R(Ω f (t); u, p, v). (32) Let f f Γ,k+1 denote te discrete ydrodynamic force at Γf,n discretization of (32), f f Γ,k+1 is calculated from: Γ f,n and sub-iteration k + 1. After time and space f f Γ,k+1 v dγ = R(Ω n f ; u,k+1, p,k+1, v ), (33) 14

15 were u,k+1 and p,k+1 are te discrete velocity and pressure at te DN or RN sub-iteration k +1, obtained from solving system (28)-(29). By using matrix notation, eq. (33) can be written as follows: M n,f Γ F f Γ,k+1 = R k+1, (34) were F f Γ,k+1 is te array of nodal values of f f Γ,k+1, M n,f Γ is te mass matrix at Γ f,n, and R k+1 corresponds to te known values of te combined residuals appearing on te rigt-and side of equation (33). Tis defines te ydrodynamic force, calculated at te fluid mes nodes along te beam. To enforce te dynamic coupling condition (13), tis ydrodynamic force needs to be set equal to te structural load f exerted onto te structure. For tis purpose, we need to assign te values of te ydrodynamic force to te structure mes nodes Γ s,n wic do not necessarily lie on te structure discretization defined by te fluid mes, see Fig. 3. To do tat, we first project te structure mes nodes of Γ s,n onto te fluid mes interface Γ f,n. At te fluid mes, te fluid loading onto te structure f f Γ,k+1 is defined by te process described above. We take tose values of f f Γ,k+1 and interpolate tem first along te edges of te fluid mes at te projected structural nodes, and ten we assign tose values back to te original structure nodes. More precisely, wenever te structural load f k+1 (x) needs to be evaluated for some x Γ s,n (for instance at te quadrature nodes needed to evaluate te rigt and side of eq. (30)), we first define te projected structure node x := arg min x y y Γ f,n and ten let f k+1 (x) = f f Γ,k+1 ( x). We use te following notation to summarize tis procedure: F s Γ,k+1 = B n sf F f Γ,k+1, (35) were we used Bsf n to denote te extrapolation of te values of te ydrodynamic quantities at te fluid nodes onto te te structure nodes. Tis defines te ydrodynamic force at te structure mes nodes, and enforces te dynamic coupling condition (13). It is important to notice tat in tis numerical implementation of te dynamic coupling condition, te power excanged between te fluid and structure is not perfectly balanced, i.e., at te discrete level, te energy imparted by te fluid onto te structure is not perfectly converted into te total energy of te structure, and vice versa. Tis is due to te non-matcing fluid and structure meses. In te case of te DN algoritm, tis mismatc can be precisely quantified as follows. At te time t n+1, after te convergence of te DN sub-iterations, te discrete power excanged at te interface from te fluid side is P f,n+1 = f f,n+1 Γ u n+1 dγ = (U n+1 Γ ) T M f,n+1 Γ F f,n+1 Γ = (Ẋn+1 ) T (B n+1 fs ) T M f,n+1 Γ F f,n+1 Γ, (36) Γ f,n+1 were for te last equation we used (31). Similarly, te discrete power excanged at te interface from te structure side is P s,n+1 = f s,n+1 Γ ẋ n+1 dγ = (Ẋn+1 ) T M s,n+1 Γ F s,n+1 Γ = (Ẋn+1 ) T M s,n+1 Γ B n+1 F f,n+1 Γ, (37) Γ s,n+1 were for te last equation we used (35). Tus, te power excanged at te interface is balanced if (B n+1 fs ) T M f,n+1 Γ = M s,n+1 Γ B n+1 sf. Since Γ f,n+1 and Γ s,n+1 are aligned but do not coincide (Γ s,n+1 is a piecewise cubic globally C 1 function, and Γ f,n+1 is a piecewise quadratic interpolation) and te fluid and structure discretizations are based on different elements, te balance equation is not necessarily fulfilled exactly. However, in Sec we 15 sf

16 will sow tat te difference between P f,n+1 and P s,n+1 is very small (0.01% of te power value) in our computations. Enforcement of Robin boundary condition (27). We first evaluate te rigt and side of te Robin boundary condition (27) on te structure mes. Denote by S k te discretization of te rigt and side at sub-iteration k. Ten S k is given by: L 0 S k yds = L 0 ρ s x n x n 1 t 2 yds EI L 0 (αx k + (1 2α)x n + αx n 1 ) y ds. (38) Boundary condition (27) can be treated similar to (31) by interpolating S k at te fluid interface nodes. Tus, te Robin boundary condition (27) in matrix form reads: ρ s t M f,n Γ U Γ,k+1 M f,n Γ F f Γ,k+1 = M f,n Γ Bn fss k. (39) 6. Numerical results A series of numerical tests is presented tat sowcase te main features and performance of te Extended ALE Metod. In all te tests, te fluid density of ρ f = 1 g/cm 3 is considered, te structure mes always consists of 45 nodes (wit cubic Hermite elements), and te stopping tolerance in (21) for te partitioned scemes (eiter DN or RN) is always set to ε = We use te SI unit system, and present all te quantities in te centimeter-gram-second (CGS) units. If te units of a certain quantity are omitted for te sake of simplicity, it is implied tat tey are in te CGS system A stationary periodic beam Te goal of tis first test is to sow te importance of accurately capturing te pressure discontinuity across te interface, and to validate our implementation of te Extended ALE Metod and RN algoritm. We coose a simple test case wit a stationary solution, wic can be calculated explicitly. We consider te periodic beam model (PB) wit EI = 1 g/(cm s 2 ), ρ s = 10 g/cm, and we take te beam to be a circle of radius 1 centered at te origin: x(s) = [cos(s) sin(s)] T, s [0, 2π], (40) immersed in a viscous fluid wit viscosity µ = 1 g/cm s, occupying a rectangular fluid domain Ω f = ( 2, 2) ( 2, 2). Te fluid is at rest, i.e. u = 0. Te coupled FSI problem (3) troug (6), plus (12) and (13), as a steady-state solution, wic is given by Eq. (40) for te structure position and { EI if x Ω 1 f u = 0 in Ω f, p(x) =, 0 if x Ω 2 f, (41) were Ω 1 f (Ω2 f ) is te fluid domain inside (outside) te beam. We fix te time step size to be t = 10 2 s and consider four different fluid meses of mes size j = j, j = 1,..., 4. We solve te time-dependent problem (3), (4), (5), (6), (12), (13), wit te initial data (40) and u = 0. We consider two different pressure approximations: a discontinuous and a continuous one. Table 1 reports te errors for te velocity in te H 1 semi-norm, and for te pressure in te L 2 norm, after 100 time steps. 16

17 continuous p discontinuous p ref. level j u u 1 p p u u 1 p p e e e e e e e e e e e e e e e e 06 Table 1: L 2 errors for te pressure and errors in te H 1 semi-norm for te velocity after 100 time steps obtained by using continuous and discontinuous pressure approximations on four meses wit different refinement levels. In te continuous pressure case, we observe a low convergence rate of about 1/2, wic is to be expected wen approximating discontinuous functions using continuous functions (see for instance [47]). Te same olds for te velocity in te H 1 semi-norm, indicating te presence of spurious velocities wic often lead to instabilities. Wen using a discontinuous approximation for te pressure, te approximation error is reduced significantly. In fact, already for refinement level 2, we reac a value of te order of Figure 4 sows a warp of te pressure obtained wit bot continuous and discontinuous pressure approximations for refinement level 2: spurious oscillations appear in te case of a continuous pressure approximation. Tis example sows tat our metodology captures te correct steady-state solution, and it empasizes te importance of accurate pressure approximation for tis class of problems, motivating te use of a discontinuous pressure approximation, wic we employ in te rest of tis work. (a) Continuous pressure approximation (b) Discontinuous pressure approximation Figure 4: Warp of te pressure solution after 100 time steps for (a) continuous and (b) discontinuous pressure approximations on te mes wit refinement level A rotating periodic beam Te second test is quasi-stationary: we consider a rotation of te periodic structure in te previous subsection. Te geometry is te same as in subsection 6.1. Te structure parameters are set to EI = 1 g/(cm s 2 ) and ρ s = 1 g/cm. Observe tat ( ) y u(t, x) = u(x, y) =, p(x, y) = 0.5 ( x 2 + y 2) in Ω x f (42) defines a quasi-stationary solution to te coupled FSI problem (3) troug (6), plus (12) and (13). A sort computation reveals tat te kinematic coupling condition u(t, x(t, s)) = ẋ(t, s) on Γ(t) (43) 17

18 implies d ρ s ẍ(t, s) = ρ s dt u(t, x(t, s)) = ρ s ( t u(t, x(t, s)) + u(t, x(t, s))ẋ(t, s)) (44) = ρ s u(t, x(t, s))u(t, x(t, s)) = ρ s x(t, s), (45) and tus ρ s ẍ + EIx = 0 = f Γ, for our coice of ρ s and EI. Observe tat even toug te fluid solution (42) is stationary wit respect to time, te kinematic boundary condition leads to a time-dependent solution of te structure problem, i.e. a counter clockwise rotation of te structure. We fix te time step size to be t = 10 4 s to keep time discretization errors as small as possible and consider four different fluid meses of mes size j = j, j = 1,..., 4. We solve te time-dependent problem (3), (4), (5), (6), (12), (13), wit te initial data for u and x corresponding to te above solution. Te errors of te pressure (in te L 2 norm) and te velocity (in te H 1 semi-norm and L 2 norm) obtained after 100 time steps are sown in Table 2. ref. level j u u 1 EOC u u EOC p p EOC e e e e e e e e e e e e Table 2: Errors in te H 1 semi-norm and L 2 norm for te velocity and L 2 errors for te pressure after 100 time steps obtained on tree different mes sizes using a fixed time step size of t = 10 4 s. Wile we observe te teoretically expected convergence rate of 2 for te pressure in te L 2 norm, te velocity errors sow a sligtly super-convergent beavior. Tis is to be expected, since te linear velocity solution is only almost contained in te velocity trial space wic is made up of isoparametric deformed elements at te structure interface wic do not necessarily contain piecewise linear functions An oscillating periodic beam Te goals of tis example are: (1) to sow tat for small structural displacements, te solution obtained using our Extended ALE Metod coincides wit te solution obtained using te standard ALE approac, and (2) to ceck te convergence beavior of our approac in time and space. We consider te same fluid domain as in te previous test, but we decrease te fluid viscosity to µ = 0.2 g/cm s. Te fluid is initially at rest. Te initial sape of te beam is elliptic: x(s) = [a cos(s), sin(s)/a] T, s [0, 2π], (46) were a > 0 is a given parameter. Te no-slip condition is imposed at te rectangular (rigid) boundary of te fluid domain Ω f. Due to elastic forces, we expect te beam to evolve towards te steady state solution of a circle, wit an area equivalent to tat of te initial ellipse (46), i.e. a circle of radius 1. We coose te parameter a = 1.5 suc tat te structure displacement is not too large, and te standard ALE metod can be applied to simulate te problem. 18

19 (a) t = 1 s (b) t = 10 s (c) t = 100 s 19 Figure 5: Pressure and (coarse) mes deformation for te oscillating beam at time (a) t = 1 s, (b) t = 10 s, and (c) t = 100 s for bot te standard ALE (left) and extended ALE metods (rigt).

20 Fig. 5 sows te computed pressure and mes deformation at times t = 1 s, 10 s, and 100 s for bot standard ALE and extended ALE metods. We see in Fig. 5(a) and (b) tat te beam, initially elliptic, oscillates. Ten, at time t = 100 s it reaces a circular sape, sown in Fig. 5(c). Tese figures also sow te difference in te mes deformation given by te standard ALE metod and our Extended ALE approac. Next, we track te maximum x-coordinate of te beam position over time for: - two mes sizes = j, wit j = 0 (coarse mes) and j = 1 (fine mes), and - four time step sizes t = j, j = 0,..., 3. Te maximum x-coordinate is expected to evolve towards 1 troug damped oscillations, given tat te stationary solution for te beam is te circle of radius 1 centered at te origin. 20

21 (a) standard ALE, j = 0 (coarse mes) (b) Extended ALE, j = 0 (coarse mes) (c) standard ALE, j = 1 (fine mes) (d) Extended ALE, j = 1 (fine mes) (e) difference between standard and Extended for j = 0 (coarse mes) (f) difference between standard and Extended for j = 1 (fine mes) Figure 6: Maximum x-coordinate of te beam position (in cm) over time (in s) for (a) standard ALE and coarse mes, (b) Extended ALE and coarse mes, (c) standard ALE and fine mes, and (d) Extended ALE and fine mes. (e) Difference in absolute value between te curves in (a) and (b), and (f) difference in absolute value between te curves in (c) and (d). Indeed, Fig. 6(a) and (b) display tese damped oscillations computed wit te coarse mes for all te 21

22 time steps under consideration by te standard and Extended ALE metod, respectively. Te corresponding figures for te fine mes are Figs. 6(c) and (d). For bot metods, bot meses, and all te time steps, te maximum x-coordinate evolves towards 1, as expected. Moreover, we see tat te curves obtained wit te different time steps are almost superimposed over te wole time interval for bot metods and bot meses, indicating tat we ave reaced time step size independence for a rater large interval of time. Fig. 6(e) sows te difference in absolute value between te curves in Fig. 6(a) and (b), wile te difference in absolute value between te curves in Fig. 6(c) and (d) is reported in Fig. 6(f). From Fig. 6(e) and (f) we see tat tere is a sligt difference in amplitude between te oscillations computed by te standard and Extended ALE metods. Suc a difference sould get smaller as te mes gets finer. In fact, te maximum difference in te maximum x-coordinate is around for te coarse mes and around for te fine mes, indicating tat te beam maximum x-coordinates computed by te two metods get closer wit mes refinement Periodic beam advected by a cannel flow Tis example is designed to sow tat for FSI problems wit large deformation, te standard ALE metod breaks down due to excessive mes distortion, wile te Extended ALE Metod keeps te mes quality under control, and gives excellent results beyond te point of break down of standard ALE. We consider te fluid domain to be a rectangular cannel Ω f = ( 12, 12) ( 2, 2) filled wit a viscous fluid of viscosity µ = 0.1 g/cm. At t = 0, te fluid is at rest, and te beam of circular sape, wit radius 1, is immersed in te fluid, wit its center located at ( 8, 0). See Fig. 7(a). Te following structure parameters are used in te simulation: ρ s = 10 g/cm and EI = 1 g/(cm s 2 ). We study FSI between te periodic beam and te flow of a viscous, incompressible fluid, wic is driven by te boundary conditions imposed on Γ in and Γ out (see Fig. 1(a)). At Γ in, we prescribe a non-zero x- component of te fluid velocity, wic is smootly increased to 1 cm/s over time interval [0, 1], and is kept equal to 1 cm/s until t = 20 s, wen it drops to 0 cm/s: [ 2t 3 + 3t 2, 0] T if 0 t 1, u(t, x) Γin = [1, 0] T if 1 t 20, [0, 0] T if t > 20. At Γ out we enforce a omogeneous Neumann boundary condition. Te no-slip condition is prescribed at te top and bottom cannel boundary. Due to te background fluid flow, te periodic beam is transported troug te cannel wile undergoing strong deformation as te fluid and structure interact via a two-way coupling. See Fig. 7(b) and (c). At t 20 s wen te inlet fluid velocity returns to zero, te viscous forces lead to a deceleration of te beam, wic stops and eventually returns to its original circular sape, as visible at around t = 100 s. See Fig. 7(f). Figure 7 sows te beam position and mes deformation computed at different times by te standard ALE metod (top panel in eac subfloat) and te Extended ALE metod (bottom panel in eac subfloat). Te standard ALE is able to follow te deformation and transport of te beam until about t s, wen mes distortion becomes too severe. For tis reason, te position of te beam in te top panels in Figs. 7(d), (e), and (f), corresponding to times t = 25, 50, and 100 s, is not updated. As long as te two simulations run, we observe very good agreement in te beam position and deformation computed by te standard and Extended ALE metods. See Fig. 7(a), (b), and (c). A more detailed comparison between te two metods is sown in Fig. 8. Tere, a comparison between te maximum x-coordinate of te beam position computed by te two metods is given in Fig. 8(a), sowing excellent agreement. Fig. 8(b) sows a close-up view of Fig. 8(a) around te time wen te standard ALE metod breaks down. Notice tat until t s te curves given by te two metods are almost superimposed. Wit te extended ALE metod we are able to carry out te simulation all te way until t = 100 s wen te fluid velocity and beam motion are almost zero, and te sape of te beam as returned to almost circular. Fig. 7 sows tat te mes obtained wit te standard ALE metod gets severely distorted, wile te quality of te mes computed by our Extended ALE metod remains ig trougout te entire time 22

23 Standard ALE Standard ALE Standard ALE Extended ALE Extended ALE Extended ALE (a) t = 0s Extended ALE (b) t = 5s Extended ALE (c) t = 15s Extended ALE (d) t = 25s (e) t = 50s (f) t = 100s Figure 7: Beam position and mes deformation for te periodic beam transported troug a cannel at (a) t = 0 s, (b) t = 5 s, (c) t = 15 s, (d) t = 25 s, (e) t = 50 s, (f) t = 100 s. Wen available, te solution computed by te standard ALE metod is above te solution computed by te Extended ALE metod. (a) Maximum x-coordinate of te beam (b) Zoomed view Figure 8: (a) Maximum x-coordinate of te beam position over time (in s) computed by te standard and Extended ALE metods and (b) close-up view around te time wen te standard ALE metod breaks down (t = s). 23

24 interval. As a furter proof of te different quality of te meses given by te standard and extended ALE metods, we report in Fig. 9 te maximum angle of te mes elements over time. We see tat te maximum angle in te mes given by te standard ALE metod increases up to nearly 170 degrees, rigt before te simulation crases. On te oter and, te maximum angle for te mes given by te Extended ALE metod never exceeds 132 degrees. Figure 9: Maximum angle of te elements in te mes given by te standard and Extended ALE metods versus time (in s) An immersed beam Tis test is aimed at assessing te performance of te DN algoritm, witout and wit Aitken s acceleration metod, and te performance of RN algoritm. We consider te immersed beam model (IB), wic is more callenging tan te periodic beam model (PB) due to te inextensibility constraint [29]. For tis test, we are going to use only te Extended ALE Metod Comparison between te DN and RN algoritms We consider a cannel Ω f = [ 3, 3] [ 0.5, 0.5] filled wit a viscous, incompressible fluid of viscosity µ = 0.01 g/m s. A beam of lengt 0.5 cm, wit ρ s = 5 g/cm and EI = 0.05 g/(cm s 2 ), is immersed in te fluid, clamped at te mid-point x = 0 bottom of te cannel Γ down. Te beam is initially straigt and vertical, and te fluid is initially at rest. We consider a time-dependent FSI problem wic is driven by te time-dependent inlet velocity data: a time-dependent Poiseuille velocity profile, wit maximum velocity U(t) = 1 4 ( ( π )) 1 cos 2 t is prescribed at Γ in, wic corresponds to x = 3. A omogeneous Neumann condition is enforced at Γ out, i.e., at x = 3. Te no-slip condition is imposed on Γ down, and a symmetry condition is imposed on Γ up. See Fig. 10. Te Stroual number for tis problem is 0.5. Te inlet boundary condition and te structural parameters were cosen to generate a moderate -amplitude oscillatory motion of te beam around its initial configuration. Fig. 10 sows two snapsots of te velocity magnitude togeter wit te beam position at te time of maximum deflection. To sow tat bot DN and RN algoritms yield numerical results tat are very close, we compare te x coordinate of te beam tip over time. Te results, sown in Figure 11(a), indicate tat te two curves are superimposed. To zoom into te difference between te two results, we plot te absolute value of te difference between te two curves in Fig. 11(b). One can see tat te maximum difference over time interval [0, 10] s is , wic occurs after a first ALE reparameterization at around t = 4.8. cm/s 24

25 Γ up Γ up Γ in Γ out Γ in Γ out Γ down (a) t = 7 s Γ down (b) t = 9 s Figure 10: Velocity magnitude and beam position at time (a) t = 7 s and (b) t = 9 s. (a) DN and RN (b) Difference Figure 11: (a) Comparison of te x-component of te beam tip position computed wit Diriclet-Neumann and Robin-Neumann algoritm and (b) difference in absolute value between te curves in (a). In order to evaluate te performance of te DN and RN algoritms, we let te structure density vary: ρ s = 16, 8, 4, 2, 1 g/cm. Te convergence properties of te DN algoritm are known to depend eavily on te added-mass effect, wic becomes worse as te structure density gets closer to te density of te fluid, wic is ρ f = 1 g/cm 3 [22]. Tus, we expect to see an increase in te number of iterations required by te DN metod as te ratio ρ s /ρ f approaces one (from above). Indeed, Fig. 12 (a) sows tat te number of DN iterations witin eac time step over time interval [0, 10] increases dramatically to 56 as te density ratio ρ s /ρ f approaces 1. However, wen relaxation based on Aitken s acceleration is used, te number of DN iterations decreases significantly. Indeed, Fig. 12 (b) sows tat te maximum number of DN iterations wit Aitken s acceleration equals 7 wen ρ s /ρ f = 1. 25

26 (a) DN wit no acceleration (b) DN wit Aitken s acceleration Figure 12: Number of sub-iterations required by te DN algoritm (a) witout acceleration tecniques and (b) wit Aitken s acceleration tecnique to converge over time for different for different values of te structure density. Te legend in (b) is common to bot subfloats. We next compare te performance of te DN algoritm wit Aitken s acceleration and te RN algoritm wit no acceleration. Fig. 13(a) sows te same grap as in Fig. 12(b) but on a different scale, and Fig. 13(b) sows te number of iterations required by te RN algoritm. (a) DN wit Aitken s acceleration (b) RN Figure 13: Number of iterations required by (a) te DN algoritm wit Aitken s acceleration metod and (b) te RN metod to converge over time (in s) for different for different values of te structure density. Te legend in (b) is common to bot subfloats. We see tat for all te structure densities under consideration, te RN algoritm requires less iterations to converge tan te DN metod wit Aitken s acceleration tecnique. Moreover, te RN metod is quite insensitive to variations in te structure density. In fact, we see in Fig. 13(b) tat for ρ s = 16, 8, 4, 2 g/cm, te RN algoritm converges in 2 iterations most of te time. Even wen ρ s = 1 g/cm, te number of iterations required by te RN metod is at most 5 and only for a very limited time Power excange at te interface We consider te same problem as in Sec wit structure density ρ s = 5 g/cm. Te goal is to verify ow well te power excanged at te interface is approximated. We first calculate te discrete power 26

27 excanged at te interface from te fluid side P f, defined in (36), computed wit two fluid meses of mes size = j, j = 0, 1. Fig. 14 (a) sows tat tey are in a pretty good agreement. Te occasional jumps in te discrete power tat can be seen in Fig. 14 (a) occur wenever te ALE mapping is reparametrized. Tese jumps are to be expected for dynamically canging meses, as pointed out in [16, 18]. (a) P f : coarse v.s. fine mes (b) P f and P s (c) P f P s Figure 14: Discrete power excanged at te interface from te fluid side P f, defined by (36), computed wit two different meses. Next, we quantify te unbalance in te power excange at te interface. As explained in Sec. 5.2, at eac time t n+1 te powers excanged at te interface from te fluid side P f,n+1 and from te structure side P s,n+1 are not necessarily equal. In Fig. 14(b), we plot te powers P f and P s computed wit mes j = 1 over te time interval under consideration, wile in Fig. 14 we sow te difference P f P s. One can see in Fig. 14(c )tat over a long time interval te difference between te two powers excanged at te interface is of te order of 10 6 g cm/s 3. Tis corresponds to 0.01% of te power value, wic is of te order of 10 2 g cm/s 3, as sown in Fig. 14(b). Suc a small difference between P f and P s does not endanger stability A valve near a contact (regurgitant valve) We consider te same fluid domain as in te previous two sections Ω f = [ 3, 3] [ 0.5, 0.5], but a longer immersed beam: we set te beam lengt to be L = 0.95 cm, just sort of toucing te top boundary Γ up. See Fig. 15(a). All te pysical and discretization parameters are cosen as in Sec Te initial beam configuration is vertical, as sown in Fig. 15(a), and te initial fluid velocity is zero. We study a time-dependent FSI problem in wic te fluid flow is driven by te difference in te normal stress prescribed te inlet and outlet, Γ in and Γ out. At Γ in te time-dependent normal stress is given by a time-periodic function wit period 10 s, were during eac period te normal stress is given by te step function: { [1.5, 0] T if 0 t 3, σn(t, x) = [0, 0] T for x Γ in if 3 < t < 10 were t [0, 10) is t = mod(t, 10). Terefore, a constant, normal stress in te orizontal direction is applied for te first 3 s, wen te stress is released and set to zero. Tis is repeated six times until t = 60 s. At Γ out, te normal stress is prescribed to be zero. Te no-slip condition is imposed on Γ down, and a symmetry condition is imposed on Γ up. Due to te periodic forcing of te fluid, te induced beam movement will be periodic: te beam is pused to te rigt for 0 t 3, and ten bends backward due to elastic forces for 3 < t < 10. Tis motion is repeated six times until t reaces 60 s. See Figure 15. Because of te symmetry condition imposed on te upper wall, tis test corresponds to simulating a 1D valve wic is just sort of closing, i.e., near a contact. We can see in Figure 15 tat tere is only one element between te tip of te valve and Γ up. Tis scenario is typically associated wit various problems. In particular, due to te fact tat te beam tip is very close to te upper boundary of te fluid domain, te standard ALE metod breaks down because 27

28 (a) t = 0 s (b) t = 1 s (c) t = 3 s (d) t = 6 s (e) t = 8 s (f) t = 11 s Figure 15: Beam position and velocity magnitude togeter wit mes deformation at (a) t = 0 s, (b) t = 1 s, (c) t = 3 s, (d) t = 6 s, (e) t = 8 s, (f) t = 11 s. 28

29 te mes associated wit tis simulation gets quickly severely distorted and breaks down before a single cycle of valve motion is completed. Te Extended ALE Metod, owever, does not suffer from tis problem. Indeed, Fig. 15 sows tat te mes quality remains good, and tat completing a cycle of a near-closing (regurgitant) valve is not a problem. To sow tat tis metod does not induce spurious energy or instabilities over time, we let tis simulation run for 6 cycles. Figure 16 sows te x and y coordinates of te beam tip position over time. Te movement of te beam tip exibits perfect monocromatic beavior, sowing tat our metod does not introduce spurious energy over time and tat no instabilities arise over a rater long time interval. From Fig. 16(b), we also see tat te inextensibility constraint is never violated. In fact, te beam is clamped at (0, 0.5) cm and its y coordinate never exceeds 0.45 cm, te beam being 0.95 cm long. (a) x-coordinate (b) y-coordinate Figure 16: (a) x-coordinate and (b) y-coordinate of beam tip position (in cm) over time (in s). Tis example sows tat te Extended ALE Metod deals well wit structure motion near a contact, and tat it does not introduce any spurious energy or instabilities over time. 7. Conclusions Standard ALE metods for te simulation of fluid-structure interaction problems fail wen te structural displacement is large. In tis paper, we proposed an Extended ALE Metod to overcome tis limitation witout remesing. Our metod relies on a variational mes optimization tecnique wit an additional constraint to enforce te alignment of te structure interface wit edges of te resulting triangulation. Te performance of te Extended ALE Metod was evaluated on a series of test examples involving FSI problems wit two-way coupling, and large displacements. We considered two partitioned algoritms: te classical Diriclet-Neumann metod wit and witout Aitken s acceleration, and te Robin-Neumann metod. Eac partitioned algoritm was combined wit te Extended ALE Metod to solve te FSI problems. Because te Extended ALE Metod provides a fluid mes aligned wit te interface, non-trivial boundary conditions suc as te Robin boundary conditions can be easily implemented. We sowed tat in problems were te structure density is close to te fluid density, te Robin-Neumann metod outperforms te Diriclet-Neumann algoritms bot wit and witout Aitken s acceleration. In fact, in balloon-like problems, te Diriclet-Neumann algoritms fail, wile te Robin-Neumann algoritm combined wit te Extended ALE Metod performs well, and appears insensitive to te added mass effect even wen ρ s /ρ f 1. Te test problems presented in tis manuscript were carefully cosen to study te performance of our approac in FSI problems wit large displacements. Tey involve various scenarios of an elastic, possibly inextensible beam interacting wit an incompressible, viscous fluid in 2D. Te examples clearly sow tat te 29