Department of Information Technology and Mathematical Methods. Working Paper. Exact and inexact partitioned algorithms for fluid-structure

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1 UNIVERSITÀ DEGLI STUDI DI BERGAMO DIPARTIMENTO DI INGEGNERIA DELL INFORMAZIONE E METODI MATEMATICI QUADERNI DEL DIPARTIMENTO Department o Inormation Technology and Mathematical Method Working Paper Serie Mathematic and Statitic n. 13/MS 2012 Exact and inexact partitioned algorithm or luid-tructure interaction problem with inite elaticity in haemodynamic by F. Nobile, M. Pozzoli, C. Vergara Viale Marconi. 5, I Dalmine (BG), ITALY, Tel ; Fax

2 COMITATO DI REDAZIONE Serie Inormation Technology (IT): Steano Parabochi Serie Mathematic and Statitic (MS): Luca Brandolini, Ilia Negri L acceo alle Serie è approvato dal Comitato di Redazione. I Working Paper della Collana dei Quaderni del Dipartimento di Ingegneria dell Inormazione e Metodi Matematici cotituicono un ervizio atto a ornire la tempetiva divulgazione dei riultati dell attività di ricerca, iano ei in orma provvioria o deinitiva.

3 EXACT AND INEXACT PARTITIONED ALGORITHMS FOR FLUID-STRUCTURE INTERACTION PROBLEMS WITH FINITE ELASTICITY IN HAEMODYNAMICS FABIO NOBILE, MATTEO POZZOLI, AND CHRISTIAN VERGARA Abtract. In thi paper we conider the numerical olution o the three-dimenional (3D) luidtructure interaction problem in haemodynamic, in the cae o phyiological geometrie and data, and inite elaticity veel deormation. We introduce new partitioned algorithm and compare their eiciency with that o exiting one. We alo tudy ome new inexact variant, obtained rom emiimplicit approximation, and how that they allow to improve the eiciency while preerving the accuracy o the related exact (implicit) cheme. Key word. Fluid-tructure Interaction, inite elaticity, partitioned algorithm, Robin tranmiion condition, haemodynamic. 1. Introduction. To obtain predictive accurate inormation about the blood luid-dynamic in the arterie o the cardiovacular tree, it i neceary to olve in three-dimenional (3D) realitic geometrie a luid-tructure interaction (FSI) problem, that arie rom the interaction between blood and vacular veel [41, 7, 14, 44, 12, 3, 16, 17]. To capture the complex dynamic, non-linear luid and tructure model have to be taken into account. Thi lead to the olution o a complex non-linear coupled problem, ormed by the luid and the tructure ubproblem, together with the luid domain ubproblem when the luid equation are written in Arbitrary Lagrangian- Eulerian (ALE) ormulation [25, 11]. Eicient numerical trategie are mandatory to olve uch non-linear FSI problem in 3D real geometrie and with phyiological data. Only ew work have ocued on thi apect. We mention [7, 34] among the monolithic cheme, which build the whole non-linear ytem, and [30, 21] among the partitioned cheme, which conit in the ucceive olution o the ubproblem in an iterative ramework (ee alo [12, 5, 4, 9] in the cae o ininiteimal elaticity). In thi work, we ocu on the numerical olution o the FSI problem with partitioned trategie in haemodynamic, when non-linear ub-problem and 3D real computational domain and phyiological data are conidered. Thi problem i very complex, the main diicultie being: 1. The high added ma eect, due to the imilar luid and tructure denitie, which make very diicult the olution o the FSI problem with partitioned trategie [8, 20, 40]; 2. The treatment o the phyical interace condition, which enorce the continuity o velocitie and normal tree at the luid-tructure (FS) interace between the luid and the tructure ubproblem; 3. The treatment o the geometrical interace condition, which enorce the continuity o diplacement at the FS interace between the luid and the tructure domain; MOX, Dipartimento di Matematica Briochi, Politecnico di Milano, Italy, and MATHICSE, CMCS, EPF de Lauanne, Switzerland, abio.nobile@epl.ch Dipartimento di Ingegneria dell Inormazione e Metodi Matematici, Univerità di Bergamo, Italy, matteo.pozzoli@unibg.it Dipartimento di Ingegneria dell Inormazione e Metodi Matematici, Univerità di Bergamo, Italy, chritian.vergara@unibg.it 1

4 4. The treatment o the contitutive non-linearitie in the luid and the tructure model. Regarding point 1 and 2, it ha been clearly highlighted in everal work that the phyical condition have to be treated implicitly in haemodynamic, due to the high added ma eect [8, 20, 3, 40]. In particular, in thi work we conider partitioned algorithm baed on Robin interace condition, which have good convergence propertie, independent o the added-ma eect [3, 4, 1, 22, 38]. For what concern point 3, we can ditinguih between exact and inexact algorithm. The irt group conit in thoe cheme that atiy exactly the geometrical interace condition (geometrical exact cheme [23, 7, 13]). On the contrary, in the geometrical inexact cheme thi condition i not atiied, due to an explicit treatment o the interace poition by extrapolation rom previou time tep (the o-called emi-implicit cheme [12, 6, 39]), or to an a priori ixed number o ixed-point iteration perormed over the interace poition [38]. The emi-implicit cheme have been hown to be table [37, 43, 12, 39] and accurate [36, 2] in the cae o the linear ininiteimal elaticity. Regarding point 4, we have to conider the luid and the tructure contitutive non-linearitie. We ocu here on partitioned trategie, obtained by the application o a uitable linearization o the monolithic ytem. A irt approach o thi type conit in olving the non-linear luid and tructure ubproblem in an iterative ramework until convergence o the phyical continuity condition (think or example to the claical Dirichlet-Neumann cheme) [29, 31, 24, 27, 40]. At each iteration two non-linear ubproblem have to be olved, or example with the Newton method. In thi cae, the contitutive non-linearitie are treated in an inner loop with repect to both the phyical and the geometrical interace condition. We reer to thee cheme a claical partitioned algorithm. A econd trategy conidered o ar conit in applying the Newton or the approximate-newton method (the latter obtained by approximating the tangent operator) to the monolithic non-linear ytem (approximate-newtonbaed algorithm). In [23], the author propoed a block-diagonal approximation o the Jacobian, leading to a partitioned algorithm where all the interace condition and non-linearitie are treated in the ame loop (ee alo [33, 24, 30, 10, 45]). In [38], the author conidered alternative approximation o the Jacobian, leading to dierent, mot eicient partitioned algorithm. The general tructure o uch cheme conit in an external loop to manage the geometrical interace condition and the contitutive non-linearitie and in an internal one to precribe the phyical interace condition. The tudy o the eectivene and accuracy o dierent partitioned cheme to treat the tructure non-linearity or a ull non-linear FSI problem in haemodynamic i ar to be exhautive nowaday. The preent work aim at providing ome anwer in thi direction. The irt goal o thi paper conit in comparing the perormance o dierent partitioned algorithm, to undertand which are the mot eective or 3D real haemodynamic application. In particular, we conidered three amilie o cheme: the claical algorithm, the approximate-newton-baed algorithm, a new cla o cheme, obtained by conidering ixed point iteration over the geometrical interace condition (ixed-point-baed cheme). For all the three amilie, we conidered everal alternative, all baed on the exchange o Robin condition to precribe the phyical interace condition. We ran each o the conidered cheme on a real 3D geometry with phyiological data, with the aim o tudying the eiciency o uch cheme or practical purpoe. The reported numerical reult how that the double 2

5 loop approximate-newton-baed cheme are the mot perorming, while the claical one are the lowet. All the algorithm conidered o ar olve exactly (i.e. up to a given mall tolerance) the non-linear FSI problem. A already mentioned earlier, in the context o linear elaticity, emi-implicit approache which treat explicitly the geometrical coupling have been proven to be table and accurate. It i thereore worth aking i in haemodynamic application with inite elaticity one really need to olve exactly alo the tructure non-linearity, or i the latter can be linearized around a tate uitably extrapolated rom previou time tep. In particular, here we propoe to olve at each time tep a linearized elaticity problem, coupled with the luid in a ixed domain, with both the tructure linearization point and the luid domain extrapolated rom previou time tep. Thi approach would correpond to perorm jut one approximate-newton iteration on the monolithic FSI problem, tarting rom a well-choen initial gue. Alternatively, we propoe to run jut ew Newton iteration at each time tep, with the aim o improving the accuracy. We alo conider the inexact variant o the ixedpoint-baed cheme, obtained by perorming one or ew external iteration, where again both the tructure linearization point and the luid domain are extrapolated rom previou time tep. The econd goal o thi work conit in tudying the accuracy o uch inexact cheme. When a globally third order accurate time dicretization o the FSI problem i conidered, we how numerically on a imple tet cae that perorming at leat two Newton iteration allow to recover a third order convergence in time even when tarting rom a irt order extrapolation, while one Newton iteration i enough when tarting orm a third order extrapolation. We alo how that uch cheme are very accurate in the cae o a real 3D cae, and that they allow to improve the computational eiciency up to three time. The outline o the work i a ollow. In Section 2 we preent the global FSI problem, it time dicretization and a Lagrange multiplier ormulation ueul to derive the numerical cheme. In Section 3 we preent the exact cheme. In particular, in Section 3.1 we introduce the claical partitioned algorithm, in Section 3.2 thoe baed on the approximate-newton method, and in Section 3.3 the new amily baed on a ixed-point reormulation o the global FSI problem. In Section 3.5 we tudy the eiciency o uch cheme by conidering a real cae in haemodynamic. Then, in Section 4 we introduce the inexact cheme, both thoe derived by approximate- Newton-baed method (Section 4.1) and thoe derived by ixed-point-baed method (Section 4.2). Finally, in Section 4.3 we provide the convergence rate o inexact method ued to olve an analytical tet cae, and in Section 4.4 we provide a tudy on the accuracy and eiciency o uch cheme applied to a real haemodynamic cae. 2. The FSI problem and it time dicretization. Reerring to the luid domain Ω t like the one repreented in Figure 2.1, let, we denote, or any unction v living in the current luid coniguration, by ṽ := v A it counterpart in the reerence coniguration Ω 0, where A i the ALE map. By conidering intead the tructure domain Ω t like the one repreented in Figure 2.1, right, we denote, or any unction g deined in the current olid coniguration, by g := g L it counterpart in the reerence domain Ω 0, where L i the Lagrangian map. The trong ormulation o the FSI problem, including the computation o the ALE map read then a ollow: 1. Fluid-Structure problem. Given the (unknown) luid domain velocity u m and luid domain Ω t, ind, at each time t (0,T], luid velocity u, preure p 3

6 Fig Repreentation o the domain o the FSI problem: luid domain on the let, tructure domain on the right. and tructure diplacement η uch that D A u ρ + ρ ((u u m ) )u T (u,p ) = Dt in Ω t, u = 0 in Ω t, u = η on Σ t, t T (η )n T (u,p )n = 0 on Σ t, 2 η ρ t 2 T ( η ) = in Ω 0, (2.1) where ρ and ρ are the luid and tructure denitie, µ i the contant blood vicoity, and the orcing term, n the unit normal exiting rom the tructure domain, and DA Dt denote the ALE derivative; 2. Geometry problem. Given the (unknown) interace tructure diplacement η Σ 0, ind the diplacement o the point o the luid domain η m uch that { ηm = 0 on Ω 0, η m = η on Σ 0 (2.2), and then ind accordingly the luid domain velocity ũ m := eη m t, the ALE map and the new point x t o the luid domain by moving the point x0 o the reerence domain Ω 0 : A(x 0 ) = x t = x 0 + η m. In the previou problem, T (u,p ) i the Cauchy tre tenor related to a homogeneou, Newtonian, incompreible luid, whilt T ( η ) and T (η ) are the irt Piola-Kirchho and the Cauchy tre tenor o the olid, repectively, decribing the 4

7 tructure problem. The two matching condition enorced at the FS interace are the continuity o velocitie (2.1) 4 and the continuity o normal tree (2.1) 5 (phyical interace condition), whilt condition (2.2) 2 enorce the continuity at the FS interace o diplacement o the luid and tructure ubdomain (geometrical interace condition). Equation (2.1) and (2.2) have to be endowed with uitable boundary condition on Ω t \ Σt and Ω 0 \ Σ 0, and with uitable initial condition. Let be the time dicretization parameter and t n := n, n = 0,1,... For a generic unction z, we denote with z n the approximation o z(t n ). We conider Backward Dierentiation Formulae o order p (BDFp) o the orm ( ) D p v n+1 := 1 p β 0 v n+1 β i v n+1 i ( i=1 ) Dpv 2 n+1 2 := 1 p+1 2 ξ 0 v n+1 ξ i v n+1 i i=1 = v t (tn+1 ) + O( p ), = 2 v t 2 (tn+1 ) + O( p ), or uitable coeicient β i and ξ i [38, 40]. We report here the ormulation o the time dicretization o order p o problem (2.1)-(2.2). 1. Fluid-Structure problem. Given the (unknown) luid domain velocity u n+1 m and the luid domain Ω n+1 and the olution at previou time tep, ind the luid velocity u n+1, the preure p n+1 and the tructure diplacement η n+1 uch that D p u n+1 ρ + ρ ((u n+1 u n+1 m ) )u n+1 T (u n+1,p n+1 ) = n+1 in Ω n+1, u n+1 = 0 in Ω n+1, u n+1 = u n+1 on Σ n+1, T (η n+1 )n T (u n+1,p n+1 )n = 0 on Σ n+1, Dp 2 ρ ηn+1 2 T ( η n+1 n+1 ) = in Ω 0. (2.3) In problem (2.3) we have alo introduced the tructure velocity u n := Dpηn. 2. Geometry problem. Given the (unknown) interace tructure diplacement Σ 0, olve a harmonic extenion problem η n+1 { η n+1 m = 0 in Ω 0, η n+1 m = η n+1 on Σ 0, (2.4) and then ind accordingly the dicrete luid domain velocity ũ n+1 m and the point x n+1 o the new luid domain by m := D p η n+1 m, x n+1 = x 0 + η n+1 m. (2.5) ũ n+1 We conider here an equivalent ormulation o (2.3) and (2.4) baed on the introduction o three Lagrange multiplier living at the FS interace, repreenting the luid and tructure normal tree λ and λ, and the normal derivative o the luid meh diplacement λ m [38]. For the ake o notation we remove the temporal index n+1. With Σ D, ΣD,0 and Σ D m we denote the part o the boundary where Dirichlet 5

8 boundary condition are precribed. Then, we deine the ollowing pace V := {v H 1 (Ω ) : v Σ D = 0}, Q := L 2 (Ω ) 1, V := {v H 1 (Ω 0 ) : v Σ D,0 = 0}, V m := {v H 1 (Ω 0 ) : v Σ D,0 m = 0}. Let v := (u,p ) collect the luid unknown and F : V Q V m (V Q) be the luid operator. Analogouly, or the tructure ubproblem we deine the operator S : V (V ), and or the harmonic extenion we introduce the operator H : V m (V m ). We then rewrite problem (2.3)-(2.4) a ollow H η m + γ m λ m = 0 in (V m ), γ m η m = γ η on Σ 0, F(v,u m ) + γ λ = G in (V Q), α γ v + λ D = α p eη γ λ on Σ 0 (2.6), D α p eη γ + λ = α γ v λ on Σ 0, S( η ) + γ λ = G in (V ), where γ : V Q H 1/2 (Σ 0 ), γ : V H 1/2 (Σ 0 ), γ m : V m H 1/2 (Σ 0 ) are trace operator and γ, γ, γ m are their adjoint, G and G account or the right hand ide, (2.6) 2 i the geometrical interace condition, and the interace phyical condition (2.6) 3 4 are linear combination o condition (2.3) 3 4 through coeicient α and α. Thi will allow to obtain partitioned algorithm baed on Robin interace condition, which have good convergence propertie, independent o the added-ma eect when the parameter α and α are uitably choen [3, 4, 1, 22] Outlook o iterative algorithm. A dicued above, we have to ace three ource o coupling and non-linearitie, namely (G) the geometrical interace condition; (C) the contitutive non-linearitie; (P) the phyical interace condition. We give here an outlook o the partitioned algorithm conidered in the ollowing ection. In principle, our model algorithm will conit o three neted loop, one or each o the three ource o coupling and non-linearitie ummarized above. Jut to ix the idea, we uppoe here that the external loop will manage the geometrical interace condition, the intermediate one the contitutive non-linearitie and the internal one the phyical interace condition. Our model algorithm i then o the type while (geometrical interace condition not atiied) do... while (contitutive non-linearitie not atiied) do... while (phyical interace condition not atiied) do... end end end 1 Since we olve the FSI problem in a partitioned way with Robin condition at the FS interace (ee (2.6)), the preure i alway deined and L 2 (Ω ) i the uitable preure pace or the weak ormulation. 6

9 We call thi algorithm G -C -P, where mean that we let the iteration continue until convergence. Starting rom thi model cheme, we can obtain many other algorithm a ollow. 1. The order o the loop could be exchanged, leading to dierent cheme (G - P -C,...); 2. We can merge two or more loop. For example, tarting rom the model algorithm G -C -P, we could decide to treat in the ame loop the geometrical interace condition and the contitutive non-linearitie, obtaining the algorithm GC -P. 3. The external loop could be olved not until convergence, but perorming jut ew external iteration. In thi cae, uch algorithm have to be intended a inexact, ince the external topping criterion i not checked and atiied. For example, tarting rom our model cheme, we could decide to do jut 2 external iteration, obtaining the algorithm I-G2-C -P, where we put a letter I at the beginning to emphaize that uch cheme i inexact. 4. In the cae o inexact cheme where only ew (even one) iteration are perormed in the external loop, we could conider tarting the iteration rom an initial gue obtained by a p th order extrapolation in time. In thi cae, we add the letter E ater the number o iteration o the loop involving the extrapolation. For example, tarting rom the model algorithm with jut 1 external iteration, i we decided to ue an extrapolation o the interace poition a initial gue, we name the correponding cheme a I-G1E-C -P. In thi work, we have alway ued the ame order p or the extrapolation a the order o the temporal cheme. 3. Exact cheme. We dicu here the amily o exact cheme, that i cheme which atiy exactly, up to given tolerance, the three interace condition and the contitutive non-linearitie. In particular, we decribe the claical cheme, the approximate-newton-baed cheme and the new amily o ixed-point-baed cheme. Then, in Section 3.5, the perormance o thee cheme will be compared or the irt time or a real haemodynamic cae, by uing an exponential law or the tructure train energy Claical cheme. The irt trategy correpond to imple iteration at each time tep between the luid geometry, the luid and the tructure ubproblem (ee [24, 29, 31, 27] or the Dirichlet-Neumann cae). Here, we preent the Robin- Robin verion o uch cheme introduced in [40]. In particular, we have the ollowing Algorithm 1. GP -C cheme. Given the olution at iteration k, olve until convergence 1. The luid geometric problem { H η k+1 m + γ m λ k+1 m = 0 in (V m ), γ m η k+1 m = γ η k (3.1) on Σ 0, 2. The (non-linear) luid problem in ALE coniguration with Robin interace condition { F(v k+1,u k+1 m ) + γ k+1 λ = G in ( V (η k+1 m ) Q(η k+1 m ) ), α γ v k+1 + λ k+1 D = α γ pη k λ k on Σ k+1 ; (3.2) 7

10 3. The (non-linear) tructure problem with Robin interace condition { S( η k+1 ) + γ k+1 λ = G in (V ), D α γ p eη k+1 k+1 + λ = α γ v k+1 k+1 λ on Σ 0 ; 4. Relaxation tep η k+1 = ω η k+1 + (1 ω) η k, where ω (0,1] i a relaxation parameter. At tep 2. we have highlighted the dependence o V and Q on η k+1 m. We monitor the reidual o condition (3.1) 2 and (3.2) 2 and top the iteration when uch reidual are below a precribed tolerance. At each iteration o the previou algorithm, the luid and tructure ubproblem have to be olved with a proper trategy to handle the non-linearitie, uch a with Picard iteration or the luid and Newton iteration or the tructure. Algorithm 1 ha a double-loop nature and, according to the notation introduced in Section 2.1, it will denoted in what ollow a GP -C Approximate-Newton-baed cheme. We preent here the prototype o uch amily o cheme, which combine an approximate-newton cheme or the monolithic FSI problem with Robin-Robin ubiteration or the linearized problem. Thi i given by the ollowing Algorithm 2. GC -P cheme [External loop - index k]. Given the olution at iteration k, olve until convergence 1. The harmonic extenion { H η k+1 m + γ m λ k+1 m = 0 in (V m ), γ m η k+1 m = γ η k (3.3) on Σ 0, obtaining the new luid domain and luid domain velocity; 2. The linearized FSI problem. For it olution, we conider the ollowing partitioned algorithm: [Internal loop - index l] Given the olution at ubiteration l 1, olve at current ubiteration l until convergence (a) The luid ubproblem with a Robin condition at the FS interace v F(u k uk+1 m )v k+1,l + γ k+1 λ,l = G in ( V (η k+1 m ) Q (η k+1 m ) ), α γ v k+1,l + λ k+1 λ k+1,l 1 on Σ k+1 ; (3.4) (b) The tructure ubproblem with a Robin condition at the FS interace,l = α γ D p η k+1,l 1 η S( η k )δ η k+1,l + γ δ λ k+1,l = G S( η k ) γ λ k in (V ), (c) Relaxation tep D p η k+1,l α γ k+1 λ,l = α γ ṽ k+1 k+1,l λ,l on Σ 0 ; η k+1,l = ω η k+1,l + (1 ω) η k+1,l 1, 8 (3.5)

11 where v F i the Oeen approximation o v F with the convective term highlighted in the bracket. Such algorithm i obtained by applying the approximate-newton method to the monolithic non-linear ytem (2.6), by conidering a approximation o the Jacobian H γ m γ m γ um F v F γ β α γ I I α,0 γ, β α γ I I α,0 γ γ η S the ollowing expreion Ĵ DL = H γ m γ m v F γ β α γ I I α,0 γ β α γ I I α,0 γ γ η S, which neglect the term um F involving the hape derivative and the term γ which couple the luid geometry and the tructure problem [38]. Thi algorithm ha a double-loop nature a Algorithm 1, however in thi cae the phyical interace condition are managed in the internal loop (GC -P cheme). For the external loop, we monitor the reidual o equation (3.3) 2 and the reidual related to the convergence o the non-linear term in the luid and in the tructure ubproblem. For the internal loop, we monitor the reidual o equation (3.4) 2. In any cae, we top the external and internal iteration when the related reidual are below a precribed tolerance. Another cheme o thi amily, obtained by conidering a dierent approximation o the Jacobian, i the o-called Single-loop (GCP ) cheme [23, 33, 30, 38], where all the non-linearitie and interace condition are treated in the ame loop. Such algorithm i obtained by applying the approximate-newton method to the monolithic non-linear ytem (2.6), by conidering the ollowing approximate Jacobian Ĵ SL = H γ m γ m v F γ α γ I. β α γ I I α,0 γ γ η S. Single-loop cheme i alo obtained by Algorithm 2 by perorming jut 1 internal iteration, however monitoring the reidual o equation (3.4) Fixed-point-baed cheme. In thi ection we preent new algorithm, irtly propoed in [42], to olve the coupled FSI problem The ixed-point problem. We tart rom the Lagrange multiplier ormulation (2.6) and we rewrite it a a ixed-point problem over the interace poition. 9

12 To thi aim, we introduce the variable ξ := γ η, that repreent the olid diplacement at the FS interace. Moreover, we deine the ollowing operator: - The harmonic extenion operator deined a ollow H : H 1/2 ( Σ) Ṽ m Ṽ m, ( η m,ũ m ) = H ξ, ( η m,ũ m ) = H ξ : H η m + γ m λ m = 0 in (Ṽ m), γ m η m = ξ on Σ, ũ m = D p η m in Ω ; - The operator that repreent the interaction between the luid and the olid problem in a known given luid domain, FS : V m V m Ṽ, η = FS (η m,u m ). Given η m and u m, thi operator i deined a ollow F(v,u m ) + γ λ = G in (V (η m ) Q (η m )), η = FS α (η m,u m ) : γ ṽ + λ D = α p eη γ λ on Σ, D α p eη γ + λ = α γ ṽ λ on Σ, S( η ) + γ λ = G in (Ṽ ). We can now introduce a map φ : H 1/2 ( Σ) H 1/2 ( Σ), deined a φ := γ FS (H H ( ξ )), ξ ( η m,ũ m ) FS eγ η ξ, }{{} φ and then write problem (2.6) a a ixed-point problem: Find ξ uch that ξ = φ( ξ ). (3.6) The numerical algorithm. Problem (3.6) can be olved with a ixedpoint iteration method: Given ξ 0, ξk+1 = φ( ξ k ), k 0. (3.7) Thi iterative algorithm written in extended orm read a ollow: Given the olution ξ k at iteration k, olve at the current iteration k + 1 until convergence 1. The luid geometric problem { H η k+1 m + γ m λ k+1 m = 0 in (Ṽ m), γ m η k+1 m = ξ k on Σ, obtaining the new luid domain and luid domain velocity; 10

13 2. The non-linear FSI problem deined in a known luid domain F(v k+1,u k+1 m ) + γ λ = G in (V (η k+1 m ) Q (η k+1 m )), α γ ṽ k+1 k+1 D + λ = α p eη γ k+1 on Σ, D α p eη γ k+1 + S( η k+1 ) + γ λ k+1 λ k+1 k+1 λ = α γ ṽ k+1 on Σ, k+1 λ = G in (Ṽ ) ; (3.8) 3. The olid diplacement i then retricted to the interace Σ and updated, in cae, with a relaxation tep ξk+1 = ω G γ η k+1 + (1 ω G ) ξ k, where ω G (0,1] i a relaxation parameter. The econd tep o the previou algorithm (problem (3.8)) i a coupled FSI problem olved in a known luid domain (obtained thank to η k+1 m ), but where the contitutive non-linearitie are till preent. Thereore, to olve thi problem we have to manage in an internal loop both uch non-linearitie and the phyical interace condition. To do thi, we conider an approximate-newton method. Dierent approximate Jacobian lead to dierent algorithm, which are preented in what ollow. 1. Uing a ingle internal loop - G -CP cheme. In thi cae, we apply the approximate-newton method to ytem (3.8), with the ollowing approximation o the Jacobian P 1 = Thi cheme then read a ollow v F γ α γ I β α γ I I α,0 γ γ η S. (3.9) Algorithm 3. G -CP cheme [External loop - index k]. Given the olution ξ k at iteration k, olve at the current iteration k + 1 until convergence 1. The luid geometric problem { H η k+1 m + γ m λ k+1 m = 0 in (Ṽ m), γ m η k+1 m = ξ k on Σ; (3.10) obtaining the new luid domain and luid domain velocity. 2. The FSI problem in a known given luid problem. For it olution, we conider the ollowing approximate-newton-baed partitioned algorithm: [Internal loop - index j] Given the olution at ubiteration j 1, olve at the current ubiteration j until convergence (a) The luid ubproblem with a Robin condition at the FS interace { v F(u k+1,j 1 u k+1 m )v k+1,j α γ v k+1,j 11 + γ k+1,j λ = G in (V (η k+1 m ) Q (η k+1 m )), + λ k+1,j = α γ D p η k+1,j 1 λ k+1,j 1 on Σ k+1, (3.11)

14 (b) The tructure ubproblem with a Robin condition at the FS interace η S( η k+1,j 1 )δ η k+1,j + γ δ λ k+1,j = G S( η k+1,j 1 ) γ (c) Relaxation tep D p η k+1,j α γ η k+1,j = ω P η k+1,j k+1,j λ = α γ ṽ k+1,j k+1,j λ + (1 ω P ) η k+1,j 1, where ω P (0,1] i a relaxation parameter. 3. The olid diplacement i then retricted to the interace Σ λ k+1,j 1 in (Ṽ ), on Σ. ξk+1 = ω G γ η k+1 + (1 ω G ) ξ k, where ω G (0,1] i a relaxation parameter. To top the external iteration, we monitor the reidual o condition (3.10) 2, while to top the internal iteration we monitor the reidual o condition (3.11) 2 and the reidual related to the convergence o the non-linear term in the luid and in the tructure ubproblem. Remark 1. G -CP algorithm ha a double loop tructure, a GC -P decribed in Algorithm 2. The dierence with that algorithm conit in the act that there the tructure Jacobian wa updated jut at each external iteration, while here it i updated at each internal iteration. 2. Uing two neted internal loop - G -C -P cheme. In thi cae, we conider two neted loop to olve the FSI problem (3.8): an intermediate one to manage the contitutive non-linearitie and an internal one to precribe the phyical interace condition. Thi correpond to ue v F P 2 = γ β α γ I I α,0 γ β α γ I I α,0 γ γ η S a approximate Jacobian or the approximate-newton method applied to problem (3.8). At each approximate-newton iteration, we have a ully linearized FSI problem. Thi can be olved with a block-gau-seidel preconditioner which ha ormally the ame expreion o (3.9), but where the tructure Jacobian i built dierently, a it will be clear by Remark 2. We have then the ollowing Algorithm 4. G -C -P cheme [External loop - index k]. Given the olution ξ k at iteration k, olve at the current iteration k + 1 until convergence 1. The luid geometric problem { H η k+1 m + γ m λ k+1 m = 0 in (Ṽ m), γ m η k+1 m = ξ k on Σ; (3.12), obtaining the new luid domain and luid domain velocity. 12

15 2. The FSI problem in a known given domain. For it linearization, we conider the ollowing approximate-newton iteration: [Intermediate loop - index j] Given the olution at ubiteration j 1, olve at the current ubiteration j until convergence v F(u k+1,j 1 u k+1 m α γ v k+1,j + λ k+1,j D p η k+1,j α γ η S( η k+1,j 1 )v k+1,j D = α γ p η k+1,j λ k+1,j )δ η k+1,j + γ k+1,j λ = G in (V ( η k+1 m ) Q ( η k+1 m )), = α γ ṽ k+1,j + γ δ λ k+1,j = λ k+1,j on Σ k+1, G S( η k+1,j 1 λ k+1,j ) γ on Σ, λ k+1,j 1 in (Ṽ ). (3.13) At each iteration o the intermediate loop thi problem i till coupled through the phyical interace condition. For thi reaon we conider a Robin-Robin partitioned algorithm or it olution: [Internal loop - index l] Given the olution at ubiteration l 1, olve at the current ubiteration l until convergence (a) The luid ubproblem with a Robin condition at the FS interace v F(u k+1,j 1 u k+1 m α γ v k+1,j,l + λ k+1,j,l )v k+1,j,l = α γ D p η k+1,j,l 1 + γ k+1,j λ,l = G in λ k+1,j,l 1 on Σ k+1, ( V ( η k+1 m ) Q ( η k+1 m )), (b) The tructure ubproblem with a Robin condition at the FS interace η S( η k+1,j 1 )δ η k+1,j,l + γ δ λ k+1,j,l D p η k+1,j,l α γ (c) Relaxation tep η k+1,j,l λ k+1,j,l = α γ ṽ k+1,j,l = ω P η k+1,j,l = G S( η k+1,j 1 ) γ k+1,j 1 λ in (Ṽ ), λ k+1,j,l + (1 ω P ) η k+1,j,l 1, where ω P (0,1] i a relaxation parameter. 3. The olid diplacement i then retricted to the interace Σ ξk+1 = ω G γ η k+1 + (1 ω G ) ξ k, (3.14) on Σ. where ω G (0,1] i a relaxation parameter. To top the external iteration, we monitor the reidual o condition (3.12) 2. Regarding the intermediate iteration, we monitor the reidual related to the convergence o the non-linear term in the luid and in the tructure ubproblem. Finally, to top the internal iteration we monitor the reidual o condition (3.13) 2. Remark 2. G -C -P algorithm ha a triple loop nature. We oberve that the tructure Jacobian i updated at each intermediate iteration, by evaluating it or 13

16 a tructure diplacement ( η k+1,j 1 ) which atiie exactly the phyical interace condition, dierently rom that ued to update the tructure Jacobian in G -CP algorithm, which doe not atiy them. 3. Uing two neted internal loop - G -P -C cheme. Thi cheme i obtained by exchanging the order o the loop in G -C -P cheme, that i by treating the contitutive non-linearitie in the internal one. Even i we preented uch algorithm a a ixed-point-baed cheme, in the numerical reult we will conider it a a claical one, due to it implementation baed on olving in an iterative ramework the nonlinear luid and tructure ubproblem. For the ake o brevity, we do not report here the detailed decription o thi algorithm An eicient choice o the internal tolerance. In the algorithm preented in previou ection, whenever Newton or approximate-newton iteration are conidered, the linear ytem involved at each iteration do not need to be olved until convergence when an iterative method i conidered. Indeed, a oberved or example in [28], it i enough to top the internal iteration when the reidual i below a tolerance which i proportional to the Newton reidual. Thi lead to a great aving in the computational time, without aecting the accuracy, ince at convergence o the Newton iteration the tolerance o the internal linear ytem ha become uiciently low. It i then poible to apply uch idea to our cae, in particular to GC -P and G -C -P. In both cae, the FSI linear ytem ariing at each Newton iteration (tep 2. in Algorithm 2 and tep b. in Algorithm 4, repectively) doe not need to be olved until convergence. Thi mean that the phyical interace condition are in act not atiied at each approximate-newton iteration. However, at convergence o the approximate-newton loop, they are atiied, o that thee cheme are in act exact. In [28] uch trategy i reerred to a inexact-newton. However, in order to avoid conuion with the inexact cheme preented in Section 4, we name thee algorithm exact cheme with dynamic tolerance and we add the uix DT at the end o the name. In what ollow, we detail GC -P -DT cheme. Algorithm 5. GC -P -DT cheme [External loop - index k]. Given the olution at iteration k, olve until convergence 1. The harmonic extenion { H η k+1 m + γ m λ k+1 m = 0 in (V m ), γ m η k+1 m = γ η k (3.15) on Σ 0, obtaining the new luid domain and luid domain velocity; 2. The linearized FSI problem. For it olution, given the external reidual R k+1 := γ η k+1 γ η k X + ((u k+1 u k ) )u k+1 W + G S( η k+1 ) γ k+1 λ K, we conider the ollowing partitioned algorithm: [Internal loop - index l] Given the olution at ubiteration l 1 and a uitable calar σ k+1, olve at current ubiteration l until α β ( ) 0 γ η k+1,l γ η k+1 k+1 k+1,l 1 + λ,l λ,l 1 σ k+1 R k+1, Z 14

17 (a) The luid ubproblem with a Robin condition at the FS interace v F(u k,l uk+1 m )v k+1,l + γ k+1 λ,l = G in ( V (η k+1 m ) Q (η k+1 m ) ), α γ v k+1,l + λ k+1 D p η,l = α γ k,l 1 λ k,l 1 on Σ k+1 ; (3.16) (b) The tructure ubproblem with a Robin condition at the FS interace η S( η k,l)δ η k+1,l + γ δ λ k+1,l = G S( η k ) γ λ k in (V ), (c) Relaxation tep D p η k+1,l α γ k+1 λ,l = α γ ṽ k+1 k+1,l λ,l on Σ 0 ; η k+1,l = ω η k+1,l + (1 ω) η k+1,l 1. (3.17) For the choice o σ k we ollow [28]. In particular, we et σ max ( k = 0, σ k = min σ max,γ ( R k /R k 1) ) 2 k > 0, γ(σ k 1 ) 2 0.1, ( min σ max,max (γ ( R k /R k 1) )) 2,γ(σ k 1 ) 2 k > 0, γ(σ k 1 ) 2 > 0.1. (3.18) In the numerical imulation preented in thi work we have ued σ max = and γ = 0.9. In the computation o the reidual, X, W, Z, K are uitable Sobolev pace. In particular, the right choice i X = H 1/2 (Σ 0 ), W = H 1 (Ω ), Z = H 1/2 (Σ 0 ), K = H 1 (Ω ). However, due to the complexity in the computation o thee norm, in practical implementation we conidered W = L 2 (Ω ), K = L 2 (Ω ) and X = Z = L 2 (Σ 0 ). Remark 3. In [28] it ha been hown that the choice (3.18) guarantee a econd order convergence when the exact Newton i conidered. For approximate-newton trategie, a in our cae, thi choice allow to recover irt order o convergence Numerical reult or exact cheme Generalitie. In all the numerical experiment o thi work, we conidered the nearly incompreible exponential material whoe irt Piola-Kirchho tenor read T (F ) = GJ 2/3 whoe related energy i given by W(F ) = G 2γ ( F 1 ) 3 tr(f T F )F T e γ(j 2 3 tr(f T F ) 3) + κ ( J ) ln(j ) J F T, 2 J (3.19) ( ) e 2 γ(j 3 tr(f T F ) 3) 1 + κ 4 ( (J 1) 2 + (lnj ) 2). Here F := x 0 x t, with x 0 the coordinate in the reerence coniguration and x t thoe in the current coniguration, J := det(f ), κ i the bulk modulu and G the hear modulu. For mall deormation uch material behave a a linear tructure decribed by the ininiteimal elaticity, characterized by a Poion ratio ν and a Young modulu E related to κ and G a ollow κ = E 3(1 2ν), G = E 2(1 + ν). 15

18 The parameter γ characterize the tine o the material or large diplacement. Moreover, we ued P1bubble P1 inite element or the luid ubproblem and P1 inite element or the tructure ubproblem, and the ollowing data: inal time T = 0.4, vicoity µ = 0.03g/(cm), luid denity ρ = 1g/cm 3, tructure denity ρ = 1.2g/cm 3, bulk modulu κ = 10 7 dyne/cm 2, hear modulu G = dyne/cm 2 (correponding or mall diplacement to Young modulu E = dyne/cm 2 and Poion ratio ν = 0.45), γ = 1. Moreover, i not otherwie peciied, we ued a time dicretization parameter = For the precription o the interace continuity condition, in all the imulation we have conidered the Robin-Robin (RR) cheme [3, 4], with the optimized coeicient propoed in [22] and adapted to the variou temporal cheme in [38]. To compute the optimal α we have ued the value o E = dyne/cm 2. In all the imulation o thi work, RR cheme ha converged without any relaxation, conirming it uitability or haemodynamic application. The numerical experiment have been perormed with the parallel Finite Element library LIFEV ( ee [38] or detail Eiciency o exact cheme in a real tet cae. In all the imulation o thi ection and o Section 4.4 we conidered the computational domain depicted in Figure 2.1, repreenting the real carotid o a patient, ater the removal o a plaque. The veel lumen ha been recontructed by uing the code VMTK (ee while the tructure geometry ha been obtained by extruion, by etting the ratio between the lumen radiu and the thickne equal to The number o degree o reedom i or the luid domain and or the tructure, and the luid and tructure mehe are conorming at the interace. For the harmonic extenion and or the tructure, we precribed at the artiicial ection normal homogeneou Dirichlet condition and tangential homogeneou Neumann condition, that i we let the domain move reely in the tangential direction. At the luid inlet we impoed the patient-peciic low rate depicted in Figure 3.1, meaured by mean o the Eco- Color Doppler technique and precribed through the Lagrange multiplier method (ee [15, 46, 18, 19]). At the luid outlet, we ued an aborbing reitance boundary low rate [cm 3 /] time [] Fig Patient-peciic low rate waveorm precribed at the inlet o the carotid. condition, ee [39, 38] or detail. At the external urace Σ 0 out we precribed a Robin 16

19 boundary condition with Robin coeicient α e with the aim o modeling the preence o a urrounding tiue around the veel [32, 35, 9, 38]. In particular, we et α e = dyne/cm 3. Thi value allow to recover a preure in the phyiological range. A a repreentative cae, we reported in Figure 3.2 a naphot o the treamline obtained with GC -P cheme and BDF1/BDF1 time dicretization. Fig Streamline o the velocity ield at itole (0.31, let) and at diatole (0.80, right). GC -P - BDF1/BDF1. In Table 3.1 we reported the number o iteration or dierent exact cheme. The number o external iteration reported in the table ha to be intended a an average one over the period [0,T], whilt the intermediate and the internal one a the average per outer loop (the external and the intermediate one, repectively). We reported alo the CPU time normalized over that o GC -P -DT cheme, ued here a our gold-tandard. # o external # o intermediate # o internal Normalized iteration iteration iteration CPU time GC -P GC -P -DT GCP G -CP G -C -P G -P -C GP -C Table 3.1 Average number o iteration in the external loop and average number o iteration per outer loop in the intermediate and internal one, and CPU time normalized with repect to that o GC - P -DT cheme. Exact cheme. BDF1/BDF1. Dicuion o the numerical reult. The reult reported in Table 3.1 how that 17

20 the approximate-newton-baed cheme are the mot eicient among exact method. In particular, GC -P cheme i lightly ater than GCP. We alo ran the propoed numerical experiment with GC -P -DT cheme, decribed in Algorithm 5. A oberved in Table 3.1, the CPU time needed by GC -P -DT have been almot halved with repect to GC -P. Regarding the ixed-point-baed cheme, they eem to be quite lower than approximate-newton method. The bet perormance have been obtained by G C - P cheme, the CPU time being le than two time greater than or GC -P. We point out that in any cae, ixed-point-baed cheme converged without any relaxation (ω G = 1). We ran G -C -P cheme alo with an Aitken relaxation procedure [26] over (3.14), with the hope o improving the eiciency. However, we ound that the CPU time normalized with repect to that o GC -P -DT i 2.90, againt 3.05 or the cae ω = 1, o that no ubtantial improvement i oberved with the Aitken procedure. We did not conider G -C -P -DT cheme, due to the wore perormance o G -C -P with repect to GC -P. Concerning the claical cheme, they howed a very poor eiciency in comparion to approximate-newton, their CPU time being more than our time greater with repect to that o GC -P -DT. They are alo lower than the ixed-point-baed method. Such cheme are however the mot appealing rom the computational point o view, when one ha at dipoal two black-box olver or the luid problem in ALE ormulation and or the tructure, ince they need jut to implement uitable routine or the traner o the interace condition between the two code. Intead, approximate-newton-baed and ixed-point-baed algorithm can be implemented in a modular way provided that one can acce to the luid and tructure tangent problem (alway poible by running jut 1 Newton internal iteration). In concluion, we ugget GC -P -DT a the mot uitable among exact cheme or real haemodynamic application. 4. Inexact cheme. Here, we want to extend to the cae o the inite elaticity the emi-implicit cheme [43, 12, 6, 39, 9] and, more generally, the geometrical inexact cheme [38]. A irt way to do thi, conit in conidering the claical cheme G -P -C and to perorm jut one (or ew) external iteration over the interace poition [35]. In thi cae the phyical interace condition and the contitutive nonlinearitie are both treated exactly. Here, we want to introduce a dierent amily o inexact cheme, where, beide the geometrical interace condition, alo the luid and tructure contitutive non-linearitie are not precribed exactly. In other word, we ak i it i neceary in haemodynamic application to handle exactly the contitutive non-linearitie, in particular the tructure one. In Section 4.1 we conidered the inexact verion o the approximate-newton-baed cheme, and in Section 4.2 the inexact verion o the ixed-point-baed cheme. To tudy the accuracy, we conidered both an analytical tet cae in Section 4.3 and a real tet cae in Section 4.4. In the latter ection, we alo tudy the eiciency o the inexact cheme in a real context Approximate-Newton-baed inexact cheme. The tarting point i the obervation that in the cae o the linear ininiteimal elaticity, emi-implicit cheme can be regarded a GC -P cheme where the number o external iteration i ixed and equal to 1. By perorming jut one external iteration alo in preence o the inite elaticity, we obtain a cheme where alo or the contitutive non-linearitie jut one iteration i perormed (I-GC1-P ). More in general, it i poible to perorm m external iteration or a ixed m > 1, obtaining the I-GCm-P cheme. For uch cheme, the topping criteria on the geometrical condition and on the contitutive 18

21 non-linearitie are not checked, o that they are in principle inexact alo with repect to the contitutive non-linearitie. Thi act make very intereting the tudy o the accuracy o uch cheme, ince there i no a priori evidence that the luid and, epecially, the tructure problem need to be olved exactly in order to recover a global accurate olution Fixed-point-baed inexact cheme. We introduce here the inexact variant o the ixed-point-baed cheme introduced in Section 3.3. The philoophy i the ame ued to derive the inexact cheme rom the approximate-newton-baed algorithm, that i to perorm a ixed number o iteration in the external loop. We can derive two group o inexact algorithm, one rom G -CP cheme and one rom G -C -P cheme. In any cae, a or the approximate-newton-baed inexact algorithm, the phyical interace condition are atiied exactly, due to the high added ma eect in haemodynamic. In the irt cae we obtain I-Gm-CP cheme, derived rom Algorithm 3 by perorming jut m external iteration. Thi cheme, dierently rom I-GCm-P, olve exactly the contitutive non-linearitie, and only the geometrical interace condition i not precribed correctly. Since we are here intereted in the accuracy o cheme which do not olve exactly the contitutive non-linearitie, we do not conider uch cheme in the ollowing numerical experiment. The econd group o inexact cheme i derived rom G -C -P cheme. In thi cae, they are obtained by conidering jut m iteration in the external loop and r iteration in the intermediate loop (in principle, alo the cae r = could be allowed, but it i not conidered here). We obtain I-Gm-Cr-P cheme, derived rom Algorithm 4 by perorming jut m external iteration and r intermediate iteration. Such cheme, a or I-GCm-P cheme, treat inexactly both the geometrical interace condition and the contitutive non-linearitie Numerical reult or inexact cheme: Convergence with repect to time. We conider the ame analytical tet cae propoed in [38] or the linear ininiteimal elaticity. Thi tet conit in a tranlation o a cylinder o mall thickne (the tructure) illed by the luid and in a rotation around it axi with no volume orce. Reerring to the ame data reported in [38], it i eay to check that the analytical olution o the FSI problem i given by { u = ū in Ω t, p = 0 in Ω t, where η := η = η in Ω 0, η m = η in Ω 0, x 0,1(co θ 1) x 0,2 in θ + c 1, x 0,1 in θ + x 0,2(co θ 1) + c 2, c 3,, ū := θ(c 2 x,2 ) + ċ 1, θ(x,1 c 1 ) + ċ 2, ċ 3, or given unction o time θ(t) and c(t). We oberve that with repect to the analytical olution propoed in [38], here the preure i identically zero. We conidered, in particular, the cylindrical geometry depicted in Figure 4.1, where the length i L = 5cm, the luid domain radiu R = 0.5cm, the tructure thickne H = 0.1cm. The pace dicretization parameter i h = 0.025cm and the luid and tructure mehe are conorming at the interace. The meh i compoed o about degree o reedom or the luid and about 6000 or the tructure. For 19

22 Fig Cylindrical geometry. what concern the data o the tet, we have et c = 0 and θ(t) = 0.2(1 co(50π t)). We ran all the imulation on 4 proceor or the olution o the luid problem and on 1 proceor or the tructure. In Figure 4.2 we how the convergence hitory o our elected inexact cheme, namely I-GC1-P, I-GC2-P, I-G1-C2-P and I-G2-C2-P, choen a the mot repreentative, or three elected temporal cheme, namely BDF1/BDF1, BDF2/BDF2 and BDF3/BDF3. A relative L 2 norm o the error i computed at time t = The time dicretization parameter i = , 10 3, , For BDF2/BDF2 and BDF3/BDF3 cheme, I-GC1-P and I-G1-C2-P eatured jut irt order convergence, o that we have conidered in thee cae alo the extrapolated verion I-GC1E-P and I-G1E-C2-P. Dicuion o the numerical reult. From the convergence rate depicted in Figure 4.2, we oberve that when BDF1/BDF1 i ued, all the our inexact cheme conidered recovered irt order convergence without any extrapolation o the interace poition, luid velocity and tructure diplacement. Regarding BDF2/BDF2 and BDF3/BDF3, we oberve that cheme which perorm two iteration in the loop related to the geometrical condition and to the contitutive non-linearitie (I-GC2-P and I-G2-C2-P ) eatured econd and third order convergence, repectively, without any extrapolation. For the other two cheme (I-GC1E-P and I-G1E-C2-P ) an extrapolation o order two and three, repectively, ha been needed in order to recover the right convergence order. Thi reult how that, at leat or the analytical tet cae, it i not needed to olve exactly the contitutive non-linearitie to recover an accurate olution Numerical reult or inexact cheme: Eiciency and accuracy in a real tet cae. In thi ection we report the numerical reult obtained or the ame tet cae preented in Section 3.5.2, by uing the our inexact cheme conidered above. Thi allowed to tudy the accuracy and the eiciency o uch cheme in a real context. In Table 4.1 and 4.2 we report the relative error o the inexact cheme by uing the olution obtained with GC -P cheme a the reerence one. In partic- 20

23 relative luid velocity error I-GC1-P I-GC2-P I-G1-C2-P I-G2-C2-P t relative luid preure error t I-GC1-P I-GC2-P I-G1-C2-P I-G2-C2-P relative olid diplacement error t I-GC1-P I-GC2-P I-G1-C2-P I-G2-C2-P (a) BDF1/BDF1 relative luid velocity error t I-GC1E-P I-GC2-P I-G1E-C2-P I-G2-C2-P relative luid preure error t I-GC1E-P I-GC2-P I-G1E-C2-P I-G2-C2-P relative olid diplacement error t I-GC1E-P I-GC2-P I-G1E-C2-P I-G2-C2-P (b) BDF2/BDF2 relative luid velocity error t 3 I-GC1E-P I-GC2-P I-G1E-C2-P I-G2-C2-P relative luid preure error t I-GC1E-P I-GC2-P I-G1E-C2-P I-G2-C2-P relative olid diplacement error t I-GC1E-P I-GC2-P I-G1E-C2-P I-G2-C2-P (c) BDF3/BDF3 Fig Convergence rate o three temporal cheme conidered. Relative error o the luid velocity (let), o the preure (middle) and o the tructure diplacement (right) - BDF1/BDF1 (up), BDF2/BDF2 (middle), BDF3/BDF3 (bottom) - t = ular, we report the L (L )-norm o average quantitie, namely the mean tructure diplacement η, the low rate Q and the mean preure P computed over ection perpendicular to the axial axi. To do thi, we computed quantitie a max j x j EX xj L (0,T) max j x j EX, (4.1) L (0,T) where x j i one o the average quantitie computed at dierent ection Σ j orthogonal to the axial direction, EX tand or olution computed with the exact cheme and * tand or one o the inexact cheme. The reult reported in Table 4.1 reer to BDF1/BDF1, while thoe in Table 4.2 to BDF2/BDF2. 21

24 η (%) Q (%) P(%) I-GC1-P I-GC2-P I-G1-C2-P I-G2-C2-P Table 4.1 Relative error o inexact cheme with repect to the exact olution, computed with (4.1). BDF1/BDF1. Let: diplacement. Middle: low rate. Right: mean preure. η (%) Q (%) P(%) I-GC1-P I-GC1E-P I-GC2-P I-G1-C2-P I-G1E-C2-P I-G2-C2-P Table 4.2 Relative error o inexact cheme with repect to the exact olution, computed with (4.1). BDF2/BDF2. Let: diplacement. Middle: low rate. Right: mean preure. In Table 4.3 we report the number o iteration or BDF2/BDF2. In particular, the number o iteration in the intermediate and in the internal loop ha to be intended a the average per outer loop. We alo report the CPU time normalized over that o GC -P -DT cheme, that i the atet among the exact cheme. # o external # o intermediate # o internal Normalized iteration iteration iteration CPU time I-GC1-P I-GC1E-P I-GC2-P I-G1-C2-P I-G1E-C2-P I-G2-C2-P Table 4.3 Average number o iteration per outer loop in the intermediate and internal one, and CPU time normalized with repect to that o GC -P -DT cheme. Inexact cheme. BDF2/BDF2. Dicuion o the numerical reult. The reult reported in Table 4.1 and 4.2 how that the relative error o inexact cheme with repect to the olution obtained with an exact cheme are in any cae le than 1%. In particular, the accuracy improve o one order o magnitude by perorming jut one external iteration with a uitable extrapolation in the cae o BDF2/BDF2, and o two order o magnitude by perorming two external iteration intead o one both or BDF1/BDF1 and or BDF2/BDF2, Thee reult how that alo in real application, an inexact treatment o the contitutive non-linearitie and o the geometrical interace condition i uicient to recover a atiactory olution or practical purpoe. 22