Fourth International Conference on Advanced COmputational Methods in ENgineering (ACOMEN 2008) Editors: M. Hogge, R. Van Keer, L. Noels, L. Stainier, J.-P. Ponthot, J.-F. Remacle, E. Dick

©University of Liège, Belgium, 26-28 May 2008

Use of Generalized-α Methods for Interfield Parallel Integrationof Heterogeneous Structural Dynamic Systems

O.S. Bursi1 , L. He1, A. Bonelli1 and P. Pegon2

1 Dipartimento di Ingegneria Meccanica e Strutturale, Universita di TrentoVia Mesiano 77, 38050, Trento, Italy

2 ELSA Laboratory, IPSC, JRC, European Commission21020 Ispra (Varese), Italy

e-mails: Oreste.BursiLeqia.He,Alessio.Bonelli@ing.unitn.it, pierre.pegon@jrc.it

Abstract

A novel partitioned algorithm able to solve ODEs arising from transient structural dynamicsis presented. The spatial domain is partitioned into a set of disconnected subdomains owing tocomputational or physical considerations; and continuity conditions of the velocity at the interfaceare modelled using a dual Schur formulation, where Lagrange multipliers represent reaction forces.Interface equations along with subdomain equations lead to a system of DAEs for which a a parallelinterfield procedure is developed. The algorithm first solves interface Lagrange multipliers, that aresubsequently used to advance the solution in subdomains. The proposed coupling algorithm thatenables arbitrary Generalized-α schemes to be coupled with different time steps in each subdomainis an extension of a method originally proposed by Pegon and Magonette. Thus, subcycling to dealwith stiff and nonstiff subsystems is allowed. In detail, the paper presents the convergence analysisof the interfield parallel scheme for a linear single-degree-of-freedom system as a multi-degree-of-freedom system is too difficult to analyse mathematically. However, the insight gained from theanalysis of this coupled problem and the conclusions drawn are confirmed by means of numericalexperiments on a four-degrees-of-freedom system.

1 Introduction

Systems of ODEs arising from transient structural dynamics very often exhibit high-frequency/low-frequency and stiff/nonstiff behaviour of subsets of state variables. Hence, both Runge-Kutta (RK) andLinear Multistep (LMS) algorithms integrating all state variables present limitations. In this respect, themajority of researchers have employed one of two basic approaches to solving the problem. The firstapproach can be called multirate integration, and it is primarily applicable to systems in which statevariables can be divided into high-frequency and low-frequency subsets. In this approach, stepsizes ofintegrators for different subsets of variables are different, while the integrator itself may be the same.See, among others, [1, 2]. The second approach, called multi-method approach, is applied primarilyto systems in which state variables can be partitioned into stiff and nonstiff subsets owing to computa-tional or physical considerations. In this case, step sizes of integrators for different subsets of variablesare the same, but different types of integrators are used for different subsets. See among others, thework of Weiner [3]. Nonetheless, we can identify methods where these two approaches are combined.For instance within the framework of partitioned RK integration, Weiner et al. [3] presented an algo-rithm that automatically selects stiff components in subintervals, i.e. splits the system of ODEs locallyinto stiff and nonstiff subsystems. They claimed that it works particularly well when stiff and nonstiffcomponents are weakly coupled. As a variant, Shome et al. [4] presented a method in which two dif-ferent RK integrators were employed for two subsets of variables, one of which was a stabilized RKmethod. In this algorithm, they used the fundamental assumption that linearized system eigenvalues

corresponding to the two subsystems are widely separated. In the framework of RK and LMS meth-ods Arnold et al. [5] suggested the use of different time integrators for subsystems, in order to tailoreach method to the solution behaviour of the corresponding subsystem, a so-called co-simulation. Thecommunication between subsystems was restricted to discrete synchronization points and required inter-polation/extrapolation owing to the use of different time steps. They stated that co-simulation techniquesmay suffer from numerical instability, that is further exacerbated by discretization errors introduced byinterpolation/extrapolation. Thus, several modifications of co-simulation techniques [6, 7] were made toimprove accuracy and stability also for larger coarse steps.

In transient structural dynamics, partitioned methods mainly rely on domain decomposition tech-niques and LMS methods [8]. Based on Schur complements to split the coupled mechanical systemsinto subsystems, these partitioned analysis procedures create a coarse problem with a reduced number ofunknowns by the elimination of internal subsystems unknowns. Only the original, i.e. primal unknowns-displacements, velocities or accelerations- are considered in the computation. Moreover, the develop-ment of parallel algorithms has been motivated, emphasizing the communication and synchronizationbetween subsystems in view of the improvement of computational performance. A distinct feature ofthese parallel algorithms is the use of dual unknowns, such as Lagrange multipliers, to enforce the con-tinuity between subsystems. In this paper, we consider a dual Schur domain decomposition [9]. Bypartitioning a structural domain Ω into sd subdomains, the following semi-discrete dynamic system ofequations of motion is obtained:

Miui(t) + Ciui(t) + Kiui(t) = Fie(t) + LiT Λ ∀i ∈ 1, · · · , sd, (1)

sd∑

i=1

Liui(t) = 0 (2)

where Mi, Ci and Ki are the subdomain mass, damping and stiffness matrices, respectively, Fie(t) is the

vector of applied loads on the i subdomain, ui(t) is the i subdomain displacement vector and superim-posed dots indicate time differentiation, Li are the constraint matrices that express a linear relationshipon the connected boundaries and Λ is the vector of Lagrange multipliers. The associated initial valueproblem consists in determining the function ui = ui(t) fulfilling (1) and (2) for all t ∈ [0, tf ], tf > 0,for given initial conditions ui(0) = di

0 and u(0) = vi0. In detail, we consider in (2) only the velocity

continuity on interfaces [9]. (1) and (2) result in a system of differential-algebraic equations (DAEs).By means of a general approach Gravouil and Combescure solved (1)-(2) with a structural integrator,

i.e. the Newmark scheme [10], thus obtaining a multi-time-step explicit-implicit method [9], hereafterreferred to as the GC method. In detail, they found that the GC method possesses a conservation lawand therefore it is spectrally stable [9]. The GC method is very appealing for heterogeneous (numeri-cal/physical) subsystems, as different implicit/explicit Newmark schemes with different time steps canbe used according to their complexity and characteristics. Nonetheless, the staggered solution procedureof the GC method can be considered as a drawback in real-time or parallel applications. In order to solvethis issue, Pegon and Magonette proposed an interfield parallel solution procedure to the GC method,that led to a new method, the PM method. The favourable convergence properties of this method havebeen analysed in [11].

Among the LMS methods, the Generalized-α methods [12], in brief the G-α methods, include mostof the popular structural integrators as special cases. In detail, the G-α method is well-known for its op-timal performances being able to control numerical dissipation and to damp out spurious high-frequencycomponents of the response without affecting too much low-frequency components. Moreover, it alwaysexhibits a second-order accuracy. By considering a non-partitioned dynamic system, the G-α methodexpressed in LMS form reads [13],

3∑

j=0

[Mαjui+j + ∆tCγjui+j ] + ∆t23∑

j=1

δj(Ri+j−αf− Fe,i+j−αf

) = 0 (3)

Figure 1: Comparison of G-α (dotted lines) and Optimal Blended Lobatto 2-stage RK (solid lines)method after [14].

Figure 2: A coupled problem with two subdomains A and B: a) a staggered procedure; b) an interfieldparallel procedure.

where Fe,i+1−αf= Fe[(1−αf )ti+1+αf ti] and Ri+1−αf

= K((1−αf )ui+1+αfui) are approximatedby means of the mid-point rule. Moreover, parameters in (3) read

α0 = αm α1 = 1− 3αm α2 = −2 + 3αm α3 = 1− αm

γ0 = αf (−1 + γ) γ1 = −1 + 2αf + γ − 3γαf γ2 = 1− αf − 2γ + 3γαf γ3 = (1− αf )γδ1 = 1

2 + β − γ δ2 = 12 − 2β + γ δ3 = β

(4)In agreement with Dahlquist’s barriers, it is second-order accurate if γ − 1

2 − (αf − αm) = 0, it canbe L-stable, and optimizes low/high-frequency numerical dissipation with β − 1

4(1 + αf − αm)2 = 0.Recently, Hulbert [14] has shown its good performance with respect to an optimal blended Lobatto2-stage RK method; see in this respect Figure 1, where spectral radii of the methods are compared.

Though a few studies have dealt with partitioned algorithms, there is still a paucity of publica-tions devoted to the possibility of introducing robust LMS algorithms in parallel partitioned algorithms,maintaining the favourable convergence properties of progenitor algorithms. All together, they repre-sents basic aspects of the temporal integration of partitioned algorithms and are the issues that the paperexplores further. In detail in Section 2, we introduce the concepts of staggered and interfield parallelsolution procedures in partitioned integration. In Section 3, we present a new interfield parallel methodthat relies on G-α methods and its relevant convergence properties. Numerical experiments that illustratethe performance of the method are presented in Section 4. Conclusions are drawn in Section 5.

2 Staggered/interfield parallel integration

Usually a coupled problem (1)-(2) can be solved by using either a staggered or an interfield parallel pro-cedure [15]. By using a staggered procedure, analysis of each subdomain is executed in a strictly serialmanner, whereas using an interfield parallel procedure they are executed concurrently. Both proceduresare illustrated in Figure 2 in the case of sd = 2, considering one time step from tn to tn+1. Arrows rep-

Figure 3: The interfield parallel solution procedure of the PM method.

resent the advance of time integration (solid lines) or the exchange of information between subdomains(dashed lines). Computational Steps 1 to 4 are sequential in Figure 2(a); while computational Steps A1,B1 and A2, B2 are respectively concurrent in Figure 2(b).

The GC method [9] as most of available methods, is in essence a sequential staggered algorithmthat follows the procedure depicted in Figure 2(a). Consequently, the process in one subdomain has tosystematically stop in order to wait for the process in the other subdomain. Based on the GC method,Pegon and Magonette [16] developed the interfield parallel PM algorithm, where different subdomainstates advance simultaneously and continuously. Figure 3 illustrates the interfield parallel solution pro-cedure of the PM method. In detail, subdomain A is discretized in time with a coarse time step ∆tA;and subdomain B is integrated with a fine time step ∆tB associated with ∆tA = ss∆tB , being ss thenumber of substeps. In order to send in advance information to subdomain B at the beginning of a coarsetime step, the PM method exploits a time step equal to 2∆tA in subdomain A.

Remark 1 In both the GC and PM methods, the stability of the global problem depends on the stabil-ity conditions of the Newmark schemes considered in each subdomain. Coupling of implicit/explicitmethods has no influence on the spectral stability of each subdomain (see [9, 11]).

Remark 2 Partitioned schemes preserve second-order accuracy with ss = 1 and their rate of conver-gence degrades to first order with ss > 1 owing to coupling, if Newmark schemes of second orderaccuracy are used in each subdomain (see [9, 11]).

3 Novel interfield parallel integration by using G-α methods

3.1 G-α methods with equilibrium enforced at the end of each time step

The PM method together with the Trapezium Rule and the Central Difference is numerically non-dissipative with a common time step and entails numerical dissipation only at interfaces with differenttime steps in different subdomains, that are not controllable by users. Even though the Newmark meth-ods with γ damping can be used in the PM method [10], the algorithm was found to be over-dissipative inthe low frequency range and with results poorly accurate [17]. In order to introduce controllable numer-ical dissipation, we introduced the G-α methods into the PM method. Nonetheless the original implicit[12] and explicit [18] G-α methods, which enforce equilibrium of a system using a balanced equationbetween two consecutive time steps (3), is found to be incompatible with the partitioned problems (1)-(2). In detail, problems arouse with respect to the continuity enforcement via Lagrange multipliers. Forthis reason, we employed the G-α method for the implicit subdomain enforcing the equilibrium at theend of each coarse time step [19]. Further, we proposed an explicit G-α method for the explicit subdo-main, that also enforces equilibrium at the end of each time step [17]. Within the class of LMS methods,differential equations are usually solved by a predictor-corrector (PC) approach. We consider here the

PC form of the implicit G-α method [19]:

Predictors: un = un + ∆tun + ∆t2(12− β

1− αm)an + ∆t2β

αf

1− αmun (5)

˜un = un + ∆t(1− γ

1− αm)an + ∆tγ

αf

1− αmun (6)

Equilibrium equation: Mun+1 + Cun+1 + Kun+1 = Fe,n+1 (7)

Correctors: un+1 = un + ∆t2β1− αf

1− αmun+1, un+1 = ˜un + ∆tγ

1− αf

1− αmun+1 (8)

Recursive equation: (1− αm)an+1 + αman = (1− αf )un+1 + αf un , a0 = u0 (9)

Moreover, we propose the following equilibrium equation to obtain the explicit G-α method

Mun+1 + Cun+1 + Kun = Fe,n+1 (10)

Equations (5-6) and (8-10) provide an explicit implementation of the G-α method. For the solutionprocedure, both the implicit and the explicit G-α methods can be implemented in the following u-form:

Mun+1 = Fe,n+1 −C˜un −Kun (11)

where for the implicit and explicit methods,

M = MI = M + ∆tγ1− αf

1− αmC + ∆t2β

1− αf

1− αmK and M = ME = M + ∆tγ

1− αf

1− αmC, (12)

respectively. On each time step, the solution of un+1, un+1 and an+1 are updated through (8-9), after(11) is solved. In Eqs. (5,6,9), a is an acceleration-like variable vector that is first-order accurate withrespect to the true acceleration u that exhibits second-order accuracy [19].

The integration parameters of both the novel implicit and explicit G-α methods are still those of theoriginal implicit G-α method [17]. They can be expressed in terms of the spectral radius ρ, which is ρ∞,the spectral radius at the high frequency limit, for the implicit algorithm and ρb, the spectral radius at thebifurcation point, for the explicit algorithm, respectively. In a greater detail, the integration parametersread αm = 2ρ−1

ρ+1 , αf = ρρ+1 , β = 1

4(12 + γ)2 and γ = 1

2 − αm + αf [13].

3.2 A novel interfield parallel method

The main challenge in multi-time-step integration is effectively to account for the coupling between statevariables integrated at different rates in different subsystems. We follow here a specific PC approachproposed by Gravouil and Combescure [9], known as the free-link approach . In each subdomain, theintegration by the G-α methods is decoupled into a free problem and a link problem:

Miuin+1f

= Fie,n+1 −Ci ˜ui

n −Kiuin, ui

n+1f= ui

n + αi1u

in+1f

, uin+1f

= ˜ui

n + αi2u

in+1f

(13)

Miuin+1l

= LiT Λn+1, uin+1l

= αi1u

in+1l

, uAn+1l

= αi2u

An+1l

(14)

with αi1 = ∆t2i βi

1−αif

1−αim

and αi2 = ∆tiγi

1−αif

1−αim

. Here, subscripts f and l denote the free and link problem,respectively. The original kinematic quantities are obtained by summing free and link quantities, i.e.,(·) = (·)f +(·)l. The idea is to solve in each subdomain a free problem that does not require informationof constraints between subdomains; then Lagrange multipliers are solved to define the link problem. Dueto different time scales between A and B, the projections from the coarse time step to the fine time step[9] are made by means of a linear interpolation:

uAn+j/ssf

= (1− j

ss)uA

nf+

j

ssuA

n+1f, uA

n+j/ssl= (1− j

ss)uA

nl+

j

ssuA

n+1l(15)

Λn+j/ss = (1− j

ss)Λn +

j

ssΛn+1, ∀j ∈ 1, · · · , ss (16)

Now, we rewrite the continuity relationship (2) as

LAuAn+j/ssl

+ LBuBn+j/ssl

= −(LAuAn+j/ssf

+ LBuBn+j/ssf

) (17)

By considering (14-16), one obtains from (17) a condensed global problem at interfaces:

HΛn+j/ss = −(LAuAn+j/ssf

+ LBuBn+j/ssf

) (18)

with H = αA2 LAMA−1

LAT+ αB

2 LBMB−1LBT

.As a result, we present the method advancing from tn−1 to tn+1 in subdomain A and from tn to tn+1

in subdomain B by the following pseudo-code:

1. solve the free problem (13) in subdomain A by using 2∆tA, thus advancing from tn−1 to tn+1

2. start the loop on ss substeps in subdomain B

3. solve the free problem (13) in subdomain B by using ∆tB , thus advancing from tn+(j−1)/ss totn+j/ss with j = 1, . . . , ss

4. interpolate the free velocity uAn+j/ssf

in subdomain A according to (15)

5. compute Lagrange multipliers Λn+j/ss by solving the condensed global problem (18)

6. solve the link problem (14) in subdomain B at tn+j/ss

7. compute kinematic quantities in subdomain B at tn+j/ss by summing free and link quantities

8. if j = ss, then end the loop in subdomain B

9. solve the link problem (14) in subdomain A by using 2∆tA, from tn−1 to tn+1

10. compute kinematic quantities in subdomain A at tn+1 by summing free and link quantities

Remark 3 The ongoing interfield processes in the two subdomains are inherently parallel as depictedby dash-dot lines in Figure 3. Note that the process in subdomain A is split into two independentparts, linked only through the subdomain B, which enables the parallel computation and synchronizedexchange of information.

Hereafter, the new method will be referred to as the PM-α method. Similar to the GC method [9], thePM-α method can be generalized to multiple subdomains implicit or explicit with different time scales.

3.3 Convergence properties of the novel interfield parallel method

In the following analysis, the implicit and explicit G-α methods are considered in subdomain A and B,respectively, of the PM-α method. The method uses two different schemes on two or more subdomainswith different time steps. As a result, the modal analysis approach to numerical stability is inapplicable[10]. To find a decaying norm becomes formidable owing to the large number of state variables involvedalso for simple model problems [11]. As a result, we investigate the convergence of the PM-α methodexploiting a proper approach for linear singe-degree-of-freedom (SDoF) problem.

We consider the scalar test equation mu + ku = 0 associated with the assumption mA + mB = m,kA + kB = k and b1 = mA

mB= kB

kA. The problem, being partitioned into A and B subdomains, can be

physically interpreted as a partitioned SDoF mass-spring system. By choosing different values of b1, thesystem has different frequencies in the two subdomains, while keeping the global problem unchanged[11]. When applied to linear problems, the PM-α method can be recast in a recursive form as

Xn+1 = A Xn + Ln (19)

where X is an appropriate state vector depending on the formulation of the scheme, A is the amplifica-tion matrix and L is the load vector that depends on external forces. In order to advance to the next timestep, the PM-α method requires not only the state variables at time tn but also some at tn−1. For thisreason, we consider the following state vector:

Xn = ( XAn−1 XA

n XBn )T (20)

where XAn = ( uA

n uAn uA

nfuA

n ∆tA aAn ∆tA )T collects the kinematic quantities of subdomain

A and XBn =

(uB

n uBn uB

n ∆tA aBn ∆tA

)T collects those of subdomain B. Consequently, Xn hasthe dimension 10 nA + 4 nB , being nA, nB the number of DoFs in the two subdomains.

We analyse the convergence of the PM-α method without considering the load vector Ln in (19),by assuming that the power of the loading error term of its approximation is greater than the order ofaccuracy of the method.

Definition 1 The PM-α method (13-18) is of order k, if the local truncation error τn reads τn =AX(tn)−X(tn+1) = O(∆tk+1

A ), being X (tn) the corresponding exact solution of the state vector Xn

at tn.

As τn = O(∆t2A), the PM-α method is at least first-order accurate. In particular for ss = 1, thecondition ρ∞ = ρb = 1 yields τn = O(∆t3A), that entails a second-order accuracy.

(a) (b)

Figure 4: The PM-α method: a) coupling of two subdomains; b) ρ(D) as a function of ρ and b1

With regard to stability, the PM-α method is firstly investigated using the zero-stability concept [6].

Definition 2 The PM-α method (13-18) is said to be zero-stable, if the integrators in the subdomainsare zero-stable and if un+1 = Dun + g is zero-stable under the assumption stated in [6]. Here un

is the input vector which collects all inputs to each subsystem. The matrix D and the vector g containconstant terms as a result of a vanishing time step.

The zero-stability of the PM-α method is nothing more than the minimal demand that the coupled systemis well posed [6, 13]. In order to perform the zero-stability analysis, we identify the relationship betweeninputs, Λn , and outputs, uA

nf, uB

nf, as depicted in Figure 4(a). Being ρ(D) < 1 with ρ(•) = max

i|λi(•)|,

the PM-α method is zero-stable. Moreover, ρ(D) is depicted in Figure 4(b). It is considered that ρ∞ ofthe implicit method and ρb of the explicit method can be equal to ρ ∈ [0, 1] with b1 ∈ [0.01, 100].

Further, the absolute stability of the PM-α method is investigated employing non-vanishing timesteps. Stability of a numerical scheme requires that the numerical solution remain bounded.

Definition 3 The PM-α method (13-18) is stable if ρ(A) ≤ 1.

The matrix A defined in (19) is endowed with 9 non-zero eigenvalues. Moreover, we have found linearlyindependent eigenvectors for repeated eigenvalues λi with any ∆tA > 0, 0 ≤ ρ∞ ≤ 1 and 0 ≤ ρb ≤ 1.

Figure 5: The PM-α method: a) |λi(A)| with ρ(= ρ∞ = ρb) = 0.8, b1 = 10, ss = 2; b) ss = 5; c)ρ(A) with b1 = 0.01 and ss = 5.

Figure 6: Global error of the PM-α method with ρ(= ρ∞ = ρb) = 0.5 and b1 = 10: a) ss = 1; b)ss = 2; c) ss = 20.

Hence, ρ(A) ≤ 1 suffices that the numerical solution is bounded. |λi(A)| are plotted versus ΩB =ωB ×∆tB in Figure 5(a)(b) with a few values of ss, being ω the angular frequency. The PM-α methodis conditionally stable with the stability limit equal to or higher than that of the explicit subdomain B.In detail, Figure 5(c) shows the spectral radius ρ(A) of the PM-α method. Here ρ(A) is considered byignoring one real unity eigenvalue, that contributes to a constant drift-off error of u and does not affect uand u [11]. The controllable dissipation of the PM-α method with respect to the PM method is evident.

When an algorithm is zero-stable, the local order condition implies global convergence. Nonetheless,the convergence of the PM-α method is also affected by the initialization procedure as the method is notself-starting. As shown in Figure 3, two points in subdomain A are required for starting the PM-αmethod. A staggered solution procedure, that exploits ∆tA on subdomain A from t0 to t1, can beused to start the PM method. The staggered procedure labelled as GC-α method [17], can simply beobtained by substituting ∆tA for 2∆tA in Point 1 and 9 of the pseudo-code of the PM-α method. Wedefine the the initialization error e2 as e2 = X2 −X(2∆tA) = AX1 −X(2∆tA), being the error atthe end of t2 = 2∆tA. The global error of the PM-α method is found to be en = Xn − X(tn) =An−2e2 +

∑n−1i=2 An−(i+1)τi.

Definition 4 The PM-α method is globally convergent of order k, if it is stable and the initializationerror e2 and the local truncation error τn satisfy e2 = O(∆tkA) and τn = O(∆tk+1

A ).

When ss = 1 and ss > 1, e2 = O(∆t2A) and e2 = O(∆t1A), respectively. Hence, the PM-α method isconsistent and at least first order convergent.

Herein, we present some numerical experiments concerning the global convergence of the PM-αmethod. Figure 6 shows the global error |en| versus ∆tA in a double logarithmic scale for the linearSDoF scalar problem presented above. The order of convergence with respect to ∆tA is evident in the

Figure 7: Partitioned four-DoF system: a) model problem; b) displacement response in free vibrationprovided by both the PM (upper) and PM-α (lower) methods.

figure, which indicates a second-order accuracy when ss = 1 and first-order accuracy when ss > 1.One notices that the PM-α method is second-order convergent when ss = 1 even if ρ∞ = ρb 6= 1.

4 Numerical experiments for a stiff case

In order to highlight the numerical performance of the PM-α method in a stiff case, we consider a four-DoF coupled system shown in Figure 7(a). The displacement vector of subdomain A is chosen to beUA = u2, u3, u4T while the displacement vector of subdomain B is chosen as UB = u1, u2T . Theemulated system is endowed with the following natural frequencies: f1 = 0.47 Hz, f2 = 2.90 Hz, f3 =4.18 Hz and f4 = 9.47Hz. Figure 7(b) shows the free vibration time histories that are computed by thePM and PM-α methods, respectively, considering the initial value u1, u2, u3, u4 = 0, 0, 0, 1. Thesame time step ∆tAf1 = 1/40 and ss = 20 are employed for both methods; moreover, ρ∞ = ρb = 0.5for the PM-α method. It is evident that the unwanted high-frequency components of the response aretraced by the PM method, while they are annihilated by the PM-α method in a few time steps.

5 Conclusions

In this paper, we extended the interfield parallel PM method [11] by means of the G-α methods [12]. Indetail, the G-α methods were developed for this partitioned application without employing a balancedformulation of equilibrium equations. The convergence properties of the new partitioned method hasbeen analysed by means of a SDoF linear model problem. It has been shown that the controllable nu-merical dissipation is advantageous for solving coupled and possibly heterogeneous structural dynamicsystems in which convergence and computational efficiency can be adversely affected by spurious high-frequency components of the response entailed by spatial discretization and/or kinematic constraints.

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