computational techniques for simulation of monolithic and...
TRANSCRIPT
Computational Techniques for Simulation ofMonolithic and Heterogeneous Structural
Dynamic Systems
Oreste S. Bursi *
* Department of Mechanical and Structural Engineering, University of Trento,Trento, Italy
Abstract The prediction of the transient dynamic response of mono-lithic structural systems, as well as of heterogeneous (numerical/physical) subsystems, decomposed by computational or physicalconsiderations typical of hardware-in-the-loop and pseudo-dynamictests using numerical integration, has become an accepted practicealmost to the extent that such solutions in non-linear problems of-ten are considered to be exact solutions. It is for this reason thatthis chapter is placed immediately at the beginning of the book.
In light of the large body of literature on computational methodsdeveloped for both testing and control techniques applied to linearand non-linear systems, no attempt is made to cover this subjectin greater depth. Rather the concepts upon which ad hoc compu-tational methods rely are presented in a common framework alongwith a few applications.
1 Introduction
Nowadays, there is a significant development of models and scalable com-putational techniques that enable very complex simulations in various en-gineering and scientific fields essentially in multi-physics, mechatronics anddesign and control analysis. Distinguishing features are the use of parti-tioned techniques (Felippa et al., 2001), modular and distributed simulationsof multibody systems (Kubler and Schiehlen, 2004), real-time simulations(Cuadrado et al., 1997), time-parallel frameworks for parallel simulations(Farhat et al., 2006) and reduced models (Biondi and Muscolino, 2000).Several techniques rely on a top-bottom approach which begins with the fullglobal equations of motion and then partitions them into substructures bytaking their interactions into account. Other ones employ a bottom-up ap-proach, that consists in approximating each substructure and then in assem-bling the approximated substructural equations in order to form the global
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equations of motion. These techniques do integrate cutting-edge research onthe fields of numerical analysis, linear/non-linear dynamics, multibody dy-namics, digital control, system modelling and also experimental techniques.In such a scenario, the role of the transient analysis of multi-physics dynam-ical systems is still at the forefront of computational methods for solvingproblems of non-linear dynamics. Moreover, because simulations, i.e. ex-perimentations with models, are necessarily limited, techniques such as theHardware-in-the-loop (HiL) and Pseudo-Dynamic (PsD) testing based onthe principle of Dynamic Substructuring (DS ) (Pinto et al., 2004; Missel-horn et al., 2006; Bursi et al., 2008) are becoming more and more impor-tant in testing, design and control. Thus, computational techniques for thesimulation of monolithic and heterogeneous (numerical/physical) dynamicsystems presented here reflected a shift in features from discipline-orienteddynamics to system-oriented dynamics, from sequential to parallel compu-tations, from accuracy concerns to system-model refinements. In addition,HiL is actually employed to test more complex mechanical systems (Wagget al., 2008) than just a piece of hardware, e.g. a controller; whilst PsD isused nowadays to perform tests on civil-engineering components in fast orreal time (Pegon, 2008; Shing, 2008). Hence, these developments are nec-essarily not uniform and difficult to fit into established categories and suchunified approaches, as for instance, in mechatronics (Samin et al., 2007).
Since this chapter is the first, in this section we present a brief state-of-the-art on real-time testing with DS in the context of HiL and PsDtesting. So this section introduces and interleaves particular cases of theGeneralized-α methods both in HiL and PsD testing. Up to now, thesestructural integrators represent the most common time integration methodsused in finite element FE simulations and testing. After the introduction,different forms of equations of structural dynamics are presented in Sec-tion 2 with correction and compensation techniques specific to PsD testing.Thus, Section 1 and 2 present also the notation used in this chapter and thetwo sets of equations employed by different time integration methods, andlinearly implicit Rosenbrock methods are interleaved. A major classifica-tion of time integration algorithms is then described in Section 3, followedby analysis techniques and metrics useful for structural dynamics problems.A subsection devoted to error propagation in time integration algorithmsis also included, being particularly useful for experimental techniques. InSection 4 we introduce problems and analysis techniques that arise and arefaced when coupling of numerical and physical substructures occurs. Be-cause HiL and PsD techniques require transfer systems, i.e. actuators andcontrollers for their actuation, numerical techniques used to discretize andlinearize adaptive controllers are introduced. Moreover, filter design tools
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like bilinear transformations, are employed to characterize time integrators.Successively, Section 5 and 6 present variants of general algorithms special-ized in the domain of real time HiL and PsD testing. In detail, Section5 reports time integration methods for the solution of coupled-field real-time problems in HiL, i.e. primarily methods for structure-structure andstructure-control interaction problems. In fact the analysis and design ofalgorithms for the operation of structure-control systems, multibody andmechatronic systems operating in real time still remain one of the liveliestresearch activities in recent years (Arnold et al., 2007). Conversely, Section6 presents innovative parallelized partitioned methods based on Lagrangemultipliers able to guarantee fast pseudo-dynamic tests of complex substruc-tures. Conclusions are summarized in Section 7 with future perspectives ofnumerical techniques for HiL and PsD testing with DS. This chapter is notmeant to be an exhaustive state-of-the-art of the research, but is intended toassist researchers and practitioners in selecting time integration algorithmsin a numerical/physical context.
1.1 Algorithms for HiL and Real-time Testing with DS
The aim of this section is to review some numerical methods that wereused for Simulation of Heterogeneous Systems with Dynamic substructur-ing (SHSDS ) and underline some problems derived from their application.Several properties of algorithms mentioned herein are discussed at lengthin Subsection 3.2.
SHSDS is a novel form of hybrid numerical-experimental testing, whichcan be used to test structural or mechanical systems under dynamic loading(Williams and Blakeborough, 2001). The technique involves splitting thesystem being tested into two parts: the Physical Subsystem (PS ), that con-tains a key region of interest and is experimentally tested; the NumericalSubsystem (NS ), that contains the remainder of the system and is numer-ically simulated as illustrated in Figure 1. By imposing compatibility andequilibrium conditions at the interface between the NS and the PS, re-spectively, the subsystems are made to interact in real time such that theyemulate the dynamic behaviour of the full system. In detail, a real-timesimulation of the emulated system requires a computational time per timeinterval, i.e. an integration time ∆t plus the time for display, smaller thanthe physical time taken by the actual motion of the emulated system. Inorder to interconnect the PS to the NS, a transfer system that acts onthe PS is typically controlled to follow NS interface kinematic quantitiesor other outputs. This transfer system normally consists of an electric orelectro-hydraulic actuator and a controller as shown in Figure 1.
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Figure 1. Schematic representation of a 2-DoF system with dynamic sub-structuring.
Figure 2. Block diagram representation of a heterogeneous simulation withdynamic substructuring.
The relevant block diagram representation is depicted in Figure 2 inwhich both the transfer system with delay and a delay compensator areshown. Note that coupling can be applied in the opposite sense by operatingthe transfer system in force control: however, operating a transfer system inforce control is much more involved owing to the mechanical characteristicsof the actuator itself, the load cell measurement errors as well as the relevantcontrol (Reinhorn et al., 2004).
The technique exploited herein to partition a system is called differentialpartitioning (Felippa et al., 2001) and is based on the governing differentialequations describing the NS and PS involved, respectively. Historically,differential partitioning was the original approach used in the derivationof staggered solution procedures for Fluid-Structure Interaction (FSI ), and
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remains a natural way to handle physically diverse systems. Starting fromthe governing differential equations, the system of equations of motion of adiscretised structure reads
Mx(t) + Cx(t) + Kx(t) = fe(t), (1.1)
where M, C and K are the global mass, damping and stiffness matrices andfe(t) corresponds to the externally applied time-varying forces vector. x, xand x are respectively displacement, velocity and acceleration vectors of theglobal structure. By splitting the global matrices such that M = Mn +Mp,C = Cn + Cp and K = Kn + Kp, where the superscripts ”n” and ”p” referto the NS and the PS, respectively, one gets
Mnx(t) + Cnx(t) + Knx(t) = fe(t) + fs(t), (1.2)
where the coupling force vector fs(t) reads:
fs(t) = −Mpx(t) − Cpx(t) − Kpx(t). (1.3)
Thus, by means of differential partitioning, the global system can be un-coupled using a set of two differential equations, where (1.2) defines the NSand (1.3) describes the PS, respectively. Since the modes of the structurepartition are not necessarily modes of the coupled system, the modal de-composition does not hold and a typical model problem cannot be defined.
A standard approach for discrete time integration is to formulate SHSDSin a continuous-time environment, see for instance Wallace et al. (2004),and then use Runge-Kutta integrators, like the embedded Dormand-Princemethod ODE45 (Dormand and Prince, 1980), which is implemented inSimulink (Mat, 2005). Typically, these algorithms are only conditionallystable and computationally expensive since they need many stages to achievehigh order accuracy. For instance, the ODE45 uses 7 stages and exhibits afifth-order accuracy entailing a high computational effort. Other numericalapproaches based on structural integrators were also suggested by Darbyet al. (2002). They pointed out that the Central Difference (CD) method(Hughes, 1987) introduces inaccuracies on both Single- and Multiple-Degrees-of-Freedom (MDoF) systems. Note that the CD algorithm is second-orderaccurate with no substructure coupling and does not provide any explicittarget velocity (Wu et al., 2005).
Bonnet (2006) analysed different structural integrators for SHSDS. Atthe end, only three of them were considered suitable for most applications.The Newmark explicit scheme (Hughes, 1987), which is derived from the
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Newmark scheme (Newmark, 1959) by setting β = 0 and γ = 0.5 :
Mnak+1 + Cnvk+1 + Kndk+1 = fe,k+1 + fs,k+1
dk+1 = dk + ∆tvk + ∆t2
2ak
vk+1 = vk + ∆t2
(ak + ak+1)
(1.4)
In detail, the first difference equation is used to derive explicitly thedisplacement command signal to be imposed on the specimen. The forcevector fs,k+1 is measured and the combination of the equation of motionand of the second difference equation is exploited to derive ak+1 and vk+1,necessary to carry on the scheme for k+2. In its standard form this schemeis second-order accurate and requires very little computation. When sta-bility issues become important, the modified Newmark explicit algorithmproposed by Chang (2002) was shown to be very efficient. It reads,
Mnak+1 + Cnvk+1 + Kndk+1 = fe,k+1 + fs,k+1
dk+1 = dk + β1∆tvk + β2∆t2ak
vk+1 = vk +∆t
2(ak + ak+1)
(1.5)
where
β1 = [I +∆t
2M
−1
C +∆t2
4M
−1
K0]−1[I +
∆t
2M
−1
C] (1.6)
β2 =1
2[I +
∆t
2M
−1
C +∆t2
4M
−1
K0]−1
The matrices β1 and β2 which are determined from the initial stiffnessmatrix K0, render the method unconditionally stable, with no numericaldissipation and no overshooting effects like the trapezium rule for linearstructures with K0 (Hughes, 1987). The limits of these methods lie on thefacts that the velocity vector expression is implicit and for their efficient use,the damping matrix C must be constant. Moreover, these schemes do notprovide an explicit target velocity, thus limiting the control of the transfersystem (Wu et al., 2005).
If numerical damping is needed, Bonnet (2006) suggested the use of the
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α-Operator-Splitting (OS-α) method, which can be expressed by
Mnak+1 + (1 + α)Cnvk+1 − αCnvk + (1 + α)Kndk+1 − αKndk
= (1 + α)(fe,k+1 + fs,k+1) − α(fe,k + fs,k) (1.7)
dk+1 = dk+1 + β∆t2ak+1 (1.8)
dk+1 = dk + ∆tvk + ∆t2(1
2− β)ak (1.9)
vk+1 = vk + (1 − γ)∆tak + γ∆tak+1 (1.10)
γ =1
2− α and β =
(1 − α)2
4(1.11)
Also this scheme suffers of the limits underlined for the two previous onesand Wu et al. (2006) proposed an Operator-splitting method able to dealexplicitly with systems endowed with non-linear damping. Other integrationalgorithms, originated from the control theory, were developed: see, amongothers, Lopez-Almansa et al. (1998); Zhang et al. (2005); Chu et al. (2005).Nonetheless, the first two methods were only conditionally stable, whilethe third method implied the calculation of the exponential of the systemmatrix (Moler and Loan, 1978) which is of limited use for large FE models.
In summary, we recall that all unconditionally, i.e. A-stable methods, foruncoupled problems suggested for SHSDS and derived from structural inte-grators obey to the second Dahlquist barrier (Lambert (1991), p. 243), viz.: (i) an explicit linear multistep method cannot be A-stable; (ii) the orderof an A-stable linear multistep method cannot exceed two; (iii) the second-order A-stable linear multistep method with smallest error constant is thetrapezium rule. As a result, the algorithms used so far are utmost second-order accurate for uncoupled problems. To perform real-time computation,it appears more convenient to employ unconditionally single step integrationformula with fixed time step and order, to achieve the same computationalcost in each integration step without iterations (Arnold et al., 2007). Accu-racy is the feature that must be sacrificed in conflict with other propertiesand mainly stability; in detail, the time integration method must be com-putationally inexpensive, with few evaluations in each step and withoutintroducing excessive loss of accuracy. Along these line, L-stable real-timecompatible algorithms based on Rosenbrock methods have been developed(Bursi et al., 2008). They are introduced in Subsection 3.1 and analysed atlength in Section 5 for both monolithic and heterogeneous coupled systems.
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1.2 Algorithms for PsD Testing
After decades of development, the PsD method is recognised to be asa reliable technique for testing full scale structures. A conceptual flow ofthe method is shown in Figure 3. By its very numerical nature, the PsDmethod allows substructuring and requires only a part of the structure tobe tested in the laboratory, the non-linear experimental part, while thelinearly responding part of the structure, the analytical part, can be treatednumerically (Shing et al., 1994). The time integration of multiple-degrees-of-freedom (MDoF) systems is most frequently performed using implicit orlinearly inflict time-stepping schemes. Examples of such schemes are: (i)Thewalt and Mahin’s method (Thewalt and Mahin, 1987) based on theHHT -α method of Hilber et al. (1977) and on iterations carried out ata hybrid digital-analog level; (ii) the method of Nakashima et al. (1990),based on the mixed implicit-explicit OS (Hughes et al., 1979) which doesnot require any numerical iteration; (iii) the scheme suggested by Shinget al. (1991), in short the C -α algorithm, based on the HHT -α methodwhich relies on a modified Newton-Raphson type iterative procedure; (iv)the scheme suggested by Bonelli and Bursi (2004), in short the IPC-ρ∞
algorithm, based on the CH-α method (Chung and Hulbert, 1993) and apredictor-single corrector procedure to avoids iterations.
Implicit and Explicit CH-α Algorithms Unconditional stability maynot be sufficient to ensure a robust time integration; some high-frequencycomponents of the numerical solution may damp out very slowly and showup as oscillations altering the solution. The low effectiveness of numeri-cal dissipation and the overshoot consequences on the response of HHT-α method applied to stiff dynamic systems were highlighted by Erlicheret al. (2002). Some of these issues motivated the development of the CH-αmethod. Let us consider a structure to be tested, whose dynamic behaviourcan be condensed on a reduced number of DoF, each of which is controlledby an actuator. According to the CH-α method, its motion can be analysedby means of a system of weighted dynamic equilibrium equations written inmatrix notation
Mai+1−αm+ Cvi+1−αf
+ ri+1−αf= fe,i+1−αf
(1.12)
which arises from the temporal discretization of semi-discrete equations.The solution is computed inside the time step size ∆ti = ti+1 − ti withgiven initial conditions. ri+1−αf
is the non-linear nodal restoring force vec-tor, fe,i+1−αf
denotes the external excitation vector applied to the system.Let 0 = t0 < t1 < ... < tn = tf , where n is the number of time steps,
define a partition of the time domain [0, tf ] and assume a constant time step
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Figure 3. Flow of the PsD method
size ∆t = ∆ti. The CH-α method relies on the temporal approximationof v and a proposed by Newmark (1959)
ai+1 =1
∆t2β
(di+1−di+1
)(1.13)
vi+1 = vi + ∆t
[(1 − γ) ai + γ
1
∆t2β
(di+1−di+1
)](1.14)
where ai is the numerical acceleration at time ti and
di+1 = di + ∆tvi + ∆t2(
1
2− β
)ai (1.15)
with the initial conditions d0 = d, v0 = v and a0 = M−1(f0(0)−Cv0 −r(d0)). The time discrete combinations of displacements, velocities andaccelerations read
di+1−αf= (1 − αf) di+1 + αfdi
vi+1−αf= (1 − αf) vi+1 + αfvi
ai+1−αm= (1 − αm) ai+1 + αmai
(1.16)
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where i ∈ 0, 1, ..., n − 1. With regard to the evaluation of the non-linearinternal forces ri+1−αf
= r (αf , di, di+1) of (1.12), various quadraturerules can be employed (Erlicher et al., 2002). Nonetheless, only the gener-alized trapezoidal rule, the most suitable for the PsD method is consideredhere:
ri+1−αf= (1 − αf) r (di+1) + αfr (di) (1.17)
fi+1−αf= (1 − αf) f (ti+1) + αf f (ti) . (1.18)
The PsD method is mainly exploited in the non-linear case, where al-gorithms regarded as unconditionally stable in linear dynamics can exhibitsevere numerical instabilities in the non-linear case (Erlicher et al., 2002).When applied to the non-linear regime, the algorithm defined in (1.12-1.16)can be implemented through a matrix K, which considers inertial and vis-
cous damping effects and an effective stiffness matrix K∗(k)i+1 . They read
K =1 − αm
(1−αf) β∆t2M + γ
1 − αm
β∆tC and (1.19)
K∗(k)i+1 = K + K
(k)a,i+1, (1.20)
respectively, where K(k)a,i+1 is the tangent stiffness matrix of the structure.
Defining the residual as
R(k)i+1 = fi+1 +
αf
1 − αf
fi − 1 − αm
1 − αf
Ma(k)i+1 − αm
1 − αf
Mai − Cv(k)i+1
− αf
1 − αf
Cvi − r(k)i+1 − αf
1 − αf
ri, (1.21)
the displacement increment ∆d(k)i+1 can be evaluated by means of a Newton-
Raphson method
∆d(k)i+1
(r(k)i+1, v
(k)i+1, a
(k)i+1
)=
(K
∗(k)i+1
)−1
R(k)i+1 (1.22)
and therefore
d(k+1)i+1 = d
(k)i+1 + ∆d
(k)i+1 (1.23)
a(k+1)i+1
(d
(k+1)i+1
)=
1
∆t2β
(d
(k+1)i+1 −di+1
)(1.24)
v(k+1)i+1 = vi + ∆t (1 − γ) ai + ∆tγa
(k+1)i+1 . (1.25)
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In the linear case, ri+1−αf= (1 − αf)Kdi+1 + αfKdi and the
CH − α scheme is endowed with: (i) second-order accuracy; (ii) opti-mized dissipation characteristics of higher modes, i.e. the high-frequencycomponents of the response are filtered with minimum influence on the low-frequency region. The amount of dissipation can be user-specified by meansof the high-frequency spectral radius ρ∞ (= ρ (Ω) for Ω → ∞ whereΩ represents the non-dimensional frequency); in detail, ρ∞ ∈ [0, 1] andthe choice ρ∞ = 0 corresponds to asymptotic annihilation of the high-frequency response, i.e. L-stability, while ρ∞ = 1 corresponds to no algo-rithmic dissipation. The relations between the user-specified spectral radiusat the high-frequency limit ρ∞ and the free parameters β, γ, αm and αf ,obtained from optimizations in the linear case (Chung and Hulbert, 1993)read
β =1
(1 + ρ∞)2 ∈ [
1
4, 1]; γ =
1
2
3 − ρ∞
1 + ρ∞∈ [
1
2,3
2];
αm =2ρ∞ − 1
1 + ρ∞∈ [−1,
1
2]; αf =
ρ∞
1 + ρ∞∈ [0,
1
2]
. (1.26)
As far as unforced systems are concerned, a detailed convergence analysisof the CH -α method in the non-linear case can be found in Erlicher et al.(2002). Summing up, it has been proved that the assumptions (1.26) entaila consistent second order scheme, viz.
di = u (ti) + O(∆t2
), (1.27)
where u(t) is the exact solution of the parent system of differential equa-tions. Accelerations evaluated by means of (1.24) are just first order accu-rate, while a second order approximation can be found using the followingcombination
[(1−αm)ai+1+αmai]−[(1−αf)d(ti+1)+αf d(ti)] = O(∆t2) (1.28)
This relationship was used by Lacoma and Romero (2007) and Bruls andGolinval (2008), among others, in their applications.
Moreover, it was demonstrated that the dissipative properties of thealgorithm can be still controlled in the non-linear case by the parameter ρ∞.In detail, assume ai = ra,ia0, vi = rv,iv1 and ri+1−αf
= rr,ir1−αf; if
∆τ = ∆tT
≫ 1, where T represents the period of the highest frequencycomponent, the following relations hold
limi→∞
ra,i+1
ra,i
=rv,i+1
rv,i
=rr,i+1
rr,i
= −ρ∞. (1.29)
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If C = 0, denote with Ei = Ei + ∆t2(β − γ
2
)12aT
i Mai the generalized
energy, where Ei = Ti + Ui = 12vT
i Mvi + U (di) is the total mechanicalenergy of the system at the beginning of the i-th time step, the followingrelations hold
Ei+1
Ei
= ρ2∞
(i + 1)2 (1−ρ∞)4
16+ (1−ρ∞)2
4(1+ρ∞)2
(−ρ∞ + i+1
2
(1 − ρ2
∞
))2
i2 (1−ρ∞)4
16+ (1−ρ∞)2
4(1+ρ∞)2
(−ρ∞ + i
2
(1 − ρ2
∞
))2
(1.30)
limi→∞
Ei+1
Ei
= ρ2∞. (1.31)
for a0 6= 0, i > 0. (1.31) confirms the effectiveness of the dissipativecharacteristics of the CH − α scheme in the high frequency range.
With regard to non-linear forced systems, the CH -α method is capableto limit the error near resonance both in terms of frequency location andintensity of the resonant peak (Baldo et al., 2006).
When the time step ∆t is very small, as in the continuous PsD methodintroduced in Section 6, explicit schemes can be used, owing to their reducednumerical effort and accuracy. For instance, the ECH -α method still em-ploys the Newmark approximations of velocity and acceleration (1.13-1.14)as well as (1.12). First, let us consider the undamped case and two param-eter ρp and ρs which represent the modulus of the principal roots and ofthe spurious root at the bifurcation frequency Ωb, respectively. To maxi-mize the high-frequency numerical dissipation, Hulbert and Chung (1996)impose ρs ≤ ρp. By denoting the value of the spectral radius at Ωb by ρb,the condition ρs = ρp = ρb entails the optimal balance of dissipation atlow and high frequencies. The relations among the algorithmic parametersthen become
αm =2ρb − 1
ρb + 1(1.32)
β =ρ3
b−2αf ρ3
b+α2
f ρ3
b−3ρ2
b+3αf ρ2
b+3αf ρb−3α2
f ρb+3ρb−2α2
f −1−2αf
(−2+ρb)(ρb+1)2αf
(1.33)
γ =1
2− αm + αf . (1.34)
where (1.34) entails second order accuracy and αf is a free parameter;
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moreover, Ωb and Ωcr read
Ωb = (ρb + 1)√
2 − ρb
Ωcr =
√12(ρb+1)3(ρb−2)
(ρ4
b−ρ3
b+ρ2
b−15ρb−10)
(1.35)
respectively, where Ωcr represents the critical frequency corresponding tothe stability limit.
For damped systems, Hulbert and Chung (1996) proposed both an ex-plicit and an implicit treatment of physical damping. Therefore two imple-mentations are possible. If αf = 1, the ECH -α method can be formulatedwith an explicit treatment of damping to attain second-order accuracy evenin the non-linear case, and implemented in a predictor-corrector form:Predictor phase:
vi+1 = vi + (1 − γ)∆tai (1.36)
di+1 = di + ∆tvi +
(1
2− β
)∆t2ai (1.37)
Corrector phase:
(1 − αm)M ai+1 = fex (ti) − αmM ai − Cvi − r(di
)(1.38)
vi+1 = vi+1 + γ∆tai+1 (1.39)
di+1 = di+1 + β∆t2ai+1. (1.40)
With an implicit damping treatment and αf 6= 1, Hulbert and Chung(1996) in the correction phase substitute the equilibrium equation (1.38)with the following relationship
((1 − αm)M+γ∆t (1 − αf)C) ai+1 = fex
(ti+1−αf
)
−αmM ai − (1 − αf) Cvi+1 − αf Cvi − r(di+1−αf
)(1.41)
CH -α Algorithms Applied to PsD Testing The application of animplicit scheme to the PsD method is not trivial as a consistent linearizationvia the full Newton-Raphson method is required to solve the non-linear
system (1.12). Moreover, r(k)i+1 in (1.21) is measured during a PsD test. As
a result, every time step needs generally more than one movement of eachactuator. The C -α method (Shing and Vannan, 1990) adopts a system ofmodified equilibrium equations (1.12) with αm = 0 and αf = −α andthe Newmark approximations (1.13-1.14). It is essentially endowed with
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Figure 4. Non-linear solution method for the C -α algorithm: a) modifiedNewton iterations; b) correction residual errors.
modified Newton-Raphson iterations with linear convergence and the error-correction procedure depicted in Figure (4)a and described in (1.21-1.25).
If more than one movement of each actuator is needed inside one timestep like in the C -α-method, there is a double negative effect: i) the overalltest time increases; ii) the time scaling factor between earthquake time andtest time varies in each step. To avoid this problem the OS -α algorithmwas proposed (Nakashima et al., 1990). This scheme is the synthesis of theOS technique and the C -α method (Bursi and Shing, 1996). It results inan explicit predictor-implicit corrector or linearly implicit algorithm char-acterized by the following equations:
Mai+1 + (1 + α)Cvi+1 − αCvi + (1 + α)(Kdi+1+ri+1−Kdi+1)
−α(Kdi+ri−Kdi) = (1 + α)fi+1 − αfi (1.42)
The deviation of the structural response from linearity, as depicted inFigure 5(a), is approximated as
Kdi+1 − ri+1 ≈ Kdi+1 − ri+1 (1.43)
in which K represents a predictor stiffness matrix, and ri+1 is the restoringforce vector developed by the structure when the predictor displacementsdi+1 have been imposed.
The predictor-one pass corrector scheme IPC−ρ∞ suggested by Bonelliand Bursi (2004) requires, like the OS -α algorithm, only one single move-ment of the actuator per time step. But differently from previous schemesit evaluates the movement to be imposed by the actuator by means of theimplicit formula (1.23) instead of the explicit one (1.37) employed by theOS -α method.
Let’s superscript m denote experimental measures and, for brevity, con-sider only the main steps. The solution procedure of the IPC−ρ∞ schemeis summarized in an algorithmic form in what follows.
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Figure 5. Solution methods for the OS algorithms: a) corrections for non-linearity; b) I-Modification for displacement control error.
1. Predictor phase:
a) Compute ai+1(di) and vi+1(di) ((1.24) and (1.25))
b) Compute ∆d(0)i+1
(ri, vi+1, ai+1
)(1.22) and d
(1)i+1 (1.23)
c) Impose d(1)i+1 to the structure
d) Measure dmi+1 and rm
i+1 developed by the structure
e) Compute ai+1(dmi+1) and vi+1(d
mi+1) ((1.24) and (1.25))
2. Corrector phase:
a) Compute ∆d(1)i+1(r
mi+1, vi+1, ai+1) ((1.22)) and di+1 = dm
i+1+
∆d(1)i+1
b) Compute ri+1 = rmi+1 + K∆d
(1)i+1
c) Compute ai+1(di+1) and vi+1(di+1) ((1.24) and (1.25))
d) Set i = i + 1, ti+1 = ti + ∆t and go to Step 1b
where Step 1b and 2a are the predictor and the corrector steps, respectively.Each predictor-corrector step is equivalent to the first iteration of a modifiedNewton-Raphson procedure.
For a SDoF system the predictor-one pass corrector process is depictedin Figure 6a, where the evolution of the secant stiffness Ks
i from step i tostep i + 1 is clearly illustrated. The error corrections performed in Step2a and 2b define a solution which satisfies (1.12)-(1.14), i.e. the space ofsolutions at step i + 1; but they trace only approximately, see the hatchedline in Figure 6a, the actual non-linear response of the structure, viz. thecontinuous line. In detail, the solution space corresponds to a straight linewith negative slope. In fact for a SDoF system, (1.12) by virtue of (1.13)and (1.14) can be recasted as follows
ri+1 = −Kdi+1 + Fi+1 (1.44)
15
Figure 6. IPC−ρ∞ scheme: a) non-linear solution method and correctionfor residual errors; b) zone of unconditional stability in ρ∞ − R space.
where
K =1 − αm
(1−αf) β∆t2m + γ
1 − αm
β∆tc (1.45)
Fi+1 = Kdi+1 + fi+1 +αf
1 − αf
fi − αf
1 − αf
cvi −[(1−γ)∆tc +
αm
1 − αf
m
]ai − αf
1 − αf
ri (1.46)
With regard to consistency, it can be proved that the IPC−ρ∞ methodis second order accurate. As far as stability properties are concerned, thefollowing relationships
maxi
|λi| =
|ρ∞| if ρ∞ ≥ |1 − R||1 − R| if ρ∞ < |1 − R| (1.47)
can be proved, where R = Ka
Kis the stiffness ratio and Ka the actual
stiffness of the SDoF system. They lead to an unconditional stable schemeif and only if
0 < R < 2. (1.48)
Figure 6(b) depicts the zone of unconditional stability in the ρ∞ −R space.In detail, one may observe that the condition ρ = maxi |λi| = 0, i.e. theL-stability property of the IPC − ρ∞ scheme is attained in PsD testsonly if R = 1 and ρ∞ = 0. The IPC − ρ∞ method exhibits dissi-pative favourable properties; moreover differently from the OS -α implicitimplementation (Combescure and Pegon, 1997), the scheme maintains un-conditional stability properties even with R > 1.
16
In general, the application of explicit schemes to the PsD method is eas-ier than the one of implicit schemes as movements to be imposed to actua-tors are explicitly evaluated without approximations. Along the line of thepredictor-corrector ECH -α algorithm summarized in (1.36-1.40), Bonelliand Bursi (2004) proposed the EPC -ρb method for PsD tests.
2 Forms of Equations of Structural Dynamics and
Solution Techniques
In this section two forms of the equations of structural dynamics are pre-sented, from which popular integration algorithms employed in PsD andSHSDS techniques stemmed from. Moreover, some corrections and com-pensation techniques employed in testing are commented upon.
2.1 Euler-Lagrange form
The semi-discrete dynamic system of equations of motion reads
Mx(t) + Cx(t) + Kx(t) = fe(t), (2.1)
where M, C and K are the global mass, damping and stiffness matrices,respectively, fe(t) is the vector of applied loads, x(t) is the displacementvector and superimposed dots indicate time differentiation. The associ-ated initial value problem consists in determining the function x = x (t)fulfilling (2.1) for all t ∈ [0, tf ], tf > 0, for given initial conditionsx (0) = x0 and x (0) = v0.
2.2 Hamilton form
Instead of expressing the inertial effects in terms of acceleration, theHamilton form of (2.1) is given by
x(t) = y2(t)My2(t) + Cy2(t) + Kx(t) = fe(t)
(2.2)
where y2(t) is the vector of velocity unknowns. The initial value prob-lem consists in determining the function x = x (t) and y2(t) which satis-fies (2.2) for all t ∈ [0, tf ], tf > 0, for given initial conditions x (0) =x0 and y2(t) = v0. Some integration methods used in control and multi-body dynamics are designed for first-order differential equations. For thesemethods the Hamilton form can be applied, by introducing the state vec-tors, y = x(t) x(t)T = y1(t) y2(t)T . The first-order momentumform is then written as
y = f (t, y) (2.3)
17
with y(0) = x0 v0T where
f =
[0 I
−M−1K −M−1C
]y+
0
M−1fe(t)
(2.4)
Other forms exist like, for instance, the time-discontinuous space-timeFE formulation developed by Hughes and Hulbert (1988). In these schemes,the space is discretized using conventional FE, the time interval is parti-tioned in a number of subintervals, while the response is approximated bymeans of trial functions in the time variable. The use of discontinuousdisplacement and momentum fields leads to the class of implicit schemes,named time discontinuous Galerkin (TDG) methods. A comprehensive re-view of these methods can be found in Hulbert (2004). These uncondition-ally stable schemes can damp out any undesirable high-frequency mode,without introducing excessive algorithmic damping in the low-frequencyresponse. Nonetheless, the factorization of a matrix larger than the oneexploited in standard implicit schemes is required owing to unknown extradisplacement and momentum fields (Bursi and Mancuso, 2002). To limitthe computational effort, Li and Wiberg (1996) implemented a predictor-multicorrector solution algorithm in the TDG method. Based on similarstrategies, a new class of explicit TDG methods, with a user-defined dissi-pation, was developed by Bonelli et al. (2001). The new proposed schemeswere implemented in a predictor-multicorrector form, exhibited third-orderaccuracy and accrued a stability limit higher than the one of the CD scheme.This scheme was also extended to non-linear elastodynamics (Bonelli et al.,2002) showing that both third-order accuracy and energy-decaying proper-ties can be retained. These techniques could be explored in SHSDS andPsD techniques.
2.3 Corrections and Compensation Techniques for Testing
The efficiency of a time-stepping scheme for PsD is connected to itsreduced sensitivity to errors, as in a heterogeneous environment severalerrors arise. While in Subsection 3.2 an error propagation analysis of somealgorithms is presented, herein we show modified solution procedures toreduce unfavourable error effects.
Correction techniques for PsD testing The multicorrector phase (1.21)-(1.25) used in the implicit CH -α scheme results to be very sensitive to ex-perimental errors (Shing and Vannan, 1990). To overcome these difficulties,Shing et al. (1991) proposed a numerical correction to be evaluated after
18
convergence:
di+1 = d(l)i+1 + ∆d
(l)i+1
ri+1 = r(l)i+1 + K∆d
(l)i+1
(2.5)
in which ∆d(l)i+1 represents a residual displacement vector which satisfies the
convergence criteria∣∣∣∆d
(l)i+1
∣∣∣ 6 ∆dn, where ∆dn is a preset displacement
tolerance vector. This error correction, depicted in Figure 4(b) reducesthe effect of stepwise residual errors on numerical result and eliminates thespurious higher-mode response that could otherwise be excited by residualerrors.
To mitigate the effects of displacement control errors in PsD tests similarmodifications were proposed for the OS -α algorithm introduced in Subsec-tion 1.2. In detail, the I-Modification was proposed. It simply corrects therestoring force vector as follows:
rcoi+1 = ra
i+1 − K(da
i+1 − dri+1) (2.6)
in which dri+1 denotes the exact numerical solution, da
i+1 and rai+1 rep-
resent experimental feedback quantities, and rcoi+1 denotes the corrected
restoring force vector as sketched in Figure 5(b). The relevant OSM -αalgorithm offers the main benefit of simplicity in implementation and nonumerical iteration. However, in such a scheme, each predictor-correctorstep is equivalent to the first iteration of a modified Newton procedure. Asa result, errors increase when the predictor stiffness is very different fromthe actual stiffness of a structure, or when the structural specimen exhibits asevere non-linearity (Bursi and Shing, 1996). To reduce these effects, smallintegration time intervals as well as accurate displacement measurementsand servo-controls have to be used. Moreover, to maintain the stabilityproperties of the original OS -α scheme, the predictor stiffness has to begreater than the actual stiffness in (1.42).
The IPC − ρ∞ method introduced in Subsection 1.2 is endowed withan error corrections in Step 2a and 2b along the line of (2.5) to reduce theeffects of stepwise residual errors on numerical results.
Explicit schemes are very sensitive to errors near their stability limit.Typically, they are used with small time step to avoid special correctionformulae (Bonelli and Bursi, 2004).
Substepping and Compensation in SHSDS Standard or modifiedNewton-Raphson linearizations were used in Subsection 1.2 to solve non-linear problems in structural dynamics. Other techniques to solve SHSDSuse a digital feedback technique, an implicit integration scheme based on the
19
Figure 7. The history of the coupling force fs in a substepping strategy
Newmark method and a substepping technique (Bayer et al., 2005). Sincethe coupling force fc is unknown at time step (i + 1)∆t, the response di+1
and its derivatives cannot be obtained directly. Through substepping, thecoupled system of equations of motion reads,
Mai+1 + Cvi+1 + Kdi+1 = fl,i+1, + fs,i+1,k−1 (2.7)
where the index i + 1, k − 1 denotes the substep k − 1 in order to reachi + 1 and k is the number of substeps.
In order to satisfy equilibrium, an error force compensation method wasapplied, and a compensating force ∆f was added into the right side of (2.7);thus, the equilibrium equation becomes
Mai+1 + Cvi+1 + Kdi+1 = fl,i+1, + fs,i+1,k−1 + ∆fi+1 (2.8)
To approach the correct solution, the compensated force ∆f should ap-proach the true force ∆f∗
i+1, such that,
∆fi+1 → ∆f∗i+1 = fs,i+1 − fs,i+1,k−1 (2.9)
The compensation force ∆fi+1 is strongly depending on the response d andcan be treated as
∆fi+1 = P [fu,i + I∆t
i∑
i
fu,i +D
∆t(fu,i − fu,i−1)] (2.10)
where P, I and D are the proportional, integral and differential gains ofa controller,respectively. Numerical studies on these parameters need to
20
be performed before actual experiments to gather experience for optimalparameter settings. fu is the so-called unbalanced force in substep control.
fu = (fl,i + fs,i − (Mai + Cvi + Kdi)] (2.11)
The PID error force compensation algorithm can reduce of about 50%fu when P ≈ 1, I = 0, D = 0.
3 Classification, Analysis and Properties of Time
Integration Algorithms
We provide in Subsection 3.1 an essential classification of time integrationalgorithms for solving structural dynamic problems; while we define in Sub-section 3.2 notions of accuracy, stability, numerical dissipation, dispersionand error propagation in order to understand the attributes of some inte-gration schemes.
3.1 Classification of time integration algorithms
For convenience, we deal with the Hamilton form (2.3) of the equationsof structural dynamics. If we consider the problem (2.3) in the interval[0, tf ], it admits one and only one solution y(t), t ∈ [0, tf ], if the functionf (t, y) is continuous in t and Lispschitz continuous in y, that is,
‖f (t, y) − f (t, y)‖ < M ‖y − y‖ (3.1)
for some positive constant M (Lambert, 1991). As we anticipated, thereare many numerical methods capable of solving (2.3). In a greater detail,methods may be classified as explicit, when the direct solution of one ormore of the dynamical states does not require factorizing a combination ofthe damping and stiffness matrices; or implicit, when the direct solution ofthe dynamical states requires a factorization of some form of a matrix thatcomprises a combination of the mass, damping and stiffness matrices; one-step if they rely solely on the initial conditions provided at the beginning ofa time step tk and on the equations of motion and the dynamical states attime points within the time interval [tk, tk+1]; or multistep if they employa sequence of earlier dynamical states and equation of motion evaluations;single or multistage according to the number of times the equations of mo-tion must be evaluated to advance the solution by one time step. We recalltwo such methods to orient the reader.
The simplest explicit method is the one-step forward Euler method:
yk+1 = yk + ∆tf (tk, yk) (3.2)
21
Figure 8. Classification of integration methods.
where ∆t is the step size and yk+1 is the numerical solution. The simplestimplicit method is the one-step backward Euler method:
yk+1 = yk + ∆tf (tk+1, yk+1) (3.3)
This equation is usually solved for yk+1 by Newton-Raphson lineariza-tion, see Subsection 1.2, and involves the Jacobian matrix J = ∂yf (t, y).Both the Euler and backward Euler methods are simple but not very accu-rate and may require very small step sizes. More accurate finite differencemethods were developed from Euler’s method in two streams:
1. Linear multistep methods (LMS) which combine values yk+1, yk,yk−1, . . ., and fk+1, fk, fk−1, . . ., in a linear way to achieve higheraccuracy, but sacrifice the one-step format. Linearity permits simplelocal error estimates, but makes it complicate to change step size.
2. Runge–Kutta methods which achieve higher accuracy by retaining theone-step form but sacrificing linearity. One-step form makes it easyto change step size but renders it difficult the estimation of the localerror.
Other methods use the computation of the derivatives of the function y,viz. the Taylor series techniques; or of the function f , e.g. the Rosenbrock’smethods which will be introduced hereafter. All these methods can besummarized in Figure 8 using the notation proposed by Butcher (2003); themethods based on the derivative of the f function are preferred, as theyallow more flexibility.
22
Linear multistep methods LMS methods are typically applied to theHamilton form (2.3); see, for instance, Hairer and Wanner (1996) for atreatment of these methods. For brevity, an example of LMS method isderived from the single step CH-α method (Erlicher et al., 2002) introducedin Subsection 1.2.
Eqs. (1.23)-(1.25) allow di+1, vi+1 and ai+1 to be determined startingfrom known values of di, vi and ai. If the same relations are applied to thetime intervals [ti+1, ti+2] and [ti+2, ti+3], nine equations are obtained, inwhich four velocities and four accelerations can be eliminated. Thereby, arecurrence relation is obtained among displacements di+j , j = 0, 1, 2, 3,that leads to the LMS formulation of the CH-α method. The resultingthree-step algorithm reads
3∑
j=0
[Mαjdi+j + ∆tCγjdi+j] + ∆t23∑
j=1
δj(ri+j−αf− fi+j−αf
) = 0
(3.4)where fi+1−αf
= f[(1−αf)ti+1+αf ti] and ri+1−αf= r[(1−αf)di+1+
αfdi] are approximated by means of the mid-point rule. Moreover, theparameters in (3.4) read
α0 = αm γ0 = αf(−1 + γ) δ1 = 12
+ β − γ
α1 = 1 − 3αm γ1 = −1 + 2αf + γ − 3γαf δ2 = 12
− 2β + γα2 = −2 + 3αm γ2 = 1 − αf − 2γ + 3γαf δ3 = β
α3 = 1 − αm γ3 = (1 − αf)γ.(3.5)
Runge-Kutta methods In order to introduce the Runge-Kutta methodsin structural dynamics, we employ the form (2.3). Generally, a Runge-Kutta s-stage method applied to (2.3) may be expressed by
yk+1 = yk +
s∑
i=1
biki (3.6)
where bi are the weights of the function and ki are the estimates of the right-hand side of the differential equation, function of s intermediate substepinside the time step (Hairer et al., 1987). Their values are expressed as
ki = f(tk + ci∆t, yk +
s∑
j=1
aijkj)∆t (i = 1, . . . , s) (3.7)
In practice, the method sends feelers into the solution space, to gather sam-ples of the derivatives and the solution is obtained as a linear combination
23
of these feelers. The method can be summarized in the Butcher tableau(Butcher, 2003)
c AbT
where b is the vector of the weights of the estimates ki in (3.6), c is thevector of the time estimate of the function f , while A is the matrix of thecoefficients aij that define the weights of the estimates kj in the evaluationof the estimate ki in (3.7). The matrix A, the vectors b and c are made ofreal numbers. For instance, the explicit three-stage Runge-Kutta methodcan be summarized in the Butcher tableau
0 0 0 01/3 1/3 0 02/3 0 2/3 0
1/4 0 3/4
and it relies on the following solution:
yk+1 = yk +1
4k1 +
3
4k3, (3.8)
where
k1 = f(fe,k, yk)∆t,
k2 = f(fe,k+ 1
3
, yk +1
3k1)∆t, (3.9)
k3 = f(fe,k+ 2
3
, yk +2
3k2)∆t,
in which the external forces fe,k+ 1
3
and fe,k+ 2
3
are defined at time tk+ 1
3
=
tk + 13∆t and tk+ 2
3
= tk + 23∆t, respectively. All these intermediate
evaluations are sketched in Figure 9. Note that both the forward Euler andthe explicit three-stage Runge-Kutta algorithm are real-time compatible, asthey do not require the knowledge of the value of the function f or itsderivatives at the end of ∆t, to obtain the solution of (2.3). This is animportant property for SHSDS applications.
To study the properties of time integration algorithms in Subsection 3.2,we introduce shortly the real-time compatible Rosenbrock method with onestage (Rosenbrock, 1963). Let’s consider an autonomous problem, charac-terized by
y′ = f (y) . (3.10)
24
Figure 9. Representation of the Explicit three-stage Runge-Kutta method.
The solution of the problem by means of an s-stage Rosenbrock method isgiven by the formulas:
yk+1 = yk +
s∑
i=1
biki (3.11)
ki = ∆t(f(yk +
i−1∑
j=1
αijkj) + J
i∑
j=1
γijkj), (i = 1, . . . , s)
where αij, γij and bj are the algorithm coefficients and J is evaluated atthe initial step solution yk. Each stage of this method consists of a systemof linear equations with unknowns ki and with the inversion of the matrix[I − ∆tγiiJ] . We assume that γ11 = γ22 = . . . = γss = γ, so thatwe need only one LU -decomposition per step. Non-autonomous problems,characterized by the differential equation
y′ = f (t, y) (3.12)
can be converted to autonomous form by adding t′ = 1. If the method isapplied to the augmented system, the components corresponding to the t
25
variable can be computed explicitly, thus arriving to the equations
ki = ∆t(f(tk + αi∆t, yk +
i−1∑
j=1
αijkj) + γi∆t∂f
∂t
∣∣∣∣tk,yk
+ J
i∑
j=1
γijkj)
(3.13)
ki = ∆t(f(tk + αi∆t, yk +
i−1∑
j=1
αijkj) + J
i∑
j=1
γijkj) (3.14)
yk+1 = yk +s∑
i=1
biki (3.15)
where (3.14) is a simplified form of (3.13), and the additional coefficientsare given by
αi =
i−1∑
j=1
αij and γi =
i∑
j=1
γij (3.16)
The method with one stage reads:
yk+1 = yk + k1b1 (3.17)
with
k1 = f(tk, yk)∆t + ∆t∂f
∂y(tk, yk)γ11k1
k1 =f(tk, yk)∆t
(1 − ∆tJγ11)
(3.18)
The parameters values
b1 = 1 and γ11 = 1 (3.19)
entail L-stability to this method, labelled LSRT1. Additional details areprovided hereafter.
3.2 Analysis of time integration algorithms
In order to understand the pros and cons of the plethora of time inte-gration algorithms employed both for SHSDS and PsD testing, accuracy,stability, numerical dissipation and dispersion and error propagation arediscussed herein.
26
Accuracy A time integration algorithm can be considered accurate if thecomputed numerical response approaches as much as desired to the exactresponse. Usually, the load vector fe(t) is not included in the analysisassuming that the power of the leading error term of its approximation isgreater than the order of accuracy of the method. With reference to theHamilton form (2.3) of the equations of structural dynamics and along thelines of Hulbert and Hughes (1987) we consider the error both owing to theinitialization process and to the local truncation error. Accordingly, theglobal error ek at tk is affected by the initial error e0 and by the error ateach integration step ∆t, i.e.
eN = ANe0 +
N−1∑
k=0
AN−(k+1)τk (3.20)
where ek is defined as
ek = e (tk) = yk − y (tk) (3.21)
being y(tk) the corresponding exact solution of the state vector yk at tk
and τk = τ (tk) the local truncation error defined as
τk = τ (tk) = Ay (tk) + Lk − y (tk+1) . (3.22)
If τ = O(∆tk+1
)the algorithm is said to be of the k-th order (Hairer
et al., 1987). A treatment of e0 will be performed in Subsection 6.1 withreference to PsD tests; whilst the conditions to assure a specific accuracyorder of τk for the Rosenbrock method (3.13)-(3.15) are performed herecomparing the derivative terms of the Taylor series of the exact solutionwith those of the numerical solution. Along this line, we expand the Taylorseries of the SDoF differential equation y′ = f (t, y (t)) up to the thirdorder:
yk+1 = yk + fk∆t +∆t2
2(fk
∂f∂y
∣∣∣k
+ ∂f∂t
∣∣∣k) +
∆t3
6fk(
∂f∂y
∣∣∣k)2
+∆t3
6(f2
k∂2f∂y2
∣∣∣k
+ ∂f∂y
∣∣∣k
∂f∂t
∣∣∣k
+ 2fk∂2f∂t∂y
∣∣∣k
+ ∂2f∂t2
∣∣∣k) + O(∆t4)
(3.23)where fk = f (tk, yk) and k = (tk, yk). In a similar way the Taylor seriesexpansion of the problem integrated by a three stages Rosenbrock method
27
introduced in Subsection 5.1 is obtained:
yk+1 = yk + fk∆t(b1 + b2 + b3) + ∆t2(b1α1 + b2α2 + b3α3)∂f
∂t
∣∣∣∣k
+ fk∆t2(b2β21 + b3(β31 + β32) + γ)∂f
∂y
∣∣∣∣k
+
+ fk∆t3(γ − γ2 + (α32 + γ32)(α21 + γ21)b3)(∂f
∂y
∣∣∣∣k
)2+
+f2
k∆t3
2(α2
21b2 + (α31 + α32)2b3)
∂2f
∂y2
∣∣∣∣k
+
+ ∆t3(β21b2α1 + (α1β31 + α2β32)b3 +γ
2)
∂f
∂y
∣∣∣∣k
∂f
∂t
∣∣∣∣k
+
+ fk∆t3(α21b2α2 + (α31 + α32)b3α3)∂2f
∂t∂y
∣∣∣∣k
+∆t3
2(b1α2
1 + b2α22 + b3α2
3)∂2f
∂t2
∣∣∣∣k
+ O(∆t4) (3.24)
where βij = αij + γij . Comparing each term of (3.23) with those of(3.24) the accuracy conditions for a third-order system can be obtained. Fornon-autonomous systems, by substituting the values of (3.16), the accuracyconditions read
1st order b1 + b2 + b3 = 1
2nd order
b2α2 + b3α3 = 1/2b2β21 + b3 (β31 + β32) = 1/2 − γ
3rd order
b2α22 + b3α2
3 = 1/3α2β32b3 = 1/6 − γ/2β32β21b3 = 1/6 − γ + γ2
(3.25)
For a method with lower number of stages one can use the same equation(3.25) neglecting the coefficients with higher subscripts.
The conditions to obtain third order accuracy for a three stages Rosen-brock method applied to an autonomous system can be computed neglectingthe contribution of the derivative ∂f
∂tand read:
1st order b1 + b2 + b3 = 12nd order b2β21 + b3 (β31 + β32) = 1/2 − γ
3rd order
b2α2
2 + b3α23 = 1/3
b3β32β21 = 1/6 − γ + γ2
(3.26)
28
The first equation in (3.26) justifies the first coefficient in (3.19); theother order conditions have been used to set the coefficients of the Rosen-brock methods with two and three stages introduced in Subsection 5.1.
Due to relevance of the error (3.20) in the performance of an integrationalgorithm, several papers deal with error estimators. For brevity, we men-tion among others, Wiberg and Li (1993) and Lacoma and Romero (2007).For instance, Lacoma and Romero (2007) developed an algorithm for esti-mating the error when the HHT-α method is used both in the linear andnon-linear regime. The correct formulation of the estimator θi+1, is basedon the acceleration vector
anewi+1 =
1
α[ai+1 − (1 − α)anew
i ] (3.27)
which is a third-order approximation to the exact acceleration u(ti+1) ofthe local problem, since the one estimated by the HHT-α method is not suf-ficiently accurate as shown in Subsection 1.2. The pseudo-code to evaluatethe new error estimator for the HHT-α method is summarized here.
Given the solution at ti, that is, the vectors di, vi, ai:1. Compute the solution at ti+1, di+1, vi+1, ai+1 using the HHT-α
method.2. Compute anew
i+1 at ti+1 by (3.27).3. Compute the displacement, velocity and acceleration at instant ti+1/2:
a) for displacements and velocities, use the following expressions:
d∗i+1/2 = d∗
i+1/2 =3
4di +
1
4di+1 +
∆ti
4vi or (3.28)
d∗i+1/2 = d∗
i+1/2 =1
2(di + di+1) (3.29)
+∆tn
4(vi − vi−1) +
∆t2n16
(anewi + anew
i+1 )
v∗i+1/2 = vi +
3∆ti
8anew
i +∆ti
8anew
i+1 ]; (3.30)
b) for the acceleration, first compute the vector of internal forces attn+1/2 and, subsequently, solve the following linear system of equa-tions:
Ma∗i+1/2 = f(ti+1/2) + r(d∗
i+1/2 + v∗i+1/2) (3.31)
4. Compute the quadrature rule
Qn+1n =
∆tn
6
[vi
anewi
]+
4∆ti
6
[v∗
i+1/2
a∗i+1/2
]+
∆ti
6
[vi+1
anewi+1
]
(3.32)
29
5. Compute the error estimate and its energy norm
θi+1 = zi+1 − Qi+1i − zi =⇒‖ θi+1 ‖E (3.33)
6. Set i ← i + 1.
Absolute stability and the phenomenon of stiffness Two generaltypes of stability can be considered for time integration algorithms: i) theso-called external stability, conditioned by the external excitation; ii) theso-called equilibrium stability related to the free evolution of the discretedynamical system starting from some initial conditions as with the Lya-punov stability (Nawrotzki and Eller, 2000). With reference to the externalstability, different studies on the algorithmic behaviour with a forcing termwere performed by (Preumont, 1982). He derived the transfer functions ofthe discretized SDoF equation and the comparison between the numericaland the exact transfer function provided a more complete picture of the be-haviour of the Newmark method detecting spurious resonance conditions.Similar analyses were carried out on different schemes by others (Cannilloand Mancuso, 2000; Pegon, 2001; Mugan and Hulbert, 2001; Baldo et al.,2006).
Herein we develop equilibrium stability, which assures that initial dis-tortions and round-off errors do not increase artificially in the computa-tion process, i.e. the Lyapunov stability, or eventually damp out, viz. theasymptotical and global stability. We start introducing the definition ofA-stability as the properties of an integrator of preserving the Lyapunovstability of the numerical solution when applied to a stable system, i.e. asystem with s-plane representation on the left half-plane, described by theDahlquist differential equation problem (Dahlquist, 1963)
y′(t) = λy(t)y(0) = 1
(3.34)
This is a desirable property for a numerical method and the domain A thatcorresponds to an A-stable method for any value of λ∆t < 0, is
A = λ∆t ∈ C; |R (λ∆t)| 6 1 (3.35)
where R is named the stability function defined as yk+1 = R (λ∆t) yk.For a particular type of system, called a stiff system (Lambert, 1991), the
A-stability property might not be sufficient and we invoke global stability. Indetail, a system is said to be stiff in a given interval [0, tf ], if the neighboringsolution curves approach the solution curve at a rate which is very large incomparison with the rate at which the solution varies in that interval. Thus,
30
when we have a large value of |λ∆t| with |R| really close to 1, the high-frequency (stiff) components of the response are damped out very slowly.As a consequence, the L-stability property was introduced as the conditionto be satisfied, in addition to the A-stability property, for stiff problems. Itcan be summarized as
limλ∆t→−∞
R (λ∆t) = 0 (3.36)
and entails that an L-stable integration method nearly annihilates the re-sponse of a high-frequency mode in a single time step.
For instance and with reference to the LSRT1 method introduced inSubsection 3.1, we investigate the stability property when λ is large. Thesolution with the Rosenbrock method provides:
yk+1 = yk + k1b1, k1 =λ∆t
(1 − λ∆tγ)yk
yk+1 = yk +λ∆t
(1 − λ∆tγ)yk (3.37)
R (λ∆t) =yk+1
yk
=1 − λ∆tγ + λ∆t
1 − λ∆tγ
To analyse the A-stability property, we introduce the variable z = −λ∆t,imposing that |R (z)| 6 1 with z ≥ 0
R (z) =1 + zγ − z
1 + zγ(3.38)
|R (z)| =
∣∣∣∣1 + z (γ − 1)
1 + zγ
∣∣∣∣ < 1
γ > 1/2
To induce the L-stability property, we impose that limz→+∞ R = 0, i.e.
R (z) =1 + z (γ − 1)
1 + zγ(3.39)
limz→+∞
R =z (γ − 1)
zγ=
γ − 1
γ= 0
γ = γ11 = 1
This value agrees with (3.19). The plot of the relevant stability functionR(z) for γ11 = 1 is shown in Figure 10(a).
31
(a) (b)
Figure 10. (a) Stability function plot for a first-order system with γ = 1and with a single-stage Rosenbrock method assuming b1 = 1;(b) stabilityfunction plot for a second-order system with γ = 1 and with a single-stageRosenbrock method assuming b1 = 1.
To analyse the properties of the LSRT1 method for a mechanical second-order system, we consider the model problem
y′ = pp′ = −ω2y
(3.40)
As a result, λ2 = −ω2 and the spectral radius ρ = |R(z)| of the LSRT1method is simply obtained by substituting λ1,2 = ±iω into the stabilityfunction R(z). The corresponding evolution of ρ is plotted in Figure 10b.For brevity, it is compared only with the L-stable CH -α method character-ized by ρ∞ = 0 (Chung and Hulbert, 1993); one can observe the propertyof the LSRT1 method.
Both the property of accuracy dealt with in Subsection 3.2.1 and sta-bility considered in this subsection entail convergence of a time integrationalgorithm applied to a linear system (Lambert, 1991). Nonetheless, the in-tegration of non-linear dynamic systems may impair numerical stability andconvergence of computed responses for several reasons (Soroushian et al.,2005). Some of them are listed here: i) computed responses affected by bothtruncation errors, see (3.22) and residuals of non-linearity solutions, e.g.(1.22); ii) approximations in the formulation of time integration methods,like the CH-α method that replaces the equations of motion with balanceequations, see (1.12), where the acceleration-like variables are approximateas detected by (1.28).
32
Dissipation and dispersion Other important measures which are usefulin the choice of time integration algorithms are the algorithmic dampingratio ξ and the frequency error (Ω− Ω)/Ω, where Ω = ω∆t (Hughes,1987). For instance, with reference to the LSRT1 method applied to (3.40),the response can be expressed as
yk+1 =2∑
s=1
csλk+1s
= e(−ξΩ(k+1))[d1 cos(Ω (k + 1)
)+ d2 sin
(Ω (k + 1)
)] (3.41)
being λ1,2
λ1,2 = C ± iD (3.42)
the eigenvalues of the amplification matrix of the method and di constantsdetermined from initial conditions. As a result, ξ and ω = Ω/∆t read
ξ = − ln(C2 + D2
)
2Ω(3.43)
ω∆t = Ω = arctan
(D
C
)(3.44)
Both ξ and the frequency error (Ω− Ω)/Ω are depicted in Figure 11,respectively, and compared to the corresponding values of the CH-α methodwith ρ∞ = 0. The favourable properties of the LSRT1 method in terms ofhigh algorithm damping and limited frequency error are evident.
These measures can be expressed also in an alternative form using adigital filter theory as presented in Subsection 4.3.
Error propagation in PsD testing In Subsection 3.2 we have reportedan algorithm proposed by Lacoma and Romero (2007) able to perform er-ror estimate when the HHT-α method is used to solve structural dynamicproblems. The efficiency of a time-stepping scheme applied, for instance, toa PsD testing is connected to its reduced sensitivity to errors of differentnature. In a PsD environment several errors arise (Combescure and Pegon,1997) and the integrator should mitigate their effects. In fact, the time inte-gration algorithm computes the displacement to be imposed on the structureand the digital board of the actuators generates an approach function to thedesired target. As the approach process has to be stable and gradual, it isinevitable a certain difference between the effective displacement applied tothe tested structure (superscript c) and the target displacement computedfrom the PsD algorithm, which is called stepwise control error
ed,ci+1 = dc
i+1 − di+1. (3.45)
33
(a) (b)
Figure 11. (a) Equivalent damping for a second-order system with a single-stage Rosenbrock method assuming γ = 1 and b1 = 1; (b) frequency errorfor a second-order system with a single-stage Rosenbrock method assumingγ = 1 and b1 = 1.
Accurate displacement control has been found to be of prime importanceto ensure reliable response. In fact owing to undershoot, the controllergenerates error forces which are in phase with the structure under testing; ifundershooting errors depend on the sign of the displacement increment, theyadd energy to the test piece (Combescure and Pegon, 1997). An additionaldisplacement error is the stepwise measurement error (superscript m) causedby the precision of the displacement transducers which reads
ed,mi+1 = dm
i+1 − dci+1. (3.46)
Another source of errors is the stepwise measurement error associated withthe reaction force
er,mi+1 = rm
i+1 − rci+1. (3.47)
By considering a first order approximation, the total error on the reactionforce can be recast in the following form
eri+1 = rm
i+1 − ri+1 = rmi+1 − rc
i+1 + rci+1 − ri+1 ≃ er,m
i+1 + Ka,i+1ed,ci+1
(3.48)where Ka,i+1 represents the actual stiffness of a structure. The off line anal-ysis of the error effects on the structural response can easily be performedconsidering a linear elastic structure with a single DoF (Bursi and Shing,1996; Combescure and Pegon, 1997; Bonelli and Bursi, 2004; Chang, 2005).More specifically, a single error in displacement or in force is equivalent to anexternal force applied to the structure leading to different effects dependingon the scheme employed. The analysis described herein regards the effect
34
of uncorrelated errors. Nonetheless, some errors like displacement controlerrors are correlated, and therefore analyses based on the discrete-time fre-quency response function of the scheme have been performed too (Combes-cure and Pegon, 1997; Bonelli and Bursi, 2004). Also on line analyses can beperformed based on error monitors, which can provide on line early warn-ings of unacceptable levels of experimental errors (Mosqueda et al., 2007a,and Mosqueda et al., 2007b). These error monitors based on estimates ofenergy added to the simulation allow to pause the test, and correct theproblem preferably before the specimen is irreparably damaged.
For brevity, we summarize herein the off line analysis applied to theIPC − ρ∞ method (Bonelli and Bursi, 2004). Let’s analyse the effect ofuncorrelated errors by means of an error excitation fe
j of impulsive type,such that
fej =
fe if j = i − 10 j 6= i − 1
(3.49)
Under this assumption, the response of a SDoF system evaluated with theIPC − ρ∞ algorithm amounts to
Yei = ((1−αf)A+αfI) B
[fe
i−1
0
](3.50)
and hence
Yei+r = ArYe
i = Ar ((1−αf)A+αfI) B
[fe
i−1
0
](3.51)
where A is the amplification matrix of the IPC − ρ∞, Y = di, ∆tvi,
∆t2ai, ∆t2rr
i
mT denotes its state vector and B is related to the forcing
terms. Equation (3.51) can be represented in terms of eigenvalues λk of Aand more specifically, the displacement assumes the following form
dj =
n∑
k=1
ckλjk. (3.52)
A has four eigenvalues and, retaining only the complex conjugate ones,(3.52) can be recast as
dj = e(−ξωi∆t) (c1 cos (ωi∆t) + c∗
1 cos (ωi∆t)). (3.53)
The error amplification factor C1 reads
C1 = c1 + c∗1, (3.54)
35
where c1 and c∗1 are complex conjugate. C1 and e(−ξωi∆t), allow the error
propagation characteristics of a PsD algorithm to be evaluated. As lengthyexpressions are associated with c1 and c∗
1, only a graphical representationof C1 is provided.
Consider a control error ed,ci+1 in the actuator movement with no measure-
ment error. By substituting Yi=
dri , ∆tvr
i , ∆t2ari , ∆t2
rri
m
T
, which
denotes the exact numerical solution evaluated from experimental feedbackquantities, in (3.50) and using (3.48), one obtains the following error force
fei =
K + K
K + 2K − Ka
(Ka − K) ed,ci+1 (3.55)
for the IPC − ρ∞ scheme where K is found in (1.45). Conversely, the
error force related to the measurement error ed,mi+1 without control errors
readsfe
i = Kaed,mi+1 . (3.56)
The performance of the OSM − ρ∞ scheme and of the C − ρ∞ methodis presented, where these acronyms stand for the OSM and the C − αscheme implemented in the CH −α method, for a fair comparison (Bonelliand Bursi, 2004). In these cases, the error force caused by a stepwise control
error ed,ci+1 reads
fei = (K − Ka) ed,c
i+1. (3.57)
The corresponding error amplification factors are depicted in Figure 12.The favourable behaviour of the IPC − ρ∞ method at high frequencies isevident. Moreover, the IPC − ρ∞ and the C − ρ∞ method do not showany unfavourable effect when R > 1, where R = Ka
Kis the stiffness ratio.
Conversely, the OSM − ρ∞ scheme exhibits bifurcation, i.e. the complexconjugate roots λ1,2 become real and distinct, a clear warning of instability.
A complementary way to characterize the time-stepping scheme in thepresence of errors consists in the analysis of its discrete-time frequency re-sponse function, assuming that the error force fe
i acts at each time step(Combescure and Pegon, 1997; Bonelli and Bursi, 2004). It is clear thaterrors derived from experimental data can be random but also systematicowing to significant bias or calibration errors. Therefore, probabilistic tech-niques based on vectorized Monte Carlo simulations or the like should beused to find cumulative probability distributions for output variables of anal-yses and to investigate the uncertainty propagation in numerical/physicalmodels. Analyses of this type were performed by Vasquez and Whiting(2006).
36
Figure 12. Control error amplification factor for various algorithms imple-mented in the CH-α method for ξ = 0
37
4 Coupled integration and control engineering issues
In this section we introduce problems and analysis techniques that arisewhen coupling of numerical and physical substructures occurs. Real timesituations are challenging in coupling and relevant requirements are dealtwith in Subsection 4.1. As SHSDS require the use of actuators and con-trollers to control them, in Subsections 4.2 we deal with numerical aspectsrelevant to the discretization of an adaptive controller and its linearization.Subsection 4.3 concludes this Section applying concepts from digital filteringto time integrators.
4.1 Coupling algorithms
In this subsection, we study the simulation of coupled dynamic systemsputting emphasis on partition and interconnection. The subject emergedover 30 years ago (Felippa and Park, 1980), but impulse was given to for-mulations and implementations rather recently (Park and Felippa, 2000 andFelippa et al., 2001) through the availability of inexpensive computer alge-bra systems. Borrowing the terminology from the area of FSI, see Zhangand Hisada (2004) among others, we distinguish between strong couplingor direct coupling when the variable vector of the discretized coupled sys-tem is solved and corrected simultaneously; conversely, in the weak cou-pling method an iterative method is employed to solve the same discretizedequations. Moreover, we distinguish between monolithic approaches wheresolution methods are developed ad hoc to solve coupled applications; andpartitioned methods where separate solvers are used for the fluid and thestructure, respectively. Moreover, large systems of non-linear equationsoriginated from subsystems are treated with iterative solvers, i.e. Jacobi orGauss–Seidel solvers.
Typically, the coupled system can be expressed in the following form:
y1 = f1(y1, y2, u1) (4.1)
0 = g1(y1, y2, u1), (4.2)
and
y2 = f2(y1, y2, u2) (4.3)
0 = g2(y1, y2, u2), (4.4)
with the global coupling condition
0 = h(y1, y2, u1, u2), (4.5)
38
In these conditions, an algebraic loop might exist among some of thevariables of the models. We assume that we have integrators for each sub-system separately. Hence, for Subsystem 1, we may say that given the values
(y(i)1 , u
(i)1 ) and the functions (y2, u2) over the interval [ti, ti+1], with an
implicit method we obtain
(y(i+1)1 , u
(i+1)1 ) = Φ1(y
(i+1)1 , u
(i+1)1 , y
(i)1 , u
(i)1 , y2, u2) (4.6)
Similarly, for Subsystem 2 we have a similar function,
(y(i+1)2 , u
(i+1)2 ) = Φ2(y
(i+1)2 , u
(i+1)2 , y
(i)2 , u
(i)2 , y1, u1) (4.7)
From these single system integrators, different options may be used tointegrate the global system. The simplest strategy is a staggered scheme,also referred to as weak coupling. The scheme then is for Subsystem 1 toassume the time evolution of the state variables (y2, z2) to be constant over
the interval [ti, ti+1], i.e. identically equal to (y(i)2 , u
(i)2 ), and then solve
(4.6). Similarly for Subsystem 2, we assume (y1, z1) over the time incre-
ment to be constant and identically equal to the initial values (y(i)1 , z
(i)1 ),
and with this we can solve (4.7). To sum up, information is only exchangedbetween the subsystems after the increment has been performed; inside theincrement, each subsystem is solved in parallel independently of each other.This scheme will only give first order accuracy in time, no matter how accu-rate the single system integrators are. This could be improved by a higherorder extrapolation, but this might have a negative impact on its stabilitycharacteristics (Felippa et al., 2001). Although the integrators for the in-dividual subsystems may be implicit, the staggered strategy is explicit andtherefore conditionally stable. Moreover, global constraints are not satisfiedexactly and energy conservation is not assured.
Implicit methods which use partitioned software but satisfy an implicitcondition globally are desired and are referred to as strong coupling. Nonethe-less, implicit methods usually need iterations and starting values to extrap-olate the solution into the new time step. Thus, both Subsystems (4.6) and(4.7) can be independently solved and if wanted in parallel until conver-gence.
It is evident from Subsection 1.1 that integration algorithms employed inSHSDS must take into account the strategy to couple NS and PS. Firstly,the real-time requirement for numerical integrators must be fulfilled duringthe simulation, and therefore iterations during coupling must be avoided:this leads to a weak coupling. As a result, we need to resort to prediction-substitution procedures which can be based either on static or kinematiccontrol at the interface. The relevant coupling possibilities are shown in
39
Figure 13. Coupling strategies with f force quantity vector and X kine-matic quantity vector
40
Figure 14. Coupled integration in displacement control
Figure 15. Coupled integration in displacement control with the algebraiccoupling equation
Figure 13 where f represents the vector of force quantities and X the vectorof kinematic quantities, respectively.
SHSDS does not require computations on the PS which behaves as ablack box and the Calculator has to enforce an interconnecting conditionbetween the NS and PS ; thus, the mixed f−X coupling strategy illustratedin Figure 13c is preferred. In detail, the interface kinematic vector Xn
k
of the NS and the force quantity vector fpk of the PS are used as input
to the Calculator, while the force quantity vector fnk of the NS and the
interface kinematic vector Xpk of the PS become output from the Calculator.
Clearly, the mixed strategy can be set as a X −f coupling strategy, but thedisadvantage in operating the transfer system in force control favours thef − X coupling strategy. This strategy is schematically depicted in Figure14 were the interconnecting function of the Calculator is emphasized.
With reference to a splitted-mass SDoF system illustrated in Figure 1(c)with mp
2 = kp2 = cp
2 = 0, an algebraic coupling is shown in Figure 15 wherewe assumed that the application of kinematic quantities and the relevant
41
measurements entail a delay of ∆t (Bursi et al., 2008). All the inputs unk
and upk, corresponding to the NS and the PS inputs are shown and collected
in a vector uk. The same is done with the outputs, using the vector zk.The interconnection enforces the algebraic condition
uk = Lzk, (4.8)
where L is a time invariant Boolean localization matrix representing theinput/output interconnection of the block diagram and containing only zerosand ones. As a result, one gets
L =
[0 Lp
Ln 0
], zk =
x2,k
mpx2,k−1
, uk =
un
k
upk
. (4.9)
When using this coupling strategy, the set of differential equations (1.2) cor-responding to the numerical subsystem are integrated to advance from tk−1
to tk as shown in Figure 15. The proposed procedure is rather different froma truly parallel scheme (Felippa et al., 2001), which would entail the cou-pling force as predictor at tk. In fact, in the actual prediction-substitutionprocedure the coupling force is taken at tk−1 owing to the nature of thePS. The prediction-substitution scheme used to enforce the interconnectionis summarized in algorithmic form as follows:
• Initialization procedure:
(a) t = 0; k = 0
(b) Assuming Mn, Cn, Kn and the initial conditions x0, x0 known,estimate x0 = (Mn)−1 (−Cnx0 − Knx0 + fe,0)
(c) initial coupling forces: fs,0 = Mnx0 + Cnx0 + Knx0 − fe,0(d) initial input vector: fe,0 fs,0
• For each time step:
1. perform the numerical integration: xnk ;
2. update the PS input vector: xpk = xn
k;
3. apply xpk onto the PS while measuring fs,k;
4. update the NS input vectors: fe,k fs,k ;
5. k = k + 1; tk = tk + k∆t.
Note that to perform the numerical integration of Step 1, it is neces-sary to employ explicit methods like the ones introduced in Subsection 1.1and 3.1; or linearly implicit methods like the LSRT1 method introduced inSubsection 3.1; or companion methods of the same class which will be in-troduced in Subsection 5.1. Moreover along the lines of Burgermeister et al.
42
(2006), the injection of Mp, Cp and Kp of the PS in the Jacobian matrixJ could improve the stability properties of linearly implicit methods. Asshown in Subsection 5.2, these methods might loose their stability propertywhen applied to coupled problems in HiL.
4.2 Delay and Compensation Techniques
Being the SHSDS based on the PS feedback, slightest inaccuracies trans-late into a feedback error which, in turn propagate into an actuator com-mand signal error on the following time interval. To characterise this com-plex dynamics in terms that can be analysed and acted upon during aSHSDS, a pragmatic approach is to define a time delay and an amplitudeerror, assumed to vary continuously. The time delay approach of actuatordynamics has been used by numerous researchers in the field of SHSDS (Ho-riuchi et al., 1999 and Darby et al., 2002). As the delay can be of the orderof 5-15ms, it is not negligible with respect to the ∆t considered. In general,the total time delay in a control system can be partitioned into two contribu-tions. The first part is referred to as fixed time delay owing to on-line dataacquisition, filtering, manipulation of digital data inside the digital controlprocessor, calculation of the required control force and signal transmissionfrom the computer to the actuators. The second part usually depends on theparticular dynamics of the actuators interacting with the controlled plant,which is referred to as floating delay time; it can be adjusted by explicitlycombining the actuator dynamics with that of the plant (Chu et al., 2002).For instance in real-time PsD tests a fixed time delay of the order of 3∆twas measured by Jung et al. (2007). In detail, the time delay between thecomputation of displacements in the target PC and the initiation of thecomputation of control signals in the digital controller was of the order of2∆t. An additional ∆t was required to execute the computation of controlsignals and sending signals to the valve drives. Bonnet (2006) classified thecompensation methods devoted to the delay into two main categories: i) theschemes that, from known, assumed or calculated delay and amplitude errorestimates, aim at directly correcting the command signal accordingly: theywere labelled as direct compensation schemes; ii) the algorithms that correctthe command signal from the knowledge of the real-time error to the desiredoutput using more standard control engineering techniques, referred to asouter-loop controllers. Within the former category fall the forward predic-tion schemes proposed by Wallace et al. (2005) and Horiuchi et al. (1999),which assume that the delay and amplitude factor estimates are known andaccurate; and the performance estimation algorithms suggested by Bonnetet al. (2007), which compute the delay and amplitude factor more precisely
43
Figure 16. Block diagram representation of the outer loop controller con-cept
along a test starting from rough estimates. To the second category be-long the minimal control synthesis family of outer-loop control algorithms(Stoten and Neild, 2003); instead of trying to compensate explicitly for theactuator delay, this approach considers the synchronisation error as a wholeand minimises it. This is obtained by including a model of the transfersystem as illustrated in Figure 16. With this approach all information goesthrough the controller, which computes the transfer system command signalaccordingly. Eliminating the direct influence from the NS to the PS is theaim of the outer loop controller. This concept was implemented for simpleSHSDS at the Bristol University using the shaking table as transfer systemfor the PS (Neild et al., 2005).
Different researchers also proposed to adopt adaptive control algorithms.In detail, adaptive control techniques provide a mechanism to continuouslyupdate the controller parameters with respect to the variation of the dy-namics. Therefore, a change in a substructure would not necessarily implythe retuning of the control system. Researchers from Bristol Universityworked on the minimal control synthesis (MCS) family of adaptive con-trollers (Stoten and Neild, 2003) towards their use in SHSDS testing (Neildet al., 2005). Variants of this controller were proposed: the MCSmd al-gorithm, by introducing some previous knowledge of the transfer system toinitialise the gains to typical values and an inverse of the reference model ap-plied to the external demand before the application of the controller gains(Lim et al., 2005). The MCSmd outer loop controller was discussed andtested by Bonnet et al. (2007) through a series of SHSDS including non-linearities in the PS and multi-axis tests. The stability and accuracy prop-erties of the MCS algorithm by applying techniques from numerical analysesand time-delay effects usually present in the inner dynamics are consideredhereafter.
44
4.3 The ZOH and the MCS controller
The MCS algorithm mentioned in the previous subsection belongs todissipative controllers, as its properties are based on Popov’s theory ofhyperstability (Popov, 1973), which implies the positive real property ofcontrol systems and the dissipative (passive) property of systems. If oneconsiders a square stable plant (A, B, C) , i.e. a linear system with thenumber of inputs equal to the number of outputs, an open-loop square sys-tem with simple poles is dissipative, if there exists a symmetric positivedefinite matrix P and a matrix Q that satisfy the following equations
AT P + PA = −QT Q (4.10)
BT P = C (4.11)
This system is strictly dissipative if QT Q is positive definite. The equa-tions above allow for the simple determination of a dissipative system. GivenA and B we select the matrix Q. Next we solve (4.10) for P and we find theoutput matrix C from (4.11). Some matrices introduced above are indicatedin Figure 17.
Stoten and Benchoubane (1990) proposed the MCS algorithm whichderives from the MRAS formulation: it assumes null constant gains Kand Kr and unknown plant parameters. Quantities to be known are thereference model parameters Am and Bm, the structure of the plant withthe degree of freedom and order and the sign of the coefficients of B, thatusually is assumed to be positive. Therefore the control law becomes
u (t) = δK (t) x (t) + δKr (t) r (t) (4.12)
and the output readsye (t) = Cexe (t) (4.13)
whereCe = P (4.14)
and P is the solution of the Lyapunov equation (4.10). Stoten and Neild(2003) proposed a pragmatic solution for first order, e.g. Ce = [4/ts] andsecond order Ce = [4/ts 1] SDoF systems that induces an exact pole-zerocancellation. The block diagram of a standard MCS controller is shown inFigure 17.
The MRAS formulation contains a linear controller whose coefficients aredetermined from the plant model. In order to be implemented, it needs thedevelopment of a linearized plant model, a reference model, a linear modelreference control and a hyperstable block. The MCS controller does notneed all the above elements and provides excellent responses which match
45
Figure 17. The standard Minimal Control Synthesis algorithm
Figure 18. Plant model
those of the MRAS controller. Therefore, the comparable performance ofthe MCS algorithm with the MRAS algorithm permits to understand theadvantages of the first one, since allows accurate system identification to beavoided.
The analysis of the MCS controller allows us to introduce: (i) the discretecontrol law of the MCS sampled by the Zero Order Hold (ZOH ) method(Vaccaro, 1995); (ii) two linearization techniques based on physical insightand Taylor series expansion; (iii) a stability analysis. For brevity, bothaccuracy and frequency domain analyses are not dealt with here; they canbe found in Bursi et al. (2007). Moreover, first-order linear time-invariantsystems both for the plant and the reference model are considered.
The plant is a first-order system which can be represented by means of
46
Figure 18. Its behaviour is described by the continuous-time equation
(u (t) − x (t)) k − cx (t) = 0x (t) + x (t) /T = u (t) /T
(4.15)
where u (t) is the input, i.e. the control variable, x (t) is the output, i.e.the displacement of the mass m, and T = c/k is the time constant of thesystem. Considering a state-space approach, (4.15) can be rewritten as
x (t) = Ax (t) + Bu (t) (4.16)
where A = −1/T and B = 1/T. One can note that every term is scalar.By applying a ZOH sampling, (4.16) becomes
x [k + 1] = A′x [k] + B′u [k] (4.17)
where
A′ = eA∆t = e−∆t/T (4.18)
B′ =
∫ ∆t
0
eAτ dτB =1
A
(eA∆t − 1
)B = 1 − e−∆t/T (4.19)
In a similar fashion, the discrete equivalent of the reference model be-comes
xm [k + 1] = A′mxm [k] + B′
mr [k] (4.20)
where
A′m = e−∆t/Tm = e−4∆t/ts (4.21)
B′m = 1 − e−∆t/Tm = 1 − e−4∆t/ts (4.22)
with Tm time constant of the reference model. It is assumed that Tm =ts/4, where ts is the settling time, used as a measure of the time taken forthe oscillations to die away (Vaccaro, 1995).
Figure 17 shows the block diagram of the MCS controller whose controllaw is
u (t) = K (t) x (t) + Kr (t) r (t) (4.23)
where K (t) and Kr (t) read
K (t) =
∫ t
0
αye (τ ) x (τ ) dτ + βye (t) x (t) (4.24)
Kr (t) =
∫ t
0
αye (τ ) r (τ ) dτ + βye (t) r (t) (4.25)
47
The error output is given by
ye (t) = Cexe (t) (4.26)
where Ce = P. In detail, P is the solution of the Lyapunov’s equation andxe(t) = xm(t) − x(t) is the error between the reference model and plantstates.
To generalize the numerical treatment of the controller, we introduce anad hoc delay in the control law. Thus, (4.23) becomes
u (t) = K (t − γδt) x (t − γδt) + Kr (t − γδt) r (t − γδt) (4.27)
where γδt can assume any value from zero to infinity to represent typicalsources of delay in a controlled system (Chu et al., 2002). The generalformulation of the discrete control law equation provided by the ZOH forthe MCS controller considering a fixed delay time γδt = γ∆t in (4.27)reads:
u [k] = K [k − γ] x [k − γ] + Kr [k − γ] r [k − γ] (4.28)
where the adaptive gains read
K[k − γ] =K[k − 1 − γ] + βye[k − γ]x[k − γ]
− σye[k − 1 − γ]x[k − 1 − γ](4.29)
Kr[k − γ] =Kr[k − 1 − γ] + βye[k − γ]r[k − γ]
− σye[k − 1 − γ]r[k − 1 − γ](4.30)
with σ = β − α∆t (Vulcan, 2006).By assuming no delay, i.e. γ = 0, the initial conditions of the MCS
controller read K[−1] = 0 and KR[−1] = 0, respectively. To completethe characterization of the controller, the eigenvalues of the reference model,α and β from (4.24) and (4.25) and Ce must be selected. In addition,the reference model must be stable. The values of α and β are arbitrarypositive numbers and are selected by trial and error. However, an increaseof β entails a reduction of the settling time ts of the adaptation, while areduction of the β/α ratio improves damping (Bursi et al., 2007). The ratioβ/α can be set to 0.1 as a compromise between the speed of the adaptationmechanism and the stability limit (Vulcan, 2006). The relatively arbitraryselection of the error vector weighting matrix Ce is like the one proposed forcontinuous time systems, i.e. Ce = 4/ts. Hereafter, the analysis is carriedout by considering two limit cases: i) γ = 0; we implicitly assume thatthe processing time δ is negligible with respect to the time step ∆t; and
48
therefore, the control signal u generated by the MCS algorithm is obtainedinstantaneously; ii) the fixed delay time is equal to ∆t. The second casecould be an arguably assumption, since in reality the delay may be a fractionof ∆t, i.e. 0 ≤ γ ≤ 1; however, it is always possible to manage the delayto be equal to ∆t, such that this assumption is correct. Therefore, theanalyses presented hereafter allow the effect of one ∆t in the control lawcomputation on the system to be understood. The reference input signalused for the stability analysis reads
r [k] =
0, k < 01, k ≥ 0
(4.31)
As the overall system is non-linear owing to the adaptation mechanism,the stability analysis is carried out by linearizing the system at steady statein a neighbourhood of an operating point (Vulcan, 2006). Thus, the compar-ison between the linearization by physical insight and by the more rigorousTaylor series expansion is presented herein; this comparison shows the ad-vantages of the first approach both in terms of computational burden andof an a priori selection of the control parameters α and β. In this first case,the analysis of the system is developed without delay and therefore γ = 0in (4.28).
The linearization by physical insight is summarized herein. The discrete-time equations that rule the system can be arranged as
xe [k + 1]xe [k]
x [k + 1]K [k]Kr [k]
= CIT
xe [k]xe [k − 1]
x [k]K [k − 1]Kr [k − 1]
+
B′mr [k]0000
. (4.32)
where CIT reads,
CIT =
C11IT C12
IT A′m − A′ −B′x[k] −B′r[k]
1 0 0 0 0C31
IT C32IT A′ B′x[k] B′r[k]
βx[k]Ce −σx[k − 1]Ce 0 1 0βr[k]Ce −σr[k − 1]Ce 0 0 1
(4.33)
49
with
C11IT = A′
m − B′βCex2[k] − B′βCer2[k] (4.34)
C12IT = B′σCer[k − 1]r[k] + B′σCex[k − 1]x[k] (4.35)
C31IT = B′βCex2[k] + +B′βCer2[k] (4.36)
C32IT = −B′σCer[k − 1]r[k] − B′σCex[k − 1]x[k] (4.37)
CIT is not constant owing to the adaptation mechanism; thus, to reducethe complexity of the analysis, CIT is locally linearized in a neighbour-hood of an operating point, located at the steady-state condition, such thatlimk→+∞ xe [k] = 0. The linearization by physical insight consists ofsubstituting the values corresponding to the operating point, i.e.
x [k − 1] = x [k] = r [k − 1] = r [k] = 1 and xe [k − 1] = xe [k] = 0(4.38)
in the amplification matrix CIT , which in turn, is arranged to reduce thecomplexity of the analysis. As a result, the amplification matrix at steadystate reads
CIT,ss =
A′m − 2B′βCe 2B′σCe A′
m − A′ −B′ −B′
1 0 0 0 02B′βCe −2B′σCe A′ B′ B′
βCe −σCe 0 1 0βCe −σCe 0 0 1
(4.39)where the eigenvalues of CIT,ss read
λ1 = 1, λ2 = 0, λ3 = A′m (4.40)
λ4 = 12(1 + A′ − 2B′βCe −
√(1 + A′ − 2B′βCe)2 + 8B′σCe − 4A′)
λ5 = 12(1 + A′ − 2B′βCe +
√(1 + A′ − 2B′βCe)2 + 8B′σCe − 4A′)
In the linearization by Taylor series expansion, the amplification ma-trix CIT,T,ss is obtained by linearizing the system through Taylor seriesnear the operating point at steady state conditions. Usually, the discretelinearization techniques follow those of continuous-time non-linear systems(Vaccaro, 1995). In detail, the system of state space equations of a non-linear discrete-time plant reads
x [k + 1] = f (x [k] , u [k]) (4.41)
As for continuous-time systems, it is advantageous to define variablescentred about an operating point (x0,u0)
x [k] = x0 + δx [k]u [k] = u0 + δu [k]
(4.42)
50
By substituting (4.42) in (4.41), one gets
δx [k + 1] = f (δx [k] + x0, δu [k] + u0) − x0 (4.43)
Thus, expanding the rhs of (4.43) in Taylor series about an operating pointand keeping only linear terms, one obtains
δx [k + 1] =∂f
∂x [k]|(x0,u0) δx [k] +
∂f
∂u [k]|(x0,u0) δu [k] (4.44)
Therefore, the Jacobian matrix with respect to every state and errorvariable can be computed. When we apply this linearization to the sys-tem described by (4.32), the computation of the Jacobian matrix is neededas well as the evaluation of the resulting matrix at the operating point.Moreover, the vector related to the input is obtained by differentiating theequations with respect to r [k] and then by evaluating every term at theoperating point. The operating point is defined by
xe0 [k] = xe0 [k − 1] = 0, x0 [k] = x0 [k − 1] = 1
K0 [k − 1] = K0 [k − 2] = K∗, (4.45)
Kr0 [k − 1] = Kr0 [k − 2] = K∗r , r0 [k] = r0 [k − 1] = 1
As the equations that rule the system do not depend on the variable x[k −1], K[k − 2] and Kr[k − 2], one can simply reduce the problem to a 5x5system. As a result, one gets the following relationship
xe [k + 1]xe [k]
x [k + 1]K [k]Kr [k]
= CIT,T,ss
xe [k]xe [k − 1]
x [k]K [k − 1]Kr [k − 1]
+DIT,T,ss r [k] (4.46)
51
where CIT,T,ss and DIT,T,ss at steady state read
CIT,T,ss =
A′m − 2B′βCe 2B′σCe A′
m − A′ − B′K∗ −B′ −B′
1 0 0 0 02B′βCe −2B′σCe A′ + B′K∗ B′ B′
βCe −σCe 0 1 0βCe −σCe 0 0 1
(4.47)
DIT,T,ss =
−B′K∗r + B′
m 00 0
B′K∗r 0
0 00 0
(4.48)
The eigenvalues of CIT,T,ss are listed here
λ1 = 1, λ2 = 0, λ3 = A′m (4.49)
λ4 =1
2+
1
2A′ − B′βCe +
1
2B′K∗+
− 1
2
√(1 + A′ − 2B′βCe + B′K∗)
2 − 4 (A′ + B′K∗ − 2B′σCe)
λ5 =1
2+
1
2A′ − B′βCe +
1
2B′K∗+
+1
2
√(1 + A′ − 2B′βCe + B′K∗)
2 − 4 (A′ + B′K∗ − 2B′σCe)
One can observe that CIT,T,ss is equal to CIT,ss of (4.39), apart fromthe term B′K∗. Therefore, the results provided by physical insight corre-sponds to the case B′K∗ equal to zero. This approximation entails thateither K∗ or B′ must be very small. As B′ = 1 − e−∆t/T , the two matri-ces meet if ∆t → 0, or T approaches large values. The term K∗ appearingin (4.48) is not known a priori; therefore, the analysis can be performed onlywith the support of preliminary numerical simulations (Vulcan, 2006). Here-after, the amplification matrices obtained at steady-state are analysed byintroducing the following non-dimensional variables: ∆t
Tm, β
4Tm, T
Tm, β
α1
4Tm.
The absolute stability of the discrete system is assured when the modu-lus of the eigenvalues of the amplification matrix at the steady-state regimeis less than one. For the linearization by Taylor series expansion only 4 and5 of (4.49) are relevant for stability analysis; so these eigenvalue moduli andthe eigenvalue moduli 4 and 5 of (4.40) obtained by means of the lineariza-tion by physical insight can be compared in Figure 19. In the figure, the
52
Figure 19. Eigenvalue moduli for a unit step input, β/(α4TM) = 0.1and T/TM = 10: comparison between the linearizations by physical insightand by Taylor series expansion
physical insight linearization is characterized by p.i., whereas the Taylorseries expansion by T.e.; further, it is assumed , ts = 1s, Tm = 0.25s,β/α = 0.1, and consequently, β
α1
4Tm= 0.1. Looking at the eigenvalues,
they can be complex conjugates or real: from the plots one can underlinethat as long as the eigenvalues are complex conjugates, the system remainsstable; but when they become real, the system gets quickly unstable as usualin discrete-dynamic analysis. Moreover, it is possible to determine the Sam-pling interval - Gain (SG) space, i.e. the stability region, as a function of∆tTm
and β4Tm
, for βα
14Tm
= 0.1 and r = 1; this domain is depicted in Fig-ure 20a. This space may be interpreted as operational tool that allows theoptimal combination of ∆t − β/α values to be selected. In fact being theT/Tm ratio unknown, it is usually set by trial and errors. The advantageof this space is that it permits operators to determine a priori informationon the behaviour of the sampled-data control system.
From the previous analysis we have demonstrated the effectiveness of thelinearization by physical insight for most of the practical cases, i.e. for veryshort ∆t or Tm approaches large values. Herein we employ this linearizationintroducing a fixed time step delay, i.e. γ = 1 in (4.28); thus, the controllaw reads
u [k] = K [k − 1] x [k − 1] + Kr [k − 1] r [k − 1] (4.50)
By repeating the same type of analysis performed for the case withoutdelay, we can plot the SG space for β
α1
4Tm= 0.1 in Figure 20(b). From
the figure it is evident that, if the control law u is computed with one timestep delay, the stability limit reduces. This finding underlines the need ofreducing the discrete time of the control law or to assume small ∆t
Tmratios,
53
(a) (b)
Figure 20. (a) Absolute stability regions in the SG space for a unit step in-put and β/(α4TM) = 0.1: comparison between the two linearizations;(b)Comparison of the SG space for the MCS algorithm with and without a fixedone time step delay and β/(α4TM) = 0.1.
to increase the stability of the time-delayed system. Moreover, it highlightsa greater reduction of the stability regions when the ratio T/Tm increases,i.e. when the plant is slow. Simulations and tests which confirm these andother analytical findings can be found in Vulcan (2006).
4.4 Bilinear transformation applied to time integration methods
From a signal processing point of view, each time integration formulaintroduced in Sections 2 and 3. can be viewed as a process of digitally filter-ing the input to produce the output at a discrete time k∆t, the mechanicalsystem acting as an analog filter (Preumont, 1982). Therefore, it is worth-while to analyse how the poles of a time integrator, like the novel LSRT1method appears in the discrete counterpart of the s-plane; and to comparethese representations with the relevant properties of the CH-α method in-troduced in Section 2.1. The location of the poles in the s-plane influencesand determines the absolute and the relative stability of a continuous time-invariant closed loop control system. In fact, complex closed-loop poles,close to the imaginary axis jω, determine an oscillatory behaviour of thesystem; conversely, real closed-loop poles that lie on the left-hand side of thes-plane cause an exponential decay behaviour of the system. It is noted thata closed-loop pole in the right-hand side of the s-plane entails an unstablebehaviour of the system (Sherrick, 2004).
Since the complex variables s and z are related to each other by the
54
following relationz = e∆ts (4.51)
the location of poles and zeros in the z-plane will depend on the pole andzero locations in the s-plane. This means that the stability of a lineardiscrete-time time-invariant closed-loop control system can be determined interms of poles and zeros of the closed-loop pulse transfer function. Moreover,the dynamic behaviour of a discrete-time system depends on the samplingperiod ∆t as well. As a result, a change of ∆t causes a different locationof the poles and zeros and, consequently, it also influences the stability ofthe system.
If we express the complex s variable in the following form
s = σ + jω (4.52)
then z reads
z = e∆t(σ+jω) = eσ∆tejω∆t = eσ∆t (cos ω∆t + j sin ω∆t) (4.53)
i.e. the combination of an exponential term and a periodic term. Due tothe presence of the periodic term, poles and zeros that differ in frequencyof the sampling frequency 2π/∆t in the s-plane, are mapped into the samelocations in the z-plane, see Figure 21, a clear manifestation of aliasing.Moreover, the imaginary axis jω in the s-plane corresponds to the unitcircle in the z-plane, because σ = 0 corresponds to |z| = 1. Since σ < 0in the left half of the s-plane, the latter is mapped inside the unit circle inthe z-plane, which therefore represents the stability area depicted in Figure21 (Sherrick, 2004).
From a signal processing point of view, one popular way of creating dig-ital filters from analog filters is to use a bilinear transform design (Sherrick,2004). In detail, it can be used to obtain a one-to-one mapping from thez-plane into the w-plane with the relation:
w =1
∆t
z − 1
z + 1(4.54)
obtained taking the first term of the expansion of the natural logarithm from(4.51) into continuous fractions. The factor 2 at the numerator of (4.54)has been omitted to avoid any assumption. That factor is generally used toproperly obtain the poles by using the trapezium rule (Tustin’s method, inFranklin and Powell (1980)). The w-plane with the relevant stable regionis sketched in Figure 21; and in this plane, the roots of the digital systemhave a similar representation as the roots of the continuous system in the
55
Figure 21. The s-, z- and w-plane and the relevant stability regions
prototype analog s-plane. Anew, from Figure 21 one can observe that everypoint on the w-plane is the image of an infinite number of points on thes-plane, with frequencies differing by integer multiples of ωr=2π/∆t. Byplugging (4.53) in (4.54), and defining
Ωw,d = ωw,d∆t, Ωs,d = ωs,d∆t (4.55)
one obtains
Ωw,d = tanΩs,d
2(4.56)
In detail, there is a tendency of the digitization process to shorten thestructure eigenvalues, generating a period elongation. Moreover, the verylow structure periods are transformed by the integration process to periods2∆t: this phenomenon is called spurious high frequency oscillations.
If we consider a SDoF system, for the LSRT1 method with one stage,we get,
k01
k11
=
[[1 00 1
]− γ∆t
[0 1
−m−1k −m−1c
]]−1
·[
p (t)−m−1kx(t) − m−1cp(t) + m−1fe(t)
]∆t (4.57)
yk+1 =
x [k + 1]p [k + 1]
= yk + b1
k01
k11
(4.58)
x [k + 1] = q00x [k] + q01p [k] + q02fe [k] (4.59)
p [k + 1] = q10x [k] + q11p [k] + q12fe [k] (4.60)
56
where the coefficients of (4.59) and (4.60) can be easily derived. Then, weapply the z -transform to (4.59) and (4.60),
zX [z] − zX [0] = q00X [z] + q01P [z] + q02Fe [z] (4.61)
zP [z] − zP [0] = q10X [z] + q11P [z] + q12Fe [z] (4.62)
and we can write
X [z] = n0Fe [k] + rdX [0] + rvP [0] (4.63)
Then, we apply the bilinear transformation
z =−1 − w∆t
−1 + w∆t(4.64)
and we obtain
X [w] = n0wFe [w] + rdwX [0] + rvwP [0] (4.65)
Each of the aforementioned coefficients, for instance rdw, can be expressedas
rdw =Nrdw
Drdw
(4.66)
where in this case, Drdw represents the denominator of the transfer functionbetween X [w] and Fe [w]. By using the relations
k = mω2, c = 2ξmω, Ω = ω∆t and ξ = 0 (4.67)
Drdw reads
Drdw = b1Ω2w −2b1Ω
2w(b1−2γ)w+(4+b2
1Ω2w −4b1Ω
2wγ+4Ω2
wγ2)w2
(4.68)and, by assuming b1 = 1 γ = 1 for the LSRT1 method, one gets the poles
w1,2 =−2Ω2
w ±√
4Ω2w − 4Ω2
w
(4 + Ω2
w
)
2(4 + Ω2
w
) =Ωw
∓2i − Ωw
. (4.69)
w2 can be represented in the w-plane as depicted in Figure 22(a).Note that limΩw→0 Re[w2] = 0, limΩw→0 Im[w2] = 0 identify
the continuous condition; while limΩw→+∞ Re[w2] = −1, limΩw→+∞
Im[w2] = 0 indicate that the method entail no oscillations i.e. the L-stability condition. This result can be compared in Figure 22(b) with therepresentation of the L-stable CH-α method with the assumption fe,k+1 =
57
(a) (b)
Figure 22. (a) Representation of one pole of the LSRT1 method for ξ =0.;(b) representation of one complex and one spurious pole of the CH-αmethod for ρ∞ = 0.
fe,k. One can observe a similar trend for one of the complex conjugate polesas well as the presence of the so-called spurious root which does not entailoscillations. The favourable performance of the LSRT1 method in terms ofdamping is evident from the comparison.
5 Novel schemes for simulations of monolithic and
heterogeneous stiff systems in real time
For a specific number of stages introduced in Subsection 3.1, an importantobservation is that the Runge-Kutta formulae is not unique. This meansthat exist free parameters which can be chosen for specific needs. Thesenovel L-stable real-time (LSRT) compatible algorithms derived from theRosenbrock methods (Rosenbrock, 1963) can be introduced for SHSDS. Indetail, two and three-stage methods, which are second- and third-orderaccurate, respectively, for uncoupled and linear problems, will be developedin a greater detail in Subsection 5.1 and applied to linear coupled problemsin Subsection 5.2.
5.1 L-stable methods for monolithic systems
The two-stage L-stable real-time (LSRT2) method when applied to (2.3),exploits the estimates:
k1 = [I − γ∆tJ]−1
f (tk, yk)∆t (5.1)
k2 = [I − γ∆tJ]−1
(f (tk+α2, yk+α2
) + Jγ21k1)∆t (5.2)
58
where yk+α2represents the value of the state vector at the α2 fraction
of ∆t; the external force fe,k+α2is defined at time tk+α2
= tk + α2∆tand the measured substructuring force fs,k+α2
is obtained when applyingyk+α2
at the α2 fraction of ∆t. Thus,
yk+α2= yk + α21k1 (5.3)
and the state vector reads
yk+1 = yk + b1k1 + b2k2 (5.4)
The accuracy order of the method can be obtained by comparing thederivative terms of the Taylor series expansion of the exact solution withthose of the numerical solution according with the developments in Subsec-tion 3.2. For a two-stage method, the conditions needed to achieve second-order accuracy are
b1 + b2 = 1, α2b2 = 12, b2(α21 + γ21) = 1
2− γ (5.5)
Then, we impose the condition to assure L-stability, viz. limλ→−∞
|R(λ∆t)| = 0, and the relevant γ values read
γ = 1 ±√
2
2(5.6)
To enforce the compatibility of the algorithm with the real-time dSPACEsoftware (dSP, 2001) the intermediate substeps must have a sample ratewhich is integer multiple of the base sample rate. As a result, α2 and α21
must be rational numbers and, for instance,
γ = 1 −√
2
2, α2 = α21 =
1
2, γ21 = −γ, b1 = 0 and b2 = 1 (5.7)
Figure 23(a) shows the spectral radius of the LSRT2 method for asecond-order system y = −ω2y; the spectral radius of the CH-α withρ∞ = 0 is included for comparison. One can observe how a user canachieve a favourable behaviour of the algorithm in the low-frequency range,
i.e. for γ = 1−√
22
; or a favourable dissipation in the high-frequency range
for γ = 1 +√
22
.Formulae and parameter values relevant to the three-stage L-stable real-
time LSRT3 method read
k1 = [I − γ∆tJ]−1
f (tk, yk) ∆t (5.8)
k2 = [I − γ∆tJ]−1
(f (tk+α2, yk+α2
) + Jγ21k1)∆t (5.9)
k3 = [I − γ∆tJ]−1
(f (tk+α3, yk+α3
) + Jγ31k1 + Jγ32k2)∆t (5.10)
59
(a) (b)
Figure 23. Spectral radius of the LSRT methods for a second-order system:(a) LSRT2 integrator; (b) LSRT3 integrator
where: yk+α2represents the estimate of the state vector at the α2 fraction
of ∆t:yk+α2
= yk + α21k1; (5.11)
yk+α3represents the estimate of the state vector at the α3 fraction of ∆t,
yk+α3= yk + α31k1 + α32k2 (5.12)
Both the external forces fe,k+α2and fe,k+α3
are defined at time tk+α2=
tk + α2∆t and tk+α3= tk + α3∆t, while the measured substructuring
forces fs,k+α2and fs,k+α3
are obtained when applying yk+α2and yk+α3
,respectively, to the PS. The state vector value reads
yk+1 = yk + b1k1 + b2k2 + b3k3 (5.13)
The accuracy order of the method can be obtained comparing the deriva-tive terms of the Taylor series expansion of the exact solution with those ofthe numerical solution (Bursi et al., 2008). For a three-stage method theconditions to achieve third-order accuracy read
b1 + b2 + b3 = 1
b2(α21 + γ21) + b3(α31 + γ31 + α32 + γ32) =1
2− γ
α21b2 + (α31 + α32)b3 =1
2, α2
21b2 + (α31 + α32)2b3 =
1
3(5.14)
α21(α32 + γ32)b3 =1
6− γ/2
(α32 + γ32)(α21 + γ21)b3 =1
6− γ − γ2
60
The γ values that assure L-stability can be derived from
γ3 − 3γ2 +3
2γ − 1
6= 0 (5.15)
To ensure full compatibility with dSPACE (dSP, 2001) α21 and α31 +α32 must be rational numbers and α21 < α31+α32. Therefore, we assume
γ = 0.4359 or 2.4052, γ21 = 0.1818, γ31 = −0.0428
γ32 = −0.5384, α2 =1
3, α3 =
2
3, α21 = α31 = α32 =
1
3(5.16)
b1 =1
4, b2 = 0 and b3 =
3
4
The comparison of the LSRT3 method with the CH -α method with ρ∞ =0 in terms of spectral radius is illustrated in Figure 23(b). One can observethe capabilities offered by the proposed methods through the selection of γ.
5.2 L-stable methods for heterogeneous coupled systems
From Subsection 1.1 and the related ones a real-time simulation of aheterogeneous system can be performed with a linearly implicit algorithmwithout iteration and based on kinematic control. For the system of Figure15 reduced for simplicity to a SDoF with kp
2 = cp2 = mp
2 = 0, the cou-pling algorithm is schematically depicted in Figure 14 and further detailedin Figure 15. By assuming y = x xT
= y1 y2T, it entails: (i) the
integration of the differential equation of the NS using the coupling forcempy2,k−2 = un
k−1 available at tk−1; (ii) the motion of the actuator con-
trolled to follow the predicted kinematic quantity y2,k = upk available at
tk; (iii) the measure of the coupling force mpy2,k. In a greater detail, theSHSDS with the linearly implicit LSRT2 method is summarized in algo-rithmic form as follows:(a) Compute the Jacobian matrix J(b) Compute k1 from (5.1) and evaluate yk+α2
from (5.3)(c) Impose yk+α2
to the PS, measure the coupling force fs,k+α2, compute
k2 from (5.2) and evaluate yk+1 from (5.4)(d) Impose yk+1 to the PS and measure the coupling force fs,k+1
(e) Set k = k + 1 and go to (a) for a non-linear system or to(b) for alinear system
This actual prediction-substitution procedure is rather different from atruly parallel scheme (Felippa et al., 2001), which at tk−1 would entail alsoa prediction of the coupling force mpy2,k. This might be unpractical owingto the nature of the PS.
61
SHSDS tests can lose stability owing to: (i) control stability loss, seeHoriuchi et al. (1999) among others; (ii) stability loss owing to the inte-gration of heterogeneous systems. In this subsection we focus on point (ii),i.e. the stability of coupling integrators, taking for granted that the inte-grators considered herein and applied to monolithic linear elastic systemsare L-stable. To avoid possible unwanted effects owing to the integrationprocess, we present herein both a zero-stability analysis for the interconnec-tion shown in Figure 14 and an absolute stability analysis of the discretizedcoupled system.
The zero-stability concept is nothing more than the minimal demandthat the coupled difference system is well posed. In other words, the nec-essary and sufficient conditions for the proposed LSRT integrators to beconvergent when an algebraic loop exists are that they are both consistentand zero-stable (see Lambert, 1991, p. 31). In this approach, the key toanalyse the zero-stability of the coupling integrator is based on the factthat a one-step integrator uses a vanishing step size ∆t → 0; this leads toa time invariant state vector yk+1 = yk = y0 and, as a result, the stabil-ity condition of the coupled structure does not depend on the integrationscheme. To perform this analysis, all the inputs un
k and upk, corresponding
to the NS and the PS inputs shown in Figure 14, are collected in a vectoruk. The same is done with the outputs, using the vector zk. The firststep of this discrete analysis consists in relating inputs and outputs takenat the same instant at the interconnection, see Figure 15, through (4.8).For instance, for the structure of Figure 1 reduced for simplicity to a SDoFwith kp
2 = cp2 = mp
2 = 0, where yk = xk xkT= y1,k y2,kT
, oneobtains,
L =
[0 Lp
Lp 0
], zk =
y2,k
mpy2,k−1
, uk =
un
k
upk
(5.17)
where Lp = Ln = 1. Anew for simplicity, we exploit the LSRT1 methodand the SDoF structure neglecting the gravity force on the PS, thus we get
yk+1 = A1yk + B1fk (5.18)
where
A1 =1
mn + cn∆t + kn∆t2
[mn + cn∆t mn∆t
−kn∆t mn
],
B1 =∆t
mn + cn∆t + kn∆t2
[∆t −∆t1 −1
], fk =
fe,k
unk
(5.19)
62
Note that A1 is a 2×2 matrix; hence only complex eigenvalues charac-terize the behaviour of this class of algorithms. If the output equations aretime invariant and linearly dependent on the inputs, see Figure 15, one canrelate the input at time tk+1 to the input at time tk through the followingrelation:
uk+1 = C1uk + f (5.20)
where the matrix C1 and the vector f contain constant terms as a result ofthe vanishing time step. In detail, for the SDoF structure one obtains
C1 =mp
mn
[0 mn
0 −1
], f =
01
mn(fe,k+1 − kny1,k − cny2,k)
(5.21)In this condition, the coupled system is said to be zero stable if the spectralradius of the coupling matrix C1 lies within the unit circle in the Argand-Gauss plane. As a result, the way in which a complete system can bepartitioned into an NS and a PS must follow certain rules. In practice, thisstability condition applies to the splitting of any mass mi present at thecoupling interface. The condition results to be:
mpi ≤ mn
i (5.22)
which means that an interface mass corresponding to the PS cannot begreater than its NS counterpart. This condition arises from the fact thatdealing with feed-through systems, an algebraic loop arises between theacceleration up
k = y2,k and the interface force unk = mpyn
2,k−1, see Figure15, as their magnitudes are both dependent on each other; the algebraicloop can be eliminated together with condition (5.22) through an iterativeprocess or the presence of filters (Kubler and Schiehlen, 2004). Nonetheless,iteration is not possible here owing to the real-time need for SHSDS ; andthe use of filters can modify the dynamics of the resulting emulated system;as a result, the algebraic loop is usually not eliminated owing to the presenceof unintended delay (Kyrychko et al., 2006). In this proof, a delay of ∆tbetween the acceleration at the interface up
k and the coupling force unk has
been considered:
y2,k+1 =1
mn(fe,k+1 − fs,k − cny2,k+1 − kny1,k+1) (5.23)
This assumption and the stability condition (5.22) for vanishing ∆t havebeen confirmed by numerical simulations and tests (Bursi et al., 2008).
Differently from the zero-stability analysis, herein we perform an abso-lute stability analysis relaxing the condition ∆t → 0. Being the system
63
coupled and the LSRT integrators applied to the NS only, we need to in-
troduce a state vector y =
x x fs
T=
y1 y2 fs
Twhich
comprises also the coupling force fs acting on the mass mn as illustratedin Figure 1(c).
For the LSRT1 method and the SDoF system of Figure 1 with mp2 =
kp2 = cp
2 = 0, one obtains
yk+1 = A1yk + B1f1,k (5.24)
where for stability, one is only interested to the amplification matrix A1;for γ = 1 it reads
A1 =
1
mn + cn∆t + kn∆t2
mn + cn∆t mn∆t −∆t2
−kn∆t mn −∆t−knmp −cnmp − ∆tknmp −mp
(5.25)
Note that, differently from the previous zero-stability analysis, A1 is a 3×3matrix owing to the presence of the coupling force fs,k. The investigation of
the spectral radius ρ(A1) indicates that the LSRT1 method with the zero-stability condition (5.22) satisfied is always L-stable for the coupled systemunder exam. Nonetheless the same analysis both with the LSRT2 and theLSRT3 method applied to the coupled system shows that the LSRT2 isnot A-stable for γ = 1 −
√2/2. This result illustrated in Figure 24(a)
was not expected and must warn the analyst dealing with coupled systems.Conversely, the LSRT2 method recovers at least its A-stability for γ =1 +
√2/2 and any value of ξ, at a price of adding numerical damping in
the low-frequency range components of the response. Again, the loss ofstability of the LSRT3 method for γ = 0.43586, ξ = 0 and small valuesof Ω is shown in Figure 24(b). Simulations and tests concerning both thestability and accuracy of LSRT methods can be found in Bursi et al. (2008).
6 Partitioned Methods and PM-alpha Methods
Partitioned methods were originally developed for the analysis of coupledproblems, such as FSI. Based on Schur complements to split the coupledmechanical systems into subsystems, these partitioned analysis procedurescreate a coarse problem with a reduced number of unknowns by the elim-ination of internal subsystems unknowns. Only the original, i.e. primalunknowns -displacements, velocities or accelerations- are considered in the
64
(a) (b)
Figure 24. Spectral radius of the LSRT method applied to a SDOF systemwith mn/mp = 3 and different values of ξ: (a) LSRT2 integrator; (b)LSRT3 integrator
computation. Since the late 1980s, the development of parallel algorithmshave been motivated, emphasizing the communication and synchronizationbetween subsystems which is vital to the improvement of the performance ofparallel computing. A distinct feature of these parallel algorithms is the useof dual unknowns, such as Lagrange multipliers, to enforce the continuitybetween subsystems.
In the last decade, the Finite Element Tearing and Interconnecting(FETI ) method emerged as one of the most powerful parallel algorithms forquasi-static structural mechanics problems (Farhat and Roux, 1991); and itwas later extended to dynamic problems (Farhat et al., 1995). In the FETImethod, a given spatial domain is torn into non-overlapping substructureswhere an incomplete solution of the primal field is evaluated using a directsolver, and intersubstructure field continuity is enforced via Lagrange mul-tipliers applied at substructure interfaces. The latter gluing phase generatesa small size symmetric dual problem where the unknowns are Lagrange mul-tipliers, and which is best solved with a preconditioned conjugate gradientalgorithm. Gravouil and Combescure (2001) proved that imposing velocitycontinuity at the interface leads to a stable algorithm. In detail, they con-ceived a multi-time-step coupling method, labelled the GC method, ableto couple arbitrary Newmark schemes with different time steps in differentsubdomains. In this context, they proved that the GC method is uncondi-tionally stable as long as all individual subdomains satisfy their own stabilityrequirements. Moreover, they showed that for multi-time-step cases the GCmethod entails energy dissipation at the interface, while for the case of asingle time step in all subdomains the GC method is energy preserving. TheGC method is very appealing for HiL and in particular for the continuous
65
Figure 25. The continuous PsD testing.
PsD testing as heterogeneous numerical and physical substructures can besolved with different implicit/explicit Newmark schemes in different sub-domains according to their complexity and characteristics. The possibilityof performing a large amount of small time steps on a reduced number ofDoFs at the laboratory, at about 1kHz frequency, while computing a largetime step on a large number of DoFs on a remote computer, is mandatoryfor the proper implementation of the continuous PsD technique with DS(Pegon and Magonette, 2002, 2005). Figure 25 shows how Ns intermediateinputs are created by linear interpolation between two consecutive discreteground acceleration records. The displacement command to the actuatoris then generated in ∆ts, i.e. the sampling time of controller. Thus, anoriginal accelerogram span ∆T is completed by Ns steps. The time expan-
sion factor λ is defined as the ratio λ =Ns∆ts
∆Tbetween the experimental
time and the earthquake time. For instance, Ns = 500, ∆ts = 2ms and∆T = 5ms result in λ = 200.
The GC method employed in the continuous PsD testing can maintainthe smoothness of the displacement trajectory without using any extrapo-lation/interpolation assumption and can preserve the optimum signal/noiseratio of the continuous method. Unfortunately, the GC method as mostof the available methods, is in essence a sequential staggered algorithmwhere the tasks in different subdomains are not concurrent. To solve theaforementioned problem, Pegon and Magonette (2002, 2005) developed andimplemented an interfield parallel algorithm, the PM method, based on theGC algorithm, where the NS and the PS states advance simultaneouslyand possibly continuously. Though several studies point out the benefit ofthe parallelization of partitioned methods (Felippa et al., 2001) in view of
66
real-time simulations of heterogeneous subsystems, there is still a paucityof publications devoted to this issue. Along these developments two novelmethods are presented in Subsection 6.1 and 6.2, respectively: the PM andthe PM -α method.
6.1 Analysis of the PM method
Formulation of the PM method The structure to be studied is dividedinto two subdomains: A and B, respectively, to be discretized in time withtime steps ∆tA and ∆tB assuming
∆tA = ss ∆tB (6.1)
where ss defines the number of substeps. Let MA, MB denote the symmet-ric positive-definite mass matrices of the two subdomains, RA and RB theinternal force vectors and FA
ext and FBext the vectors of applied forces, re-
spectively. With this notation to hand, the equations of equilibrium on sub-domain A at time tn+1 and subdomain B at time tn+ j
ss, j = 1, . . . , ss,
together with the kinematic interface constraints between the subdomainscan be written as
MAuAn+1 + RA
(uA
n+1, uAn+1
)= FA
ext,n+1 + LAT
Λn+1 (6.2)
MBuBn+ j
ss
+ RB(uB
n+ jss
, uBn+ j
ss
)= FB
ext,n+ jss
+ LBT
Λn+ jss
(6.3)
LAuAn+ j
ss
+ LBuBn+ j
ss
= 0 (6.4)
where the state variables u (t) are nodal quantities arising from a spa-tial discretization and their derivatives u (t) and u (t) with respect totime t are indicated with superposed dots; Λ is the vector of Lagrangemultipliers; LA and LB are the constraint matrices which express a lin-ear relationship between the two connected boundaries (Combescure andGravouil, 2002). In detail for a linear elastic system with classical damping,
RA(uA
n+1, uAn+1
)= KAuA
n+1 + CAuAn+1 and RB
(uB
n+ jss
, uBn+ j
ss
)=
KBuBn+ j
ss
+ CBuBn+ j
ss
, where KA and KB denote the stiffness matrices
of the two subdomains, respectively, and CA and CB are the damping ma-trices. In detail, the continuity relationship in (6.4) is enforced in terms ofvelocities.
The novel parallel procedure which was proposed by Pegon and Magonette(2002) is sketched in Figure 26. A time step equal to 2∆tA is exploitedin subdomain A, to anticipate information on the subdomain B at the be-ginning of a new time step. Considering arbitrary numeric schemes of the
67
Figure 26. The PM method: a parallel procedure.
Newmark family with two characteristic parameters βA, γA and βB, γB
for subdomain A and B, respectively, the flow chart of the PM methodreads:
1. solve the free problem in subdomain A using 2∆tA, thus advancingfrom tn−1 to tn+1
MAuAn+1f
= FAext,n+1 − RA
(uA
n−1, ˜uA
n−1
)(6.5)
uAn+1f
= uAn−1 + αA
1 uAn+1f
(6.6)
uAn+1f
= ˜uA
n−1 + αA2 uA
n+1f(6.7)
with
MA = MA + αA1 KA + αA
2 CA (6.8)
uAn−1 = uA
n−1 + 2∆tAuAn−1 + (2∆tA)
2(1
2− βA)uA
n−1 (6.9)
˜uA
n−1 = uAn−1 + 2∆tA (1 − γA) uA
n−1 (6.10)
αA1 = βA (2∆tA)
2and αA
2 = γA (2∆tA) (6.11)
Here RA(uA
n−1, ˜uA
n−1
)is the internal force, that for a linear system
reads RA(uA
n−1, ˜uA
n−1
)= KAuA
n−1 + CA ˜uA
n−1.
2. start the loop on ss substeps in subdomain B
68
3. solve the free problem in subdomain B at tn+ jss
with j = 1, . . . , ss
MBuBn+ j
ss f
= FBext,n+ j
ss
− RB(uB
n+ j−1
ss
, ˜uB
n+ j−1
ss
)(6.12)
uBn+ j
ss f
= uBn+ j−1
ss
+ αB1 uB
n+ jss f
(6.13)
uBn+ j
ss f
= ˜uB
n+ j−1
ss+ αB
2 uBn+ j
ss f
(6.14)
with
MB = MB + αB1 KB + αB
2 CB (6.15)
uBn+ j−1
ss
= uBn+ j−1
ss
+ ∆tBuBn+ j−1
ss
+ ∆t2B(1
2− βB)uB
n+ j−1
ss
(6.16)
˜uB
n+ j−1
ss= uB
n+ j−1
ss
+ ∆tB (1 − γB) uBn+ j−1
ss
(6.17)
αB1 = βB∆t2B and αB
2 = γB∆tB (6.18)
In a PsD test, the internal resisting force RB(uB
n+ j−1
ss
)can be di-
rectly measured by imposing the displacement uBn+ j−1
ss
on the sub-
structure.
4. interpolate the free velocity in subdomain A
uAn+ j
ss f
=
(1 − j
ss
)uA
nf+
(j
ss
)uA
n+1f(6.19)
5. compute the Lagrange multipliers Λn+ jss
by solving the condensed
interface problem
HΛn+ jss
= −(LAuAn+ j
ss f
+ LBuBn+ j
ss f
) (6.20)
where
H = αA2 LAMA−1
LAT
+ αB2 LBMB−1
LBT
(6.21)
6. solve the link problem in subdomain B at tn+ jss
MBuBn+ j
ss l
= LBT
Λn+ jss
(6.22)
uBn+ j
ss l
= αB1 uB
n+ jss l
and uBn+ j
ss l
= αB2 uB
n+ jss l
(6.23)
69
7. compute the kinematic quantities of subdomain B at tn+ jss
, by sum-
ming the free quantities (Point 3) and link quantities (Point 6)
(·) = (·)f + (·)l (6.24)
8. if j = ss, then end the loop in subdomain B9. solve the link problem in subdomain A using a time step 2∆tA from
tn−1 to tn+1
MAuAn+1l
= LAT
Λn+1 (6.25)
uAn+1l
= αA1 uA
n+1land uA
n+1l= αA
2 uAn+1l
(6.26)
10. compute the kinematic quantities of subdomain A at tn+1 by sum-ming the free problem (Point 1) and the link problem (Point 9)
(·) = (·)f + (·)l (6.27)
The strategy described above drives the algorithm illustrated in Figure26. The dashed line describes the ongoing parallel process in the two subdo-mains. Note that the process in subdomain A is split into two independentparts, linked through the subdomain B, which enables the parallel com-putation and synchronized exchange of information. The PM method isreferred to as a simplified interfield modification of the basic GC method.Extensions to the non-linear case can be found in Pegon and Magonette(2005); Pegon (2008).
Amplification matrix of the PM method The PM method uses amesh partition with two different schemes on two or more subdomains.For this reason, the modal analysis approach to stability is inapplicable.As a result, the GC method as other methods of this class was analysedthrough the energy approach (see Hughes, 1987, p. 564). Although thePM method is spectrally stable as it will be shown herein, the well-knownenergy approach is not easily extendible to this algorithm, owing to thelarge number of system variables. For this reason, consistency and stabilityproperties of the PM method are examined exploiting a standard approachfor linear model problems (Bonelli et al., 2008).
In PsD applications, it is typically preferred to use an implicit schemein subdomain A for the NS and an explicit scheme in subdomain B for thePS. Thus, it is avoided to have Newton iterations or correction formulationsin the procedure devoted to the subdomain B, which require an estimate ofthe stiffness matrix of the PS. In what follows, we exploit the trapezoidalrule, i.e. β = 1
4, γ = 1
2in subdomain A, and the CD method, i.e. β =
70
0, γ = 12
in subdomain B. Nonetheless, as the original GC method, the PMscheme employed in this section is convergent with any Newmark schemewith γ 6= 1/2.
Time stepping schemes applied to linear problems can normally be recastin a recursive form as
Xn+1 = A Xn + Ln (6.28)
where X is an appropriate state vector depending on the formulation ofthe scheme, A is the amplification matrix and L is the load vector whichdepends on external forces. The main difficulty in the characterization ofthe PM method is the choice of sufficient state variables. To evaluate thestate of the PM method at time tn+1, not only the variables at time tn butalso some variables at tn−1 are needed. This feature transforms the schemein a second-degree method, which can be analysed as proposed in Young,1971, p. 486, to define the amplification matrix. To proceed to tn+1, theknowledge of variables on subdomain B at time tn is sufficient, whereassubdomain A needs also the knowledge of variables at time tn−1. Hence,the following state vector is considered:
Xn=(
XAn−1 XA
n XBn
)T(6.29)
where XAn collects the kinematic quantities of subdomain A
XAn =
(uA
n uAn uA
nfuA
n
)T
(6.30)
and XBn those of subdomain B
XBn =
(uB
n uBn uB
n
)T(6.31)
As a result, Xn has the dimension 8 nA +3 nB with nA and nB the DoFsin the two subdomains. For homogeneous linear model problems, consideredwithout damping for the sake of simplicity, (6.28) turns out to be
XAn
XAn+1
XBn+1
= A
XAn−1
XAn
XBn
=
0 I 0AAA AAA AAB
ABA ABA ABB
XAn−1
XAn
XBn
(6.32)where I is the identity matrix, and A is expressed using the block matrixform determined through the symbolic analysis provided in Bonelli et al.(2008). For a general MDoF system, the state vector (6.29)-(6.31) can bereduced by removing uA
n−1f. Indeed (6.9)-(6.11) show that uA
n−1fhas no
contribution from the scheme to pass from tn−1 to tn. It also conforms
71
to the fact that the corresponding row and column vectors are zero in Aexcept the submatrix part in I, that entails uA
nf= uA
nf. By summing up,
the state vector Xn reads
Xn =(
uAn−1 uA
n−1 uAn−1 uA
n uAn uA
nfuA
n uBn uB
n uBn
)T
(6.33)with a dimension of 7nA + 3nB.
Accuracy To begin the time integration process, the PM algorithm needsan estimate of free velocities and accelerations at an initial time as well asquantities’ values of the previous or first step. As a result, an accuracyanalysis needs to be addressed both for the initialization process and forthe local truncation error as suggested in Hulbert and Hughes (1987). Toobtain more seemingly convenient results along the lines of Hulbert andHughes (1987), we introduce an alternative form of (6.28)
Xn+1 = A Xn + Ln (6.34)
where
Xn = (uA
n−1 uAn−1 uA
n−1∆tA uAn uA
n uAnf
uAn ∆tA
uBn uB
n uBn ∆tA
)T
(6.35)
A = MAM−1
, Ln = MLn (6.36)
M = block diagonal[
I I ∆tAI I I I ∆tAI I I ∆tAI]
(6.37)
with I being the identity matrix. In detail, ∆tA has been introduced in theacceleration terms of (6.35). The load vector Ln in (6.34) is not includedin the following analysis, assuming that the power of the loading error termof its approximation is greater than the order of accuracy of the method.
The local truncation error τn is defined as
τn = τ (tn) = AX (tn) − X (tn+1) (6.38)
being X (tn) the corresponding exact solution of the state vector Xn at tn.The exact solution of the free velocity, uA
f (tn) reads
uAf (tn) = uA(tn) − αA
2 MA−1 (MAuA(tn) + KAuA(tn)
)(6.39)
which is derived from (6.25) and (6.27).
72
With regard to the SDoF model problem described at length at the endof this subsection, from symbolic calculations one gets
τn = O(∆t3A
)and τn = O
(∆t2A
)(6.40)
for ss = 1 and ss = 2, respectively. Moreover, for the Two-DoF modelproblem one gets
τn = O(∆t3A
)and τn = O
(∆t2A
)(6.41)
for ss = 1 and ss = 2, respectively. In general, it has been obtained thatthe order of accuracy of τn for ss > 2 is identical to the one for ss = 2.
From (6.29), one observes that two starting points on subdomain A arenecessary to initialize the PM method. To obtain the additional point onsubdomain A, one may use the GC method for the first step followed bythe PM method in the remaining steps, i.e.
XA0 , XB
0 → XA1 , XB
1 , through the GC method; (6.42)
which, entails the following initialization error
e2 = X2 − X (2∆tA) = AX1 − X (2∆tA) (6.43)
For the SDoF model problem, the initial condition of the exact solutionentails u(0) = d0, u(0) = v0. By taking
XA0 = d0, v0, vf,0, a0T
, XB0 = d0, v0, a0T
(6.44)
with vf,0 = v0 and a0 determined by the initial equilibrium condition, oneobtains, respectively, when ss = 1 and ss = 2
e2 = O(∆t3A
)and e2 = O
(∆t1A
)(6.45)
For the Two-DoF model problem, the results for e2 are similar.
Stability Since the PM method couples two or more subdomains, a zero-stability analysis involving (6.20) has preceeded the absolute stability anal-ysis (He, 2008) along the line of Subsection 5.2. This analysis does notentail any special system parameter condition.
With regard to a SDoF model problem, the absolute values of the eigen-values of the amplification matrices are plotted in Figure 27 with respectto ΩB = ωB × ∆tB employing few values of ss. The number of nonzeroeigenvalues is found to be 6. Among them only one pair of the complex con-jugate eigenvalues are the principal eigenvalues while the remaining ones are
73
Figure 27. |λi| of the SDoF problem for the PM method when b1 = 10:(a) ss = 1; (b) ss = 2; and (c) ss = 20.
spurious. In detail, the non-zero eigenvalues λi (i = 1, ..., 6) of the ampli-fication A in the limit ∆tA = 0 read
[1 1 1 −1 −b1−2
√−1−b1
2+b1
−b1+2√
−1−b1
2+b1
](6.46)
if ss = 1 and
[1 1 1 −1
−8b1−√
−256−256b1+b4
1
(4+b1)2−8b1+
√−256−256b1+b4
1
(4+b1)2
]
(6.47)if ss = 2. It can be shown that |λ5| and |λ6| are equal to unity in (6.46)and are strictly less than one unless b1 = mA
mB= KB
KAdefined in (6.52) is
either 0 or tends to +∞ in (6.47). Linearly independent eigenvectors existfor the given nonzero eigenvalues in (6.46) and (6.47). In general, we verifiedthat A has linearly independent eigenvectors for all nonzero eigenvalues. Asa result, the spectral radius of A, max
i
∣∣λi(A)∣∣ ≤ 1 suffices the stability .
The stability limit is ΩB = 2 when ss > 1. When ss = 1 the stabilitylimit is higher, which is found to be related to the system parameters b1.Nevertheless, the scheme is always stable when ΩB ≤ 2. Similar resultswere found with respect to the Two-DoF model problem (Bonelli et al.,2008). In summary, the PM method preserves the same property as theGC method, i.e. it is stable if the stability requirement in each subdomainis satisfied.
Convergence With regard to the convergence of the PM method, theglobal error en at tn is affected by both the local truncation error and the
74
Figure 28. Global error of the SDoF model problem (b1 = 10) for the PMmethod with: (a) ss = 1; (b) ss = 2; and (c) ss = 20.
initialization error,
en = An−2e2 +
n−1∑
i=2
An−(i+1)τi (6.48)
withen = e (tn) = Xn − X (tn) (6.49)
A method is convergent of the order k, if it is stable and consistent of the
order k, i.e. τi = O(∆tk+1
A
)and e2 = O
(∆tk
A
). In detail, k = 2 when
ss = 1 and k = 1 when ss > 1 for the PM method.Figure 28 shows the global error |en| = |Xn − X (tn)| versus ∆tA in
a logarithmic scale for the SDoF model problem, being X (tn) the corre-sponding exact solution of the state vector Xn at tn. Simulations confirmanalytical results: the PM scheme is second-order accurate for ss = 1 andfirst-order accurate for ss > 1.
To feature more the spectral properties of the PM method, we write thediscrete solution as (Hulbert and Chung, 1994)
ujn =
m∑
i=1
ciλni , uj
n =
m∑
i=1
ciλni , uj
n =
m∑
i=1
ciλni (6.50)
where λi’s are the nonzero eigenvalues of the amplification matrix, ci, ci
and ci serve as generic constants and j represents the subdomain A orB. Figure 29 shows both the |λi|’s and |ci|’s of uA
n , uAn , uA
n of the SDoFmodel problem with b1 = 10, ss = 50. The number of nonzero eigenvaluesplotted in the figure is 6; among them, λ2, λ3 are the principal complexconjugate eigenvalues being the relevant constants different from zero at
75
∆tA → 0, while the others are spurious ones with the relevant constantsbeing zero at ∆tA → 0. In detail, a spurious real eigenvalue |λ1| = 1always exists. Figure 29 shows how c1 6= 0 and c1 = c1 = 0 for ∆tA 6= 0.This spectral behaviour is responsible of a drift-off error of displacementsfor finite time steps, typically of the partitioned algorithms with constraintsonly on velocities. Conversely, the remaining spurious eigenvalues do notentail this behaviour being less than 1. Nonetheless, the drift-off errorstend to zero as ∆tA → 0. For on-line simulations, the drift-off effectsare found to be negligible by using a small time step which is typical inthe application of continuous PsD tests with substructuring. Moreover, foruA the constants relevant to λ5 and λ6 do not vanish for ∆tA → 0; andthe same phenomenon is observed for uB. As a result, if we analyse thelocal truncation error τn = AX (tn) − X (tn+1), one order lower errorsare found in the terms related to the acceleration compared to the onesrelated to displacement and velocity. We verified numerically that those arethe effects of λ5 and λ6. As the moduli of λ5 and λ6 are less than theprincipal ones, these contributions will damped out and eventually will noteffect the global convergence; on this basis, we introduced the alternativestate vector Xn and the amplification matrix A in (6.34).
Representative Model Problems The single-DoF mass-spring emu-lated system considered herein is split into two parts as illustrated in Figure30(a). The total mass m and the total stiffness k are the sum of the massand the stiffness of each part, i.e.
m = mA + mB , k = kA + kB (6.51)
The idea is to compare various cases keeping the natural frequency f of theemulated system unchanged. As a result, in the analyses and simulations itis assumed that
f =1
2π
√k
m= 1Hz,
mA
mB
= b1 andkA
kB
=1
b1
(6.52)
A Two-DoF mass-spring emulated system can be split into two partsas depicted in Figure 30 (b). The natural frequencies f1 and f2 of theemulated system are left unchanged. As a result, for the different casesexamined we have assumed
f1 =1
2π
√k
m= 1Hz, f2 =
√3f1 and
mA
mB
= r with mA +mB = m
(6.53)
76
Figure 29. Discrete solution of the PM method: (a) |λi|; (b) |ci| of uAn ;
(c) |ci| of uAn ; and (d) |ci| of uA
n .
Figure 30. Model problems: (a) partitioned single-DoF system and (b)partitioned two-DoF system.
77
6.2 The PM-α method
Variants of the CH -α Methods Arnold and Bruls (2007) proposeda new form of the CH -α method introduced in Subsection 1.2 where thedynamic equilibrium for the time discrete response is enforced at the end ofeach time step, instead of using a weighted formulation of the equilibriumequation (Chung and Hulbert, 1993). Moreover, it was proved that second-order accuracy of the new algorithm holds also in the accelerations. Byusing a predictor-corrector form, the novel algorithm suggested by Arnoldand Bruls for a linear problem readsPredictors:
un = un + ∆tun + ∆t2(1/2 − β
1 − αm
)an + ∆t2βαf
1 − αm
un
(6.54)
˜un = un + ∆t(1 − γ
1 − αm
)an + ∆tγαf
1 − αm
un (6.55)
Equilibrium equation:
Mun+1 + Cun+1 + Kun+1 = Fn+1 (6.56)
Correctors:
un+1 = un + ∆t2β1 − αf
1 − αm
un+1 (6.57)
un+1 = ˜un + ∆tγ1 − αf
1 − αm
un+1 (6.58)
Recursive equation:
(1 − αm)an+1 + αman = (1 − αf)un+1 + αf un , a0 = u0 (6.59)
The state variables u are nodal quantities arising from a spatial discretiza-tion and their derivatives u and u with respect to time t are indicated withsuperposed dots. a are the auxiliary acceleration-like variables, which areidentical to the computational accelerations in Chung and Hulbert (1993).By contrast, u, which are the true accelerations keep the property of second-order accuracy as stated in (1.28)(Erlicher et al., 2002).
For the solution procedure, we implemented the CH -α method in a u-form.
MI un+1 = Fn+1 − C˜un − Kun (6.60)
where
MI = M + ∆tγ1 − αf
1 − αm
C + ∆t2β1 − αf
1 − αm
K (6.61)
78
After (6.60) is solved, (6.57) and (6.58) may then be used to compute un+1
and un+1, respectively. In addition, the recursive equation (6.59) needs tobe solved for an+1. As a result four quantities, i.e. u, u, u and a have tobe solved for each time step. The new implementation of the CH -α methodby Arnold and Bruls (2007) is no more a one-step three-stage method asindicated in (3.4) but a one-step four-stage method.
Based on the implicit form of the CH -α method by Arnold and Bruls(2007), we propose a new explicit form of the CH -α method enforcing theequilibrium at the end of each time step. Like its implicit progenitor, theexplicit CH -α method preserves the properties of second-order accuracy inthe accelerations, being a one-step four-stage method. Hence, the followingequilibrium equation is proposed for the new explicit form of the CH -αmethod:
Mun+1 + Cun+1 + Kun = Fn+1 (6.62)
where un is defined by (6.54). Equation (6.62) together with (6.54), (6.55)and (6.57) - (6.59) provide an explicit implementation of the CH -α method.For the solution procedure, one may use the following u-form:
MEun+1 = Fn+1 − C˜un − Kun (6.63)
where
ME = M + ∆tγ1 − αf
1 − αm
C (6.64)
On each time step, after (6.63) is solved, un+1, un+1 and an+1 are cal-culated using (6.57), (6.58) and (6.59). Moreover, the contribution of K in(6.61) is neglected to keep an explicit form.
It has been proved by He (2008) that the new explicit form of the CH -α method is numerically equivalent to the explicit-α method proposed byDaniel (2003). Moreover, the integration parameters of the new algorithmcan be obtained using those of the implicit CH -α method of Chung andHulbert (1993), with the spectral radius ρ∞ substituted by ρb, the spectralradius at the bifurcation point of the explicit algorithm. In terms of ρ,which is either ρ∞ for the implicit algorithm or ρb for the explicit one, thealgorithmic parameters of the new implicit and explicit CH -α method read
αm =2ρ − 1
ρ + 1, αf =
ρ
ρ + 1(6.65)
β =1
4(1
2+ γ)2 , γ =
1
2− αm + αf (6.66)
79
In detail, the bifurcation limit Ωb and the stability limit Ωs of the explicitalgorithm, in the case of C = 0 are defined as
Ωb = (1 + ρb)√
2 − ρb (6.67)
Ωs =
√12(1 + ρb)
3(2 − ρb)
10 + 15ρb − ρ2b + ρ3
b − ρ4b
(6.68)
where b and s subscript are used to denote bifurcation and stability.
Formulation of the algorithm By adopting the CH -α methods whichenforce equilibrium at the end of each time step, a novel interfield parallelmethod is obtained, which is referred to as the PM -α method. We considersubdomain A to be integrated with a coarse time step ∆tA using the implicitCH -α method and subdomain B to be integrated with a small time step∆tB using the explicit CH -α method. Hence, the flow-chart of the PM-αmethod reads:
1. solve the free problem in subdomain A using 2∆tA, thus advancingfrom tn−1 to tn+1
MAuAn+1f
= FAn+1 − CA ˜uA
n−1 − KAuAn−1 (6.69)
uAn+1f
= uAn−1 + αA
1 uAn+1f
(6.70)
uAn+1f
= ˜uA
n−1 + αA2 uA
n+1f(6.71)
where MA, uAn−1 and ˜uA
n−1 are given by (6.61), (6.54) and (6.55),respectively; and
αA1 = 4∆t2AβA
1 − αAf
1 − αAm
and αA2 = 2∆tAγA
1 − αAf
1 − αAm
(6.72)
2. start the loop on ss substeps in subdomain B at the j-th subsetp withj = 1, . . . , ss
3. solve the free problem in subdomain B at tn+ jss
by
MBuBn+ j
ss f
= FBn+ j
ss
− CA ˜uB
n+ j−1
ss− KBuB
n+ j−1
ss
(6.73)
uBn+ j
ss f
= uBn+ j−1
ss
+ αB1 uB
n+ jss f
(6.74)
uBn+ j
ss f
= ˜uB
n+ j−1
ss+ αB
2 uBn+ j
ss f
(6.75)
80
where MB, uBn+ j−1
ss
and ˜uB
n+ j−1
ssare given by (6.64), (6.54) and
(6.55), respectively; and
αB1 = ∆t2BβB
1 − αBf
1 − αBm
and αB2 = ∆tBγB
1 − αBf
1 − αBm
(6.76)
In case of a PsD test, RB(uB
n+ j−1
ss
)can be directly measured by
imposing the displacement uBn+ j−1
ss
on the physical substructure B.
4. interpolate the free velocity in subdomain A by
uAn+ j
ss f
= (1 − j
ss)uA
nf+
j
ssuA
n+1f(6.77)
5. compute the Lagrange multipliers Λn+ jss
by solving the condensed
interface problem
HΛn+ jss
= −(LAuAn+ j
ss f
+ LBuBn+ j
ss f
) (6.78)
withH = αA
2 LAMA−1
LAT
+ αB2 LBMB−1
LBT
(6.79)
6. solve the link problem in subdomain B at tn+ jss
MBuBn+ j
ss l
= LBT
Λn+ jss
(6.80)
uBn+ j
ss l
= αB1 uB
n+ jss l
(6.81)
uBn+ j
ss l
= αB2 uB
n+ jss l
(6.82)
7. Compute the kinematic quantities u, u and u of subdomain B attn+ j
ssby
(·) = (·)free + (·)link (6.83)
and then compute the auxiliary acceleration-like variables aBn+ j
ss
of
subdomain B at tn+ jss
by
(1 − αBm)aB
n+ jss
+ αBmaB
n+ j−1
ss
= (1 − αBf )uB
n+ jss
+
αBf uB
n+ j−1
ss
with aB0 = uB
0
(6.84)
8. if j = ss, then end the loop in subdomain B
81
9. solve the link problem in subdomain A using a time step 2∆tA fromtn−1 to tn+1
MAuAn+1l
= LAT
Λn+1 (6.85)
uAn+1l
= αA1 uA
n+1l(6.86)
uAn+1l
= αA2 uA
n+1l(6.87)
10. Calculate the kinematic quantities u, u and u of subdomain A at tn+1
by
(·) = (·)free + (·)link (6.88)
and then compute the auxiliary acceleration-like variables aAn+1 of
subdomain A at tn+1 by
(1−αAm)aA
n+1 +αAmaA
n−1 = (1−αAf )uA
n+1 +αAf uA
n−1 , aA0 = uA
0
(6.89)
Regarding the initialization of the PM -α method, the problem is similar tothat of the PM method as two starting points are required on subdomainA. We suggest to use a starting condition similar to that of the PM method,see (6.42)), which has been discussed before.
Amplification matrix of the PM-α method In the following analysis,the implicit CH -α method in subdomain A and the explicit CH -α methodin subdomain B are characterized by ρ∞ and ρb, respectively, according to(6.65) and (6.66).
When applied to linear problems, the PM-α method can be recast in arecursive form as
Xn+1 = A Xn + Ln (6.90)
with X an appropriate state vector depending on the formulation of thescheme, A the amplification matrix and L the load vector which dependson external forces. To proceed to the next time step the PM-α methodrequires not only the state variables at time tn but also some ones at tn−1.For this reason, we consider the following state vector:
Xn=(
XAn−1 XA
n XBn
)T(6.91)
where XAn collects the kinematic quantities of subdomain A
XAn =
(uA
n uAn uA
nfreeuA
n aAn
)T
(6.92)
82
and XBn those of subdomain B
XBn =
(uB
n uBn uB
n aBn
)T(6.93)
Consequently, Xn has the dimension 10 nA +4 nB with nA, nB the DoFsin the two subdomains. For homogeneous linear model problems, consideredwithout damping for the sake of brevity, (6.91) turns out to be
XAn
XAn+1
XBn+1
= A
XAn−1
XAn
XBn
=
0 I 0AAA AAA AAB
ABA ABA ABB
XAn−1
XAn
XBn
(6.94)where I is the identity matrix, and A is expressed using the block matrix.
Accuracy of the PM-α method In order to perform the accuracy anal-ysis, we introduce an alternative form of (6.91)
Xn+1 = A Xn + Ln (6.95)
where
Xn = (
uAn−1 uA
n−1 uAn−1f
uAn−1∆tA aA
n−1∆tA uAn
uAn uA
nfuA
n ∆tA aAn ∆tA
uBn uB
n uBn ∆tA aB
n ∆tA
)T (6.96)
A = MAM−1
, Ln = MLn (6.97)
M = block diagonal[I I I ∆tAI ∆tAI I I
I ∆tAI ∆tAI I I ∆tAI ∆tAI]
(6.98)
In particular, ∆tA has been introduced in the acceleration terms of (6.96).Given the definition of local truncation error τn in (6.38), the accuracy
analysis results of the PM-α method for the SDoF model problem intro-duced above turn out to be similar to those stated in (6.40) and (6.41).Moreover, given the definition of the initialization error e2 in (6.43), thePM-α method exhibits on the SDoF model problem similar results to thoseobtained in (6.45).
Stability The stability of the PM-α method is investigated herein em-ploying the spectral analysis approach on the SDoF model problem. It isobtained that A defined in (6.90) is endowed with 9 non-zero eigenvalues.
83
Figure 31. |λi| of the SDoF problem for the PM -α method when b1 = 10:(a) ss = 1; (b) ss = 2; and (c) ss = 5.
In detail, with ρ∞ = ρb = 1 and ∆tA = 0 the non-zero eigenvalues read
[1 1 1 −1 −1 i −i −b1−2
√−1−b1
2+b1
−b1+2√
−1−b1
2+b1
]
(6.99)if ss = 1, and
[1 1 1 1 −1 i −i
−8b1−√
−256−256b1+b4
1
(4+b1)2−8b1+
√−256−256b1+b4
1
(4+b1)2
]
(6.100)if ss = 2. The modulus of the eigenvalues are less than or equal to 1,provided any value of b1 = mA
mB= kB
kA. Moreover, the repeated eigenvalues
have linearly independent eigenvectors. We verified that the aforementionedfindings are valid for any ∆tA > 0, 0 ≤ ρ∞ ≤ 1 and 0 ≤ ρb ≤ 1. Asa result, the method is stable if max
i
∣∣λi(A)∣∣ ≤ 1. The absolute values
of the eigenvalues of A are plotted in Figure 31 versus ΩB = ωB × ∆tB
employing few values of ss. The global problem has the stability limit alwaysequal to or higher than that of the explicit subdomain B.
Convergence The global error en at tn of the PM-α method is alsoaffected by both the local truncation error and the initialization error, asindicated by (6.48)). Analytical findings of He (2008) indicate that the PM-α method is second-order accurate when ss = 1 and first-order accuratewhen ss > 1. Figure 32 shows the global error |en| = |Xn − X(tn)|versus ∆tA in a logarithmic scale for the SDoF model problem, whichconfirms the convergence analysis results.
84
Figure 32. Global error of the SDoF problem (b1 = 10) for the PM -αmethod with: (a) ss = 1; (b) ss = 2; and (c) ss = 20.
Numerical dissipation Figure 33 (a) shows the numerical damping ra-tio induced by the PM -α method on the SDoF model problem, whereρ = ρ∞ = ρb. In detail, the algorithmic damping ratio ξ and the dis-torted angula frequency Ω have been defined in (3.43) and (3.44), respec-tively. Figure 33(a) shows the high-frequency dissipation capabilities of thePM-α method which are controllable by the user. Conversely, 33(b) showsthe spectral radius of the scheme, ignoring the real unity eigenvalue, whichcontributes to a constant drift-off displacement error as indicated in Figure29) for the PM method. One can observe from Figure 33 that the equiva-lent damping generated at the shared node by the PM method is modesteven for large time steps. In contrast with that, the PM -α method pro-vides adequate numerical dissipation of the higher modes components of theresponse.
Some Numerical Simulations The Four-DoF emulated system consid-ered herein can be split into two parts as shown in Figure 34(a). The
displacement vector of subdomain A is chosen to be UA= u2, u3, u4T
while the displacement vector of subdomain B is chosen as UB= u1, u2T.
The mass matrices of each subdomain read
MA = m
r1+r
0 0
0 1 00 0 1
and MB = m
[1 00 1
1+r
](6.101)
85
Figure 33. Numerical dissipation of the PM -α method for the SDoF prob-lem when b1 = 0.01 and ss = 5: (a) algorithmic damping ratio and (b)spectral radius.
while the stiffness matrices of each subdomain read
KA = 106
35.5 −32.6 9.7−32.6 32.3 −5.6
9.7 −5.6 7.4
and KB = 106
[12.4 −6.6−6.6 3.9
]
(6.102)where the condensed mass m = 20, 000kg for each DoF and r = mA
mB
being m = mA+mB. As a result, the natural frequencies for the emulatedsystem read
f1 = 0.47 Hz, f2 = 2.90 Hz, f3 = 4.18 Hz and f4 = 9.47 Hz(6.103)
Figure 34(b) and (c) show the results provided by the PM and PM -αmethods, respectively, using the same time step ∆tA/T1 = 1/40 andss = 20. It is evident that the unwanted high-frequency components ofthe response are traced by the PM method, while the PM -α method dampsout them after a few time steps.
7 Conclusions and Future Perspectives
This section summarizes the main conclusions of the material presented inthe chapter which is not intended to cover all of the numerical methods andtechniques presented or all of the types of systems covered. Moreover, thesection offers recommendations for future directions.
86
Figure 34. Partitioned four-DoF system: (a) the model problem; the dis-placement response in free vibration integrated by (b) the PM method and(c) the PM -α method.
7.1 Conclusions
Although analytical or symbolic solutions provide the greatest insightregarding the influence of various parameters in the transient dynamic re-sponse of monolithic structural systems as well as of heterogeneous (numer-ical/physical) subsystems, the researchers frequently resort to numericalstudies or testing techniques or fusion of both in order to discern trends oftheir dynamic behaviour. Thus, long and expensive trials with simulationsand experiments are often the only avenue left to understand the behaviourof complex non-linear mechanical and civil-engineering systems. Since thischapter has been the first, we have presented a brief state-of-the-art onreal-time testing with DS in the context of HiL and PsD testing. We haveintroduced the CH -α methods whose special cases even today represent themost common structural integrators used in FE codes and testing. In detail,
87
we have shown applications both in monolithic and innovative parallelizedpartitioned methods based on Lagrange multipliers able to guarantee fastpseudo-dynamic tests of complex substructures. We have also introducedfew variants of the Rosenbrock methods conceived to be real-time compat-ible, linearly implicit and thus rather powerful for real-time HiL applica-tions. These time integration methods have been used for the solution ofcoupled-field real-time problems, primarily for structure-structure interac-tion problems occurring in HiL. However, they can be easily extended tostructure-control problems. Because HiL and PsD techniques require ac-tuators and therefore controllers to control them, for this reason numericalanalyses applied to integrate adaptive controllers and to linearize them inorder to understand their properties have been illustrated.
7.2 Future perspectives
As fully illustrated in the previous sections, research on numerical inte-gration is ongoing and continuing to develop and up to now it has studied indepth only a few aspects of real-time HiL and PsD with DS. Other aspectsonly mentioned in the chapter do deserve full scrutiny and in-depth research:we believe that this effort is worthy, being the possibilities offered by thesimulation of heterogeneous systems with DS both unique and attractive.In fact, other HiL techniques such as shaking tables or PsD testing withDS are evolving towards it. Future developments of the LSRT integratorsapplied to the structure-structure interaction shall include the proper simu-lation of damping, especially for emulated structures with non-proportionaldamping. Moreover, the application of the Rosenbrock-based LSRT al-gorithms to structure-control interaction, including the exploitation of thevelocity through a derivative-feedback control to assure a better control ofthe acceleration of the transfer system, deserves further studies. Basingon the current literature, it is highly probable that each future successfulgeneralized control strategy is going to contain adaptive elements. Thus,the analysis described in this chapter that was performed on a first-ordersystem controlled by a discrete MCS controller, could be further developedfor second-order systems that model structural systems. Finally, since theeffect of the delay reduces the stability limit and the effectiveness of the nu-merical damping generated by the controller or other numerical techniques,this important characteristic appears to be worth while in-depth study andresearch in the simulation of heterogeneous systems with DS even in viewof the presence of a delay either multiple of the time step or variable.
88
Bibliography
M. Arnold and O. Bruls. Convergence of the generalized-α scheme forconstrained mechanical systems. Multibody Syst Dyn, 18:185–202, 2007.
M. Arnold, B. Burgermeister, and A. Eichberger. Linearly implicit timeintegration methods in real-time applications: DAEs and stiff ODEs.Multibody System Dynamics, 17:99–117, 2007.
G. Baldo, A. Bonelli, O. Bursi, and S. Erlicher. The accuracy of the gener-alized method in the time integration of non linear single and two DoFforced systems. Computational Mechanics, 38:15–31, 2006.
V. Bayer, U. E. Dorka, U. Fullekrug, and J. Gschwilm. On real-time pseudo-dynamic sub-structure testing: algorithm, numerical and experimentalresults. Aerospace Science and Technology, 9:223–232, 2005.
B. Biondi and G. Muscolino. Component-mode synthesis method variantsin the dynamics of coupled structures. Meccanica, 35:17–38, 2000.
A. Bonelli and O. S. Bursi. Generalized-α methods for seismic structuraltesting. Earthquake Engineering and Structural Dynamics, 33(10):1067–1102, 2004.
A. Bonelli, O. S. Bursi, and M. Mancuso. Explicit predictor-multicorrectortime discontinuons Galerkin methods for linear dynamics. Journal ofSound and Vibration, 246(4):625–652, 2001.
A. Bonelli, O. S. Bursi, and M. Mancuso. Explicit predictor-multicorrectortime discontinuons Galerkin methods for non-linear dynamics. Journalof Sound and Vibration, 256(4):659–724, 2002.
A. Bonelli, O. S. Bursi, L. He, P. Pegon, and G. Magonette. Convergenceanalysis of a parallel interfield method for heterogeneous simulations withdynamic substructuring. International Journal for Numerical Methodsin Engineering, 2008. DOI: 10.1002/nme.2285.
P. A. Bonnet, C. N. Lim, M. S. Williams, A. Blakeborough, S. A. Neild,D. P. Stoten, and C. A. Taylor. Real-time hybrid experiments with New-mark integration, MCSmd outer-loop control and multi-tasking strate-gies. Earthquake Engineering and Structural Dynamics, 36(1):577–588,2007.
P.A. Bonnet. The development of multi-axis real-time substructure testing.PhD thesis, University of Oxford, 2006.
O. Bruls and J.C. Golinval. On the numerical damping of time integra-tors for coupled mechatronic systems. Computer Methods in AppliedMechanics and Engineering, 197:577–588, 2008.
B. Burgermeister, M. Arnold, and B. Ester. DAE time integration for real-time applications in multi-body dynamics. ZAMM Z. Angew. Math.Mech., 86(10):759–771, 2006.
89
O. S. Bursi and M. Mancuso. Analysis and performance of a predictor-multicorrector time discontinuous Galerkin method in non-linear elas-todynamics. Earthquake engineering and structural dynamics, 31:1793–1814, 2002.
O. S. Bursi and P. B. Shing. Evaluation of some implicit time-stepping algo-rithms for pseudodynamic tests. Earthquake engineering and structuraldynamics, 25:333–355, 1996.
O. S. Bursi, D. P. Stoten, and L. Vulcan. Convergence and frequency-domain analysis of a discrete first-order model reference adaptive con-troller. Structural Control and Health Monitoring, 14:777–807, 2007.
O. S. Bursi, B. A. Gonzalez, L. Vulcan, N. A. Neild, and D. J. Wagg. Novelcoupling Rosenbrock-based algorithms for real-time dynamic substruc-ture testing. Earthquake Engineering and Structural Dynamics, 37(3):339–360, 2008.
J. C. Butcher. Numerical Methods for Ordinary Differential Equations.Wiley, 2003.
V. Cannillo and M. Mancuso. Spurious resonances in numerical time inte-gration methods for linear dynamics. Journal of Sound and Vibration,238(3):389–399, 2000.
S. Y. Chang. Error propagation in implicit pseudodynamic testing of non-linear systems. Journal of Engineering Mechanics, 131(12):1257–1269,2005.
S.-Y. Chang. An improved on-line dynamic testing method. EngineeringStructures, 24:587–596, 2002.
S. Y. Chu, T. T. Soong, C. C. Lin, and Y. Z. Chen. Time-delay effectand compensation on direct output feedback controlled mass dampersystems. Earthquake Eng. and Struct. Dyn., 31(1):121–137, 2002.
S. Y. Chu, T. T. Soong, and A. M. Reinhorn. Active, Hybrid, and Semi-active Structural Control: A Design and Implementation Handbook. Wi-ley, Chichester, 2005.
J. Chung and G. M. Hulbert. A time integration algorithm for structuraldynamics with improved numerical dissipation: the Generalized-alphamethod. Journal of Applied Mechanics, 60:371–375, 1993.
A. Combescure and A. Gravouil. A numerical scheme to couple subdo-mains with different time-steps for predominantly linear transient anal-ysis. Computer Methods in Applied Mechanics and Engineering, 191:1129–1157, 2002.
D. Combescure and P. Pegon. α-Operator Splitting time integration tech-nique for pseudodynamic testing error propagation analysis. Soil Dy-namics and Earthquake Engineering, 16:427–443, 1997.
90
J. Cuadrado, J. Cardenal, and E. Bayo. Modeling and solution methods forefficient real-time simulation of multibody dynamics. Multibody SystemDynamics, 1:1897–1913, 1997.
G. Dahlquist. A special stability problem for linear multistep methods. BITNumerical Mathematics, 3(1):27–43, 1963.
W. J. T. Daniel. Explicit/implicit partitioning and a new explicit form ofthe generalized alpha method. Communications in numerical methodsin engineering, 19:909–920, 2003.
A. P. Darby, M. S. Williams, and A. Blakeborough. Stability and delaycompensation for real-time substructure testing. Journal of EngineeringMechanics, 128(12):1276–1284, 2002.
J. R. Dormand and P. J. Prince. A family of embedded Runge-Kutta for-mulae. Journal of Computational and Applied Mathematics, 6:19–26,1980.
Real-time interface, Implementation guide, Release 3.2. dSPACE GmbH,Paderborn, Germany, 2001.
S. Erlicher, L. Bonaventura, and O. S. Bursi. The analysis of generalized-αmethods for non-linear dynamic problems. Computational Mechanics,28:83–104, 2002.
C. Farhat and F.-X. Roux. A method of finite element tearing and inter-connecting and its parallel solution algorithm. International Journal forNumerical Methods in Engineering, 32:1205–1227, 1991.
C. Farhat, P.S. Chen, and J. Mandel. A scalable Lagrange multiplier baseddomain decomposition method for time dependent problems. Inter-national Journal for Numerical Methods in Engineering, 38:3831–3854,1995.
C. Farhat, J. Cortial, C. Dastillung, and H. Bavestrello. Time-parallel im-plicit integrators for the near-real-time prediction of linear structuraldynamic responses. International Journal for Numerical Methods in En-gineering, 67:697–724, 2006.
C. A. Felippa and K. C. Park. Staggered transient analysis proceduresfor coupled dynamic systems: formulation. Comp. Meths. Appl. Mech.Engrg., 24:61–112, 1980.
C. A. Felippa, K.C. Park, and C. Farhat. Partitioned analysis of coupledmechanical systems. Computer Methods in Applied Mechanics and En-gineering, 190:3247–3270, 2001.
G. F. Franklin and J. D. Powell. Digital Control of Dynamic Systems.Addison-Wesley Publishing Company, New York, 1980.
A. Gravouil and A. Combescure. Multi-time-step explicit-implicit methodfor non-linear structural dynamics. International Journal for NumericalMethods in Engineering, 50:199–225, 2001.
91
E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiffand Differential-Algebraic Problems. Springer-Verlag, 1996.
E. Hairer, S. P. Norsett, and G. Wanner. Solving Ordinary DifferentialEquations I. Nonstiff Problems. Springer-Verlag, 1987.
L. He. Development of Partitioned Time Integration Schemes for ParallelSimulation of Heterogeneous Systems. PhD thesis, University of Trento,2008.
H. M. Hilber, T. J. R. Hughes, and R. L. Taylor. Improved numerical dis-sipation for time integration algorithms in structural dynamics. Earth-quake Engineering and Structural Dynamics, 5:283–292, 1977.
T. Horiuchi, M. Inoue, T. Konno, and Namita Y. Real-time hybrid experi-mental system with actuator delay compensation and its application to apiping system with energy absorber. Earthquake Eng. and Struct. Dyn.,28(10):1121–1141, 1999.
T. J. R. Hughes. The Finite Element Method, Linear Static and DynamicFinite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1987.
T. J. R. Hughes and G. M. Hulbert. Space-time finite element methods forelastodynamics: formulations and error estimates. Computer Methods inApplied Mechanics and Engineering, 66:339–363, 1988.
T. J. R. Hughes, K. S. Pister, and R. L. Taylor. Implicit-explicit finiteelements in non-linear transient analysis. Computer Methods in AppliedMechanics and Engineering, 17/18:159–182, 1979.
G. M. Hulbert. Space-time finite element methods: 1970’s and Beyond,chapter An Overview of Variational Integrators, pages 116–123. CIMNE,Barcelona, Spain, 2004.
G. M. Hulbert and J. Chung. The unimportance of the spurious root oftime integration algorithms for structural dynamics. Communicationsin Numerical Methods in Engineering, 10:591–597, 1994.
G. M. Hulbert and T. J. R. Hughes. An error analysis of truncated startingconditions in step-by-step time integration: Consequences for structuraldynamics. Earthquake Engineering and Structural Dynamics, 15:901–910, 1987.
G.M. Hulbert and J. Chung. Explicit time integration algorithms for struc-tural dynamics with optimal numerical dissipation. Computer Methodsin Applied Mechanics and Engineering, 137:175–188, 1996.
R. Y. Jung, P. B. Shing, E. Stauffer, and B. Thoen. Performance of areal-time pseudodynamic test system considering nonlinear structuralresponse. Earthquake Engineering and Structural Dynamics, 32:1785–1809, 2007.
R. Kubler and W. Schiehlen. Modular simulation in multibody systemdynamics. Multibody System Dynamics, 4:107–127, 2004.
92
Y. N. Kyrychko, K. B. Blyuss, A. Gonzalez-Buelga, S. J. Hogan, andD. J. Wagg. Real-time dynamic substructuring in a coupled oscillator-pendulum system. Proceedings of the Royal Society A, 462:1271–1294,2006.
L. M. Lacoma and I. Romero. Error estimation for hht method in non-linearsolid dynamics. Computers and structures, 85:158–169, 2007.
J. D. Lambert. Numerical Methods for Ordinary Differential Systems: TheInitial Value Problem. Wiley, 1991.
X. D. Li and N. E. Wiberg. Structural dynamic analysis by a time-discontinuous galerkin finite element method. International Journal forNumerical Methods in Engineering, 39:2131–2152, 1996.
C. N. Lim, S.A. Neild, D. P. Stoten, and D. Drury. Real-time dynamic sub-structure testing via an adaptive control strategy. 1st International Con-ference on Advances in Experimental Structural Engineering, Nagoya,Japan, pages 393–400, 2005.
F. Lopez-Almansa, A. H. Barbat, and J. Rodellar. SSP algorithm for linearand non-linear dynamic response simulation. International Journal forNumerical Methods in Engineering, 26(12):2687 – 2706, 1998.
Simulink: simulation and model-based design. The MathWorks, Natick,MA, 2005.
W. E. Misselhorn, N. J. Theron, and P. S. Els. Investigation of hardware-in-the-loop for use in suspension development. Vehicle System Dynamics,44:65–81, 2006.
C. Moler and C. Van Loan. Nineteen dubious ways to compute the expo-nential of a matrix. SIAM Review, 45:1–46, 1978.
G. Mosqueda, B. Stojadinovic, and S.A. Mahin. Real-time error monitoringfor hybrid simulation. part I: Methodology and experimental verification.Journal of Structural Engineering, ASCE, 133(8):1100–1108, 2007a.
G. Mosqueda, B. Stojadinovic, and S.A. Mahin. Real-time error monitor-ing for hybrid simulation. part II: Structural response modification dueto error. Journal of Structural Engineering, ASCE, 133(8):1109–1117,2007b.
A. Mugan and G. M. Hulbert. Alternative characterization of time integra-tion schemes. International Journal for Numerical Methods in Engineer-ing, 51(3):351–376, 2001.
M. Nakashima, T. Kaminosomo, M. Ishida, and K. Ando. Integration tech-niques for substructuring pseudodynamic test. In fourth U.S. NationalConference on Earthquake Engineering, volume 2, Palm Springs, Cali-fornia, 1990.
P. Nawrotzki and C. Eller. Numerical stability analysis in structural dy-namics. Comput. Methods Appl. Mech. Engrg., 189:915–929, 2000.
93
S. A. Neild, D. P. Stoten, D. Drury, and D. J. Wagg. Control issues relatingto real-time substructuring experiments using a shaking table. Earth-quake Engineering and Structural Dynamics, 34(9):1171 – 1192, 2005.
N. M. Newmark. A method of computation for structural dynamics. Journalof the Engineering Mechanics Division ASCE, 85:67–94, 1959.
K. C. Park and Carlos A. Felippa. A variational principle for the formulationof partitioned structural systems. Int. J. Numer. Meth. Engng., 47:395–418, 2000.
P. Pegon. Alternative characterization of time integration schemes. Com-puter Methods in Applied Mechanics and Engineering, 190(20):2707–2727, 2001.
P. Pegon. Continuous psd testing with substructuring. In O. S. Bursi andD. J. Wagg, editors, Modern Testing Techniques for Structural Systems,Dynamics and Control. CISM, 2008.
P. Pegon and G. Magonette. Continuous PSD testing with non-linear sub-structuring: Presentation of a stable parallel inter-field procedure. Tech-nical Report I.02.167, E.C., JRC, ELSA, Ispra, Italy, 2002.
P. Pegon and G. Magonette. Continuous PsD testing with non-linear sub-structuring: using the operator splitting technique to avoid iterative pro-cedures. Technical Report SPI.05.30, E.C., JRC, ELSA, Ispra, Italy,2005.
A. V. Pinto, P. Pegon, G. Magonette, and G. Tsionis. Pseudo-dynamic test-ing of bridges using non-linear substructuring. Earthquake Engineeringand Structural Dynamics, 33:1125–1146, 2004.
V.M. Popov. Hyperstability of Automatic control systems. Springer-Verlag,New York, 1973.
A. Preumont. Frequency domain analysis of time integration operators.Earthquake Engineering and Structural Dynamics, 10(5):691–697, 1982.
AM. Reinhorn, MV. Sivaselvan, Z. Liang, and X. Shao. Real-time dynamichybrid testing of structural systems. In Proceeding of the 13th WorldConference on Earthquake Engineering, Vancouver, Canada, 2004.
H. H. Rosenbrock. Some general implicit processes for the numerical solutionof differential equations. Computer Journal, 5:329–330, 1963.
J. C. Samin, O. Bruls, J. F. Collard, L. Sass, and P. Fisette. Multiphysicsmodeling and optimization of mechatronic multibody systems. MultibodySystem Dynamics, 18:345–373, 2007.
J. D. Sherrick. Concepts In Systems and Signals. Prentice Hall, 2004.
P. B. Shing. Real-time hybrid testing techniques. In O. S. Bursi and D. J.Wagg, editors, In Modern Testing Techniques for Structural Systems,Dynamics and Control. CISM, 2008.
94
P. B. Shing and M.T. Vannan. On the accuracy of an implicit time inte-gration for pseudodynamic tests. Earthquake Engineering and StructuralDynamics, 19:631–651, 1990.
P. B. Shing, M.T. Vannan, and E. Cater. Implicit time integration forpseudodynamic tests. Earthquake Engineering and Structural Dynamics,20:551–576, 1991.
P. B. Shing, O. S. Bursi, and M.T. Vannan. Pseudodynamic tests of aconcentrically braced frame using substructuring techniques. J. of Con-struct. Steel Research, 29:121–148, 1994.
A. Soroushian, P. Wriggers, and J. Farjoodi. On practical integration ofsemi-discretized equations of motion part 1: reasons for probable insta-bility and improper convergence. Journal of Sound and Vibration, 18:705–731, 2005.
D. P. Stoten and H. Benchoubane. Empirical studies of an mrac algo-rithm with minimal controller synthesis. International Journal of Con-trol, 51(4):823–849, 1990.
D. P. Stoten and S. A. Neild. The error-based minimal control synthesisalgorithm with integral action. Proc. of the Institetion of MechanicalEngineers, Part I: Journal of systems and Control Engineering, 217:187–201, 2003.
C. R. Thewalt and S. A. Mahin. Hybrid solution techniques for general-ized pseudodynamic testing. Technical Report No. UCB/EERC-87/09,Earthquake Engineering Research Center, University of California Berke-ley, CA, 1987.
R. J. Vaccaro. Digital Control - A State-Space Approach. McGraw-Hill,New York, 1995.
V. R. Vasquez and W. B. Whiting. Accounting for both random errorsand systematic errors in uncertainty propagation analysis of computermodels involving experimental measurements with monte carlo methods.Risk Analysis, 25(6):1669–1681, 2006.
L. Vulcan. Discrete-time analysis of integrator algorithms applied to S.I.S.Oadaptive controllers with minimal control systhesis. PhD thesis, Univer-sity of Trento, Italy, 2006.
D. J. Wagg, S. A. Neild, and P. Gawthrop. Real-time testing with dynamicsubstructuring. In O. S. Bursi and D. J. Wagg, editors, Modern TestingTechniques for Structural Systems, Dynamics and Control. CISM, 2008.
M.I. Wallace, J. Sieber, S.A. Neild, D.J. Wagg, and B. Krauskopf. Stabil-ity analysis of real-time dynamic substructuring using delay differentialequation models. Earthquake Engineering and Structural Dynamics, 34(15):1817 – 1832, 2004.
95
M.I. Wallace, D.J. Wagg, and S.A. Neild. An adaptive polynomial basedforward prediction algorithm for multi-actuator real-time dynamic sub-structuring. Proceedings of the Royal Society A, 2064(461):3807–3826,2005.
N. E. Wiberg and X.D. Li. A post-processing technique and an a posteriorierror estimate for the Newmark method in dynamic analysis. EarthquakeEngineering and Structural Dynamics, 22(6):465–489, 1993.
M.S. Williams and A. Blakeborough. Laboratory testing of structures underdynamic loads: an introductory review. Philosophical Transactions of theRoyal Society of London, Series A, 359(1786):1651–1670, 2001.
B. Wu, H. Bao, J. Ou, and S. Tian. Stability and accuracy analysis of thecentral difference method for real-time substructure testing. EarthquakeEngineering and Structural Dynamics, 34(7):705–718, 2005.
B. Wu, G. Xu, Q. Wang, and M. S. Williams. Operator-splitting method forreal-time substructure testing. Earthquake Engineering and StructuralDynamics, 35(3):293–314, 2006.
D. M. Young. Iterative Solution of Large Linear System. Academic Press,Orlando, 1971.
Q. Zhang and T. Hisada. Studies of the strong coupling and weak couplingmethods in fsi analysis. International Journal for Numerical Methods inEngineering, 60:2013–2029, 2004.
Y. Zhang, R. Sause, J. M. Ricles, and C. J. Naito. Modified predictor-corrector numerical scheme for real-time pseudo dynamic tests usingstate-space formulations. Earthquake Engineering and Structural Dy-namics, 34(3):271–288, 2005.
96