A-posteriori error estimation for second order mechanical systems

Acta Mech. Sin. (2012) 28(3):854–862DOI 10.1007/s10409-012-0114-7

RESEARCH PAPER

A-posteriori error estimation for second order mechanical systems

Thomas Ruiner ··· Jorg Fehr ··· Bernard Haasdonk ··· Peter Eberhard

Received: 29 September 2011 / Revised: 27 February 2012 / Accepted: 1 April 2012©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2012

Abstract One important issue for the simulation of flexiblemultibody systems is the reduction of the flexible bodies de-grees of freedom. As far as safety questions are concernedknowledge about the error introduced by the reduction of theflexible degrees of freedom is helpful and very important.

In this work, an a-posteriori error estimator for linearfirst order systems is extended for error estimation of me-chanical second order systems. Due to the special secondorder structure of mechanical systems, an improvement ofthe a-posteriori error estimator is achieved. A major advan-tage of the a-posteriori error estimator is that the estimatoris independent of the used reduction technique. Therefore, itcan be used for moment-matching based, Gramian matricesbased or modal based model reduction techniques.

The capability of the proposed technique is demon-strated by the a-posteriori error estimation of a mechanicalsystem, and a sensitivity analysis of the parameters involvedin the error estimation process is conducted.

Keywords A-posteriori error estimator · Flexible multibodysystems ·Model reduction

1 Introduction

The simulation of technical systems nowadays has to beconsidered as an essential part of the development processof many new products. For the description of the dynamicalbehavior of mechanical subsystems the multibody systems

T. Ruiner · J. Fehr · P. EberhardInstitute of Engineering and Computational Mechanics,University of Stuttgart,Pfaffenwaldring 9, 70569 Stuttgart, Germanye-mail: [email protected]

B. HaasdonkInstitute of Applied Analysis and Numerical Simulation,University of Stuttgart,Pfaffenwaldring 57, 70569 Stuttgart, Germanye-mail: [email protected]

method (MBS) [1] is frequently used.Nowadays, the use of lighter structures and an increased

operating speed demand the consideration of elastic effectsfor many components. This approach is called flexible multi-body systems or elastic multibody systems (EMBS), seeRef. [2]. Mostly, the modeling and numerical simulationof elastic bodies is achieved with the finite element method,which often leads to a high number of degrees of freedom.Therefore, one essential step for an efficient simulation ofEMBS is the reduction of the elastic degrees of freedom,which usually introduces an error. Existing a-priori respec-tively a-posteriori error estimators investigate the error in thefrequency-domain, see Ref. [3]. Efficient a-posteriori errorestimators, which estimate the error in the time-domain dur-ing the simulation of the reduced system, are used in thereduced basis community. For example, a formulation forfirst order systems is given in Ref. [4]. However, mechani-cal systems are usually described as second order dynamicalsystems and it is very important to keep the second orderstructure in the model reduction step.

For the model reduction only the linear elastic part ofthe equation of motion of a flexible body is considered, seeRef. [5]. Therefore, in the model reduction procedure lineartime-invariant second order multiple-input multiple-output(MIMO)-systems need to be considered. The linear elasticpart written as a linear second order system reads

MMMe · qqq(t) +DDDe · qqq(t) + KKKe · qqq(t) = BBBe · uuu(t),yyy(t) = CCCe · qqq(t),

(1)

where qqq ∈ RN corresponds to the displacements andMMMe,DDDe,KKKe ∈ RN×N denote the flexible mass, damping, andstiffness of the system, respectively. The input uuu ∈ Rp relatesto the states by BBBe ∈ RN×p, while the output yyy ∈ Rr is relatedto the system’s behavior with CCCe ∈ Rr×N .

Using the ansatz qqq(t) ≈ VVVmqqq(t), where qqq ∈ RN , qqq ∈ Rn,VVVm ∈ RN×n and n � N, the flexible coordinates qqq can bereduced from N to n degrees of freedom. We introduce theresidual RRRm(t)

RRRm(t) = MMMe ·VVVm · qqq(t) +DDDe ·VVVm · qqq(t)

A-posteriori error estimation for second order mechanical systems 855

+ KKKe ·VVVm · qqq(t) − BBBe · uuu(t), (2)

which represents the error induced by the reduction of theoriginal system (1). The reduced system is obtained by mul-tiplying with a second matrix WWWm from the left and requir-ing WWWT

mRRRm(t) = 000 with WWWm ∈ RN×n. Overall, this Petrov–Galerkin ansatz yields the following reduced second orderMIMO system

WWWTm ·MMMe ·VVVm︸��︷︷��︸

MMMe

· qqq(t) +WWWTm ·DDDe ·VVVm︸��︷︷��︸

DDDe

· qqq(t)

+WWWTm · KKKe ·VVVm︸��︷︷��︸

KKKe

· qqq(t) =WWWTm · BBBe︸��︷︷��︸

BBBe

· uuu(t), (3)

yyy(t) = CCCe ·VVVm︸��︷︷��︸

CCCe

· qqq(t).

It is very important to keep the second order structurein the model reduction step because in multibody systemtools like Adams, Simpack or Neweul-M2, the reduced elas-tic body (3) must be described as a second order system forsimulations. Usually, the reduction process introduces an er-ror. As far as safety questions are concerned, the knowledgeabout the error introduced by the reduction of the flexibledegrees of freedom is helpful and very important, for exam-ple in crash simulations, or optimization based on damagevalues. The error in the time domain in the position statesis defined as eeem(t) = qqq(t) − VVVmqqq(t). This error needs to beestimated with an evaluation of the full original system (3).Therefore, in the next section the a-posteriori error estima-tor from Ref. [4] is recited and tested for a simple secondorder MIMO system written as a first order system to ex-plain the hump phenomenon. Afterwards, in Sect. 3 the a-posteriori error estimator is extended for error estimation ofmechanical second order systems. A major advantage of thea-posteriori error estimator is that the estimator is indepen-dent of the used reduction technique. Therefore, the estima-tor can be used for moment-matching based on Refs. [6, 7],Gramian matrices based on Refs. [8, 9] or modal based onRefs. [10] model reduction techniques. In Sect. 4 the ca-pability of the proposed technique is demonstrated by thea-posteriori error estimation of a mechanical system and thepaper finishes in Sect. 5 with a conclusion.

2 A-posteriori error estimation for first order systems

The error eeem(t) needs to be measured in an adequate norm.Standard norms, such as the 2-norm, are not feasible if el-ements in a vector or matrix have different units, since e.g.the units of displacement and rotation are hardly compara-ble. Under these circumstances it is necessary to normalizethe diversified entries, which is achieved by using a scalingmatrix GGG ∈ RN×N in the inner product 〈aaa,bbb〉GGG = bbbT ·GGG · aaa.Additionally this matrix can be used to normalize differentunits or to emphasize important values and weakening the

influence of unimportant ones. Therefore, the vector norm isdefined as

‖zzz‖GGG =√

zzzT ·GGG · zzz, (4)

where the inner product matrix GGG needs to be symmetric pos-itive definite. Note thatGGG could also be chosen trivially as theidentity matrix III, which then yields the usual 2-norm ‖·‖2. Infinite element systems the inner matrix GGG is usually chosenas the mass matrix. This vector norm can be used to definean induced matrix norm forΦΦΦ ∈ RN×N , which reads

‖ΦΦΦ‖GGG = max‖zzz‖GGG=1

‖ΦΦΦ · zzz‖GGG = maxzzz�0

‖ΦΦΦ · zzz‖GGG‖zzz‖GGG . (5)

Assuming that the usual 2-norm ‖ · ‖2 is used to com-pare output values, the corresponding norm for the outputmatrix CCC ∈ Rr×N is given by

‖CCC‖GGG,2 = max‖zzz‖GGG=1

‖CCC · zzz‖2 = maxzzz�0

‖CCC · zzz‖2‖zzz‖GGG . (6)

2.1 Relations between first order and second order systems

In order to extend the original error estimator from Ref. [4]directly onto second order MIMO systems, certain relationsare clarified in this section. The original error estimatorgiven in Ref. [4] was developed for first order state-spacemodels,

xxx(t) = AAAs · xxx(t) + BBBs · uuu(t),

yyy(t) = CCCs · xxx(t),(7)

where xxx ∈ RNs is called the state vector, AAAs ∈ RNs×Ns thestate matrix, BBBs ∈ RNs×p the input matrix, and CCCs ∈ Rr×Ns

the output matrix. Furthermore, the model must be reducedby bi-orthonormal projection matrices VVV s and WWW s. Then thereduced system remains in non-descriptor form and reads

xxx(t) = AAAs · xxx(t) + BBBs · uuu(t),

yyy(t) = CCCs · xxx(t),(8)

where AAAs =WWWTs ·AAAs ·VVV s, BBBs =WWWT

s ·BBBs, and CCCs = CCCs ·VVV s. Forthe derivation of the error estimator the error eees and resid-ual RRRs of the state-space model are defined as

eees(t) = xxx(t) −VVV s · xxx(t), (9)

RRRs(t) = AAAs · VVV s · xxx(t) + BBBs · uuu(t) −VVV s · xxx(t). (10)

The residual RRRs(t) has the remarkable property that it is zeroif no reduction error is introduced. Utilizing these defini-tions, an error bound Δx(t) for the state variable xxx is devel-oped in Ref. [4], which reads

‖eees(t)‖GGGs � Δx(t) = C1‖eees,0‖GGGs + C1

∫ t

0‖RRRs(τ)‖GGGs dτ, (11)

where eees,0 denotes the initial error, the constant C1 is givenby C1 � maxt ‖ΦΦΦ(t)‖GGGs , andΦΦΦ corresponds to the fundamen-tal matrixΦΦΦ(t) = eAAAst. In this article the error bound Δx(t) is

856 T. Ruiner, et al.

optimized for second order mechanical systems. In particu-lar, the constant C1 is improved and a more accurate integralis proposed.

In order to utilize the error estimator for first order sys-tems, the second order system from Eq. (1) is transformedinto the first order system from Eq. (7) according to

[

qqq(t)qqq(t)

]

︸︷︷︸

xxx(t)

=

[

000 III−MMM−1

e · KKKe −MMM−1e ·DDDe

]

︸��︷︷��︸

AAAs

·[

qqq(t)qqq(t)

]

︸︷︷︸

xxx(t)

+

[

000BBBe

]

︸︷︷︸

BBBs

· uuu(t),

yyy(t) =[

CCCe 000]

︸��︷︷��︸

CCCs

·[

qqq(t)qqq(t)

]

︸︷︷︸

xxx(t)

. (12)

Consequently, the dimensions of the first order and secondorder systems are related by Ns = 2N. The required bi-orthogonality of WWW s and VVV s can be ensured for any WWWm andVVVm by using the projection matrices

WWWTs =

⎡

⎢⎢⎢⎢⎣

(WWWTm ·VVVm)−1 ·WWWT

m 000

000 MMM−1e ·WWWT

m ·MMMe

⎤

⎥⎥⎥⎥⎦,

VVV s =

[

VVVm 000000 VVVm

]

.

(13)

Furthermore, the error eees(t) and residual RRRs(t) from Eqs. (9)and (10) are related to the second order mechanical systemby

eees(t) =

[

eeem(t)eeem(t)

]

=

[

qqq(t) −VVVm · qqq(t)qqq(t) −VVVm · qqq(t)

]

, (14)

RRRs(t) =

[

000˜RRRm(t)

]

=

[

000−MMM−1

e · RRRm(t)

]

. (15)

2.2 Hump phenomenon for second order systems

The error estimator derived for first order systems, Eq. (11),can deliver impractical results for second order mechani-cal systems, as a very large over-prediction of the error canbe observed. This problem originates from the hump phe-nomenon, explained e.g. in Ref. [11], which leads to extremevalues of the constant C1 = maxt ‖ΦΦΦ(t)‖GGGs = maxt ‖eAAAst‖GGGs .However, this is partly due to the fact that the error estimatordetermines an error bound Δx(t) for the state variable xxx and,therefore, a single error bound for both, qqq and qqq. The largehump is actually related to the velocity states qqq. Luckily forelastic body simulations, only the position states qqq are rele-vant for an error estimation. Therefore, an error bound Δq(t)will be derived in Sect. 3 to improve the output error estimateignoring the velocities.

If the system xxx(t) = AAAsxxx(t) is asymptotically stable,then all eigenvalues of AAAs have nonpositive real parts and‖eAAAst‖ → 0 with t → ∞. However, this does not necessar-ily mean that ‖eAAAst‖ decreases monotonically as t increases.If AAAs is not normal, meaning AAAH

s AAAs � AAAsAAAHs , then ‖eAAAst‖ can

grow arbitrarily large for small but nonzero t and decreaseafterwards. A mathematical example for this phenomenonis provided e.g. in Ref. [11]. The phenomenon can be il-lustrated with a simple mass spring damper system with onedegree of freedom, depicted in Fig. 1. The equation of mo-tion written as a second order linear time-invariant systemreads

mq(t) + dq(t) + kq(t) = F(t), (16)

where the force F(t) is the input into the system. Analo-gously to Eq. (12), this system can be rearranged into[

q(t)q(t)

]

︸︷︷︸

xxx

=

⎡

⎢⎢⎢⎢⎢⎢⎣

0 1

− km− d

m

⎤

⎥⎥⎥⎥⎥⎥⎦

︸��︷︷��︸

AAAs

·[

q(t)q(t)

]

︸︷︷︸

xxx

+

⎡

⎢⎢⎢⎢⎢⎢⎣

01m

⎤

⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸

BBBs

F(t)︸︷︷︸

uuu

. (17)

Fig. 1 Simple mass spring damper system

If no external force is applied, then F(t) = 0 and thesolution of this linear time-invariant differential equation iscompletely described by the fundamental matrixΦΦΦ(t) = eAAAst

and the initial condition xxx0 and reads

xxx(t) = ΦΦΦ(t) · xxx0 =

[

ΦΦΦ11(t) ΦΦΦ12(t)ΦΦΦ21(t) ΦΦΦ22(t)

]

·[

q0

q0

]

. (18)

Plots of ‖ΦΦΦ(t)‖ exhibit the hump phenomenon for certainparameters of m, d, and k. In Fig. 2 the fundamental matrixnorm ‖ΦΦΦ(t)‖GGGs is depicted and the hump phenomenon can beobserved. For the GGGs-norm the mass matrix is used accord-ing to Eq. (24). The stiffness k = 10 kg/s and damping valuesd = 0.2 kg/s2 are constant and the mass is varied. Even withthis simple system the hump phenomenon can be explainedvisually, e.g. by assuming an initial displacement. Oncereleased, the displacement decays due to the damping andthe mass eventually reaches its equilibrium position, possi-bly after a few oscillations. In order to reach the equilibriumstate, however, the velocity must increase. Thus, in contrastto the displacement, the velocity increases at first and causesthe hump phenomenon. This connection between initial dis-placement and velocity is represented by the entry ΦΦΦ21(t) inEq. (18) and plots of all entries of ΦΦΦ(t) confirm that ΦΦΦ21(t)is responsible for the hump. Exemplary, the entry ΦΦΦ21(t) isdepicted in Fig. 2. This knowledge is utilized to refine theerror estimator in the present article. Furthermore, it can beconcluded that the higher the velocity, the more the hump isemphasized. This means that the stiffer the spring and thelighter the mass, the more the mass is accelerated and the


faster it oscillates before reaching the equilibrium state. Thisis exactly what can be observed in Fig. 3, where the humpincreases dramatically when a smaller mass is used. Overall,it can be concluded that the hump is particularly pronouncedin case of high-frequency oscillations.

Fig. 2 Norm of the fundamental matrix over time

Fig. 3 EntryΦΦΦ21(t) of the fundamental matrix

3 A-posteriori error estimator for second order mechan-ical systems

After reciting the error estimator for first order systems theerror estimator is now extended to second order systems.Subtracting the full system, Eq. (7), from the reduced sys-tem, Eq. (10), yields a differential equation for the error

eee(t) = AAAs · eee(t) +RRRs(t). (19)

The explicit solution for this differential equation can befound in literature such as Ref. [12], and reads

eees(t) = ΦΦΦ(t) · eees,0 +

∫ t

0ΦΦΦ(t − τ) ·RRRs(t)dτ, (20)

where ΦΦΦ(t) = eAAAst is the fundamental matrix of the linearsystem xxx = AAAs · xxx and eees,0 denotes the initial error. This dif-ferential equation is the basis for the refined error estimatorfor second order mechanical systems. Plugging Eqs. (14),(15) and (18) into Eq. (20) yields[

eeem(t)eeem(t)

]

=

[

ΦΦΦ11(t) ΦΦΦ12(t)ΦΦΦ21(t) ΦΦΦ22(t)

]

·[

eeem,0

eeem,0

]

+

∫ t

0

[

ΦΦΦ11(t − τ) ΦΦΦ12(t − τ)ΦΦΦ21(t − τ) ΦΦΦ22(t − τ)

]

·[

000˜RRRm(t)

]

dτ, (21)

where the fundamental matrix ΦΦΦ(t) = eAAAst, ∈ R2N×2N is de-composed into four submatrices ΦΦΦi j ∈ RN×N . Since only anerror bound for eeem(t) is sought, Eq. (21) is separated and just

eeem(t) = ΦΦΦ11(t) · eeem,0 +ΦΦΦ12(t) · eeem,0

+

∫ t

0ΦΦΦ12(t − τ) ·˜RRRm(τ)dτ (22)

is considered in the following. Note that only half of thefundamental matrix, namelyΦΦΦ11 andΦΦΦ12, is relevant for thecomputation of the error bound for the displacements qqq. Theentry ΦΦΦ21, which causes the large hump, is no longer re-quired. In the next step, two error bounds ˜Δq(t) and Δq(t)are derived, which fulfill

‖eeem(t)‖GGGm � ˜Δq(t) � Δq(t), (23)

where ˜Δq(t) is more accurate but requires more computationtime than Δq(t). In order to maintain consistency to the errorestimator for first order systems, the relationship

GGGs =

[

GGGm 000000 GGGm

]

(24)

is compulsory, because this yields the equivalence‖RRRs(t)‖GGGs = ‖ ˜RRRm(t)‖GGGm . For the following derivation theabbreviations ˜C11(t) ≥ ‖ΦΦΦ11(t)‖GGGm and ˜C12(t) ≥ ‖ΦΦΦ12(t)‖GGGm

are introduced. Applying the triangle inequality and the def-inition of the matrix norm from Equation (5) yields the errorbound

˜Δq(t) = ˜C11(t)‖eeem,0‖GGGm +˜C12(t)‖eeem,0‖GGGm

+

∫ t

0

˜C12(t − τ)‖˜RRRm(τ)‖GGGm dτ. (25)

High computational cost is caused by the integral whichmust be reevaluated for each time t. However, the evalua-tion of the integral could also be limited to the final instantor certain relevant instants of time. For all other instants oftime the factors C11 � maxt ˜C11(t) and C12 � maxt ˜C12(t) canbe used to obtain less accurate but also computationally lessexpensive error bounds from the altered estimate

Δq(t) = C11‖eeem,0‖GGGm + C12‖eeem,0‖GGGm

+C12

∫ t

0‖˜RRRm(τ)‖GGGm dτ. (26)

In this estimate there is no need to reevaluate the integral, be-cause after every time step the new information can simplybe added to the integral obtained at the previous time. Botherror bounds for the state variable ˜Δq(t) and Δq(t) can be usedto obtain upper bounds for the output. Due to the relation

yyy(t) − yyy(t) = CCCm · (qqq(t) −VVVm · qqq(t)) = CCCm · eeem(t), (27)


upper bounds for the output can be computed from

‖yyy(t) − yyy(t)‖2 � ˜Δy(t) � Δy(t), (28)

where ˜Δy(t) = C2˜Δq(t), Δy(t) = C2Δq(t) and C2 ≥ ‖CCCe‖GGGm,2.For the sake of simplicity, only Δy(t) is used in the follow-ing, even though all derivations and conclusions are equallyvalid for ˜Δy(t). A further improvement can be achieved bysplitting the output matrix CCCe into r submatrices for each ofthe r output dimensions. This results in one error bound peroutput dimension

‖y(1)(t) − y(1)(t)‖2 � Δy(1)(t) = C2(1)Δq(t),...

......

‖y(r)(t) − y(r)(t)‖2 ≤ Δy(r)(t)︸�︷︷�︸

Δyyy(t)

= C2(r)︸︷︷︸

CCC2

Δq(t),(29)

which can be summarized into the vector-valued function

Δy(t) = CCC2Δq(t), where CCC2 =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

‖CCCe(1,:)‖GGGm,2...

‖CCCe(r,:)‖GGGm,2

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (30)

This improved output estimate is consistent withEq. (28) since CCC2 = C2 if only one output dimension is con-sidered.

3.1 Initial error

One part of the error estimation are the initial errors eeem,0 andeeem,0 in Eqs. (25) and (26), which are eliminated if all com-ponents of the initial condition xxx0 are in the reduced space.However, if this condition is not fulfilled then the initial errormust be accounted for. The initial condition xxx0 = WWWT

s xxx0 andthe definition of the state-space error from Eq. (9) yield theinitial error

eees,0 = xxx0 −VVV s ·WWWTs · xxx0. (31)

It can be further analyzed using the block matrix struc-ture of the projection matrices VVV s and WWW s from Eq. (13). Thisyields the initial errors of the reduced second order system

eeem,0 =[

III −VVVm · (WWWTm ·VVVm)−1WWWT

m

]

qqq0,

eeem,0 =

[

III −VVVm ·MMM−1e ·WWWT

m ·MMMe

]

qqq0.(32)

3.2 Residual norm

One further essential part in the calculation of the errorbound is the calculation of the residual norm ‖˜RRRm(t)‖GGGm . Forthe efficient calculation of the residual norm an offline/onlinedecomposition is helpful. Therefore, besides of the directcomputation, which is derived directly from the matrix normdefinition equation (4) and defined as Method 1

‖˜RRRm(t)‖2GGGm= ˜RRR

Tm(t) ·GGGm ·˜RRRm(t) (Method 1) (33)

it is also possible to expand this multiplication into a sumof ten terms using Eqs. (2) and (10), which is defined asMethod 2. For the sake of simplicity this sum can be re-arranged into Eq. (34). Consequently,

‖˜RRRm(t)‖2GGGm=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

qqq(t)qqq(t)qqq(t)−uuu(t)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

·

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

˜MMM11˜MMM12

˜MMM13˜MMM14

˜MMMT12˜MMM22

˜MMM23˜MMM24

˜MMMT13˜MMM

T23˜MMM33

˜MMM34

˜MMMT14˜MMM

T24˜MMM

T34˜MMM44

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

·

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

qqq(t)

qqq(t)

qqq(t)

−uuu(t)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(Method 2), (34)

where the matrices ˜MMMi j denote abbreviations for

˜MMM11 = VVVTm · KKKT

e ·MMM−Te ·GGGm ·MMM−1

e · KKKe ·VVVm,

˜MMM22 = VVVTm ·DDDT


e ·DDDe ·VVVm,

˜MMM33 = VVVTm ·GGGm ·VVVm,

˜MMM44 = BBBTe ·MMM−T

e ·GGGm ·MMM−1e · BBBe,



e ·DDDe ·VVVm, (35)˜MMM13 = VVVT

m · KKKTe ·MMM−T

e ·GGGm ·VVVm,



e · BBBe,


e ·MMM−Te ·GGGm ·VVVm,



e · BBBe,

˜MMM34 = VVVTm ·GGGm ·MMM−1

e · BBBe.

The offline/online decomposition of the error estimatorbecomes obvious, since the matrices ˜MMMi j are independentof time. Note that formally Method 1 and Method 2 yieldthe same quantity for the squared residual norm. Howevernumerically, the sum of 10 vector/matrix/vector products inMethod 2 yields different numerical annihilation effects thanthe single vector/matrix/vector product in Method 1. Hence,the methods practically result in different numerical values.

4 Example of applications

The application example is the model of a stabilization link-age of a car front suspension, which was introduced inRef. [13]. The model consists of 19 3D beam elements and20 nodes. A fixed displacement boundary condition is ap-plied to the first node, as depicted in Fig. 4. This leaves a to-tal of 114 degrees of freedom for the constraint model, sinceevery node has 6 degrees of freedom, namely three displace-ments and three rotations. In this example, the displacementof node 20 in the z-direction is chosen as the output of thesystem. The linkage is excited at node 20 by a sinusoidalforce F with a frequency of 1 Hz and an amplitude of 100 Nin the z-direction. The model is reduced to twelve degrees


of freedom using Krylov subspaces and the mass matrix isused for the GGGm-norm, thus GGGm = MMMe. Furthermore, the inte-gration is performed with an ODE45 integrator from 0 to 2 sin the [m, kg, s] unit system. The output error bounds ˜Δy(t)and Δy(t) are computed for the output with Eq. (30) and theresults are depicted in Fig. 5. The results clearly show thesuperiority of ˜Δy(t) over Δy(t) for larger times. This is due tothe fact that ‖˜RRRm(τ)‖GGGm and ˜C12(t − τ) are fairly large in thebeginning, but decay over time and consequently, attenuateeach other due to the convolution in the error estimator, seeEq. (25).

Fig. 4 Constrained stabilization linkage of a car

Fig. 5 Displacement of node 20 in the z-direction with error bounds

This example also demonstrates the advantage of thenew estimator for second order mechanical systems overthe original error estimator. This becomes obvious from acomparison between the different norms of the fundamen-tal matrix, which are required for both error estimators. Forthe original error estimator the norm of the full fundamen-tal matrix is required and the constant C1 = maxt ‖ΦΦΦ‖GGGs =

4.349 2 × 104 is used. For the new error estimator, how-ever, only the upper half of the fundamental matrix is con-sidered and the norms C11 = maxt ‖ΦΦΦ11‖GGGm = 1.000 0 andC12 = maxt ‖ΦΦΦ12‖GGGm = 0.019 6 are utilized. Since C11 andC12 are significantly smaller than C1 the results are substan-tially better. Computations confirm that C1 = maxt ‖ΦΦΦ‖GGGs =

maxt ‖ΦΦΦ21‖GGGm = 4.349 2×104 and, therefore, that the subma-trix ΦΦΦ21 causes these extremely high values, which is anal-ogous to the pictorial example provided in Sect. 2.2. More-over, the constant C2 = ‖CCCe‖GGGm,2 = 6.915 1 is determined.

4.1 Sensitivity analysis of the derived error estimator

The error estimator has been found to be very sensitive tothe finite precision of computers. This becomes obviousif the accuracy is reduced from double to single precision,or if the standard SI unit system with [m, kg, s] is changedinto [km, g, s] to work with larger values. Even though thecomputed displacement is still correct, the error estima-tion yields unusable error bounds. Furthermore, the errorbounds depend significantly on the computation method forthe residual norm, which can be computed in both the waysproposed in Eqs. (33) and (34), respectively. This problemis illustrated in Fig. 6, where the residual norm ‖RRRm(t)‖GGGm iscomputed for various reduction levels using both Methods 1and 2. It is obvious that Method 1 yields better values once

Fig. 6 Results for different reduction levels, computed with Meth-ods a 1 and b 2


the residual norm drops below 10−2. Furthermore, both com-putation methods seem to converge to a certain limit whichmay be a numerical problem as well. Reductions leavingmore degrees of freedom only reveal better initial results,which can be seen from the difference between 30 and 84 de-grees of freedom. Moreover, numerical errors become crit-ical at 84 degrees of freedom and for the full model. Theformer yields a residual norm that shows random peaks inthe otherwise smooth curve. The latter encounters problemswith Method 2, which yield a pretty large and consequentlywrong residual for the full model.

Since Method 1 seems to yield the most accurate results,it is used to calculate the output error bounds as depicted inFig. 7 along with the exact errors, which are computed usingthe result of the full model. The graph demonstrates that theerror bounds are larger than the exact error throughout all theexamples provided.

Fig. 7 Exact errors and error bounds computed with Method 1

4.2 Approximation of the norms of the fundamental matrix

One weakness of the derived error estimator is the needof the fundamental matrix ΦΦΦ(t), which is extremely ex-pensive in computation. Consequently, the performanceof the error estimator can be improved significantly byapproximating the fundamental matrix norm. This isachieved by using a second low-dimensional reducedmodel. Let VVVm, WWWm denote projection matrices andΦΦΦ(t) ≡ eAAAt the fundamental matrix of the reduced system,

in which AAA = WWWTm · AAA · VVVm. Considering that the funda-

mental matrix ΦΦΦ(t) = eAAAst entirely describes the system’sbehavior, it can be concluded that ΦΦΦ(t) = eAAAst completely de-scribes the reduced system’s behavior. For the error estima-tion of second order systems the full fundamental matrix ΦΦΦis not needed explicitly since only the norms ‖ΦΦΦ11(t)‖GGGm and‖ΦΦΦ12(t)‖GGGm are required. Therefore, we propose an approxi-mation by

‖ΦΦΦ11(t)‖GGGm ≈ ‖ΦΦΦ11(t)‖GGGm, (36)

‖ΦΦΦ12(t)‖GGGm ≈ ‖ΦΦΦ12(t)‖GGGm, (37)

where the reduced matrix GGGm = WWWTm · GGGm · VVVm is used for

the norm, which equals the definition of the reduced massmatrix MMMe in the common case GGGm = MMMe. For Eqs. (36)and (37) to be good approximations, we chose VVVm and WWWm

as modal projection matrices. In Fig. 8 we use the mostdominant eigenmode (eigenmode 1) to illustrate the result-ing values of ‖ΦΦΦ11(t)‖

GGGmin comparison with ‖ΦΦΦ11(t)‖GGGm . We

see that eigenmode 1 is not sufficient for all times, eventhough the approximation is very accurate for the local max-ima. The discrepancy between these maxima is due to theeigenmodes 2-114, and using more eigenmodes leads to animprovement of the approximation. However, it is difficultto find the required number of eigenmodes and, therefore,we choose as ˜C11(t) the exponential envelope of the curveof the most dominant eigenmode, illustrated by the dashedred line. As the most dominant eigenmode 1 is also the leastdamped eigenmode, it seems obvious that the local maximayield an exponential envelope which is larger than what theless dominant eigenmodes 2-114 could yield. The same ap-proach holds for ˜C12(t) as well. This approximation withthe exponential envelope slightly worsens the results of theerror estimator, depicted in Fig. 7 and labeled y ± ˜Δenv

y , butsignificantly saves computation time. The computation of˜C11(t), ˜C12(t),C11 and C12 using the computer system givenin the next section took 56.251 s with full model and only0.077 s with the approximation. Consequently, the approxi-mation with eigenmode 1 is computed more than 700 timesfaster in this example.

Fig. 8 Results of the full system and an approximations with a onedegree of freedom system using the first dominant eigenmodes

4.3 Runtime comparison

All results were obtained using a standard computer(Intel Core 2 Duo CPU P8400, 2.27 GHz, 4 GB RAM) andMethod 1 for equally distributed 201 timesteps, the integra-tion was performed with the Matlab ODE45 integrator. Thenorms of the fundamental matrix ˜C11(t), ˜C12(t), C11 and C12

were computed in an offline phase, the required computationtimes are given in the previous section. All other compu-tations were performed in an online phase and the runtime


results are summarized in Table 1. The runtime for the in-tegration and the offline phase are averaged over 5 runs andthe results of the online phase over 100 runs.

Table 1 Online runtime comparison

Dimension Integration/sError estimation

Δy/s ˜Δy/s

6 1.8 0.073 8 0.080 4

12 4.3 0.078 4 0.084 7

24 19.3 0.084 7 0.091 1

30 29.6 0.088 2 0.095 7

36 58.4 0.091 9 0.097 5

84 1 753.1 0.124 4 0.132 2

114 (full) 4 940.9 − −

First of all, it is obvious that the online phase of the er-ror estimator is extremely fast and makes up only a fractionof the integration time. Additionally, the computation timeof the online phase hardly increases for larger systems, indi-cating that most of the time is used for those calculationsthat are independent of the system’s dimension. Further-more, there is hardly any difference between the fast errorestimation Δy and the more accurate ˜Δy. This indicates thatmost computational time is spend on other necessary calcu-lations and the difference might be more pronounced withan optimized code. The most significant improvement of theerror estimator is gained by approximating the norms of thefundamental matrix. By reducing the system to one degreeof freedom using the most dominant eigenmode, the compu-tation is more than 700 times faster.

5 Conclusion

In this article the error estimator for first order state-spacemodels from Refs. [4, 14] has been applied to second ordersystems. Therefore, the relationship between reduced firstand second order systems has been derived so that the pro-jection matrices from standard reduction techniques can beused. It has been found that the original error estimator deliv-ers impractical results for second order systems representingflexible bodies. This originates from a large hump of the fun-damental matrix norm ‖ΦΦΦ(t)‖, which yields extremely highvalues for small but nonzero t. This problem was traced backto the norm of submatrixΦΦΦ21(t) and is found to be related tohigh frequency oscillations with very low amplitudes, wherethe hump represents the high velocity in these oscillations.And thus a modified error estimator has been derived for thesecond order system of flexible bodies, which does not re-quire this submatrix. This refined error estimator yields good

error bounds for the model of a stabilization linkage, whichwas successfully used as an illustrative example to reproduceevery step of the error bound computation. Furthermore, anoffline/online decomposition has been presented by dividingthe required residual norm into a sum of 10 terms. How-ever, this revealed problems with the accuracy of the finitecomputer precision, leading to worse results.

The required submatrices ΦΦΦ11(t) and ΦΦΦ12(t) of the fun-damental matrix ΦΦΦ(t) are extremely expensive in computa-tion, but can be computed during the offline phase prior tothe simulation. However, the fundamental matrix is not onlytime-consuming but also very difficult to compute, especiallyfor large systems. Depending on the structure and conditionnumber of the matrix, the computation may be very inac-curate or fail completely. In this article an approximationtechnique for these norms has been developed on the basisof modal analysis. If the envelope of the resulting curves isused, then the fundamental matrix can be reduced to one sin-gle degree of freedom using the most dominant eigenmode.This solves both problems and the necessary norms can becomputed fast and reliably.

Acknowledgement

The authors would like to thank the German ResearchFoundation (DFG) for financial support of the project withinthe Cluster of Excellence in Simulation Technology (EXC310/1) at the University of Stuttgart.

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A-posteriori error estimation for second order mechanical systems