review and assessment of model updating for non-linear, transient dynamics

Mechanical Systems and Signal Processing (2001) 15(1), 45}74doi:10.1006/mssp.2000.1351, available online at http://www.idealibrary.com on

REVIEW AND ASSESSMENT OF MODEL UPDATINGFOR NON-LINEAR, TRANSIENT DYNAMICS*

FRANhOIS M. HEMEZ AND SCOTT W. DOEBLING

Engineering Analysis Group (ESA-EA), Los Alamos National Laboratory, P.O. Box 1663,M/S P946, Los Alamos, NM 87545, U.S.A.

E-mails: [email protected], [email protected]

(Accepted 1 September 2000)

The purpose of this publication is to motivate the need of validating numerical modelsbased on time-domain data for non-linear, transient, structural dynamics and to discusssome of the challenges faced by this technology. Our approach is two-fold. First, severalnumerical and experimental testbeds are presented that span a wide variety of applications(from non-linear vibrations to shock response) and di$culty [from a single-degree-of-freedom (sdof ) system with localised non-linearity to a three-dimensional (3-D), multiple-component assembly featuring non-linear material response and contact mechanics]. Thesetestbeds have been developed at Los Alamos National Laboratory in support of theAdvanced Strategic Computing Initiative and our code validation and veri"cation program.Conventional, modal-based updating techniques are shown to produce erroneous modelsalthough the discrepancy between test and analysis modal responses can be bridged. Thisconclusion o!ers a clear justi"cation that metrics based on modal parameters are not wellsuited to the resolution of inverse, non-linear problems. In the second part of this work, thestate-of-the-art in the area of model updating for non-linear, transient dynamics is reviewed.The techniques identi"ed as the most promising are assessed using data from our numericalor experimental testbeds. Several di$culties of formulating and solving inverse problems fornon-linear structural dynamics are identi"ed. Among them, we cite the formulation ofadequate metrics based on time series and the need to propagate variability throughout theoptimisation of the model's parameters. Another important issue is the necessity to satisfycontinuity of the response when several "nite element optimisations are successively carriedout. An illustration of how this problem can be resolved based on the theory of optimalcontrol is provided using numerical data from a non-linear Du$ng oscillator. The publica-tion concludes with a brief description of current research directions in inverse problemsolving for structural dynamics.

( 2001 Academic Press

1. INTRODUCTIONs

Inverse problem solving is at the core of engineering practices as such work generallyinvolves designing a system to target a given performance or to satisfy operating con-straints. Increasingly, designers are faced with shorter design cycles while their testingcapabilities are reduced and the physics they must understand becomes more sophisticated.The consequence is the need for larger-size computer models, coupled-"eld calculationsand more accurate representations of the physics. To improve the predictive quality of

*Authorized for unlimited, public release on January 14, 2000. LA-UR-00-091.sThe &&Standard Notation for Modal Testing & Analysis'' is used throughout this paper, see reference [1].Symbols not commonly used in the modal testing and structural dynamics communities are de"ned in thetext.

0888}3270/01/010045#30 $35.00/0 ( 2001 Academic Press

46 F. M. HEMEZ AND S. W. DOEBLING

numerical models and enhance the capability to extrapolate the response of a system, it isoften necessary to formulate and solve inverse problems where simulations are compared to"eld measurements [2]. In addition, it has been recognised that non-deterministic ap-proaches must be employed to alleviate our lack of test data and incomplete understandingof mechanics [3].

The work presented in this publication deals with the formulation of inverse problems forcorrelating transient dynamics to responses obtained from non-linear "nite element models.The application targeted is clearly structural dynamics although most of the techniquesdiscussed here either originate or "nd their counterparts in physics and other engineering"elds. The reason why time series rather than modal parameters are considered is essentiallybecause the increasing complexity and sophistication of engineering applications makes itnecessary to analyse systems with arbitrary sources of uncertainty and non-linearity.Another motivation is the modelling of highly transient dynamic events for which modalsolutions make no sense due to the clustering of high-frequency modes and because modaldamping cannot capture with a su$cient degree of accuracy the energy dissipation mecha-nism involved.

For clarity, this paper is organised according to three main parts. First, a brief discussionof current trends in testing, modelling and computing for structural dynamics applicationsis provided (Section 2). The purpose of this discussion is to provide an insight regarding theissues that, to our opinion, inverse problem solving will have to address in the near future.Then, several numerical and experimental testbeds are presented (Section 3). These are usedfor generating reference data or test data that feed various test-analysis reconciliationtechniques. Our intent is to introduce a series of problems that feature various sources ofnon-linearity and modelling di$culties and that challenge the existing model updatingstrategies. The second part of this publication reviews the state-of-the-art in model updatingand error control algorithms for non-linear dynamics (Sections 4 and 5). Several techniquesare illustrated using our numerical or experimental testbeds. This demonstration is notintended as a thorough comparison between di!erent model updating algorithms nor as anexhaustive assessment of each method's advantages and limitations. The objective is ratherto identify the critical de"ciencies of the technology currently available with the ultimategoal of developing tools for assessing the predictive quality of a stochastic, non-linearnumerical model. Finally, the third part of this publication presents some of the key aspectsthat, we believe, will drive the development of test-analysis correlation and model validationtechnology in the years to come (Section 6).

2. CURRENT CHALLENGES IN INVERSE PROBLEM SOLVING

In this section, some of the current trends in testing, modelling and computing forstructural dynamics applications are brie#y discussed. Our purpose is to introduce thechallenges faced by inverse problem solving and to motivate the particular choice of thetestbeds presented in Section 3.

2.1. TRENDS IN STRUCTURAL DYNAMICS

The main reason why numerical models have become so popular is because it is much lessexpensive to use computational time than it is to run a sophisticated experiment. Manypractical situations also occur where the phenomenon of interest cannot be measureddirectly. Hence, the scienti"c community has turned to numerical models that can beparametrised and used to study a wide variety of situations. This argument has beenreinforced in recent years by the increasing e$ciency of processors, the greater availability

47ASSESSMENT OF MODEL UPDATING

of memory, the breakthrough of object-oriented data structures together with the growingpopularity of parallel processing whether it involves computers with massively parallelarchitectures or networks of single-CPU workstations. Interestingly enough, the miniatur-isation of CPUs and their greater e$ciency have in#uenced greatly testing procedures,making it possible to instrument structures with hundreds of transducers. Powerful dataanalysis and friendly computer graphics are also a driving force behind the development ofnon-intrusive, optical measurement systems such as holography and laser vibrometry.These technological breakthroughs are not without major consequences on the wayengineers analyse structural systems and on their conception of test-analysis correlationand inverse problem solving.

An illustration of this evolution is the rapid development of modelling procedures fornon-linear dynamics. It is reasonable to foresee that the bottleneck of computing power willbe removed in the near future, at least when it comes to engineering applications.tConsequently, research and development e!orts in recent years have been mostly focusedon improving the representation of the geometry and the physics. Examples are thederivation of small-scale, statistical models for contact dynamics; the implementation ofhigh-"delity, non-linear material models; or the e!orts to expand our current modellingcapabilities to the high-frequency range for predictive acoustics. In spite of these undeniableadvances, very rarely have the issues of uncertainty quanti"cation and predictability beenraised. These are central questions when it comes to assessing whether a numericalsimulation is capable of reproducing with acceptable accuracy the experiment it is supposedto replace. It is therefore reasonable to state that, no matter how powerful computersbecome, there will always be some degree of uncertainty in the numerical models due tounknown interfaces, unknown physics, environmental variability, parameter and assemblyuncertainty, idealisation errors, discretisation errors, numerical errors, etc. Increased com-putational power allows to bound some of these sources of uncertainty (such as discretisa-tion and numerical errors), but not all of them can be reduced to desirable levels. Becausethe trend of replacing expensive laboratory or "eld experiments with numerical simulationsis not going to change, systematic model validation and inverse problem-solving strategiesare needed that must be capable of handling large-size numerical models geared atcapturing time-dependent, stochastic and non-linear phenomena.

2.2. SPECULATIVE OUTLOOK

When analysing the dynamic response of a complex system using the "nite element (FE)method, it is not acceptable to neglect the contribution of an important component, joint orinterface. In the past, neglected dynamics were accounted for by tuning parameters in themodel to agree with the experimental data. For example, the damping (modal or other) wasdetermined &ad hoc' using test data obtained from testing the fully assembled system. Then,the identi"ed damping properties were added to the model to improve its predictiveaccuracy. At present, some of the full-scale testing capabilities which formerly existed at theU.S. national laboratories and many other facilities in the automotive, aerospace and civilengineering communities are no longer functional. Therefore, it is no longer possible toreconcile a model with experimental data for all environments. In the future, models will beconstructed with limited use of these expensive, full-scale test data sets. In addition, it is our

tThe Accelerated Strategic Computing Initiative (ASCI) platforms &BlueMountain' and &Red' at Los AlamosNational Laboratory and Sandia National Laboratories routinely perform over three Tera#ops distributed overseveral thousand computational nodes. This power may not yet enable high-energy physics simulations withenough accuracy but it is considered su$cient for most engineering applications. For reference, see:www.sgi.com/newsroom/press

}releases/2000/may/blue mountain.html.


opinion that structural dynamics in the 21st century will become increasingly:

(1) Non-linear.(2) Non-structural.(3) Non-modal.(4) High bandwidth.(5) Multi-physics.

In these conditions, can the concept of FE model updating that has been developed forlinear, modal dynamics be generalised? Is FE model updating the correct answer to modelvalidation? What &features' other than the conventional mode shapes and resonant frequen-cies can be extracted from the data to characterise the response of a non-linear system? Howto quantify the total uncertainty of an experiment? How to propagate the parametricuncertainty of a numerical simulation? These are some of the long-term questions that, webelieve, should be addressed by the structural dynamics community at large in order todevelop inverse problem-solving techniques that would be relevant to tomorrow's struc-tural dynamics applications.

3. TESTBEDS FOR THE VALIDATION OF NON-LINEAR STRUCTURAL MODELS

In this Section, several numerical and experimental testbeds are presented. They havebeen found in the literature or developed in support of the Advanced Strategic ComputingInitiative and our code validation and veri"cation program. Aside from programmaticwork, they are also used to answer some of the questions mentioned in the previousSection 2. The examples discussed span a wide variety of applications and di$culty. The"rst three involve non-linear vibrating systems (Sections 3.1.1, 3.1.2 and 3.2). The fourth onerepresents a high-frequency, transient shock response (Section 3.3). The "fth one representsthe response of a full, assembled system to an explosive load (Section 3.4). The sources ofnon-linearity considered are either purely analytical (cubic springs), undetermined butlocalised (impact mechanism) or parametric and distributed (hyper-foam material andcontact dynamics). These examples also feature various mechanisms for the dissipation ofenergy that cannot, we believe, be reduced to the conventional, modal representation ofdamping.

3.1. NUMERICAL TESTBEDS WITH A CUBIC NON-LINEARITY

The "rst two examples are purely analytical. They represent point masses and linearsprings. Damping is considered in the "rst case only. The "rst system exhibits a sdof and thesecond system has four dof. In both cases, the source of non-linearity considered is a cubicspring, that is, the non-linear, internal force is proportional to the third power of a displace-ment. Advantages of these numerical simulators is that the modelling error, the non-linearperturbation introduced and the noise-to-signal ratio can be controlled explicitly.

3.1.1. Single-degree-of-freedom oscillator

Our "rst numerical testbed is the sdof system pictured in Fig. 1. This system is also knownas the Du$ng oscillator and it has been studied extensively for the ability of its response x (t)to transition from ordered to chaotic behaviour. The equation of motion, the harmonicforcing function considered here and the non-linear, internal force are de"ned in equation(1). In the numerical tests performed, reference data are generated by integrating equation(1) over a period of 30 s. Then, the internal force is assumed to be unknown which

Figure 1. Single dof mass/spring system with a cubic non-linearity.

Figure 2. Test-analysis correlation of the sdof system before model updating (top-half, displacement time-history; bottom-half, velocity time-history): **, test; } } }, simulation.


introduces a modelling error. The purpose of test-analysis correlation and FE modelupdating would typically be to identify this missing term as a function of time.

mL2x

Lt2(t)#d

Lx

Lt(t)#kx(t)#F

*/5(t)"F

%95(t)

F%95

(t)"F0cos(Xt), F

*/5(t)"k

nlx3(t)

m"2.56, d"0.32, k"1, knl"0.05, F

0"2.5, X"1. (1)

In the example discussed in Section 5, simulated &measurements' are assumed to beavailable only at 11 equally spaced samples. They consist of displacement and velocity data.Although this testbed is probably the simplest one that could be imagined, the amount ofcomputation required for solving the inverse problem may be enormous (see Section 5).Figure 2 shows the di!erence between test and analysis time histories before the model isoptimised. Displacements are shown on the top-half and velocities are shown on thebottom-half. Clearly, the response predicted by the linear model is inconsistent with testdata.


3.1.2. Multiple-degree-of-freedom (mdof ) oscillator

The second system investigated is a 4-dof, mass/spring system with cubic non-linearity.Figure 3 provides an illustration and the non-linear, internal force added to the equation ofmotion is de"ned in equation (2). This system generalises the Du$ng oscillator to a mdofcase. A higher dimensionality is required to study the e!ect of spatial incompleteness, thatis, the e!ect of measuring the response at a subset of the model's dofs. The system shown inFig. 3 is numerically excited with a random excitation and implicit time integration is usedfor obtaining the acceleration response at the 4 dofs.

Although these results are not presented here, we performed several numerical experi-ments where the objective was to recover a 15% sti!ness reduction of the third linear springwhen one of the system's output was unknown.A The overall performance of time-domainmodel updating was studied using various con"gurations where, among other things,(1) features used to characterise the data were varied; (2) the noise-to-signal ratio was varied;and (3) the non-linear force was known or unknown. The performance of three FE modelreduction techniques was also studied and whether or not model reduction could becombined to parametric model updating for non-linear dynamics. The results obtained sofar indicate that some of the model reduction techniques tested provide a valuable alterna-tive to the conventional modal expansion even when the dynamics of interest are signi"-cantly non-linear. The reader is referred to references [4, 5] for further details about theseinvestigations.

3.2. EXPERIMENTAL TESTBED FOR NON-LINEAR VIBRATIONS

Our testbed for model validation in the context of non-linear vibration is theLANL 8-dof system (which stands for Los Alamos National Laboratory eight-degrees-of-freedom system) illustrated in Fig. 4. It is formed with eight translatingmasses connected by springs. Each mass is a disc of aluminium 25.4 mm thick and76.2 mm in diameter with a centre hole. The hole is lined with a Te#on bushing. There aresmall steel collars on each end of the discs. The masses all slide on a highly polished steel rodthat supports the masses and constrains them to translate along the rod. The masses arefastened together with coil springs epoxied to the collars that are, in turn, bolted to themasses. The support rod introduces a source of friction that constitutes the "rst modellingchallenge.

Vibration tests are performed on the nominal system and on a damaged version wherethe sti!ness of one of the springs is reduced by 14%. The excitation applied at the drivingpoint measurement consists of a random signal at various force amplitudes. The time-domain acceleration responses are collected at all eight masses over a period of 8 s witha total of 213"8192 samples. Figure 5 illustrates the response obtained at the "rst and "fthsensors during one of the tests performed with the linear, undamaged con"guration of theLANL 8-dof system. The "rst step of this experiment is to identify the system's modalparameters using a classical frequency-domain curve "tting algorithm. Table 1 shows thatthe original, linear and undamped FE model is in good agreement with the identi"ed modalparameters prior to any parametric adjustment. The more-or-less uniform shift in frequencyseen in Table 1 for the "rst three modes is caused by the driving point attachment systemwhose mass was originally overestimated in the FE representation. This error was sub-sequently corrected by performing a few iterations of modal-based, parametric adjustment.(The model-updating technique employed for this purpose is the force-based residual

AData provided to the FE model updating algorithms consisted of the acceleration &measurements' at locations 1,3 and 4 only.

Figure 3. Four-dof mass/spring system with a cubic non-linearity.

Figure 4. LANL 8-dof testbed. Illustration of the system and its support stand.

Figure 5. Typical acceleration response at sensors 1, 5. (Linear, undamaged LANL 8-dof system.)


approach reviewed in Section 4.) This update reduces the frequency errors to 1% in averageand brings the modal assurance criterion (MAC) over 99% for the "rst "ve modes.

A contact mechanism can also be added between two masses to induce a source ofcontact/impact. The hardware used for this purpose is pictured in Fig. 6. A bumper is placed

TABLE 1

¹est-analysis correlation of the ¸AN¸ 8-dof testbed (¹he table shows the correlation betweenthe nominal, undamaged system and the corresponding linear model.)

Identi"ed Modal assuranceMode Identi"ed damping ratio FE model criterion Frequencyno. frequency (Hz) (%) frequency (Hz) (MAC) (%) error (%)

1 22.6 8.5 21.8 99.7 !3.52 44.5 4.3 43.0 99.4 !3.43 65.9 3.3 63.0 99.4 !4.44 86.6 5.0 80.8 93.2 !6.75 99.4 2.6 95.6 98.5 !3.8

Figure 6. LANL 8-dof testbed. Contact mechanism between masses 5 and 6.


on one mass that limits the amount of motion that mass may move relative to the adjacentmass. When the distance between the ends of the rods is equal to the initial clearance,impact occurs. This impact simulates damage caused by spring deterioration to a degreewhich permits contact between adjacent masses, or in a simpli"ed manner, the impact fromthe closing of a crack during vibration. The degree of damage is controlled by changing theamount of relative motion permitted before contact, and changing the hardness of thebumpers on the impactors. This testbed is designed to simulate situations where a mostlylinear system is connected to a secondary component. A common example is the dynamicsof a delivery vehicle-payload assembly. Non-linearity and energy dissipation mechanismsare generally introduced at the interface between the two by the attachment system.Although the dynamics of the whole system may remain mostly linear, accounting for thenon-linear perturbation introduced by the interface may be important for predictabilitypurpose. Likewise with the LANL 8-dof system, the sources of non-linearity (friction,impact) have a sensible e!ect on the mostly linear dynamics. This is illustrated in Figs. 7and 8 that show how the resonant frequencies and modal damping ratios identi"ed fromthe time-domain acceleration data vary as the impact mechanism is activated and the inputforce level is increased. As expected, these sources of non-linearity do not modify signi"-cantly the distribution of mass and sti!ness. As a result, the identi"ed frequencies shown inFig. 7 are relatively insensitive to the impact and input force level. On the other hand,

Figure 7. E!ect of the non-linearity (friction, impact) on the identi"ed resonant frequencies: , u1

(noimpact); , u

2(no impact); , u

3(no impact); , u

4(no impact); , u

5(no impact); , u

6(no

impact); , u1

(w/impact); , u2

(w/impact); , u3

(w/impact); , u4

(w/impact); , u5

(w/impact); , u6

(w/impact).

Figure 8. E!ect of the non-linearity (friction, impact) on the identi"ed modal damping ratios: , m1

(noimpact); , m

2(no impact); , m

3(no impact); , m

4(no impact); , m

5(no impact); , m

6(no

impact); , m1

(w/impact); , m2

(w/impact); , m3

(w/impact); , m4

(w/impact); , m5

(w/impact); , m6

(w/impact).


activating the bumping mechanism has the e!ect of promoting more energy dissipationwhich translates into higher damping (for a given mode) as shown in Fig. 8. Increasing theinput force level overall decreases damping because less &stick and slip' friction is obtained athigher forcing levels.

The test discussed in this publication consists of identifying the damaged spring (14%sti!ness reduction) using a linear model that does not account for the friction nor the source


of contact/impact. The performance of conventional model-updating approaches is evalu-ated when the distance between test data and modelling is de"ned using modal parameters(resonant frequencies and mode shapes). Time-domain techniques are also tested for theirability to identify the (unknown) non-linearity when less measurements than dofs areavailable. Finally, we mention that test data from the LANL 8-dof testbed can be madeavailable to assess model validation strategies. We believe that this is an interesting testbedbecause small dimension, linear models can be developed that provide a good starting pointfor test-analysis correlation. Nevertheless, some of the con"gurations tested allow todiscriminate between numerical techniques that would perform well when the dynamicsremains linear but that fail when non-linear e!ects dominate the response.

3.3. IMPACT TEST EXPERIMENT

The next testbed presented deals with transient responses rather than non-linear vibra-tions. The application targeted is a high-frequency shock experiment that features a com-ponent characterised by a non-linear, visco-elastic material behaviour. Results of severalexplicit FE simulations are compared to "eld measurements and parameters of the numer-ical models are optimised to improve the correlation.

An illustration of the set-up is provided in Fig. 9 and the actual hardware used duringimpact testing is pictured in Fig. 10. The system tested is composed of two components(steel impactor, foam pad) assembled to the carriage (impact table) in a rather straightfor-ward manner. The centre of the steel cylinder is hollow and has been "xed with a rigid collarto restrict the motion of the impactor to the vertical direction. This assures perfectly bilinearcontact between the steel and foam components, allowing the structure to be modelledaxisymmetrically. In spite of this, a full 3-D model is developed to verify this assumption'svalidity. Besides the material modelling, another important parameter in this study is theamount of preload applied by the bolt used to hold the assembly together. The torqueapplied was not measured during testing. It is very likely to have varied from test to testwhich introduces a signi"cant source of uncertainty in the numerical simulation.

The analysis program used for these calculations is HKS/Abaqus-Explicit, a general-purpose package for "nite element modelling of non-linear structural dynamics [6]. Itfeatures an explicit time integration algorithm, which is convenient when dealing withnon-linear material behaviour, potential sources of impact or contact, and high-frequency

Figure 9. Description of the assembly of the cylindrical impactor and carriage.

Figure 10. LANL impact test set-up.

Figure 11. 3-D model of the LANL impact testbed.


excitations. In an e!ort to match the test data, several FE models are developed by varying,among other things, the constitutive law and the type of modelling. Therefore, optimisationvariables consist of the usual design variables augmented with structural form parameterssuch as kinematic assumptions, geometry description (2-D or 3-D), contact modelling andnumerical viscosity. Figure 11 illustrates one of the discretised models used for numericalsimulations.

During the actual tests, the carriage that weights 955 lbm (433 kg) is dropped fromvarious heights and impacts a rigid #oor. The input acceleration is measured on the topsurface of the carriage and three output accelerations are measured on top of the steelimpactor that weights 24 lbm (11 kg). The location of the four sensors can be seen in Fig. 10.Impact tests are repeated several times to collect multiple data sets from which therepeatability of the experiment can be assessed. Table 2 gives the number of data setscollected for each con"guration tested. Dropping the carriage from two di!erent heightsresults into di!erent velocities at the time of impact. The values obtained by integrating theacceleration signal match the analytical formula xR

*.1!#5"J2gh

$301and provide 60.4 in/s

(1.5 m/s) and 222.9 in/s (5.7 m/s) for the low- and high-velocity impacts, respectively. The

TABLE 2

Data collected with the impact testbed

No. of data sets Low-velocity impact High-velocity impactcollected (13 in/0.3 m drop) (155 in/4.0m drop)

Thin layer (0.25 in/6.3 mm) 10 tests 5 testsThick layer (0.50 in/12.6 mm) 10 tests 5 tests

Figure 12. Input and output accelerations measured during a low-velocity impact. (Top: low-velocity impact ofa thin foam pad; Bottom: low-velocity impact of a thick foam pad.): } } }, input 1;==, output 1;00, output 2;}}}, output 3.


reason why less data sets are available at high impact velocity is because these tests provedto be destructive to the elastomeric pad and could not be repeated to study the variability.

Upon impact, the steel cylinder compresses the foam pad to cause elastic and plasticstrains during a few ls. Typical accelerations measured during the impact tests are depictedin Fig. 12. Both data sets are generated by dropping the carriage from an initial height of13 in (0.33 m). Figure 12 shows on top the acceleration response when a 1/4 in-thick(6.3 mm) foam pad is used and it shows on the bottom the acceleration response witha 1/2 in-thick (12.6 mm) foam pad. It can be seen that over a thousand g's are measured ontop of the impact cylinder which yields large deformations in the foam pads. The time scalealso indicates that the associated strain rates are important. Lastly, the variation in peakacceleration observed in Fig. 12 suggests that non-zero angles of impact are involved,making it necessary to model this system with a 3-D discretisation. Clearly, modalsuperposition techniques would fail because of the following reasons: (1) contact cannot berepresented e$ciently from linear mode shapes; (2) non-linear hyper-foam models, thatpossibly include visco-elasticity, are needed to represent the foam's hardening behaviour;and (3) very re"ned meshes would be required to capture the frequency content over


10 000 Hz. Although it may be possible to compute the solution via modal superposition fora single design, repeating this procedure in the context of inverse problem solving would betotally unfeasible.

Since several data sets are available for the same con"guration tested (see Table 2), thetotal uncertainty of the experiment can be estimated. Figure 13 illustrates the variability inpeak acceleration obtained when the same test is repeated 10 times (low impact, thin-padcon"guration). Although the environment of this particular experiment was controlled verythoroughly, Fig. 13 shows that a signi"cant spread in the acceleration response is obtained.This variability is believed to be caused essentially by random changes in the boundarycondition, the angles of impact and the preload applied by the tightening bolt. The purposeof test-analysis correlation should therefore not be to reproduce a particular data set butrather to predict the distribution of experimental results at a given level of accuracy. Theimpact testbed also serves the purpose of assessing the performance of statistical samplingstrategies and fast probability integration algorithms used to propagate uncertaintythrough the forward analysis. Statistical analysis techniques used for this application relyon the orthogonal array sampling and the Latin Hyper-cube sampling presented inreference [7]. Fast probability is provided via the software package NESSUS and interfacedto our explicit FE analysis and parallel processing capabilities [8].

3.4. FORWARD MOUNT IMPULSE TESTING

The last testbed discussed represents a fully integrated system as opposed to the phenom-enological or component experiments presented in Sections 3.1}3.3. It involves a complexengineering application performed at Los Alamos National Laboratory for the AcceleratedStrategic Computing Initiative (ASCI). The U.S. Department of Energy's ASCI program isaimed at developing massively parallel hardware and software environments for enhancingthe modelling, predictive quality and reliability of computer simulations for high-energyphysics and engineering applications.

Figure 13. Variability of the acceleration response at output sensor 1. (Impact test considered: 13 in/low-velocity impact of a 0.25 in/thin foam pad.): , test 08; , test 09; , test 10; , test 11;

, test 12; , test 13; , test 14; , test 15; , test 16; , test 17.


The system tested is an assembly of sub-components (inside shell; outside shell; supportstructure; electronics package and payload simulators) that feature complex joints subjectedto an impulse load. The impulse is generated by detonating an explosive grid on the surfaceof the structure as shown in Fig. 14.

A total of 33 strain gauges and six accelerometers instrument the response of this systemwith sampling rates as high as 20 MHz. Figure 15 shows typical strain and accelerationresponses obtained during one of the tests performed. Figure 16 illustrates the shockresponse spectrum obtained from the sixth acceleration response. Shock response analysesallow designers to estimate, for example, the peak acceleration transmitted to an electronicscomponent as a function of the component's fundamental resonant frequency. Figure 16

Figure 14. Instrumented test item with the explosive grid ready to detonate.

Figure 15. Measured strain and acceleration responses. (Top: strain at sensor 1; Bottom: acceleration at sensor 1).

Figure 16. Shock response spectrum obtained from acceleration sensor 6: **, sensor 6 (test 1).


clearly shows that most of the system's dynamics is identi"ed in the 5}30 kHz range whichwould render the development of a high-accuracy modal model a prohibitively computa-tionally expensive task.

A fully 3-D model is developed that includes a very detailed representation of thegeometry, material non-linearities, preloading and contact surfaces. Di!erent materials areinvolved in this system (aluminium, stainless steel, carbon steel and titanium) which makesit more di$cult to implement accurate models for the contact dynamics. The resultingnon-linear, explicit model features in the order of 1.4 million elements and 5 million dofs. Itis analysed on the ASCI platform &BlueMountain'with 504 dedicated processors. Typically,1 hour of CPU time is required to simulate 10~3 s of response. As before, several tests arecarried out to assess the degree of variability of the assembly. Figure 17 compares two strainmeasurements obtained at the same location when di!erent torque values are applied to theassembly between the inside shell and the outside shell. Although the two curves o!er manysimilarities in terms of peak response and frequency content, the initial time-delay indicatesa variation in the contact mechanics of the main threaded joint. Therefore, modelling andanalysing this system in the time domain requires the analysts to address the stochasticnature of the experiment.

The ultimate goals of the impulse experiment are to validate our modelling methods,identify speci"c joint behaviours and quantify the degree of variability of the assembly.Because of the computational cost involved, the ability to analyse this system relies stronglyon the capability to propagate e$ciently uncertainty and variability through the forwardcalculation. Again, statistical sampling and analysis methods are being evaluated to meetthis "rst objective. Also, such large computer simulations tend to generate enormousamounts of output that must be synthesised into a small number of indicators for theanalysis. This step is referred to as feature extraction [2] in the following. The main issue infeature extraction is to de"ne indicators that provide meaningful insight regarding theability of the model to capture the dynamics investigated. Several features are used foranalysing this non-linear, transient data sets that include: conventional modal analysis;Fourier transforms; the principal component decomposition; the shock response spectrum;

Figure 17. Comparison of strain time histories measured at sensor 1: **, test 1 (tight assembly); } } }, test3 (loose assembly).

Figure 18. Correlation between measured and predicted responses: **, test data (test 3); } } }, explicit FEsimulation.


AR/ARX- and ARMA-based features; the power spectral density; higher-order statisticalmoments; and probability density functions.

Figure 18 illustrates a typical correlation obtained between the test data and the explicitFE simulation after a few critical parameters of the numerical model have been adjusted.The emphasis of this preliminary update was to match the peak strain at a few sensorlocations and it can be seen that this objective has been reached. As a result, the frequencycontent and damping characteristics of the measured signals are also reproduced with good


accuracy. Finally, we mention that the correlation illustrated in Fig. 18 is within the range ofexperimental uncertainty inferred from the multiple test data sets.

It is emphasised that these results are preliminary. The impulse testbed is not discussedfurther in the following because it is currently being analysed in more detail. In particular,more testing is planned in the near future to characterise the response of the system undervarious excitations (modal testing; impulse testing) and input levels.

4. ASSESSMENT OF MODEL UPDATING TECHNIQUES

In the review of model updating techniques presented in this Section, the discussion isfocused around methods that can: (1) handle any type and source of non-linearity; and(2) enable both parametric and non-parametric updating to be carried out simultaneously.Very few techniques have been found in the published literature that could meet theseconstraints. As an illustration of this lack of techniques relevant to the non-linear world, thereader is invited to review from references [9, 10] the state-of-the-art in model updatingtechnology. Among the earliest and most promising work in test-analysis correlation fornon-linear dynamics, we cite the work by Hasselman et al. [11] and the work by Dipperyand Smith [12].

4.1. THEORY

Generally, parametric optimisation is achieved by minimising a &distance' between testdata and predictions of the numerical model, whether this distance is evaluated in the timeor frequency domain. The optimisation problem can be formulated as the minimisation ofthe cost function shown in equation (3) where the "rst contribution represents the metricused for test-analysis correlation and the second serves the purpose of regularisation

minMdpN

+j/1,2, N5%45

MRj(p#dp)N*[S

RRj]~1MR

j(p#dp)N#MdpNT[S

pp]~1MdpN. (3)

Constraints such as p.*/

)(pe#dp

e))p

.!9are added to the formulation to "lter out local

minima that would not be acceptable from a physical standpoint. In the test discussed inSection 4.3 for example, the parameters of interest MpN are the seven spring sti!nessparameters allowed to vary of $30% about their nominal value. Weighting matrices inequation (3) are generally kept constant and diagonal for computational e$ciency. Theycan also be de"ned as covariance matrices which formulates a Bayesian correction proced-ure, as shown in reference [3]. The only real di$culty is then to track the evolution of thecovariance coe$cients as parameters in the model are adjusted.

Obviously, many choices for the metric MRj(p)N are available, the simplest of all being the

di!erence between test and analysis modal parameters. It is also known in the modelupdating community as the output error residue, see reference [13], because quantitiesbeing compared are outputs as opposed to the excitation inputted to the system.

MRj(p)N"G

u5%451

!u1(p)

F

u5%45j

!uj(p)

/5%4511

!/11

(p)

F

/5%45ij

!/ij(p)H . (4)


Formulations based on optimisation problems may not be as e$cient as otherapproaches, such as formulating the adjoint problem [14], but they o!er some advantagesbesides constituting the overwhelming majority of inverse problem-solving techniques.Among others, formulation (3) is independent of the set of partial di!erential equations ornumerical solver used, independent of the optimisation solver and de"ned only by thechoice of metric MR

j(p)N. Nevertheless, it is shown in Section 5 that more relevant inverse

problem-solving strategies are available when the correlation takes place in the timedomain.

4.2. LINEAR MODEL UPDATING STRATEGIES

In addition to using the output residue documented in many publications among whichwe cite reference [13], the results summarised in Section 4.3 involve the de"nition of theforce and hybrid modal residues de"ned in references [15, 16], respectively. The "rst one isreferred to as a &force' vector because it estimates the out-of-balance forces obtained whenthe (erroneous) model fails to match the measured dynamics

MRj(p)N"([K(p)]!(u5%45

j)2[M(p)])M/5%45

jN. (5)

The second one is referred to as a &hybrid' vector and it can be explained, in a simpli"edversion, as being the displacement "eld produced by applying the force vector (5) to thestatic system. It can be de"ned as

MRj(p)N"M/5%45

jN!Mu

jN (6)

where vector MujN is obtained by solving the following static system:

[K(p)]MujN"(u5%45

j)2[M(p)]M/5%45

jN. (7)

The actual hybrid modal residue can handle incomplete measurements as well as noisydata. Reference [16] o!ers a complete derivation and addresses some of the implementationconcerns. Clearly, the residue vectors (5) and (6) are dual and both vanish when the FEmodel reproduces the measured data with accuracy. The "rst metric o!ers the advantage ofbeing very easy to compute and the second one features nice properties in terms ofnumerical conditioning. The important issue of incomplete measurement which mayintroduce a mismatch between sensor locations and degrees of freedom of the model isaddressed in both references [15, 16]. An alternative to dealing with the problem of spatialincompleteness is to apply model reduction and the three metrics (4)} (6) accommodatecondensation techniques with no di$culty. We emphasise that our purpose is not tocompare various "gures of merit for their e$ciency to identify sources of modelling errorbut rather to illustrate the danger of modal-based updating when the system is character-ised by a source of non-linearity not accounted for by the numerical model. Here, forexample, friction in the LANL 8-dof system is not represented. Although the linear modelprovides a good agreement with identi"ed modal parameters before and after parametriccorrection, the update fails to yield a positive identi"cation of damage.

4.3. VALIDATION OF MODAL-BASED METRICS

To illustrate that modal parameters are unable to characterise non-linear dynamicsproperly, the numerical model associated with the LANL 8-dof system is re"ned using threedi!erent cost functions based on metrics (4)}(6). The objective of this test is to make theupdated, linear model match the identi"ed mode shapes and frequencies with a betterdegree of con"dence than the one illustrated in Table 1. In doing so, the main modellingerror (a 14% sti!ness reduction located at the "fth spring) should be identi"ed. Each time,


the seven sti!ness parameters are optimised based on modal parameters extracted from theacceleration data. Measurements are assumed to be available at locations 1, 5 and 6 onlyand the mismatch between the identi"ed and numerical mode shapes is resolved using themodal expansion procedure relevant to each metric. To decouple the de"nition of test-analysis correlation metrics from algorithmic considerations, several optimisation solversare used to minimise the three cost functions. In this study, the order-0 Simplex algorithm,the order-1 conjugate gradient algorithm and the order-2 BFGS and Levenberg}Mar-quardt methods are implemented. These classical solvers are described in many publica-tions and manuscripts among which we cite reference [17].

A typical correlation after updating is illustrated in Table 3. It shows that the originalfrequency errors have been reduced to very small amounts and that the correlation of modeshape vectors has been improved. Based on such good correlation, most analysts wouldagree on the success of the updating procedure. However, the adjustment brought to each ofthe spring sti!ness values in Fig. 19 proves it impossible to identify the precise location andextent of structural damage. Worse, false positives (that is, sti!ness reductions predicted atlocations where no damage was originally introduced) are obtained. In the context ofa practical engineering application, such false positives would result into a wrong diagnosisand/or erroneous stress predictions.

The top half of Fig. 19 illustrates the correction brought to each of the seven springs whenthe force residue metric (5) is implemented. Similarly, results of the hybrid residue metric (6)and (7) are shown on the bottom half of Fig. 19. The correlation listed in Table 3 corres-ponds to metric (5) and similar trends are obtained with metric (6) and (7). The output errorresidue (4) fails to provide any meaningful model and these results are not commentedfurther.

We conclude from these results that situations may arise where modal-based metrics failto estimate the &true' correlation between two data sets when one of them or both representa non-linear system. In the example presented, the three metrics considered seem to indicatethat various models match the test data when, in fact, the actual modelling error is notidenti"ed. Since various optimisation solvers were implemented, we believe that these poorresults are to blame on the metrics themselves and not on potential convergence di$culties.Our conclusion illustrates the need to de"ne other metrics. Next, test-analysis correlationbased on time series is discussed.

4.4. TIME-DOMAIN CORRELATION METRICS

Two time-domain metrics are introduced brie#y. The "rst one correlates the measuredand simulated signals directly while the second one correlates the subspaces to which these

TABLE 3

¹est-analysis correlation of the ¸AN¸ 8-dof testbed for non-linear vibrations.(Correlation between the damaged, linear system and the adjusted linear model.)

Mode Identi"ed FE model Modal assurance Frequencyno. frequency (Hz) frequency (Hz) criterion (MAC) (%) error (%)

1 22.3 22.4 99.9 0.32 43.9 44.2 99.9 0.63 64.8 65.9 98.2 1.74 85.9 85.1 97.2 !0.95 99.7 99.8 99.6 !0.1

Figure 19. Adjustment brought to the spring sti!ness parameters of the LANL 8-dof model (top: force modalresidues (5) are minimised; bottom: hybrid modal residues (6) and (7) are minimised).


signals belong. This is achieved by making the numerical model match the singular valuesand vectors obtained by decomposition of the test data matrix, a procedure generallyreferred to as principal component decomposition (PCD). Reference [11] o!ers a completedescription of this procedure. This presentation is not intended as an exhaustive discussionof time-domain metrics. Obviously, many choices are possible depending on the applicationtargeted. Our purpose is to introduce the main correlation indicators and to assess theire$ciency using the LANL 8-dof testbed de"ned in Section 3.

For clarity, we assume that the system of interest is modelled by a set of partialdi!erential equations that can be integrated in time to provide the time history of variousstate variables. In structural dynamics, the second-order equation of motion is described by

[M(p)] GL2x(p, t)

Lt2 H#[K (p)]Mx (p, t)N#MF*/5

(p, t)N"MF%95

(t)N. (8)

Equation (8) states that the system is in equilibrium when the applied loading on theright-hand side matches the combination of inertia and internal forces on the left-hand side.The non-linear internal force vector MF

*/5(p, t)N accounts for any non-linear function of the

system's state variables. Since the numerical model can be parametrised by a set of designvariables MpN, time-domain responses such as the position vector Mx (p, t)N are shown todepend implicitly on these parameters. Next, we assume that time-varying quantitiesdenoted by My (t)N are obtained from the state variables. These can represent arbitrarycombinations of displacement, velocity and acceleration or any other quantity (strain,pressure, etc.) that may be needed for engineering analysis. In the example below, acceler-ation data are employed. Since our objective is to generate a more accurate model, thesimplest and most natural metric is the root mean square (RMS) error between test andsimulation data. The corresponding residue vectors are de"ned as

MR(t)N"My5%45(t)N!My (p; t)N. (9)


The computational procedure consists of the following steps: (1) for a parametric modelde"ned by a design MpN, the FE response is simulated via numerical integration of equation(8); (2) residues (9) are calculated at prescribed dofs and time samples; and (3) the costfunction J (p) is optimised where

J(p)" +j/12Nt

+i/12No

(y5%45i

(tj)!y

i(p; t

j))2#a +

k/12Np

(pk!p/0.*/!-

k)2. (10)

For simplicity, equation (10) assumes a deterministic form. Nevertheless, generalisation toa recursive Bayesian formulation o!ers no di$culty provided that the covariance matricescan be updated at low computational cost.

The PCD method validated in references [11, 18], among others, generalises the notion ofmode shape for non-linear systems. Rather than using a direct comparison between timeseries, test and simulation data are "rst assembled into matrices

[>(ti: t

f)]"

u1(ti) u

1(ti`1

) 2 u1(tf)

u2(ti) u

2(ti`1

) 2 u2(tf)

F F } F

uN0

(t1) u

N0(ti`1

) 2 uN0

(tf)

. (11)

Then, the subspaces spanned by the rows (or columns) of such test and analysis, rectangularmatrices are estimated using a singular value decomposition (SVD) expressed in terms ofleft, orthogonal singular vectors MU

jN, right, orthogonal singular vectors MW

jN and singular

values pj:

[>5%45]"[U5%45][R5%45][W5%45]T [> (p)]"[U (p)][R(p)][W(p)]T. (12)

Finally, the cost function used for optimising the model's parameters MpN is de"ned asa combination of the distance between the pseudo-subspaces (represented by the leftsingular vectors MU

jN), the energy di!erence of each signal (represented by the singular

values pj) and the error in time series (represented by the right singular vectors MW

jN)

J(p)"1

N20

EdU(p)E22#

1

N0

EdR(p)E22#

1

N0N

t

EdW(p)E22#aEp!p/0.*/!-E2

2(13)

where the normalised errors between test and analysis components of the SVDs are de"nedas

[dU(p)]"[U5%45]T[U (p)]![I]

[dR(p)]"[R5%45]~1([R5%45]![R(p)]) (14)

[dW(p)]"[W5%45]T[W(p)]![I].

Since the singular vectors are orthogonal, they provide a time-varying basis of themulti-dimensional manifolds to which the non-linear signals belong [19]. In addition, theSVD o!ers a practical way of "ltering out any measurement noise or rigid-body modecontribution because these are typically associated with singular values much smaller thanthose characteristic of the dynamics. However, we emphasise that the SVD along cangenerally not be used for detecting the presence of a non-linearity.

4.5. VALIDATION OF TIME-DOMAIN METRICS

The results are brie#y discussed of applying the PCD time-domain correlation metric toa test data set collected using the LANL 8-dof testbed. This illustration consists of locating


structural damage with a linear con"guration of the system. The con"guration used for thisexample is identical to the one described in Section 4.3 (same linear system, sensingcon"guration and damage scenario). The only di!erence is that the PCD cost function isnow implemented instead of metrics based on modal data. The results can be comparedwith those of Section 4.3 (see Fig. 19).

The updating procedure implemented performs an optimisation of the (constant) designparameters when a set of incomplete, time-domain measurements are provided. Onlyacceleration outputs at locations 1, 5 and 6 are assumed available. In addition to updatingthe model's seven spring sti!ness parameters, the procedure also optimises the eighttime-varying components of the internal force vector MF

*/5(p, t)N in equation (8). The reason

for optimising the internal force vector is because our original model does not account forany source of non-linearity that we know to be a!ecting the measurements (friction, &stickand slip', etc.). The overall procedure goes as follows. Unknowns of the optimisation are theseven spring sti!ness parameters and eight force components. The PCD correlation is basedon the "rst 90 acceleration measurements at sensors 1, 5 and 6 that span a [0; 0.168] s timewindow. For the numerical simulation, the FE matrices and force vectors are reduced to thesize of the test model (locations 1, 5 and 6 only) and the response of the condensed model isintegrated in time using 10 sampling points between any two measurements. As theresponse is integrated in time, the sti!ness values and internal force vector are optimised.A total of 89 optimisations are therefore carried out in the time window [0; 0.168] s. Aftera few optimisations, the spring sti!ness parameters converge to the values shown below. Noclear interpretation of the identi"ed forcing function (non-linear term) can be made andthese results are not further discussed here.

The adjustment brought to the seven springs is represented in Fig. 20 in percentage of theoriginal sti!ness value. The only spring that witnesses a signi"cant reduction is the "fth one,as expected. In addition, the 17% loss of sti!ness predicted is in very good agreement withthe 14% reduction in#icted during the test. The sti!ening of springs 1, 2 and 6 cannot beexplained other than by evoking the implicit least-squares nature of the solution procedurethat tends to smear the adjustment throughout the set of parameters. Nevertheless, these

Figure 20. Adjustment brought to the spring sti!ness parameters of the LANL 8-dof model. (The PCD time-domain cost function (13) is minimised.)


corrections remain small in magnitude and inconsistent with a damage. As mentionedpreviously, we believe that friction plays an important role in the dynamics of themass/spring system. The reason is that large modal damping ratios (over 10%) are identi"edfrom the test data which is inconsistent with the oscillations observed (the accelerationresponse does not feature an exponential decay anywhere comparable to 10%). The factthat friction is not accounted for in the numerical model may also contribute to thedistortion of the solution.

This example illustrates that time-domain metrics outperform modal-based metrics whenthe dynamics investigated is signi"cantly non-linear and/or modal parameters are not wellsuited to its description. They, however, introduce problems of their own that are addressedin Sections 5 and 6.

5. OPTIMAL ERROR CONTROL

One problem of time-domain model validation that has not been addressed previously isthe reconstruction of continuous solution "elds during the optimisation. This issue isfundamental because, if the inverse problem is not formulated correctly, the numericalmodel yields discontinuous acceleration, velocity and displacement "elds which contradictsthe laws of mechanics for the class of problems investigated here. After explaining wherediscontinuous solution "elds originate from, a new formulation of the inverse problem inthe time domain is proposed. It is based on the theory of optimal control, as explained inreferences [12, 20], and relies on the resolution of multiple two-point boundary valueproblems (BVP). When satisfactory solutions to the two-point BVPs are obtained, thenumerical model is guaranteed to match the measured data at the beginning and at end ofthe time window considered. We emphasise that the idea of optimal error control is not anoriginal one. Full credit must be given to the authors of references [12, 20] although theirmotivation was somewhat di!erent. Our contribution is to provide a complete derivation ofthe algorithm which is needed for the discussion in Section 5.5.

5.1. DISCONTINUITY OF THE SOLUTION FIELDS

The strategy of implementing successive optimisations produces several optimisedmodels, one for each time window considered. This is necessary not only for computationalpurposes but also because some of the parameters being optimised may vary in time andfollowing such evolution as it is occurring may be critical to model validation. However,nothing in the formulation of the inverse problem enforces continuity between the solution"elds obtained from models optimised within the ith and (i#1)th time windows. Since theoptimisation variables can converge to di!erent solutions Mp(i)N and Mp(i`1)N in twosuccessive time windows, the discontinuity of the solution can be written, for example, interms of the displacement "eld as

limt?tit)ti

x (p(i), t)Olimt?tit*ti

x (p(i`1), t). (15)

In the following, it is explained how optimal control strategies can be implemented tosolve the inverse test-analysis correlation problem while reconstructing continuous solution"elds and identifying the source of modelling error. An application to a sdof system showsthat the optimal control approach does indeed resolve the discontinuity but also thatpractical applications remain out-of-reach because of the extreme computational require-ment of the algorithm.


5.2. TEST-ANALYSIS CORRELATION VIA OPTIMAL ERROR CONTROL

For clarity, the evolution of the system investigated is recast as a set of "rst-order partialdi!erential equations. In structural dynamics, this would typically be achieved using thestate-space form. We emphasise that the "rst-order formulation is used for simplicity only;the algorithm can be implemented in terms of second-order systems if computationale$ciency dictates so, as we show in Section 5.3. Our starting point is a time-domain solverthat provides a solution Mx (t)N to the set of non-linear equations

GLx

Lt(t)H"MF(p, x, t)N#Me (t)N. (16)

In equation (16), vector Me (t)N represents the contribution of a generic modelling error. Itincludes everything that the numerical model does not account for. It is important to realisehow general this de"nition is: the error contribution can be de"ned as a combination ofmodelling errors, discretisation errors, etc., but it can also account for experimental ornumerical variability. We also introduce the notation My (t)N that represents a vector offeatures, that is, quantities obtained from the solution vectors Mx (t)N and that can also beassessed from the measurements. This vector can, for example, represent an arbitrarycombination of acceleration and pressure data at various locations.

The inverse problem is re-formulated as follows. The objective is to reconstruct continu-ous time series such as Mx (t)N in such a way that the features in My (t)N match the test data.Simultaneously, the procedure must provide a parametric adjustment of the model's designvariables MpN and identify the unknown, non-parametric error Me(t)N. For simplicity, the costfunction is formulated as the rms error between test and simulated data but it can be veri"edthat any cost function may be used instead. Since the unmodelled dynamics plays a roleidentical to that of a controller in the theory of optimal control, a &minimum e!ort' penaltyterm must be added to avoid generating unrealistic error levels. Hence, the basic costfunction can be stated as

J (p)" +k/12Nt

(My5%45(tk)N!My (p, t

k)N)T[S

yy(tk)]~1(My5%45(t

k)N!My (p, t

k)N)

# +k/12Nt

Me (tk)NT[S

ee(tk)]~1Me (t

k)N. (17)

The resolution procedure is a direct application of the theory of optimal control. First,the cost function (17) is augmented with the evolution equation (16). This is achieved byintroducing a vector Mj(t)N of Lagrange multipliers (also referred to as the system's &co-state'variables) that penalise the total cost in the event where the state equation is not satis"ed.The augmented cost function is de"ned as

JM (p; x; e)" +k/12Nt

(My5%45(tk)N!My (p, t

k)N)T[S

yy(tk)]~1(My5%45(t

k)N!My (p, t

k)N)

# +k/12N1

Me (tk)NT[S

ee(tk)]~1Me (t

k)N!Mj (t

k)NTAG

Lx

Lt(tk)H

!MF (p; x; tk)N!Me(t

k)NB. (18)

Adding a set of variables Mj (t)N may seem inconsistent with the objective of simplifyingthe resolution procedure. The reason is that these new unknowns provide additionalequations that can be solved for to ensure that the numerical solution matches the test data


at the beginning and at the end of each time window. The numerical algorithm is obtainedin a classical manner by: (1) converting JM into a continuous functional; (2) performing anintegration over the time window [t

i; t

i`1] considered; and (3) applying the necessary

conditions of stationarity LJM /Lx"0 and LJM /Le"0. It yields the following set of governingequations that must be solved simultaneously in each time window [t

i; t

i`1]

MxR (t)N"MF (p, x, t)N#Me(t)N

MjQ (t)N"!CLF

Lx(p; x; t)D

TMj (t)N

Me(t)N"!

1

2[S

ee(t)]Mj(t)N (19)

Mj(t`k

)"Mj(t~k

)N#2 CLy

Lx(p, t

k)D

T[S

yy(tk)]~1(My5%45(t

k)N!My(p, t

k)N)

Mx(ti)N"Mx5%45(t

i)N or Mj (t~

i)N"0

Mx(ti`1

)N"Mx5%45(ti`1

)N or Mj(t`i`1

)N"0.

It can be observed that the fourth equation above represents the discontinuity of theco-states. As expected, it is shown to be a direct consequence of how well test data arereproduced by the numerical model. To the extent where the observation vector matchesthe test data exactly, no discontinuity is propagated to the co-states. The last two ofequations (19) ensure that the test data are reproduced at the beginning and at the end ofeach time window. The only remaining step is to integrate this procedure to a parametricoptimisation solver. The solver itself may be any &black-box' optimiser that the user "ndsconvenient to implement. The basic requirement is that the value of a cost function J(p) beobtained for any arbitrary design MpN. This calculation is summarised below.

5.3. NUMERICAL IMPLEMENTATION

To illustrate how the optimal control formulation can be applied to structural dynamics,equations are transformed back into their second-order form. The "ve-step procedureoutlined below is encapsulated within the optimisation solver. For a given set of variablesMpN, the cost function J(p) de"ned in equation (17) is calculated by:

Step 1. Choice of the initial condition Mj0N: Given a particular design MpN, the co-state's

initial condition must be obtained that minimises the di!erence between test and numericaldata. The cost function to minimise can, for example, be de"ned as the rms error

J (j0)" +

k/i, i`1

(My5%45(tk)N!(y(j

0, t

k)N)T(My5%45(t

k)N!My(j

0, t

k)N). (20)

This optimisation problem introduces the two-point boundary value problem mentionedpreviously because, once the optimum is found, the observation vector is guaranteed tomatch test data at the time increments considered. The optimisation problem de"ned by thecost function (20) would typically be embedded into the outer parametric optimisation loop.In addition, calculating the feature My(j

0, t)N for a given initial condition Mj

0N requires the

evaluation of Steps 2}5 below which renders the entire procedure extremely time-consuming.


Step 2. ¹ime integration for the co-state: Once the initial condition has been optimised toensure continuity of the solution "elds, the co-state equation (19) is integrated. Theequivalent second-order form is

[M(p)] GL2j(t)

Lt2 H#[K (p)] GLj (t)

Lt H"0. (21)

Note that this system is unstable: it can be veri"ed that some of its eigenvalues are complexwith a positive real part. Very short time windows [t

i; t

i`1] must therefore be considered so

that oscillations do not have time to grow unstable. The obvious inconvenience is thatshorter windows require more two-point BVPs to be solved.

Step 3. Reconstruction of the non-parametric, unmodelled dynamics:

Me (t)N"!

1

2[S

ee(t)]Mj (t)N. (22)

Step 4. Integration of the equation of motion: The second-order equation of motion isintegrated with the contribution of the modelling error identi"ed at Step 3 and starting withthe initial condition Mx(t

i)N"Mx5%45(t

i)N and My (t

i)N"My5%45(t

i)N

[M(p)]MxK (t)N#[K (p)]Mx(t)N#MF*/5

(p, t)N"MF%95

(t)N#Me(t)N. (23)

Adding the error term in equation (23) guarantees that the displacement, velocity andacceleration "elds match the measured data at the end of the current time window, t

i"t

i`1.

Potential discontinuities are &absorbed' by the co-state and error variables. Note that thetolerance used for monitoring the convergence of the two-point BVP (20) de"nes themaximum allowed discontinuity of the solution "elds. A trade-o! between the continuityconstraint and the computational requirement can therefore be achieved.

Step 5. Calculation of the cost function: Finally, the feature vector My(t)N is estimated usingthe solution "elds obtained by integrating equation (23). The cost function J(p) is calculatedfrom one of the de"nitions (10), (13) or (17) or any other if more appropriate metrics can bede"ned.

So far, the choice of matrices [Syy

(t)], [See

(t)] and [Spp

(t)] (if a minimum-changecontribution is added to the cost function) has not been discussed. If kept constant, it ismost convenient to adopt diagonal matrices. However, reference [12] reports that theselection of weights in [S

ee(t)] can be critical to the overall convergence, in which case

a second inner loop should be implemented to adjust these weights as needed. Theweighting can also be de"ned in terms of covariance matrices for implementing a maximumlikelihood approach or a Bayesian parameter identi"cation method. Variance andcovariance data in matrix [S

yy(t)] are obtained directly from test data. Similarly, variance

and covariance data in matrix [Spp

(t)] are derived from the model's uncertainty character-isation. The only computational di$culty is the evaluation and updating of the errorcovariance matrix [S

ee(t)]. The analyst must generally compromise between computational

e$ciency (that favours uncorrelated statistics and diagonal matrices) and accuracy (thatrequires higher-order approximations and multiple matrix factorisations).

To illustrate the optimal error control approach, a numerical example is presented thatinvolves forced vibrations of the sdof Du$ng oscillator. This simple, non-linear systemprovides a good understanding of the method's capability and limitation, as discussed below.

5.4. NUMERICAL ILLUSTRATION USING A SINGLE-DEGREE-OF-FREEDOM SYSTEM

To illustrate how the optimal control algorithm works for solving an inverse test-analysiscorrelation problem, the Du$ng oscillator presented in Section 3.1 is investigated. First, the


set of equations (1) is integrated in time to generate the so-called reference data in the[0; 30] s window. Two &experimental' data sets are obtained by sampling the reference dataand adding noise to the discrete measurements. The "rst set of displacement and velocitymeasurements consists of 11 points while the second set consists of 151 points. The noiseintroduced in both cases is Gaussian with a noise-to-signal ratio set to 5%. Then, theinternal force is removed. Therefore, the starting model is linear and the (unknown)modelling error is simply e (t)"F

*/5(t). Ten successive optimisation windows [t

i; t

i`1] are

de"ned over the 30 s time period and a total of 75 increments are used for integrating thesolution within each window. The feature vector My (t)N considered collects the displacementand velocity &measurements' at the 11 (respectively, 151) samples t

iwhere test data are

assumed available. The inverse problem is "rst solved using the set of 11 measurements,then a second resolution is performed using the set of 151 measurements. Both results arecompared below. In both cases, the error weighting matrix is kept constant with weightsequal to one-half:

My5%45(ti)N"G

x5%45(ti)

xR 5%45(ti)H My(p, t

i)N"G

x (p, ti)

xR (p, ti)H [S

ee(tk)]"C

0.5

0

0

0.5D . (24)

Figure 21 illustrates the correlation of the updated models. The procedure outlined inSection 5.3 identi"es the internal force as a function of time and this result is shown inFig. 22. It can be observed from Fig. 21 that the optimised models are in good agreementwith reference data especially when the test-analysis correlation is based on 151 samples.The dashed lines that represent the response of the models after optimisation providea signi"cant improvement over the original response in Fig. 2. We emphasise that only the11 or 151 sampling points are used to "t the model; the baseline, noise-free solution picturedby the solid line is shown only for the purpose of assessing whether or not the modelsreproduce the continuous solution. This example illustrates that correlated models cangenerate continuous displacement, velocity and acceleration "elds even though 10 success-ive optimisations are carried out independently from one another.

Figure 21. Test-analysis correlation of the sdof system after model updating:**, reference; } ) } ) }, 11 samples(and 5% noise); } } }, 151 samples (and 5% noise).

Figure 22. Non-parametric identi"cation of the modelling error with the sdof system:**, reference; } ) } ) }, 11measurements; } } }, 151 measurements.


When analysing the solution obtained from 11 measurements, it can be concluded fromFig. 22 that the error identi"ed from this procedure approaches the &true' cubicnon-linearity with a fair accuracy. Both the periodicity and overall amplitude of thenon-linear force are captured. At the very least, this solution could provide some insightinto the nature of the non-linearity and the approximate value of the non-linear spring. Forexample, if a simple parametric model such as e(t)"a(x3 (t)) is curve-"tted in the least-squares sense to the values of e (t) obtained in Fig. 22, the non-linear spring sti!ness isestimated to be equal to a"0.065 instead of the value k

nl"0.05 used (30% error). Since the

error vector is estimated directly from the co-states in equation (22), discontinuitiesobtained when the solution progresses from one time window to the next are visible inFig. 22. The theory of optimal control states that the controller becomes continuous if anin"nite number of observations are available. For our application, this means that the errorvector should converge to the &true' error if more measurements are available. This isveri"ed in Fig. 22. For the case where 151 measurements are available instead of theprevious 11, it shows that the optimal error control delivers a very accurate estimation ofthe unknown dynamics. A direct result is that the displacement, velocity and acceleration"elds predicted by the optimised models, while remaining continuous, match the test datawith even greater accuracy.

5.5. DISCUSSION OF THE RESULTS

The important conclusion that we emphasise in this Section is that extreme care must bebrought to the formulation of model validation when time series are employed. Uncon-strained optimisation may result in discontinuous solution "elds. This di$culty can bealleviated by formulating an optimal controller of the modelling error. This is a veryattractive technique because not only does it handle parametric and non-parametricidenti"cations simultaneously but it also propagates uncertainty using the Bayesian theoryof information and it provides a rigorous framework for generating continuous solutionsfrom an arbitrary number of optimisations.


However, this improvement comes with the additional cost of formulating a two-pointBVP to guarantee continuity of the solution. Since the procedure is embedded within anoptimisation solver, multiple two-point BVPs must be solved for. Unfortunately, the impacton the computational requirement is enormous. For example, solving the Du$ng oscillatorproblem requires a total of 16}20 hours of CPU time depending on the number ofmeasurement points available. The 4-dof system shown in Fig. 3 was also investigated. Thisapplication consisted in calculating the sti!ness of each of the four springs and identifyingthe cubic internal force (assumed unknown). The resolution required about 70 hours ofCPU time. These timings were obtained on a dedicated R10 000/250 MHz RISC processorwhen the algorithm is programmed within the environment provided by MatlabTM. Clearly,these "gures prohibit any application of the technique to practical engineering problems. Itis for this reason that other avenues must be explored.

6. CONCLUSION

The work presented in this publication addresses the correlation of non-linear modelswith test data. One of the speci"city is that modal superposition can generally not beconsidered as a valuable solution technique due to its inability to represent arbitrarynon-linearity and to capture the dynamics of transient events. For this reason, test-analysiscorrelation must rely on time series. Several testbeds developed at Los Alamos NationalLaboratory in support of our health monitoring and code validation and veri"cationprograms are presented. Their purpose is to provide experimental data for validating thestrategies implemented for test-analysis correlation and inverse problem solving. Linearmodel updating techniques are reviewed and their ine$ciency is illustrated using a testbeddeveloped to analyse non-linear vibrations. Then, time-domain techniques are discussedand applied with greater success. Finally, we show that a careless formulation of the inverseproblem in the time domain yields discontinuous solution "elds which violates the mostbasic laws of mechanics. This di$culty is alleviated by formulating an optimal controller ofthe modelling error. This technique is capable of simultaneously: (1) handling parametricand non-parametric identi"cations; (2) propagating uncertainty and variability using theBayesian theory of information; and (3) providing a rigorous framework to generatecontinuous solutions from an arbitrary number of optimisations. Its computational require-ment, however, makes it impractical for solving meaningful engineering problems.

To bypass some of the di$culties identi"ed in this work, our current emphasis is onreplacing inverse problems with multiple forward, stochastic problems. After a metric hasbeen de"ned for comparing test and analysis data, response surfaces are generated that canbe used for: (1) assessing in a probabilistic sense the quality of a particular simulation withrespect to &reference' or test data; and (2) optimising the model's design parameters toimprove its predictive quality. One critical issue to be investigated in future research is thede"nition of adequate metrics for correlating transient, non-linear data. Rather thanattempting to de"ne deterministic distances, future work will emphasise dealing with&clusters' of test and analysis data that must be compared in a statistical sense.

To conclude, it is our opinion that test-analysis correlation and inverse problem solvingfor non-linear structural dynamics must address the following issues: (1) How to character-ise the variability of an experiment/design? (2) How to generate additional or surrogate datasets that can increase the knowledge about the experiment/design? (3) How to select featuresthat best characterise a non-linear data set? (4) How to assess statistically the consistencybetween multiple data sets? (5) How to develop fast probability integration capabilities andinterface them to our analysis tools? These problems and the overall issue of modelvalidation are further discussed in reference [21].


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review and assessment of model updating for non-linear, transient dynamics

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