Regularizationinmodelupdating - Friswellmichael.friswell.com/PDF_Files/J147.pdf · 2008. 7. 14. · et al. [16] presented an iterative algorithm based on Gauss–Newton (GN) sequential

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2008; 75:440–478Published online 27 December 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2257

Regularization in model updating

B. Titurus and M. I. Friswell∗,†

Department of Aerospace Engineering, University of Bristol, Queen’s Building, University Walk,Bristol BS8 1TR, U.K.

SUMMARY

This paper presents the theory of sensitivity-based model updating with a special focus on the propertiesof the solution that result from the combination of optimization of the response prediction with a prioriinformation about the uncertain parameters. Model updating, together with the additional regularizationcriterion, is an optimization with two objective functions, and must be linearized to obtain the solution.Structured solutions are obtained, based on the generalized singular value decomposition (GSVD), andspecific features of the parameter and response paths as the regularization parameter varies are explored.The four different types of spaces that arise in the solution are discussed together with the characteristicsof the regularized solution families. These concepts are demonstrated on a simulated discrete exampleand on an experimental case study. Copyright q 2007 John Wiley & Sons, Ltd.

Received 25 May 2007; Revised 22 October 2007; Accepted 24 October 2007

KEY WORDS: sensitivity; model updating; non-linear regression; regularization; optimization; GSVD

1. INTRODUCTION

The goal of model updating is to improve a mathematical model of an existing structure usingmeasurements performed on this structure. Model updating is an inverse problem, and a charac-teristic feature of inverse problems is that they may be ill posed. A problem is well posed [1] if itssolution exists, is unique, and continuously depends on errors present in the problem formulation.If the problem fails to fulfil any of these conditions, then it is said to be ill posed. The existenceand uniqueness are often assured by the introduction of additional assumptions, leading to somegeneralized solutions of the problem, such as the normal pseudosolution [2], i.e. a minimum normsolution. The solution stability, that is the dependence of the solution on perturbations in the data,can be violated even when the solution exists and is unique [3]. Research in this area has built on

∗Correspondence to: M. I. Friswell, Department of Aerospace Engineering, University of Bristol, Queen’s Building,University Walk, Bristol BS8 1TR, U.K.

†E-mail: [email protected]

Contract/grant sponsor: Royal Society

Copyright q 2007 John Wiley & Sons, Ltd.

REGULARIZATION IN MODEL UPDATING 441

the early work of Tikhonov and Arsenin [4], leading to applications in parameter identification [5]and non-linear regression [6]. Hansen [7] provided a detailed account of linear ill-posed problemsand Ben-Israel and Greville [8] summarized the theory and applications of generalized inverses.Aster et al. [9] discussed a number of well-established methods and approaches within the fieldand Tarantola [10] described statistical approaches to inverse problems with a strong emphasis onthe application of Monte Carlo methods. There is also an overlap with the methods and algorithmsin the field of optimization (e.g. [11]), concerned with estimation, regularization and parameteridentification.

Many researchers apply regularization methods to their specific problems. The field of geophysicshas considered regularization extensively [12]. Tarantola and Valette [12] presented a non-linearinverse problem, formulated as least-squares (LS) discrete and continuous problems, and demon-strated the approach on three examples from geophysics. Constable et al. [13] introduced theconcept of Occam’s inversion for electromagnetic sounding data formulated as a non-linear inverseproblem. Haber and Oldenburg [14] developed two algorithms for medium- and large-scale non-linear inverse problems, again applied to geophysics. Stabilization of inverse ill-posed problems isoften achieved via regularization, by introducing one free parameter. The two most popular methodsused to determine the value of this free parameter are generalized cross validation (GCV) andthe L-curve method, and Farquharson and Oldenburg [15] compared these methods for a sequen-tially linearized non-linear inverse problem using data from synthetic and field experiments. Doicuet al. [16] presented an iterative algorithm based on Gauss–Newton (GN) sequential linearizationand regularization, with an application to a non-linear ill-posed problem in atmospheric remotesensing. Special consideration was given to the selection of the regularization matrices and param-eters within this scheme. Henn and Witsch [17] considered a non-linear inverse problem in imagematching. Maciejewski [18] used Tikhonov regularization for a problem in inverse kinematics andPiccolomini et al. [19] applied regularization techniques in magnetic resonance imaging.

The mathematics community usually focuses on properties such as convergence conditions, errorestimation and parameter selection. Hanke et al. [20] provided a generalization of Landweber’salgorithm for non-linear inverse problems, and the authors also proposed a stopping rule for theiterative algorithm and analysed the convergence properties. Kaltenbacher [21] also addressed theproblem of non-linear inverse problems in a functional analysis setting. She modified the standardlinearized equation into iteratively regularized GN, Newton–Landweber and Newton-like methods,using the truncated singular value decomposition. For these methods, local convergence was provedand an a priori stopping rule was proposed, ensuring the convergence to a solution when the noiselevel decreased to zero. Tautenhahn [22] analysed a general regularization scheme for non-linearill-posed problems, which may be understood as a generalization of the GN method with Tikhonovregularization, in a functional analysis context. This work was extended using a generalization ofthe weighting [23]. Numerical analysts have taken an alternative view, for example, the work onthe automatic selection of regularization parameters using the L-curve method [7].

Early work on model updating in structural dynamics was based on statistical system identi-fication [24], using measured modal data and a parameterized finite element model to improvethe model of the structure. The method minimized the variance of the estimated parameters andprovided an iterative algorithm that included a priori information in the form of an estimated orassumed parameter covariance matrix. Recently, Leclere et al. [25] considered the identification ofthe forces exciting a structure from acceleration measurements and used regularization based onthe truncated singular value decomposition along with the L-curve concept. Regularization has alsobeen applied to the non-linear inverse problem of unbalance reconstruction in rotor dynamics [26].

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2008; 75:440–478DOI: 10.1002/nme

442 B. TITURUS AND M. I. FRISWELL

Figure 1. Essential components and structure of the updating procedures.

The solution was based on the minimization of the Tikhonov functional, and Morozov’s discrep-ancy principle was used to define the stopping rule. Nordberg and Gustafsson [27] also consideredinput force identification from measured accelerations using a Tikhonov-based formulation ofthe inverse problem. Friswell and Mottershead [28] summarized early research in the field ofmodel updating. Link [29] considered an effective parameterization, where the objective functioncontained a regularization term that constrained insensitive parameters. Ahmadian et al. [30] andFriswell et al. [31] gave further examples of regularization methods within model updating, basedon truncated decompositions, the GCV method and the L-curve concept, that were evaluated on asimulated system and a real frame structure.

In model updating, two models with compatible responses need to be available, namely themathematical model to be improved, usually obtained by the finite element method (FEM), andthe experimental model that serves as a reference for updating, for example, natural frequenciesand mode shapes. These two models have to be specified for identical environments, such asboundary conditions, loads, operating history, etc. The measured outputs may be modal properties,inertial properties, static displacements, accelerations, strains, etc. and the parameterization of themathematical model may take a variety of forms. Alternative types or classes of parameterizationsare available for the same model based on substructure parameters (SPs) and physical parameters(FPs) [28], geometric parameters [31] and generic elements (GEs) [32]. A key assumption inmodel updating is that the comparison of the measured responses to the corresponding numericalpredictions has the capacity to characterize the model quality. Figure 1 shows a schematic of themodel updating procedure, and further clarification of this figure is provided in Section 2.

The motivation of this paper is to provide a qualitative treatment of Tikhonov-type regularization,within the context of sensitivity-based finite element model updating. In particular, the effect of thedecomposition of a non-linear problem into a sequence of regularized linear inverse problems willbe explored. The paper is organized as follows: Section 2 introduces the notation and main conceptsof model updating. In Section 3, alternative methods to decompose a non-linear inverse probleminto iterative linear steps are given, together with methods to regularize each step. Section 4 analysesthe influence of the regularizing information on the linear sub-problem. Section 5 summarizes thevector spaces that are relevant for sequential regularization and discusses the qualitative behaviour



of the relevant quantities within these spaces during the regularization process. Section 6 presentstwo case studies, one simulated and one experimental.

2. MODEL UPDATING AS AN INVERSE PROBLEM

The unknown parameters p and the responses z are assembled into column vectors

p∈DP ⊆RNP , z∈RNZ (1)

The parameter vector p belongs to the parameter space of dimension NP and the response vectorz belongs to the response space of dimension NZ . The mathematical model acts as operator Zbetween these spaces

Z(p) :DP ⊆RNP →RNZ (2)

The inverse operator may be defined as a map from the response space to the parameter space,and corresponds to the model updating objective, where a measured response vector is used tocompute the parameter values of the mathematical model. The inverse operator is denoted by

W(z)=Z−1(z)=p (3)

The actions of forward and inverse operators are shown in Figure 1. Suppose the exact responseze is measured and the model is such that ze∈ range(Z), where range(Z) is the range of Z(p).Then the parameter vector pe is symbolically computed as

pe=Z−1(ze) (4)

Equation (4) represents the idealized case of model updating, which is, in general, an ill-posedproblem [2].

In reality, the response is measured with only finite accuracy. The measured response vectorwill be denoted by zm and assumed to be of the following form:

zm=ze+eZ (5)

where eZ is an unknown experimental deviation from the exact response ze due to experimentalnoise and other influences. eZ is often assumed to represent only non-systematic experimentalerrors, where each element is given by uncorrelated normal probability distributions. Also, theresponse vector in Equation (4) may not capture all of the essential aspects of the modelledproblem. Finally, it is usually not possible to construct the inverse operator Z−1 explicitly. Withinthis paper it is assumed that Z is not fundamentally wrong.

An important conceptual step in inverse problems is the definition of the difference between thetwo models represented by the response vectors zA and zB , leading to the concept of the responseresidual

�z=zA−zB (6)

The goal of model updating is to find parameter values for a certain parameterization such thatthe distance between the response from the mathematical and experimental models is minimized.



pr

ps

pt

pr

ps

prpr

ps

pt

zm

zmzm zm

pu

Figure 2. The forward response model and its capacity for model updating, for NZ =3 and NP ={4,3,2,1}.

The definition of a scalar measure of distance between models is implemented as the vectornorm, [33],

d(zA,zB)2W =‖�z‖2W =�zTW�z≡‖W1/2�z‖22 (7)

where �z=zA−zB ; ‖◦‖W and ‖◦‖2 are weighted and Euclidean norms, respectively. The matrixW is the weighting matrix and is symmetric and positive definite [34], and hence d(zA,zB)W�0.This distance measure is used to quantify the distance between the predicted and the measuredresponse vectors as

‖�z‖2WZ=�zTWZ�z=(z−zm)TWZ (z−zm)=(Z(p)−zm)TWZ (Z(p)−zm) (8)

This measure will be the basis for discussion of the model updating algorithms. The definition ofthe weighting matrix WZ in Equation (8) allows for response scaling, usually based on nominalvalues, and the assignment of the relative importance of the responses.

It is also assumed that for dim(DP)=NP , where NP is the dimension of the parameter space,dim(◦) is the dimension of a vector space, and the following general assumption is valid

dim(range(Z))=NP , range(Z)⊆RNZ (9)

Two different scenarios may be recognized, based on the vector space dimensions, namely theoverdetermined case, NP�NZ , and the underdetermined case, NP>NZ . These scenarios areillustrated in Figure 2, for NZ =3 and NP ={4,3,2,1}. In all scenarios it is seen that ze /∈range(Z), either due to the finite extent of DP or range(Z), or possibly because NP<NZ . If ze∈range(Z), the problem may still exist due to use of the measured response zm instead of the exactresponse ze.

The problem of model updating can then be formulated as follows:

‖�z(p)‖WZ →min for p∈DP ⊆RNP (10)

Equation (10) belongs to the class of non-linear LS problems. The structure of the residual �z=z−zm allows further simplification of the problem. A general form of the inverse problem, notconsidered in this paper, is based on non-linear residual vectors �z=r(z,zm).



3. MODEL UPDATING PROCEDURES AND SEQUENTIAL REGULARIZATION

3.1. Standard procedures used in model updating

Equation (10) defines the model updating problem, where the measured data zm are used toimprove the model Z by estimating the parameter vector p, starting from the initial/nominalparameter vector p0. The model Z is assumed to represent a linear conservative elastic structure.However, damping may be included in the model. Undamped structures are characterized by a pairof structural matrices {M,K} or via its modal representation {X,U}, where M,K∈RND×ND arethe mass and stiffness matrices, respectively; X=diag(�0,i ) is a diagonal matrix of ND undampednatural frequencies andU=[u1,u2, . . . ,uND

], ui ∈RND , is a matrix of ND mode shapes. Althougha variety of response quantities may be considered, this paper concentrates on the modal model.Owing to the relationship between the pairs {M(p),K(p)} and {K(p),U(p)}, the model Z isconsidered as a non-linear mapping. The solution to the model updating problem given in Equation(10) is usually based on local linearization or approximation, utilizing the truncated Taylor seriesexpansion around the current parameter vector. The resulting linear subproblems are solved toprovide parameter updates, and these two steps are joined together in an iterative algorithm.

The non-linear LS problem of Equation (10) is equivalent to minimizing the cost function, [28],JLS(p)=‖�z(p)‖2WZ

for p∈DP ⊆RNP (11)

The cost function of Equation (11) is approximated by a second-order local approximation usinga Taylor series expansion about the current parameter vector. The gradient vector g of JLS is

g(p)=∇ JLS(p)=[�JLS(p)

�pi

]=2STWZ (z−zm)∈RNP (12)

and the Hessian matrix HJ of JLS is

HJ (p)=∇2 JLS(p)=[

�2 JLS(p)

�pi�p j

]=2

[NZ∑i=1

Hi (wTZ ,i (z−zm))+STWZS

]∈RNP×NP (13)

where S is Jacobian matrix of Z(p), wZ ,i is i th row of the matrix WZ and Hk is Hessian matrixof kth response in the vector z. The matrices S and Hk are defined as

S(p)=∇z=[�z(p)

�pi

]∈RNZ×NP , Hk(p)=∇2zk =

[�2zk(p)

�pi�p j

]∈RNP×NP (14)

3.2. Sequential linearizations and approximations in model updating

The Newton–Raphson (NR) procedure [6] may be used to approximate JLS in Equation (11), usinga truncated Taylor series expansion to the second order at pk , as follows:

JNR(�pk)= JLS,k+gTk �pk + 12�p

TkHJ,k�pk (15)

where �pk =p−pk , JLS,k = JLS(pk), gk =g(pk) and HJ,k =HJ (pk). The stationary point of JNR,found by solving �JNR/�(�pk)=0, is given by

HJ,k�pk =−gk (16)



or using the definitions in Equations (12) and (13) as(STkWZSk+

NZ∑i=1

Hk,i (wTZ ,i (z−zm))

)�pk =−STkWZ (zk−zm) (17)

where Sk =S(pk), Hk,i =Hi (pk) and zk =Z(pk). The solution to Equation (17) provides theparameter update vector �pNR,k .

The omission of the terms involving the matrices Hk,i in the NR formulation gives the GNmethod, which amounts to local linearization of Z, [28],

zL =zk+Sk�pk (18)

The GN method does not require the computation of the second-order derivatives of the responseswith respect to the parameters, at the expense of a reduced approximation quality around pk . Inpractice, the approximation quality is sufficient. The GN procedure will be used throughout thispaper in its extended regularized form. From Equation (17), the GN parameter update formula is

(STkWZSk)�pk =−STkWZ (zk−zm) (19)

corresponding to the cost function JGN=‖zL −zm‖2WZ.

The model Z may contain symmetries, for example, topological or parametric, leading toclose linear dependencies in Sk for some or all iteration steps. This situation may render theinverse problem at that particular iteration as insufficiently defined causing non-uniqueness and illconditioning in the inversion processes. Insufficient data may lead to underdetermined problemsat a specific iteration. A solution to these problems was suggested by Levenberg and Marquardt,who modified the diagonal of the system matrix in the normal Equations (17) and (19) [6]. In theLevenberg procedure Equation (19) is modified as follows:

(STkWZSk+ϑkI)�pk =−STkWZ (zk−zm) (20)

providing Levenberg’s parameter update �pLev,k . In the Marquardt procedure Equation (19) ismodified to give

(STkWZSk+ϑk diag(STkWZSk))�pk =−STkWZ (zk−zm) (21)

producing Marquardt’s parameter update �pMar,k . In both cases ϑk>0 is assumed. The purpose ofthe diagonal modifications in Equations (20) and (21) is the stabilization of the nominal problem,Equation (19). In statistics, a similar approach is called ridge regression. Link [29] used an identicalapproach to Equation (21) to regularize a modal updating problem.

The final procedure reviewed here uses the gradient vector g of JLS, Equation (12). The unitvector of steepest descent is specified as eG,k =−∇ JTLS,k/‖∇ JTLS,k‖2. The optimum parametervector update �pSD,k along this direction is

�pSD,k =�keG,k =−�k(STkWZ (zk−zm))/‖STkWZ (zk−zm)‖ (22)

where �k ∈(0, R�] is usually obtained by a one-dimensional optimization, R�>0 being the upperlimit for each step.

The application of the methods surveyed in this section implies an iterative algorithm, where ateach step the quantities, such as Sk , Hk,i and zk , have to be calculated, while advancing towards



the minimum of JLS. After each parameter step, a new updated value pk+1=pk+�pk is computedwhere �pk can be based on �pNR,k , �pGN,k , �pLev,k , �pMar,k or �pSD,k . These are standard methodsused in nonlinear regression, for instance [5, 6].

3.3. Regularization in model updating

The linear equations associated with the normal Equations (17) and (19) provide the basis for theanalysis of a single step of a regularized updating algorithm. For the GN method, Equation (19)is equivalent to the solution of the problem

[W1/2Z Sk]�pk =[W1/2

Z (zk−zm)] (23)

which is the weighted version of the equation Sk�pk =�zk and �pGN,k is minimum norm solutionof this equation.

The associated linear problem for the NR method may be derived from Equation (17) by thedecomposition

QH =NZ∑i=1

Hk,i 〈wZ ,i ,zk−zm〉=UHDHUTH (24)

where QH =QTH , U

THUH =I and DH =diag(dH,i ). The corresponding linear equation is then⎡

⎣W1/2Z Sk

D1/2H UT

H

⎤⎦�pk =

[W1/2

Z (zk−zm)

0

](25)

The LS solution of Equation (25) is the parameter step �pNR,k . The block form of Equation (25)suggests information augmentation in the NR method exploiting additional available information.The new information provides a better estimate of the parameter step by including more informationabout the original cost function, Equation (11). Linear equations similar to Equation (25) maybe derived for the Levenberg and Marquardt methods given by Equations (20) and (21). Varyingthe coefficient ϑk in Equation (20), the solution �pLev,k(ϑk) starts at �pGN,k for ϑk =0, andthe parameter vector then moves towards the direction of ∇ JLS, effectively giving �pSD,k while‖�pLev,k‖→0, for ϑk →∞. Such qualitative features in the behaviour of �pk(ϑk) will be the focusof our interest later in this paper.

An approach to the ill conditioning in inverse problems is to add new, regularizing information[35], preferably independent of the measured data set. The aim is to promote certain regions ofparameter space where, it is believed, the model realization should be. Tikhonov [4] achieved acontrolled influence of the regularizing information by forming a weighted sum of the originalcost function JLS and a new cost function JReg based on the regularizing information, given in theform

C(p)=0 (26)

where C contains NC equations. In this paper, we assume that C takes the form C(p)=Cp−dP , with C∈RNC×NP and dP ∈RNC , which represents the general linear problem Cp=dP . Forimplementation in iterative model updating, the incremental form of this equation is C�pk =d,



where d=dP −Cpk . The residuals corresponding to this equation are used to give the second costfunction

JReg(p)=‖C(p)‖2WP=‖Cp−dP‖2WP

=(Cp−dP)TWP(Cp−dP) (27)

where WP is a symmetric positive-definite weighting matrix. There are two cost functions, JLSand JReg, that need to be minimized concurrently. A composite cost function JLSR is formulatedas [35]

JLSR= JLS+�JReg=‖�z(p)‖2WZ+�‖Cp−dP‖2WP

(28)

where ��0 is a regularization parameter and JReg is the stabilizing or regularization part.The gradient of the cost function in Equation (27) is gReg=2CTWP(Cp−dP) and the Hessian

matrix is HReg=2CTWPC, based on Equations (12) and (13). The composite cost function forthe GN approach is then

JGNR= JGN+�JReg=‖Sk�pk−�zk‖2WZ+�‖C�pk−d‖2WP

(29)

and minimizing this cost function gives the regularized parameter step as the solution of theequation

(STkWZSk +�CTWPC)�pk =−(STkWZ (zk−zm)+�CTWP(Cpk−dP)) (30)

This approach generates the regularized GN approximation of JLS and will be the basis for allfurther investigations presented in this paper.

The regularization expressions JReg usually used in model updating are based on engineeringassumptions, concerning the parameter variations during iterations. The most frequently usedconditions, [28], are: (a) p→0, i.e. that the parameter values will be small; (b) p→p0, i.e. thatthe parameter changes with respect to reference model will be small and (c) �p→0, i.e. that theparameter step between iterations will be small. The incremental forms of these conditions are:(a) �pk →−pk , (b) �pk →p0−pk and (c) �pk →0. Condition (a), and to some extent condition(b), represents physical assumptions, while condition (c) acts mainly as a stabilizing condition inhighly non-linear problems. The conditions may be unified into the cost function

JReg=‖p−pref‖2WP=‖W1/2

P (p−pref)‖22 (31)

where pref is the reference parameter vector, and is equal to 0, p0 or pk for scenarios (a), (b) and(c), respectively. Since p=pk +�pk , Equation (31) at the kth iteration becomes

JReg,k =‖W1/2P (�pk−(pref−pk))‖22=‖I�pk−(pref−pk)‖2WP

=‖C�pk −d‖2WP(32)

where C=I and d=pref−pk .The weighting of the response residual or the parameters in Equation (29) introduces subjective

a priori information into the problem. These weighting matrices may be obtained from the statisticalprocessing of the raw data or may be based on a subjective assessment and/or previous experience.Also, these weighting matrices can be used to scale the problem. The stopping rule is an importantpart of the algorithm, for both numerical and physical considerations. Standard optimization-basedcriteria are usually used, which are based on the relative changes of parameters, responses, costfunction gradients and/or the cost function values themselves, between subsequent iterations. Oncethe variations of one or more of these criteria are less than a certain preset threshold value, theiterative procedure is stopped.



4. REGULARIZATION OF THE LOCAL LINEARIZED MODEL

The previous sections have provided the necessary conceptual and technical framework on whichfurther analyses will be based. It has been established that model updating consists of a series oflinearized problems that requires regularization. Improved understanding of each step is an impor-tant prerequisite to understand the whole updating process. To simplify the analytical treatmentin this section, the standard updating notation will be changed from Equation (29) to the genericform

J (x)= J1(x)+�J2(x)=‖Ax−b‖22+�‖Cx−d‖22=∥∥∥∥∥[

A

�1/2C

]x−

[b

�1/2d

]∥∥∥∥∥2

2

(33)

where A∈RmA×n , b∈RmA , C∈RmC×n , d∈RmC , x∈Rn and ��0. Note that the iteration index hasbeen dropped. Also, for simplicity, the Euclidean norm of the residual vectors is used instead ofthe weighted norm. It can be shown that minimizing the cost function in Equation (33) correspondsto finding the LS solution of the associated linear problem[

A

�1/2C

]x=

[b

�1/2d

](34)

The normal equation corresponding to Equation (34) is

(ATA+�CTC)x=ATb+�CTd (35)

with the solution symbolically formulated as

x(�)=(ATA+�CTC)−1{ATb+�CTd} (36)

Substitution of the regularized solution x(�) into both of the constituent equations in (34) leadsto the output residual vectors

r1(�)=Ax(�)−b, r2(�)=Cx(�)−d (37)

The residual vectors r1(�) and r2(�), and their associated norms, serve as important quantitativeindicators of the changes introduced by the application of regularization term J2(x)=‖Cx(�)−d‖22.4.1. The solution of a single set of linear equations

This section considers the general form of the solution of the system of linear equations Ax=b[33]. Three qualitatively different solution scenarios occur for this problem. The vector xb is thesolution of the system if Axb−b=0. The system of equations can have (a) a single unique solutionif b∈ range(A) and rank(A)=n, (b) infinitely many solutions if b∈ range(A) and rank(A)<n or (c)no solutions if b /∈ range(A). rank(◦) is the rank of a matrix, range(A)=span(A), where span(◦)

is the linear space spanned by the vector basis formed from the columns of a matrix.In the context of model updating any of these three situations can occur. In the case of over-

parameterization (mA<n) an infinite number of solutions is most likely to occur, particularly atthe final updating stages with very fine parameterization structure, for example, element-levelparameterization. In the case of under-parameterization (mA>n) the most likely result is for nosolutions to exist and this often occurs for coarse parameterization, for example, substructure



parameters. The case mA=n can lead to a unique solution; however, this scenario is unlikely to beachieved, and even in this case the most likely outcome is for no solutions to exist. The remainderof this section will describe some qualitative features of the three scenarios.

The problem Ax=b may be reformulated as the solution of the optimization problem

J1(x)=‖Ax−b‖22→min (38)

This alternative definition effectively joins the scenarios (a) and (c), where now there is a uniquesolution in both cases. Suppose the solution to Equation (38) is a vector xb∈Rn . The generalsolution of the system Ax=b may be expressed as [7]

xb=xP +xH =A†b+xH (39)

where A† is the pseudo-inverse of A [8], xP is a particular solution and xH ∈SN ,1≡null(A) is ahomogeneous solution, such that AxH =0. If rank(A)=min(mA,n), then the solution to Equation(38) is unique if the null space is trivial, SN ,1={0}, which usually occurs for mA�n. Oftenin modal updating, the null space SN ,1 is not trivial and we then define the nullity of A asnN =dim(null(A)) [33]. The numerical determination of the null space and the nullity nN may bea challenging task for inverse problems, due to possible numerical errors. In this case, we includevectors x such that Ax≈0, introducing the concept of the effective nullity and the effective rankof A. Any homogeneous solution xH in Equation (39) may be expressed as [33]

xH =A⊥n where range(A⊥)=null(A), n∈RnN (40)

and A⊥ is the orthogonal complement of A.The general form of the solution of the problem Ax=b may be defined using the concept of

an affine subspace (AS) [36]. The AS A is a generalization of the concept of a linear subspace,being effectively a translated linear subspace. The general solution (39) of Ax=b is then

xb∈A0≡A(A†b,null(A))=A†b+null(A) (41)

4.2. Refined solution via the GSVD

The regularized solution may be analysed using the generalized singular value decomposition(GSVD) of the matrix pair {A,C} [37]. The GSVD is used in many applications due to its abilityto simultaneously diagonalize two rectangular matrices with an equal number of columns andarbitrary numbers of rows. The GSVD of the matrix pair {A,C}, in a compact form, is [38][

UTA 0

0 UTC

][A

C

]=[SA

SC

]X−1 (42)

where

A ∈ RmA×n, UA∈RmA×mA , SA∈RmA×n, X∈Rn×n

C ∈ RmC×n, UC ∈RmC×mC , SC ∈RmC×n(43)

The matrices UA and UC are orthogonal, so that

UTAUA=I(mA×mA), UT

CUC =I(mC×mC ) (44)



and the matrices SA and SC are block diagonal, fulfilling the following constraint:

STASA+STCSC =I (45)

The assumption that X is a regular matrix gives the following condition for the pair {A,C} [37]:

rank

[A

C

]=min(mA+mC ,n) (46)

The matrices SA and SC contain the square blocks RA and RC , where

RA=diag(sA,i ), RC =diag(sC,i )

1�sA,1�sA,2 · · ·�sA,n��0, 0�sC,1�sC,2 · · ·�sC,n��1 (47)

and sA,i and sC,i are the generalized singular value pairs, and n� is the number of these pairs.Their ratios are called generalized singular values

gi =sA,i/sC,i (48)

The size of the submatrices RA and RC , and their positions in SA and SC , depend on the relationshipbetween mA, mC and n, and also on the rank deficiencies of A and C.

GSVD decomposition (42) may be applied to Equation (35) or (36), which is the least normsolution of Equation (33). The solution may be derived in terms of the GSVD components (43) as

x(�)=X(STASA+�STCSC )−1{STAUTAb+�STCU

TCd} (49)

Solution (49) can be expressed as x(�)=xb(�)+xd(�), where xb(�) is due to non-zero b andxd(�) is due to non-zero d. Further structure of x(�) may be observed by considering all relevantrow size combinations of the pair {A,C}. The relevant size combinations are

1. mA�n and mC�n,

2. mA�n and mC<n,

3. mA<n and mC�n,

4. mA<n and mC<n and mA+mC�n.

The case mA+mC<n is not considered here, since this always has a non-unique solution. Doicuet al. [16] considered case 2 and Friswell et al. [31] investigated case 4.

To ensure the existence of a non-trivial x(�), the following complementary condition has to befulfilled [7]

null(A)∩null(C)=0 (50)

Two functions that are useful in describing the solution are now introduced. These are

fi (�)= 1

1+(�/g2i ), f ∗

i (�)= fi (�)�

g2i=1− fi (�) (51)

where fi (�) is called the filter function and it occurs in standard derivations of regularizationmethods, e.g. [7]; f ∗

i (�) corresponds to the second term in Equation (33) and its form correspondsto the specification of the regularization part Cx=d, in an analogous way to the filter function forthe nominal physical part Ax=b of the problem in Equation (33).



The basic behaviour of fi (�) is as follows [7]: Assuming some gi >0, fi decreases slowlyfrom fi (0)=1 with increasing � until �≈gi , when fi decreases sharply towards zero. For ��gi ,as �→∞, the filter function fi decreases slowly and asymptotically to zero. This view helpsto understand the behaviour of x(�) as � is varied to find the optimal value �opt. Alternatively,for a fixed �∗, a small gi , such that gi ��∗, results in a small value of fi , thus suppressing thecontribution of i th term in the solution x(�). On the other hand, for large gi , such that gi ��∗,fi approaches 1 and the contribution of the i th member in x(�) is effectively unchanged. Thisview helps to understand the effect of fi once �opt is chosen.

The following sections derive solution formulae for cases 1 and 2 from the above list. Thesecases correspond to the examples given later, while formulae for cases 3 and 4 are not given inthis paper.

4.2.1. The regularized solution for case 1. The numbers of the rows in this case are specifiedas mA�n and mC�n. This results in the following structure of the GSVD [38, 39], based onEquations (42) and (43):

A=[UA,1 UA,2][

0

RA

]YT, C=[UC,1 UC,2]

[0

RC

]YT (52)

UA,1 ∈ RmA×(mA−n), UA,2∈RmA×n, RA∈Rn×n, Y=X−T∈Rn×n

UC,1 ∈ RmC×(mC−n), UC,2∈RmC×n, RC ∈Rn×n(53)

Based on the structure given in Equation (52), the solution for x(�) given in Equation (49) maybe cast into the form

x(�)=xb(�)+xd(�)=Xcol

(fi (�)

uTA2,ib

sA,i

)+Xcol

(f ∗i (�)

uTC2,id

sC,i

)(54)

where UA,2=[uA2,i ], UC,2=[uC2,i ] and col(◦) represents a column vector built from the elementsin the brackets where the range of the indices in the brackets determines the size of this vector.

4.2.2. The regularized solution for case 2. The numbers of rows in this case are specified as mA�nand mC<n. This results in the following structure of the GSVD [38, 39], based on Equations (42)and (43):

A=[UA,1 UA,2 UA,3]⎡⎢⎣0 0

I 0

0 RA

⎤⎥⎦[YT1

YT2

], C=UC [0 RC ]

[YT1

YT2

](55)

UA,1 ∈ RmA×(mA−n), UA,2∈RmA×(n−mC ), UA,3∈RmA×mC , RA∈RmC×mC

UC ∈ RmC×mC , RC ∈RmC×mC , Y1∈Rn×(n−mC ), Y2∈Rn×mC(56)



Based on the structure given in Equation (55), the solution for x(�) given in Equation (49) maybe cast into the form

x(�)=xb(�)+xd(�)=(X1col(uTA2, jb)+X2 col

(fi (�)

uTA3,ib

sA,i

))+X2 col

(f ∗i (�)

uTC,id

sC,i

)(57)

where X=[X1 X2], X1∈Rn×(n−mC ), X2∈Rn×mC , UA,2=[uA2, j ], UA,3=[uA3,i ], UC =[uC,i ] andthe size of the column vectors defined by col(◦) are specified by the sizes of correspondingsubmatrices in Equation (56).

4.3. Discussion of the decomposed solutions

For cases 1 and 2, from Equations (54) and (57), the general structure of x(�) is

x(�)= ∑i1∈I1

ci1xi1 +∑

i2∈I2

ci2(�)xi2 (58)

where

I1∪I2={1, . . . ,n}, I1∩I2=∅ci contains Fourier coefficients [7], xi is the i th column of X, and the index sets I1 and I2correspond to the unfiltered and filtered subspaces of Rn , respectively. Note that the coefficientsci2(�), for i2∈I2, are functions of � through the filter factors fi (�). The solution x(�) for �∈(0,∞)

is specified by the AS, A�,

x(�)∈A� ≡A

( ∑i1∈I1

ci1xi1,span(xI2)

)⊆Rn (59)

where the filter functions fI2(�) control the path of the solution for �∈(0,∞). For �→∞, thesolution is x(�)=∑i1∈I1

ci1xi1 . The dimension ofA� is dim(A�)=dim(span(xI2))=nI,2, wherenI,2=n for case 1 and nI,2=mC for case 2.

The application of Equation (58), which is based on the GSVD of the matrix pair {A,C}, is thesubject of ongoing investigation, particularly concerning the properties of the subspace span(xI2)

and its relation to the subspaces range(AT) and range(CT). However, to advance the presentanalysis, the use of the SVD of the individual matrices A and C is considered. This allows furtherinsight into the properties of separate subproblems, namely physical J1 and regularizing J2, and itwill also provide concrete insight into the nature of the one-parameter trajectory of the regularizedsolution x(�).

The solution x(�) belongs to A0 for �=0, to A� for �∈(0,∞) and to A∞ for �→∞. Thespace A0 is defined solely by {A,b} and is a unique vector xb if null(A)={0}. Similarly, A∞is defined solely by {C,d} and is a unique vector xd if null(C)={0}. Further, the complementarysubspaces A0 and A∞ are defined such that A0={x∈Rn|x /∈A0} and A∞ ={x∈Rn|x /∈A∞}.The cost functions J1 and J2 introduce quadratic structures on A0 and A∞. This is demonstratedhere only for J1, since J2 is qualitatively the same. The cost function J1 is expressed as follows:

J1(x)=‖Ax−b‖22=(Ax−b)T(Ax−b)=xTATAx−2xTATb+bTb (60)



The minimum of this function is any element from A0. An arbitrary element from A0 is writtenin the form x=x∗+r, where r∈Rn so that r /∈null(A) and x∗ ∈A0. Substituting this expressionfor x into Equation (60) leads to

J1(r)=rTATAr+K (61)

where K ∈R is a constant due to the shift x∗ and K�0. The SVD [37] of the matrix in Equation(61) may be expressed as A=USVT and the new cost function J ∗

1 (r)= J1(r)−K is defined. Thiscost function can be expressed as

J ∗1 (y)=rTVSTUTUSVTr=rTV(STS)VTr=yT(STS)y (62)

where y=VTr represents the rotation of the coordinate system, so that the new coordinate systemis aligned with the principal directions given by V. This specifies a quadratic structure around A0in Rn . For the general case of a matrix with nS,1 non-zero, nS,2 near-zero and nS,3 zero singularvalues, where nS =nS,1+nS,2+nS,3, S=diag(si ), i=1, . . . ,nS and nS =min(mA,n), then

J ∗1 (y)=yT(STS)y=

n∑i=1

y2i s2i =∑ y2j s

2j +∑

y2k s2k +∑ y2l s

2l +∑ y2u0

j ∈{1, . . . ,nS,1}, k∈{nS,1+1, . . . ,nS,1+nS,2}l∈{nS,1+nS,2+1, . . . ,nS}, u∈{nS+1, . . . ,n} (63)

where the elements with indices l and u specify null(A), and the set for u is non-empty if mA<n.The case with a non-empty set of u indices corresponds to the over-parameterized updating scenarioand the corresponding subset of principal directions in the matrix V can be joined with the principaldirections of the index subset l. These principal directions will be identified later as uncertaintydirections in the parameter space.

The modified cost function J ∗1 (y) represents the generalized ellipsoidal surface. When the value

of J ∗1 (y) is fixed at an arbitrary value, for example J ∗

1 (r)=1, then the surface is described by

nS,1+nS,2∑i=1

(yi1/si

)2

=1 (64)

where 1/si is the size of the semi-axis of the ellipsoid along the i th principal direction definedby the transformation y=VTr. Equation (64) contains only elements corresponding to non-zerosingular values. The index sets u and l will correspond to degenerated, infinitely long semi-axesof the resulting object. This will have further implications on the trajectory x(�).

Equation (64) specifies a quadratic iso-surface in Rn for J ∗1 or J1. The singular values, si ,

and the existence of a non-trivial null space of A characterize the shape of the object. Largevalues of si lead to small semi-axes along the corresponding principal directions, and converselysmall values of si lead to large semi-axes. The increase in the size of the semi-axis may beviewed as the increase in the uncertainty for x(�) along corresponding principal direction in Rn .A smaller semi-axis indicates a decreased uncertainty, or an increased certainty, in the correspondingprincipal direction. The nS,2 singular values with si ≈0 will therefore correspond to relatively longsemi-axes. This concept may be further extended to the indices l with zero singular values andto the indices u with zero contribution to the value of J ∗

1 (y), Equation (63), where the lengths of



the corresponding semi-axes will go to infinity. These two index sets can be used to define theuncertainty subspace.

The above discussion applies to both {A,b} and {C,d}. In the regularization set-up the formerpair is based on physical information gained from the model to be updated, while the latter pair isusually based on a priori assumptions. Consequently, the parametric solution path x(�) is specifiedby the interaction of the two objects J1(y) and J2(y) defined on Rn , around A0 and A∞, usingthe minimization condition ∇ J1(y)=−�∇ J2(y). This path tends to start and end along the leastcertain, i.e. the longest, principal directions of these objects. Mathematically, this is due to lowestnumerical weights associated with corresponding directions, proportional to the low si as shownin Equation (63), therefore allowing the largest changes along this direction. Thus, the preferredmovement of x(�) is along the directions corresponding to the low singular values. The preferredchanges, when modifying the reference physical model using the regularization model, are in thedirections of the least knowledge or certainty, represented by low si that give the longest semi-axes1/si . In terms of the quantities defined in Section 3, the solution �pk(�) for the kth iteration steptraces a one-parameter path between the two affine spaces A0 and A∞, where the solutions of theinitial physical problem and the regularization problem reside, respectively. The initial physicalproblem is defined by the pair {Sk,�zk} and the regularization problem is specified by the pair{C,d}. The purpose of regularization is to move from �pk(0)∈A0 along �pk(�) to a new position�pk(�opt)∈A�, that is close to �pk(0), so that the effect of the a priori condition is reduced.

5. VECTOR SPACES AND EVALUATION CRITERIA IN UPDATING

This section considers the vector spaces that are involved in model updating and discusses thevarious criteria that are used to evaluate the success of the estimation and regularization process.

5.1. The space of the regularization parameters: �-space

The contribution of J2(x) to the overall cost function (33) is measured by one free regularizationparameter �>0. This is a scalar parameter, although more regularization terms, Ji (x), could beused to specify the composite cost function J (x). In this case, more regularization parameters, �i ,would have to be used to weight each regularization contribution independently. If the number ofregularization terms is N�, then N� parameters with �i>0 are needed, specified in the space ofregularization parameters in RN� . In the single parameter case, the problem of regularization isdefined by the two subproblems described by their associated cost functions J1 and J2, with therelative weight between the two given by the regularization parameter �.

5.2. Parameter space: P-space

For model updating the mathematical model is parameterized, p∈DP ⊆RNP , where the vector pmay contain different quantities (such as material density or shell thickness), different classes ofparameters (such as substructure, physical, generic or geometric) and possibly spatially overlappingparameters (for example, Young’s modulus and a nodal offset for the same finite element). Thesolution of the model updating problem is formulated as a sequence of linearized subproblems,each of them regularized separately, as shown in Section 3.3. A sequence of parameter steps �pk(�)

is thus obtained as a one-parametric family of solutions of the problem given in Equation (29).Four different configurations of the general problem were listed in Section 4, and for practical



Figure 3. Single parameter regularization in parameter space.

implementation the selection of the optimal �opt is vital. Figure 3 illustrates the variation of �pk(�)

due to the minimization of the composite cost function J specified in Equation (33). The quantitiesshown in Figure 3 are described in detail in Section 4, particularly in Sections 4.1 and 4.3.

5.3. Response space: Z-space

Response space is defined as the space containing response quantities of interest and its relationshipwith the parameter space is realized by the map Z(p) given in Equation (2). A geometric analysisof the domain RNZ and related quantities provides qualitative insight into the updating problem.The two cases considered here are: (a) NP<NZ and (b) NP�NZ .

If there are less parameters than response quantities, NP<NZ , then the response surface maybe viewed as NP -dimensional manifold embedded in RNZ . This scenario is illustrated in Figure 2.The model Z(p) is a non-linear finite dimensional map with sensitivity matrix S(p) given byEquation (14). This may be visualized for any response point, Z(p j ), with NP one-parametriccurves z(�), where �= pi with the remaining NP −1 parameters held constant. In this case thegradient �z/�� represents the tangent vector to z(�). The tangent response space at z0 is defined as

AZ ,0={z0+u;z0=Z(p0),u∈ range(S(p0))} (65)

and the dimension of AZ ,0 is dim(AZ ,0)= rank(S)�NP . The special case rank(S)<NP mayhappen, for instance, when the modelled structure is both topologically and parametricallysymmetric.

The second case with more or equal number of parameters than responses, NP�NZ , representsan over-parameterized model. Hence, there is a potential for a solution pm such that Z(pm)=zm.However, such a solution is not required, or sometimes not even desired, as the measured vector zmwill be contaminated by noise and the exact solution can be sensitive to noise. The tangent responsespace AZ ,0 is also defined by Equation (65), with the most frequent case dim(AZ ,0)=NZ .

Model updating is defined as the minimization of the non-linear response residual ‖Z(p)−zm‖for p∈DP ⊆RNP . This problem is reformulated into a sequence of steps where the minimizationof the linear response residual ‖Sk�pk−�zk‖ is required. In this case the linear operator Sk mapsthe parameter increment �pk to the response space, with the objective that the simulated andmeasured responses are close, i.e. to minimize �zk =zm−zk .

The response space quantities in case of one step of iterative model updating are illustratedin Figure 4 for the case of NP =2 and NZ =3. The response surface Z(p)∈R3 is a function ofmodel parameters p∈DP ⊆R2. The response surface in the response space is a two-parametersurface in R3. The surface is locally approximated by a linear model based on S(p)∈R3×2, which



Figure 4. The process of parameter step computation on the response surface, NP<NZ .

represents a tangent plane at the point zk =Z(pk). The two columns of the sensitivity matrix,�z/�p1 and �z/�p2, define this plane. These vectors define a non-orthonormal basis for the tangentplane. The null space null(S(p)) also provides further information about the model. These vectors,among other things, indicate the local quality of the parameterization. The case with three locallyindependent parameters is also shown, which would lead to the third basis vector �z/�p3. Thiswould allow a new basis to span the whole response space and the only limitations would beimposed by DP and possibly by the non-linear map Z.

The figure presents three consecutive steps, zk−1, zk and zk+1 as they progress towards a discretemeasured realization of the response vector zm. A detailed investigation of the non-linear stepbetween zk and zk+1 is shown. Without regularization, the current response residual �zk =zm−zkis projected onto the Sk plane, symbolically as �zpk =Sk(S

+k �zk). However, the non-linearly mapped

value corresponding to the orthogonal projection �zpk is the new position zk+1=Z(pk+�pLN,k),where �pLN,k represents a least norm estimate, in this case LS, of the parameter step. The presenceof the regularization terms alters the parameter update �pk(�) and therefore also the mapped value�zk+1(�) on the Sk plane and the non-linear predicted response zk+1(�)=Z(pk+�pk(�)). Theone-parameter path, �pk(�), in parameter space and the resulting path, �zk+1(�), in response spaceare investigated in this paper. The left part of Figure 4 shows the tangent plane itself with quantitiesthat are defined in the response space. The orthogonal projection �zpk corresponds to the casewithout regularization, i.e. when �=0. Two examples of the paths Sk�pk(�) are shown for the twodifferent regularization cases. The case I�pk =0, corresponding to Equation (20), converges to thevalue zk . The more general case C�pk =0 is also shown and the corresponding analysis will bepresented in the case study in Section 6.1.

5.4. Space of residual norms: r-space

The regularization parameter expresses the degree of influence of the regularization term on theoriginal problem. However, this is a free parameter requiring a method to select its optimum value,�opt, as a separate subtask in the estimation process. The choice of the regularization parameterfixes the value of the computed parameter step �pk(�opt) for kth iteration, leading to the updatedparameter vector pk+1=pk +�pk(�opt) and response model zk+1=Z(pk+1). The operational pointon the response surface moves to a new position zk+1 and all relevant quantities are recomputedto obtain the matrix Sk+1 and other required quantities. The choice of �opt significantly affects thedirection and the magnitude of the parameter update. The two main types of methods to choose



�opt [7] are those that require some knowledge of the structure of the measurement noise, andthose that try to identify �opt without such knowledge.

The residual norms can be computed for the whole range of the regularization parametersand this information may be used to choose �opt [7]. Consider the original cost function (28),JLSR= JLS+�JReg, which is the regularized LS-based non-linear cost function. Here, this problemis transformed to a sequence of linearized GN problems, given by the quadratic cost function (29),where closed-form solution is available. The choice of a specific �c provides �pk(�c) and this inturn can be used to compute the residual norms

JGNR(�c)= JGN(�c)+�c JReg(�c)=‖Sk�pk(�c)−�zk‖2WZ+�c‖Cp(�c)−d‖2Wc

(66)

The two terms JGN(�c) and JReg(�c) represent a point in R2 corresponding to �c. Hence, thetrade-off curve may be obtained for the range �c∈(0,∞) and this will represent trade-off betweenthe response and regularization residuals

t (�) : [JGNR,1(�), JGNR,2(�)], �∈[0,∞) (67)

where in this specific case JGNR,1= JGN and JGNR,2= JReg.The concept is usually applied to inverse problems by considering the logarithms of the residual

norms, resulting in the L-curve [7]l(�) : [log(JGNR,1(�)), log(JGNR,2(�))], �∈[0,∞) (68)

The shape of the trade-off curve is always convex [40], and it depends on the specific natureof the inverse problem and is influenced by the noise in the data. The logarithmic scale doeshighlight the L-shape of l(�). Often the curve has one convex corner, which may be used toobtain �opt≈�corner. In this case, there are two regions of the L-curve, one corresponding toover-smoothing for ��opt and the second corresponding to under-smoothing for ��opt. Thesearch for �opt is thus a search for the optimal balance between regularization (based on a prioriassumptions about certain aspects of the problem) and the physical description of the problem(influenced by the noisy and incomplete data zm). These features of the L-curve are theoreticallypresented in References [7, 40], where certain criteria are specified to be fulfilled for the L-curveto have a distinctive corner indicating �opt. The main assumptions are related to the structure ofthe noise and the ability of the regularized solution to approximate the real solution.

6. CASE STUDIES

6.1. A simulated spring–mass system

6.1.1. Model description. Figure 5 shows the five degree-of-freedom (DOF) discrete spring–masssystem used in this study. This simple structure allows the model updating problem to be treated ina controlled environment and also to visualize the theoretical concepts. The two attachment points,represented by the springs k1 and k4, have finite flexibility with nominally equal properties. Oftenthe primary sources of uncertainty in the modelling of structural systems are joints and interfaces.The objective is to improve the mathematical model of the uncertain structural interfaces by usingartificially generated experimental data. The same model will be used to generate the ‘measured’responses, and in this study no additional assumptions are placed on these data, as the purpose isto illustrate the regularization concepts in non-linear model updating.



k1

m4

k6

k5

k4k3k2

m5

m3m2m1

+X

x1

x4

x2

x3

x5

Figure 5. Spring–mass system with 5 degrees of freedom.

Table I. Five natural frequencies of the initial and ‘measured’ models.

1 2 3 4 5

Initial (Hz) 1.07 4.38 6.02 8.52 9.63Measured (Hz) 4.65 5.51 6.03 12.36 12.97Difference (%) −76.99 −20.51 −0.33 −31.07 −25.75

Spring elements are characterized by stiffness parameters, k j , and mass elements are describedby their mass mi . These parameters are assembled into the vectors m and k, with nominal valuesmnom and knom,

m= [m1,m2,m3,m4,m5], k=[k1,k2,k3,k4,k5,k6], mi ,k j>0 (69)

mnom = [1.00,2.00,1.08,0.50,0.50]kgknom = [1.00,1.50,1.50,1.00,0.70,0.70]kNm−1 (70)

The model is parameterized by the two stiffness parameters of the structural interfaces, k1 andk4, and it is assumed that they are nominally equal, k1≈k4. Initial estimates of the parameter valuesare k1=k4=117.4Nm−1, the parameter values of the measured case are k1=k4=3958.2Nm−1.The large ratio between the actual and initial values is chosen to represent an arbitrary choice ofinitial parameter values for highly uncertain areas of the model. It will also allow the observationof a range of behaviours during the updating. Table I shows all five natural frequencies of theinitial and ‘measured’ models.

The parameterization and parameter domain used in this study are defined as

p = [p1, p2]T=[k1,k4]T∈DP ⊂R2

DP = [k1,min,k1,max]×[k2,min,k2,max], k1,min,k1,max,k2,min,k2,max>0(71)

where DP is specified such that ki,min=0.01×ki,nom,ki,max=8×ki,nom and ki,nom is given inEquation (70).

Three responses are chosen, namely the first, second and fourth natural frequencies. This responsevector, along with the two parameters, allows for an easy visualization of the concepts introduced



in this paper. Thus, the response vector is

z=Z(p)=⎡⎢⎣z1(p)

z2(p)

z3(p)

⎤⎥⎦=

⎡⎢⎣

f1(k1,k4)

f2(k1,k4)

f4(k1,k4)

⎤⎥⎦∈R3 (72)

6.1.2. General model properties and sensitivity analysis. An initial qualitative analysis isperformed, together with a sensitivity analysis. The symmetric topology of the structure waschosen to correspond to the experimental study in Section 6.2. Four types of symmetries arerecognized in this problem: (a) structural topology symmetry, (b) parametric symmetry, (c)physical symmetry and (d) algebraic symmetry. Structural topology symmetry is represented bythe symmetric arrangement of the structural components, as shown in Figure 5 by the symmetryin the x direction relative to m2. Parametric symmetry occurs when the parameter valuesdefined on a topologically symmetric system are symmetrically chosen from Dp. A physicallysymmetric structure is topologically symmetric and also symmetric over the parameterized andnon-parameterized parts of its model. Algebraic symmetry indicates that the structural matricesare symmetric. Parametric and physical symmetries have serious implications on the sensitivityand related properties of the system, and often creates rank-deficient sensitivity matrices. Similarproblems arise for nominally symmetric structures with symmetric parameterizations. A nominallysymmetric structure is almost symmetric, where the lack of symmetry arises from factors suchas production inaccuracies and errors, asymmetric wear, sensor placement, etc. In this case, thestructure will have ill-conditioned sensitivity matrices with a detrimental impact on updatingalgorithms.

These issues are demonstrated on the current system with parameterization (71). Figure 6 showsthe surfaces in response space for three different choices of m3, where m3=m3,sym+�m3 andm3,sym∈{1.00,0.08,0.00}. This figure demonstrates the effect of increasing symmetry in the systemon the response surfaces and the resulting sensitivities. The two intersecting dashed lines representcases with a single parameter variation, the dotted line represents the symmetric parameterizationcase and the intersection point of the dashed lines is chosen for the sensitivity matrix evaluation,with arrows representing unit sensitivity vectors with respect to the parameters k1 and k4, �z/�k1and �z/�k4. Case (a) represents a physically asymmetric structure because of the asymmetry in thenon-parameterized part of the structure, (b) represents nearly physically symmetric structure forthe choice p1= p2 and (c) represents fully physically symmetric structure for a symmetric choiceof parameters. Case (b) will be considered further in detail. The shape of the response surfaceis characterized by two parts separated by the dotted line representing the parametric symmetryconfiguration. The distance between these two parts decreases as �m3 decreases. The two parts ofthe response surface in the non-zero �m3 case have mutually corresponding points for symmetricchoices of the parameters from Dp, in this case chosen symmetrically with respect to the linek1=k4. Also note that the sensitivity vectors provide, in general, a non-orthogonal basis for thetangent plane.

Figure 7 shows the fully normalized sensitivity matrices for the three cases in Figure 6, andthe sensitivities of all five natural frequencies to all six spring stiffness parameters are shown.The sensitivity values are represented by squares and in shades of grey colour. Large sensitivitiesare represented by large squares and light grey colour, and small sensitivities by small, darksquares. For visualization all components are normalized relative to the largest absolute sensitivity



Figure 6. Model response surfaces and local sensitivity directions in the response space: (a) �m3=1.00;(b) �m3=0.08; and (c) �m3=0.00.

1 2 3 4 5 6

1

2

3

4

5

1 2 3 4 5 6

1

2

3

4

5

1 2 3 4 5 6

1

2

3

4

5

(a) (b) (c)

Figure 7. The normalized sensitivity matrices, with increasing symmetry in the spring–masssystem. The rows represent the natural frequencies and the columns the spring stiffnesses: (a)

�m3=1.00; (b) �m3=0.08; and (c) �m3=0.00.

component in each matrix separately. All sensitivity values are positive as an increase in anystiffness parameter will increase the natural frequencies. The increasing symmetry in this examplemeans that the columns of the sensitivity matrix become nearly linearly dependent along the dottedline in Figure 6(b) and fully linearly dependent in the case presented in Figure 6(c). This situationtranslates into rank deficiency and ill conditioning in the sensitivity matrix, as demonstrated inFigure 7. Figure 7(c) clearly indicates three pairs of symmetric parameters for �m3=0.0, leadingto the reduction in the rank of the sensitivity matrix from five to three.

Figure 7 also shows that the third natural frequency is not sensitive to k1 or k4, which is whythis frequency was not chosen as a response. To understand this low sensitivity the correspondingmode shapes for case (b) are shown in Figure 8. The modes are visualized as a stacked bargraph, with each stack corresponding to one physical coordinate. Each stack contains five bars,each corresponding to one mode shape. The third mode shape is highlighted and shows that theamplitudes at coordinates x1 and x3 are small. Consequently, any physical modification localized



0

0.5

1

x1

x2

x3

x4

x5

scal

ed m

odal

am

plitu

des

Figure 8. Mode shapes of the spring–mass system.

to this position will have a small influence on the third mode. Similar arguments could be usedfor other sensitivities in Figure 7. For example, the relatively low sensitivity of the first naturalfrequency with respect to parameters k2, k3, k5 and k6 arises because the modal amplitudes andphases for the coordinates neighbouring these parameters are similar.

6.1.3. Model updating problem definition and analysis. The choice of three responses and twoparameters constitutes an over-determined problem with sufficient experimental data to computeparameter values for the specified parameterization. However, further regularization terms haveto be included to ensure convergence for highly non-linear problems. This configuration providesvisual interpretation of the intermediate processes and the whole convergence procedure, and thusprovides insight into the nature of this inverse problem. The regularizing component JReg usedassumes nominal equality of the two interface parameters, k1 and k4. This component is added tothe nominal part of the GN problem containing the sensitivity information and response residuals.The regularizing condition has the form

JReg=‖C�pk −d‖2, d=dp−Cpk, C=[1 −1], dP =0 (73)

The regularization parameter � will be held constant at �C =2.5×10−5 for all iterations. Thisvalue is simply chosen to ensure that the regularization condition, k1≈k4, is reasonably fulfilled.The purpose here is to investigate the influence of the regularization condition on the parameterupdate process, rather than to select �opt. The next section will investigate the regularizationsubproblem at each iteration. Finally, no scaling and weighting are applied here due to the relativelysimple nature of the problem. The convergence is established on the basis of the stabilization ofboth the response and parameter variations.

Figure 9 shows convergence of the absolute values of the parameters and responses, wherethe true values are indicated with dashed lines. Figure 10 shows the development of the modalassurance criterion (MAC) [28] between the ‘measured’ and computed mode shapes correspondingto the first four iterations. The MAC criterion is also used to pair the modes at each iteration. Theconvergence to the ‘measured’ responses is stable and the ‘true’ values are reached quickly. Further-more, the relationship between the two parameters is retained, as required by the regularizationcondition.



0 1 2 3 4 5 6 7 8 90

5

10

15

iteration

resp

onse

s [H

z]

0 1 2 3 4 5 6 7 8 90

1000

2000

3000

4000

iteration

para

met

ers

[Nm

]

p1

≈ p2

z1

z2

z3

Figure 9. The convergence of the parameters and response for the discrete spring–mass system.

1 2 3 4 5

1

2

3

4

5

Iteration 1

exp

1 2 3 4 5

1

2

3

4

5

Iteration 21 2 3 4 5

1

2

3

4

5

Iteration 31 2 3 4 5

1

2

3

4

5

Iteration 4

model

Figure 10. MAC matrices for first four updating iterations.

6.1.4. Definition of the regularization conditions. A comprehensive analysis is provided for tworegularization conditions. The first regularization condition is the nominally equal parameterscondition used for model updating in the previous section. This condition is based on the equation

C�pk =−Cpk or [1 −1][

�p1,k

�p2,k

]=−[1 −1]

[p1,k

p2,k

](74)

where the relationship between the two parameters has been transformed into incremental form.The second regularization condition is a standard minimum norm regularization applied to theparameter increments. This regularization is based on the equation

I�pk =0 (75)

and aims to reduce the magnitude of the parameter change at the kth iteration step towardszero. This condition results in a smooth transition between the non-regularized LS solution and



�pk(�)→0 in the direction of −∇ JLS as �→∞. Both regularization conditions will be combinedwith a GN procedure given by

Sk�pk =�zk (76)

The first regularization scenario may be expressed in the form[Sk

�1/2C

]�pk =

[�zk

−�1/2Cpk

](77)

and the second regularization scenario is[Sk

�1/2I

]�pk =

[�zk

0

](78)

In both cases �∈[1×10−10,1] and logarithmic spacing is used. Both regularization scenariosare applied without any pre-processing, such as weighting or scaling of the intermediate equations.

6.1.5. Regularization analysis in the response space. Figure 11 shows the updating processbetween the initial model and the measured/true state of the structure, for both regularizationscenarios and the first few iterations. The process of model updating is affected by the shape ofthe response surface and the regularization conditions. The measured responses of the true stateare assumed to be or close to Z(p). The point of interest here is how the regularization conditionsinfluence the intermediate solutions, and hence this example does not have modelling errors. Theresponses corresponding to the condition k1=k4 are represented by the dashed line. By definition,the initial model z0 and the measured/true state zm are located on this line. The responses usedhere are defined in Equation (72).Z(p) is computed for the range of parameters k1 and k4 given byEquation (71). Figure 11 shows the response surface (the narrow and curved general surface in R3),the tangent planes (the two triangular patches for the first two iterations), two regularization pathsfor each iteration (one straight and one curved line localized in the tangent plane) and the mappingdue to the operator/model Z (indicated by the white arrows between the linear approximationpoint and the point on the response surface).

At the initial parameters, the sensitivity matrix S(p0) is evaluated and its columns form thetangent AS AZ ,0 at z0. At the kth iteration, zk and AZ ,k (denoted ASZ ,k in Figure 11) are definedso that

AZ ,k =zk+range(Sk) (79)

The non-regularized solution is an orthogonal projection of zm onto AZ ,k , and is expressed as

zlin,k+1(0)≡zLS,k+1=SkS+k �zk+zk, �zk =zm−zk (80)

where S+k represents the pseudo-inverse of Sk [8]. This linear prediction corresponds to the non-

linear response prediction based on the forward operator Z,

zk+1=Z(S+k �zk+pk) (81)

The addition of the regularization conditions introduces two new parameter solution paths, �pk(�),that correspond to linear response predictions, zlin,k+1(�), at each iteration, as shown in Figure 11



Figure 11. First three iterations on the response surface with local regularization.

by solid lines with dot and circle markers. The lines with dot markers correspond to the nominallyequal parameters regularization and the lines with circle markers correspond to the minimum normregularization. Both lines start at zLS,k+1, for �=0, and continue along their own characteristictrajectories or linear response prediction paths, converging towards their own final values. Thelinear and non-linear response predictions are

zlin,k+1(�)=Sk�pk(�)+zk, zk+1(�)=Z(pk+�pk(�)) (82)

For the nominally equal parameters regularization, C=[1,−1], the values of zlin,k+1(�) arelocated along the straight line (dot markers) starting at zLS,k+1 in the direction of the line lC . lC isshown in Figure 11 for the first iteration and it represents null(C) transformed into response space

lC ={Sk�pC +zk;�pC ∈null(C)} (83)

This behaviour is specific for the first iteration if equal parameters are used, C�p0=−Cp0=0.However, a similar process occurs for other iterations with non-zero Cpk defining an AS based onthe solutions of C�pk =−Cpk . An additional feature of this linear response prediction path is thatit is orthogonal to lC since the problem is a minimum distance problem. This will be discussedfurther in connection with Figures 12 and 13.

In the case of minimum norm regularization, zlin,k+1(�) has a more complicated path (circlemarkers). The parameter update starts at �pLS,k+1 and then changes its direction towards −∇ JLSwith its magnitude decreasing to 0. The linear response prediction zlin,k+1(�) has a similar trend,starting at zLS,k+1 and continuing towards the orientation in the response subspace specified bythe translated Sk∇ JLS vector at zk . The magnitude of the linear response prediction decreases to0 so that zlin,k+1(∞)=zk .

The linear response prediction paths, zlin,k+1(�), lie in AZ ,k . The points located on the responsesurface, identified as Z(pk+1), correspond to zlin,k+1(�) and this is indicated in Figure 11 by whitesolid arrows. The model updating presented in Section 6.1.3 was based on the nominal parameterequality condition, with �k chosen so that zlin,k+1(�k)→ lC , where Sk�pk(�k) starting at zk areindicated by the grey solid arrows.



Figure 12. Parameter spaces and tangent affine subspaces for the first three iterations of model updating.

6.1.6. Regularization analysis in parameter space and the response AS. Figure 12 shows thedetailed analysis of the influence of the regularization during the first three iterations, presented inparameter space and the subspace AZ ,k . The three subplots on the first row shows the parametersolution paths, �pk(�), for both regularization scenarios and for each iteration. The nominally equalparameters regularization is given by dotted lines and the minimum norm regularization is givenby solid lines. The other objects present are the non-regularized LS solutions �pLS,k , representedby grey circle markers located in the centres of the ellipses characterizing operators Sk accordingto the theory presented in Section 4.3. The ellipses represent a single contour of the cost functioncorresponding to the pair {Sk,�zk}. The circles in these subplots are based on the same concept,however, they correspond to the minimum norm regularization characterized by the pair {I,0}.The circle markers in the centre of the circles represent trivial solution of the problem specifiedby {I,0}.

The �pk(�) path for the minimum norm regularization varies continuously between these twomarkers, starting at the centre of the ellipse, which is defined uniquely due to the over-determined



0 0.3 0.6

0

1

2

3

4

5

0 0.6

0

1

2

3

4

5

[x,y]=[log(norm(Ax b)),log(norm(Cx d))]0

0

1

2

3

4

5

0

0

1

2

3

0

0

1

2

3

[x,y]=[log10

α,κL]

0

0

1

2

3

0 2 4 6

0

1

2

3

4

x 104 Iteration 1

0 2 4 60

2000

4000

6000

8000

10000

12000

[x,y]=[norm( ),norm( )]

Iteration 2

0 2 4 60

200

400

600

800

1000

1200Iteration 3

Figure 13. Trade-off curves, L-curves and L-curve curvatures for the first three iterations of themass–spring example: the solid lines represent minimum norm regularization, the dashed lines

represent nominally equal parameters regularization.

nature of the problem {Sk,�zk}. The path then moves along the least certain direction represented bythe longest semi-axis of the ellipse and later turns towards the second marker, so that ‖�pk(�)‖→0along the direction −∇ JLS. Note that this regularization is effectively a continuous version of theminimum norm solution of the under-determined problem with a degenerate characteristic ellipse,



i.e. at least one singular value is zero. The minimum norm solution corresponds to the projectionof the origin of the coordinate system onto the affine solution subspace due to {Sk,�zk}. This trendis shown in the parameter space plots in Figure 12, where the second part of the parameter pathtends to be orthogonal to the longest semi-axis of the ellipse. This trend is more obvious for themore degenerate ellipses, i.e. those with higher aspect ratios corresponding to more ill-conditionedoperators Sk .

The other features to observe in the parameter space plots in Figure 12 arise from the nominallyequal parameters regularization. Equal parameters are given by the thick dashed lines correspondingto the affine solution subspace of {C,−Cpk}. Also shown is a thick dotted line, which representsthe orthogonal complement of this AS. This model may be viewed as degenerate ellipse, whereonly the line of equal solutions is shown. The regularization subproblem is under-determined withsolutions represented by null(C) translated from the origin. For the problem of Equation (74),null(C)=[1,1]T, and for the first iteration, due to the initial choice of k1=k4, the solutions aregiven by null(C). For subsequent iterations the equality condition is only fulfilled approximately,and the zoomed region in Figure 12 around the origin for the third iteration shows that the solutionsubspace is a general AS, i.e. translated null(C).

For nominally equal parameters regularization, the parameter path, �pk(�), represented by thedotted line, starts at the marker representing the centre of the characteristic ellipse of the pair{Sk,�zk}, and ends on the subspace of the solutions to the regularization subproblem {C,−Cpk}.Thus, there is no change in the direction of the parameter path, �pk(�), because there is no singleunique solution of the regularization subproblem {C,−Cpk}. The direction of �pk(�) is set by theleast certain direction as defined by the quadratic form based on Sk , and is therefore not orthogonalto the affine solution subspace of {C,−Cpk}. The selected solutions at each iteration are not shownin the parameter space, since the regularization parameter, �k , was chosen large enough to give asolution with k1≈k4.

Now consider the lower row of plots in Figure 12 that represents the development of linearpredictions in the response space. The operator Sk maps the parameter space to the subspacesAZ ,k . The parameter path, �pk(�), is mapped to zlin,k+1(�) in the response space. The dottedarrows indicate non-regularized solutions, given by the orthogonal projection of zm onto AZ ,k .The linear response prediction paths are qualitatively similar to the associated parameter paths.The circle markers represent points, zk , that lie on the response surface, and of course the spaceAZ ,k is tangent to the response surface. The light grey arrows represent scaled sensitivity vectorscorresponding to the columns of the sensitivity matrix Sk , thus constituting a basis for AZ ,k . Theinfluence of near physical symmetry causes these vectors to be almost linearly dependent, and thecorresponding Sk is ill conditioned. The two paths zlin,k+1(�) for the two regularization approachesare both represented by solid lines, and in both cases the solution path zlin,k+1(�) starts at zLS,k+1.

The path zlin,k+1(�) for the minimum norm regularization is shown as the thick solid line,and starts at zLS,k+1 and then moves towards the line defined by Sk∇ JLS. The initial directionis close to the path of the second regularization scenario as, at this early stage, the influence of{Sk,�zk} still dominates in both cases. Later, zlin,k+1(�) continues towards zk as the parameters‖�pk(�)‖→0 for �→∞.

The path zlin,k+1(�) for the nominally equal parameters regularization, shown as the thin solidline, also starts at zLS,k+1 and is a straight line moving towards the line lC,k , correspondingto the solution subspace of the problem {C,−Cpk}. This is due to the fact that the updatingproblem is defined in the response space via minimum distance formulation. Therefore, the linearresponse prediction path, zlin,k+1(�), is orthogonal to the line lC,k . The specific response zlin,k+1(�k)



corresponding to the actual solution �pk(�k) used in the updating scheme is indicated by the solidarrows.

6.1.7. Regularization analysis based on the trade-off and L-curves. It has been demonstrated thatthe parametric solution �pk(�) and the associated quantities in the response and parameter spaceshave certain characteristic features. For instance, the direction of the parameter path, �pk(�), hasbeen considered, where the change in its direction is often used to choose �k for each iteration.

Figure 13 shows the trade-off curves, L-curves and L-curve curvatures for the first threeiterations of the system specified in Section 6.1.1. The nominally equal parameters regularization isshown by the thick dashed lines and the minimum norm regularization is shown by thick solid lines.A sequence of linear problems is created, each of which is based on the composite cost functioncontaining the nominal and regularizing parts. The trade-off curves show the balance betweenthe two terms in the cost function for a range of the regularization parameter, �∈[0,∞).A characteristic feature of the trade-off curve, see [7, 40], is that it is monotonic for the whole rangeof the regularization parameter and the existence of the corner on this curve allows the selectionof the optimum value, �k ≡�corner≈�opt, leading to a specific updated parameter vector �pk(�k).The character of the curve before and after the corner highlights the influence of the originalphysical sub-problem represented by J1, e.g. JGN, or the influence of additional regularizingcriterion represented by J2, e.g. JReg. The trade-off curve expressed in the logarithmic scale isoften called the L-curve.

A further argument for choosing �k at the corner of the trade-off curve is based on the minimummodification of the original problem that gives a new, more stable problem description. Thevector �pk(�) initially moves along the direction corresponding to the least certain directionsin the parameter space, given by the {Sk,�zk} model. The first significant change in direction,due to the effect of the regularizing term, may be considered as the first acceptable, regular-ized model of the problem. This change in direction corresponds to the corner of the trade-offcurve.

Finding the corner of the trade-off curve may be applied at the first iteration for the minimumnorm regularization, as shown in Figure 13. However, there is no corner for the nominally equalparameters regularization. Returning to the parameter subspace plots of Figure 12 helps to under-stand the shape of these curves. For minimum norm regularization the sharp turn in parameterspace, particularly at the first iteration, produces an L-shaped trade-off curve with a distinctiveconvex corner. This behaviour arises because initially the parameter path causes significant changesto J2, and after the change of the direction the change in J1 is more significant. However, for thenominally equal parameters regularization there is no change in direction of the parameter path,and this produces a trade-off curve without a corner.

Another feature of the trade-off curves is the decreasing presence of a clearly identifiable cornerwith increasing iteration number. For the nominally equal parameters regularization, the trade-offcurve does not have a corner for any iteration, although the residual norm decreases, suggestingconvergence towards the solution. For the minimum norm regularization, the initial trade-off curvehas a clear corner, and as the iterations progress the corner becomes less obvious, although theresidual norm continues to decrease. This effect can be observed in different forms in Figures 4 and12. Figure 4 shows how the solution path for least norm regularization, {I,0}, varies between twolimiting lines. The first represents the non-regularized, LS solution via the projection of zm ontoAZ ,k , given by Sk(S

+k �zk), and the other line corresponds to the transformed gradient direction,

Ske(∇ J ), towards which, while decreasing in magnitude, �pk(�) will converge. The character of



this change will be defined by the relationship between these two lines and by the properties ofSk . This is demonstrated in a real context in Figure 12. The initial combination of previouslymentioned factors results in a path with a clear corner. With increasing iteration number, the mutualposition of these two lines and the character of Sk change so that the corner is less clear. Theparameter space plot of the third iteration in Figure 12 also supports these reasons behind thedecreasing observability of the corner. An extreme scenario corresponds to the situation where bothparticipating operators have characteristic circles instead of ellipses with no preferred directionleading to a straight line solution path.

For the L-curve, the convex corner corresponds to the maximum curvature, and this may be usedto find the optimal regularization parameter algorithmically [7]. This concept complements theanalysis using the trade-off curves. The L-curves are also shown in Figure 13. Both the trade-offcurves and L-curves have unique non-zero points for �=0 since the physical problem is over-determined. When �→∞, the trade-off curves approach 0, as the regularization conditions for bothcases have at least one exact solution with zero residual norm. For the first iteration, the L-curvefor the minimum norm regularization is a typical L-curve shape with the two concave corners,due to the logarithmic transformation, and a convex corner between them, due to the corner inthe trade-off curve. The corresponding trade-off curve has an analogous interpretation. The seconditeration, for the same regularization scenario, shows a similar situation although the convex corneris less obvious. By the third iteration the corner has effectively disappeared in both the L-curve andthe corresponding trade-off curve. This effect is described in the previous paragraph. No corner,for any iteration, is observed on L-curves for the nominally equal parameters regularization.

Figure 13 also shows the curvatures of the L-curves, �L(�), which are often used for the selectionof the maximum curvature points corresponding to the convex corners of the L-curves. Theseplots provide clear indications of both convex and concave corners of the L-curves. The maximumpositive curvature indicates the convex corner, which for the minimum norm regularization is moredifficult to detect for higher iterations. Another characteristic feature is the region close to the LSsolution, �→0 where the L-curve becomes almost vertical with a corresponding increase in thecurvature. The detail of this behaviour is provided for the first iteration of the minimum normregularization. The L-curve curvature for the nominally equal parameters regularization clearlyindicates the presence of a single concave corner.

6.2. An experimental updating study using a frame structure

The simulated case study illustrated the theoretical concepts within a fully controlled environment.The dimensions in the problem were adjusted to allow a visual representation of the regularizationprocesses during updating. This example will use these concepts on a real-world example withphysically based parameterization and subsequent regularized model updating.

6.2.1. Model description and problem definition. Figure 14 shows the steel frame structure inves-tigated in this example, which consists of four thin-walled tubes joined by four welded joints.The dimensions of the structure and the cross-section properties of the tubes are specified inthe figure. The boundary conditions are assumed to be free-free. A finite element model of thestructure is built using baseline Euler–Bernoulli beam elements with three DOFs at each node(two linear orthogonal displacements and one rotation around the axis perpendicular to the planeof the structure). The model consists of 32 beam elements with 32 nodes and 96 DOFs, shownin Figure 14. The joints are modelled by rigid nodal connections of the corresponding DOFs.



+X

+Y

W#1

W#4

W#2

W#3

500

mm

290 mm

1100 mm

60×20×2

40×20×2

Acc#2,+X

Acc#1,+Y

(a) (b)

Figure 14. The welded frame structure and its discretization.

Table II. The measured natural frequencies and the corresponding frequencies from the nominal model.

1 2 3 4 5 6 7 8 9

EMA (Hz) 60.57 126.53 147.05 175.89 280.76 320.56 360.70 437.72 566.52FEM (Hz) 98.15 135.95 182.65 184.93 293.68 405.33 510.47 561.67 608.82� f (%) 62.06 7.44 24.21 5.14 4.60 26.44 41.52 28.32 7.47

Young’s modulus was assumed to be E=2.1×1011 Pa and the material density, �=7850kgm−3.Although this represents a very crude model of the structure, the situation represents the practiceof using equivalent models, together with model updating, for joints. Furthermore, this case studywill primarily focus on the qualitative features of the solutions obtained throughout the iterativeprocess.

The measured data were obtained using experimental modal analysis [41]. The structure wastested in free–free conditions, by mounting the frame on soft springs, with the response measuredby two accelerometers whose locations are shown in Figure 14. The translational inertias of thetwo accelerometers were included in the model, thus breaking the physical symmetry of thestructure. The structure was excited by an instrumented impact hammer, and the excitation pointscorresponded to the finite element nodes shown in Figure 14. The direction of excitation waslimited to be perpendicular to the surface of the structure. In total, 64 experimental frequencyresponse functions were obtained in the frequency range 0–625Hz, which contained the first nineglobal modes of the structure. Table II gives the natural frequencies obtained from the experimentand the finite element model. The correlation between the modes is ensured by the standardMAC criterion. Titurus et al. [41] gave a more detailed description of the experiment, the modeland the potential sources of uncertainty. The experimental and analytical natural frequenciesare significantly different, indicating inappropriate modelling assumptions that were localizedprimarily to the joints. The objective was to improve the model of the joints using physically basedparameterization and model updating.

6.2.2. Parameterization of the model using GEs. The parameterization of the structure is based onthe concept of GEs [32], and this particular application is described in [41]. The parameterizationis shown in Figure 15, where the 32 elements were divided into four groups of GEs. The first



Figure 15. Parameterization of the welded frame structure.

GE (GE I) represents a joint substructure and consists of three beam elements located in theweld joint region. The same GE is used for all four welded joints, after necessary positionaltransformations. The remaining three GEs represent beam elements of different parts of the model.The GE parameterization used here focuses on stiffness modification, as the mass distribution islikely to be accurately modelled. Table II shows that the initial finite element model is much stifferthan the real structure. This difference is attributed to the incorrect modelling of the joints.

The basis for the parameterization in this case is the spectral decomposition of the initial stiffnessmatrices of the GEs. The GE I is a substructure with four nodes, 12 DOFs, three zero eigenvaluesand nine non-zero eigenvalues of the initial substructure stiffness matrix. The first two substructureeigenvalues are chosen for parameterization, and the substructure stiffness matrix is parameterizedas

KI(�I1,�I2)=

2∑j=1

�Ij�I0, j (�

I0, j )

T+9∑j=3

�I0, j�I0, j (�

I0, j )

T=KI1(�

I1,�

I2)+KI

2 (84)

where �Ij and �I0, j are the j th eigenvalue and eigenvector of the (initial) substructure stiffness

matrix, KI0�

I0, j =�I0, j�

I0, j , (�

I0,i )

T�I0, j =�i j and initially �Ij =�I0, j .

An analogous approach is used for the remaining three sets of GEs, where the initial GE has sixDOFs but only three non-zero eigenvalues. For these cases only the first substructure eigenvalueis used in the parameterization. The stiffness matrix is parameterized as follows:

K�(��1)=��

1��0,1(�

�0,1)

T+3∑j=2

��0, j�

�0, j (�

�0, j )

T=K�1(�

�1)+K�

2, �∈{II, III, IV} (85)

The resulting set of parameters is

pT=[p1, p2, p3, p4, p5]=[�I1,�I2,�II1 ,�III1 ,�IV1 ]∈R5, pi>0 (86)

A selection of substructure eigenvectors is shown in Figure 16. The first row in the figure showsthe first three stiffness eigenvectors of GE I, of which the first two are used for model updating.The second row shows all three stiffness eigenvectors of the three GEs, GE II, GE III and GE IV,of which only the first is selected for updating.

6.2.3. Sensitivity and updating analysis. The minimum norm regularization approach, �pk =0,with the fixed value of regularization parameter �=0.1, is used for model updating. The fiveparameters given in Equation (86) are updated using the first seven natural frequencies. Scaling is



. . .

...

Figure 16. First three eigenvectors of generic element GE I and all three eigenvectorsof generic elements GE II, III, IV.

used, based on the initial estimates of the parameters, p0, and measured responses, zm. The algo-rithm therefore uses fully normalized sensitivity matrices SN ,k at each iteration step. No additionalweighting factors are applied since no statistical information about the responses and parameterswas available. Correlation between the computational and experimental modes is ensured by theMAC criterion.

The equation for parameter updates with minimum norm regularization and scaling is based onthe equation [

SN ,k

�1/2D−1P

]�pN ,k =

[�zN ,k

0

], pk+1=pk +D−1

P �pN ,k (87)

SN ,k = DZSkD−1P , �pN ,k =DP�pk, �zN ,k =DZ�zk

DP = diag(1/|p0,i |), DZ =diag(1/|zm, j |), p0=[p0,i ], zm=[zm, j ](88)

Figure 17 shows the convergence of the response residuals based on the natural frequencies,parameter changes and MAC correlation during the model updating process. The first nine iterationsare shown, indicating that the convergence is achieved very quickly. The converged model givesa significant reduction in the response residuals, reducing the maximum residual from 62.06% toapproximately 2.00%. The largest parameter change is almost −70.0%, and corresponds to thewelded joint regions with the largest modelling errors. The correlation between the mode shapedata sets also improves significantly to over 95% for all seven modes. Furthermore, the correlationfor the eighth and ninth modes, not used in the updating, also improves suggesting physicallybased parameter modifications. Titurus et al. [41] updated this structure in a similar way, exceptthey used the regularization p≈p0. The two approaches give almost identical parameter updates,which further increases confidence in the results.

6.2.4. The analysis of the regularization procedure. The regularization process is now analysedin a similar way to the simulated study. Figure 18 shows the trade-off curves, L-curves andL-curve curvatures for the first three iterations for regularization parameters in the range�∈[1×10−12,1×102]. Figure 19 shows the summary plots for the first seven iterations, when



Figure 17. Model updating of the welded frame structure.

convergence is achieved. The high dimensions of the parameter and response spaces, characterizedby the sensitivity matrix with dimensions Sk ∈R7×5, make direct visualization of the parameterand response paths impossible. The regularization is identical to that used for the updating inSection 6.2.3, and is characterized by the regularization problem {D−1

P ,0}. As in the simulatedcase, the trade-off curves have their characteristic convex shape. The convex corner is initiallyclearly visible, but gradually becomes less observable by the third iteration. This effect wasanalysed in Section 6.1.7.

The nominal problem characterized by Sk ∈R7×5 is over-determined and thus the upper endof the trade-off curves and L-curves, �=0, corresponds to the LS solution with non-zero normsof the response residuals. Increasing the regularization parameter causes the corresponding normof the regularization term to decrease towards zero, while the response residual norm increases.This regularizing condition is similar to the case {I,0}, and provides trade-off curves and L-curveswith clearly identifiable convex corners. This is also shown by the L-curve curvature plots. Alsocurvature increases when �→0, as in the previous simulated case study.

Figure 19 summarizes the whole updating and regularization process. As the parameters convergethe range of the norm of the response residuals, J1(�pk(�)), approaches its lower bound, givingthe non-linear LS residual at convergence. For this over-determined problem, this indicates that thelinear prediction after convergence cannot be influenced significantly by the direction or magnitude



0 0.5 1 1.5 2

0

1

2

3

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3

[x,y]=[log(norm(Ax b)),log(norm(Cx d))]0 0.5 1 1.5 2

0

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[x,y]=[log10

α,κL]

0 5

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0 50 100

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200Iteration 1

0 7 14

0

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40

[x,y]=[norm( ),norm( )]

Iteration 2

0 2 40

10

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40Iteration 3

Figure 18. Summary of the first three updating iterations with local regularization sweeps.

of the parameter path, �pk(�), induced by the regularization applied. This situation can occur whenthe measured response, zm, is not located directly on the response surface, Z(p), either because ofmodel limitations or experimental errors. Consequently, the trade-off curve and L-curve becomealmost vertical on convergence, and so the curvature of the L-curve tends to zero, except for theregions close to �pLS,k as �→0.



0 0.5 1 1.5 2

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1

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3

0 2

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15

log10

(α)

κ L

0 20 40 60 800

50

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Iter.1

Iter.2

Iter.3Iter.4

Iter.7Iter.1Iter.2

Iter.2

Iter.7

Iter.1Iter.2

Iter.3Iter.4

Figure 19. Summary process of regularization during updating iterations.

7. CONCLUSION

The use of regularization in model updating to incorporate a priori information about the parametersis very common. The trade-off curve and L-curve are often used to determine the optimumvalue of the regularization parameter that balances the information and errors in the measureddata and the a priori information. This approach usually works very well, although for somecommon regularization terms, such as the case where parameters are nominally equal, the trade-offand L-curves do not take the expected forms. Similarly, as the parameters converge the shapeof these curves change and the expected convex corner disappears, which makes the choice ofregularization parameter difficult. This paper has explored the updating and regularization processfrom a geometric perspective, using the parameter and response spaces, and considering the pathsthe solution takes in these spaces as the regularization parameter changes. The improved insightis significant in the determination of why the L-curve takes the shape it does, and why convexcorners may not occur. The detailed understanding so obtained may be used to choose better valuesof the regularization parameter and therefore to improve the model updating process.

ACKNOWLEDGEMENTS

B. T. gratefully acknowledges the support of the Royal Society through a NATO Postdoctoral Fellowship.M. I. F. gratefully acknowledges the support of the Royal Society through a Royal Society-WolfsonResearch Merit Award.

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