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A variational approach for solving an inverse vibration problem

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Inverse Problems, Design and Optimization Symposium Rio de Janeiro, Brazil, 2004
A Variational Approach for Solving anInverse Vibration Problem
Leonardo D. Chiwiacowsky and Haroldo F. de Campos Velho Paolo Gasbarri
Laborat´ orio Associado de Computac¸˜ ao Aplicada - LAC Dipartimento di Ingegneria Aerospaziale e Astronautica Instituto Nacional de Pesquisas Espaciais - INPE Universit`a degli Studi di Roma, La SapienzaS˜ ao Jos´ e dos Campos, SP, Brazil Rome, Italy
[leodc,haroldo]@lac.inpe.br paolo.gasbarri@uniroma1.it
Abstract.
The present investigation is focused on thesolution of a dynamic inverse problem which is con-cerned with the assessment of damage in structures bymeans of measured vibration data. This inverse problemhas been presented as a optimization problem and hasbeen solved through the use of the iterative regularizationmethod, i.e. the Conjugate Gradient method. The resultshave been presented in a satisfactory form when a smallstructure with few degrees-of-freedom (DOF) is consid-ered, however when a higher DOF-structure is consid-ered, the simple application of the iterative regularizationmethod is not more satisfactory, being necessary the ap-plication of an additional methodology. To solve this dif-ﬁculty, in this paper a new approach, based on the use of the Genetic Algorithm (GA) method has been proposed.The GA method is used to generate a primary solutionwhich is employed as the initial guess for the conjugategradient method. The application of this new approachhas been showed that better results can be achieved, al-though the computational time for the application hereanalyzed could be increase. The damage estimation hasbeen evaluated using noiseless and noisy synthetic exper-imental data, and the reported results are concerned witha trussand a beam-like structures both modeled through aﬁnite element technique. Moreover, in order to take intoaccount the reduced set of experimental data to be em-ployed in the optimization algorithm, a Guyan reductiontechnique has been adopted on the ﬁnite element formu-lation.
INTRODUCTION
Considerable research and effort over the lastfew decades has taken place in the ﬁeld of systemidentiﬁcation problem, for different reasons. Oneof the most interesting applications involves themonitoring of structural integrity through the iden-tiﬁcation of damage. It is well known that damagemodiﬁes the dynamic response of a structure and,at the same time, that changes in its behavior maybe associated with the decay of the system’s me-chanical properties [1].The damage identiﬁcation problem is displayedas an inverse vibration problem, since the damageevaluationis achievedthroughthe determinationof the stiffness coefﬁcient variation, or the stiffnesscoefﬁcient by itself. The inverse problem solutionis generallyunstable,therefore,small perturbationsin the input data, like random errors inherent to themeasurements used in the analysis, can cause largeoscillations on the solution. In general the inverseproblem, i.e. the ill-posed problem, is presentedas a well-posed functional form, whose solution isobtained through an optimization procedure.Based on these considerations, various papershave examined the use of measured variations indynamic behavior to detect structural damage. Avariety of experimental, numerical and analyti-cal techniques has already been proposed to solvethe damage identiﬁcation problem, and have re-ceivednotable attentiondue to its practical applica-tions [2]. These methods are usually classiﬁed un-der several categories, such as frequency and timedomain methods, parametric and non-parametricmodels,deterministicandstochasticapproaches[3,4].Among the classical methods, recently the useof the conjugate gradient method with the adjointequation[5, 6], or VariationalApproach,which hasbeenusedsuccessfullyinthermalsciences, hasalsobeen presented as a satisfactory choice to face thedamage identiﬁcation problem. Some papers re-garding to the use of the Alifanov’s method in in-verse vibration problems can be found in the litera-ture, for instance, Huang [7, 8] has been estimatedthe time-dependent stiffness coefﬁcients consider-ing spring-mass systems with one and multiple de-grees of freedom. Also, Castello and Rochinha[9] have been identiﬁed the elastic and dampingparameters of a bar-like structure using the Ali-fanov’s method. On the other hand, among thenon-classical methods the stochastic methods, rep-resented by the GA method, represent a powerfulchoice to face non trivial optimization problems.
Inverse Problems, Design and Optimization Symposium Rio de Janeiro, Brazil, 2004
GAs are search algorithms based on the mechan-ics of nature selection and natural genetics [10],which are design to efﬁciently search large, non-linear, discrete and poorly search space where ex-pert knowledge is scarce or difﬁcult to model andwhere classical optimization techniques fail. Somepapersregardingto the use of the GA methodalonecan be found in the literature, for instance Bar-bosa and Borges [11] have been identiﬁed damagescenarios in a framed structure, while Mares andSurace [12] have been used the GA method for thesimultaneouslocationandquantiﬁcationofdamagein a truss and a beam structures.It has been noticed that when the system consid-ered presents a slightly high number of DOFs, theconjugate gradient method becomes sensible in re-lation to some parameters, such as the initial guess.By virtue of the above considerations, the simpleapplication of the iterative regularization methodcould be not more satisfactory, being necessary theapplication of an additional methodology. In thiswork, a hybrid approach is proposed to solve theinverse structural vibration problem for the dam-age identiﬁcation, which can be estimated throughthe determination of some stiffness parameters of the structure. The genetic algorithm is applied ini-tially in order to determine a better initial guess fora following application of the standard conjugategradient method. This hybrid approach has alreadybeen used for an inversedamage problemconsider-ing spring-mass systems and it has produced goodresults [13, 14] and in this work more complicatedstructures will be studied and also the effect of thenoisy in the experimentaldata has been considered.
THE INVERSE ANALYSIS
The inverse vibration problem of estimation of the stiffness coefﬁcients has been considered inthis work. The unknown stiffness coefﬁcients havebeen recoveredfrom the synthetic system displace-ment measurements of a forced dynamical systemwith
-DOF. The inverse analysis with the conju-gate gradient method involves the following steps[5, 6]:(i) the solution of direct problem;(ii) the solution of sensitivity problem;(iii) the solution of adjoint problem and the gradi-ent equation;(iv) the conjugate gradient method of minimiza-tion;(v) the stopping criteria;(vi) the solution algorithm.Next, a brief description of basic procedures in-volved in each of these steps is presented.
The Direct Problem
The
-DOF dumped systems considered in thiswork are presented in the Figs. 1-2 and the math-ematical formulation of this forced vibration sys-tems is given by
(1)with initial conditions
and
(2)Here
representsthesystemmassmatrix,
the time-dependentstiffness matrix,
the time-dependent damping matrix,
the external forcesvector, and
the displacements vector. Thereexists no analytical solution for Eq. (1) for any ar-bitrary functions of
,
, and
, obtainedthrough the ﬁnite element method. For this reason,the numerical solution with the
Newmark
method[15], will be applied to solve Eq. (1). The directproblem calculates the system displacement
,if initial conditions, system parameters
,
and
, and the time-dependent external forces
are known.
Figure1:
The three-bay truss structure considered in thiswork.
Figure2:
The
-DOF beam structure considered in thiswork.
The Sensitivity Problem
The problem involves
unknown time-dependent stiffness parameters, which constitutethe elements of the stiffness matrix
℄
, where
℄
and theparameters
,
represent the struc-tural stiffness parameters of the ﬁnite element; forinstance for a bar-like structure
, fora beam-like structure
, where
is
Inverse Problems, Design and Optimization Symposium Rio de Janeiro, Brazil, 2004
the Young’s module,
is the inertial moment,
is the cross section area and
is the length of the ﬁnite element. In order to derive the sensitiv-ity problemfor each unknownfunction
, eachunknown stiffness parameter should be perturbedat a time. Supposing that the
is perturbedby a small amount
Æ
, where the
Æ
is the Dirac-delta function and
, itresults in a small change in displacements by theamounts
. The sensitivity problem is ob-tained by replacing in the direct problem, Eqs. (1)-(2),
by
Æ
,
by
, andbysubtractingfromthe result-ing expression the srcinaldirect problem,and alsoby neglecting the second-order terms. Therefore,
sensitivity problems have been obtained, since
, i.e., a different sensitivity problemfor each perturbed stiffness parameter. The sensi-tivity problem is deﬁned by the following systemof differential equations
(3)
where
and with initial conditions
and
(4)
The Adjoint Problem and the Gradient Equa-tion
In general the inverse problem does not satisfythe requirements of existence and uniqueness, thenit must be solved as an optimization problem re-quiring that the unknown function
shouldminimize the functional vector
℄
deﬁned by
℄
℄
T
℄
(5)
where
is the ﬁnal time,
and
arethe computed and measured displacements at time
, respectively. The adjoint problem is developedby multiplying Eq. (1) by the
Lagrange
multi-plier vector
, integrating the resulting expres-sion over time domain and then adding this resultto the functional given by Eq. (5). The resultingexpression is given by
℄
℄
T
℄
T
(6)The variation
℄
of the functionalis ob-tained by perturbing
by
and
by
in Eq. (6), and subtracting from it the src-inal Eq. (6). Neglecting the second-order terms,the resulting expression is given by
℄
℄
T
T
(7)
When the second term of the right-hand side of this expression is integrated by parts and the nullinitial conditions of the sensitivity problem are em-ployed, the following adjoint problem is obtainedfor the determination of the
Lagrange
multipliervector
. Since the adjoint problem has no de-pendence on the perturbed stiffness (
), thesubscript
hasbeenneglectedandtheadjointprob-lem is deﬁned by the expression
℄
(8)
with ﬁnal conditions
and
(9)The adjoint problem is different from the standardinitial value problems because a ﬁnal time condi-tion at time
is speciﬁed instead of the classi-cal initial conditionat
. However,the problem(8) can be transformed to an initial value problemby introducing a new time variable:
.Then the standard technique of
Newmark
methodcan be applied for the solution of the transformedproblem.During the process for obtaining the adjointproblem, the following integral term was used
℄
T
(10)In this work the inverse vibration problem of stiff-ness estimation has been solved as a parameter es-timation problem where the stiffness parametershave been assumed constants; i.e.
const.,during the time. Therefore, the integral term leftcan be written as
℄
T
(11)By deﬁnition, the directional derivative of
℄
inthe direction of a vector
is given by [6]
℄
(12)where
is the gradient direction of the functional
. A comparison of Eqs. (11) and (12) reveals thatthe
component of the gradient direction,
, isgiven by
℄
T
(13)
Inverse Problems, Design and Optimization Symposium Rio de Janeiro, Brazil, 2004
where
refers to the
perturbed stiffnessmatrix, i.e.
℄
.
The Conjugate Gradient Method of Minimiza-tion
The iterative procedure based on the conjugategradient method is used for the estimation of theunknown stiffness parameters
given in the form[5, 6]:
(14)where
is the step size vector and
is the di-rection of descent vector at the step
deﬁned as[5]
with
(15)where
is the conjugate coefﬁcient vector. Notethat Eq. (14) is a iterative procedure for whichthe stopping criteria will be discussed in next sec-tion. It should be noticed that the special case
, for any
, corresponds to the steepest de-scent method. Different deﬁnitions of the conju-gate coefﬁcient
are reported in the literature [5,16]. In the present work the conjugate coefﬁcientvector has been adopted as
℄
℄
(16)The step size vector
, appearing in Eq. (14), isdetermined by minimizing the functional
℄
given by Eq. (5) with respect
, i.e.
℄
℄
(17)
By performing a
Taylor
-series expansion of the in-tegrand of Eq. (17) the value of
for the mini-mum can be evaluated analiticaly:
℄
T
℄
℄
T
℄
(18)
The DiscrepancyPrinciplefortheStoppingCri-teria
In the absence of measurement errors, due to theexperimental devices, one can use the customarystopping criteria
℄
(19)where
℄
is deﬁned by Eq. (5) and
is asmall speciﬁed number. However, in practical ap-plications, measurement errors are always present;therefore the
Discrepancy Principle
[17, 5] as de-scribed below should be used to establish the stop-ping criteria.Let the standard deviation
of the measurementerrors be the same for all sensors and measure-ments, that is,
(20)Introducing this result into Eq. (5), we obtain
(21)Then the discrepancyprinciplefor the stoppingcri-teria is taken as
℄
(22)
The Solution Algorithm
The standard computational procedure of theconjugate gradient method is summarized in thefollowing algorithm:
Step 1:
Choose an initial guess
.
Step 2:
Solve the direct problem, Eqs. (1)-(2), toobtain
.
Step 3:
Solve the adjoint problem, Eqs. (8)-(9), toobtain the
Lagrange
multiplier vector
.
Step 4:
Knowing
, computethe gradientfunc-tion vector
from Eq. (13).
Step 5:
Compute the conjugate coefﬁcient vector
from Eq. (16).
Step 6:
Compute the direction of descent vector
from Eq. (15).
Step 7:
Setting
[6], solve the sensitiv-ity problem, Eqs. (3)-(4), to obtain
.
Step 8:
Compute the step size vector
fromEq. (18).
Step 9:
Compute
from Eq. (14).
Step 10:
Test if the stopping criteria, Eq. (22), issatisﬁed. If not, go to step 2.As mentionedbefore, the inverse vibrationprob-lem of damage identiﬁcation, considered in thiswork, have already been solved through the use of the iterative regularization method, i.e. the Con- jugate Gradient method [7, 8, 9]. The resultshave been presented in a satisfactory form whenbothlumped-parameterandbar-likeforcedsystems
Inverse Problems, Design and Optimization Symposium Rio de Janeiro, Brazil, 2004
have been considered using a small number of un-known stiffness parameters. However, it has beenobserved by the authors that when a slightly highernumber of unknown parameters is considered, theapplication of the iterative regularization method,in the standard form is not more satisfactory, beingnecessary the application of an additional method-ology [13, 14].It has also been noticed that the initial guesschoice becomes more decisive when more stiffnessparameters are sought to be estimated. For avoid-ing this difﬁculty, it has been proposed a new ap-proach where the GA method is used to generatea primary solution which is employed as the ini-tial guess for the conjugate gradient method. Thisnew approach could be inserted in the above pro-cedure as the new
Step 1
. The stochastic approachhas alreadybeenused in an inverseheat conductionproblem,as uniquemethodology,andhas producedgood results [3], and also this proposed hybrid ap-proach has already been used in an inverse vibra-tion problem where a 10-DOF lumped parametersystem has been considered [13].
The Stochastic Method – Genetic Algorithm
Genetic algorithms are essentially optimizationalgorithms whose solutions evolve somehow fromthe science of genetics and the processes of natu-ral selection - the Darwinian principle. They differfrom more conventional optimization techniquessince they work on whole populations of encodedsolutions, and each possible solution is encoded asa gene.The most important phases in standard GAsare selection (competition), reproduction,mutationand ﬁtness evaluation. Selection is an operationused to decide which individuals to use for re-production and mutation in order to produce newsearchpoints. Reproductionorcrossoveris thepro-cess by which the genetic material from two parentindividuals is combined to obtain one or more off-springs. Mutation is normally applied to one indi-vidualin orderto producea new versionofit wheresome of the srcinal genetic material has been ran-domly changed. Fitness evaluation is the step inwhich the quality of an individual is assessed [10].The application of GA method to solve the prob-lem of damage identiﬁcation is also a minimiza-tion problem, as well as the gradient conjugatemethod. The same functional form, or ﬁtness func-tion, given by Eq. (5), is employed
℄
℄
T
℄
where
is the experimental system displace-ment vector and
is the system displacementvector obtained obtained by Eqs. (1)-(2) using up-dating values for the stiffness matrix.In this GA implementation, the algorithm oper-ates on a ﬁxed-sized population which is randomlygeneratedinitially. The membersof this populationare ﬁxed-lengthand real-valued strings that encodethe variable which the algorithm is trying to opti-mize (
). Next, the evolutionary operators em-ployed are presented: Tournament Selection [18],Geometrical Crossover [19], Non-uniform Muta-tion [19], Epidemical Operator [3].
REDUCED MODEL
A properly comparison of the real structuralmodel and the ﬁnite element model is possiblewhen the completeness of the experimental datais available. However, the number of measuredDOFs use to be smaller than the number of DOFswhich describe the complete ﬁnite element model.Usually, this problem is faced applying a reductiontechnique: the srcinal system is reduced to a sys-tem whose new dimension is equal to the numberof measured DOFs, identiﬁed as principal DOFs.In this worktwodifferentforcedsystems havebeenconsidered: a 3-bay truss structure and a 20-DOFbeam structure. For the second one the Guyan Re-duction technique [20] have been applied in orderto reduce the complete ﬁnite element model, of di-mension
, toareducedmodelwhichpresentsade-pendence on the measurable vertical displacementsonly. In the Guyan reduction method the relation-ship between the principal (
) and secondary (
)DOFs is establishedusingthe static analysis,whereit is assumed that no external loads are applied tothe secondary DOFs. The relationship between thecomplete and the principal measured DOFs is ex-pressed as
(23)where
is the
Guyan reduction matrix,where the relevant expression is not reported herefor the sake of brevity. Now, the problems identi-ﬁed by the Eqs. (1)-(2), (3)-(4) and (8)-(9) havebeen reduced and show an explicit dependencyonly by the measured DOFs (the vertical displace-ments of the beam). The new reduced system ma-trices
,
and
are computed as following
T
T
(24)
T
RESULTS AND DISCUSSIONS
In this work it has been presented an alternativehybrid approach to solve the damage identiﬁcationproblem involving the estimation of the unknownstiffness parameters. In order to evaluate the ca-pability of this new approach two different struc-tures have been considered, both modeled through

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