A dual adaptive filtering approach for nonlinear finite element model updating accounting for modeling uncertainty

Mechanical Systems and Signal Processing 115 (2019) 782–800

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

A dual adaptive filtering approach for nonlinear finite element model updating accounting for modeling uncertainty Rodrigo Astroza a,⇑, Andrés Alessandri a, Joel P. Conte b a b

Facultad de Ingeniería y Ciencias Aplicadas, Universidad de Los Andes, Santiago, Chile Department of Structural Engineering, University of California San Diego, La Jolla, CA, USA

a r t i c l e

i n f o

Article history: Received 25 December 2017 Received in revised form 23 April 2018 Accepted 11 June 2018

Keywords: FE model updating Modeling uncertainty Bayesian method Parameter estimation Nonlinear FE model Structural health monitoring

a b s t r a c t This paper proposes a novel approach to deal with modeling uncertainty when updating mechanics-based nonlinear finite element (FE) models. In this framework, a dual adaptive filtering approach is adopted, where the Unscented Kalman filter (UKF) is used to estimate the unknown parameters of the nonlinear FE model and a linear Kalman filter (KF) is employed to estimate the diagonal terms of the covariance matrix of the simulation error vector based on a covariance-matching technique. Numerically simulated response data of a two-dimensional three-story three-bay steel frame structure with eight unknown material model parameters subjected to unidirectional horizontal seismic excitation is used to illustrate and validate the proposed methodology. Geometry, inertia properties, gravity loads, and damping properties are considered as sources of modeling uncertainty and different levels and combinations of them are analyzed. The results of the validation studies show that the proposed approach significantly outperforms the parameter-only estimation approach widely investigated and used in the literature. Thus, a more robust and comprehensive identification of structural damage is achieved when using the proposed approach. A different input motion is then considered to verify the prediction capabilities of the proposed methodology by using the FE model updated by the parameter estimation results obtained. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Calibration of models of structural systems is an important topic with application in different problems in the fields of structural engineering and engineering mechanics. Improving the predictive capabilities of models, providing a tool for damage identification, and verifying and improving modeling techniques are some significant problems that are assisted by model calibration. Although most efforts in model calibration have been dedicated to linear systems, as linear finite element (FE) model updating represents an important field of research [1], calibration of nonlinear models has also attracted significant attention from the community. The work by Distefano and co-workers [2–4] represents, to the authors’ knowledge, the first attempt to calibrate nonlinear models under dynamic loading. They employed numerically simulated data [2,3] and experimental data from a shake table test [4] to calibrate single-degree-of-freedom (DOF) and 3-DOFs shear-type building models with restoring force versus relative displacement at the story level characterized by cubic polynomials. Yun and Shinozuka [5] used the Extended Kalman filter (EKF) and the iterated linear filter-smoother to identify, using simulated data, ⇑ Corresponding author. E-mail address: [email protected] (R. Astroza). https://doi.org/10.1016/j.ymssp.2018.06.014 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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the hydrodynamic matrix of a 2-DOF linear system subjected to wave forces. Hoshiya and Saito [6] proposed a weighted global iteration method with the EKF (EK-WGI) to solve identification problems of structural systems under seismic excitation. They used numerical examples of 2-DOFs and 3-DOFs linear systems and a single DOF system with bi-linear hysteretic response to verify the proposed approach. Bittani et al. [7] also used the EKF to estimate parameters of a twodimensional (2D) frame with localized nonlinearities at the column ends characterized by elasto-plastic cross-sectional behavior. In [8], Hoshiya and Sutoh applied a Kalman filter-weighted local iteration procedure (EK-WLI) to FE models of geotechnical systems. They considered numerical examples of homogeneous and nonhomogeneous stochastic fields in plane strain problems of geomechanics. The FE method, originally introduced in the 1950’s [9,10], is still an active area of research that has made significant progresses in the modeling and simulation of large and complex structures by developing high-fidelity nonlinear models. From experimental-analytical correlation studies, these models have proven to predict with reasonable accuracy the response of civil structures when properly calibrated (e.g., [11,12]). All the developments in the area of FE modeling and simulation can benefit from the field of model calibration, which provides more insights into the modeling aspects of complicated nonlinear phenomena and also offers a powerful tool for damage identification of structures. In recent years, a number of researchers have developed methods to calibrate mechanics-based nonlinear FE models of civil structures. Nasrellah and Manohar [13] proposed to combine the FE method with particle filtering to identify unknown model parameters. They verified the approach by estimating three parameters of a rubber sheet model, the stiffness parameters of a simple supported beam tested in the laboratory, and the modulus of elasticity and spring parameters of a 2D FE model of an arched bridge using field test data. In [14], the authors used numerical data to calibrate the nonlinear FE model of a reinforced concrete shear wall. Astroza et al. [15] and Ebrahimian et al. [16] used the Unscented Kalman filter (UKF) and EKF, respectively, to calibrate mechanics-based nonlinear FE model of 2D and three-dimensional (3D) frame structures. Astroza et al. [17] proposed a hybrid method combining simulated annealing with the UKF for the identification of parameters of linear and nonlinear FE models. In [18], the adaptive quadratic sum-square error with unknown inputs was utilized to update a FE model with lumped plasticity of a reduced-scale frame structure tested in laboratory conditions. In [19], the authors proposed a methodology to jointly estimate the parameters and input excitation of mechanics-based nonlinear FE models subjected to earthquake base excitation. Eftekhar et al. [20] proposed a dual Kalman filter approach for joint input-state estimation of linear systems. The approach proposed in [15] was extended in [21] to reduce the computational cost of the updating procedure, allowing its use in hybrid simulation applications. Since every FE model is an idealized representation of the structure to be modeled, the selected classes of models never contain the real structure [22,23]. The sources of uncertainty are usually categorized into model parameter uncertainty, model structure uncertainty (referred to as modeling uncertainty hereafter), and measurement error/noise. Different methods and comprehensive studies related to model parameter uncertainty and measurement errors are available in the literature (e.g., [15,24]). However, although it has been recognized that modeling uncertainty is the most critical aspect when updating FE models [25–27], only limited studies of its effects have been performed and only for linear structural models (e.g., [27–30]). It is noted that modeling uncertainty can be due to, for example, the presence of nonstructural components, inappropriate modeling of energy dissipation mechanisms and boundary conditions, etc. For mechanics-based nonlinear FE models, Astroza and Alessandri [31] carried out parametric analyses to investigate the effects of modeling uncertainty in parameter estimation results and also in measured (observed) and unobserved response quantities predicted using stateof-the-art distributed plasticity FE models of building structures updated using the parameter estimation results. They found that in the presence of modeling uncertainty, a good match between the measured and FE-predicted response can be achieved after the updating process, but unobserved response quantities (at both the global and local levels) can be subjected to significant errors because the estimated parameters are biased (in agreement with the results reported in [27]) due to modeling uncertainty, even achieving non-physical parameter values. Recognizing modeling uncertainty as one of the main challenges in FE model updating, this paper proposes a novel approach based on adaptive filtering to account for this source of uncertainty when updating mechanics-based nonlinear FE models. In the proposed approach, the diagonal terms of the covariance matrix of the simulation error vector are considered time variant and estimated together with the model parameters of the nonlinear FE model. Similar approaches for estimating time-invariant variances of state and measurement noises have been proposed in the literature for linear (e.g., [32–35]) and nonlinear systems (e.g., [36–40]). In this paper, the covariance matrix of the simulation error vector is considered time variant because the modeling uncertainty may vary in time and a covariance-matching approach is used to estimate its diagonal terms, similarly to the idea presented in [33] and [37]. 2. Problem statement The dynamic response of a structural system can be modeled using a discrete-time equation of motion of a nonlinear FE model. This allows to predict the dynamic behavior of the structure and also to detect, localize, and quantify the potential damage it can suffer during an extreme event (e.g., earthquake, hurricane). The equation of dynamic equilibrium of a nonlinear FE model (which is constructed based on the best estimation of the actual properties of the structure) at time t kþ1 ¼ ðk þ 1ÞDt, where k ¼ 0 ; 1 ; . . . and Dt is the time step, is usually expressed as

€ kþ1 ðpÞ þ CðpÞ q_ kþ1 ðpÞ þ rkþ1 ðqkþ1 ðpÞ; pÞ ¼ f kþ1 ðpÞ ¼ MðpÞLðpÞu € kþ1 MðpÞ q

ð1Þ

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where p 2 Rnp 1 = vector of model parameters with np = number of model parameters, M 2 Rnn = mass matrix, _ q € 2 Rn1 = nodal displacement, velocity, and acceleration vectors, rðqðpÞ; pÞ 2 Rn1 = C 2 Rnn = damping matrix, q; q; history-dependent internal resisting force vector, f 2 Rn1 = external dynamic force vector which takes the form € for the case of rigid base seismic excitation, L 2 Rnnu = influence matrix of the base excitation, n = number of DOFs f ¼ MLu € 2 Rnu 1 = input ground acceleration vector with nu = number of acceleration components of the base of the FE model, and u excitation. It is noted that the case of earthquake excitation is considered in this paper because it corresponds to a common situation of a nonlinear structural response to a known input. However, the proposed approach can be readily applied or extended to other situations in which a structural system is subjected to known external forces, including static, quasistatic (e.g., lab. experiments), time-dependent (e.g., creep, shrinkage, corrosion), and dynamic loading. The model parameters are defined using information available from blueprints, material specifications, loading conditions, etc. Because of both aleatory and epistemic uncertainties [41] associated with that information, an accurate estimate of the model parameters is not possible for real-world structures. Data recorded from the structure of interest may help to reduce the epistemic uncertainty by calibrating/updating some model parameters characterizing the FE model. For notational convenience, in this
T paper the model parameter vector is denoted as p ¼ hT uT , where h 2 Rnh 1 = vector of unknown time-invariant model parameters (referred to as unknown model parameters hereafter) to be estimated, and u 2 Rnu 1 = vector of model parameters associated with the modeling uncertainty (referred to as modeling uncertainty parameters hereafter). It is worth mentioning that in real-world conditions, modeling uncertainty is always present since the FE model to be updated is never a perfect representation of the actual structure of interest. Effects of the modeling uncertainty on the structural response prediction, especially the unobserved global and local response quantities, may be significant [31]. With the above definition of the model parameter vector, Eq. (1) can be rewritten as

€ kþ1 ðh; uÞ þ Cðh; uÞ q_ kþ1 ðh; uÞ þ rkþ1 ðqkþ1 ðh; uÞ; h; uÞ ¼ f kþ1 ¼ Mðh; uÞ Lðh; uÞ u € kþ1 Mðh; uÞ q

ð2Þ

b kþ1 2 Rny 1 , can be viewed as a nonlinear According to Eq. (2), the FE model-predicted structural response at time tkþ1 , y function of the model parameters (h and u), the time-history of earthquake ground acceleration   € 1:kþ1 ¼ ½u € T1 ; . . . ; u € Tkþ1 T 2 R½nu ðkþ1Þ  1 , and the displacement and velocity initial conditions, q0 and q_ 0 , as u

€ 1:kþ1 ; q0 ; q_ 0 Þ ^kþ1 ¼ hkþ1 ðh; u; u y

ð3Þ

where hkþ1 ðÞ = nonlinear response function of the nonlinear FE model. The actual (measured) dynamic response of the structure, ykþ1 2 Rny 1 can be recorded by using a heterogeneous array of sensors (e.g., accelerometers, strain-gauges, LVDT’s, long-gauge fiber optic sensors, GPS, etc.) and is related to the FE modelpredicted dynamic response, at time tkþ1 , as

vkþ1 ¼ ykþ1  y^kþ1

ð4Þ

where v kþ1 2 Rny 1 = simulation error vector assumed to be Gaussian white with zero mean and covariance matrix Rkþ1 2 Rny  ny , i.e., v kþ1  Nð0; Rkþ1 Þ. The simulation error vector accounts for measurement noise/errors and modeling errors, such as model parameter uncertainty and modeling uncertainty itself (due to non-physical modeling assumptions and/or physical features not considered in the FE model) [30,42]. Following the work by Astroza and Alessandri [31], real-world conditions will be mimicked by defining a nonlinear FE model class M0 to numerically simulate the true dynamic response of the structure of interest, ytrue . Measurement noise is then added to ytrue in order to define the recorded response, y. The FE ^ , will be calibrated by updating a nonlinear FE model belonging to a model class Mj different model-predicted response, y than M0 , i.e., Mj –M0 . Different scenarios of modeling uncertainty will be represented by defining different model classes Mj (j = 1, 2, . . .). To minimize the discrepancy between the measured and FE-predicted structural responses, the unknown model parameters can be estimated using a Bayesian approach for parameter estimation. Here, the uncertainty in h is modeled by a probability density function, representing the unknown parameter vector as a random process vector, whose evolution in time is governed by a random walk. Then, the following state-space model can be defined

hkþ1 ¼ hk þ wk

€ 1:kþ1 Þ þ v kþ1 ykþ1 ¼ hkþ1 ðhkþ1 ; u; u

ð5Þ

where the initial conditions have been omitted for notational convenience and wk 2 Rnh 1 denotes the process noise assumed to be uncorrelated with v kþ1 , and Gaussian white with zero mean and covariance matrix Q k 2 Rnh nh , i.e., wk  Nð0; Q k Þ. The following sections describe the usual approach for nonlinear FE model updating, through parameter-only estimation (Section 2.1), and also present a new approach for FE model updating accounting for modeling uncertainty in the estimation phase (Section 2.2).

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2.1. Unscented Kalman filter (UKF) for nonlinear FE model updating through parameter-only estimation At least the first two statistical moments of h in Eq. (5) can be estimated by employing a Bayesian filtering method, given € ) and the measured response of the structure (y). Previous research has the records of the seismic ground acceleration (u shown that the UKF slightly outperforms other Kalman-based filters [16,43] at a comparable computational cost. On the other hand, the use of particle filters for large and complex civil structures with nonlinear inelastic behavior is prohibitive because of the unaffordable computational resources required. Therefore, the UKF has been commonly selected as a tool to ^ hh , respectively) of the unknown parameters of the nonestimate the mean vector and covariance matrix (denoted as ^ h and P linear FE model [15,44]. The UKF uses the unscented transform (UT) to deterministically choose a set of samples of the unknown parameters h, called sigma-points (SPs) and denoted by 0 herein, in order to avoid analytical linearization of the nonlinear state-space model defined in Eq. (5) as required by the EKF. If the scaled UT [45] is used in the implementation of the UKF, ð2nh þ 1Þ FE models (each one corresponding to a SP) must be run at each estimation step. This parameter-only estimation approach (referred to as parameter-only approach hereafter) has proven robust when uncertainty in the model is negligible and when measurement noises are present, but its convergence properties may be significantly affected when moderate to high levels of modeling uncertainty are present [31]. ^ hh = initial estimates Fig. 1 depicts the parameter-only approach for FE model updating using the UKF. In this figure, ^ h0j0 ; P 0j0

^ hh = prior estimates of the mean vector and covariance matrix of h at t kþ1 , of the mean and covariance matrix of h; ^ hkþ1jk ; P kþ1jk ðiÞ ^kþ1jk = pre= vector of FE-predicted response quantities at t kþ1 corresponding to the SP 0 kþ1jk ; y yy hy ^ ^ ; P = predicted response covariance and cross-covariance matrix estimates; dicted response vector at time t kþ1 ; P kþ1jk kþ1jk

respectively; y

ðiÞ kþ1

2 Rny 

1

W m , W c = weighting coefficients of the SPs for the mean and covariance matrix estimates, respectively; Kkþ1 = Kalman gain ^ hh matrix; and ^ hkþ1jkþ1 ; P kþ1jkþ1 = posterior estimates of the mean vector and covariance matrix of h at t kþ1 . More details about Bayesian model parameter estimation of nonlinear FE model can be found in [15] and [43]. 2.2. Dual adaptive Kalman filter for nonlinear FE model updating accounting for modeling uncertainty When non-negligible levels of modeling uncertainty are present, a parameter-only estimation approach is affected through compensation effects, i.e., the modeling uncertainty manifests itself through estimation bias in the unknown model parameters, when minimizing the misfit between the measured and FE model-predicted responses. These compensation effects arise from not properly accounting for model uncertainties, and although a good agreement for measured (observed) response quantities can be attained, unphysical values for unknown model parameters and a very bad agreement for

Fig. 1. Framework for parameter estimation of nonlinear FE models using the UKF (adapted from [43]).

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unobserved response quantities (at global and local levels) are usually obtained [31]. To remedy this problem, a novel dual adaptive filtering approach is proposed in this paper to appropriately calibrate the diagonal terms of the covariance matrix of the simulation error vector, Rkþ1 , in addition to calibrating h, thus accounting for modeling uncertainty in the estimation phase. There are different adaptive filtering approaches (e.g., [32]), and in this paper a covariance-matching technique is adopted because of its simple implementation, low computational cost, and its successful previous applications [32,33,37]. If the simulation errors of the various response measurements are assumed uncorrelated, Rkþ1 is a diagonal matrix with the variances of the simulation errors as diagonal entries, i.e., Rkþ1 ¼ diagðrkþ1 Þ where rkþ1 denotes a vector containing the variances of the simulation errors at tkþ1 . By employing a dual adaptive filtering approach, at every time step an UKF can be used as master filter (MF) to estimate the unknown model parameter vector h and a linear Kalman filter (KF) can be used as slave filter (SF) to estimate the diagonal entries of the covariance matrix of the simulation error (rkþ1 ). Fig. 2 shows the block diagram of the proposed dual adaptive filtering approach for FE model updating, while Fig. 3 depicts the pseudo-code of the proposed approach. In these figures, the highlighted section corresponds to the additional calculations required to incorporate the estimation of the diagonal entries of Rkþ1 when updating the nonlinear FE model. Here, T and U are the timeinvariant covariance matrices of the process and measurement noises, respectively, of the state-space model corresponding to the SF, both assumed Gaussian white with zero mean. It is noted that this approach of adaptive filtering corresponds to a covariance matching technique, in which the goal of the SF is to make the innovations (v kþ1jk ), which can be employed to provide a measure of the optimality of the filter, compatible with their expected covariance matrix [32,33]. If the diagonal terms of the actual covariance matrix of v kþ1 j k are larger than those of the covariance matrix obtained by the UKF (lkþ1 in Fig. 2), Rkþ1 should be increased to avoid divergence and consequently the Kalman gain (Kkþ1 ) decreases, thus increasing ^ hh (see Fig. 3). This means that at time step ðk þ 1Þ the filter gives less emphasis to the measurements, implying that P kþ1jkþ1

only data for which the FE model can accurately predict the observed response are used in the filter, i.e., the proposed dual adaptive filtering approach is able to account for modeling uncertainty. In addition, larger estimates of the diagonal entries of ^ hh R imply smaller Kalman gains, and consequently, the covariance matrix of the model parameters (P ) decreases less. As kþ1jkþ1

discussed later in the application example, this prevents convergence of model parameter estimates to unphysical values when modeling uncertainty is significant and maintains the estimation uncertainty of the parameters about which limited information is contained in the measurements. 3. Numerical application: 2D SAC-LA3 building A nonlinear FE model class M0 will be used to represent a realistic building and to numerically simulate seismic response data, while different modeling uncertainty scenarios (i.e., considering modeling uncertainties in the representation of the structure) will be introduced by defining nonlinear FE models belonging to different model classes Mj –M0 . Different

Fig. 2. Block diagram of the proposed dual adaptive filtering approach.

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Fig. 3. Pseudo-code of the proposed dual adaptive filtering approach for FE model updating.

nonlinear FE models belonging to model classes Mj (j = 1, 2, . . .) will be updated by estimating their unknown model parameters h. In each case, both the parameter-only estimation approach and the dual adaptive filtering estimation approach will be employed to update the nonlinear FE model. A performance comparison of both approaches and a posterior verification study (i.e., using an input motion different from the one used to update the FE model) are presented and discussed. 3.1. Building structure and FE model The steel moment-resisting frame building known as SAC-LA3 [46] is a three-story structure designed as a standard office building for Los Angeles, California, according to the 1994 UBC [47]. Soil conditions were assumed as stiff soil and the seismic masses and loads on beams were computed from the design dead and live loads reported in FEMA 355C [48]. In this study, the exterior north–south frame is modeled and subjected to horizontal seismic excitation. Columns are materialized with A572 steel, while beams are made of A36 steel. The geometry of the structure and element cross-sections are described in Fig. 4.

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Fig. 4. (a) Model of the SAC-LA3 steel moment resisting frame building; (b) Modified Giuffre-Menegotto-Pinto steel constitutive model.

The building is modeled in the software framework OpenSees [49] as a two-dimensional frame using mechanics-based nonlinear force-based fiber-section Bernoulli-Euler beam-column elements. A single force-based element is used for each beam and column, with seven and six integration points along its length, respectively. The joints are assumed rigid and rigid-end zones are considered at the ends of columns and beams. The sources of energy dissipation beyond the material hysteretic behavior are modeled using mass- and tangent stiffness1-proportional Rayleigh damping, assuming a critical damping ratio of 2% for the first two initial2 modes of periods T 1 ¼ 1:06 ½s and T 2 ¼ 0:35 ½s. In terms of section fiber discretization, the webs of column elements are discretized into 6  1 fibers across their length and width, respectively, and each column flange discretized as a single fiber. The webs of beams are discretized into 16  1, 14  1 and 11  1 fibers across their length and width, at the second, third and roof level, respectively, and each beam flange is discretized as a single fiber. Further details of the structure and nonlinear FE model can be found in [15]. The nonlinear uniaxial stress–strain behavior of the steel longitudinal fibers is modeled through the modified GiuffreMenegotto-Pinto (MGMP) steel constitutive model [50] (referred to as Steel02 in OpenSees). The MGMP material model characterizes the cyclic inelastic behavior of the steel fibers and is defined by ten time-invariant parameters that control the shape of the elastic and plastic curves, the shape of the transition between these curves, and the isotropic hardening. Four of these parameters will be considered as primary unknown model parameters and will be estimated. These four parameters consist of: the steel modulus of elasticity ðEs Þ, the initial yield stress ðf y Þ, a parameter defining the curvature of the elastic to plastic transition during the first cycle ðR0 Þ and the strain-hardening ratio ðbÞ. The remaining six (secondary) parameters are considered known and fixed in this application example. Since beams and columns members are made of different types of steel, two independent sets of primary parameters are considered to define the MGMP models of the steel fibers of the beams and columns, respectively. Therefore, the vector of unknown model parameters is defined as

2 6 h ¼ 4h

1 2

h

Ecol s

Ebeam s

col

fy

beam fy

Rcol 0 Rbeam 0

b

col

iT

beam

3 7 81 iT 52

b

Based on the tangent stiffness matrix at the last converged step of the analysis. Obtained using the structure tangent stiffness matrix after application of the gravity loads and before the seismic excitation.

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The modeling uncertainty parameter vector, u, includes geometry variables, mass properties (nodal masses), gravity loads, and damping properties, and is used to define different model classes. A given set of modeling uncertainty parameter values will be used to numerically simulate the seismic response data and then different sets of u parameter values will be defined to represent different scenarios of modeling uncertainty, thus mimicking real world conditions where the nonlinear FE model is never a perfect representation of the actual structure, i.e., the real structure does not belong to the nonlinear FE model class used to represent it. 3.2. Cases of modeling uncertainty Previous research has shown that modeling uncertainty may cause adverse effects in the parameter-only estimation approach for nonlinear FE model updating, compensating for the uncertainty in the model with biased and even physically unrealistic estimates of the unknown model parameters [31]. Although time histories of measured response quantities are properly matched after the updating process, true unobserved response quantities (at global and local levels) and the corresponding predicted responses from the updated FE model can be in poor agreement [31]. In this paper, specific cases representing realistic low to high levels of modeling uncertainty are considered to investigate and compare the performance of the proposed approach (dual adaptive filtering) against the usual approach (parameteronly estimation) for nonlinear FE model updating. The selected cases are taken from [31] as those that presented large discrepancies between the true observed responses and the corresponding FE predictions based on the updated FE models when using the parameter-only estimation approach (i.e., using the final estimates of the unknown model parameters). As previously mentioned, parameters defining the geometry, inertia properties, gravity loads, and damping properties of the structure are considered as sources of modeling uncertainty. Different cases of modeling uncertainty obtained by introducing errors in the modeling uncertainty parameters u define different FE model classes Mj (j = 1, 2, . . .) to be used to estimate the unknown model parameters h. The modeling uncertainty parameter vector considered in this study is defined below and in Table 1 (see Fig. 4).

2 6 6 ½ NM11 u¼6 6 4

½ H0 H1 H2 H3 V1 V2 V3  NM12

NM13

NM14

NM21

NM 22

NM23

NM24

½ DGL1 DGL2 DGL3  ½ aM

3

T

NM 31

NM32

T

NM33

T 7 NM34  7 72241 7 5

bK T

Table 1 describes the modeling uncertainty parameters and the magnitudes of the error considered for each of them, while Table 2 reports the modeling uncertainty cases considered to define different model classes Mj (j = 1, 2, . . ., 28). Parameters Ha (a ¼ 0; 1; 2; 3) and Vb (b ¼ 1; 2; 3) define the locations of the vertical and horizontal axes of the FE model, i.e., axes of the columns and beams of the building; the term NM i k denotes the nodal mass at the joint located at the i  th floor and k  th column, where i ¼ 1; 2; 3 and k ¼ 1; 2; 3; 4; DGLc denotes the distributed gravity load on the beams at levels c ¼ 1; 2; 3; and aM and bK are the mass and stiffness proportional coefficients defining the Rayleigh damping, respectively. It is noted that the actual structure (or ‘‘true” model) does not consider error in u. In practice, errors related to the geometry of a structure are not expected to be large and here they are defined as a percentage (%) of the bay-width and story-height, for the horizontal and vertical coordinates of the column and beam axes, respectively (e.g., if a coefficient of 0.97 is considered for V3, the modified roof coordinate is ðV3  0:03  story heightÞ). The coefficients associated to the nodal masses evenly modify all the nodal masses along a given level (e.g., NM2 is intended to be applied to all the nodal masses at the third level). Larger errors are considered for nodal masses, distributed gravity loads, and damping coefficients because, in real world applications, their values are significantly more uncertain than for geometric parameters.

Table 1 Types and magnitudes of errors considered for the modeling uncertainty parameters. Parameter type

Parameters

Variation Magnitude

Error

Geometry

H0; H1; H2; H3 V1; V2; V3

Low Low

±3% of bay-width (0.27 [m]) ±3% of story-height (0.12 [m])

Nodal Masses

NMi k

Low High

+5% +30%

Distributed Gravity Loads

DGL1; DGL2; DGL3

Low High

+5% +30%

Damping Coefficients

aM ; bK

Low High

+15% +50%

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Table 2 Cases of modeling uncertainty considered for the SAC-LA3 building (defined by coefficients applied to the modeling uncertainty parameters).

a b

Case ID

H0a

H1a

H2a

H3a

V1b

V2b

V3b

NM1

NM2

NM3

DGL1

DGL2

DGL3

aM

bK

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.03 1.00 1.00 1.00 1.00 1.00 1.03 1.00

1.00 0.97 1.00 1.00 0.97 1.03 1.03 0.97 0.97 1.03 0.97 1.00 1.00 1.00 1.00 1.03 1.00 1.00 1.03 0.97 1.00 0.97 1.03 1.03 1.03 0.97 1.00 1.03

1.00 1.03 1.00 1.00 1.03 0.97 0.97 1.03 1.03 0.97 1.03 1.00 1.00 1.00 1.00 0.97 1.00 1.00 0.97 1.03 1.00 1.03 1.00 0.97 1.00 1.03 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.03 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.03 1.00 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.00 1.03 1.00 1.00 1.00 1.03 1.00 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03

0.97 1.00 0.97 0.97 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.97 1.00 1.00 1.00 1.03 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.03 1.00 1.03 1.30 1.03 1.00 1.03 1.03 1.03 1.00 1.03 1.03 1.03 1.03 1.00 1.00 0.97 1.00 1.00 1.03 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.05 1.00 1.05 1.00 1.00 1.00 1.00 1.00 1.00 1.30 1.00 1.30 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.05 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.05 1.05 1.05 1.30 1.30 1.00 1.00 1.00 1.00 1.00 1.30 1.30 1.00 1.00 1.30 1.30 1.30 1.30 1.30 1.30

1.05 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.05 1.05 1.05 1.30 1.30 1.00 1.00 1.00 1.00 1.00 1.30 1.30 1.00 1.00 1.30 1.30 1.30 1.30 1.30 1.30

1.00 1.05 1.00 1.00 1.05 1.00 1.00 1.00 1.05 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.30 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.05 1.05 1.00 1.05 1.00 1.00 1.00 1.05 1.00 1.00 1.00 1.00 1.30 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.30 1.30 1.00 1.00 1.30 1.30

1.00 1.05 1.05 1.00 1.05 1.00 1.00 1.00 1.05 1.00 1.00 1.00 1.00 1.30 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.30 1.30 1.00 1.00 1.30 1.30

1.00 1.00 1.00 1.15 1.00 1.15 1.15 1.15 1.00 1.15 1.15 1.00 1.00 1.00 1.50 1.50 1.50 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.50 1.50 1.50 1.50

1.00 1.00 1.00 1.15 1.00 1.15 1.15 1.15 1.00 1.15 1.15 1.00 1.00 1.00 1.50 1.50 1.50 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.50 1.50 1.50 1.50

Error considered as percentage (%) of the bay-width. Error considered as percentage (%) of the story-height.

3.3. Earthquake input motions Two actual earthquake (EQ) ground motions are used in this study. The strong motion portions of records from the 1989 Loma Prieta EQ (MW = 6.9) and from the 1994 Northridge EQ (MW = 6.7) (see Fig. 5) are selected as input motions for the validation and verification studies, respectively. The time history of the 360° component of the ground acceleration recorded at Los Gatos station during the 1989 Loma Prieta EQ (station located 20.6 [km] from the epicenter and with a peak ground acceleration of 0:45½g) is used in the validation study. The SAC-LA3 building is first subjected to this record to numerically simulate the seismic response of the actual structure ðyÞ, represented by model class M0 , which is then used to update different nonlinear FE models belonging to model classes Mj (j = 1, 2, . . ., 28) by using both estimation approaches (i.e., parameter-only and dual adaptive filtering estimation approaches). It is noted that no input noise (i.e., the recorded input ground motion is assumed noiseless) is considered in this application example. Then, the FE models updated using both estimation approaches are subjected to a different seismic input motion in order to investigate the capability of the calibrated models to accurately predict the dynamic response of the building for an input excitation very different from that used for parameter estimation purposes. In this verification study, the time history of the 90° component of the ground acceleration recorded at Tarzana station during the 1994 Northridge EQ (station located

Fig. 5. Ground acceleration time histories recorded at Los Gatos station during the 1989 Loma Prieta EQ and at Tarzana station during the 1994 Northridge EQ.

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5.5 [km] from the epicenter and with a peak ground acceleration of 1.78 [g] is used to simulate the seismic response of the actual structure (represented by M0 ) and to predict the structural response by using the FE models including modeling uncertainty with the parameter estimates obtained from the validation study, in which the unknown modeling parameters are estimated using the parameter-only and dual adaptive filtering approaches with the Los Gatos input motion. 3.4. FE model updating results 3.4.1. Dynamic response simulation The horizontal absolute acceleration time histories at every level of the building structure are taken as response measurements (i.e., observed response quantities). They are denoted as y1 , y2 , and y3 in Fig. 4a. Following Fig. 4, model class M0 provides the nonlinear FE model used to numerically simulate the true response of the structure (ytrue ), i.e., this nonlinear FE model represents the actual or real structure. M0 is defined by a set of model parameters, referred to as true model parameters, which are taken as

" htrue ¼

½ 200 GPa 345 MPa 20 0:08 T ½ 200 GPa 250 MPa 18 0:05 T

2

u

true

#

3

½ 0 9:14 18:28 27:43 3:96 7:92 11:88 T ½m

h i7 6 6 ½ 64:94 129:88 129:88 194:81 64:94 129:88 129:88 194:81 69 138 138 207 T kN s2 7 7 6 m ¼6 7
 7 6 ½ 26:01 25:99 23:09 T kN 5 4 m ½ 0:1782 0:00167 T

In the estimation phase, each component of the true response vector of the structure (ytrue ) is polluted with an independent Gaussian white noise with zero-mean and 0:5%½g root-mean-square (RMS), to produce the measured response, y. Thus, 2

the measurement noise exact covariance matrix is 0:24  102 I3 ½ðm=s2 Þ , where Ii denotes the i  i identity matrix. 3.4.2. Estimation of unknown model parameters The model parameters characterizing the steel material constitutive models are considered unknown and to be estimated. Random initial values for the unknown model parameters are considered and then both estimation approaches are applied to calibrate the FE models (belonging to different model classes M j , j = 1, . . ., 28) using the measured response (y). The unknown model parameters are initially assumed statistically uncorrelated, and thus the initial estimate of their 2 ^ hh , is assumed diagonal, with entries computed as ðp  ^ hi Þ , where i ¼ 1; . . . ; nh ¼ 8 and p denotes covariance matrix, P 0j0

0j0

the initial coefficient of variation of the unknown model parameters. Then, the initial estimates of the mean vector and covariance matrix of the unknown model parameters are taken as

2 ^h0j0 ¼ 6 4h

h

1:3Ecol s

0:8Ebeam s

^ hh ¼ diag P 0j0

col

0:8f y

beam

0:75f y

0:7Rcol 0

1:25b

1:3Rbeam 0

col

iT

beam

1:4b

3 7 iT 5 ¼

"

½ 260 GPa 276 MPa 14 0:1 T

#

½ 160 GPa 187:5 MPa 23:4 0:07 T

 2  2 R88 0:2  ^h0j0

h i 2 In the estimation phase, a diagonal process noise covariance matrix Q ¼ diag ðq  ^ hi0j0 Þ with q ¼ 1  105 is taken (see [16,51] for more details). As mentioned earlier, the estimation is strongly dependent on the variances of the simulation error vector, and in a real world situation, the level of uncertainty present in the model cannot be known exactly or it may change significantly in time (e.g., a model can be a good representation of the linear elastic behavior of the structure but a poor representation of its nonlinear behavior), therefore it is important to have a good estimate of the amplitudes of the simulation errors. In this study, in the parameter-only estimation approach, the simulation error covariance matrix is assumed fixed and 2

equal to R ¼ 0:87  103 I3 ½ðm=s2 Þ , i.e., a standard deviation of 0.3% [g] is assumed. It is noted that previous studies [51] have shown that in the absence of modeling uncertainty, excellent estimates of model parameters are obtained for similar structures and for the same level of measurement noise and assumed values of R as in this study. In the dual filtering approach, this covariance matrix is considered as the initial estimate for the simulation error covariance matrix, i.e., ^ 0 ¼ diagð^r0j0 Þ ¼ 0:87  103 I3 ½ðm=s2 Þ2 . The initial covariance matrix of r for the SF, P ^ rr , is assumed diagonal with entries R 0j0

^ rr ¼ diag½ð0:2  ^r0j0 Þ2 . This coefficomputed assuming a coefficient of variation of 20% of the initial estimate ^r0j0 of r, i.e., P 0j0 hh ^ [16,51]. The time-invariant covariance matrices of cient of variation is selected based on the investigations conducted for P 0j0

the process and measurement noises used in the SF are taken as T ¼ U ¼ 1  1020 I3 . The selection of U and T is expected to

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influence the performance of the proposed procedure; however, a comprehensive study of the sensitivity of the estimation results to T and U is beyond the scope of this paper. As the SAC-LA3 building design satisfies the weak beam-strong column requirement of the UBC, it is expected that the fibers of the beams experience more hysteretic cycles and higher ductility demands than the fibers of the columns. Therefore, the measured response should contain more information about the yield- ðf y Þ and post yield- (R0 and b) related parameters of the beams than about those of the columns. Table 3 reports, for each modeling error case shown in Table 2, the final estimates of the eight unknown model parameters normalized by their corresponding true parameter values, and the associated final coefficient of variation (CV) for each Table 3 Normalized final estimates and coefficient of variation (%) (in parentheses) of the unknown model parameters obtained for the studied cases using parameteronly (N) and dual adaptive filtering (D) approaches. Case ID

1 (N) 1 (D) 2 (N) 2 (D) 3 (N) 3 (D) 4 (N) 4 (D) 5 (N) 5 (D) 6 (N) 6 (D) 7 (N) 7 (D) 8 (N) 8 (D) 9 (N) 9 (D) 10 (N) 10 (D) 11 (N) 11 (D) 12 (N) 12 (D) 13 (N) 13 (D) 14 (N) 14 (D) 15 (N) 15 (D) 16 (N) 16 (D) 17 (N) 17 (D) 18 (N) 18 (D) 19 (N) 19 (D) 20 (N) 20 (D) 21 (N) 21 (D) 22 (N) 22 (D) 23 (N) 23 (D) 24 (N) 24 (D) 25 (N) 25 (D) 26 (N) 26 (D) 27 (N) 27 (D) 28 (N) 28 (D)

Ecolest s

fy

colest

R0colest

Escoltrue

coltrue fy

Rcoltrue 0

b

1.03 1.10 0.95 1.00 1.11 1.13 1.11 1.15 1.05 1.06 1.03 1.02 0.99 1.06 0.99 1.06 1.06 1.10 1.04 1.02 1.09 1.10 1.17 1.10 1.05 1.13 1.03 1.03 0.97 1.01 1.11 1.00 0.97 0.95 1.14 1.22 1.02 1.07 1.29 1.35 1.03 1.03 1.00 1.03 0.86 1.08 1.01 1.11 1.07 1.09 1.08 1.08 1.04 1.09 1.08 1.12

1.31 1.17 2.18 0.99 1.18 1.02 0.77 1.19 1.11 1.13 0.98 0.95 1.41 1.09 2.34 1.08 1.02 1.08 1.22 0.99 1.09 1.05 1.02 1.21 1.22 1.31 1.19 1.52 1.67 0.95 0.28 0.95 0.68 0.92 0.84 1.30 1.40 1.57 1.53 1.38 0.87 0.95 0.61 0.97 0.87 1.02 1.77 1.11 1.12 1.22 1.18 1.24 1.35 1.16 1.38 1.16

0.42 0.54 0.16 1.02 0.58 0.72 0.92 0.57 0.70 0.63 0.96 1.08 0.54 0.74 0.19 0.65 0.52 0.87 0.27 1.11 0.63 0.79 0.63 0.69 0.36 0.48 0.49 0.34 0.17 0.97 1.62 1.05 1.34 1.25 0.28 0.67 0.35 0.92 0.41 0.66 0.55 1.08 0.21 0.94 0.23 1.33 0.26 1.03 0.53 0.80 0.72 0.43 0.49 1.18 0.46 1.10

1.87 0.55 1.37 1.01 0.10 1.17 1.00 0.31 0.54 0.38 1.44 1.36 0.16 0.59 0.63 0.41 1.34 0.98 0.48 1.30 0.10 1.21 1.22 0.33 1.61 1.09 0.31 0.77 2.22 1.07 3.30 1.20 1.49 1.30 1.44 0.33 0.71 1.04 0.13 1.00 0.34 1.30 4.01 1.07 3.53 1.55 0.28 1.18 1.45 0.53 1.55 0.89 0.20 1.34 1.23 1.17

(0.06) (0.32) (0.07) (0.08) (0.06) (0.32) (0.07) (0.30) (0.07) (0.19) (0.09) (0.12) (0.06) (0.20) (0.07) (0.19) (0.08) (0.16) (0.11) (0.11) (0.07) (0.21) (0.07) (0.13) (0.07) (0.35) (0.09) (0.18) (0.08) (0.14) (0.11) (0.15) (0.09) (0.22) (0.11) (0.19) (0.07) (0.19) (0.08) (0.31) (0.10) (0.17) (0.12) (0.21) (0.08) (0.17) (0.06) (0.17) (0.10) (0.27) (0.11) (0.36) (0.09) (0.23) (0.10) (0.22)

(0.30) (1.60) (1.14) (0.57) (0.19) (1.81) (0.13) (0.75) (0.26) (0.70) (0.20) (0.40) (0.15) (1.00) (2.26) (0.87) (0.32) (0.88) (0.32) (0.40) (0.33) (1.66) (0.26) (1.22) (0.19) (2.27) (0.29) (2.23) (0.54) (0.22) (0.05) (0.62) (0.13) (0.86) (0.14) (0.57) (0.22) (1.88) (0.25) (2.08) (0.12) (0.66) (0.16) (1.00) (0.27) (1.26) (0.23) (1.34) (0.32) (0.84) (0.35) (1.14) (0.28) (0.69) (0.27) (0.91)

(0.43) (1.15) (0.09) (1.21) (0.20) (2.14) (0.33) (0.86) (0.37) (0.81) (1.08) (0.98) (1.02) (1.28) (0.21) (0.97) (0.26) (1.22) (0.15) (1.08) (0.27) (1.53) (0.25) (1.35) (0.11) (1.22) (0.30) (1.77) (0.08) (0.82) (0.69) (1.26) (0.70) (2.52) (0.06) (0.79) (0.08) (9.96) (0.07) (2.09) (0.12) (1.72) (0.06) (1.56) (0.10) (2.92) (0.17) (2.17) (0.23) (1.19) (0.67) (0.71) (0.15) (2.37) (0.18) (1.86)

colest

Ebeamest s

fy

Rbeamest 0

coltrue

Ebeamtrue s

beamtrue fy

Rbeamtrue 0

b

1.27 1.14 1.06 1.00 1.05 1.01 1.10 0.99 1.02 1.04 0.96 1.00 1.16 1.03 1.11 1.03 1.18 1.09 1.04 1.06 1.18 1.09 1.23 1.40 2.06 1.54 1.06 1.04 1.03 0.97 0.89 0.98 0.98 0.95 1.11 0.95 1.70 1.38 1.51 1.38 1.00 0.97 1.09 0.97 2.04 1.39 1.87 1.36 1.41 1.36 1.37 1.37 1.50 1.33 1.39 1.31

0.98 0.92 1.02 0.99 0.99 0.89 1.02 0.89 1.23 0.95 1.16 0.94 1.11 0.93 0.97 0.94 1.21 1.04 0.86 1.02 1.17 1.01 0.87 1.02 0.99 0.93 1.12 0.71 0.98 0.96 1.36 0.97 1.23 1.00 1.24 1.02 1.03 0.94 1.12 0.87 1.14 0.95 0.91 0.97 1.18 1.08 0.78 1.11 1.51 1.25 1.50 1.26 1.32 1.28 1.57 1.25

0.75 1.07 0.88 1.01 0.82 1.08 0.67 1.09 0.56 1.09 0.56 1.04 0.58 1.04 0.88 1.10 0.56 0.95 0.50 1.01 0.63 0.95 0.32 1.50 0.68 1.36 0.46 1.36 0.99 1.00 0.47 0.97 0.50 0.96 0.52 0.97 0.83 1.27 0.97 1.50 0.74 0.99 1.31 0.96 0.45 1.30 0.60 1.27 0.50 1.09 0.41 0.96 0.67 0.98 0.57 1.02

0.58 1.32 0.49 1.03 0.85 1.17 0.73 1.43 0.10 1.09 0.10 1.18 0.10 1.25 0.78 1.20 0.10 1.04 0.10 1.10 0.41 1.15 0.40 1.53 0.59 1.66 0.10 1.31 0.57 1.22 0.14 1.18 0.10 0.94 0.10 0.82 0.68 1.03 0.96 1.96 0.31 1.19 0.97 1.16 0.12 1.65 0.61 1.54 0.10 1.32 0.10 1.12 0.51 1.15 0.10 1.23

b

(2.62) (8.38) (2.05) (3.91) (0.51) (9.39) (0.74) (3.90) (1.33) (3.02) (2.21) (2.90) (1.15) (6.24) (5.74) (2.03) (2.78) (5.07) (1.16) (2.92) (1.94) (8.16) (1.83) (4.02) (1.59) (13.54) (2.26) (15.30) (2.00) (1.61) (0.69) (3.77) (1.00) (5.70) (0.83) (1.54) (1.22) (15.72) (0.77) (11.01) (0.53) (3.84) (1.04) (5.25) (2.55) (7.96) (2.51) (6.55) (2.17) (3.79) (2.41) (5.53) (1.44) (5.57) (1.76) (4.98)

(0.13) (0.41) (0.11) (0.14) (0.11) (0.45) (0.12) (0.37) (0.13) (0.30) (0.14) (0.21) (0.14) (0.33) (0.14) (0.31) (0.15) (0.28) (0.21) (0.21) (0.15) (0.35) (0.18) (0.35) (0.20) (0.77) (0.17) (0.29) (0.12) (0.20) (0.11) (0.24) (0.13) (0.34) (0.13) (0.21) (0.20) (0.46) (0.16) (0.53) (0.13) (0.28) (0.12) (0.32) (0.27) (0.39) (0.23) (0.37) (0.24) (0.47) (0.27) (0.66) (0.23) (0.47) (0.22) (0.44)

beamest

(0.22) (0.64) (0.18) (0.34) (0.25) (0.59) (0.23) (0.67) (0.17) (0.48) (0.15) (0.31) (0.15) (0.50) (0.23) (0.43) (0.20) (0.47) (0.18) (0.28) (0.30) (0.70) (0.22) (0.44) (0.23) (0.76) (0.18) (0.46) (0.19) (0.24) (0.14) (0.49) (0.20) (0.63) (0.18) (0.42) (0.19) (0.61) (0.24) (0.62) (0.17) (0.59) (0.11) (0.65) (0.17) (0.53) (0.17) (0.43) (0.26) (0.58) (0.28) (0.70) (0.31) (0.60) (0.20) (0.74)

(0.12) (1.16) (0.20) (0.52) (0.19) (1.25) (0.11) (1.26) (0.12) (0.96) (0.11) (0.59) (0.07) (0.84) (0.21) (0.92) (0.10) (0.61) (0.10) (0.45) (0.12) (0.83) (0.03) (1.50) (0.11) (2.15) (0.08) (1.72) (0.27) (0.41) (0.05) (0.62) (0.07) (0.83) (0.09) (0.65) (0.14) (1.32) (0.24) (2.74) (0.19) (0.87) (0.60) (0.83) (0.04) (1.19) (0.07) (1.01) (0.09) (0.94) (0.07) (1.06) (0.12) (0.76) (0.11) (0.90)

b

beamest

beamtrue

(0.48) (3.01) (0.45) (1.38) (0.67) (2.63) (0.54) (2.94) (0.12) (2.17) (0.12) (1.22) (0.21) (2.10) (0.60) (1.92) (0.21) (1.79) (0.22) (1.13) (0.59) (2.48) (0.32) (1.93) (0.34) (3.62) (0.12) (2.32) (0.56) (0.94) (0.16) (1.65) (0.26) (2.19) (0.11) (1.54) (0.31) (1.94) (0.55) (3.47) (0.26) (2.17) (0.48) (2.48) (0.16) (1.90) (0.29) (1.65) (0.19) (2.41) (0.12) (2.75) (0.59) (1.88) (0.14) (2.24)

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parameter estimate, with (D) and without (N) calibrating the covariance matrix of the simulation error vector. Here D denotes the dual adaptive filtering approach and N the normal (parameter-only) estimation approach. Results provided by the dual adaptive filtering approach (D) demonstrate a great improvement not only in the parameter estimates, but also in the final coefficients of variation, showing that the uncertainty in a parameter estimate is maintained when the information about this parameter is limited. The dual adaptive filtering approach prevents divergence of the parameter values and convergence to unphysical values and is able to detect the compensating effects that the beam parameter-only approach is unable to control. The initial stiffness-related parameters (Ecol ) are estimated by the s and Es parameter-only estimation approach in the range (85.91–205.84%) of the true corresponding values, and the estimation using the dual adaptive filtering approach narrows that range to (95.05–153.95%). Results for the yield-related parameters col

beam

(f y and f y ) are consistent; using the parameter-only approach the estimates vary from 27.50 to 233.88% of the true values, and employing the dual adaptive filtering approach the estimates vary in the range (70.85–157.19%). Significant differences are also observed in the estimation of the post yield-related parameters, which are the least sensitive parameters for the acceleration response measurements considered [31], and therefore, in general poor estimates are obtained for these parameters (despite a good agreement between the measured and the updated FE-predicted response). For instance, the col

parameter-only approach estimates b in the range (10.01–400.51%), while with the dual adaptive filtering approach the estimates are in the range (30.56–155.09%) of the corresponding true parameter values, demonstrating a significant col

improvement. The final coefficient of variation estimates for b are between 0.51 and 5.74% for the parameter-only approach and between 1.54 and 15.72% for the dual adaptive filtering approach. Similar results are obtained for other post beam

beam yield-related parameters (Rcol and b ). From Table 3, it is observed that as the magnitude of the modeling uncertainty 0 , R0 increases (cases are sorted from lower- to higher-magnitude modeling uncertainty), the relative errors in the parameter estimates tend to be higher. In general, the parameter-only estimation approach achieves estimating 16.96% of the unknown model parameters with a relative error lower than 5%, while the dual adaptive filtering approach is able to estimate 29.02% of the unknown parameters within the same error bounds. It is important to emphasize that the modeling uncertainty cases considered in this paper are taken as the most critical cases from [31], and therefore high relative errors are expected because of the high levels of modeling uncertainty. Fig. 6 shows the time histories of the mean estimates of the unknown model parameters, normalized by the corresponding true parameter values for case 11 (see Table 2), obtained with (dual adaptive filtering) and without (parameter-only) estimating the simulation error covariance matrix. At the beginning of the time histories, when the amplitude of the seismic response is low, the structure remains in the linear-elastic range of behavior and the measured responses contain informa-

tion about the initial stiffness-related parameters only, therefore, Ecol and Ebeam are the only parameters that are updated s s from the beginning. When the seismic response starts to increase at about 3 s, the structure enters its nonlinear range of beam

behavior and some fibers of beam elements start to yield, thus f y

starts to be updated. At around 4 s, the measured seismic col

beam

col

, Rcol , and b ). response starts to be sensitive to the other unknown model parameters (f y , Rbeam 0 0 , b When R is also estimated in the FE model updating framework (i.e., dual adaptive filtering approach), the convergence of the unknown model parameters tends to be considerably more stable and smoother, not exhibiting abrupt changes in the

Fig. 6. Comparison of the parameter estimation time-histories for Case 11 (low-magnitude multi-parameter modeling uncertainty) obtained using the parameter-only and dual adaptive filtering estimation approaches.

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R. Astroza et al. / Mechanical Systems and Signal Processing 115 (2019) 782–800

mean estimates. The high variability of the mean estimates of the strain-hardening ratios of fibers of columns and beams obtained with the parameter-only approach can be explained by the low sensitivity of the acceleration response measurements to these parameters. Thus, when the filter tries to match the measured and FE-predicted responses, estimates of model parameters for which limited information is contained in the measured response data may diverge or converge to non-physical values. When the dual adaptive filtering approach is used, the estimates of the unknown model parameters are much more stable and the final values have physical sense. In addition, the dual adaptive filtering approach recognizes when the information contained in the measured response is not sufficient to perform a good estimation, maintaining the uncertainty in the parameter estimate, which is shown in Fig. 6 as dashed lines representing plus/minus two standard deviations (2r). It is observed that both the strain-hardening ratios of columns and beams are the model parameters with the largest estimation uncertainty. It is noted that heterogeneous sensor arrays might allow to obtain better estimates of those parameters that are not well estimated if only acceleration responses are measured [19]. 3.4.3. Errors in observed and unobserved responses As described in Section 3.2, model M0 ðhtrue ; utrue Þ is used to simulate numerically the true dynamic response of the SAC-LA3 building, while model classes Mj (j = 1, 2, . . ., 28) are used to define nonlinear FE models belonging to different model classes in order to consider different modeling uncertainty scenarios. To this end, a specific u is defined for each model class and the unknown model parameters h are estimated for this model class. For each case of modeling uncertainty (see Table 2), the final estimates of the unknown model parameters are used with the FE model belonging to the corresponding model class Mj to predict the response of the structure when it is subjected to an earthquake base excitation. The response of each updated model is compared in terms of observed and unobserved responses with the true response of the structure (obtained from M0 ðhtrue ; utrue Þ). True and FE-predicted responses are compared using the relative root-mean-square error (RRMSE). The RRMSE provides a measurement of the error between two signals, a (reference) and b, and is defined as

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h P iffi 2 Ns 1 ða  b Þ i¼1 i i Ns RRMSE ða; bÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h P i  100ð%Þ Ns 2 1 i¼1 ðai Þ Ns

ð6Þ

where N s denotes the total number of data samples in the signals. Fig. 7 displays the time histories of the observed acceleration response measurements at the second ðy1 Þ, third ðy2 Þ and roof ðy3 Þ levels. The figure compares, at each floor, the true absolute acceleration response time history with its FEpredicted counterparts obtained using the nonlinear FE model defined by Case 11 together with the initial and final parameter estimates provided by the parameter-only and the dual adaptive filtering approaches, respectively. It is observed that both approaches (parameter-only and dual adaptive filtering) for FE model updating properly reduce the discrepancies between the true and FE-predicted observed responses, despite some unknown model parameters having final estimates with non-physical values (Fig. 6). It is noted that for the observed responses, the proposed dual adaptive filtering approach outperforms the parameter-only estimation approach, predicting more reliably response measurements when compared to the true ones, with an average RRMSE of 8.97% against 23.23% for the parameter-only approach. Fig. 8 compares various dynamic responses, at the global, section, and fiber structural levels, corresponding to the true response, initial FE-predicted response (before FE model updating), and final FE-predicted response (after FE model updating) for both estimation approaches for Case 11 (shown in Table 3 and Fig. 6). The global responses shown include the time history of the roof acceleration ðaccroof Þ and the total base shear ðVÞ versus roof drift ratio (relative displacement at the roof level with respect to the base normalized by the building height) ðDÞ. The local level responses consist in the moment versus curvature at sections 1-1 (M 11 vs: j11 ) and 2-2 (M 22 vs: j22 ) indicated in Fig. 4, and the stress versus strain at corner fibers in sections 3-3 (r33 vs: e33 ) and 4-4 (r44 vs: e44 ) indicated in Fig. 4. The time history of the roof acceleration response corresponds to y3 in Fig. 7 and is included in Fig. 8 to compare the performance of the two approaches for both observed and unobserved responses. For the observed quantities both approaches show a good agreement between the true and final FE-predicted responses. However, for the unobserved responses, the prediction capabilities of the FE models updated using the parameter-only and dual adaptive filtering approaches differ significantly. The compensating effects of the parameter-only estimation approach have detrimental impact on the prediction capabilities of unobserved responses, especially at the local section and fiber levels. This confirms that poor estimation of unknown model parameters about which the measured response contains limited information, which is observed in the parameter-only approach, negatively affects the prediction of the unobserved responses, a critical aspect when using a nonlinear FE model for damage identification purposes. On the other hand, FE models calibrated with the proposed dual adaptive filtering approach allow to predict accurately unobserved responses at global and local levels. Fig. 9 reports the RRMSEs between the true and FE-predicted measured (observed) acceleration responses for all cases studied (see Table 2) using the initial and final estimates of the unknown model parameters obtained from both approaches. The RRMSEs of the initial (non-updated) FE models range from 30.11% to 86.33%. The parameter-only approach reduces this misfit between the true and FE-predicted observed responses, with relative errors in the final FE-predicted responses ranging

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Fig. 7. Comparison of the true time histories of the observed absolute acceleration responses with the corresponding initial and updated FE-predicted responses for modeling uncertainty Case 11.

from 11.76% to 85.08%, but with none of the updated models achieving RRMSEs lower than 10% for the observed responses. On the contrary, when using the dual adaptive filtering approach, the RRMSEs decrease significantly, reaching values ranging between 0.68% and 25.02%, with 32% of the cases having relative errors below 10% for all measured responses, and 89% of the cases with relative errors below 20%, demonstrating an excellent performance for matching the observed responses. Fig. 10 reports the RRMSEs for selected unobserved responses following the same comparison scheme as in Fig. 9. Relative errors of representative responses at the global, section and fiber structural levels are shown. These are the roof drift ratio ðDÞ and total base shear ðVÞ, the moment ðM11 Þ and curvature ðj11 Þ at section 1-1, and the stress ðr44 Þ and strain ðe44 Þ in a corner fiber of section 4-4. In agreement with Fig. 8, the FE model updated using the parameter-only approach is not able to properly predict the unobserved responses. The relative errors often even increase after the FE model updating process, for both global and local responses. On the other hand, FE models updated using the dual adaptive filtering approach predict very well the unobserved responses of the structure, at the global and local levels. The relative errors of the FE-predicted global responses (D and V) using the initial estimates of the unknown model parameters, ^ h0j0 , range from 17.97% to 70.88%, with higher errors for the base shear than for the roof drift ratio. The parameter-only approach shows a bad performance for almost all cases at the global level, predicting responses with RRMSEs ranging from 9.62% to 87.82%, generally with better predictions for the base shear than for the roof drift ratio. The proposed dual adaptive filtering approach considerably reduces the relative errors in the global response, with values of RRMSE in the range 0.51–28.48%. This difference is even clearer for the dynamic responses at local levels (section and fiber structural levels). FE models with the initial estimates predict local dynamic responses with relative errors between (19.45–124.16%), while the local responses simulated using the FE models with the final estimates obtained with the parameter-only approach have RRMSEs ranging from 9.41% to 375.26% for displacement-related responses ðD; j11 ; e44 Þ and from 9.06% to 92.69% for force-related responses ðV; M11 ; r44 Þ. On the other hand, the responses simulated from FE models with the final estimates obtained with the proposed dual adaptive filtering method achieve RRMSEs in the range 2.10–85.09% for displacement-related responses and

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Fig. 8. Comparison of the true and the initial and final FE-predicted observed and unobserved responses of the SAC-LA3 building subjected to the Los Gatos earthquake motion obtained using the parameter-only and dual adaptive filtering estimation approaches for Case 11.

Fig. 9. RRMSEs between the true observed absolute acceleration responses and the corresponding initial and final FE-predicted responses for low- (1 to 11) and large-magnitude (12 to 28) modeling uncertainty cases for the SAC-LA3 building subjected to the Los Gatos earthquake record.

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Fig. 10. RRMSEs between the true responses and the initial and final FE-predicted responses for low- (1 to 11) and large-magnitude (12 to 28) modeling uncertainty cases for the SAC-LA3 building subjected to the Los Gatos earthquake record: (a) global unobserved responses, (b) local unobserved responses.

0.62–29.41% for force-related responses. Based on the results obtained, the proposed approach proves to significantly outperform the parameter-only estimation approach, when modeling uncertainties are non-negligible. 3.5. Verification analysis As a measure of robustness and as a verification analysis, the updated nonlinear FE models obtained for each modeling uncertainty case are subjected to a different seismic input motion in order to investigate their response prediction capabilities and also to study the performance of the FE model updating methodologies when a different input excitation to that used for the calibration of the model is considered. Thus, fifty-six FE models updated in the previous section using the Los Gatos earthquake record as input excitation (28 models for the parameter-only and 28 for the dual adaptive filtering approaches) are now subjected to the ground acceleration time history recorded at the Tarzana station during the 1994 Northridge earthquake (see Fig. 5). Then, the dynamic responses of these models are compared to the actual (true) response of the SAC-LA3 building provided by model M0 ðhtrue ; utrue Þ subjected to the Tarzana seismic input. Fig. 11 shows the RRMSEs between the true absolute acceleration responses and their counterparts predicted using the nonlinear FE models updated in Section 3.4 using the parameter-only and dual adaptive filtering approaches. The FE models derived from the parameter-only approach show RRMSEs in the floor acceleration responses varying from 18.16% to 81.62%, being generally higher at the roof level (only 3% of the cases have RRMSEs below 20% for the three floor acceleration responses). The FE models derived from the proposed dual adaptive filtering approach provide lower RRMSEs, ranging from 0.81% to 37.25%, showing very good response prediction capabilities for most cases (64% of the cases have RRMSEs below 20% for the three floor acceleration responses). Analogously to Fig. 10, Fig. 12 reports the RRMSEs between true representative responses (unobserved in the model calibration process of Section 3.4) and their counterparts predicted by the FE models derived from the parameter-only and the dual adaptive filtering approaches. The selected response quantities are the roof drift ratio ðDÞ and total base shear ðVÞ at the global structural level, the moment ðM11 Þ and curvature ðj11 Þ at section 1-1, and the stress ðr44 Þ and strain ðe44 Þ of a corner fiber at section 4-4. The RRMSE results obtained for these responses are in agreement with those discussed in Section 3.4, i.e., the FE models updated using the parameter-only approach show higher relative errors, for both global and local responses, than the FE models updated using the dual adaptive filtering approach. In addition, in all cases considered, the size of the RRMSE differs for displacement- and force-related response quantities. The nonlinear FE models obtained from the parameter-only approach yield RRMSEs for the global responses (D and V) varying from 14.82% to 146.25%, while the FE models derived from the proposed dual adaptive filtering approach produce RRMSEs between 0.8% and 36.38% for the same response quantities. At local structural level, the difference is even more significant: while most of the FE models calibrated using the parameter-only approach show RRMSEs larger than 20% (up to 207%) for all the response quantities considered, the FE models updated using the dual adaptive filtering scheme have RRMSEs in the range (0.88–74.96%). In general, the response prediction is very good for FE models updated through the dual adaptive filtering methodology, with an average RRMSE below 30%, verifying the robustness of the proposed approach for several different cases of modeling uncertainty.

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Fig. 11. RRMSEs between the true absolute acceleration responses (observed in the FE model updating process) and the FE model-predicted responses for low- (1 to 11) and large-magnitude (12 to 28) modeling uncertainty cases for the SAC-LA3 building subjected to the Tarzana earthquake record.

Fig. 12. RRMSEs between the true responses and the FE model-predicted responses for low- (1 to 11) and large-magnitude (12 to 28) modeling uncertainty cases for the SAC-LA3 building subjected to the Tarzana earthquake record: (a) global and (b) local responses (unobserved in the FE model updating process).

4. Conclusions This paper presented, validated, and verified a novel methodology to update mechanics-based nonlinear finite element (FE) models in the presence of modeling uncertainty. Using the input–output measured (observed) data, a dual adaptive filtering approach was employed to estimate the unknown FE model parameters accounting for the fact that the real structure does not belong to the nonlinear FE model class used to represent it. The Unscented Kalman filter (UKF) was adopted as estimation tool for the FE model parameters and a linear Kalman filter (KF) was used to estimate the diagonal terms of the covariance matrix of the simulation error vector based on a covariance-matching technique. The proposed method was

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validated using numerically simulated structural response data for a realistic two-dimensional three-story three-bay steel frame structure subjected to horizontal earthquake excitation. Eight parameters characterizing the nonlinear constitutive laws of the steel materials of the structure were considered unknown and estimated using the proposed dual adaptive filtering estimation approach and also the conventional parameter-only estimation approach for comparison purposes. Geometry, inertia properties, gravity loads, and damping properties were considered as sources of modeling uncertainty and different levels and combinations of them were studied. In these case studies, it was observed that the proposed dual adaptive filtering estimation approach outperforms significantly the parameter-only estimation approach, providing better estimates for the unknown model parameters of the FE model, avoiding estimating unphysical values of the unknown model parameters and reducing the discrepancies between the observed and FE-predicted responses, after updating the FE model, for both observed and unobserved response quantities at global and local structural levels. Finally, the prediction capabilities of the model updated using the novel approach proposed was verified by using an earthquake base excitation significantly different from that used for the parameter estimation. For the above reasons, the proposed approach provides a more robust and precise tool than the parameter-only estimation approach to perform structural damage identification through mechanics-based nonlinear FE model updating. Although the proposed approach was applied to mechanics-based nonlinear FE models of civil structures subjected to earthquake excitation, it can be readily applied or extended to nonlinear FE models of any system subjected to known forces, including static, quasi-static, and dynamic loading. Acknowledgments R. Astroza acknowledges the financial support from the Chilean National Commission for Scientific and Technological Research (CONICYT), FONDECYT project No. 11160009, and from the Universidad de los Andes, Chile through the research grant Fondo de Ayuda a la Investigación (FAI). References [1] M. Friswell, J. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995. [2] N. Distefano, A. Rath, System identification in nonlinear structural seismic dynamics, Comput. Methods Appl. Mech. Eng. 5 (1975) 353–372. [3] N. Distefano, A. 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