Review of finite element model updating methods for structural applications

Structures 41 (2022) 684–723 Available online 20 May 2022 2352-0124/© 2022 The Author(s). Published by Elsevier Ltd on behalf of Institution of Structural Engineers. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Review of finite element model updating methods for structural applications Suzana Ereiz a , * , Ivan Duvnjak a , Javier Fernando Jim´ enez-Alonso b a University of Zagreb, Faculty of Civil Engineering, Zagreb 10000, Croatia b Universidad de Seville, Escuela Superior de Ingeniería, Seville 4109, Spain ARTICLE INFO Keywords: Finite element model updating (FEMU) Static testing Dynamic testing Structural Maintenance Finite element modelling Civil Engineering structures ABSTRACT At the time of designing structures up to date, the density and magnitude of the load have increased, and the requirements for regulation have also become more stringent. To ensure the essential requirements, especially the mechanical resistance and stability, the numerical modelling of the structure is carried out according to the current regulations. Due to various assumptions, idealization, discretization, and parameterizations that are introduced numerical modelling, obtained numerical model may not always reflect the actual structural behavior. It is known that these structures have a hidden resistance that can be determined by combining experimental investigations (static or/and dynamic tests) and finite element model updating methods to minimize the differences between the actual and predicted structural behavior. This paper provides a review of the FEMU process and methods used and summarizes the FEMU approach to help future engineers to select the appropriate method for solving some discussed issues. First, the main terms important for understanding FEMU are introduced. The whole process of model updating is described step by step: selection of updating parameters (design variables), definition of the model updating problem, its solution using different FEMU methods. An overview of the following methods is given: sensitivity-based, maximum likelihood, non-probabilistic, probabilistic, response surface and regularization methods. Each of the method is presented with the corresponding mathematical background, implementation steps, and examples of studies from the literature. 1. Introduction Numerical models are an effective modern tool for continuous monitoring of structures, damage detection, prediction of service life, and determination of an optimal maintenance strategy. The growing number of new and the advance of current numerical modelling methods have led to the need for numerical models to fulfil stringent requirements related to the accuracy and reliability of model and the results. Therefore, the errors and uncertainties associated with model assumptions that most often lead to inaccuracies and uncertainties must be quantified. Their evaluation is important to determine the degree of reliability and accuracy of the numerical model. This has led to the development of finite element model updating (FEMU) methods that aim to calibrate the numerical model based on the actual behavior of the structure determined as a part of static and/or dynamic testing of structure. In the context of different types of structures, numerical modelling is usually performed using finite element (FE) models [1]. This type of models is used to analyse the internal forces, stresses, displacements, and structural dynamic parameters [2]. While updating the finite element models, there are two possible uncertainties, one related to the predicted FE model and one related to the experimentally obtained data. Uncertainties associated with the FE model include differences between the predicted behaviour (numerical model) and the actual behaviour of the structure. In practise, this error can be reduced but never eliminated. Modelling uncertainties can be generally divided into the uncertainties of the model parameters, the model structure, and the model code [3]. Uncertainty in model parameters is usually due to incorrect assumptions of model parameters such as material properties; section properties, and thickness of shell or plate elements [4]. The uncertainty of the model structure arises from incorrect assumptions about the mechanical properties and physical behaviour of the structure. Such erroneous assumptions arise from different idealization and simplification of the structure, inaccurate assumption of mass distributions, incorrect modelling of mesh connections, boundary conditions, joints [5-7], and so on. Incorrect assumptions of loads, geometric shape, and structural behaviour (nonlinear/linear) can also lead to obvious uncertainty in the model structure [8]. These types of errors can be eliminated by introducing appropriate modelling assumptions. Some differences and unreliability can be minimized by developing a more * Corresponding author. Contents lists available at ScienceDirect Structures journal homepage: www.elsevier.com/locate/structures https://doi.org/10.1016/j.istruc.2022.05.041 Received 1 April 2022; Received in revised form 10 May 2022; Accepted 10 May 2022 Structures 41 (2022) 684–723 685 Nomenclature  M experimental data sets θ structural model parameters z output model M model operator Mm class of models PM model parameter space PO response output space ε model uncertainty μ measurement uncertainty θopt optimal value of structural model parameters fi i th natural frequency ϕi n th mode shapes displacement nf size of the frequency residual vectors wf i weighting factor of the ith element of the natural frequency residuals rf i(θ)the ith element of the natural frequency residual vector nm size of the mode shape residual vectors wm i weighting factor of the ith element of the mode shape residuals rm i(θ) − ith element of the mode shape residual vector FRF Frequency response function Hexp(f)FRFs of experimental model Hnum(f)FRFs of FE model MF Modal flexibility MFnum(θ)Modal flexibility of FE model to be updated MFexp(θref )Modal flexibility of experimental model MFnum(θini)Initial modal flexibility of numerical model ‖ • ‖fro Frobenius norm MSE Modal strain energy MSEnum,i modal strain energy for ith mode of the FE model MSEexp,i modal strain energy for ith mode of the experimental model ACC acceleration TH time history k node number nn total number of nodes nt time step of measured acceleration a* kl measured acceleration at k th node at l th time step akl simulated acceleration at k th node at l th time step IL influence line t total number of influence line test points v total number of load step ZCk calculated influence line ZTk measured influence line ws weight factor of the kth test point ε strain n ε size of the strain residual vector ε exp i actual measured strain time histories ε num i estimated strains time histories nδ size of the displacement residual vector δ displacements δexp i actual measured deflection δnum i numerically estimated deflection ‖ • ‖- 2 norm P(θ|Mm)probability density function of the design space in the absence of any data P(θ| M,Mm)posterior probability density function after the data have been observed P( Mθ,Mm)likelihood function of data  M in the presence of parameters θ and the model class Mm P( MMm)normalisation function in the presence of the model class Mm E(f(θ) |  M)posterior expectation of function of the mean value of the updated parameters μ x membership function x fuzzy set L(x) left reference membership function R(x) right membership reference function m average value p, q constants value in membership function Si sensitivity matrix OA Optimization Algorithm NNLSS Non-Negative Least Square Solution GA Genetic Algorithm NSGA-II Non-Dominated Sorting Genetic Algorithm-II HGA Hybrid Genetic Algorithm SGA Simple Genetic Algorithms CCGA Cooperative Coevolutionary Genetic Algorithm ELD Encoding by locations and damage factor SQP Sequential Quadratic Programming IRR GA Implicit Redundant Representation Genetic Algorithm SAA Simulated Annealing Algorithm GAHA Genetic Annealing Hybrid algorithm ML-GA Multi-Layer Genetic Algorithm EnKF Ensemble Kalman filter Xn i position of swarms m number of swarm particle n number of the iteration s number of variables xn ij position of the particle in the iteration n vn ij the velocity of the particle in the iteration n C1, C2 learning factors rand1, rand2random numbers between the zero and one Pbest ij, Gbestjthe best positions achieved by the i-th agent closest to the target since the beginning of the process HPSO-NT Hybrid Particle Swarm Optimization with Sequential Niche technique MWFEM Multivariable Wavelet Finite Element Method PS-NM Hybrid Particle Swarm–Nelder–Mead MPSO Modified Particle Swarm Optimization MOPSO Multi-Objective Particle Swarm Optimization IEPSO Immunity Enhanced Particle Swarm Optimization PSO-NN Particle Swarm Optimization-based approach to train the Neural Network PSO-t-IRS Particle Swarm Optimization algorithm and an enhanced instantaneous response surface SA Simulated annealing ΔE energy change Pr Metropolis Hastings acceptance ratio K B Boltzmann constant T temperature in Kelvin HS Harmony search HM Harmony memory matrix MI Maximum improvisation parameter HMCR Harmony Memory Consideration Rate PAR Pitch Adjustment Rate UKF-HS Unscented Kalman FilterHarmony Search RS response surface based method GRSMU Generalized Response Surface Model Updating MLSM Moving Least-Squares Method DEA Differential Evolution Algorithm r number of factors b number of samples v number of design variables MLP Multi layer perceptron S. Ereiz et al. Structures 41 (2022) 684–723 686 detailed FE model or so called a high fidelity finite element model [9]. Detailed modelling can minimize the degree of uncertainty in the model and the number of parameters that need to be updated. Most often this type of models and large scale models require performing the parallel computing in order to reduce the time required to perform model updating. Some of the parallel computing methods are bisection [1013], domain [14] or loop based [15] decomposition, preconditioned conjugate gradient [16], Davidson algorithm [17], graphics processing unit GPU [18], Branch and Bound [19]. To reduce the error between the experimentally obtained structural dynamic properties and numerically obtained one, a sensitivity-based function with multiple variables is used. To find optimal structural parameters that minimize the chosen objective function, an iterative optimization process is performed. The development of a high-fidelity model FE can help to simplify the process of calibrating large structural models, which is contrary to common ideas about it [20]. The main problem on this type of models is the computational time require to perform any type of analysis, but this problem can be overcome using multi-fidelity FEM [21]. On the other hand, the symmetry and regularity of structures and their numerical models often challenge the solution of large-dimensional matrices, the computation of eigenvalues and eigenvectors. Methods such as graph coloration, group theory [22-24] bisection [25] and so on are used to solve this kind of problems. The uncertainty of the model code is related to numerical uncertainties or technical model uncertainties. These uncertainties are mostly the result of software and hardware errors [3]. The experimental methods and their results most used to update the finite element model include static and dynamic structural tests or data and results obtained as a part of structural health monitoring. The errors that are most common in this field include those due to the imperfection of measuring equipment, random measurement noise, signal processing, and, in general, the problem of post-processing the measurement data [26-28]. In order to obtain a numerical model that represents the real structural behaviour as well as possible, its quality must be evaluated [29]. This assessment consists of three steps. In the first step, the assessment of the idealisation and numerical method errors is performed in order to eliminate or reduce these two types of errors. Then, the correlation analysis between the numerical model predictions and the experimental test results is performed to determine the differences and the extent of correlation between the predictions and the test results, and to determine which design parameters of the numerical model affect the output results. In the third step of the assessment, the quality of the numerical model is assessed after updating the selected design parameters. The definition of FEMU is not uniformly established in the literature. Marwala et al., [30] write in their study that model updating is developed to correct and improve the FE model of the structure according to the actual behavior. Shahbaznia et al., [31] define FEMU as the process of updating the original numerical model of a structure to better reflect the measured response of the actual structure. Schommer et al., [32] defined model updating as an optimization method. In this method, the objective function defined during the FEMU procedure minimizes the deviation between the structural behavior predicted by the numerical model and the actual structural behavior. Mottershead and Friswell [33] define FE model updating as a procedure to update the numerical model to better reproduce the measured response of the actual structures. In another paper [34], the same authors define model updating as the process by which the response of a FE model gradually approximates the response of the real structure by gradually updating the physical parameters. So, the definition is not uniform, but more or less all authors have the common basis of the definition: updating the numerical model based on the experimentally obtained test results to obtain the actual structural behaviour numerically. General division of FEMU methods divide them into the manual and automated methods. Although this is a very general classification, it is still very important, because often a combination of these methods leads to much better results and they are often used together [35-39]. This combination of automated and manual methods is usually used to bring the initial numerical model as close as possible to the actual behavior of the structure using manual updating, while automated model updating is performed to further reduce these differences and obtain a more reliable estimate of the unknown parameters. In addition, this combination can improve complete process of model updating and speed up computational time [36,37]. Generally, manual methods rely on trial and error in the selection of structural parameters such as geometry, material properties and boundary conditions. They are used when the number of parameters to be updated is small [40]. This method is not able to provide a reasonable physical explanation for the changes in the results. This can lead to inefficient results despite its simplicity [41]. When more parameters are considered, it is recommended to use automated methods. These methods are commonly used to reduce idealization errors [5] and they are introduced with two sub-methods. The first one is a global FEMU and the second one is a local FEMU. The global model update assumes that the uncertainty parameters in the overall model have a single value for each selected element. The local model updating assumes that each mesh element has its own value for the uncertainty parameters [42]. The second classification is a bit more concrete. It divides FEMU methods into non-iterative (direct) and iterative (indirect). Direct model updating methods are the oldest methods used to update numerical models [43,44]. They are used to directly update the structure FEM by changing the structural stiffness matrix and the mass matrix. Without the use of iterative procedures, these methods can reproduce accurate experimental data, which makes them computationally efficient. These methods include the matrix update methods [45], the optimal matrix methods [46], the eigenstructure assignment methods [47], and the Langrage multiplier method [48]. As there are no direct changes in physical parameters of FEM when the model updating is performed under direct methods, the importance of the numerical model decrease, i.e., its ability for simulation decreases [49]. Despite the computational efficiency demonstrated in numerous studies [45,50,51], the use of the direct model updating method has decreased, and it has been replaced by indirect (iterative) methods [50]. The iterative methods are the most commonly used method for performing FEMU of civil engineering structures. They are further divided into deterministic (maximum likelihood) and stochastic (uncertainty quantification) methods. This paper gives a review of the most used approaches in finite element model updating of civil engineering structures, focusing on the iterative stochastic and deterministic methods and the process of their application. First, the main terms important for understanding the finite element model updating procedure are introduced. Then, the process of selecting the updating parameters is discussed. The mathematical formulation of finite element model updating problem for deterministic and stochastic iterative methods are discussed in the section 4. Sections 5-11 discuss the deterministic and stochastic iterative methods most ANN Artificial neural network t-IRS instantaneous response surface method for time-dependent reliability analysis MCMC Markov Chain Monte Carlo TMCMC Transitional Markov Chain Monte Carlo MAP Maximum a posterior ACO Art Colony Optimization FKH FuzzyKrill Herd β regularization parameters ‖{θ}‖2 2 l 2 regularization term or norm solution MPC Minimum Product Criterion S. Ereiz et al. Structures 41 (2022) 684–723 687 often used for FEMU. Each of the methods is presented with the appropriate mathematical background and examples of studies in which it has been used to improve simple and complex numerical models and structures. 2. Main terms in FEMU procedure and their relationship Before starting with the procedure and methods of FEMU, definition and explanation of some main terms is required. Those terms include the model, model class, measured data and model updating. In the following their definition and mathematical description is given. Experimental datasets  M is a q-component vector that can compress one type of dataset (homogeneous) or multiple types of datasets (heterogeneous). This vector is based on the output quantities such as the natural frequency and mode shapes, while it can also be based on the strains and displacements. Considering the structural properties as input variables and the results of numerical analysis as the output variables. The model can be generally described as the input–output function between the updating design variables and output results. Input variables considering the structural model parameters θ, while the output response z is defined as the output (i.e., displacements, strains, natural frequency, mode shapes…) due to the any input vector x. To sum previous, can be generally defined by the equation that connects the input and the output variables in the following form: z=M(x,θ)(1) where M is model operator which describes the input–output behaviour. In the finite element model updating procedure it is often working with the output results which are independent of vector £, and the previous relation can be expressed as: z=M(θ)(2) Model parameter vector θ represents a class of models Mm and ranges over a subset PM. The model in the structural model class can be defined as: Mm= {Mm(θ) |θ∊PM}(3) Each of the associated model in model class maps the model parameter space PM into the model output response space PO. After defining the experimental data sets, model, model class, the model updating can be defined as a process of parameter estimation of specific model class. If considering the vector of model uncertainty ( ε ) and vector of measurement uncertainty ( μ ), the vector of measured data sets can be defined as follows:  M=M(θ) + ε + μ (4) For model parameter optimal value θopt, the outputs of numerical model M(θopt)represent a model Mm(θopt)for the experimentally obtained data sets  M. The next section discussed about the selection of design variables whose values are updated when the finite element model is updated. 3. Selection of the updating parameters and model class The selection of an appropriate set of parameters of the numerical model, whose values are updated during the model updating is a nontrivial procedure. The selected parameters of the numerical model should represent the unknown structural properties, but their number is also be limited to avoid ill-conditioned problems. However, in addition to adjusting the number of updating parameters, ill-conditioned problems can also be solved by conditioning the structural stiffness matrices using the cut-set basis [52] and preconditioned conjugate gradients methods [53], flexibility matrices using the methods for generating cycle bases of networks [54] and other methods [55]. Regardless of the method used to perform model parameterization, problems arise during updating that lead to non-unique solutions. Parameter estimation is constrained when the amount of measured data is insufficient. This leads to an underdetermined system of equations in deterministic methods or unidentified parameters in stochastic iterative methods [2956]. Regularisation is often used to update the deterministic finite element model, but parameterization is also preferred [57]. Regardless of the simplicity or complexity of finite element models, they often have many parameters, including the material properties and cross section of the element, the connection of model and boundary condition properties, and the model geometry, which can be selected as updating parameters. Model parameterization has a significant impact on reducing errors and simplifying the finite element models. According to Mottershead and Friswell [33] in order to meet the requirements for the accuracy and reliability of the numerical model and the performance of the model updating procedure, the parameterization procedure should meet the following criteria: •To overcome the ill-conditioned problems, a limited number of parameters should be selected for updating. •The uncertainties model should be corrected by model parameterization. •The outputs of the numerical model must be sensitive to selected updating parameters. Model parameterization that includes sensitivity analysis of the model has the great advantage of providing sensitive parameters and suppressing the problem of inadequacy. This group of parameterization methods includes the subset selection method [58] and the parameter clustering method [59]. In the subset selection method, a reduced number of parameters of the finite element model are selected to be used as updating parameters. The parameters that do not affect the output results are excluded from the model updating process. Originally, this approach was used in regression analysis [60]. Since it is not practical or possible to test all possible subsets of parameters for a large number of parameters, heuristics are used [61]. In these approaches, parameters are selected by an orthogonalization process based on the similarity of their sensitivity vector corresponding to the columns of the sensitivity matrix. The orthogonalization process ensures that each parameter has a different effect on the residual reduction. In addition, there are methods based on the decomposition of the sensitivity matrix [62] and the method that uses global sensitivity analysis for subset selection in model updating [58]. The second sensitivity-based method, the clustering method [59], is based on grouping the parameters of a numerical model with similar sensitivity into a cluster, each of which changes with an update parameter [63]. Selected updating parameters from the same cluster have the same effect on the model updating process. To link similar sensitivities, the unweighted pair group method (UPGMA) is used along with the arithmetic mean [64]. This method allows grouping parameters into binary clusters and then all uncertain parameters are normalized to specific values based on their physical values. To obtain an updated model for further analysis, the updated parameters are multiplied by their initial value. In addition to the previously described parameterization methods based on sensitivity analysis, some other iterative methods are also used for parameterization, such as Bayesian parameterization [65] and particle swarm parameterization [66]. Despite the proposed techniques, the selection of updating parameters depends mainly on the understanding of structural principles, good engineering judgment, and test objectives [29,67]. In order to obtain a physically accurate model, avoid convergence difficulties and illconditioned problems, the number of updating parameters must be limited and correspond to the test objective. Ultimately, it should provide an updated analytical model that represents a real structure and its actual behavior [38,68]. In addition to properly defining parametrization and selection of the S. Ereiz et al. Structures 41 (2022) 684–723 688 updating parameters, properly performing the selection of the class of the structural model is also important for the successful and efficient updating procedure. The class of numerical model represents a set of probable input–output models of the modelled system with respect to the various parameterizations of the structure [69]. To perform the model class selection different methods can be used such as the sensitivity-based method [70], the Bayesian approach [71], and the Particle Swarm Optimization [66]. Most often, the model class selection is performed using the Bayesian approach due to the fact that it gives a quantitative expression that can be used to set those simpler models are to be preferred over unnecessarily complicated [65]. According to this method, the model class with the highest probability is selected for further use. It often happens that a complex model class is better than one that has less adjustable uncertain parameters, which is a problem. If the selected model class, which is considered optimal in each class, minimizes the rate of fitting error between the output data and the corresponding predictions, the choice of model will tend to those that have more efficient free parameters. Therefore, in choosing the optimal model class, it is very important to penalize the complicated model, which is a great challenge [71]. This topic is very important in order to select the numerical model which best describes the actual structure without compromising the computational efficiency of the model updating process and the accuracy of the adjustment [72]. This procedure is very important in model updating and the closely related procedure of selecting a model class that most accurately describes the actual behavior of the structure without compromising the complexity of the model, the computational efficiency of the improvement method, and ultimately the results of the model updating. Therefore, this topic will be discussed through the some of the following chapters that deal with the model updating methods that are also used for model class selection. 4. Definition of finite element model updating problem The finite element model updating problem is generally defined as the difference between the structural behavior predicted by the numerical model and the actual behavior. Depending on the method used, whether iterative stochastic or deterministic, this problem is defined as an optimization or statistical problem. This section gives an overview of the definition of finite element model updating problem using the iterative deterministic or stochastic (probabilistic and non-probabilistic) method. 4.1. Iterative deterministic maximum likelihood method In practical engineering applications, FEMU is performed using the maximum likelihood method (MLM), which transforms the model updating problem into an optimization problem. As part of the transformation, an objective function is defined in terms of residuals between different types of numerically and experimentally obtained data sets (Tables 1 and 2). These data sets include the structural dynamic properties [73-80], static data sets [81-84] or their combination [32,85-88]. The most often, as a first indicators the structural dynamic properties - natural frequencies and mode shapes are used. These data sets are the best indicators of the actual behaviour of the structure, because when there are changes to the structure, this leads to a change in the structural stiffness (structural flexibility), which in turn leads to a change in the structural dynamic properties. These changes are not large and emphasize. Therefore, it is very important to achieve high accuracy when performing the experimental tests in the field. In addition to the natural frequencies and mode shapes, the formulation of the objective using the frequency response function (FRF) in FEMU is also very popular [75,89-95] and offers some advantages in the application. These advantages are related to the fact that the FRF can adequately reproduce the dynamic properties. Moreover, by using the FRF, the FEMU avoids the error caused by modal fitting and does not require any fitting between the predicted and measured mode shapes [75]. Other widely used forms of the objective function that have also been successfully used in FEMU are the modal flexibility residuals (MF) [77,78,96-98]. Comparing the influence of different possible residuals (frequency, mode shapes, and modal flexibility and their combination), the authors conclude that [77] the objective function that considers all three residuals shows the best performance in model updating. In addition to the Table 1 Examples of objective function defined using dynamic data sets. Data sets Examples of related studies Example of objective function fi, ϕi Jim´ enez-Alonso et al., [110] SINGLE OBJECTIVE f(θ) = 1 2[∑nf iwf i•rf i(θ)2]1/2 +1 2[∑nm iwm i•rm i(θ)2]1/2 θ∊[θl,θu]&∑wi=∑wf i+∑wm i=1wi≥0 MULTI OBJECTIVE minf (θ) = min (f1(θ)&f2(θ) )⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ f1(θ) = 1 2[∑nf irf i(θ)2]1/2 f2(θ) = 1 2[∑nm irm i(θ)2]1/2 FRF Pu et al., [91] Hexp(f) − Hnum(f) MF Cui et al., [97] ‖MFnum(θ) − MFexp(θref )‖2 fro ‖MFnum(θini) − MFexp(θref )‖2 fro MSE Jaishi and Ren [79] ∑nm i=1(MSEnum,i MSEexp,i−1)2 ACC and TH Feng and Feng [104] ∑nn k=1(∑nt l=1[a* kl −akl]2) Table 2 Examples of objective function defined using static data sets. Data sets Examples of related studies Example of objective function IL Liao et al., [81] ∑t s=1ws∑v k=1(ZCk − η ZTk ZTk )2 ε Tchemodanova et al., [82] ∑n ε i=1(1−‖ ε exp i− ε num i‖ ‖ ε exp i− ε inum‖) δ, ε Kim et al., [83] Sanayei et al., [84] ∑nδ i=1(1−‖δexp i−δnum i‖ ‖δexp i−δinum‖)∑n ε i=1(1−‖ ε exp i− ε num i‖ ‖ ε exp i− ε inum‖) S. Ereiz et al. Structures 41 (2022) 684–723 689 previously mentioned dynamic properties and their derivatives, the objective function can also be defined using the modal strain energy (MSE) [79,80,99-101]. Measured acceleration is most commonly used for damage detection and estimation of remaining capacity in combination with some of the FEMU methods for structures subjected to traffic-induced vibrations [102-104]. In addition to the use of structural dynamic properties, which are more suitable for modelling complex structures, displacements and strains obtained from in-situ static tests have also been successfully used for FEMU [81-84] performing the FEMU. These types of data sets are also combined with the structural dynamic properties to perform model updating [6,32,36,85,88,105109]. Since different types of data sets with different nature can be used to perform the model updating, a problem how to weigh the influence of each residuals arise. There are two possible approaches: single (5) and multi (6) objective approach. min f (θ) = min 1 2∑ n i=1 wiri(θ)2=min(1 2∑ nf i=1 wf irf i(θ)2+1 2∑ nm i=1 wm irm i(θ)2) (5) min (f1(θ),f2(θ) ) = min (1 2∑ nf i=1 rf i(θ)2,1 2∑ nm i=1 rm i(θ)2)(6) In a single objective approach (5), the objective function is defined by the weighted differences (referred to as residuals) between the numerical and experimental properties under consideration. To account for the relative contributions and uncertainties associated with an experimental estimate of a dynamic or static structural parameter to the objective function, the residuals must be weighted in this approach. The weighting of the residuals is very important to obtain more accurate FEMU-a results. When the natural frequencies are taken into account, their values can be determined experimentally very easily and with high accuracy. Therefore, weighting factors with a high value are assigned to them. On the other hand, compared to the natural frequencies, the mode shapes are less sensitive to changes in structural stiffness and have about 10 times greater influence due to noise [51]. In order to achieve a possible correlation between the experimentally and numerically obtained data sets, the weighting factors of the mode shapes must be analysed to obtain their optimal values [111]. Since the optimal value of the weighting factors is not known in advance, they can be obtained by the trial-and-error method [108] or by statistical criteria [33]. Usually, it is assumed that the optimal value is between 0 and 1 [112]. The use of these values ensures the best correlation between experimentally and numerically determined mode shapes. In another work, it was assumed that the optimal value of the weighting factors is different [36]. In damage detection based on FEMU, it is also difficult to define an objective function and choose appropriate weighting factors for mode shapes or natural frequencies when structural dynamic parameters are involved, since it is not known which of them are important for a particular damage detection problem [113]. On the other hand, multiobjective approach (6) uses different objective functions with respect to the different residuals [114]. The general aim of this approach is to find the optimal solution in the Pareto optimal front [115]. To determine the best solution, a reasonable criterion must be defined. In a FEMU problem defined with two sub-objective functions (bi-criterion problems), an additional constraint is applied in most cases, mainly based on the decision-making strategy [116]. This approach tries to find good compromises or “trade-offs” between conflicting objective functions in an optimal manner. Moreover, the most commonly used criteria consider the edge knee point [117]. This criterion is based on finding a solution where a small improvement in one objective would lead to a large deterioration in at least one other objective [117]. Comparing the single and multi-objective approach their advantages and disadvantages can be pointed out. The single-objective approach depends on weighting factors given by subjective preferences, experience, or engineering judgement [118]. In addition, multiple optimization runs may be required to validate the potential models. All alternative updated models are searched in one step and the best model is selected in a single optimization run using a decision strategy, reducing the time required to find the best FE model. To compare the effectiveness of single and multi-objective functions, authors usually perform FEMU using both approaches. To overcome the disadvantage of the computational cost and the unique dependence between the updated model and the objective function considered, Jim´ enez-Alonso et al., [110] performed a study on a laboratory model of footbridge using both a single and a multi-objective function. Based on the research performed, they concluded that the multi-objective approach is the best option for the FEMU, since it allows a large search space, reduces the computational time, and provides a better balance of the influence of two sets of considered residuals based on the natural frequencies and mode shapes. Naranjo-P´ erez et al., [119] validated the performance of the new hybrid algorithm by comparing it with three different computational intelligence algorithms performed for the same real structure as in [110]. The comparison was based on the speed of convergence and accuracy of matching, using both single and multi-objective functions. They also concluded that the multi-objective approach is better than the single-objective approach. Jin et al., [118] performed the comparison between the single and multiobjective approaches and concluded that all the updated models of the single objective approach are behind the optimal Pareto front (far from the origin). On the other hand, it is also found that the weighting factors should balance the sub-objective functions, but in some cases deviate from this expectation. In addition, the updated parameters of the multi-objective approach appeared to contain physical significance with fewer objective function values, while the single-objective approach resulted in about 50% of the updated parameters being close to constraints. Based on the literature review conducted to define the FEM problem as an optimization problem, it can be concluded that the multi-objective method shows better performance in solving the updating problem. 4.2. Iterative stochastic methods Using the iterative stochastic method to update the finite element model, the FEMU problem is considered as a statistical problem focusing on the quantification of uncertainty. This quantification can be divided into two categories: probabilistic and non-probabilistic. The first category represents the classical approach to modelling uncertainty, which is based on probability theory and in which uncertainty is modelled using the probability density function (PDF). The probability distribution can be defined in different forms: Bernoulli distribution [120], uniform distribution [63], binomial distribution [121], normal distribution [122], Poisson distribution [123] or exponential distribution [124]. In civil engineering applications and finite element model updating, the uniform distribution and the normal distribution are most commonly used [20]. Each set of parameters is assigned a priori probability density function. The assigned functions incorporate prior knowledge or information about the value of the structural parameters and are subject to bias due to the quality and uncertainty of the information. The quantified uncertainty of the prior PDF is updated using Bayes’ theorem, which is mathematically represented as follows (Eq (7)): P(θ| M,Mm)= P( Mθ,Mm)P(θ|Mm) P( MMm)(7) where P(θ|Mm)is the probability density function of the design space in the absence of any data (prior distribution function) when the model class Mm is known and the data  M is absent; P(θ| M,Mm)is the posterior probability density function after the data are observed (in the presence S. Ereiz et al. Structures 41 (2022) 684–723 690 of data  M and the model class Mm; P( Mθ,Mm)is the likelihood function of data  M in the presence of parameters θ and model class Mm; P( MMm)is the normalisation function in the presence of model class Mm. In the situation where experimentally obtained data sets are constant, the marginal distribution of the data  M does not depend on the model parameters θ and the previous equation (Eq.7) can be rewritten as follows (Eq (8)): P(θ| M)∝P( Mθ)P(θ)(8) To obtain the mean value of the updated parameters the following equation (Eq (9)) is used: E(f(θ) |  M)=∫f(θ)P( Mθ)P(θ) P(θ)(9) where E(f(θ) |  M)is the posterior expectation of the function of the mean of the updated parameters f(θ) = θ, which depends on the posterior PDF. For complex systems, which are usually very large, this integral may not be available. In general, several methods are used to solve this integral. They can be generally divided into numerical evaluation, analytical approximations, and sampling methods. When the posterior distribution function P(θ| M)is very large, numerical evaluation may not be accurate [125]. On the other hand, analytical approaches are suitable and converge in local minima [126]. They considers maximum likelihood [127], maximum a posterior MAP [126] and Laplace’s method [128]. Third, sampling methods are most popular to solve the complex posterior distribution. This method considers method such as Markov Chain Monte Carlo method [128]. A variety of methods have been developed to obtain a priori probability density function. One of the most popular approaches is the conjugate prior [129]. In this approach, the prior PDF is chosen based on the likelihood function such that the posterior PDF is assigned to the same family distribution [130]. The main advantage of this approach is that it greatly facilitates the accurate calculation of the full posterior PDF. The conjugate prior approach is still relevant because it provides better insight into exactly how the data change or update the prior PDF. Other approaches [131,132] assume that the prior PDF should add as little as possible to the available prior information. In addition, some more formal methods for avoiding the addition of information by the prior PDF have been developed, mainly based on information theoretic criteria [133-135]. The most commonly used prior PDF selection approach is based on maximum entropy [135]. This approach is based on expressing the probability distribution as the best representative of the current knowledge about a parameter that provides the largest information entropy. Thus, the method itself leads to a uniform prior PDF when it is known that the parameters in the final range of values are nonzero within a certain interval. If the prior available information contains a mean and a finite variance, the maximum entropy leads to a normal distribution. A likelihood function is a function describing the probability of the observed data parameterized by θ. It can be determined by applying the law of total likelihood and according to the equations with the probabilistic model of measurement error and modelling error. The likelihood function can be calculated as the convolution of the PDFs of measurement error and modelling error. If the data sets on the individual errors are not available, the likelihood function can be determined using the probabilistic models of the total prediction error, parameterized by θ. Since there is often very little information available about the error properties, the impact of the likelihood function on the process of model updating using the Bayesian method is very large. Therefore, the definition of this function has attracted much attention in recent decades. In terms of the likelihood model, a realistic estimate can be made. The probability model represents a prediction error. It is usually assumed that the probabilistic prediction error model is known and fixed, so that the set of parameters reduces to θ= {θM}∊RNM. To represent the prediction error in structural dynamics application, an uncorrelated Gaussian model with zero mean is usually used, which is important for better understanding of the expression and calculation. In this approach, entropy is maximised in terms of prediction errors rather than model parameters. This model is not applicable in situations where errors have correlation [136] or in situations with system components [137]. To prevent the influence of incorrect and inappropriate assumptions on the results of Bayesian model updating, identification methods are used [138]. One of them incorporates error parameters (variance of the uncorrelated Gaussian model with zero mean or correlation parameters in terms of correlation length) into the Bayesian scheme and a reasonable assumption about the model class [71]. The second method uses Bayesian model class selection to determine the class that is most appropriate based on the available information. This category is divided into two subcategories. The first subcategory is used to distinguish several alternative classes of model predictions to eliminate or at least reduce model errors. The second subcategory applies model class selection to determine the most appropriate probabilistic model class that represents the prediction error based on the available information. Once the prior probability density function and the likelihood function have been determined, the following equation allows the PDFs of the model parameters to be updated based on the results of the experimental investigation. When dealing with real constructions and real systems, the computation of joint and marginal PDFs involves a large number of parameters. This leads to high-dimensional integrals for which approximate measures or sampling methods, such as the Markov Chain Monte Carlo method, are used to solve. If a conjugate prior is used, the posterior probability density function can be determined numerically. If the number of parameters is limited, the posterior PDF can be determined analytically. After the posterior PDF is calculated, estimated, or approximated, it can provide information on how much the uncertainty of the parameters decreases relative to the observed data and the available prior information. The posterior probability density function can be approximated in a different way, including the Gaussian distribution, asymptotic approximations, or sampling techniques. If both the prior PDF and the likelihood function are Gaussian, the posterior PDF will also have a Gaussian distribution [127]. Asymptotic approximations are used when a large amount of data is available. Here, the posterior PDF is approximated by a Gaussian PDF at the point of maximum posterior or MAP and characterized by a covariance matrix [139]. The most popular Markov Chain Monte Carlo method is used to sample the posterior PDF and improve the convergence speed. On the other hand, the non-probabilistic approach uses a random matrix theory [140] to construct the prediction operator [141]. Most of the proposed non-probabilistic approaches are mainly based on interval analysis [142], where the uncertainties of the variables are represented by a certain range of values. The defined ranges of values are transferred to the outputs of interest (interval analysis). One of the most popular extensions of interval analysis is fuzzy set theory [143]. Fuzzy set theory offers the possibility to model uncertainty if, in addition to the interval bounds, the desired reliability values or confidence levels are available with respect to uncertain quantities. In this approach, the classical binary concept of a set, according to which an element either belongs or does not belong to a set, is replaced by a more intuitive description of the set, where the membership is determined stepwise by the so-called membership function ( μ x). The strength of the application of fuzzy set theory can be seen in the stepwise description of membership, which can be interpreted differently depending on the application. Fuzzy finite element model updating (FFEMU) considers the fuzziness in a design variable in the finite element framework [144,145]. It uses the fuzzy theory based on classical set theory [146]. The difference between these S. Ereiz et al. Structures 41 (2022) 684–723 691 two theories is that classical set theory distinguishes between members and non-members. Fuzzy theory, on the other hand, introduces the degree of membership. The degree of membership weights the possibility of members belonging to the set and is described by the membership function. The membership function is constructed based on some probability measures, e.g., estimated means and standard deviations or confidence intervals, or even based on a complete (estimated) probability density function. In either case, it must be clear that the resulting membership value cannot be interpreted as probabilities. The strength of the fuzzy method lies in the stepwise description of the membership [3]. It can be described and interpreted differently depending on the application [3]. The fuzzy set in which the membership of a particular element ×is observed can be defined as follows in Eq (10): x={(x, μ x(x))(x∊X, μ x(x)∊[0,1])} (10) where x is the fuzzy set, μ x is the membership function defined for all elements ×in the domain X. If the membership function is equal to 1 ( μ x(x) = 1), it means that ×is a member of the set x. On the other hand, if the membership function is zero ( μ x(x) = 0), it means that the element ×is not a member of the set x. If the membership function takes values between 0 and 1, (0< μ x(x)<1)it means that the element ×is a member of the set x with a certain degree of membership. The membership function defined using the left and right reference functions, the left–right membership function, can be defined as follows in equation Eq (11). μ x(x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ L(m−x p),(x≤m,p>0) R(x−m q),(x>m,q>0) (11) where L(x) is the left reference function; R(x) is the right reference function; m is the average value; p and q are a constant value. The most common forms in which a membership function can be defined are trapezoidal [147], triangular [145], singleton [148], Gaussian [149] and piecewise linear [150] shapes (Fig. 1). In the analysis of fuzzy FEMU problems, the fuzzy membership function is divided into different sublevels, the most commonly used being the α -sublevel technique [145]. In the α -sublevel technique, a membership function is divided into a set of sublevels. For each sublevel, the lower and upper bounds are defined, and the fuzzy uncertainty propagation is defined as the application of the series to the interval analysis at multiple α -sublevels. An important problem associated with standard fuzzy set theory is the inability to account for the dependence or interaction between fuzzy input variables and outputs. This means that fuzzy modelling always specifies the maximum range of output variables at each α -sublevel because it is implicitly assumed that any combination of values of the input variables is equally probable. Fuzzy uncertainty propagation is performed using two methods: the intervalbased and the global optimization approaches [151]. The first, interval-based approach is based on standard interval arithmetic and treats the fuzzy variables as an interval for each α -sublevel. The second, global optimization approach is based on two steps. In the first step, the minimum value of the output vector is determined, after which its maximum value is determined in the second step. By combining the obtained results for all defined α -sublevels, the membership function for the output variables is obtained. Thus, the uncertainties associated with the modal parameters can be defined as membership functions, while the model updating procedure is defined as an optimization problem. In this optimization problem, the objective function is defined as the difference between the upper and lower bounds of the experimentally and numerically obtained data sets. The defined objective function must be minimised to obtain interval values for the modal parameters. The obtained modal parameter values are combined to obtain the final fuzzy parameters. In the following section follows an overview of the most commonly used finite element model updating methods: sensitivity-based FEMU, FEMU using a nature-inspired computational algorithm, Response Surface FEMU, FEMU under the non-probabilistic and probabilistic approach, and FEMU using the regularization method. All of these methods are presented with the appropriate mathematical background and examples of studies and research conducted in the field of FEMU and their application to damage detection, reliability analysis, model class selection, optimal sensor placement, and so on. 5. Sensitivity based model updating The main idea of the sensitivity-based method is based on the linearization of the generally non-linear relationship between the measurable outputs and the structural model parameters that require correction. It is developed from a Taylor series expansion truncated after Fig. 1. Graphical representation of a) triangular b) trapezoidal c) singleton d) Gaussian e) piecewise linear membership functions. S. Ereiz et al. Structures 41 (2022) 684–723 692 Fig. 2. Sensitivity based model updating method flowchart. S. Ereiz et al. Structures 41 (2022) 684–723 699 approach for training Neural Network (NN-PSO). In this method, PSO was used to select the optimal weights for the neural network classifier. The proposed method was evaluated on a multi-storey RC building and was found to be very effective in predicting structural failure. Previous studies which have dealt with PSO-based finite element model updating showed that it is a very efficient method to solve the optimization problem of finite element model updating. Moreover, it does not require any knowledge of the function, or its derivatives and it can explore multiple possible solutions in parallel manner in the same sequence. Since it is a global method, its performance does not depend on the initial population solutions. Notwithstanding the foregoing, these methods have also their drawbacks. The first is related to the solution obtained, i.e. there is no guarantee that it is more optimal than the other solution, nor is there convergence to the overall optimal value. The second one is related to the definition of the parameters, because all other algorithms require the definition of parameters that ultimately affect the final performance. The following table (Table 5) represents the sum of the previously mentioned research and studies on the implementation of the finite element model update for damage detection, condition assessment, model selection, selection of optimal weights, etc., using the different types of particle swarm optimization algorithms. Previous studies dealing with PSO-based finite element model updating have shown that it is a very efficient method to solve the optimization problem of finite element model updating. Moreover, it does not require any knowledge of the function or its derivatives and can explore multiple possible solutions in parallel manner in the same sequence. Since it is a global method, its performance does not depend on the solutions of the initial population. Notwithstanding the above, these methods also have their drawbacks. The first one is related to the obtained solution, i.e., there is no guarantee that it is more optimal than the other solution, nor is there convergence to the overall optimal value. The second one is related to the definition of the parameters, because all other algorithms require the definition of parameters that ultimately affect the final performance. 6.3. Other optimization algorithms In addition to the aforementioned computer intelligence optimization algorithms, GA and PSO, which are due to their availability in computational software, most commonly used, there are a number of algorithms developed for the purposes of FEMU. Most of them are nature inspired, such as harmony search [206], simulated annealing [207], grey wolf [208], colliding bodies [209], gravitational search [210], and several others [211]. In this paper, only some of them will be discussed and presented in detail, due to the simplicity, such as simulated annealing and harmony search. For some other nature inspired algorithms’ researches and studies, readers are referred to the following reference [211]. 6.3.1. Simulated annealing Simulated annealing is an optimization method proposed by Kirkpatrick et al., [207]. It is a probabilistic algorithm used to approximate the global optimum of a function by searching for the global extrema of a constrained function around a certain configuration range. The basic concept of this approach comes from annealing in metallurgy. The metal is slowly heated and cooled under controlled conditions until the desired properties are achieved. The process of controlled slow cooling is called annealing. Through a cooling process that promotes diffusion, the metal progresses toward a state of thermal equilibrium, reaching a state of minimum energy. Rapid cooling keeps the metal in a metastable state and prevents the metal’s phase transition. Metropolis et al., [212] described how to simulate a group of molecules in thermal equilibrium at a constant temperature T. In a simulation, a randomly selected molecule is randomly shifted, after which the energy change ΔE [J] of the entire group is calculated. The most important result is the Metropolis acceptance criterion, which defines the probability of acceptance of a simulated energy change, P r . Pr{ΔE} = {1,&ΔE≤0 e−ΔE/KBT,&ΔE≥0(17) According to the above equation (17), if the energy gain is negative the total energy of the system is accepted. On the other hand, if the change increases the total energy of the system, then it is accepted with the probability e−ΔE/KBT, where KB is the Boltzmann constant. If the Fig. 5. Flowchart of Simulated annealing optimization algorithm. S. Ereiz et al. Structures 41 (2022) 684–723 700 simulation is performed for a sufficient number of random motions, the final arrangement of the molecules is close to that in thermal equilibrium or steady state. This is the global minimum at temperature T[K]. The formal proof of convergence models the described simulation as a homogeneous Markov chain whose steady state is shown to correspond to thermal equilibrium. Theoretically, convergence is achieved only for an infinite number of simulations. The three assumptions under which thermal equilibrium is achieved according to the Metropolis algorithm are (1) Reversibility (symmetry) - the probability of choosing the next state is the same as the probability of returning from the next state to the current state (2) Ergodicity - the random displacements of the molecules are such that the molecules can reach any position in their configuration space (3) Convergence to the canonical distribution - the probabilities in the acceptance criterion are such that the ensemble weighs on average in the Boltzmann (Gibbs) distribution. The simulated cooling process is performed using the Metropolis algorithm which is performed through following steps: (1) Melt a system optimized to a high temperature; (2) At very high temperatures, all energy states are almost equally likely; (3) Slowly lower the temperature until the system freezes and there are no more changes; (4) At each temperature, the Metropolis simulation must be run until the system reaches a steady state at that temperature. The process of the simulated annealing optimization algorithm is shown on Fig. 5. Levin and Lieven [210] performed a comparison of two powerful optimizations genetic algorithms and simulated annealingto perform the model updating in the frequency domain. Based on the performed comparison they found that the SA is better than traditional GA and that the accuracy of model updating is dependent on the appropriate selection of the updating parameters. Marwala [211] has done the comparison of computational efficiency of Simulated Annealing, genetic algorithms and Response Surface method on an H-shaped structure. The comparison shows that the response surface method is 2.5 times faster than the genetic algorithm and 24 times faster than simulated annealing. Zhou et al., [82] presented ambient vibration measurements to develop a baseline modal for a newly constructed arch bridge over rives using the structural dynamic properties and performing the updating of numerical model with three algorithms including the simple genetic algorithm, the simulated annealing algorithm and genetic annealing hybrid algorithm. The SAA converged on an infeasible design because it began from a random point and then worked its way toward the minimum, meaning that a local minimum is more likely to be reached. Kourehli [212] proposed a damage detection method based on the simulated annealing using 3 different objective functions based on static and dynamic measurements, which is verified on a four-story steel frame (IASC-ASCE benchmark structure). Some of the other authors work on the improving the computation efficiency of the finite element model updating process under the simulated annealing. Zimmerman and Lynch [213] increased the computational efficiency of SA, by dividing the annealing into a series of steps, each of which is executed on each computer node. The efficiency of the algorithm was tested on three-story structures, and it was found that the larger number of sensors in the network resulted in efficiency gains. There are studies in which the simulated annealing is used in combination with stochastic Bayesian model updating. Thus, Lam et al., [214] used the simulated annealing to propose a Bayesian finite element model updating damage detection based on structural dynamic properties. Huang et al., [215] use simulated annealing to obtain maximum a posteriori values of posterior PDF of design variables for characterizing damage and quantifying uncertainty. Green [216] proposes in his work a new MCMC algorithm: “Data Annealing”, which is based on the input of the likelihood of training data, so that its effects on the posterior are introduced gradually. Moreover, the proposed approach reduces the computational effort probability that local search will get stuck in local traps. Chiu and Lie [19] developed an algorithm based on simulated annealing to cope with the problem of finding the optimal sensor placement under the minimum cost limitation. Based on the literature review conducted on the use of Simulated Annealing (Table 6) to perform model updating and closely related processes such as damage detection and optimal sensor placement, it can be highlighted that this method has not received as much attention as other computational optimization algorithms such as GA and PSO. This is mostly due to the SA requirements of a large number of annealing cycles and a slow convergence speed. These problems question the limitations of the applicability of SA to complex types of structures. However, it can be seen in the literature that authors are making efforts to solve this problem and combine the advantages of SA to develop some hybridization optimization algorithms by taking advantage of SA, such as its strength in solving combinational problems and good performance in hill climbing for the optimal solution. 6.3.2. Harmony search Harmony search [221] is an optimization algorithm proposed by Geem et al., [222], primarily intended to imitate a simplified model of improvisation where there are no chords or modes, only notes or tones. Each tone represents a value of a design variable, and each musician represents a design variable. The vector for optimizing a particular objective function forms the entire harmony. Given the notes that the musician has already played, he chooses a new note to change the harmony. These changes can also be made by pitch or by playing an adjacent tone. The harmony search algorithm consists of three steps. In Table 6 Review of the using computational intelligence SA optimization algorithm. Reported application Examples of related study Type of simulated annealing optimization Structure FEMU Levin and Lieven [213] SA and GA 2D cantilevered beam Marwala [214] SA, GA, RSB Unsymmetric H shaped structure Zhou et al., [85] SG, SA, GA Steel tubular arch bridge Haung et al [218] SA IASC-ASCE Phase II benchmark problems Kourehli [215] SA IASCASCE Lam et al., [217] Bayesian FEMU (SA for sampling scheme) Shear building model under laboratory condition Improving computational efficiency Green [219] Data Annealing Nonlinear dynamical system consist of aluminium rod with centre magnet and two outer magnet Zimmerman and Lynch [216] Parallel SA Three story steel structure Optimal sensor placement Chiu and Lie [220] SA Smaller and larger rectangular sensor fields S. Ereiz et al. Structures 41 (2022) 684–723 701 the first step, a random population of possible solutions is created, which is stored in the Harmonic Memory Matrix. In the second step, an objective function is evaluated for each of the possible solutions. In the third step, a new harmony is created at each iteration, the maximum number of which is determined by the maximum improvisation parameter (MI). The evaluation of the objective function for each new harmony is performed in the fourth step. By comparing the original and the new harmonies, the harmony memory matrix is updated in the fifth step. Until the convergence criteria are met, steps 3–5 are repeated. For the third step and the development of a new harmony, three mechanisms can be used. These include random selection, memory selection, and pitch adjustment [206]. Each design variable of the new vector can be determined from previous values stored in the harmony memory matrix HM or from a new random value. The probability of selecting the previous element of the harmony memory matrix is determined by the Harmony Memory Consideration Rate, HMCR. When the value is taken from the previous value, it changes according to the pitch adjustment rate, PAR, taking into account a predefined range of possible values. The complete process of harmony search is shown in the Fig. 6. This optimization process is characteristic of single-objective optimization, while in multi-objective optimization, both the harmony memory consideration and the pitch adjustment rate are used in each iteration to define a new value for the design variables. To rank the solutions of multiobjective optimization, non-dominant sorting [223] and crowding distance [224] are used. As for the application of the Harmony search algorithm in solving the finite element model updating optimization problem and the related global problem, there are few studies [119,225-228] in which it is used for this purpose. Most of the papers [119,226,227] focus on combining harmony search with other algorithms to improve the computational efficiency. When compared to the traditionally used optimization algorithms (GA, PSO) [110] as standalone algorithms, it can be seen that HS is the most efficient among them when comparing the computational cost and accuracy of the adjustment. This subsection has served only as introduction for mathematical background and description of steps of the harmony search algorithm, while its application and examples of studies will be more discussed in the following section. 6.4. Hybrid local–global optimization algorithms As can be seen from the previous subsections dealing with the implementation of model updating and closely related processes such as damage detection, the use of the discussed algorithms as stand-alone algorithms is very computationally intensive due to the solution of very complex mathematical problems. Moreover, the search for solutions can often get stuck in local traps. To solve the above problems, many studies and research have proposed (Table 7, 8 and 9) hybrid local–global algorithms that can successfully solve this problem by combining the advantages and disadvantages of the standalone algorithms. As for the Genetic algorithm, their hybridization in order to perform model updating more efficiently, Jung and Kim [229] proposed a hybrid genetic algorithm (HGA) by combining GA with Nelder-Mead’s modified simplex method to improve the FE model of the bridge structure. Using a Kriging model, Qin et al., [169] proposed a hybrid algorithm to perform the FEMU of complex bridge structures. To increase the chance of finding the global minima and finding the minimum that best describes the system Shabbir and Omenzetter [172] used a combination of genetic algorithms and sequential niche technique which was tested to Fig. 6. Flowchart of Harmony search optimization algorithm. Table 7 Hybridization of the finite element model updating genetic algorithm. Reported Application Examples of related study Type of GA Structure FEMU Jung and Kim [229] HGA Simply supported bridge Qin et al., [169] GA in combination with Kriging model Tied basket arch bridge Tran-Ngoc et al., [168] SGA, GAHAA Steel truss bridge Shabbir and Omenzetter [172] GA and sequential niche technique Cable stayed footbridge Damage detection Boonlong [192] CCGA beam Cha and Buyukozturk [113] HGA 3D steel structures Shallan et al., [230] HGA +SQP +Inter point +Active Set Beam and Frame Raich et al., [170] IRR GA Beam and Frame S. Ereiz et al. Structures 41 (2022) 684–723 702 perform FEMU of simple laboratory structures and a full-scale pedestrian bridge. To deal with the damage detection, Boonlong [192] proposed a cooperative coevolutionary genetic algorithm (CCGA) capable of solving an optimization problem with a large number of decision variables, as an optimizer for beam damage detection. In the proposed method, each population contains several types of subpopulations. Using the populations, the proposed method explores the search space with a smaller number of generated solutions. As in the classical genetic algorithm, each species is independently involved. Cha and Buyukozturk [113] proposed a damage detection approach using a multi objective hybrid genetic algorithm based on MSE to determine the exact location and extent of damage in 3D steel structures. Shallan et al., [230] used Hybrid GA in combination with sequential quadratic programming, interior point, and active set to minimize the objective function, and performed the localization and quantification of damage to beams and simple frames using static datasets from a limited number of sensors. Raich et al [170] presented the FRF-based damage detection method using the implicit redundant representation (IRR) GA, which identifies both the location and severity of the damage using the limited amount of measurement information. The effectiveness of the proposed method was demonstrated on cantilever beams, two-span continuous beam, and frame structures. Since the application of the optimization-based approaches to damage detection is slow to converge and requires a large Table 8 Hybridization of the finite element model updating particle swarm optimization algorithm. Reported application Examples of related study Type of PSO Structure FEMU Shabbir and Omenzetter [231] HPSO-NT Pedestrian cable stayed bridge Damage detection Luo and Yu [232] PSO based sparse regularization Beam Vakil-Baghmisheh et al., [233] PS-NM Saada et al., [234] Modified PSO Jena and Parhi [233] MPSO Kang et al., [235] IEPSO Beam, frame, truss Kaveh et al., [227] HPSO Alkayem et al., [171] PSO and a social version of sine–cosine optimization algorithm 3D irregular shape structure Time-variant reliability-based design optimization Li and Chen [236] PSO-t-IRS Simply supported beam, two bar frame, six-bar indeterminate truss structure, 23-bar truss structure. Table 9 Hybridization of the finite element model updating simulated annealing and harmony search optimization algorithms. Optimization algorithm Reported application Examples of related study Type of simulated annealing optimization Structure Particle swarm optimization Damage detection He and Hwang [237] New hybrid algorithm combining the GA and SA Damaged clamped beam, mild steel truss Rong and Shun [238] Hybrid algorithm that combines the GA and SA Helical spring optimization design case, auto regressive and moving average exogenous model Astroza et al., [239] Hybrid global optimization algorithm (SA +UKF) 3D 5-story 3-by-2 bay steel frame building Harmony search FEMU Lee and Geem [225] Harmony search meta-heuristic Different type of objective function Pressure vessel design Welded beam design 10/18 bar plane truss Naranjo-P´ erez et al., [119] Hybrid Unscented Kalman Filter-Harmony Search Laboratory footbridge Naranjo-P´ erez et al., [226] Collaborative machine learning algorithm Bormujos footbridge Damage detection Kaveh et al., [227] Particle swarm-ray optimization with harmony search (HRPSO) Five story and four span frame, 52 bar space truss S. Ereiz et al. Structures 41 (2022) 684–723 703 amount of computation. The Particle swarm optimization algorithm is also successfully used to propose a hybrid local–global optimization algorithms to solve, most often, FEMU and damage detection method. As a combination of GA and sequential niche technique, Shabbir and Omenzetter [231] also proposed a method that combine sequential niche technique with PSO. In this way, the authors made a possible a systematic search for multiple minima and confidence in finding the global minima was increased. In order to solve the damage detection problem, Luo and Yu [232] proposed a sparse regularisation method based on particle swarm optimization to detect structural damage. The proposed method consisted of two steps. In the first step, the FEM is updated based on the sensitivity analysis while the damage location is determined by the PSO. In the second step, the possible damage location and PSO are used to determine the extent of damage in subsequent iterations. Numerical simulations on a cantilever beam show the robustness and applicability of the proposed method. Vakil-Baghmisheh et al., [233] used a hybrid particle swarm Nelder Mead algorithm to perform damage detection in a cantilever beam by minimizing the objective function based on the differences in natural frequencies. Saada et al., [234] proposed a modified particle swarm optimization algorithm for damage detection in beam structures to facilitate the performance of FEMU in accordance with experimentally determined natural frequencies. The main idea of the modified method was to identify multiple optima by running the algorithm a predetermined number of times, each time identifying an optimal location. Jena and Parhi [233] modified the PSO technique (MPSO) with the strategy of squeezing the physical domain of the search space to perform an inverse analysis of damage identification based on natural frequency. Kang et al., [235] improved a PSO by combining it with the artificial immune system and developed a new immunity-based particle swarm optimization (IEPSO) algorithm for model updating damage detection. Compared with the classical PSO algorithm and various evolutionary algorithms, the proposed method showed better performance in determining the location and extent of damage to simply supported beams and truss structures. Kaveh et al., [227] proposed a mixed particle swarm ray optimization combined with harmony search for model updating damage detection of the 3D structure of a five-story frame and space truss structure. To solve the damage prediction problem for structures with irregular shape, Alkayem et al., [171] combined PSO and the sine–cosine (SCO) optimization algorithm and developed a new hybrid optimization algorithm. Using the social interaction between PSO and SCO, the highly nonlinear and multimodal optimization problem of FEMU -based damage detection was overcome. The reliability of the developed approach was tested on irregularly shaped 3D modular structures and proved to be very effective and efficient. Li and Chen [236] proposed a PSO-tIRS to study time-varying reliabilitybased design optimization problems, which are also associated with high computational cost and difficulty in modelling. This method combines the PSO and the enhanced instantaneous response surface (tIRS). Enhanced instantaneous response surface was used to construct the extended surrogate model for the instantaneous response, while the PSO was integrated with the extended surrogate model and used to find the optimal solution for the time-varying reliability-based design optimization. The effectiveness of the proposed approach was demonstrated in several examples, including a simply supported beam, a two-bar frame, a six-bar indeterminate truss structure, and a 23-bar truss structure. The other nature inspired computation algorithms advantages are also used to propose and develop some better, more computational effective hybrid local–global optimization algorithms. Thus, He and Hwang [237] combined Genetic algorithm and Simulated annealing in order to propose a new hybrid algorithm for finding actual damage. The results of the validation of the proposed hybrid method showed its efficiency when the measured data are free of error. When the measured data have an error that is acceptable, the proposed method provides less accurate but still acceptable and reasonable results. In order to increase the quality of the solution and the speed of convergence, Rong and Shun [238] proposed a new algorithm that combines the advantages of genetic and the simulated annealing algorithm. The efficiency was tested on a helical spring optimization design case and system identification problem described by auto regressive and moving average exogenous model. Astroza et al., [239] combine simulated annealing with the unscented Kalman filter in their work to reduce the computational cost. The results show that the proposed combination saves significant computational time without affecting the estimation performance. To reduce the high computational time for model updating of complex structures under the harmony search optimization algorithm, Naranjo-P´ erez et al., [119] proposed a novel hybrid Unscented Kalman FilterHarmony Search (UKF-HS). The performance of the proposed algorithm was tested on the benchmark footbridge under the context of single and multi-objective optimization and compared with three computational optimization algorithms. In another work, Naranjo-P´ erez et al., [226] combine multi-objective harmony search, active set algorithms, artificial neural network and principal component analysis to solve the problem of high computational time and uncertainties associated with selecting the best updated model among all Pareto-optimal solutions. Kaveh et al., [227] proposed a new optimization algorithm for damage detection that combines mixed particle swarm ray optimization with harmony search. Miguel et al., [228] proposed a new vibration based method that combines a time-domain modal identification technique with the evolutionary harmony search algorithm. The proposed method was verified on a numerical example and three cantilever beams with different damage scenarios and noise levels. The results show that the proposed method can be efficiently used for structural damage detection and remaining life prediction. Fig. 7. FEMU using surrogate based method flowchart. S. Ereiz et al. Structures 41 (2022) 684–723 704 7. Surrogate based finite element model updating To reduce computational efficiency in finite element model updating problems, the finite element model was replaced by a mathematical model that approximates the relationship between the preselected inputs and outputs of the FE model. The main objective in developing the surrogate-based model updating method is to replace the original finite element model in order to obtain a surrogate that is analytically more practical and computationally more convenient. The use of surrogatebased optimization (SBO) replaces direct model optimization with an iterative process of creating, optimizing, and updating a fast and analytically tractable surrogate model. A locally defined surrogate model should have at least reasonable accuracy in representing a numerical model. By optimizing the design of the surrogate model, a design is obtained that is verified by evaluating a numerical model, and its data is further used to update the surrogate model. Until the termination criterion is satisfied, the optimization process continues by applying a predictor–corrector [240]. Performing all operations on a defined surrogate model reduces the duration and computational effort of the optimization compared to direct optimization. The procedure for performing surrogate-based model updating is presented below (Fig. 7). In the first step, an initial surrogate model is generated. In the second step, an approximate solution to the nonlinear minimization problem is obtained by optimizing the surrogate. Then, the model is evaluated based on the approximate solution computed in the previous step. In the fourth step, the new model is used to update the surrogate model. If the termination criteria are satisfied, the procedure is terminated. If the termination criteria are not satisfied, we proceed to the second step of the procedure. Main terms in surrogate-based optimization are surrogate models and surrogate based optimization technique (Fig. 8). For developing the surrogate model design experiments, surrogate modelling technique, model validation and surrogate criterion are necessary. Surrogate based optimization technique include the approximation model management optimization, space mapping, manifold mapping, and surrogate management framework. Surrogate model is a key component of surrogatebased method, and it can be classified as the physical and functional surrogate models [241]. Surrogate model developing include strategy of design of experiments, model data acquisition, data fitting and model validation (Fig. 9). Design of the experiment (DOE) is a strategy of assigning samples (points in the design space) that aim to maximise the amount of information collected. Model estimation is performed at the assigned points in the design space to create a data set that is later used to construct a surrogate model. To explore a large region of the search space, classical DOE techniques such as factorial designs [242] applied to discrete design variables are used. By applying samples of possible combinations, called factorial design, once discretized continuous variables can be easily analysed. In a fully factorial design, the number of samples increases exponentially with the number of design variables. When the number of design variables is large and model evaluation is expensive, fractional factorial designs [243] are used. In addition, to evaluate the effect and interaction among design variables, their quadratic effects and interactions, full two-level (2 r ) and three level (3r) design is applied such as a central composite design [106], star design [244] and Box Behnken design [245]. During the construction of the initial surrogate model, there is usually non prior knowledge about the objective function. Therefore, some of the DOE approaches mainly aim at uniform distribution of samples within the design space [246]. Therefore, Latin hypercube sampling (LHS) [247-250] is mostly used. In this approach, the design space is divided into b v bins, where b is the number of samples and v is the number of design variables. The samples are selected according to the following criteria: (1) each sample is within a bin, and (2) in each bin there is exactly one sample for all one-dimensional projections of the p samples and the bin. The standard LHS may lead to nonuniform distribution. Therefore, approaches that provide a uniform distribution of samples have been proposed. There are several other approaches to sampling, including Monte Carlo sampling [251], Hammersley sampling [132], and orthogonal array sampling [246]. The common DOE method used in the RS method refers to fullfactorial experimental design [169], central composite experiments [252], orthogonal design [183], uniform design [253], etc. For low-order Fig. 8. Classification of the surrogate based models and their characteristics and main advantages. Fig. 9. Surrogate model construction flowchart. Fig. 10. Five requirements that design of experiment methods should fulfil. S. Ereiz et al. Structures 41 (2022) 684–723 705 RSM, orthogonal and uniform design are usually used. Full factorial experiment design requires too many calculations, although it could lead to relatively more accurate results. Central composite experiment [254] design and D-optimality [255] are applied to model updating of large RSMs, and they achieve almost the same accuracy as when a polynomial response surface is created [254]. Regardless of the DOE method chosen, the ideal DOE should fulfil the requirements shown in Fig. 10. After the design of the experiment has been appropriately selected and the data collected, an approximate model and fitting methodology are established below. The most popular surrogate modelling techniques include polynomial regression [252,256], radial basis function [252,257,258], Kriging predictor [184,249,259,260], neural network [261-263] and other methods based on the above [264-267]. For additional descriptions of previous methods, the authors refer the reader to some of the following references [264,266,268,269]. Some of these methods define a surrogate model and an estimate of the approximation error built into the process. These include the Kriging or Gaussian regression methods. Separately, there are methods that are used only to assess the predictability of a particular model. One of the simplest methods for validating a model is the split sample method [246]. In this method, the sample is divided into two subsets: a training subset and a testing subset. The training subset contains the points from which the surrogate model is created. On the other hand, there is the cross-validation method [270]. In this method, the data sets are divided into L subsets. Each subset is used in turn to test the surrogate model developed based on the other L-1 subsets. In the case where the number of subsets L is equal to the sample size p, the crossvalidation is called leave-one-out cross-validation [271]. The advantage of this approach is that it is less biased compared to the split-sample method. However, the main disadvantage is that it requires the construction of a surrogate model. Nevertheless, the robustness of the surrogate model and the validation approach can be improved since all data are used simultaneously as training data and as test data. 7.1. Response surface-based method In order to avoid model updating being trapped in local solutions that yield an unacceptable value of the objective function, in the initial phase of the surrogate-based model updating method, it is recommended to use the surrogate model that is valid in the global search space [272]. Therefore, regardless of the method used, a correction of the surrogate model is performed to avoid the possibility that the global accuracy of the model may be useless for the further optimization process. To improve the surrogate locally, two methods are used. The first one refers to the correction of the objective function [273] and the other one refers to the concept of space mapping concept [274]. The following is an example of studies in which some of the surrogate-based methods mostly the response surface based method applied to perform model updating, damage detection, or reliability analysis. Ren et al., [253] proposed a FEMU based on the RSB method using measured static structural responses. In addition, a technique to reduce the parameters for constructing the RSB model was proposed. The method was verified on a numerical beam and a full-scale continuous box girder bridge. It was found that the proposed method has the advantages of simple implementation, highly efficient cost, reasonable updating accuracy, and does not require FE calculation in each iteration of the optimization procedure during updating. Ren and Chen [154] proposed an application of the response surface-based finite element model updating for the simulation of a simply supported beam and a full-size continuous box beam, and discussed the main aspects of its implementation. Using simulation data from FEM, a quadratic polynomial response surface was constructed and used to reduce computational costs. Comparison of FEMU with RSB using the sensitivity-based method showed that the RSB method is efficient and converges faster. The application of model updating using the response surface method is characterized by the difficulty of finding a suitable design, followed by a series of trials and errors with different dimes and subset models. To overcome this limitation, Shahidi et al., [245] proposed a generalized response surface based method. To extract as much data as possible from the measured signals and to compensate for the error that often occurs in regression models, the formulation of the MU problem in the time domain was also proposed. The efficiency of the proposed method was demonstrated on a steel frame structure. Marwala [214] presented a finite element model updating method in which the response surface equation of the finite element model is approximated by a multilayer perceptron. The updating parameters were determined by optimizing the objective function using a genetic algorithm. Verification of the proposed approach was performed on an asymmetric H-shaped structure. It showed that the proposed approach is 2.5 times faster than the genetic algorithm and 24 times faster than simulated annealing. To increase the influence of the sample points near the prediction point on its prediction value and solve the coefficients of the response surface polynomial, Chakraborty and Sen [275] combined the moving least square methods and response surface methods. The effectiveness of the proposed method was verified on a 10-member truss and a masonry culvert. Zong et al., [254] presented an application of the RS method on model updating of a bridge structure. By implementing the third order polynomial function, the response surface model of the bridge was created. In their work, the authors considered important aspects for the implementation of model updating, such as experimental design, screening of parameters, construction of a high-order polynomial RS model, optimization methods, and verification of the accuracy of the RS model. Compared with the traditional sensitivity-based model updating method, the proposed method was shown to be more efficient and converge quickly. Zhou et al., [252] validated the effectiveness of the radial basis function (RBF) based response surface method where an RS model was constructed using quadratic polynomials. The model updating procedure was demonstrated on a theoretical structure, a laboratoryscale bridge model, and a real structure using static and dynamic test results and SHM data. The focus was on the selection of updating parameters and responses for updating. The results showed that RMS can be easily implemented for updating complex structures such as long span cable-stayed bridges. Yu and Ou [189] combined the substructure method with the response surface model updating method using SHM data to reconstruct the actual operating condition of the Aizhai suspension bridge. The time-domain based RSM was obtained by comparing the characteristics of the time-domain response of the tests and their FEM counterparts and establishing the functional relationship between the time-domain response and the structural parameters. The unknown structural parameters of FEM are modified by optimization calculations based on the response surface method. Deng and Cai [276] proposed the response surface based method combined with GA to update the bridge model to obtain the explicit relationship between the structural responses and the parameters from the simulation results. By using second or higher order polynomials, the RSM can model the curvature effects between the responses and the parameters that the sensitivity-based methods cannot represent. In the proposed method, the first step is to select the updating parameters. The experimental design for the selected parameters was optimized using RSM. The structural responses were selected according to the purpose of model updating, while the minimization of the objective function was performed using GA. Using the regression method, the RSF for structural responses can be obtained, while the objective function can be developed using the residuals between the measured and predicted responses from the generated RSF and optimised using GA to obtain the updated structural parameters. To investigate the accuracy of the RSM based on the time domain, Han and Yang [277] conducted an experiment on a simply supported beam where the frequency response and modal parameters were obtained. Han and Yang compared the results with those obtained by applying the FEMU based on the frequency response function and modal characteristics and concluded that the time-domain based RSM can reduce the computation time and improve the efficiency of the S. Ereiz et al. Structures 41 (2022) 684–723 706 FEMU and can be used for further damage detection and condition assessment. Kriging model prediction is a modelling method based on the Gaussian process, which has been shown to be compact and effective in solving optimization problems. Su et al., [278] developed a Gaussian Process (GP) surrogate model to assess probabilistic seismic performance of a pile-supported wharf structure. GP was used as a substitute for the computationally intensive model. In terms of computational cost, the approach based on the surrogate model GP is far superior to the brute-force MCS. To address the challenge from the perspective of the Kriging model, Shan et al., [279] proposed a new method to update the finite element model by combining the substructure with the response surface method. The effectiveness of the proposed method was verified on a laboratory-scale cable-stayed suspension bridge. Moravej et al., [280] integrated a modular Bayesian approach to structural reliability analysis and proposed a structural performance evaluation approach using Gaussian process surrogate models. By substituting the finite element model and the associated discrepancy function for the Gaussian process surrogate model, efficient calculation and comprehensive uncertainty quantification were ensured. The proposed method was tested on a large box girder bridge at laboratory scale in undamaged and damaged conditions. On this basis, FEMU has been shown to be a robust and computationally efficient tool for calibrating structural properties under uncertainty. Wu et al., [247] proposed an update of the finite element model based on the combination of the Kriging model and Latin hypercube sampling to perform the model update for different bridge types. The Kriging model was used as a surrogate model, while the Latin hypercube sampling was applied to the preselected samples defined to predict the relationship between the predicted and actual structural behaviour. Comparing the computational cost of the proposed method with that of the Genetic Algorithm, the proposed method shows better performance. To reduce the computational cost of the large number of iterations required for Bayesian model updating, Mao et al., [249] used the Kriging predictor to build the surrogate model of the suspension bridge. The Latin hypercube sampling method was used to generate the experimental data sets. In addition, the authors investigated four types of correlation functions: the Gaussian, exponential, linear, and spline functions. The comparison of the correlation functions showed that the spline and linear correlation functions generate a much larger validation error than the Gaussian and exponential functions, while the error of the Gaussian correlation function is smaller than that of the exponential function. Based on the performed model update of the cable-stayed bridge, it was proved that using the Kriging predictor to update the large complex structure reduces the computational cost and ensures the accuracy. Wang et al., [184] proposed a multi-scale finite element model updating method using the kriging metamodel as a surrogate for the multi-scale model to perform the finite element model updating of the laboratory model of a transmission tower. The review of the proposed method has shown that it can improve the accuracy of the multiscale model in both local and global structural response. Bassoli et al., [166] performed the update of the finite element model of a masonry tower using a surrogate-assisted differential evolution algorithm to reduce the computational cost. To further reduce the number of objective function evaluations, a filling sampling strategy was introduced in the applied algorithm. The accuracy in the optimal region through local and global exploration was increased by selecting candidate points. Based on the research conducted, it was found that a fully characterised structure can be used to achieve a consistent description of the dynamic behaviour of the structure using numerical modelling. Aruna and Ganguli [21] performed a model update on a cantilever beam to solve the problem of understanding a reaction surface based multi-fidelity model and quantifying the uncertainty associated with free vibrations. Based on the conducted study, it was found that the responses obtained with the multi-fidelity model are very close to the responses obtained with the high-fidelity model. Moreover, when compared with the Monte Carlobased quantification of uncertainty, it was found that the multifidelity model requires little computational time and is as accurate as the high-fidelity model. Qin et al., [169] combined the Kriging model with a genetic algorithm in a hybrid finite element model updating method. Based on sampled data regression, a Kriging model was developed between the updating parameters, frequencies and displacements. The analytical values of frequency and displacement in the objective function are predicted by the kriging model and solved by the genetic algorithm. In addition to the previous study, where the surrogate model is used to update the finite element model, it can be successfully used for damage detection and reliability analysis. Fang and Perera [255] proposed a FEMU method for damage detection based on the RSB method and the optimal D-design using only the natural frequency values. The authors used the D-design to establish response surface models and perform the screening out non-significant updating parameters because it requires a smaller number of samples than a standard design such as CCD and allows for an irregular design space. The proposed method was verified on the numerical model of a beam, a RC frame, and a full-scale bridge, and it was concluded that the linear RS model proved to be suitable for the purposes of SDI. Bucher and Bourgund [281] proposed a new adaptive interpolation scheme that provides a factual and accurate representation of the system behaviour through the response surface. To obtain the desired reliability estimates, the response surface was used in conjunction with advanced Monte Carlo simulation techniques. The effectiveness of the proposed method was verified using two examples that included an SDOF system and a three-bay five-story frame. Das and Zheng [282] proposed a method for cumulative formation of the response surface proposed a cumulative response surface function method to perform reliability analysis of a stiffened plate structure, which is very time consuming. The proposed method includes three main steps: search, improvement, and verification. In the first two steps, the iteration process reduces the distance between the central point and a sample point. The third step, the verification step, is performed based on the sample points already obtained. The proposed method has been verified on several examples with a 3 and 12-story frame. To determine reliability and obtain moderate computation time, Gaspar et al., [248] combined a response surface model based on second-order polynomials with a first-order reliability method. By combining the adaptive interpolation scheme and the Latin hypercube sampling technique, an iterative definition of the response surface model was performed in the space of basic random variables that contribute most to the probability of failure. The case study was performed on plate elements, and the effects of considering constrained or restrained boundary conditions and corroded plate elements on the reliability analysis results were observed. Li et al., [258] proposed a sequential surrogate reliability method based on radial basis functions. In the proposed method, the optimization problem is solved iteratively to update a surrogate model of the limit state function. By using new points and updating the surrogate model, the surrogate model of the limit state function becomes more accurate in the important region that has a high probability of failure and at the boundary of the limit state function. The main objective of the optimization is to find a new point that maximizes the probability density function. The proposed method was verified on several numerical examples and showed the accuracy of the surrogate model in the important regions with a smaller number of samples. To estimate the failure probability in each iteration, Monte Carlos simulation was used to obtain a sequence of approximate failure probabilities. Li et al., [250] proposed a new instantaneous response surface method (tIRS) for timedependent reliability analysis. The proposed method does not need to build and update the surrogate models separately for each time node. It uses the expansion optimal linear estimation method to discretize the stochastic process into a set of independent standard normal variables along with some deterministic functions of time. The initial samples are then generated by applying Latin Hypercube Sampling (LHS). The corresponding response values are used to construct the Kriging surrogate model of the instantaneous response. To update the Kriging surrogate model, the active learning method is used until satisfactory accuracy is S. Ereiz et al. Structures 41 (2022) 684–723 707 achieved. Using Monte Carlo simulation, the surrogate model for the instantaneous response is used to calculate the time-dependent reliability. Rutherford et al., [283] presented the possibility of using response surface metamodels to identify damage in the form of changes in stiffness and damping coefficient. The advantage of the proposed approach is that it requires a relatively small data set and can detect changes in parameters and locations in some nonlinear problems. The effectiveness of the proposed method for damage detection was demonstrated on a simple linear and nonlinear 5DOF system. Dey et al., [284] proposed a method to predict the crack parameters (location and depth) from the measured natural frequencies based on the integration of the finite element response surface method and the genetic algorithm. The verification of the proposed method was performed on a thin-walled channel section cantilever beam. The sum of the previously mentioned research works based on the application of the surrogate-based method to solve finite element model updating problems, indicating the application and the type of structures and models to which it was applied, can be found in Table 10. Based on the literature review conducted, it can be concluded that replacing the finite element models with an analytically feasible and computationally cheaper surrogate can successfully reduce computation time. In addition to updating the finite element model to determine the unknown structural parameters, surrogate models can be successfully used for damage detection and reliability analysis. They can also handle complex optimization problems, which usually have a computationally expensive objective function. The application of model updating using the response surface method is characterized by difficulties in finding a suitable design, followed by a series of trials and errors with different dimensions and submodel. 7.2. Artificial neural network Artificial neural networks are an artificial replica of the human brain that attempts to simulate the learning process and tries to implement simplified models that make up the biological neural network. It consists of a series of interconnected simple process elements, units, or nodes whose functionality is based on a biological neuron. The processing power of the network is stored in the strength of the connections between its individual connections, i.e., neurons. Data processing is performed by parallel operation of neural network nodes. The most Table 10 Review of the using surrogate based method for FEMU, damage detection and reliability analysis. Reported Application Examples of related study Types of SB method Structure FEMU Ren et al., [253] RS based on the measured static structural response Numerical beam, full-scale box-girder bridge Ren and Chen [154] RS Beam, full-size precast continuous box girder bridge Shahidi et al., [245] GRSMU Steel frame structure Marwala [214] RS +GA H-shaped structure Chakraborty and Sen [275] MLSM - RS 10 member truss, masonry culvert Zong et al., [254] A third order polynomial RS Long span prestressed continuous rigid frame bridge Zhou et al., [252] RS method based on the RBFs Cable stayed bridge Yu and Ou [189] RS method using SHM Suspension bridge Deng and Cai [276] RS +GA Simply supported concrete beam, Prestressed concrete slab on girder highway bridges Han and Yang [277] RS based on time domain Beam Shan et al., [279] Substructure method in combination with RS Cable stayed suspension bridge Moravej et al., [280] Gaussian Process surrogate model Large lab-scale box girder bridge Su et al., [278] Gaussian Process surrogate model Pile-supported wharf structure Wu et al., [247] Kriging Model and Latin Hypercube Sampling method Truss bridge and an arch bridge Mao et al., [249] Kriging predictor Cable stayed bridge Qin et al., [169] Kriging model +GA Complex bridge structures Wang et al., [184] Multi scale model updating using Kriging-meta model Transmission tower Bassoli et al., [166] Second order surrogate +DEA Historical masonry structures Aruna and Ganguli [21] Multi fidelity response surfaces Beam Damage detection Fang and Perera [255] RS (First order response surface models) Simply supported beam, RC frame, full scale bridge Rutherford et al., [283] RS meta-models Simple linear and nonlinear 5DOF system Dey et al., [284] RS +GA Thin-walled channel section cantilever beam. Reliability analysis Bucher and Bourgund [281] RS SDOF system and three bay five story frame Das and Zheng [282] Improved RS 3 bay 12 story frame Gaspar et al., [248] RS Deck plate element Li et al., [258] Sequential surrogate reliability method beam, circular pipe structure, speed reducer shaft, cantilever tube, nonlinear oscillator Li et al., [250] t-IRS Corroded beam structure, Cantilever Tube Structure, 2D Truss structure S. Ereiz et al. Structures 41 (2022) 684–723 708 commonly used network topology, the multilayer perceptron (MLP), consists of an input layer, an output layer, and a number of hidden neuron layers. Its parts are interconnected in such a way that the transmission of information is one-way - a neuron sends the information to other neurons without receiving information from them. Such a general configuration of a neural network and a multilayer perceptron is shown in the figure. To ensure and maintain the accuracy of the neural network, it is important to choose the right number of hidden layers and neurons in each layer. The selection of the number of neurons is done by the trial-and-error method [285] or by applying empirical relations [286]. To obtain the output of the neurons, a nonlinear procedure of transforming the weighted sum of the inputs is performed. The backpropagation is determined by the connections between the neurons, the activation function they use, and the learning algorithm that determines the weight adjustment process. During backpropagation, the learning algorithm goes through two phases. In the first phase, the training input pattern is displayed by the input layer of the network. The grid is distributed from layer to layer by the input pattern, and the output layer is generated by the output pattern. If the output pattern deviates from the desired output, an error is calculated and then propagated backward through the network from the output layer to the input layer. According to the error propagation, the value is modified. Based on the above, it can be concluded that learning with multilayer perceptrons (Fig. 11 b)) can be conceptualised as a nonlinear minimization problem that can be solved by applying a gradient-based algorithm. On this basis, the set of the optimal weights are obtained that provide a minimum error value between the neural network outputs and the actual values [286]. The procedure for using artificial neural network in performing finite Fig. 11. Neural network a) flowchart b) graphical representation of multi-layer perceptron. S. Ereiz et al. Structures 41 (2022) 684–723 715 The most important thing when performing finite element model updating or damage detection using the regularization method is to choose the right regularization parameter β. The best way to define it correctly is to find a suitable compromise between data fidelity and sparsity solutions. In the Tikhonov regularization, the L-curve criterion is used, which satisfies the small norm of the solution and the small norm of the residual. For the ℓ 1-regularization problem, the curves of the residual norms versus the solution norm do not have the form L and are therefore plotted in a linear scale, unlike the ℓ 2-norm where they are plotted in the log–log scale. Since the ℓ 1-norm does not have a smooth solution and the curvature of the L-curve is not identified, it cannot be expressed explicitly [316]. The main differences between the Tikhonov regularization and the sparse regularization are based on linearity, convergence to zero, regularization path, and sparsity [317]. Moreover, the l1 minimization requires more computational effort when comparing the computational efficiency and the obtained solution, and there is no closed-form solution. The Tikhonov regularization ( ℓ 2-norm) is a common approach to the insufficiency problem and provides an acceptable and smooth solution. It is somewhat more widely used due to its computational efficiency and ease of implementation. However, since the ℓ 2-norm promotes smoothing of the solution, this solution is sometimes over-smoothed, especially when the number of sensors is limited. On the other hand, the solution of the inverse problem in the context of damage detection usually has sparse properties, since typically only a small number of structural components are damaged compared to the whole structure. In the following, an example is given where the regularization method is used to solve the problem of updating the finite element model in the context of damage detection and determining the optimal value of the structural parameters. In the context of the finite element model, the regularization method has been successfully used [318]. To solve ill-posed invers problem of finite element model updating of a bridge with two continuous spans, Rezaiee-Pajand et al., [99] proposed a new iterative and hybrid regularization method based on the Krylov subspace theory [319] and the bidiagonalization process. The proposed method was compared with the least square minimum residual method (LSMR), Reg-LSMR (further development of the LS problem by adding regularization parameters), and hybrid LSMR (LSMR combined with the Tikhonov regularization method). The comparison showed that the proposed sensitivity-based strategy and the regularized solution method are influential and successful in model updating under incomplete noisy data. Luo and Ling [232] proposed a particle swarm optimization based sparse regularization approach for structural damage detection. Classical first-order sensitivity analysis and ℓ 1/2-norm regularization were introduced to define an objective function solved by PSO. The proposed method was verified on a 10-element cantilever beam and was shown to be capable of detecting locations and accurately quantifying the extent of damage. Hernandez [317] provided a new theoretical basis for the identification of localized damage in structures in terms of stiffness reduction using incomplete modal information. Based on the study conducted, it was concluded that the l1-based sensitivity approach can accurately identify damage using noisy data. Zhou et al., [320] proposed a new l1 regularization approach to identify damage using data from the first few frequencies. The proposed method is mainly based on the sparse vector theory, since a sparse vector can be successfully recovered with a small number of measured data. The advantage of the proposed method is that the first few natural frequencies can be measured more accurately than the mode shapes. In addition, the authors conducted a study on the influence of damage severity, number of damages, number of measurement data, and noise on the damage detection results and came to the following conclusions. The number of measurement data, measurement noise, and damage severity affect the accuracy of damage detection. More severe damage, less measurement noise, and more measurement data generally improve damage detection. If the frequency changes caused by the damage are significantly higher than the frequency changes caused by the noise, the damage can be detected correctly. Hou et al., [191] proposed a l1 regularization-based technique for model updating by utilizing the sparsity of structural damage and developing a strategy to select the regularization parameters for the l1 regularization problem. The effectiveness of the proposed method was demonstrated on a truss structure and a three-story steel frame. Hou et al., [321] used the l1-norm regularized damage detection for a cantilever beam and a three-story frame using optimal sensor data. The optimal sensor placement was determined using the OSP method based on GA. To provide the best conditioned model updating scenario for the target structure, Garcia-Palencia et al., [322] proposed a regularization technique and frequency point selection protocol for updating the university of central Florida benchmark structure. In the initial phase, the model updating Table 13 Review of the using non-probabilistic method for FEMU, quantification of the uncertainties, and damage detection. Examples of related study Type of non -probability method Reported application Structure Khodaparast et al., [142] Fuzzy finite model updating Finite element model updating 3-DOF mass spring system; DLR AIRMOD structure Liu and Duan [144] Consideration the effect of the measured uncertainty of modal parameters on the updated model. Continuous prestressed concrete bridge Erdogan and Bakir [310] Quantification of uncertainties of the measured uncertainties of differently type of measured parameters Reinforced frame structure Bulkaibet et al., [145] 5-DOF mass-spring system Moens and Vandepitte [313,314] Plate with uncertain boundary conditions; Garteur benchmark problem; solar panel; COROT baffle cover Erdogan et al., [106] Fuzzy set-based uncertainty quantification approach Investigation the effect of uncertainties on the predicted response of structures Two-span benchmark structure Sun et al., [181] Two-phase FEMU method based on sensitivity analysis and fuzzy outranking method Assess the mechanical state of structure Cable-stayed bridge Dominik and Iwaniec [311] Fuzzy logic method based on the natural frequency comparison Finite element model updating for damage detection Electric pylon Mojtahedi et al., [312] Fuzzy Krill Herd algorithm Finite element model updating for damage detection Offshore jacket platforms S. Ereiz et al. Structures 41 (2022) 684–723 716 was performed by changing the stiffness, mass, and damping parameters to obtain the initial undamaged numerical model. Then, the process was repeated in the damaged state, using the updated parameters of the initial model and the subsequent damaged state for damage detection. The updating of the model was also performed in two steps: In the first step, the stiffness was calibrated and in the second step, the damping was calibrated. According to the proposed approach, the frequency points for the first step of the procedure should be outside the resonance, anti-resonance and noisy regions in the experimentally obtained FRFs, while for the second step, when the identification of the damping is performed, the frequency points in the resonance are needed because the damping causes the significant changes in the response. Zahn and Xu [323] proposed an alternative method based on the sparse regularization ( ℓ 1-norm regularization) to solve the problem related to the illposedness in response sensitivity based damage detection. The effectiveness and superiority of the proposed method was tested on an overhanging beam. Shahbaznia et al., used sensitivity analysis and Tikhonov regularization methods to solve the inverse problem of model update and damage detection in the time domain of railway bridge without knowledge of the moving load. Pan and Yu [324] proposed a sparse regularization based method for damage detection in a beam bridge using only structural responses caused by an unknown moving force. To improve the ill-conditioned problem, Lp-norm sparse regularization was used in the proposed method. Ding et al., [325] proposed a new non-probabilistic method based on Hybrid C-Jaya-TSA. In the proposed method, the authors used sparse regularization technique to solve the problem of limited number of measured data and performed damage detection on TV tower, truss model and cantilever beam. Entezami et al., [160] proposed a new regularization method, Regularized Least Squares Minimal Residual (RLSMR), to solve the problem of damage detection based on sensitivity using incomplete data contaminated by noise. The proposed approach is based on the Krylova subspace and uses bidiagonalization and iterative algorithms to solve systems of linear equations. The comparison of the proposed approach with Tikhon’s regularization method has shown that the proposed approach gives better results in damage detection. Hua et al [326] addressed the determination of regularization parameters in the implementation of the Tikhonov regularization technique in finite element model updating. They formulated an adaptive strategy that allows varying the value of regularization parameters in different iteration steps. The optimal value of the parameters is determined based on the minimum product criterion. The comparison of the proposed method with the L-curve method and the generalized cross-validation of the truss bridge has shown that the proposed strategy is more effective than the method that uses a constant value for the regularization parameters. Based on the literature review and previous studies (Table 14), it can be seen that model updating can be successfully applied using the regularization method. As has been previously reported in the literature, the regularization method is used particularly successfully when the data is contaminated with noise. For these datasets, the main problem is that they are sensitive to small fluctuations in outputs that lead to unreasonably large fluctuations in the values of the damage and design variable values during model updating. Another problem that can also be successfully solved by applying regularization methods is the problem of insufficient definition of a system of equations whose solution leads to an infinite number of solutions. A number of recent studies have shown that regularization methods can be successfully applied even in cases where the amount of vibration data is limited. They can provide a satisfactory solution for damage detection requiring a small number of measurement points. In addition, it was shown that the regularization method requires a lot of computational effort which is not justified by improving accuracy. 11. Conclusion To better develop the numerical model of actual structural behaviour and make its predictions as credible as possible, it is increasingly common to combine numerical modelling with the results of experimental investigation of the structure. This article provides an overview of the process of updating finite element models based on the results of experimental investigations and the most used methods. Several relevant conclusions can be drawn from the literature review and based on the wrote paper: 1. Considering the actual behavior of the structure and the parameters that most credibly represent it, one can conclude that the numerical model of the structure can be improved based on the experimentally determined dynamic properties - natural frequencies and mode shapes. The change in structural dynamic parameters is most associated in the changes of structural stiffness in form of the damage. This prove that these parameters are in fact the first indicators of the occurrence of a damage on the structure. 2. In addition to the use of the structural dynamic properties, the static data sets can also be used for updating the finite element model, especially for numerical modelling of complex structures. Table 14 Review of the using regularization method for FEMU and damage detection. Reported application Examples of related study Type of regularization method Structure FEMU Rezaiee-Pajand et al., [99] Krylov subspace method Two story concrete frame, two span continuous steel truss Damage detection Luo and Ling [232] PSO based sparse regularization Cantilever beam Hernandez [317] L1 norm regularization Shear-beam, plate in bending Zhou et al., [320] Cantilever beam Hou et al., [191] Cantilever beam, three story frame Hou et al., [321] Planer truss, three story steel frame Garcia-Palencia et al., [322] Tikhonov regularization USF-Benchmark Structure Zahn and Xu [323] Tikhonov regularization and sparse regularization Planer truss, Overhang beam Shahbaznia et al., [31] Tikhonov regularization using L-curve Railway Bridges Pan and Yu [324] Sparse regularization Two span continuous bridge Ding et al., [325] TV Tower, truss model, cantilever beam structure Entezami et al., [160] Regularized Least Squares Minimal Residual Truss bridge Output error based FEMU Hua et al., [326] Tikhonov regularization in conjunction with MPC Truss bridge S. Ereiz et al. Structures 41 (2022) 684–723 717 The main problem with static measurements is the placement of the sensors and the errors they are subject to (size, position, orientation, thermal expansion, reading technique, or measurement accuracy). While, in combination with the dynamic data sets and with the proper definition of the weighted factor values, their combination can give a very accurate and reliable numerical model of structure. 3. Selecting the appropriate design variables have a significant influence on reducing errors and simplifying the finite element model. The best way to perform model parameterization is sensitivity analysis, which results in suppressing the problem of inadequacy. Based on this, the parameters that do not affect the output results are excluded from the model updating process. It is very important that the selected design variables represent the real structural behavior as well as possible. 4. The finite element model updating methods currently represented in literature are divided in two main groups: direct (non-iterative) and indirect (iterative) methods. Direct methods straightly update the elements of the stiffness or mass matrix in one step and provides dynamic parameters corresponding to those obtained by experimental tests. The study addresses the limitation of the direct FEMU, although it reproduces the structural dynamic parameters without the guarantee that it accurately reproduces the actual values of the physical parameters of the structure 5. The most studies have relied on iterative maximum likelihood method that is performed using the single or multi objective approach. The single objective approach obtains only a single solution, which is a subset of the set of possible solutions. Moreover, in this approach, the effect of the weighted factors on the defined single objective function had to be analysed. In contrast, in the multi-objective approach, one obtains a set of possible solutions, called the Pareto optimal front. Determination of the Pareto optimal front is very computationally intensive and time consuming. Moreover, the best possible solution can be found by solving the decision-making problem (best balanced solution), which allows updating the numerical model to better reflect the actual structural behavior. Finally, compared to the single-objective approach, the multi-objective approach shows better performance in solving the finite element model updating problem. 6. Iterative sensitivity-based methods allow many updating parameters and measured outputs and many of them require a high computational effort. Moreover, the sensitivity equation is generally a nonlinear problem linking the input parameters of the numerical model and its output, so an iterative procedure must be performed. The literature identifies the convergence problem in determination of the updating parameters values. 7. The iterative FEMU methods currently represented in the literature are divided into stochastic (Bayesian) and deterministic (maximum likelihood). The existing research has highlighted the advantage of the stochastic method, which provides the overall probability of the distribution of the physical parameters under consideration, while the main drawback is the time required to perform FEMU of complex structures. Compared to stochastic methods, deterministic method provides the point of estimation of the expected value. On the other hand, the reduced time required to calculate the FEMU of a complex structure has led to the deterministic method being widely used for practical applications in civil engineering. Therefore, several maximum likelihood methods optimization algorithms have been proposed to deal with the FEMU problem. However, the Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) are the most commonly used, since their code is usually integrated into numerical computation software. 8. The paper reviews the Harmony Search (HS) and Simulated Annealing (SA) optimization algorithm, where the previous studies have shown that the harmony search algorithm is the most efficient without compromising accuracy compared to GA and PSO. In addition to using optimization algorithms as standalone algorithms, their drawbacks are minimised by combining them with other methods in order to improve the computational efficiency. 9. Surrogate-based finite element model updating shows good performance by replacing the finite element model with a mathematical model that is analytically more practical and computationally more effective. In order to improve the computational efficiency in FEMU, the literature discussed the importance of sampling in creating effective surrogate models. The main problem is to find a suitable design, followed by a series of trials and errors with different dimensions and submodels. 10. The capabilities of the Artificial Neural Network method are also successfully used for solving various types of FEMU problems, especially when consider nonlinear relationship between the damage detection (location and severity) and the measured structural dynamic properties. Moreover, this method can also be used for Structural Health Monitoring by using the Frequency Response Function instead of the traditionally used structural dynamic parameters (natural frequencies and mode shapes). The main problem with this method is the data sets needed to properly train the network. 11. The Fuzzy approach is a kind of stochastic approach in which the updating of model is represented as a non-probabilistic problem, in contrast to the Bayesian method (probabilistic problem). Therefore, the results obtained with these two methods could only be compared qualitatively. The fuzzy finite element model updating can provide results that are mostly easy and intuitive to interpret, and their further processing can provide information about the resulting uncertainties. In addition, the fuzzy approach does not take into account the dependence between the model parameters and/or the experimental data, while the Bayesian approach automatically incorporates the interaction between the two data sets. 12. The regularization method is used to solve the problem of illconditioning and overfitting when the number of available measurement data is limited. The most important thing in this method is the proper definition of the regularization parameter. This method is especially successfully used in damage detection. On the other hand, when the number of sensors used for the experimental study is limited, the solution obtained by the regularization method can sometimes be over-smoothed. Also, the problem presents a situation where a small number of structural components are damaged compared to the whole structure which results in sparse properties of the solution. Although it can be successfully applied in improving the accuracy of the solution, it did not justify the required computational effort. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This research was funded by the European Union through the European Regional Development Fund’s Competitiveness and Cohesion Operational Program, grant number KK.01.1.1.04.0041, project “Autonomous System for Assessment and Prediction of infrastructure integrity (ASAP). S. Ereiz et al. 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