1 Introduction

Bridge supports are often designed to match the idealized boundary conditions such as rollers and hinges in structural mechanics to facilitate structural analysis. However, the boundary constraints of a constructed bridge may largely deviate from the original designs due to many reasons: friction between girders and abutments [1], damage of expansion joints [2], aging or deterioration of bridge bearings [3], installation error of girders [4], and restraining effect of non-structural components (such as rails and ballasts) [5], to name but a few. Relative to the local damage (e.g. cracks) emerging in the girder, the deviation of boundary constraints can lead to a more significant change in the dynamic characteristics [6, 7]. Thus, identifying the boundary constraints of a real bridge is of paramount importance for subsequent finite element (FE) model-based behavior prediction or condition assessment [8].

The bridge modal parameter-based finite element model updating (FEMU) has been testified as an effective and practical method to identify the bridge boundary conditions [9], in which the discrepancies of the bridge modal parameters (e.g. frequencies and mode shapes) between the ones predicted by FE model and those identified from the measurement of a real bridge are minimized through the gradual adjustment of bridge end constraints. The related studies featuring the boundary updating of a bridge are briefly summarized in the following. Dilena et al. [10] updated the boundary conditions of a damaged reinforced concrete bridge by adopting a longitudinal spring to simulate the constraining effect of bridge supports, in which the accuracy of the updated FE model was evaluated via frequency deviation. Zhang and Huang [11] updated the boundary conditions of a maglev guideway using the bridge frequencies to construct the objective function and manually adjusting the constraint stiffness. Shi et al. [12] updated the rotational and horizontal constraints of a monorail bridge by simultaneously utilizing the frequency and mode shape of the first vibration mode. Hester et al. updated the boundary conditions of a steel girder short-span highway bridge through the simultaneous use of the data recorded in the ambient vibration tests and static loading tests [1], in which the bridge frequencies were adopted for constructing the objective function; for low-level vibration amplitude, it was shown that the behavior of the supports was pinned–pinned due to the friction on the bearings, as opposed to the pinned-roller supports in the design drawing. Salehi and Erduran [5] conducted a sensitivity analysis of the end-constraint influence on the bridge frequencies, and they found that the rotational stiffness and vertically translational stiffness were the influential parameters. They further updated the boundary conditions of a railway bridge using artificial neural networks and setting the frequencies and constraint stiffness as the input and output variables, respectively.

Relative to the bridge frequency, mode shape can provide more information about the distribution of both structural stiffness and mass [13, 14]. However, the mode shapes are rarely adopted for such a task since the differences in the mode shapes under different boundary conditions are generally nuanced and hard to be distinguished by the mode shapes of low spatial resolution. The difficulty arises greatly when large errors are introduced in modal parameter identification. Conventionally, the sensors should be densely deployed along the bridge to ensure the resolution of the identified mode shapes, but it is hard to meet this requirement due to the need for massive sensors or multiple runs of sensor installation [15]. In recent years, vision-based health monitoring techniques have been actively studied and verified to be effective methods for acquiring high-resolution structural mode shapes as every pixel can be viewed as a sensor [16, 17]. Nevertheless, these techniques are restricted by the field of view of the camera and are sensitive to illumination conditions and background interference. Moreover, image or video processing often requires high-performance computers. Therefore, inferring the high-resolution mode shapes from the measurement data recorded by a few contact-type sensors (e.g. LVDTs or accelerometers) is more practical. In this vein, two strategies have been proposed so far. The first one uses polynomials to interpolate the sparse mode shape coordinates into the dense structural geometry [5, 18]. Apparently, this strategy is an empirical method with limited accuracy. The second one first estimates the full-field bridge responses using full-state estimation techniques such as Kalman filter-based methods [19, 20] and then identifies the high-resolution mode shapes from the reconstructed responses. As this strategy requires the use of an FE model of the bridge, an inaccurate model leads to an ineffective prediction of bridge responses.

In mathematics, FEMU is an optimization problem that generally involves a large number of iterations to seek the optimal structural parameters. For a high-fidelity FE model with dense discretization, the optimization time is often intolerably long, even if parallel and pooling computing is employed. Recently, the surrogate-assisted optimization technique has been developed and offers a practical solution [21, 22]. Specifically, the surrogate models are first constructed to approximate the original FE model by explicitly fitting the relations between the structural parameters and responses using simple functions. Then, the surrogate models (not FE models) are employed for updating the structural parameters. In contrast to the FE model, the computational time required by a surrogate model for a run of iteration is negligibly small, and lots of time can be saved. The commonly constructed surrogate models are for the scalars such as frequencies [23] and static deflections [24]. In this study, the bridge mode shapes in vector form will be adopted for FEMU, in which the issue of reasonably constructing the surrogate models for bridge mode shapes will be resolved.

Based on the above literature review, this study proposes a numerical framework for quantitatively updating the bridge boundary constraints using the high-resolution mode shapes constructed from the responses measured by a few accelerometers. Compared with the low-resolution mode shapes, the high-resolution ones can tell the difference in the bridge end constraints, and a more accurate updated result can therefore be expected. Particularly, the frequency and MAC surrogate models are constructed to reduce the CPU time in FEMU, and the MAC models are especially sensitive to boundary constraints. To the best of the authors’ knowledge, the current study is the first one to fully use the high-resolution mode shapes in an efficient way for identifying bridge boundary constraints.

The rest of the paper is organized as follows: Sect. 2 details the procedure for constructing the high-resolution bridge mode shapes. Section 3 presents the surrogate-assisted updating method for optimization. Section 4 illustrates the proposed method via a detailed numerical example. The experimental validation is given in Sect. 5. Section 6 concludes this study.

2 Compressive sensing-based construction of high-resolution mode shapes

2.1 Compressive sensing in spatial domain

Compressive sensing (CS) is an efficient and cost-effective signal acquisition and reconstruction technique based on the sparse representation of signals and the theory of solving underdetermined linear systems [25,26,27]. It can recover a potentially sparse signal from far fewer samples than that required by the Shannon-Nyquist sampling theorem, hence the storage space and transmission bandwidth can be drastically reduced. To date, the overwhelming majority of CS applications are performed in the time domain, where the structural response at a single measurement point is sparsely and randomly sampled, and the original signal is recovered by the CS theory [27]. To construct the high-resolution mode shapes, the present study applies the CS theory in the spatial domain. Namely, a few sensors are randomly distributed on the bridge while the response of each measurement channel is uniformly sampled; the bridge response field is then reconstructed by the CS theory. This new application paradigm can be implemented in the same way as those for conventional bridge health monitoring [28] without changing the basic theoretical framework of CS theory to be introduced below.

Let \({\varvec{y}}={\left[\begin{array}{ccc}\begin{array}{cc}w({x}_{1})& w({x}_{2})\end{array}& \cdots & w({x}_{m})\end{array}\right]}^{T}\in {\mathbb{R}}^{m}\) be the response (such as displacement or acceleration) of a bridge at a given time, in which \({x}_{1}\sim {x}_{m}\) are the coordinates of the densely and uniformly distributed points (i.e. virtual measurement points) along the bridge. Assuming that a signal \({\varvec{y}}\in {\mathbb{R}}^{m}\) can be sparsely represented in the domain spanned by the orthonormal basis vectors \({{\varvec{d}}}_{j}\in {\mathbb{R}}^{m}\) as

$${\varvec{y}}=\sum_{j=1}^{n}{h}_{j}{{\varvec{d}}}_{j}={\varvec{D}}{\varvec{h}}$$
(1)

where \({\varvec{D}}\in {\mathbb{R}}^{m\times n}\) is the basis matrix; \({{\varvec{d}}}_{j}\) is the jth column of D; \({\varvec{h}}{\in {\mathbb{R}}}^{n}\) is a sparse vector, i.e. the majority of its elements (i.e. \({h}_{j}\)) are zeros or near zeros.

Assuming that only \(p\) positions (i.e. real measurement points) of the bridge can be monitored by the available sensors as schematically shown in Fig. 1, the measured response \({\varvec{z}}\in {\mathbb{R}}^{p}\) can be linked to y via the following equation:

$${\varvec{z}}=\boldsymbol{\Theta }{\varvec{y}}$$
(2)

where \(\boldsymbol{\Theta }\in {\mathbb{R}}^{p\times m}\) (\(p\ll m\)) is the measurement matrix or binary sensing matrix, i.e. each row of \(\boldsymbol{\Theta }\) has an element equal to 1 depending on the location of the sensor while the rest are zeros. To consider the measurement noise \({\varvec{e}}\in {\mathbb{R}}^{p}\), Eq. (2) can be recast as

Fig. 1
figure 1

Sketch of virtual and real measurement points

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$${\varvec{z}}=\boldsymbol{\Theta }{\varvec{D}}{\varvec{h}}+{\varvec{e}}=\boldsymbol{\Phi }{\varvec{h}}+{\varvec{e}}$$
(3)

CS attempts to recover \({\varvec{h}}\) from \({\varvec{z}}\) using the optimization algorithm. Mathematically, this goal can be achieved in the framework of the least absolute shrinkage and selection operator (LASSO) [29] presented by

$$\underset{{\varvec{h}}}{{\text{argmin}}} \, \| \boldsymbol{\Phi }{\varvec{h}}-{\varvec{z}}{\| }_{2}+\lambda \| {\varvec{h}}{\| }_{1}$$
(4)

where \({\Vert \Vert }_{1}\) and \({\Vert \Vert }_{2}\) respectively stand for the \({{\ell}}_{1}\) and \({{\ell}}_{2}\) norms. \(\lambda \ge 0\) is a hyperparameter that controls the trade-off between the data-fitting and sparsity-promoting regularization terms. When \(\lambda =0\), Eq. (4) reduces to the ordinary least-squares regression. As \(\lambda\) increases, stronger penalization is imposed on the absolute values of the entries in \({\varvec{h}}\), forcing some of them to shrink towards zero. This feature makes the LASSO regression effective in solving the underdetermined linear system of equations where the most influential components in \({\varvec{h}}\) need to be selected [30]. This study adopts the interior point method [31] to solve the optimization problem as presented in Eq. (4). Then, \({\varvec{h}}\) is substituted into Eq. (1) to obtain the dense measurement \({\varvec{y}}\). By combing the calculated responses at \(s\) measurement instants, the full-field response \({\varvec{Y}}\in {\mathbb{R}}^{m\times s}\) can be obtained, which is finally analyzed by the system identification algorithm to extract the bridge mode shapes. Compared with the mode shapes directly extracted by the real measurements having values only at real measurement points, those extracted from \({\varvec{Y}}\) have a much higher spatial resolution, i.e. the values at all virtual measurement points can be provided. Then, the extracted high-resolution mode shapes are used to update the bridge end constraints.

2.2 Construction of basis matrix

From the above derivation, the construction of a basis matrix D that can sparsely represent bridge response is the key step for identifying high-resolution mode shapes. This problem will be solved in this section.

The flexural behavior of a bridge is generally described by the Euler–Bernoulli beam theory [32] as

$$\frac{{\partial }^{2}}{\partial {x}^{2}}\left[EI(x)\frac{{\partial }^{2}w(x,t)}{\partial {x}^{2}}\right]+\rho A(x)\frac{{\partial }^{2}w(x,t)}{\partial {t}^{2}}=f(x,t)$$
(5)

where \(w(x,t)\) is the vertical vibration of the beam; E and \(\rho\) are the elastic modulus and density; I(x) and \(A\left(x\right)\) are the moment of inertial and cross-sectional area at position x; \(f(x,t)\) is the external force. By applying the principle of modal superposition, \(w(x,t)\) can be expressed as

$$w(x,t)=\sum_{i=1}^{N}{\phi }_{i}(x){q}_{i}(t)$$
(6)

where \({q}_{i}(t)\) is the ith modal coordinate; \({\phi }_{i}(x)\) is the corresponding mode shape; \(N\) is the largest mode in consideration.

For a bridge in service, its vibration is generally controlled by the first few vibration modes. That is, the bridge vibration can be sparsely represented in a space spanned by the mode shapes. Following this direction, Ref. [33] constructed a basis matrix using the theoretical expression of the mode shapes of a single-span Euler–Bernoulli beam with a uniform cross-section. In reality, continuous and variable cross-sectional bridges are ubiquitous, especially for medium to large-span bridges, owing to their better mechanical properties than those of single-span uniform bridges. But it is difficult to derive their theoretical mode shapes. To circumvent this difficulty, the mode shape approximation strategy is adopted herein. Specifically, the bridge mode shapes are approximated by the Fourier series as follows:

$${\phi }_{i}(x)=\sum_{j=1}^{\infty }{c}_{ij}{\text{sin}}\frac{j\pi x}{L}\approx \sum_{j=1}^{n}{c}_{ij}{\text{sin}}\frac{j\pi x}{L}$$
(7)

where n is the number of truncated terms in the approximation; L is the length of the bridge; \({c}_{ij}\) is the coefficient of Fourier series expansion. For bridges with commonly encountered boundary conditions, their mode shapes can be well approximated by the first few terms of the sinusoidal series. As an example, Fig. 2(a) and (b) compare the theoretical mode shapes and those approximated by the first four significant terms (i.e. having the largest values of the expansion coefficients) of the sinusoidal series for the first mode of the fixed-pinned and fixed–fixed bridges, respectively. Visually, the boundary features of the bridge mode shapes are nicely captured by the four sinusoidal terms.

Fig. 2
figure 2

The mode shapes of the first mode approximated by four sinusoidal terms for different boundary conditions: a fixed-pinned bridge; b fixed–fixed bridge

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Then, the basis matrix can be expressed as

$${\varvec{D}}=\left[\begin{array}{cccc}{\text{sin}}(\frac{\pi {x}_{1}}{L})& {\text{sin}}(\frac{2\pi {x}_{1}}{L})& \cdots & {\text{sin}}(\frac{n\pi {x}_{1}}{L})\\ {\text{sin}}(\frac{\pi {x}_{2}}{L})& {\text{sin}}(\frac{2\pi {x}_{2}}{L})& \cdots & {\text{sin}}(\frac{n\pi {x}_{2}}{L})\\ \vdots & \vdots & \ddots & \vdots \\ {\text{sin}}(\frac{\pi {x}_{m}}{L})& {\text{sin}}(\frac{2\pi {x}_{m}}{L})& \cdots & {\text{sin}}(\frac{n\pi {x}_{m}}{L})\end{array}\right]$$
(8)

The above basis matrix is a function of spatial coordinates for CS in the spatial domain, as opposed to the conventional CS in the time domain (i.e. functions of time). The extraordinary merit of the proposed basis matrix lies in the ease of construction (only \({x}_{1}\sim {x}_{m}\) and \(n\) need to be specified) and applicability to any beam-type structures.

The corresponding sparse vector is given by

$${\varvec{h}}=\left[\begin{array}{cccc}{c}_{11}& {c}_{21}& \cdots & {c}_{N1}\\ {c}_{12}& {c}_{22}& \cdots & {c}_{N2}\\ \vdots & \vdots & \ddots & \vdots \\ {c}_{1n}& {c}_{2n}& \cdots & {c}_{Nn}\end{array}\right]\left[\begin{array}{c}{q}_{1}(t)\\ {q}_{2}(t)\\ \vdots \\ {q}_{N}(t)\end{array}\right]$$
(9)

The sparsity of \({\varvec{h}}\) comes from the sparsity both in the spatial and temporal domains; that is, the bridge mode shapes can be nicely approximated by the first few significant sinusoidal terms, and the bridge vibration is dominated by the first few vibration modes.

3 Surrogate-assisted updating of bridge boundary constraints

3.1 Objective functions

In the literature, the bridge end constraints are generally modelled by the translational and rotational springs [4, 5, 12, 34]. As shown in Fig. 3, the current study focuses on the vertical vibration plane, on which the most sensitive bridge vibration modes to boundary constraints are located [5]. For the bridge model with two end constraints, \({k}_{lt} ({k}_{lr})\) and \({k}_{rt} ({k}_{rr})\) respectively stand for the translational (rotational) spring stiffness on the left and right ends. The spring stiffness is arranged in a to-be-updated vector \({\varvec{X}}={\left[\begin{array}{ccc}{k}_{lt}& \begin{array}{cc}{k}_{lr}& {k}_{rt}\end{array}& {k}_{rr}\end{array}\right]}^{T}\). To simply focus on updating the bridge boundary constraints, the other model parameters (such as elastic modulus and density) are assumed to be unchanged, although they can be updated by incorporating them in \({\varvec{X}}\). In other words, the discrepancies in the bridge frequencies and mode shapes between the FE model predicted values and those identified from the measurement data merely arise from the bridge boundary constraints.

Fig. 3
figure 3

Bridge model with two end constraints

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In this study, both bridge frequencies and mode shapes are adopted to update \({\varvec{X}}\). To be specific, the bridge frequencies and high-resolution mode shapes identified from the vibration tests are regarded as the targets, and the discrepancies in the bridge frequencies and mode shapes between the FE model-predicted values and those identified from the measurement data need to be minimized by adjusting \({\varvec{X}}\). To account for the difference in frequencies, the following objective function is adopted:

$${f}_{F,i}\left({\varvec{X}}\right)={\left(\frac{{f}_{i}^{FEM}({\varvec{X}})-{f}_{i}^{EXP}}{{f}_{i}^{EXP}}\right)}^{2}, i=1, 2,\cdots ,{n}_{f}$$
(10)

where \({f}_{i}^{EXP}\) and \({f}_{i}^{FEM}\left({\varvec{X}}\right)\) are the ith bridge frequencies identified from the measurement data and predicted by the FE model, respectively; \({n}_{f}\) is the number of objective functions for bridge frequencies.

For the difference in mode shapes, the objective function defined below is adopted:

$${f}_{M,j}({\varvec{X}})=\frac{{\left(1-\sqrt{{MAC}_{j}({\varvec{X}})}\right)}^{2}}{{MAC}_{j}({\varvec{X}})}, j=1, 2, \cdots ,{n}_{m}$$
(11)

with

$${MAC}_{j}({\varvec{X}})=\frac{{|{{\varvec{\phi}}}_{FEM, j}^{T}({\varvec{X}}){{\varvec{\phi}}}_{EXP, j}|}^{2}}{{{\varvec{\phi}}}_{FEM,j}^{T}({\varvec{X}}){{\varvec{\phi}}}_{FEM,j}({\varvec{X}}){{\varvec{\phi}}}_{EXP,j}^{T}{{\varvec{\phi}}}_{EXP,j}}$$
(12)

where \({n}_{m}\) is the number of the objective functions for mode shapes; \({{\varvec{\phi}}}_{EXP, j}\) and \({{\varvec{\phi}}}_{FEM,j}({\varvec{X}})\) are the high-resolution mode shapes of the jth vibration mode, as respectively identified from the measurement data and predicted by the FE model. \({MAC}_{j}({\varvec{X}})\) is the so-called modal assurance criterion (MAC) that measures the similarity between \({{\varvec{\phi}}}_{EXP, j}\) and \({{\varvec{\phi}}}_{FEM,j}({\varvec{X}})\). Theoretically, \({MAC}_{j}({\varvec{X}})\) ranges from 0 to 1, and a larger value indicates a higher correlation between the two vectors. However, the bridge mode shapes under different boundary constraints do not have much difference in general; to improve the sensitivity of the MAC value to \({\varvec{X}}\), the gradients of \({{\varvec{\phi}}}_{EXP, j}\) and \({{\varvec{\phi}}}_{FEM,j}({\varvec{X}})\) are adopted to calculate \({MAC}_{j}({\varvec{X}})\) [35, 36]. To simply verify this idea, Fig. 4 compares the normalized mode shapes (left panel) and their gradients (right panel) for the first two vibration modes of bridges under different boundary conditions, while Table 1 lists the corresponding MAC values. By observation, the MAC values of the mode shape gradients are smaller than those of the mode shapes, indicating that the difference among the mode shape gradients is greater than that among the mode shapes. In this context, the MAC values between the theoretical and the approximated mode shapes are 1 and 0.9998 respectively for the fixed-pinned and fixed–fixed bridges as shown in Fig. 4, while the MAC values for their mode shape gradients are 0.9953 and 0.9935. MAC values close to 1 quantitatively denote good accuracy of the Fourier series-approximated mode shapes [13, 14].

Fig. 4
figure 4

Comparison of mode shapes (left panel) and their gradients (right panel) under different boundary conditions for the first two modes: a first; b second

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Table 1 MAC values of mode shapes and their gradients (in brackets) for the first two modes of bridges under different boundary conditions
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3.2 Optimization methods

Conventionally, the differences in frequencies and mode shapes between the FE model-predicted and measured values are arranged in a single objective function as a weighted sum of \({f}_{F,i}({\varvec{X}})\) and \({f}_{M,j}({\varvec{X}})\), and the minimum value of the objective function is sought by the single-objective optimization algorithm [37]. However, it is hard to determine the weighting factors before updating the model. Moreover, different factors often lead to significantly different updated results [38, 39]. To seek the proper weighting factors, the trial-and-error strategy should be adopted [39], which imposes a huge computational burden. In this study, the above difficulties are circumvented by directly minimizing both \({f}_{F,i}({\varvec{X}})\) and \({f}_{M,j}({\varvec{X}})\) using the multi-objective algorithm.

Genetic algorithm (GA) [40] is an optimization algorithm that is inspired by the process of natural selection. It is used to find the optimal solution to a problem by creating a population of potential solutions and then using operations such as mutation, crossover, and selection to evolve and improve the solutions over multiple generations. Compared with other commonly adopted optimization methods such as the Gauss–Newton algorithms [37] and interior point method [41], GA has the following eminent merits: derivative-free, high efficiency, noise resistance, parallelism, and liability. The non-dominated sorting genetic algorithm II (NSGA-II) is a variant of the classic GA for multi-objective optimization [42, 43], and it is an improved version of the original NSGA algorithm in terms of computational efficiency and diversity of the solutions, with the following working principles: non-dominated sorting, elite preserving operator, crowding distance, and selection operator. Specifically, in the step of non-dominated sorting, the population members are sorted into different fronts based on the concept of Pareto dominance. In the step of the elite preserving operator, the elite solutions of a population are directly transferred to the next generation. For the step of crowding distance calculation, the density of solutions surrounding a particular solution is found. In the selection operator step, the population for the next generation is selected through a crowded tournament selection operator by ranking the population members and their crowding distances. The solutions with non-dominated members are obtained to constitute the Pareto front.

For a multi-objective optimization problem, the objective functions cannot be optimized simultaneously in general. Instead, a series of non-dominated (Pareto-optimal) solutions are given, which optimize one objective without sacrificing the others and can be viewed as the best trade-off among the conflicting objectives. Accordingly, a preferred solution should be selected from the non-dominated solutions as the final updated parameters. Ref. [44] introduced the commonly adopted selection criteria for the preferred Pareto-optimal solution, while Ref. [14] compared the performance of several criteria and concluded that the criterion of the minimum distance from the equilibrium point can yield satisfactory results. Given this, the minimum distance criterion is adopted in this study. The distance between the kth non-dominated solution and the equilibrium point Q is defined by the minimum values of each objective function as

$${d}_{Qk}=\sqrt{{\left({f}_{1}^{k}-{f}_{1}^{{\text{min}}}\right)}^{2}+\left({f}_{2}^{k}-{f}_{2}^{{\text{min}}}\right)+\cdots +{\left({f}_{{n}_{0}}^{k}-{f}_{{n}_{0}}^{{\text{min}}}\right)}^{2}}$$
(13)

where \({n}_{o}={n}_{f}+{n}_{m}\) denotes the total number of frequency and mode shape objective functions; \({f}_{1}^{{\text{min}}}\)~\({f}_{{n}_{o}}^{{\text{min}}}\) are the minimum values of the objective functions \({f}_{1}\sim {f}_{{n}_{0}}\) (consisting of \({f}_{F,i}({\varvec{X}})\) and \({f}_{M,j}({\varvec{X}}))\). The preferred solution is the one having the minimum \({d}_{Qk}\). The geometry notation of this criterion for bi-objective optimization is shown in Fig. 5.

Fig. 5
figure 5

Schematic diagram of the criterion of minimum distance from the equilibrium point for bi-objective optimization

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3.3 Kriging surrogate models

Due to the complexity of FE models and lack of analytic gradients of the objective functions with respect to the parameters to be updated, it often takes a long time to complete FEMU. Fortunately, surrogate methods provide a feasible approach to significantly reduce the computational time in FEMU [21, 22]. To be specific, FE models can be viewed as a black box with inputs and outputs as structural parameters and modal parameters, respectively. The surrogate methods try to approximate FE models by explicitly fitting the relations between the input and output sampling data generated from the FE models. The fitted explicit functions, i.e. the so-called surrogate models, are the mathematical approximations of the FE models and can be leveraged for FEMU. In this study, Kriging surrogate modeling, developed by Krige [45] and Matheron [46], is adopted. It predicts the output by combining the global trend (described by polynomials) and the local variation (described by a stochastic process) of data, and the deviation between the output response and the polynomial regression is due to modeling error only, regardless of measurement errors and other random factors. Instead of sticking to the numerical precision of the output polynomials, it focuses on constructing an appropriate surrogate model by effectively filling the stochastic process, making it particularly suitable for dealing with nonlinearity. Meanwhile, this method can provide not only predictions of the output response but also measures of uncertainty associated with those predictions. This advantage allows the surrogate models to be gradually updated with improved accuracy by adding sampling points. More details on Kriging surrogate modeling can be referred to Refs. [14, 21].

For the present study, the bridge frequency and MAC Kriging surrogate models are constructed for FEMU. The frequency surrogate models can be simply constructed using the FE model [23]. To construct the MAC models, the high-resolution mode shapes \({{\varvec{\phi}}}_{EXP, j}\) need to be first constructed by the method presented in Sect. 2; then, \({{\varvec{\phi}}}_{FEM,j}({\varvec{X}})\) are calculated from the FE model by varying X; next, the gradients of \({{\varvec{\phi}}}_{EXP, j}\) and \({{\varvec{\phi}}}_{FEM,j}({\varvec{X}})\) are substituted into Eq. (12) to calculate the MAC values; finally, the MAC surrogate models are constructed by following the standard procedure of Kriging modeling [21]. To avoid the indeterminacy of FEMU, the number of modal parameters should not be smaller than that of the to-be-updated parameters [14].

The procedure for updating bridge end constraints is summarized in Fig. 6, including the following five steps:

  1. (1)

    Conducting the ambient/forced vibration tests for the to-be-updated bridge. The sensor positions can be determined by the Latin hypercube sampling algorithm [47], a sample generation method for experimental design.

  2. (2)

    Using the method in Sect. 2 to construct the high-resolution mode shapes.

  3. (3)

    Constructing the frequency and MAC surrogate models for the first two vibration modes. Note that the gradients of the mode shapes, rather than the mode shapes, are adopted to calculate the MAC values.

  4. (4)

    Optimizing the objective functions in Eqs. (10) and (11) by the NSGA-II algorithm.

  5. (5)

    Selecting the preferred Pareto-optimal solution by the criterion of the minimum distance from the equilibrium point as the final updated bridge end constraints.

Fig. 6
figure 6

Flowchart for updating bridge boundary constraints

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4 Illustrative example

A numerical example is presented to illustrate the proposed method in this section. As shown in Fig. 7(a), a two-span continuous and variable cross-sectional bridge is investigated, in which the rectangular cross-section has linearly varying height and constant width. The support constraints are modelled as the translational and rotational springs. The bridge is uniformly discretized into 160 elements with the stiffness and mass matrices of the tapered beam element given in Ref. [48]. The theoretical frequencies and mode shapes are calculated by the FE model-based modal analysis [49]. The frequencies and other bridge model parameters are listed in Table 2. The number of the virtual measurement points is assigned as m = 161 with an interval of 0.5 m so that the virtual measurement points coincide with the nodes of the FE model.

Fig. 7
figure 7

Problem description: a to-be-updated bridge model; b sensor deployment

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Table 2 Parameters of bridge
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In bridge modal parameter identification, the ambient vibration is usually adopted as an excitation source; to simply simulate such excitation [28, 50, 51], a random force F(t) with an amplitude of 5 kN and frequency content of 0 ~ 20 Hz is applied at 2L/5 from the left end, as shown in Fig. 7(b). The bridge responses are calculated by the Newmark-\(\beta\) method with a time step of 0.001 s and a duration of 100 s. The above procedure is called the forward analysis for the numerical study. The bridge accelerations calculated in seven positions are regarded as the measured responses, where these positions (i.e. S1-S7) are determined by the Latin hypercube sampling algorithm, coinciding with the 4th, 31st, 58th, 86th, 110th, 133rd, and 159th virtual measurement points, as denoted by the stars in Fig. 7(b). The bridge frequencies (together with mode shapes) and spring stiffness values listed in Table 2 are regarded as the target values, which will be estimated by FEMU from the bridge accelerations measured by the seven sensors. This procedure is referred to as the backward analysis.

The full-field bridge response is first reconstructed. To construct the sparse vector x, a proper value of the regularization parameter \(\lambda\) in Eq. (4) needs to be assigned. Thus, the full-field responses Y reconstructed from a range of \(\lambda\) are calculated and the corresponding reconstruction errors are compared. The reconstruction error is defined by [52]

$${e}_{j}=\frac{1}{{R}_{j, max}}\sqrt{\frac{1}{s}\sum_{i=1}^{s}{\left[{\widehat{R}}_{j}(i)-{R}_{j}\left(i\right)\right]}^{2}}\times 100\mathrm{\%}$$
(14)

where \({R}_{j}\left(i\right)\) and \({\widehat{R}}_{j}(i)\) are the ith measured (by forward analysis) and reconstructed (by backward analysis) accelerations in the position of the jth sensor, respectively; s is the number of sampling instants; \({R}_{j, max}\) is the maximum value of \({R}_{j}\left(i\right)\). The optimal \(\lambda\) corresponds to the minimum of the average error e defined below:

$$e=\frac{1}{p}\sum_{j=1}^{p}{e}_{j}$$
(15)

Figure 8 presents the curve of e with respect to \(\lambda\), and the optimal \(\lambda\) is selected as 10–7.

Fig. 8
figure 8

The curve of the average error e versus the regularization parameter λ

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To examine the quality of the reconstructed full-field bridge acceleration, Fig. 9 compares the reconstructed accelerations and those calculated by the forward analysis in two different positions. Specifically, Fig. 9(a) shows the accelerations at the measurement point S1, while Fig. 9(c) shows the results obtained at the middle point of the left span where no sensor is installed. Figure 9(b) and (d) are the close views of the gray-box regions in Fig. 9(a) and (c), respectively. From Fig. 9, it is observed that the reconstructed accelerations generally coincide well with those obtained by the forward analysis at both points with and without measurement, although the reconstruction error is slightly larger for the latter. Figure 10 displays the corresponding power spectral densities (PSDs) of the bridge accelerations at the two observation points, in which the PSDs of the reconstructed accelerations match nicely with those of the forward analysis while small errors are observed in the high-frequency region. This infers that the CS theory is generally more effective in the reconstruction of the components of low-order vibration modes [53].

Fig. 9
figure 9

Comparisons of the reconstructed and forward-analysis accelerations: a accelerations at the measurement point S1; b close view of the gray region in (a); c accelerations at the middle point of the left span; d close view of the gray region in (c)

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Fig. 10
figure 10

The PSDs of the reconstructed and forward-analysis accelerations at the two observation points: a at the measurement point S1; b at the middle point of the left span

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Then, the reconstructed full-field acceleration is utilized for modal parameter identification, and the covariance-driven SSI (SSI-cov) algorithm is adopted for its inherent signal-denoising ability [28]. Figure 11 displays the stabilization diagram with the model order ranging from 0 to 100, and the following criteria have been adopted to remove the spurious modes: (1) The change in the frequency and damping ratio of the two consecutive model orders are within 1% and 5%, respectively. (2) The MAC of the mode shapes is higher than 99%. To further unveil the overall spectral characteristics of the reconstructed acceleration field, the average normalized power spectral density (ANPSD) [28] defined below is also superimposed in Fig. 11:

$$ANPSD(f)=\frac{1}{m}{\sum }_{i}^{m}\frac{PSD{(f)}_{i}}{{\sum }_{j}^{s}PSD{({f}_{j})}_{i}}$$
(16)

where m is the number of the measurement channels, and it is equal to the number of virtual measurement points, i.e. m = 161; \(PSD{(f)}_{i}\) is the PSD of the ith channel; s is the number of data points of the ith channel in PSD. By comparing the frequencies of the stable poles with the theoretical ones presented in Table 2, it is concluded that the first three bridge frequencies are successfully identified from the reconstructed acceleration field. The average values of the stable poles are calculated for FEMU.

Fig. 11
figure 11

Stabilization diagram generated by SSI-cov algorithm

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Figure 12 presents the identified bridge mode shapes for the first three vibration modes. In this figure, the blue solid lines stand for the mode shapes calculated from FE model-based modal analysis [49], and the red stars denote the mode shapes directly identified from the accelerations obtained at seven real measurement points, while the black circles represent the mode shapes extracted from the reconstructed acceleration field. Noticeably, the mode shapes identified from the reconstructed acceleration field match well with the other two results, especially for the first two modes. A relatively larger identification error of the mode shape for the third mode can be mainly attributed to the larger reconstruction error of the full-field bridge response in the high-frequency region as shown in Fig. 10. Still, the mode shapes are successfully identified by the proposed method using seven sensors with a resolution of 0.5 m, and they are ready for surrogate-based updating of bridge boundary constraints.

Fig. 12
figure 12

Comparison of the mode shapes identified from measured and reconstructed accelerations and those obtained by modal analysis for the first three vibration modes: a first; b second; c third

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For the present example, the number of the to-be-updated boundary stiffness values is five; to control the degree of illness of FEMU, six objective functions (i.e. no = 6) of the frequencies and mode shapes for the first three modes are established. The bridge frequency and MAC Kriging surrogate models are constructed to reduce CPU time in FEMU. Particularly, 250 bridge boundary constraint vectors (i.e. X) are first generated by the Latin hypercube sampling algorithm, and the lower and upper boundary values of the elements in X are specified as \({10}^{3}\) and \({10}^{12}\) (units in N/m or N \(\bullet\) m/rad), respectively. X is then substituted into the FE model for modal analysis, which generates \({{\varvec{\phi}}}_{FEM,i}\left({\varvec{X}}\right)\) and \({f}_{i}^{FEM}\left({\varvec{X}}\right)\) (i = 1, 2, 3). Next, the gradients of \({{\varvec{\phi}}}_{FEM,i}\left({\varvec{X}}\right)\) and \({{\varvec{\phi}}}_{EXP, i}\) are computed and their MAC values are obtained. The frequency and MAC Kriging surrogate models are finally constructed using the input (i.e. \({\varvec{X}}\)) and output data (frequencies and MAC values) [21]. The constructed surrogate models are the functions of bridge end constraints (i.e. \({k}_{lt}\), \({k}_{lr}\), \({k}_{mt}\) \({k}_{rt}\), and \({k}_{rr}\)). Figure 13 graphically presents the surrogate models and their predicted mean square errors (MSEs) of the first vibration mode with respect to the stiffness values of the left (i.e. \({k}_{lt})\) and middle (i.e. \({k}_{mt}\)) translational springs; the rest stiffness values are specified as \({k}_{lr}={k}_{rr}=2.4\times {10}^{10}\) N \(\bullet\) m/rad and \({k}_{rt}=2.4\times {10}^{10}\) N/m. In these plots, the meshed surfaces denote the Kriging predicted values, while the red dots present the sampling points adopted in the construction of Kriging models. As can be seen from the right column of Fig. 13, the largest predicted MSEs of the surrogate models are generally less than 10–4, which indicates that the constructed surrogate models are of the desired accuracy to approximate the FE models.

Fig. 13
figure 13

The Kriging surrogate models (left column) and predicted MSEs (right column) of the first vibration mode with respect to the stiffness values of the left and middle translational springs: a frequency surrogate model; b MAC surrogate model

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The surrogate models are subsequently utilized for updating bridge boundary constraints. The objective functions presented in Eqs. (10) and (11) are optimized by the NSGA-II algorithm using the following parameters to ensure both accuracy and efficiency: the population size of 250, and the maximum number of generations (i.e. iterations) equal to 300. Figure 14 shows the distribution of the non-dominated solutions obtained by the NSGA-II algorithm, in which the red lines stand for the target values. Afterward, the preferred solution is selected by the criterion of the minimum distance from the equilibrium point as X = [1 \(\times\) 109, 4.09 \(\times\) 109, 9.64 \(\times\) 1010, 3.30 \(\times\) 1010, 1 \(\times\) 109]T and is represented by the red stars in Fig. 13. By comparing this preferred solution with the target value X = [1 \(\times\) 109, 5 \(\times\) 109, 5 \(\times\) 1010, 2.4 \(\times\) 1010, 2 \(\times\) 108]T, it is found that the updated solution is very close to the target value. The estimation errors mainly arise from the identification errors of the high-resolution mode shapes and the construction errors of surrogate models. The updated stiffness values are passed to the FE model, and the first three frequencies (relative errors) are computed as 1.97 Hz (0.50%), 3.15 Hz (1.25%), and 7.51 Hz (1.62%), respectively. Figure 15 presents the corresponding mode shapes and their gradients. The corresponding MAC values between the updated and target mode shapes for the first three modes are 0.989, 0.981, and 0.994 for the mode shapes; for the gradients of the mode shapes, the MAC values are 0.968, 0.977, and 0.982. The above observations show that the updated stiffness values have satisfactory accuracy.

Fig. 14
figure 14

The distribution of the non-dominated solutions obtained by the NSGA-II algorithm: a \({k}_{lt}\); b \({k}_{lr}\); c \({k}_{mt}\); d \({k}_{rt}\); e \({k}_{rr}\)

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Fig. 15
figure 15

The mode shapes (left column) and their gradients (right column) of the updated model for the first three modes: a first; b second; c third

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On the other hand, the updating process is computationally efficient. The total CPU time from the construction of surrogate models to the selection of the preferred solution is 2,057.9 s by a laptop equipped with an Intel Core i5-12500H CPU at 4.5 GHz and a RAM of 16 GB. In contrast, the result updated by directly using the FE model (without using the surrogate models) is X = [1 \(\times\) 109, 4.8 \(\times\) 109, 3.4 \(5\times\) 1010, 4.6 \(7\times\) 1010, 5.2 \(\times\) 108]T, and the corresponding frequencies (relative errors) are 2.00 Hz (1.01%), 3.21 Hz (0.63%), and 7.41 Hz (0.27%); the MAC values for the mode shapes are 0.999, 0.999, and 0.999, while those for the mode shape gradients are 0.995, 0.996, and 0.996. Although these results have slightly better accuracy than those obtained by the surrogate-assisted updating, the CPU time is 465,279.2 s, which is about 226 times longer than that of the surrogate-assisted updating. The above results demonstrate that surrogate-assisted optimization can greatly reduce CPU time with competitive accuracy.

To highlight the gain of the high-resolution mode shapes in the identification of bridge boundary constraints, the results obtained using the low-resolution mode shapes (denoted by the red stars in Fig. 12) are also presented. Since the resolution of the mode is low and the measurement points are not uniformly distributed, the MAC values are directly calculated using the mode shapes (instead of the mode shape gradients). By following the same procedure and parameter setting, the preferred solution is obtained as X = [9.38 \(\times\) 1010, 1.75 \(\times\) 109, 8.42 \(\times\) 1010, 3.11 \(\times\) 1010, 2.49 \(\times\) 109]T. The frequencies (relative errors) of the updated model are 2.12 Hz (7.07%), 3.31 Hz (3.76%), and 7.69 Hz (4.06%). The corresponding MAC values for the mode shapes are 0.971, 0.974, and 0.965 and those for the gradients of the mode shapes are 0.962, 0.971, and 0.962. Noticeably, the identified X is much worse relative to the result obtained by using the high-resolution mode shapes.

5 Experimental validation

In this section, the proposed method will be validated by the laboratory test data provided by a research team from Kyoto University [54]. The scaled bridge was modeled as an H-shaped steel beam with a uniform cross-section as shown in Fig. 16(a). Five accelerometers (denoted as S1 ~ S5), installed at \(\frac{2}{12}\), \(\frac{3}{12}\), \(\frac{6}{12}\), \(\frac{9}{12}\), and \(\frac{10}{12}\) of the bridge length from its left end, were used to record the bridge response, as schematically shown in Fig. 16(b). The boundary conditions were set to simulate the ideal hinge and roller, as displayed in Fig. 16(c). To identify the bridge modal parameters, the bridge was pressed downward at midspan and then released five times. The bridge accelerations were sampled at a frequency of 500 Hz, as displayed in Fig. 16(d). The tests were originally conducted for the indirect identification of bridge frequencies [55]. Herein, only the first free-decay responses (marked by the dashed orange box) of the bridge without a vehicle are adopted. The rest bridge parameters are listed in Table 3.

Fig. 16
figure 16

The setting and responses of the scaled bridge: a cross-section of the beam; b sensor deployment; c close views of beam supports [55]; d recorded accelerations

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Table 3 Parameters of H-shaped beam
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To construct the high-resolution mode shapes, the beam is discretized by 108 elements, and the number of virtual measurement points is m = 109 with an interval of 0.05 m. As the number of the virtual measurement points is significantly larger than that of the real measurement points (p = 5), the sensor deployment can be viewed as sparse. The overall average error e with respect to the regularization parameter \(\lambda\) is shown in Fig. 17(a), from which the minimum e is achieved when \(\lambda =0.01\), and it is adopted for reconstructing the acceleration field. In Fig. 17(b), the reconstructed accelerations in the sensor positions are compared with the measured values, and good agreement between the reconstructed and measured results is observed.

Fig. 17
figure 17

The responses of the bridge: a the overall average error e with respect to the regularization parameter \(\lambda\); b comparison of the reconstructed and measured accelerations

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Subsequently, the reconstructed full-filed responses are adopted to identify the high-resolution mode shapes. Figure 18 shows the identified mode shapes using the measured (by five accelerometers) and reconstructed full-field responses for the first two vibration modes. For comparison, the theoretical mode shapes of the simply supported beam (\({\text{sin}}\frac{i\pi x}{L}\), i = 1, 2) are also superimposed in this figure. Obviously, the three kinds of mode shapes agree with each other very well for both modes. This observation indicates that the high-resolution mode shapes are very accurate and the boundary conditions of a scaled bridge can be viewed as the simply supported one.

Fig. 18
figure 18

Comparison of the identified mode shapes for the first two modes: a first; b second

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Next, the FE model is constructed using the parameters listed in Table 3, and the high-resolution mode shapes are adopted for constructing the MAC surrogate models. As the scaled bridge is a single-span beam, the to-be-updated bridge boundary constraint vector is \({\varvec{X}}={\left[\begin{array}{ccc}{k}_{lt}& \begin{array}{cc}{k}_{lr}& {k}_{rt}\end{array}& {k}_{rr}\end{array}\right]}^{T}\) (see Fig. 3). The frequency and MAC surrogate models for the first two vibration modes are constructed to update \({\varvec{X}}\). The preferred solution is found as X = [7.14 \(\times\) 1010, 22.54, 7.12 \(\times\) 1010, 27.66]T, and the first two frequencies (relative errors) predicted by the updated FE model are 3.62 Hz (0.23%) and 14.48 Hz (0.91%), respectively. Figure 19 compares the gradients of the mode shapes predicted by the updated FE model and those identified from the reconstructed acceleration field for the first two vibration modes. The corresponding MAC values of the mode shape gradients for the two modes are 0.999 and 0.998, respectively. The above results demonstrate that the boundary conditions have been greatly identified.

Fig. 19
figure 19

Comparison of the identified mode shape gradients for the first two modes: a first; b second

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6 Conclusions

A numerical framework for updating bridge boundary constraints based on the high-resolution mode shapes as constructed from the sparse measurement is newly proposed in this study. Particularly, the high-resolution mode shapes are constructed by the compressive sensing theory relying on the sparse nature of the bridge responses in the spatial and temporal domains. To reduce the CPU time, the frequency and MAC Kriging surrogate models are constructed and adopted in FEMU. The NSGA-II algorithm is applied to optimize the multi-objective functions, while the criterion of minimum distance from the equilibrium point is adopted for selecting the preferred solution. In the experimental validation, the merit of using a simply supported beam provides the analytical solutions for both mode shapes and their gradients, thereby facilitating the validation of the updated boundary constraints. More complex boundary identification will be part of future work. From the numerical investigation, the following conclusions can be drawn:

  1. (1)

    Using the compressive sensing theory, the full-field response can be constructed from the measured responses by few sensors.

  2. (2)

    The high-resolution mode shapes can be accurately identified from the reconstructed full-field responses by the SSI-cov algorithm.

  3. (3)

    The surrogate-assisted FEMU significantly reduces the CPU time while providing a satisfactory updated result.

  4. (4)

    The accuracy of the updated bridge end constraints can be dramatically improved by using the high-resolution mode shapes in contrast to the low-resolution ones.