Bayesian model updating method based on multi-sampling strategy integration

Abstract

The Bayesian model updating method is widely applied in structural health monitoring. Traditional Bayesian model updating methods suffer from limitations such as dimensional constraints, slow convergence, and low computational efficiency. To enhance the convergence speed and computational efficiency of Bayesian model updating methods, this paper proposes a multi-sampling strategy integrated Bayesian model updating method based on the Differential Evolution Adaptive Metropolis (DREAM) algorithm. First, the Kalman-inspired distribution and sampling difference vectors from past states are introduced into the DREAM algorithm, addressing the issue of parallel chain limitations in DREAM and improving the exploration efficiency of the posterior distribution. Second, drawing inspiration from clustering algorithms that use centroids for data point clustering, a novel centroid update sampling strategy is proposed and integrated with other sampling strategies to increase sampling diversity among different chains, thereby avoiding local optima and accelerating the convergence process. Finally, the proposed method’s effectiveness is validated through numerical examples of simply supported beams and experimental examples of a three-story frame structure. The results demonstrate that the proposed Bayesian model updating method based on multi-sampling strategy integration achieves high updating accuracy and faster convergence. The updated model shows improved ability to simulate the inherent properties and response characteristics of actual structures, making it suitable for use as a benchmark model in structural health monitoring.

Keywords

Kalman-inspired distribution sampling difference vectors from past states centroid update multi-sampling strategy integration Bayesian model updating

Introduction

The accuracy of the structural baseline finite element model (FEM) is crucial in structural health monitoring (Kamariotis et al., 2022; Zhou et al., 2016). However, due to factors such as model simplifications and structural damage, there exists a certain discrepancy between the FEM and the actual structure, rendering the established FEM unsuitable for use as a baseline model in structural health monitoring (Rosati et al., 2022). Therefore, finite element model updating (FEMU) techniques that can reduce the error between the FEM and the actual structure have garnered significant attention from many researchers (Caicedo and Zárate, 2016; He et al., 2024; Lin et al., 2022). Among the various FEMU methods, deterministic FEMU methods were developed earliest. However, these methods treat the parameters to be updated as constants without considering the uncertainties that are prevalent in practical engineering, which leads to a reduction in the reliability of the updated FEM (Simoen et al., 2015). To address this issue, many researchers have proposed probabilistic FEMU methods. These methods quantify multi-source uncertainties in practical engineering by treating the parameters to be updated as probability distributions, and have become a mainstream direction in the research of FEMU methods (Liu and Jiang, 2022).

In probabilistic FEMU methods, Bayesian inference is widely applied due to its advantages in statistical inference and parameter estimation, gradually forming the Bayesian model updating framework (Xu et al., 2022). Within this framework, the assumptions of prior information, the construction of the likelihood function, and the estimation of posterior probability density have become the three key issues in Bayesian model updating research. The assumptions of prior information are typically determined based on the researcher’s experience, multiple experimental tests, and the prior uncertainty quantification of parameters (Faes et al., 2019; Lam et al., 2024). The likelihood function is a conditional probability density function (PDF) that provides a measure of consistency between the model output and test data (Goller et al., 2012). Over the years, the forms of likelihood functions have become increasingly diverse. Examples include normal likelihood functions (Goller et al., 2011), Gaussian likelihood functions (Shi et al., 2024), combined normal-lognormal distribution likelihood functions (Das and Debnath, 2018), approximate Bayesian computation likelihood functions (Bi et al., 2019), and likelihood-free Bayesian inference (Ni et al., 2022). Although different likelihood functions have demonstrated their respective advantages in specific problems, the Gaussian likelihood function remains the most widely used due to its efficiency and versatility. Once the prior information and likelihood function are determined, the estimation of the posterior PDF of the parameters to be updated becomes a critical issue that affects the accuracy of Bayesian model updating. Since the estimation of the posterior PDF of the parameters involves high-dimensional integration, it is generally challenging to directly obtain the posterior PDF of the parameters to be updated (Beck and Au, 2002). To address this issue, various sampling-based posterior PDF estimation methods have been developed. Lam et al. (2015) used the Markov chain Monte Carlo (MCMC) simulation method to sample a set of models in high-probability regions to resolve the unresolvable posterior PDF of parameters in complex parameter spaces, and validated their approach through a coupled floor system of a building structure. Sengupta and Chakraborty (2023) proposed a Bayesian model updating method based on the improved transitional Markov chain Monte Carlo (TMCMC), which enhances the posterior PDF estimation of the parameters by considering different dimensional observations in the sampling space. This method was validated through numerical and experimental examples. Ding et al. (2022) applied the Delayed Rejection Adaptive Metropolis-Hastings (DRAM) algorithm for Bayesian model updating, and validated the proposed FEMU method through a three-story frame structure. However, the aforementioned Bayesian model updating methods based on MCMC and its improved forms are single-chain methods, whose convergence performance depends on the prior distribution, and they exhibit certain limitations when addressing high-dimensional problems with large data samples (Vrugt et al., 2008; Zeng and Kim, 2022). In contrast, multi-chain methods, which involve the parallel operation of multiple Markov chains combined with different sampling strategies, have emerged as a powerful approach for handling complex posterior PDF. Sherri et al. (2019) proposed a multi-chain FEMU method based on the Differential Evolution Markov Chain Monte Carlo (DE-MC) algorithm, combining the Differential Evolution (DE) algorithm with the MCMC algorithm, and validated the proposed FEMU method through two real-world examples. Chen et al. (2022) compared the performance of the DE-MC algorithm with traditional MCMC algorithms, demonstrating the advantages of DE-MC in multimodal problems and parameter estimation. Vrugt (2016) proposed the Differential Evolution Adaptive Metropolis (DREAM) algorithm, which performs a global search through multiple chains and automatically adjusts the scale and direction of sample distribution in the random subspace during the search process. Jin et al. (2019) compared the DRAM, adaptive hybrid MCMC, and DREAM algorithms, finding that DREAM outperforms the other two methods in terms of parameter estimation convergence. However, in the standard DREAM algorithm, the number of parallel chains N must be set to at least d/2 (where d is the number of parameters to be updated), and each chain needs to converge to achieve the posterior PDF of the parameters, which presents challenges such as outlier chains, slow convergence, and high computational cost in practical engineering applications (Qin et al., 2023).

Based on the above analysis, this paper addresses the issues of multi-chain MCMC methods in FEMU by proposing a multi-sampling strategy integrated Bayesian model updating method based on the standard DREAM algorithm. First, the Kalman-inspired distribution and differential vector sampling strategy from past states are introduced into the DREAM algorithm to resolve the limitation on the number of parallel chains in the DREAM algorithm, thereby improving the exploration efficiency of the posterior distribution. Second, inspired by the concept of centroid-based data clustering in clustering algorithms, a centroid update sampling strategy is proposed and integrated with other sampling strategies to increase sampling diversity among different chains, avoiding local optima and accelerating the convergence process of the algorithm. Finally, the effectiveness of the proposed Bayesian model updating method based on multi-sampling strategy integration is validated through numerical examples of simply supported beams and experimental examples of a three-story frame structure.

Bayesian model updating theory

The core of the Bayesian model updating theory is to combine the prior information of the parameters to be updated with experimental observation data, using Bayes' theorem to solve the posterior distribution of the parameters. It is worth noting that the experimental observation data used in this paper are structural modal parameters. Therefore, assuming that in structural dynamic modeling and simulation analysis, the parameter to be updated is θ , and the experimental modal parameters are x ^exp, the Bayes' formula for solving the posterior distribution of the parameter θ can be expressed as:

p (θ | x^{\exp}) = \frac{p (x^{\exp} | θ) p (θ)}{\int p (x^{\exp} | θ) p (θ) d θ}

(1)

where

p (θ | x^{\exp})

and

p (θ)

are the posterior and prior PDF of the parameters θ , respectively;

p (x^{\exp} | θ)

is the likelihood function constructed in the FEMU, used to represent the consistency between the FEM’s modal parameters and the experimental modal parameters, given the known uncertainty parameters;

\int p (x^{\exp} | θ) p (θ) d θ

is the normalization constant. If we denote the normalization constant by c, then equation (1) can be expressed as:

p (θ | x^{\exp}) = c \cdot p (x^{\exp} | θ) p (θ) \propto p (x^{\exp} | θ) p (θ)

(2)

where the posterior PDF

p (θ | x^{\exp})

of the parameter θ is proportional to the likelihood function. Therefore, in Bayesian model updating, once the prior information of the parameter θ is determined, the larger the value of the likelihood function, the closer the parameter θ is to the true value. However, in practical engineering applications, the parameter θ to be updated, often involves section dimensions of structural elements, material properties, boundary conditions, etc., whose prior distribution is usually unknown. Nevertheless, these parameters have clear physical significance, allowing for an approximate determination of their most likely range. In this paper, the Bayesian prior information is assumed to follow a normal distribution as described below:

p (θ) = N (μ, cov (θ)) = \exp (- \frac{1}{2} {(θ - μ)}^{T} cov {(θ)}^{- 1} (θ - μ))

(3)

where µ represents the prior mean of the parameter θ to be updated; cov( θ ) is the prior variance of θ . Once the prior information of θ is determined, the FEM modal parameters x ^fem can be calculated based on different prior information. The likelihood function can then be constructed based on the error residual between x ^fem and the experimental modal parameters x ^exp, where the error residual ε can be expressed as:

ε = x^{f e m} (θ) - x^{\exp}

(4)

where the error residual ε represents numerous uncertainty factors present in practical engineering applications, such as measurement errors and environmental fluctuations. Using the joint PDF from equation (4) to construct the likelihood function, and assuming a multi-source Gaussian distribution, the constructed likelihood function can be expressed as:

p (x^{\exp} | θ) = \prod_{i = 1}^{n} \frac{1}{{(2 π)}^{d / 2} {| σ |}^{1 / 2}} \exp (- \frac{1}{2} (ε_{i}^{T} σ^{- 1} ε_{i}))

(5)

where n is the number of test data samples; d is the dimensionality of the data; σ is the covariance matrix of the experimental modal parameters, obtained through statistical analysis of multiple measurement results. Generally, to simplify calculations and ensure numerical stability, equation (5) is often transformed into its logarithmic form:

L (x^{\exp} | θ) = - \frac{n d}{2} \log (2 π) - \frac{n}{2} \log (| σ |) - \frac{1}{2} \sum_{i = 1}^{n} ε_{i}^{T} σ^{- 1} ε_{i}

(6)

where

L (x^{\exp} | θ)

represents the logarithmic form of the likelihood function. After determining the likelihood function, substituting equations (3) and (5) into equation (2) gives:

p (θ | x^{\exp}) \propto \exp (- \frac{1}{2} J (θ))

(7)

J (θ) = ε^{T} σ^{- 1} ε + {(θ - μ)}^{T} cov {(θ)}^{- 1} (θ - μ)

(8)

From equation (7), it can be seen that the posterior PDF of the structural parameter θ to be updated is maximized when J( θ ) approaches zero. However, for complex models, it is not possible to directly obtain the posterior PDF of θ , and precise numerical methods are required. In this paper, a multi-chain MCMC algorithm with multi-sampling strategy integration is used to obtain the structural parameter samples corresponding to the maximum posterior PDF of θ . By using all the parameter samples from multiple Markov chains after convergence, the mean, variance, and other statistics of each parameter to be updated can be calculated, thereby achieving the purpose of FEMU.

Bayesian algorithm with multi-sampling strategy integration

DREAM algorithm

The DREAM algorithm is a multi-chain MCMC algorithm that initially generates N starting sample points, corresponding to N Markov chains, based on the prior information of the parameters. At the t-th iteration of the i-th chain, a $d^{'}$ -dimensional subset is generated within the d-dimensional parameter space using a subspace sampling method. This $d^{'}$ -dimensional subset is then used to generate the candidate sample $θ_{d^{'}, t}^{i p}$ as follows:

θ_{d^{'}, t}^{i p} = (I_{d^{'}} + ψ_{d^{'}}) γ (δ, d^{'}) [\sum_{m = 1}^{δ} θ_{d^{'}, t - 1}^{r_{1} (m)} - \sum_{n = 1}^{δ} θ_{d^{'}, t - 1}^{r_{2} (n)}] + ξ_{d^{'}} + θ_{d^{'}, t - 1}^{i}

(9)

where

θ_{d^{'}, t}^{i p}

represents the candidate sample point generated by the i-th chain at the t-th iteration;

I_{d^{'}}

is a

d^{'}

d^{'}

identity matrix;

ψ_{d^{'}}

and

ξ_{d^{'}}

are small numbers randomly generated, following the uniform distribution U _d (-v, v) and the Gaussian distribution N _d (0, β), respectively, where typically v = 0.05 and β = 10⁻¹²;

γ (δ, d^{'}) = 2.38 / \sqrt{2 δ d^{'}}

is the scaling factor;

δ

is the number of parallel chain pairs used to generate the candidate sample;

r_{1} (m), r_{2} (n) \in {1, \dots, i, \dots, N}

are the randomly selected indices of the parallel chains, and

r_{1} (m) \neq r_{2} (n)

m, n \in {1, \dots, δ}

. After sampling, the acceptance of the candidate sample

θ_{d^{'}, t}^{i p}

is determined based on the crossover probability CR:

θ_{d^{'}, t}^{i} = {\begin{cases} θ_{d^{'}, t - 1}^{i}, if U \leq 1 - C R, d_{t}^{'} = d_{t - 1}^{'} - 1 \\ θ_{d^{'}, t}^{i p}, otherwise \end{cases}

(10)

where

U \in [0, 1]

is a random number that follows a uniform distribution;

C R = m / n_{C R}

m = 1, \dots, n_{C R}

n_{C R} = 3

. Once the candidate sample is determined, its posterior probability density and acceptance rate

α (θ_{d^{'}, t}^{i p}, θ_{d^{'}, t - 1}^{i})

are calculated as follows:

α (θ_{d^{'}, t}^{i p}, θ_{d^{'}, t - 1}^{i}) = {\begin{cases} \min (\frac{p (θ_{d^{'}, t}^{i p} | y)}{p (θ_{d^{'}, t - 1}^{i} | y)}, 1), if p (θ_{d^{'}, t - 1}^{i} | y) > 0 \\ 1, if p (θ_{d^{'}, t - 1}^{i} | y) = 0 \end{cases}

(11)

where

p (θ_{d^{'}, t}^{i p} | y)

is the posterior probability density of the candidate sample

θ_{d^{'}, t}^{i p}

; y represents the actual observations. According to equation (10), the acceptance of the new candidate sample is determined, continuing until a complete Markov chain is obtained, and the Inter-Quartile-Range (IQR) method is used to detect and remove outlier chains (Vrugt, 2016). Then, a quantitative convergence criterion, namely the scale reduction factor

R_{c}

, is used to assess the convergence of the parallel chains in the DREAM algorithm. The scale reduction factor

R_{c}

is calculated using the following formula:

R_{c} = \sqrt{\frac{L - 1}{L} + \frac{N^{*} + 1}{N^{*}} \frac{B}{L} \frac{1}{W}}

(12)

where L represents the evolution generations of the Markov chains;

N^{*}

is the number of Markov chains used to assess convergence;

B / L

is the variance of the means of the

N^{*}

Markov chains used for convergence assessment;

W

is the mean variance of the

N^{*}

Markov chains. When the convergence diagnostic R_c < 1.2 for all parameters, it can be concluded that the Markov chains have entered the stationary phase and converged to a stable posterior distribution (Vrugt, 2016). Otherwise, the number of iterations needs to be increased to improve convergence.

Kalman-inspired distribution

The Kalman-inspired distribution is motivated by the state analysis steps of the Kalman filter and is designed to enhance the exploration efficiency of the chain towards the posterior distribution when dealing with high-dimensional target distributions. The candidate sample $θ_{p}$ is calculated based on the distance between the model output $x^{f e m} (θ)$ and the measured response $x^{\exp}$ :

\begin{array}{l} θ_{p} = C_{θ x} {(C_{x x} + σ)}^{- 1} [x^{\exp} (θ_{t - 1}) + ε_{t - 1} - x^{f e m} (θ_{t - 1})] + θ_{t - 1} \\ = θ_{t - 1} + K r_{t - 1} + K ε_{t - 1} \\ = θ_{t - 1} + Δ θ_{t - 1} \end{array}

(13)

K = C_{θ x} {(C_{x x} + σ)}^{- 1}

(14)

where

C_{θ x} = C o v (θ, x)

is the cross-covariance matrix between the model parameters and the model output;

C_{x x} = C o v (x, x)

is the autocovariance matrix of the model output;

σ

is the measurement error covariance matrix from equation (5);

ε_{t - 1}

represents the modeling and measurement errors from equation (4);

r_{t - 1} = x^{\exp} (θ_{t - 1}) - x^{f e m} (θ_{t - 1})

is the residual vector of the parameter

θ_{t - 1}

;

K

is the Kalman gain.

Sampling difference vectors from past states

To reduce the number of parallel chains and improve the convergence speed of the algorithm, the differential vector in the update process of equation (9) can be sampled from past states of the Markov chains. This approach avoids the requirement that the number of parallel Markov chains N must be greater than the parameter dimension d. The main steps for sampling differential vectors from past states of the Markov chains are as follows:

(1) Generate an initial set of past state vectors $Z = {X^{1}, \dots, X^{M_{0}}}$ based on the prior distribution with M₀ > max(d, N), where the first N rows of Z are set as the initial population $X = {X^{1}, \dots, X^{N}}$ , and set M = M₀.

(2) Define the sparsification rate K, where the set X is updated every K iterations.

(3) Add the updated set X to Z , resulting in M = M + N.

(4) If convergence is achieved (R_c< 1.2) or the final iteration number T is reached, proceed to step 5; otherwise, return to step 2.

(5) Discard the initial samples and burn-in samples from Z , resulting in the posterior samples of the parameters.

Centroid update

The strategy of estimating the optimal solution direction through centroids is widely used in data science and machine learning, particularly in clustering algorithms (Kong et al., 2024; Xu and Jiang, 2022). Inspired by the concept of using centroids for data point clustering in clustering algorithms, this paper proposes a centroid update sampling strategy, which is integrated with other sampling methods and applied to the DREAM algorithm. The core of the proposed centroid update sampling strategy is to use the centroid to update the state of the Markov chain. In each iteration, the centroid of multiple Markov chains is used as the target position to guide each chain’s movement, ensuring that all chains gradually converge toward the center of the overall data distribution. The specific main steps are as follows:

(1) Randomly select N′ chains $Z^{a}, Z^{b}, \dots, Z^{N^{'}}$ from the initial N Markov chains.

(2) Calculate the centroid G of the N′ chains $Z^{a}, Z^{b}, \dots, Z^{N^{'}}$ using the centroid calculation formula: $G = \frac{Z^{a} + Z^{b} + \dots + Z^{N^{'}}}{N^{'}}$ .

(3) Define the step size coefficient λ, and update the sample $X^{i p}$ using the formula: $X^{i p} = X^{i} + λ (G - X^{i})$ .

(4) Determine whether to accept the new candidate sample $X^{i p}$ based on equation (11).

From the above steps, it can be seen that the choice of step size coefficient λ is crucial. An appropriate step size coefficient λ helps guide all chains to gradually converge toward the center of the overall data distribution. However, if the step size is too large, it may cause excessive jumps during the update process, affecting the effectiveness of the sampling. Conversely, if the step size is too small, the convergence may be too slow. Therefore, it is usually necessary to continuously adjust the step size based on the specific problem to achieve optimal sampling efficiency. Compared to the Snooker update strategy (Qin et al., 2023), the centroid update sampling strategy does not require correction when calculating the acceptance rate, which improves the computational efficiency of the algorithm. Additionally, by moving the current state of each Markov chain toward the centroid position of multiple Markov chains, the strategy helps ensure that all chains move closer to the overall data distribution center, making them more aligned with the global probability distribution. This approach helps avoid getting trapped in local optima and accelerates the convergence process of the algorithm. Moreover, for multi-chain algorithms, the centroid update sampling strategy increases diversity among different chains, thereby enhancing sampling efficiency.

Multi-sampling strategy integration

The Bayesian algorithm with integrated multi-sampling strategies is based on the DREAM algorithm, combining the Kalman-inspired distribution, sampling difference vectors from past states, and the centroid update sampling strategy. Based on the theories discussed in the previous sections, the calculation formula for the candidate sample $θ_{p}^{i}$ in the i-th chain at iteration t is as follows:

θ_{p}^{i} = θ_{t - 1}^{i} + Δ θ_{t}^{i}

(15)

where

Δ θ_{t}^{i}

represents the update distance of the candidate sample at iteration t. The calculation formula for

Δ θ_{t}^{i}

at iteration t is as follows:

{\begin{cases} \begin{array}{l} Δ θ_{t}^{i} = γ_{K} (K_{t - 1} r_{t - 1}^{i} + K_{t - 1} ε_{t - 1}^{i}), if u \in [0, p_{K}] \\ Δ θ_{t}^{i} = λ (G - X^{i}), if u \in (p_{K}, p_{K} + p_{z}] \end{array} \\ Δ θ_{t}^{i} = (I_{d^{'}} + ψ_{d^{'}}) γ (δ, d^{'}) [\sum_{m = 1}^{δ} Z_{t}^{r_{1} (m)} - \sum_{n = 1}^{δ} Z_{t}^{r_{2} (n)}] + ξ_{d^{'}}, if u \in (p_{K} + p_{z}, 1] \end{cases}

(16)

where

p_{K}

and

p_{z}

represent the probabilities of using the Kalman-inspired distribution and the centroid update sampling strategy, both set to 0.3 to achieve a balanced advantage of the three sampling methods and improve FEMU efficiency;

u \sim U (0, 1)

is a random number that follows a uniform distribution;

r_{1} (m)

and

r_{2} (n)

are the indices of the parallel chains

Z_{t}^{r_{1} (m)}

and

Z_{t}^{r_{2} (n)}

, randomly selected from the past state vectors at iteration t; other parameters have the same meanings as described in the previous sections. It should be noted that, to compare the performance of the proposed algorithm with that of the algorithm presented in the reference (Zhang et al., 2020) and to ensure the fairness of the comparison, parameters shared between the two algorithms are set to the same values in both numerical and experimental cases. This approach aims to make performance differences stem mainly from the algorithms themselves rather than parameter settings. Specifically, the initial number of Markov chains N is set to 10, the maximum number of iterations T is set sufficiently high at 10,000 to allow the algorithm to reach convergence, the initial row count M₀ of the past state vector Z is set to 10d, the sparsification rate K is set to 10, and the step size coefficient λ is set to 0.3.

After updating the candidate samples, the acceptance rate of the candidate samples is calculated according to equations (11) and (12). The outlier chains are detected, and the convergence is assessed, ultimately resulting in an accurate and stable posterior distribution of the parameters. The flowchart of the Bayesian model updating method based on multi-sampling strategy integration is shown in Figure 1.

Figure 1.

Flowchart of the Bayesian model updating method based on multi-sampling strategy integration.

Numerical example

In this section, a simply supported beam numerical example is used to validate the proposed Bayesian model updating method based on multi-sampling strategy integration. Table 1 provides the dimensional and material parameters of the simply supported beam, including length l, width w, height h, density ρ, elastic modulus E₀, and Poisson’s ratio

μ

. The left end of the beam is fixed with a hinge support, while the right end has a sliding hinge support. The beam is divided into 10 elements along its length, each element being 200 mm long, with element numbers ranging from 1 to 10. 9 measurement points are set along the length of the beam. The schematic diagram of the simply supported beam structure is shown in Figure 2.

Table 1.

Simply Supported Beam Parameters.

l (mm)	w (mm)	h (mm)	ρ (kg⋅m⁻³)	E₀ (Pa)	$μ$
2000	100	10	7800	2.1 × 10¹¹	0.3

Figure 2.

Simply supported beam structure (Unit: mm).

To simulate actual structural damage, the elastic modulus of six out of the 10 elements are selected as the parameters to be updated. The selected element numbers are 2, 4, 6, 7, 9, and 10. The initial values of the elastic modulus for these elements are taken as E₀, the elastic modulus provided in Table 1. The true values of the elastic modulus E for the selected elements are determined by randomly decreasing or increasing the initial values E₀. During the updating process, the parameters θ to be updated are the normalized values of the selected elements' elastic modulus relative to the initial value E₀, i.e.,

θ = {[θ_{1} = E_{2} / E_{0}, \dots, θ_{6} = E_{10} / E_{0}]}^{T}

. The initial and true values of the selected elements' parameters are shown in Table 2.

Table 2.

Initial and True Values of the Updating Parameters.

Element number	θ ₁	θ ₂	θ ₃	θ ₄	θ ₅	θ ₆
Initial value	1	1	1	1	1	1
True value	1.1	0.9	0.8	1.2	1.3	1.1

To simulate actual measured responses, the true values of θ are assumed to follow a normal distribution N _i(θ_i, (0.1θ_i)²) (Qin et al., 2023), and the experimental modal parameters corresponding to the generated θ values are treated as the actual measured responses. To demonstrate that the proposed algorithm can avoid getting trapped in local optima, the initial sampling range for the parameters to be updated is set sufficiently large, specifically [0.1, 2.5]. The first six natural frequencies and mode shape data of the simply supported beam are selected as the experimental modal parameters to construct the likelihood function. The Bayesian algorithm with multi-sampling strategy integration is then used to approximate the posterior PDF of the parameters to be updated. During the model updating process, the total iteration process is divided into 10 stages, with the mean posterior PDF of the parameters at each iteration stage shown in Figure 3.

Figure 3.

Mean posterior PDF of the parameters to be updated at each iteration stage.

As shown in Figure 3, the initially generated values of the parameters to be updated (at stage 0) are far from the true values. However, as the number of iterations increases, the mean posterior PDF of the parameters rapidly decrease and gradually converge to values near the true values, indicating that the model has reached a stable state. To illustrate this process in detail, one of the Markov chains from the initially generated N chains is projected onto the θ₁ and θ₂ plane. The convergence plots of θ₁ and θ₂ samples at different stages are shown in Figure 4.

Figure 4.

Convergence plots of θ₁ and θ₂ samples at different stages.

In Figure 4, stage 0 represents the 60 random samples uniformly generated within the interval [0.1 2.5], which serves as the initial past state vectors. According to the theory in sampling difference vectors from past states, the posterior samples of the parameters to be updated are obtained by continuously discarding the initial past state vectors and burn-in samples, as illustrated in stages 2, 6, and 10 in the figure. It is important to note that the true values of θ₁ and θ₂ are 1.1 and 0.9, respectively. In the final stage (i.e., stage 10), the mean posterior PDF of θ₁ and θ₂ concentrate towards (1.0972, 0.8979), and similar results are obtained for the other parameters to be updated, which are not elaborated here. The joint and one-dimensional marginal posterior distributions of the parameters to be updated are shown in Figure 5.

Figure 5.

Joint and one-dimensional marginal posterior distributions of the parameters to be updated.

In Figure 5, the upper triangular blocks display scatter plots of the parameters to be updated, which can be used to determine the range of the posterior sample distributions. The lower triangular blocks show the density cloud plots of the posterior sample distributions (with colors ranging from purple to red indicating increasing point density). These plots reveal that, as the number of iterations increases, the posterior samples of the parameters eventually converge to the red areas with higher point density, indicating that the parameter values have stabilized. The diagonal blocks present histograms of the posterior distributions, corresponding to the scatter plots and density cloud plots, which reflect the range and convergence status of the posterior sample distributions. The posterior PDF of the parameters can be obtained from these histograms to estimate the parameter values. After running the program 10 times, the estimated means and errors of the parameters are presented in Table 3.

Table 3.

Estimated Means and Errors of the Parameters to Be Updated.

Parameters	Initial value	True value	Initial errors (%)	Updated value	Updated variance (%)	Updated error (%)
θ ₁	1	1.1	10	1.0972	0.0701	0.2545
θ ₂	1	0.9	10	0.8979	0.0964	0.2333
θ ₃	1	0.8	20	0.8010	0.0899	0.1250
θ ₄	1	1.2	20	1.2038	0.1172	0.3167
θ ₅	1	1.3	30	1.3071	0.1111	0.5461
θ ₆	1	1.1	10	1.1004	0.1487	0.0364

As shown in Table 3, the updating errors of the parameters after being updated by the proposed algorithm are relatively small, all reduce to below 0.6%, with the posterior distribution variances of the parameters being less than 0.4%. This demonstrates that the proposed algorithm has high updating accuracy and good stability. To further verify the proposed algorithm, the errors in the first six natural frequencies of the simply supported beam before and after updating, as well as the first six normalized mode shapes, are compared, as presented in Table 4 and Figure 6.

Table 4.

Errors in the first six Frequencies of the simply Supported Beam Before and After Updating.

Frequency	Initial value (Hz)	True value (Hz)	Updated value (Hz)	Updated error (%)
f ₁	5.8821	5.8082	5.8093	0.0189
f ₂	23.5284	24.0860	24.0907	0.0195
f ₃	52.9388	53.8258	53.8435	0.0329
f ₄	94.1137	96.6014	96.6220	0.0213
f ₅	147.0532	150.4275	150.4850	0.0382
f ₆	211.7582	214.1404	214.1733	0.0154

Figure 6.

Normalized modal shapes of the first six modes of the simply supported beam.

As shown in Table 4, the errors between the first six natural frequencies of the simply supported beam after updating using both algorithms and their true values are minimal, all within 0.04%. Figure 6 illustrates that while there is some deviation between the initial mode shape curves and the true values, the updated mode shape curves after applying the proposed algorithm almost completely overlap with the true mode shapes. Both Table 4 and Figure 6 demonstrate the effectiveness of the proposed algorithm. However, the data used in the aforementioned validation process is structural modal data, which represents the inherent characteristics of the structure and is independent of external excitation. These factors lead to a low reliability of the updated FEM. Therefore, to verify the reliability of the updated FEM, a unit excitation is randomly applied to the simply supported beam, and the frequency response function (FRF) curve at a random point on the beam is measured. The FRF curves of the initial FEM, updated FEM, and true FEM of the simply supported beam are shown in Figure 7.

Figure 7.

The FRF curves of the simply supported beam.

From Figure 7, it can be seen that the FRF curve of the initial FEM significantly deviates from that of the true FEM, indicating that the initial FEM has a weak capability to simulate the response characteristics of the true FEM. However, the FRF curve of the updated FEM coincides with that of the true FEM, demonstrating that the updated FEM has improved in simulating the true FEM. However, the Bayesian algorithm often struggles with computational efficiency, particularly when dealing with high-dimensional data or complex models. The need to compute posterior distributions, often through methods like MCMC, can lead to significant computational overhead, resulting in slow convergence and long processing times. To highlight the higher computational efficiency of the proposed algorithm, the R _c convergence curve of the proposed algorithm is compared with that of the improved DREAM algorithm from the reference (Zhang et al., 2020). The R _c convergence curves of the two algorithms are shown in Figure 8.

Figure 8.

R_c convergence curves: (a) Algorithm from reference (Zhang et al., 2020); (b) Proposed algorithm.

In Figure 8, the vertical axis represents the scale reduction factor R_c value, and the horizontal axis represents the number of iterations of Markov chains. The red horizontal dashed line represents the threshold R_c = 1.2, while the blue vertical dashed line indicates the iteration number at which all Markov chains reach R_c = 1.2. The convergence time can be calculated based on the required number of iterations. According to the theory discussed in DREAM algorithm, when the R_c values for all parameters to be updated are to the right of the blue dashed line(i.e., R_c < 1.2), it can be considered that the Markov chains for these parameters have converged. As shown in the figure, the improved DREAM algorithm from the reference (Zhang et al., 2020) and the proposed algorithm reach convergence at the 18,000th and 11,000th iterations, respectively. This indicates that the proposed algorithm requires 7000 fewer iterations to achieve convergence compared to the improved DREAM algorithm in the reference (Zhang et al., 2020), demonstrating that the proposed algorithm has higher computational efficiency.

Experimental example

In this section, the proposed algorithm is validated using experimental data from the three-story frame structure at Los Alamos National Laboratory (LANL). This structure primarily consists of four aluminum plates and four aluminum columns, and is mounted on a track that allows only longitudinal sliding. Excitation is applied via a shaker connected at the base. Experimental modal parameters are measured using accelerometers installed on each floor, with a sampling interval of 3.1 ms, totaling 8192 data points, and a sampling frequency of 322.58 Hz. To avoid rigid body modes, the shaker was randomly excited with frequencies set between 20 and 150 Hz. Further details on the tests can be found in the experimental report on the LANL website and in the literature (Figueiredo et al., 2009). The actual three-story frame structure and the corresponding FEM are shown in Figure 9. In this study, the theoretical model of the structure is modeled as a 4-degree-of-freedom system, with A1, A2, A3, and A4 as acceleration response measurement points. The stiffness k₁ and damping c₁ are used to simulate the friction between the track and the structure.

Figure 9.

Three-story frame structure: (a) Actual structure; (b) FEM.

In Figure 9(b), the mass, stiffness, and damping parameters of the initial theoretical FEM of the three-story frame are calculated based on the publicly available structural dimensions and material information. The dynamic analysis of the initial theoretical FEM, along with the experimentally obtained 2nd, 3rd, and 4th natural frequencies and their corresponding errors, are presented in Table 5 (the first natural frequency corresponds to a rigid body mode).

Table 5.

Frequencies and Errors of the FEM and the experimental Structure.

Frequency	Experimental value (Hz)	Initial value (Hz)	Errors (%)
f ₂	30.7	24.8504	19.0541
f ₃	54.2	45.8098	15.4817
f ₄	70.7	59.9426	15.2156

As shown in Table 5, there is a significant error of frequency between the FEM and the experimental structure, indicating the need to update the FEM to reduce the discrepancy between the FEM and the experimental structure. Since the material properties of the structure do not vary significantly, but additional mass factors such as the sensors and bumpers mounted on the aluminum plates were not considered in the modeling process, and the stiffness of each floor did not account for overall coupling effects, the normalized values of the mass parameters m₁, m₂, m₃, and m₄, and the stiffness parameters k₂, k₃, and k₄ are selected as the parameters to be updated. Since the literature (Figueiredo et al., 2009) only provides frequency data without mode shape data, the mode shape data identified in the literature (Dessena et al., 2024) are used as experimental data. The 2nd, 3rd, and 4th natural frequencies of the structure, along with their corresponding mode shapes, are selected as the experimental modal parameters for model updating. Because the parameters to be updated are normalized values and their true values are unknown, a sufficiently large range of [0.1, 2.5] is set for each parameter. The total iteration process is divided into 10 stages, and the changes in the mean posterior PDF of the parameters at each iteration stage are shown in Figure 10.

Figure 10.

Mean posterior PDF of the parameters to be updated at each iteration stage.

As shown in Figure 10, the initially generated values of the parameters to be updated (at stage 0) are far from the true values. However, as the number of iterations increases, the mean posterior PDF of the parameters gradually converge. To illustrate this process in more detail, one of the Markov chains from the initially generated N chains is projected onto the m₁ and m₂ plane. The convergence plots at selected stages are shown in Figure 11.

Figure 11.

Convergence plots of m₁ and m₂ samples at different stages.

In Figure 11, stage 0 represents the 70 random samples uniformly generated within the interval [0.1, 2.5], which serves as the initial past state vectors. According to the theory discussed in sampling difference vectors from past states, the posterior samples of the parameters to be updated are obtained by continuously discarding the initial past state vectors and burn-in samples, as illustrated in stages 2, 6, and 10 in the figure. The joint and one-dimensional marginal posterior distributions of all the parameters to be updated are shown in Figure 12.

Figure 12.

Joint and one-dimensional marginal posterior distributions of the parameters to be updated.

As shown in Figure 12, the posterior samples of the parameters to be updated eventually converge near a single value, as indicated by the red areas with higher point density. The diagonal line displays the histograms of the posterior distributions for the parameters to be updated. These histograms can be used to determine the posterior PDF of the parameters, which can then be used to estimate their values. After running the program 10 times, the mean estimates and errors of the updated parameters are presented in Table 6.

Table 6.

Estimated Means and Errors of Parameters to Be Updated.

Parameters	Initial value	Updated value	Updated variance (%)	Change (%)
m ₁	1	0.9705	0.0632	2.950
m ₂	1	1.1566	0.1525	15.66
m ₃	1	1.2558	0.0954	25.58
m ₄	1	1.1558	0.0790	15.58
k ₂	1	1.4771	0.1795	47.71
k ₃	1	1.7143	0.1462	71.43
k ₄	1	1.6845	0.1436	68.45

As shown in Table 6, the variance of the updated parameters is less than 0.2%, indicating that the posterior distributions of the parameters are quite stable. In terms of the change rates of the updated parameters, the stiffness parameters exhibit a significant change, while the mass parameters show varying degrees of increase, except for a slight decrease in the mass of the first aluminum plate. It is important to note that these changes do not imply that the material properties of the structure have changed after the update, but rather reflect the adjustments made to the structure’s stiffness and mass due to model simplifications and errors in the finite element modeling process. For example, the mass values of the four aluminum plates, after denormalization, are found to be m₁ = 6.2334 kg, m₂ = 7.5942 kg, m₃ = 8.2457 kg, and m₄ = 7.7161 kg. From the structural modeling perspective, the mass of each aluminum plate was calculated based on its publicly available dimensions and material properties. However, the existing literature does not provide specific mass values for the experimental equipment, such as sensors, bumpers, and bolts, which were not accounted for in the theoretical modeling process. The updated FEM shows that the maximum mass is located between the 2nd and 3rd floor, which is reasonable since that is where the central column and bumper are placed. Therefore, the updated mass parameters effectively reduce the structural errors caused by model simplifications and enhance the FEM’s ability to simulate the actual structure. To further validate the updated FEM, the errors in the 2nd, 3rd, and 4th natural frequencies of the three-story frame structure before and after updating, as well as the corresponding normalized mode shapes, are compared in Table 7 and Figure 13.

Table 7.

Frequency Errors Before and After Updating of Three-story Frame Structure.

Frequency	Experimental value (Hz)	Initial value (Hz)	Initial errors (%)	Updated value (Hz)	Updated errors (%)
f ₂	30.7	24.8504	19.0541	30.6947	0.0173
f ₃	54.2	45.8098	15.4817	54.2062	0.0114
f ₄	70.7	59.9426	15.2156	70.7306	0.0433

Figure 13.

Normalized modal shapes of the 2nd, 3rd, and 4th modes of the three-story frame structure.

As shown in Table 7, the errors between the updated 2nd, 3rd, and 4th natural frequencies of the three-story frame structure and their experimental values are minimal, all within 0.05%. Figure 13 illustrates that while the initial mode shape curves of the 2nd, 3rd, and 4th modes deviate somewhat from the true values, the updated mode shape curves almost completely overlap with the experimental values. Both Table 7 and Figure 13 demonstrate the effectiveness of the proposed algorithm. However, the modal shape data used were identified from the reference (Dessena et al., 2024), and both the frequencies and modal shapes are inherent properties of the structure. These factors contribute to the limited reliability of the updated FEM. Therefore, to further verify the reliability of the updated FEM, a comparison is made between the experimentally measured FRF curves and the FRF calculated from the updated FEM. The FRF curves for the four measurement points of the three-story frame structure, including the initial FEM, updated FEM, and experimental structure, are shown in Figure 14.

Figure 14.

FRF curves of the three-story frame structure: (a) A1 measurement point; (b) A2 measurement point; (c) A3 measurement point; (d) A4 measurement point;

As shown in Figure 14, the FRF curves of the initial FEM significantly deviate from those of the experimental structure, indicating that the initial FEM has a limited ability to simulate the response characteristics of the experimental structure. In contrast, although the updated FEM’s FRF curve shows some deviation from the experimental structure’s FRF curve due to factors such as measurement errors, noise, model simplification, and rigid body modes, the updated FEM’s FRF curve nearly overlaps with the experimental structure’s curve compared to the initial FEM, indicating a significant improvement in the updated FEM’s ability to simulate the experimental structure. This evidence validates the accuracy of the updated FEM in terms of both the inherent structural properties and the response characteristics, proving that the proposed Bayesian model updating method achieves high updating accuracy. However, in practical engineering applications, Bayesian algorithms often encounter challenges related to computational efficiency. To demonstrate the computational efficiency of the proposed algorithm, the R _c convergence curve of the proposed algorithm is compared with that of the improved DREAM algorithm from the reference (Zhang et al., 2020). The R _c convergence curves of the two algorithms are shown in Figure 15.

Figure 15.

R_c convergence curves: (a) Algorithm from reference (Zhang et al., 2020); (b) Proposed algorithm.

As shown in Figure 15, the improved DREAM algorithm from the reference (Zhang et al., 2020) reaches convergence at the 17,000th iteration, while the proposed algorithm reaches convergence at the 9,000th iteration. This indicates that the proposed algorithm requires 8000 fewer iterations to achieve convergence compared to the improved DREAM algorithm in the reference (Zhang et al., 2020), demonstrating that the proposed algorithm maintains a high level of computational efficiency even in the experimental example.

Conclusion

This paper addresses the issues with multi-chain MCMC methods by improving the standard DREAM algorithm and proposing a Bayesian model updating method based on multi-sampling strategy integration. The proposed algorithm is validated using a simply supported beam numerical example and a three-story frame structure experimental example, leading to the following conclusions:

(1) The introduction of the Kalman-inspired distribution and sampling difference vectors from past states into the DREAM algorithm addresses the limitation on the number of parallel chains in the DREAM algorithm, thereby improving the exploration efficiency of the posterior distribution.

(2) The integration of the proposed centroid update sampling strategy with other sampling strategies increases the sampling diversity among different chains in the DREAM algorithm, and accelerating the convergence process.

(3) The proposed Bayesian model updating method, which integrates multiple sampling strategies, demonstrates high updating accuracy and fast convergence speed. The updated FEM shows improved capability in simulating the inherent properties and response characteristics of the experimental structure, making it suitable for use as a benchmark model in structural health monitoring.

Footnotes

Acknowledgments

The authors would like to thank the Engineering Institute at Los Alamos National Laboratory for providing the experimental data used in this article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the National Natural Science Foundation of China (Grant No. 51768035) and the Natural Science Foundation of Gansu Province (Grant No. 24JRRA172).

ORCID iD

Zhenrui Peng

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