Quantifying Uncertainties in Model Updating Following Bayesian Approach Using a Parameter Space-Search Algorithm

Abstract

Model updating aims to provide accurate models to reveal possible structural damage information for structural health monitoring (SHM). Uncertainties always exist in model updating due to incomplete information in measurement and modeling. These uncertainties usually cause the problem of non-uniqueness, i.e., multiple equivalent models can fit the experimental data the same well. Locating all these equivalent models and including them for representing structural dynamics is a challenging task. This work employs a Bayesian probabilistic framework for model updating, so that the uncertainties can be quantified, and thus all of the multiple equivalent models can be considered naturally. A parameter space-search algorithm is proposed to systematically locate all the equivalent models. A transmission tower under laboratory conditions with limited modal parameters was used to valid the proposed method.

1 Introduction

Model updating is an important research topic, because it is useful in downstream research of structural health monitoring. The updated models can be used to determine damage location and intensity of structures. Model updating has successfully assisted monitoring the fatigue, corrosion and earthquake effects of bridges, large-scale building structures and aerospace structures [1-3]. Therefore, A great deal of research has aimed at establishing efficient local and global model updating methods for structural health monitoring in the past four decades. Despite of numerous methods developed, technical challenges still remain and effective model updating methods for complex structures have not yet been established.

One classical idea of model updating is to match the model-predicted modal parameters with the experimental ones, i.e., the experimental natural frequencies and mode shapes identified using measured structural responses. Model updating is to solve the inverse problem of determining structural parameters given some modal data [4, 5]. The hypothesis is that the change in model parameters related to different structural parts causes the change in model-predicted modal parameters, so theoretically adjusting model parameters can make the model-predicted modal parameters match with the experimental ones, giving the updated model that can reflect actual structural behaviors. Most of the deterministic methods pinpoint one model, even though there may be other equivalent models. The problem is the lack of theoretical basis for considering all the equivalent models.

There are inherent difficulties in model updating of complex structures in practice. The solution of the inverse problem is uncertain and not unique due to incomplete information, e.g., measurement noise and modeling errors. Meanwhile, how to incorporate all the equivalent models to represent the actual structure is difficult. This paper follows the Bayesian probability framework for quantifying the uncertainties in model updating [6-10], so that multiple models can be considered with a theoretical basis. An improved parameter space-search algorithm [11, 12] is developed to locate all the equivalent models.

2 Method

2.1 Bayesian Probabilistic Framework

According to Bayes’ theorem, the posterior probability density function (PDF) of the model parameters ${{\varvec{\uptheta}}}$ given some measured data ${\mathbf{D}}$ can be expressed as follows:

$$ \begin{array}{*{20}c} {p\left( {{{\varvec{\uptheta}}}|{\mathbf{D}}} \right) = \frac{{p\left( {{\varvec{\uptheta}}} \right)p\left( {{\mathbf{D}}|{{\varvec{\uptheta}}}} \right)}}{{p\left( {\mathbf{D}} \right)}}} \\ \end{array} $$

(1)

Here, $p\left( {{\varvec{\uptheta}}} \right)$ is the prior PDF of model parameters, $p\left( {\mathbf{D}} \right)$ is a normalizing constant, and $p({\mathbf{D}}|{{\varvec{\uptheta}}})$ is the likelihood function that need to be determined. When the prior PDF is considered as a uniform PDF, the posterior PDF can be rewritten as:

$$ \begin{array}{*{20}c} {p\left( {{{\varvec{\uptheta}}}|{\mathbf{D}}} \right) = cexp\left( { - L\left( {{\varvec{\uptheta}}} \right)} \right)} \\ \end{array} $$

(2)

where $c$ is constant and $L\left( {{\varvec{\uptheta}}} \right)$ is the negative log likelihood function.

$$ \begin{array}{*{20}c} {L\left( {{\varvec{\uptheta}}} \right) = - lnp\left( {{\mathbf{D}}|{{\varvec{\uptheta}}}} \right) = N_{d} \ln \left( {2\pi \sigma^{2} } \right) + \frac{{J_{0} \left( {{\varvec{\uptheta}}} \right)}}{{2\sigma^{2} }}} \\ \end{array} $$

(3)

where $J_{0} \left( {{\varvec{\uptheta}}} \right)$ is the error function between calculated and measured responses and can be expressed in the following formula:

$$ \begin{array}{*{20}c} {J_{0} \left( {{\varvec{\uptheta}}} \right) = \mathop \sum \limits_{n = 1}^{{N_{d} }} (\left( {\frac{{\omega_{n} \left( {{\varvec{\uptheta}}} \right) - \hat{\omega }_{n} }}{{\hat{\omega }_{n} }})^{2} + \left( {1 - \left| {{\mathbf{\varphi }}_{n}^{T} \left( {{\varvec{\uptheta}}} \right){\hat{\mathbf{\varphi }}}_{n} } \right|^{2} } \right)} \right)} \\ \end{array} $$

(4)

where the $\omega_{n}$ is calculated frequency and $\hat{\omega }_{n}$ is measured frequency, similarly, the ${\mathbf{\varphi }}_{n}$ is calculated mode shape and ${\hat{\mathbf{\varphi }}}_{n}$ is measured mode shape.

However, the posterior PDF of model parameters is usually unknown. The second-order Taylor series expansion is adopted to approximate the posterior PDF where the negative log likelihood function $L\left( {{\varvec{\uptheta}}} \right)$ can be rewritten as:

$$ \begin{array}{*{20}c} {L\left( {{\varvec{\uptheta}}} \right) \approx L\left( {{\hat{\mathbf{\theta }}}_{i} } \right) + \frac{1}{2}\left( {{{\varvec{\uptheta}}} - {\hat{\mathbf{\theta }}}_{i} } \right)^{T} H\left( {{\hat{\mathbf{\theta }}}_{i} } \right)\left( {{{\varvec{\uptheta}}} - {\hat{\mathbf{\theta }}}_{i} } \right)} \\ \end{array} $$

(5)

where ${\hat{\mathbf{\theta }}}_{{\text{i}}}$ is an optimal parameter of ${{\varvec{\uptheta}}}$ such that the first derivation of $L\left( {{\varvec{\uptheta}}} \right)$ with respect to ${{\varvec{\uptheta}}}$ is equal to 0 and omitted, and ${\mathbf{H}}\left( {{\hat{\mathbf{\theta }}}_{i} } \right)$ is the Hessian matrix of $L\left( {{\varvec{\uptheta}}} \right)$ evaluated at ${\hat{\mathbf{\theta }}}_{i}$.

Then, one Gaussian PDF can be used to approximate the posterior PDF centered at each optimum ${\hat{\mathbf{\theta }}}_{i} :$

$$ \begin{array}{*{20}c} {p\left( {{{\varvec{\uptheta}}}|D} \right) \approx \mathop \sum \limits_{i = 1}^{{N_{d} }} \omega_{i} {\mathcal{N}}\left( {{{\varvec{\uptheta}}};{\hat{\mathbf{\theta }}}_{i} ,{\mathbf{H}}^{ - 1} \left( {{\hat{\mathbf{\theta }}}_{i} } \right)} \right)} \\ \end{array} $$

(6)

Here $\omega_{i}$ is the normalized weight of ${\hat{\mathbf{\theta }}}_{i}$ that can be calculated as:

$$ \begin{array}{*{20}c} {\omega_{i} = \frac{{\omega_{i}{\prime} }}{{\sum \omega_{i}{\prime,} }}\omega_{i}{\prime} = p\left( {{\hat{\mathbf{\theta }}}_{i} } \right)\left| {{\mathbf{H}}\left( {{\hat{\mathbf{\theta }}}_{i} } \right)} \right|^{\frac{1}{2}} } \\ \end{array} $$

(7)

2.2 The Parameters Space-Search Algorithm

It is difficult to find the multiple optimums of $L\left( {{\varvec{\uptheta}}} \right)$ since when the known information for identification is limited, there is a series of structural models (equivalent models) that can output the same modal parameters, i.e., natural frequency ${{\varvec{\Phi}}}^{1} \left( {{\varvec{\uptheta}}} \right) = \left[ {\omega_{1} \left( {{\varvec{\uptheta}}} \right),\omega_{2} \left( {{\varvec{\uptheta}}} \right),\omega_{3} \left( {{\varvec{\uptheta}}} \right), \ldots , \omega_{{N_{d} }} \left( {{\varvec{\uptheta}}} \right)} \right]$ and mode shape ${{\varvec{\Phi}}}^{2} \left( {{\varvec{\uptheta}}} \right) = \left[ {{\mathbf{\varphi }}_{1} \left( {{\varvec{\uptheta}}} \right),{\mathbf{\varphi }}_{2} \left( {{\varvec{\uptheta}}} \right),{\mathbf{\varphi }}_{3} \left( {{\varvec{\uptheta}}} \right), \ldots , {\mathbf{\varphi }}_{{N_{d} }} \left( {{\varvec{\uptheta}}} \right)} \right]$. In order to find those equivalent models $S\left( {{\varvec{\uptheta}}} \right)$ with different model parameters {${{\varvec{\uptheta}}}_{s} :s = 1,2,3, \ldots , N_{s}$}, the parameters space-search algorithm is adopted and its strategy is to search lower-dimensional manifolds in the parameter space, i.e., curves, that are subset of the parameter space of interest. More specifically, it minimizes $L\left( {{\varvec{\uptheta}}} \right)$ along different curves by generating a sequence of points. Defining $C_{k} \left( {{{\varvec{\uptheta}}};{\tilde{\mathbf{\theta }}}} \right)$ that passes through an optimal point ${\tilde{\mathbf{\theta }}}$ as a curve in the parameter space of interest, and the curve satisfies that all of the modal frequencies remain unchanged except the $k^{th}$ modal frequency. By iteratively searching on these curves, minimization of $L\left( {{\varvec{\uptheta}}} \right)$ is achieved such that the target equivalent models $S\left( {{\varvec{\uptheta}}} \right)$ can be found: $S\left( {{\varvec{\uptheta}}} \right) = C_{1} \left( {{{\varvec{\uptheta}}};{\tilde{\mathbf{\theta }}}} \right) \cap C_{2} \left( {{{\varvec{\uptheta}}};{\tilde{\mathbf{\theta }}}} \right), \ldots ,C_{{N_{d} }} \left( {{{\varvec{\uptheta}}};{\tilde{\mathbf{\theta }}}} \right) = \mathop \cap \limits_{k = 1}^{{N_{d} }} C_{k} \left( {{{\varvec{\uptheta}}};{\tilde{\mathbf{\theta }}}} \right)$where $C_{k} \left( {{{\varvec{\uptheta}}};{\tilde{\mathbf{\theta }}}} \right) = \left\{ {{{\varvec{\uptheta}}}:\omega_{r} \left( {{\varvec{\uptheta}}} \right) = \omega_{r} \left( {{\tilde{\mathbf{\theta }}}} \right),r = 1,2, \ldots ,k - 1,k + 1, \ldots ,N_{d} } \right\}$;

Minimization along the current curve $C_{k} \left( {{{\varvec{\uptheta}}};{\tilde{\mathbf{\theta }}}} \right)$ passing through ${\tilde{\mathbf{\theta }}}$ is done by searching from the start point ${{\varvec{\uptheta}}}^{1} = {\tilde{\mathbf{\theta }}}$. Given the current point ${{\varvec{\uptheta}}}^{i}$, the next point ${{\varvec{\uptheta}}}^{i + 1}$ will be generated by the following steps.

$$ \begin{array}{*{20}c} {{{\varvec{\uptheta}}}^{i + 1} = {{\varvec{\uptheta}}}^{i} + \delta {{\varvec{\uptheta}}}^{i} } \\ \end{array} $$

(8)

where vector $\delta {{\varvec{\uptheta}}}^{i}$ is calculated by following linear algebra system.

$$ \begin{array}{*{20}c} {\nabla \omega \left( {{{\varvec{\uptheta}}}^{i} } \right)\delta {{\varvec{\uptheta}}}^{i} = \delta {{\varvec{\upomega}}}^{i} } \\ \end{array} $$

(9)

where $\nabla {{\varvec{\upomega}}}\left( {{{\varvec{\uptheta}}}^{i} } \right)$ is the first derivative of $\omega$ at ${{\varvec{\uptheta}}}^{i}$; $\delta {{\varvec{\upomega}}}^{i}$ is also a vector and the $k^{th}$ element of $\delta {{\varvec{\upomega}}}^{i}$ is calculated by Eq. (10), and other elements are calculated by Eq. (11).

$$ \begin{array}{*{20}c} {\delta \omega_{k}^{i} = \gamma \alpha \omega_{k} \left( {{\tilde{\mathbf{\theta }}}} \right)} \\ \end{array} $$

(10)

$$ \begin{array}{*{20}c} {\delta \omega_{j}^{i} = \omega_{j} \left( {{\tilde{\mathbf{\theta }}}} \right) - \omega_{j} \left( {{{\varvec{\uptheta}}}^{i} } \right);j = 1, \ldots ,k - 1,k + 1, \ldots ,N_{d} } \\ \end{array} $$

(11)

where the $\gamma$ is direction parameter that takes 1 or -1 to decide the iteration direction, and $\alpha$ is the fractional step. After that $\delta {{\varvec{\uptheta}}}^{i}$ is calculated, the following condition need to be checked.

$$ \begin{array}{*{20}c} {\left( {\delta {{\varvec{\uptheta}}}^{i} } \right)^{T} \delta {{\varvec{\uptheta}}}^{i - 1} > 0} \\ \end{array} $$

(12)

This constraint helps avoid an undesired change in the iteration along curve $C_{k}$. And if the condition is not satisfied, then $\delta {{\varvec{\uptheta}}}^{i}$ is set equal to $- \delta {{\varvec{\uptheta}}}^{i}$ and the sign of $\gamma$ is also changed. After $\delta {{\varvec{\uptheta}}}^{i}$ satisfying Eq. (12), the candidate ${{\varvec{\uptheta}}}^{i + 1}$ is calculated by Eq. (8). Next, it is checked if ${{\varvec{\uptheta}}}^{i + 1}$ satisfies the constraints imposed by Eq. (13).

$$ \begin{array}{*{20}c} {\left| {\frac{{\omega_{j} \left( {{{\varvec{\uptheta}}}^{i + 1} } \right) - \omega_{j} \left( {{\tilde{\mathbf{\theta }}}} \right)}}{{\omega_{j} \left( {{\tilde{\mathbf{\theta }}}} \right)}}} \right| < \varepsilon ;j = 1,2, \ldots k - 1,k + 1, \ldots N_{d} } \\ \end{array} $$

(13)

where the $\varepsilon$ is a tolerance. If Eq. (13) in not satisfied, which mean ${{\varvec{\uptheta}}}^{i + 1}$ is not close enough to the curve, set ${{\varvec{\uptheta}}}^{i + 1}$ to a new start point and do the internal iterations with the same procedure until the above condition is satisfied.

Following the curves, the released natural frequency $\omega_{k} \left( {{\varvec{\uptheta}}} \right)$ is checked for the following equation.

$$ \begin{array}{*{20}c} {\left( {\omega_{j} \left( {{{\varvec{\uptheta}}}^{i + 1} } \right) - \omega_{k} \left( {{\tilde{\mathbf{\theta }}}} \right)} \right)\left( {\omega_{j} \left( {{{\varvec{\uptheta}}}^{i} } \right) - \omega_{k} \left( {{\tilde{\mathbf{\theta }}}} \right)} \right) < 0,j = 1, \ldots ,k - 1,k + 1, \ldots ,N_{d} } \\ \end{array} $$

(14)

When the Eq. (14) is satisfied, it indicates that there is a new point ${\hat{\mathbf{\theta }}}^{i + 1} \in S\left( {{\varvec{\uptheta}}} \right)$ lies between ${{\varvec{\uptheta}}}^{i}$ and ${{\varvec{\uptheta}}}^{i + 1}$. The algorithm needs to search the curve between ${{\varvec{\uptheta}}}^{i}$ and ${{\varvec{\uptheta}}}^{i + 1}$ with reduced steps until another point is found.

Ensure all the curves $C_{k} \left( {{{\varvec{\uptheta}}};{\tilde{\mathbf{\theta }}}} \right),k = 1, \ldots ,N_{d}$ are searched. For target set $S\left( {{\varvec{\uptheta}}} \right)$, all the points will be checked the condition:

$$ \begin{array}{*{20}c} {\frac{{\left| {{\mathbf{\varphi }}_{j} \left( {{\tilde{\mathbf{\theta }}}} \right) - {\mathbf{\varphi }}_{j} \left( {{\varvec{\uptheta}}} \right)} \right|}}{{\left| {{\mathbf{\varphi }}_{j} \left( {{\tilde{\mathbf{\theta }}}} \right)} \right|}} < \in ;j = 1,2, \ldots k - 1,k + 1, \ldots ,N_{d} } \\ \end{array} $$

(15)

If this condition satisfied, it indicates that the model at ${{\varvec{\uptheta}}}$ has equivalent mode shapes and natural frequencies, so the point ${{\varvec{\uptheta}}}$ belongs to the target set $S\left( {{\varvec{\uptheta}}} \right)$. Finally, all the output equivalent points are found.

3 Results and Discussion

A transmission tower model is used to validate the proposed method numerically. The structure is shown in Fig. 1 and the material information is shown in Table1. We treated each floor as a substructure such that in total 8 sub-structures with 8 independent stiffness parameters are used for parameterization in model updating.

Fig. 1

Transmission tower model

Table 1 Material information

Using measured natural frequencies and mode shapes of 5 modes of this tower as the experimental data for model updating. Then, model updating was conducted with the stiffness of the 8 sub-structures treated unknown. To conduct model updating, the parameter-space search algorithm started from an initial model parameters and finally located 7 equivalent sets of model parameters, i.e., input any set of these identified parameters to the finite element model of the transmission tower can produce the model-predicted modal parameters that can fit the experimental ones the same well. The error between the calculated frequencies from initial parameter and measured modal frequencies is shown in Table2. And the Fig. 2 comparing the calculated and experimental mode shapes. The results of the search algorithm by the searching that starts from one optimum are shown in Table 3.

Table 2 The error of frequencies between calculated from initial parameter and measured

Fig. 2

The comparing between the initial parameter calculated and experimental mode shapes

Table 3 The search results of parameters space-search algorithm

4 Conclusions

This paper proposed an improved parameters space-search algorithm for Bayesian model updating, considering the non-uniqueness and uncertainty in model updating. The advantages of this algorithm are verified by the example of a transmission tower. The algorithm located all the output-equivalent models for the transmission tower, and all the equivalent models can be considered following the proposed Bayesian farmwork. Complex problems for full-scale structures caused by incomplete information can be solved using the proposed method.