A surrogate model for uncertainty quantification and global sensitivity analysis of nonlinear large-scale dome structures

Abstract

Full-scale dome structures intrinsically have numerous sources of irreducible aleatoric uncertainties. A large-scale numerical simulation of the dome structure is required to quantify the effects of these sources on the dynamic performance of the structure using the finite element method (FEM). To reduce the heavy computational burden, a surrogate model of a dome structure was constructed to solve this problem. The dynamic global sensitivity of elastic and elastoplastic structures was analyzed in the uncertainty quantification framework using fully quantitative variance- and distribution-based methods through the surrogate model. The model considered the predominant sources of uncertainty that have a significant influence on the performance of the dome structure. The effects of the variables on the structural performance indicators were quantified using the sensitivity index values of the different performance states. Finally, the effects of the sample size and correlation function on the accuracy of the surrogate model as well as the effects of the surrogate accuracy and failure probability on the sensitivity index values are discussed. The results show that surrogate modeling has high computational efficiency and acceptable accuracy in the uncertainty quantification of large-scale structures subjected to earthquakes in comparison to the conventional FEM.

Graphical abstract

Keywords

large-scale dome structure / surrogate model / global sensitivity analysis / uncertainty quantification / structural performance

1 Introduction

There are numerous uncertainties in engineering structures, which mainly include any of the following [1,2]: 1) the shape of the actual structure may not be completely consistent with the design owing to human errors during execution; 2) the structural members may have minor deformations owing to manufacturing errors; 3) the actual material properties may differ slightly from those considered in the design; 4) the idealized loading considered for numerical modeling may not reflect the actual loading pattern experienced in a real-life situation; 5) the damping characteristics that affect the structural dynamic behavior are difficult to quantify with certainty. The sources of uncertainties could be either aleatoric or epistemic. The effects of some uncertainties on structural performance are considered indirectly in the current specifications, and a deterministic analysis strategy is adopted in the uncertainty quantification. Because these uncertainties are random variables, they cannot be explicitly characterized in the results. Quantifying their possible effects on structural performance is necessary as they have not been fully understood yet.

This study focuses on aleatoric uncertainties. Probability theory is an excellent tool for describing the characteristics of the aleatoric sources of uncertainties utilizing probability density functions (PDFs). Two modeling techniques that are currently in use for stochastic structural modeling are completely random modeling and globally random modeling [1–3]. The former requires more complex modeling procedures, whereas the latter is a more efficient way to evaluate the performance of a structure with uncertainties because each variable is considered for the entire structure. Global sensitivity analysis (GSA) plays a critical role in quantifying the effects of the sources of uncertainty on structural performance, and the globally random modeling method is suitable for sensitivity analysis.

The GSA has been used for model simplification, importance ranking, risk reduction, and data process management lately [4]. Applications of GSA for engineering structures can be summarized as [5–8]: 1) understanding the input-output relationship to explore causalities; 2) determining the contribution of variables to the results to reduce potential uncertainty; 3) identifying important variables for dimensionality reduction; and 4) support decisions to guide future designs.

There are several methods to conduct GSA [9], including the derivative-based, variance-based, distribution-based, and regression-based approaches. In the last decade, the variance-based approach has received increasing attention in the field of civil engineering. Nariman et al. [10] applied the variance-based method to assess the effects of variables on the tensile damage and stability of a tunnel structure subjected to earthquakes. Zoutat et al. [11] used the variance-based method to determine the contributions of variables to the lateral displacement of a frame structure. Menz et al. [12] developed a method for quantifying the sensitivity of the failure probability to variables in a framework for variance-based analysis. To quantitatively analyze the effects of variables on the results, Zhang et al. [13] performed an analytical derivation of global sensitivity for models with variables based on the variances of the variables, indicating that the analytical solution could be used in practice. Javidan and Kim [14] validated the efficiency of variance-based GSA for fuzzy structural systems. Other developments in GSA can be found in the literature [15–17]. However, it has been reported that the distribution-based techniques may be more robust than variance-based techniques in GSA when the sensitivity is described by the differences in the output distributions rather than only the variances [18]. By combining variance-based and distribution-based approaches, Baroni and Francke [19] developed an effective strategy for the GSA. However, the computational burden of the GSA for complex structures is enormous which limits its practical applications, particularly for nonlinear structures.

The meta-learning method being one of the fastest-growing research areas in machine learning has been used to reduce the computational burden of the GSA. Wei et al. [20] developed a GSA method based on the Karhunen–Loeve (KL) expansion, which was used to identify the contribution of variables to structural reliability. Ni et al. [21] presented a sensitivity analysis and uncertainty quantification for the natural frequencies and dynamic responses of a bridge structure based on the polynomial chaos expansion (PCE). Another popular method for quantifying uncertainty in design optimization and model updating is the Kriging model [22,23]. To select the parameters for stochastic model updating, Yuan et al. [24] integrated the Kriging surrogate model with GSA. Wan and Ni [25] investigated the effects of the variables on the dynamic demands of a time-varying structural system using an analytical Gaussian regression model. Amini et al. [26] investigated the superiority of the Polynomial Chaos-Kriging (PCK) metamodel over other conventional methods for quantifying the uncertainty of dam structures. The use of the meta-learning method for cases with complex structures and excitations remains a significant challenge.

In previous studies, statistical sensitivity analyses, including GSA, were performed in large-scale dome systems using sampling methods. Considering the uncertain sources in a large-scale single-layer dome, Zhang et al. [1,2] used the Monte Carlo sampling (MCS) method to construct a sensitivity analysis of the structural seismic performance based on the variance of structural demands. Xian et al. [27] performed a stochastic sensitivity analysis of structures with viscous dampers during earthquakes using Monte Carlo simulations. Because the collapse behavior of double-layer dome structures varies depending on random factors, Vazna and Zarrin [28] conducted a sensitivity analysis and used the first-order second-moment method to assess the importance of uncertain parameters. Using an MCS method and a Gaussian process metamodel, Wan and Ren [29] proposed a GSA method to solve parameter selection problems for an arch bridge. All the aforementioned studies were based on the finite element model, and the computational burden is a challenge for practical applications. Recently, Sun and Dias [30] used a sparse PCE to build a surrogate model to perform the GSA of the seismic deformation of a soil tunnel.

In previous studies, the finite element method (FEM) was mostly used to perform GSA of large-scale structures as well as elastic structures and the computational efficiency was low. This study is extended to elastoplastic structures subjected to complex stochastic excitations using an efficient Regression-Kriging (RK) surrogate model method. Variance- and distribution-based approaches are used to quantify the importance of several variables. The effects of key factors on the accuracy of the surrogate model and sensitivity index values are also discussed.

2 Surrogate modeling and global sensitivity analysis method for structures

2.1 Surrogate modeling

A surrogate model of a structure can be represented mathematically as

(1)\hat{\mathit{y}}=\phi (\mathit{x};[{\mathit{x}}_{\mathrm{t}},{\mathit{y}}_{\mathrm{t}}\left]\right),

where \mathit{x}\in {\mathrm{\Re }}^n represents the n input variables, {\mathit{x}}_{\mathrm{t}}\in {\mathrm{\Re }}^{m\times n} is the training input vector, {\mathit{y}}_{\mathrm{t}}\in {\mathrm{\Re }}^m is the training output vector, \hat{\mathit{y}} is the prediction for the true value vector \mathit{y}, and \phi (\cdot ) is the surrogate model constructed using {\mathit{x}}_{\mathrm{t}} and {\mathit{y}}_{\mathrm{t}}. To construct the surrogate model of a structure, training samples [{\mathit{x}}_{\mathrm{t}},{\mathit{y}}_{\mathrm{t}}] must be obtained using physics-based methods such as the FEM. After constructing the surrogate model, \mathit{y} can be quickly estimated for a continuous spatial field using this surrogate model by allowing efficient uncertainty quantification and GSA.

The RK method [31], a hybrid interpolation technique, was introduced in this study to construct the mapping function of a structure with variables obeying normal distributions because it can describe the global and local characteristics of the sample data well and is easily implemented.

According to the RK model, the output y can be written as

(2)y={\sum }_{i=1}^p{\beta }_i{f}_i\left(x\right)+ϵ\left(x\right),

where {\beta }_i and f_{\mathit{i}}\left(x\right) are the i\mathrm{t}\mathrm{h} regression coefficient and regression function, respectively, {\displaystyle {\sum }_{i=1}^p{\beta }_i{f}_i\left(x\right)} represents global approximations for y that can be estimated using a quadratic model to improve regression accuracy, and ϵ\left(x\right) reflects the local deviations of y, which is defined as a stochastic process with a mean of zero and variance of {\sigma }^2. The covariance matrix of ϵ\left(x\right) is written as

(3)\mathrm{c}\mathrm{o}\mathrm{v}\left[ϵ\left(x_i\right),ϵ\left(x_j\right)\right]={\sigma }^2\mathit{R}\left[R\left(\theta ;x_i,x_j\right)\right],

where R(\cdot ) is the correlation function, \theta is a parameter, and \mathit{R}\in {\mathrm{\Re }}^{m\times m} is the correlation function matrix. Many correlation functions, such as the exponential, Gaussian, Matérn linear, and Matérn cubic functions, are available, but the Gaussian correlation function is the one that is most commonly used.

(4)R\left(\theta ;x_i,x_j\right)=\prod \mathrm{e}\mathrm{x}\mathrm{p}(-{\theta }_k{(x_{ik}-x_{jk})}^2),\mathrm{\forall }{\theta }_k\in {\mathrm{\Re }}^{+},

where {\theta }_k is the k\mathrm{t}\mathrm{h} unknown parameter to be determined. Here, the Matérn 3/2 correlation function is used for surrogate modeling of elastoplastic structures, and the correlation function is expressed as [32]

(5)\begin{array}{rl}R\left(\theta ;x_i,x_j\right)=& \prod (1+\sqrt{3}{\theta }_k\left|x_{ik}-x_{jk}\right|)\\ & \cdot \mathrm{e}\mathrm{x}\mathrm{p}(-\sqrt{3}{\theta }_k\left|x_{ik}-x_{jk}\right|),\mathrm{\forall }{\theta }_k\in {\mathrm{\Re }}^{+}.\end{array}

We assume that the known sample points are denoted as [{\mathit{s}}^{\mathrm{m}},{\mathit{Y}}^{\mathrm{m}}] and \mathit{s} follows the standard normal distribution. According to the RK model, the prediction \hat{y}\left(\mathit{u}\right) for an untried sample dataset \mathit{u} can be written as [31]

(6)\hat{y}\left(\mathit{u}\right)={\beta }^{\ast }+{r\left(\mathit{u}\right)}^{\mathrm{T}}{\mathit{R}}^{-1}(\mathit{Y}-\mathit{f}{\beta }^{\ast }),

where {\beta }^{\ast } is a constant, r\left(\mathit{u}\right) is the vector of correlation values between \mathit{u} and \mathit{s}, \mathit{R} is the correlation function matrix of the known sample vector \mathit{s}, and \mathit{f} is the regression function vector. {\beta }^{\ast } can be calculated using the following formula:

(7){\beta }^{\ast }={\left({\mathit{f}}^{\mathrm{T}}{\mathit{R}}^{-1}\mathit{f}\right)}^{-1}{\mathit{f}}^{\mathrm{T}}{\mathit{R}}^{-1}\mathit{Y}.

The estimated variance is written as follows:

(8){\sigma }^{\ast 2}=\frac{1}{m}{(\mathit{Y}-\mathit{f}{\beta }^{\ast })}^{\mathrm{T}}{\mathit{R}}^{-1}(\mathit{Y}-\mathit{f}{\beta }^{\ast }).

The optimum estimate for {\theta }_k can be obtained by maximizing the likelihood estimate expressed as

(9)\mathrm{l}\mathrm{n}L\approx \frac{m\mathrm{l}\mathrm{n}{\sigma }^{\ast 2}+\mathrm{l}\mathrm{n}\left|R\right|}{2}.

More information on this is available in Ref. [31].

A surrogate model was constructed using the aforementioned method. Given the high nonlinearity of the structures, the prediction performance of the surrogate models was evaluated using an error metric (Q^2) which is expressed as follows:

(10)Q^2=1-\frac{{\sum }_{i=1}^k{(y_i-{\hat{y}}_i)}^2}{\sum _{i=1}^k{(y_i-\overline{y})}^2},

where k is the sample size and {\hat{y}}_i and \overline{y} are the prediction and mean of y_i, respectively. The closer the value of Q^2 is to 1, the better the regression effect and prediction accuracy of the surrogate model. When the value of Q^2 exceeds 0.8, the surrogate model is usually appropriate for GSA. The mean square error (MSE) is another metric that is used for assessing the accuracy of the surrogate model.

(11)MSE=\frac{{\sum }_{i=1}^k{(y_i-{\hat{y}}_i)}^2}{k}.

2.2 Global sensitivity analysis methods

The output \hat{y}\left(\mathit{x}\right) can be obtained once a sample set of \mathit{x} is determined using the Latin hypercube sampling method. \hat{y}\left(\mathit{x}\right) can be written as

(12)

where {\hat{y}}_0 is the mean value of \hat{y}\left(\mathit{x}\right), and {\hat{y}}_i\left(x_i\right), {\hat{y}}_{ij}(x_i,x_j), and {\hat{y}}_{1,2,\dots ,n}(x_1,x_2,\dots ,x_n) represent the first-, second-, and higher-order components, respectively. Because of orthogonality, the unconditional variance of \hat{y}\left(\mathit{x}\right) can be decomposed as follows:

(13)

where V_i is the contribution of the single variable x_i to the variance of \hat{y}\left(\mathit{x}\right), V_{ij} is the contribution of the interactions between x_i and x_j, and V_{1,2,\dots ,n} is a higher-order contribution. Here, V_i and V_{ij} can be expressed as [7,33]

(14)\{\begin{array}{l}{V}_i=V\left[\mathbb{E}\right(\hat{\mathit{y}}\left|x_i\right)],\\ V_{ij}=V\left[\mathbb{E}\left(\hat{\mathit{y}}|x_i,x_j\right)\right]-V\left[\mathbb{E}\left(\hat{\mathit{y}}|x_i\right)\right]-V\left[\mathbb{E}\right(\hat{\mathit{y}}\left|x_j\right)].\end{array}

Thus, the decomposition of sensitivity indices can be expressed as

(15)

where S_i = V_i/V\left(\hat{\mathit{y}}\right) is the first-order sensitivity index and S_{ij}=V_{ij}/V\left(\hat{\mathit{y}}\right) is the second-order sensitivity index. The total sensitivity index can be written as [34]

(16)S_{\mathrm{T}i}=1-\frac{V_{\sim i}\left[\mathbb{E}\right(\hat{\mathit{y}}\left|x_{\sim i}\right)]}{V\left(\hat{\mathit{y}}\right)},

where \mathbb{E}\left(\hat{\mathit{y}}\right|x_{\sim i}) is the mean value of \hat{\mathit{y}} when all parameters except x_i is fixed and V_{\sim i}\left[\mathbb{E}\right(\hat{\mathit{y}}\left|x_{\sim i}\right)] is its variance.

The total-order index of a variable reflects the effects of this variable and its interactions with all other variables on the output [34]. The first-order index evaluates the main effect of the variable on the output, the second-order index evaluates the interactive effect between two variables, and so forth. The difference in S_{\mathrm{T}i}-S_i represents the degree of interaction between the variable x_i and the other variables. Higher-order coupling effects were not considered because the index values were small.

Another GSA method that has attracted the attention of researchers recently is based on probability distributions, including probability distributions based on the variables and probability distributions based on the outputs. In engineering structures, the structural failure function G\left(\mathit{X}\right) is expressed as follows:

(17)G\left(\mathit{x}\right)=D_0-{\left|A_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}},

where D_0 and {\left|A_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}} denote the critical value and maximum demand of the i\mathrm{t}\mathrm{h} sampling structure, respectively. Once D_0 is defined, it is obvious that the values of G\left(\mathit{x}\right) may be divided into two subsets: G\left(\mathit{x}\right)\le 0 and G\left(\mathit{x}\right)>0. The effect of a variable on the output can be determined by comparing the difference between the unconditional PDF f_{x_i}\left(x_i\right) and conditional PDF f_{x_i}\left(x_i|\mathrm{F}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{u}\mathrm{r}\mathrm{e}\right) in the sample set of variables. Consequently, the index S_i of a variable is defined as [35]

(18)S_i=\frac{1}{2}\int \left|f_{x_i}\left(x_i\right)-f_{x_i}\left(x_i|\mathrm{F}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{u}\mathrm{r}\mathrm{e}\right)\right|{\mathrm{d}x}_i,

where S_i represents the first-order global sensitivity index which varies from zero to one. The high value of S_i indicates that x_i has a significant effect on structural failure. Because it can directly estimate the importance ranking of variables [35,36], only this index is discussed in the distribution-based method. It should be noted that owing to different theoretical viewpoints the variance- and distribution-based methods are numerically incomparable; however, the importance ranking of variables provided by the two is comparable.

The conditional PDF is non-parameterized and cannot be estimated easily using existing parameterized density functions. In this paper conditional PDF was estimated using the kernel-smoothing method, and the kernel-smoothed PDF is expressed as follows [37]:

(19)\hat{p}\left(x\right)=\frac{1}{Nh}{\sum }_{i=1}^N{K}_{\mathrm{e}}\left(\frac{x-x_i}{h}\right),

where K_{\mathrm{e}}(\cdot ), N, x_i, and h are the univariate kernels, sample size, i\mathrm{t}\mathrm{h} sample value, and bandwidth, respectively, and \int K_{\mathrm{e}}\left(x\right)\mathrm{d}x=1. The kernel function K_{\mathrm{e}}\left(x\right) is a symmetric PDF. In this study, the optimal bandwidth was determined using the plug-in bandwidth selection method, and the fitting effect was controlled using the asymptotic mean integrated square error criterion.

3 Validation

The aforementioned analytical strategy was developed in Python. To validate the methods, the Ishigami function from Ref. [38] was used to investigate nonlinear problems.

(20)f\left(x\right)=\mathrm{s}\mathrm{i}\mathrm{n}\left(x_1\right)+a{\mathrm{s}\mathrm{i}\mathrm{n}}^2\left(x_2\right)+b{x}_3^4\mathrm{s}\mathrm{i}\mathrm{n}\left(x_1\right),

where x_i\in [-\text{π},\text{π}] for i=1,2,3. x_1, x_2, and x_3 are independent and exhibit a uniform marginal distribution. Furthermore, a and b were set as 7 and 0.1, respectively. This function is highly nonlinear and non-monotonic. It also has a peculiar dependence on x_3. The theoretical values of the first-order Sobol indices are as follows [39]:

(21)S_1=\frac{V_1}{V},S_2=\frac{V_2}{V},S_3=0,

where V={\displaystyle \frac{1}{2}}+{\displaystyle \frac{a^2}{8}}+{\displaystyle \frac{b^2{\text{π}}^8}{18}}+{\displaystyle \frac{b{\text{π}}^4}{5}}, V_1={\displaystyle \frac{1}{2}}{\left(1+b{\displaystyle \frac{{\text{π}}^4}{5}}\right)}^2, V_2={\displaystyle \frac{a^2}{8}}, and V_{1,3}=b^2{\text{π}}^8{\displaystyle \frac{8}{225}}. The total order indices are expressed as [39]

(22)S_{\mathrm{T}1}=\frac{V_1+V_{1,3}}{V},S_{\mathrm{T}2}=S_2,S_{\mathrm{T}3}=\frac{V_{1,3}}{V}.

The sample size was set to 1000 in the current analysis, with a training set of 700 samples used to assess statistical uncertainty and construct the surrogate model of the Ishigami function, and a test set of 300 samples was used to validate the surrogate model. Fig.1 shows the Q^2 values of the training and test sets. Because the points are almost diagonal to the first quadrant, these values are very close to 1 thereby indicating that the constructed surrogate model is excellent and has good predictability.

The GSA for the Ishigami function was performed using variance- and distribution-based methods based on this surrogate model. Tab.1 lists the theoretical values, the values obtained using the two methods, and the importance ranking of the variables. The first- and total-order Sobol index values in the variance-based method were in good agreement with the theoretical values, and both methods provided the same results in terms of the importance ranking of the variables.

The values in the samples were preprocessed using the min–max normalization to improve the accuracy of the estimated index values in the distribution-based method. Fig.2 depicts the histograms of the variables in the failure set, fitted conditional PDFs, and original PDFs. The fitted conditional PDFs obtained using the kernel-smoothing method agreed well with the histograms and described the characteristics of the variables in the failure set.

4 A large-scale single-layer dome

4.1 Model

A large-scale single-layer dome [40] with a span length of 121.5 m, height of 24.2 m, and span-to-height ratio of 5:1, as shown in Fig.3, was the model chosen for this study. The structure comprised 756 pipe members that were connected by 271 joints. The external diameter of each pipe member was 0.245 m and the wall thickness was 0.01 m. The yield strength and strain-hardening ratio of the steel material were set to 345 MPa and 0.015, respectively. The elastic modulus was set to 206 GPa. According to the equivalence, a uniformly distributed roof mass was applied to each joint as a concentrated mass, and each joint had a mass of 9471 kg. The joints at the base of the structure were completely fixed.

The structural finite element model with variables was developed in a probabilistic framework using the OpenSees software, which is based on a command-driven format and parameterized modeling method. Each pipe member was modeled using a displacement-based beam-column element with a fiber cross-section model. In the circumferential and radial directions, the cross-section was divided into 20 and three fibers, respectively. Consequently, the finite element model had 756 elements and 271 nodes. The Giuffre–Menegotto–Pinto model with isotropic strain hardening was used to model the elastoplastic stress-strain relationship of the steel material. A large deformation effect was considered in the modeling because of the shape and size of the structure. The structural damping ratio was 0.02. To model the damping force in the structure, the Rayleigh damping model was applied by setting the circular frequencies of the two modes for determining the damping coefficients \alpha and \beta to 0.667 and 3 times, respectively, the structural first-order circular frequency. A static analysis was performed before the dynamic analysis. Compared to previous studies, this study considered both geometric and material nonlinearity thereby making high-accuracy surrogate modeling difficult.

Tab.2 summarizes the statistical characteristics of the dome variables based on related studies [1,2,29]. The structural failure function for single-layer domes can be expressed as [1]

(23)G\left({\mathit{x}}_1,\dots ,{\mathit{x}}_n\right)=L/n_{\mathrm{D}}-{\left|D_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}},

where {\mathit{x}}_1,\dots ,{\mathit{x}}_n are n variables in the dome; L is the span length; {\left|D_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}} is the maximum vertical deformation (MVD) in the i\mathrm{t}\mathrm{h} sampling dome during an earthquake; and n_{\mathrm{D}} is a constant defined by the limit states.

4.2 Ground motion records

Ground motion data are critical for structural dynamic analyses, and those used in numerical modeling currently are primarily based on natural earthquake records obtained from strong ground motion databases, such as the PEER Strong Ground Motion Databases. Because natural ground motion records differ in duration, time step, spectrum characteristic, peak acceleration, etc., an artificial method for generating ground motions was used to fully reflect the spectral characteristics of all chosen natural ground motions. Tab.3 lists ten typical far-field and near-field natural earthquake records from the PEER strong earthquake databases. Fig.4 shows the spectra in the three directions, as well as their mean spectra (target spectra).

The following formula was proposed by Kaul [41] to convert the target response spectrum into the corresponding power spectral density:

(24)S_x\left({\omega }_k\right)=-\frac{\xi }{\text{π}{\omega }_k}\cdot \frac{{\left[S_{\alpha }^{\mathrm{T}}\left({\omega }_k\right)\right]}^2}{\mathrm{l}\mathrm{n}\left({\displaystyle \frac{-\text{π}}{{\omega }_k{T}_{\mathrm{d}}}}\mathrm{l}\mathrm{n}\left(1-e\right)\right)},

where S_{\alpha }^{\mathrm{T}}\left({\omega }_k\right) is the target response spectrum, \xi is the damping ratio, T_{\mathrm{d}} is the ground motion duration, and e is the exceeding probability of the target response spectrum; typically, the value of e is less than 15%. Furthermore, \xi and T_{\mathrm{d}} were set to 2% and 20 s, respectively.

The stationary Gaussian process \alpha \left(t\right) can be obtained using the trigonometric series method [42],

(25)\alpha \left(t\right)={\sum }_{k=1}^N{C}_k\mathrm{c}\mathrm{o}\mathrm{s}({\omega }_{k}t+{\phi }_k),

where C_k=\sqrt{4S_x\left({\omega }_k\right)\mathrm{\Delta }\omega }, \mathrm{\Delta }\omega =({\omega }_{\mathrm{u}}-{\omega }_{\mathrm{l}})/N, {\omega }_k={\omega }_{\mathrm{l}} + (k-1/2)\mathrm{\Delta }\omega, {\phi }_k is the random phase angle uniformly distributed in (0,2\text{π}), and {\omega }_{\mathrm{u}} and {\omega }_{\mathrm{l}} are the upper and lower limit values of \omega, respectively.

To simulate the transient characteristics of the ground motions, the envelope function f\left(t\right) of the intensity was used as follows:

(26)

where c_{\mathrm{d}} is the attenuation control factor, which ranges between 0.1 and 1.0; and t_1 and t_2 are the first and last moments of the smooth segment, respectively. In this study, they were set to 2 and 12 s to ensure that the large acceleration values were within this interval. The duration T_{\mathrm{d}} is set to 20 s. The time-history acceleration series is then obtained by combining f\left(t\right) with the stationary Gaussian process \alpha \left(t\right) [43],

(27){\ddot{x}}_{\mathrm{g}}\left(t\right)=f\left(t\right)\alpha \left(t\right).

Finally, the target spectrum is used to iteratively adjust C_k to obtain the final earthquake acceleration series.

(28)C_{k,i+1}=C_{k,i}\frac{S_{\alpha }^{\mathrm{T}}\left({\omega }_k\right)}{S_{\alpha }^{\mathrm{C}}\left({\omega }_k\right)},i=0,1,\dots ,

where S_{\alpha }^{\mathrm{C}}\left({\omega }_k\right) is the response spectrum of the generated artificial ground motion.

Based on the preceding process, Fig.5 shows a group of randomly generated ground motions with peak accelerations of −2.04, −1.5, and 0.95 m/s² in the X, Y, and Z directions, respectively. Only this group of ground motions was used in the GSA for the dome because it comprehensively reflected the statistical characteristics of the spectra of natural ground motions.

4.3 Uncertainty quantification for structural performance states

During earthquakes, the members of a lattice dome are subjected to large axial forces and bending moments. The dome members are in an elastic state when subjected to minor earthquakes. However, during major earthquakes, some members may be in an elastoplastic state while others may be in an elastic state as well. The proportion r_{\mathrm{p}} of the elastoplastic members in the dome was defined to describe the structural performance state.

(29)r_{\mathrm{p}}=\frac{N_{\mathrm{p}}}{N_{\mathrm{T}}},

where N_{\mathrm{p}} and N_{\mathrm{T}} represent the number of elastoplastic and total members, respectively. During an earthquake, a structural member is defined to be in the elastoplastic state when the maximum mean strain of the cross-section in the middle of the member exceeds the yielding strain of the steel material. Based on the aforementioned criteria N_{\mathrm{p}} is determined. Two elastoplastic states were selected by adjusting the scale factor of the ground motion. Fig.6 shows the quantified distributions of {\left|D_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}} for the three performance states. The mean deformations of the sample domes are 0.0614, 0.297, and 0.5214 m, respectively. Because the mean deformation of 0.5214 m approaches 1/200 of the span length, which is considered as a critical state of collapse in dome structures, the domes in elastoplastic state 2 are on the verge of collapse. Fig.7 shows the statistical properties of r_{\mathrm{p}}. This demonstrates that in the defined elastoplastic state 1, some members in almost every dome entered the elastoplastic state, and the mean and standard deviation of r_{\mathrm{p}} were approximately 0.0223 and 0.0155, respectively. For the elastoplastic state 2, r_{\mathrm{p}} had a mean value of 0.1022 and standard deviation of 0.0229. Here, the sample size was set to 4000, which was larger than that in elastoplastic state 1 because of the potentially higher nonlinearities in this state.

5 Global sensitivity analysis for the dome

5.1 Accuracy of the surrogate model

The accuracy of the surrogate models constructed using the RK method in the three states was evaluated, as shown in Fig.8–Fig.10. In the elastic state, the Q^2 values for both the training and test sets are greater than 0.95 while using only 600 samples, the MSE values of the data are very small, and the regression model is excellent. According to the test set, the predicted values were in good agreement with the observed values because the mean value of the error was only 1.26%. Consequently, although geometric nonlinearity exists in the elastic state, a highly accurate surrogate model can be obtained with a small sample size.

For the elastoplastic state 1, Q^2 values are greater than 0.9 for both the training and test sets, indicating that the structural nonlinearities reduce the predictability of the dynamic response when compared with the elastic structure. For elastoplastic state 2, Q^2 value on the test set was close to 0.9, and it was extremely difficult to obtain a more accurate surrogate model in this case, which was close to structural collapse with high unpredictability. The mean errors in elastoplastic states 1 and 2 were 5.03% and 5.78%, respectively, which were acceptable for engineering analyses. As previously stated, surrogate models can be used in structural GSA when the value of Q^2 is greater than 0.8.

Using the constructed RK surrogate models and probabilistic distributions in Tab.2, a resampling size of 10000 for maximum demand was used for each future generation. The variance-based sensitivity indices were calculated according to Eqs. (14)–(16). In the distribution-based method, the failure samples in 50000 samples generated must be determined according to Eq. (17). The kernel PDFs of the unconditional and failure samples were obtained via fitting. The sensitivity index was calculated using Eq. (18). The following sections present the results.

5.2 Elastic state

The sensitivity index values were estimated using the variance- and distribution-based methods with a surrogate model for the elastic state, as shown in Fig.11 and Fig.12, respectively.

(1) Variance-based sensitivity analysis

Fig.11(a) presents the estimated values of the first- and total-order Sobol indices for a single variable using the variance-based method. According to these values, the damping ratio x has the greatest effect on the MVD of the structure in this state, followed by the node load N_{\mathrm{l}}, elastic modulus E, and wall-thickness t_{\mathrm{w}}. As their index values are close to zero, the yield strength f_{\mathrm{y}} and strain-hardening ratio b have almost no effect on the MVD. The values of the first-order indices of the elastic modulus, wall thickness of the member, and node load were significantly lower than those of the total-order indices, thus indicating that they interacted with other variables.

A single variable may not have a significant effect on the target output of a dome; however, in a nonlinear structure, it may have a significant effect through interactions with other variables. The interaction effects between the variables were investigated, as illustrated in Fig.11(b). The interaction of t_{\mathrm{w}} with N_{\mathrm{l}} had the largest index value S_{ij} of all interactions, and it was followed by the interaction of E with N_{\mathrm{l}}. Other interactions between variables did not influence the MVD because of the small index values.

(2) Distribution-based sensitivity analysis

The estimated values of the indices for the single variables using the distribution-based method are shown in Fig.12. To determine the failure sample sets of the variables, structural failure should first be determined using Eq. (23). Parameter n_{\mathrm{D}} was set to 1800 in this case, and the failure probability of the structure was approximately 15.69%. According to the index values, the damping ratio \xi has the greatest effect on failure among all the variables for the elastic structure, followed by the node load N_{\mathrm{l}}, wall thickness t_{\mathrm{w}}, and elastic modulus E. Because their index values are very small, the yield strength f_{\mathrm{y}} and strain-hardening ratio b do not influence structural failure.

5.3 Elastoplastic state

5.3.1 Elastoplastic state 1

(1) Variance-based sensitivity analysis

Fig.13 shows the estimated values of the first-order indices, total-order indices, and interaction effects using a variance-based method. In this state, the damping ratio \xi is still the variable with the greatest effect on the MVD, followed by the node load N_{\mathrm{l}}, elastic modulus E, and wall thickness t_{\mathrm{w}}. The yield strength f_{\mathrm{y}} and strain-hardening ratio b had the least effect on the MVD. According to the values of S_{ij}, the interaction effects of many variables on the MVD begin to increase in this state in comparison with the elastic state. Comparatively, the effects of t_{\mathrm{w}} and E on N_{\mathrm{l}} remained more predominant than those of the other interactions.

(2) Distribution-based sensitivity analysis

The index values estimated using the distribution-based method are shown in Fig.14; the constant n_{\mathrm{D}} was set to 375 and this resulted in a failure probability of 31.48%. The following variables were ranked in order of importance: damping ratio \xi, node load N_{\mathrm{l}}, elastic modulus E, wall-thickness of the member t_{\mathrm{w}}, strain-hardening ratio b, and yield strength f_{\mathrm{y}}. The damping ratio and node load were the two variables with the greatest influence on structural failure, which is consistent with the conclusion obtained using the variance-based method.

5.3.2 Elastoplastic state 2

(1) Variance-based sensitivity analysis

Fig.15 shows the estimated index values under the elastoplastic state 2 using a variance-based method. According to these values, the effects of the variables on the MVD were completely different from those in the first two structural states. Among all variables, the node load N_{\mathrm{l}} had the most significant effect. This was followed by the wall thickness t_{\mathrm{w}}, damping ratio \xi, elastic modulus E, yield strength f_{\mathrm{y}}, and strain-hardening ratio b. Despite its low index value, the yield strength f_{\mathrm{y}} had a greater influence on the MVD in elastoplastic state 2 rather than in elastoplastic state 1. Simultaneously, it was found that the interaction of t_{\mathrm{w}} on N_{\mathrm{l}} was the most significant thereby indicating that when the dome enters a higher nonlinear state, the effect of this interaction on MVD should be considered.

(2) Distribution-based sensitivity analysis

Fig.16 shows the index values estimated using the distribution-based method. The constant n_{\mathrm{D}} was set to 200, resulting in a structural failure probability of 20.12%. The importance ranking of the variables is as follows: node load N_{\mathrm{l}}, wall thickness t_{\mathrm{w}}, elastic modulus E, damping ratio \xi, yield strength f_{\mathrm{y}}, and strain-hardening ratio b, which differ slightly from the importance ranking obtained using the variance-based method. Because of its negligible value, the effect of the strain-hardening ratio b on structural failure can be ignored.

5.4 Identification of importance and characteristics of variables

The parameter S_{\mathrm{p},i} for variable {\mathit{x}}_i is defined to better quantify the effects of each variable on the structural performance in different states.

(30)S_{\mathrm{p},i}=S_{\mathrm{T}i}/{\sum }_{i=1}^n{S}_{\mathrm{T}i}.

Based on the previous results, Fig.17 presents the index S_{\mathrm{p},i} of each variable under the three states using the variance-based method. Statistically, the elastic modulus E, wall thickness t_{\mathrm{w}}, node load N_{\mathrm{l}}, and damping ratio \xi all play important roles in the structural performance in all three states, with the damping ratio \xi and node load N_{\mathrm{l}} dominating in the first two states. The following are the characteristics of the four variables that affect the structural performance.

Elastic modulus, E: the importance of this variable in the structural performance is significant initially but subsequently decreases significantly.

Wall thickness, t_{\mathrm{w}}: the importance of this variable increases slightly as the state transitions from elastic to elastoplastic.

Node load, N_{\mathrm{l}}: it is extremely important in a highly elastoplastic state.

Damping ratio, \xi: the damping ratio plays a critical role in both the elastic and near-elastoplastic states.

5.5 Time-history sensitivity analysis

The sensitivity analysis presented above was based on the MVD ({\left|D_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}) in the structure. However, there is an MVD at time t for the i\mathrm{t}\mathrm{h} sample structure, denoted as {\left|d_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right), and this physical quantity is time-dependent during an earthquake. The sensitivity of {\left|d_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right) to the variables has been discussed in this study. The time history of the GSA has not been investigated in previous studies.

Fig.18 shows the values of the time-dependent first- and total-order indices estimated using the variance-based method. For all sample structures, these values are estimated from {\left|d_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right) at time t. Because constructing a large number of new surrogate models is a time-consuming task, each estimated index value is output at an interval of 0.05 s to illustrate the evolution of indices. The total time is 20 s.

Several interesting findings have been revealed and they are summarized below.

1) Because of their large total-order index values, the damping ratio \xi, node load N_{\mathrm{l}}, elastic modulus E, and wall thickness t_{\mathrm{w}} have noticeable effects on {\left|d_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right) in the first two states, whereas the other variables have insignificant effects. The elastic modulus E and wall thickness t_{\mathrm{w}} have nearly identical effects on {\left|d_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right). The values of the first-order indices of node load N_{\mathrm{l}}, elastic modulus E, and wall thickness t_{\mathrm{w}} decreased over time, indicating that they interacted with the other variables as well.

2) The index values of the damping ratio \xi increased over time in all the three states, especially in elastoplastic state 1, indicating that this variable had a significant effect on {\left|d_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right) in the later stages of the earthquake. This variable had no obvious interactions with the other variables in any of the three states because the first-order index values were close to the total-order index values. In elastoplastic state 2, the node load N_{\mathrm{l}} has the most significant effect on {\left|d_i\right|}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right) and contributes the most to structural deformation.

6 Discussion

6.1 Sensitivity of the surrogate model to sample size

The construction of a surrogate model requires a certain number of samples. A large sample size imposes a significant computational burden, whereas a small sample size would not fully reflect the relationship between the input and output. This section discusses the effect of the sample size on the surrogate model using Q^2 values from the training and test samples, as shown in Fig.19. To better investigate the sensitivity of the surrogate model to the sample size, the scale factor of the ground motions was set to 1 and the structure remained elastic.

According to Fig.19, the Q^2 value increases with the sample size, indicating that the surrogate model is sensitive to the sample size, particularly in the test set. When the sample size is greater than 400, the surrogate model performs better in terms of generalization. Currently, there are no effective criteria for determining the sample size as it can only be defined by the convergence of statistical quantities. It can be observed that RK has a good regression ability for the elastic structure when the sample size is larger than 400.

6.2 A comparison of the accuracy of the surrogate model using different correlation functions

In the RK method, the correlation function influences the accuracy of the surrogate model. The Gaussian correlation function can only be used to obtain high-precision surrogate models for elastic structures. In this study, the Gaussian, Matérn 3/2, and Matérn 5/2 correlation functions were compared in the surrogate modeling of elastoplastic structures. Fig.20 shows the Q^2 values and mean errors of the two surrogate models on the test sets for elastoplastic state 1. The results show that in comparison with the Gaussian correlation function the Matérn 3/2 correlation function can improve the surrogate model to some extent. As the Q^2 value increased from 0.9042 to 0.9207, the MSE value decreased from 7.34 × 10⁻⁴ to 4.27 × 10⁻⁴, and the mean prediction error decreased from 5.57% to 5.03%. However, the Matérn 5/2 correlation function does not improve the accuracy of the surrogate model.

6.3 Effect of the accuracy of the surrogate model on sensitivity index values

As previously stated, for GSA the accuracy of the surrogate model should be greater than 0.8. The effect of the accuracy of the surrogate model on the sensitivity index values was investigated in this study. Four surrogate models with different accuracies were compared in the GSA, and their index values are shown in Fig.21. The surrogate models with Q^2 values greater than 0.85 present nearly identical index values, thereby indicating that these values are stable when the Q^2 values are greater than 0.85. In contrast, the index values from the surrogate model with the Q^2 value of 0.7110 differ from those of the other three models. However, it is important to note that the importance ranking of the variables remains the same for all the four models, thus indicating that the importance ranking of the variables is not sensitive to the accuracy of the surrogate model.

6.4 Effect of failure probabilities on distribution-based sensitivity indices

The conditional PDFs of the variables must be determined on a failure set using a distribution-based method. In Eq. (23), parameter n_{\mathrm{D}} should be assumed in advance. When the probability of structural failure is so low that the kernel-smoothing technique cannot accurately fit the conditional PDFs of variables with small samples, the structural GSA may have low accuracy. The effect of the failure probability on the sensitivity index values was investigated.

Fig.22 shows the estimated index values of an elastic structure using the distribution-based method for different failure probabilities based on n_{\mathrm{D}} values. Although P_{\mathrm{f}} has no obvious effect on the identification of the importance of variables in the failure probability range of 1.67%–33.22%, it has a noticeable effect on the estimated index values which decrease as the failure probability increases. Different failure probabilities result in different target functions (Eq. (23)), the variables behave differently, and the index values do not remain constant in this case.

7 Conclusions

GSA plays a critical role in quantifying the structural uncertainty. Conventional FEM leads to a heavy computational burden in quantifying the effects of sources of uncertainty on the structural dynamic performance. In this study, the RK algorithm was used to construct a surrogate model for a large-scale dome structure that was subjected to earthquake ground motions, and the global dynamic behavior of the nonlinear structure was investigated based on the variance- and distribution-based methods. The results show that the computational efficiency of the proposed method significantly improved. Based on the results, the following conclusions were drawn.

1) Among the sources of uncertainty related to structures, the damping ratio and roof load are the two most important variables that influence the structural performance in both elastic and slightly elastoplastic structures. As the structural plasticity develops, the roof load plays a more predominant role rather than the damping ratio. Moreover, according to the time-history sensitivity analysis, the index values of the roof load were generally maintained at higher levels for all the three performance states.

2) The accuracy of the surrogate model is sensitive to the sample size required for model construction and is affected to some extent by the correlation function used in the RK method. A high-accuracy surrogate model can provide stable sensitivity index values in the variance-based method, whereas the sensitivity index values in the distribution-based method depend on the selection of the failure probability. However, they did not affect the importance rankings of the variables.

References

[1] Zhang H D, Zhu X Q, Yao S. Nonlinear dynamic analysis method for large-scale single-layer lattice domes with uncertain-but-bounded parameters. Engineering Structures, 2020, 203: 109780

[2] Zhang H D, Zhu X Q, Liang X, Guo F Y. Stochastic uncertainty quantification of seismic performance of complex large-scale structures using response spectrum method. Engineering Structures, 2021, 235: 112096

[3] Bhattacharyya B. Global sensitivity analysis: A bayesian learning based polynomial chaos approach. Journal of Computational Physics, 2020, 415: 109539

[4] Wei P, Lu Z, Yuan X. Monte Carlo simulation for moment-independent sensitivity analysis. Reliability Engineering & System Safety, 2013, 110: 60–67

[5] Gupta H V, Razavi S. Revisiting the basis of sensitivity analysis for dynamical earth system models. Water Resources Research, 2018, 54(11): 8692–8717

[6] Partington D, Knowling M J, Simmons C T, Cook P G, Xie Y, Iwanaga T, Bouchez C. Worth of hydraulic and water chemistry observation data in terms of the reliability of surface water-groundwater exchange flux predictions under varied flow conditions. Journal of Hydrology, 2020, 590: 125441

[7] Sobol I M, Tarantola S, Gatelli D, Kucherenko S S, Mauntz W. Estimating the approximation error when fixing unessential factors in global sensitivity analysis. Reliability Engineering & System Safety, 2007, 92(7): 957–960

[8] TarantolaSGiglioliNJesinghausJSaltelliA. Can global sensitivity analysis steer the implementation of models for environmental assessments and decision-making? Stochastic Environmental Research and Risk Assessment, 2002, 16(1): 63–76

[9] Razavi S, Jakeman A, Saltelli A, Prieur C, Iooss B, Borgonovo E, Plischke E, Lo Piano S, Iwanaga T, Becker W, Tarantola S, Guillaume J H A, Jakeman J, Gupta H, Melillo N, Rabitti G, Chabridon V, Duan Q, Sun X, Smith S, Sheikholeslami R, Hosseini N, Asadzadeh M, Puy A, Kucherenko S, Maier H R. The future of sensitivity analysis: An essential discipline for systems modeling and policy support. Environmental Modelling & Software, 2021, 137: 104954

[10] Nariman N A, Hussain R R, Mohammad I I, Karampour P. Global sensitivity analysis of certain and uncertain factors for a circular tunnel under seismic action. Frontiers of Structural and Civil Engineering, 2019, 13(6): 1289–1300

[11] Zoutat M, Elachachi S M, Mekki M, Hamane M. Global sensitivity analysis of soil structure interaction system using N2-SSI method. European Journal of Environmental and Civil Engineering, 2018, 22(2): 192–211

[12] Menz M, Dubreuil S, Morio J, Gogu C, Bartoli N, Chiron M. Variance based sensitivity analysis for Monte Carlo and importance sampling reliability assessment with Gaussian processes. Structural Safety, 2021, 93: 102116

[13] Zhang K C, Lu Z Z, Wu D Q, Zhang Y L. Analytical variance based global sensitivity analysis for models with correlated variables. Applied Mathematical Modelling, 2017, 45: 748–767

[14] Javidan M M, Kim J K. Variance-based global sensitivity analysis for fuzzy random structural systems. Computer-Aided Civil and Infrastructure Engineering, 2019, 34(7): 602–615

[15] Arwade S R, Moradi M, Louhghalam A. Variance decomposition and global sensitivity for structural systems. Engineering Structures, 2010, 32(1): 1–10

[16] Boscato G, Russo S, Ceravolo R, Fragonara L Z. Global sensitivity-based model updating for heritage structures. Computer-Aided Civil and Infrastructure Engineering, 2015, 30(8): 620–635

[17] Zhang X F, Pandey M D. An effective approximation for variance-based global sensitivity analysis. Reliability Engineering & System Safety, 2014, 121: 164–174

[18] Cucurachi S, Borgonovo E, Heijungs R. A protocol for the global sensitivity analysis of impact assessment models in life cycle assessment. Risk Analysis, 2016, 36(2): 357–377

[19] Baroni G, Francke T. An effective strategy for combining variance- and distribution-based global sensitivity analysis. Environmental Modelling & Software, 2020, 134: 104851

[20] Wei P F, Wang Y Y, Tang C H. Time-variant global reliability sensitivity analysis of structures with both input random variables and stochastic processes. Structural and Multidisciplinary Optimization, 2017, 55(5): 1883–1898

[21] Ni P, Xia Y, Li J, Hao H. Using polynomial chaos expansion for uncertainty and sensitivity analysis of bridge structures. Mechanical Systems and Signal Processing, 2019, 119: 293–311

[22] Ni P H, Li J, Hao H, Zhou H Y. Reliability based design optimization of bridges considering bridge−vehicle interaction by Kriging surrogate model. Engineering Structures, 2021, 246: 112989

[23] Ni P H, Li J, Hao H, Han Q, Du X L. Probabilistic model updating via variational Bayesian inference and adaptive Gaussian process modeling. Computer Methods in Applied Mechanics and Engineering, 2021, 383: 113915

[24] Yuan Z X, Liang P, Silva T, Yu K, Mottershead J E. Parameter selection for model updating with global sensitivity analysis. Mechanical Systems and Signal Processing, 2019, 115: 483–496

[25] Wan H P, Ni Y Q. An efficient approach for dynamic global sensitivity analysis of stochastic train-track-bridge system. Mechanical Systems and Signal Processing, 2019, 117: 843–861

[26] Amini A, Abdollahi A, Hariri-Ardebili M A, Lall U. Copula-based reliability and sensitivity analysis of aging dams: Adaptive Kriging and polynomial chaos Kriging methods. Applied Soft Computing, 2021, 109: 107524

[27] Xian J H, Su C, Spencer B F Jr. Stochastic sensitivity analysis of energy-dissipating structures with nonlinear viscous dampers by efficient equivalent linearization technique based on explicit time-domain method. Probabilistic Engineering Mechanics, 2020, 61: 103080

[28] Vazna R V, Zarrin M. Sensitivity analysis of double layer diamatic dome space structure collapse behavior. Engineering Structures, 2020, 212: 110511

[29] Wan H P, Ren W X. Parameter selection in finite-element-model updating by global sensitivity analysis using Gaussian process metamodel. Journal of Structural Engineering, 2015, 141(6): 04014164

[30] Sun Q Q, Dias D. Global sensitivity analysis of probabilistic tunnel seismic deformations using sparse polynomial chaos expansions. Soil Dynamics and Earthquake Engineering, 2021, 141: 106470

[31] Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96: 520–535

[32] Tuo R, Wang W J. Kriging prediction with isotropic Matérn correlations: Robustness and experimental designs. Journal of Machine Learning Research, 2020, 21(1): 7604–7641

[33] Vu-Bac N, Lahmer T, Zhang Y, Zhuang X, Rabczuk T. Stochastic predictions of interfacial characteristic of polymeric nanocomposites (PNCs). Composites. Part B, Engineering, 2014, 59: 80–95

[34] Liu B, Vu-Bac N, Zhuang X, Rabczuk T. Stochastic multiscale modeling of heat conductivity of Polymeric clay nanocomposites. Mechanics of Materials, 2020, 142: 103280

[35] Xiao S N, Lu Z Z. Structural reliability sensitivity analysis based on classification of model output. Aerospace Science and Technology, 2017, 71: 52–61

[36] Fenwick D, Scheidt C, Caers J. Quantifying asymmetric parameter interactions in sensitivity analysis: Application to reservoir modeling. Mathematical Geosciences, 2014, 46(4): 493–511

[37] Sheather S J, Jones M C. A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society. Series B. Methodological, 1991, 53(3): 683–690

[38] Marrel A, Iooss B, Laurent B, Roustant O. Calculations of sobol indices for the Gaussian process metamodel. Reliability Engineering & System Safety, 2009, 94(3): 742–751

[39] Sudret B. Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety, 2008, 93(7): 964–979

[40] Liu T T, He Z, Yang Y. Vertical earthquake vulnerability of long-span spherical lattice shells with low rise-span ratios. Engineering Structures, 2020, 207: 110181

[41] Kaul M K. Stochastic characterization of earthquakes through their response spectrum. Earthquake Engineering & Structural Dynamics, 1978, 6(5): 497–509

[42] Scanlan R, Sachs K. Earthquake time histories and response spectra. Journal of the Engineering Mechanics Division, 1974, 100(4): 635–655

[43] Bani-Hani K A, Malkawi A I. A multi-step approach to generate response-spectrum-compatible artificial earthquake accelerograms. Soil Dynamics and Earthquake Engineering, 2017, 97: 117–132

RIGHTS & PERMISSIONS

Higher Education Press