Abstract
Structural reliability analysis is usually realized based on a multivariate performance function that depicts failure mechanisms of a structural system. The intensively computational cost of the brutal-force Monte-Carlo simulation motivates proposing a Gegenbauer polynomial-based surrogate model for effective structural reliability analysis in this paper. By utilizing the orthogonal matching pursuit algorithm to detect significant explanatory variables at first, a small number of samples are used to determine a reliable approximation result of the structural performance function. Several numerical examples in the literature are presented to demonstrate potential applications of the Gegenbauer polynomial-based sparse surrogate model. Accurate results have justified the effectiveness of the proposed approach in dealing with various structural reliability problems.
1. Introduction
Structural reliability analysis needs to recursively run a performance function for an accurate estimation result of the structural failure probability. Herein, the vector comprises all input random variables, and each pair of and are assumed as independent for the sake of simplicity. Or else, statistical transformations in the literature are required to obtain equivalently independent random variables [1–3]. Note that the performance function is highly possible to be implicitly expressed and computationally expensive, which makes the corresponding structural reliability analysis be not an easy but highly demanding task in reality.
Analytical derivation for stochastic characteristics of the multivariate performance function is only feasible in rare cases. The Monte Carlo simulation (MCS) and its variants allow one to alleviate the difficulty to some extent [4–6]. However, a simulation-based approach might become tedious due to high computational efforts [7]. Approximation methods, for example, the first-order reliability method (FORM) were widely investigated [8]. This paper primarily focuses on the utility of the Gegenbauer polynomials for a sparse surrogate model to mimic the true but computationally demanding performance function in structural reliability analysis. To achieve this, the orthogonal matching pursuit (OMP) in the signal processing domain is introduced to determine the principle explanatory polynomials for a sparse model, and the chaotic Gegenbauer polynomials are Directional MCS00 introduced in this paper to realize the sparse regression, rather than ordinary polynomials in conventional response surface methods (RSMs).
In reality, the RSM has received considerable attention over the past decades [9–11]. A general procedure to build the surrogate model usually comprises two aspects: (1) to locate feasible combinations of explanatory variables and their coefficients for the optimized mean response and (2) to estimate the model response in the vicinity of the optimized location for better understanding the of the “local” effect of factors [12]. Once a group of explanatory variables are available, the least-square regression and the error backprorogation mechanism are combined to determine unknown parameters [13–16].
Consider that all explanatory variables need to be fully assessed in the conventional regression analysis. The total number of model terms will be exponentially increased with the dimensionality of input variables. The curse of dimensionality makes the conventional RSM be usually applicable to low-dimensional problems. Instead, the sparse regression analysis aims to use a fairly small number of regressors to overcome the disadvantage. Combined with the Galerkin projection [17] or the stochastic collocation method [18] to calculate the coefficients, the determined surrogate models have been widely used for uncertainty quantification of various structural systems [19–21].
The paper proposes to build a sparse surrogate model for structural reliability analysis via the Gegenbauer polynomials. The univariate Gegenbauer polynomials (as ) are specified by the weight function and are constituted as a complete basis to represent an arbitrary function defined on [22]. Given various realizations of the polynomial parameter , define a group of polynomials for the model approximation. For instance, defines the Legendre polynomial, whereas and are related to the first and the second kinds of the Chebyshev polynomials. Flexility motivates to utilize the Gegenbauer polynomials to develop surrogate models for the structural reliability analysis in this paper.
Following the theorem of the generalized Furious expansion, a real-valued function can be exactly represented via a complete basis set. Results from the variance-based global sensitivity analysis have vividly demonstrated utilizing a small number of principle terms for an accurate estimation of the true model. This motivates developing a sparse approximation of the multivariate performance function. Once the multivariate basis set based on the Gegenbauer chaos polynomials is constituted, the method of the orthogonal matching pursuit is further introduced to select primary functions to maintain the high sparsity of the surrogate model. Similar algorithms include the basis pursuit [23] and the matching pursuit methods [24], or the method via the numerical approach of the orthogonal matching pursuit (OMP) [25].
The OMP algorithm has been widely recognized to select principle regressors that are correlated with residual errors of an engineering model. Relying on the orthogonal projection of residual errors to the linear space spanned by orthogonal polynomials, the OMP algorithm can iteratively minimize the global model error sparsely [26]. Due to its advantages in terms of simple implementation and fast convergence, the OMP has been successively used for data compression, signal recovery, image recognition, and so on. Besides, advanced algorithms such as the stagewise orthogonal matching pursuit (StOMP) and the regularized orthogonal matching pursuit (ROMP) were developed in [27, 28]. The paper introduces the Gegenbauer polynomials to constitute the basis functions, and a variety of stopping criteria are further investigated for a robust OMP-based sparse regression model in structural reliability analysis. The approach is nonintrusive, which implies the corresponding structural reliability analysis is only based on a small number of deterministic model evaluations, rather than requiring high order responses, for example, the gradients in the first-order reliability method (FORM) [29].
An objective of the paper is to utilize the multivariate Gegenbauer polynomials as the basis function for an effective surrogate model for the structural reliability analysis. The high-dimensional Gegenbauer polynomials are first presented to define the multivariate chaos terms. Combined with a variety of stopping criteria to realize the sparse regression, an effective surrogate model that mimics the true but computationally demanding performance function is determined for the structural reliability analysis.
The rest of the manuscript is organized as follows: Section 2 briefly summarizes the mathematic properties of the Gegenbauer polynomials, and the multivariate basis functions are further formulated by the chaotic multiplication of univariate functions. Section 3 presents several stopping criteria for a robust sparse regression result. Numerical examples in Section 4 are presented to demonstrate potential applications of the proposed sparse regression model, and conclusions are summarized in Section 5.
2. The multivariate Gegenbauer Polynomials
A reliable sparse regression result for a structural performance function depends largely on the multivariate basis functions to span the approximation space. The paper presents utilizing the chaotic multivariate Gegenbauer polynomials. Therefore, the section starts to introduce mathematical characteristics of the univariate Gegenbauer polynomials, and multivariate basis functions are further derived via the chaotic multiplication of one-dimensional polynomials.
2.1. The Gegenbauer Polynomial
The Gegenbauer polynomials have been widely treated as a group of basis functions for the numerical approximation in engineering realities [30]. In mathematics, () are defined as pairwise orthogonal polynomials for . They are particular solutions of the Gegenbauer differential equation [31]:which can be further degenerated as the Legendre and Chebyshev differential equation, and the Legendre and the Chebyshev polynomials are two special cases of the Gegenbauer polynomials as shown in numerical simulations.
Besides, the Gegenbauer polynomial can be expressed via the generating function as well [32]:and the recurrence relation of the orthogonal polynomials is given as
Mathematically, the orthogonality of is governed byHerein, denotes the weight function , and represents the Kronecker delta symbol. This is further used to define the normalization constant :in which the symbol denotes the Gamma function.
Given the parameter , and 3.0, the first six realizations of the Gegenbauer polynomial (i.e. ) are presented in Figure 1. The Legendre and the Chebyshev polynomials are two special cases of the Gegenbauer polynomial, which will be utilized to build multivariate basis functions for a sparse approximation result of the performance function in structural reliability analysis.
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2.2. The multivariate Gegenbauer polynomial
Structural reliability analysis is always evaluated based on a multivariate performance function , and the multivariate Gegenbauer polynomials will be derived for a numerical approximation of the performance function.
Define an index vector with each integral . Then, the degree of a -variate chaotic Gegenbauer polynomial can be measured by the length of the index vector, that is, . Specifically, the total number () of chaotic terms in the polynomial set is determined by parameters and the polynomial order asand the vector-indexed multivariate Gegenbauer polynomials are defined via the chaotic multiplication of the univariate functions:wherein represents the th-order univariate polynomial presented in Section 2.1.
Figure 2 presents the bivariate Gegenbauer polynomials with parameters and the polynomial degree . For other high-dimensional cases, one can refer to the tensor-product formulation in equation (7). Note that each two pair of multivariate Gegenbauer polynomials are orthogonally defined, given thatwherein represents a -dimensional weighting function as whereas denotes a normalization constant.
To illustrate the orthogonal characteristic, the original index vector of an arbitrary two orders of the chaotic polynomials are recovered as and , respectively. Following this notation, the inner product in equation (8) is realized as
Note that and as and .
Define the normalized univariate Gegenbauer polynomial:
Substituting to equation (8), one can obtain the normalized multivariate set , whose elements have the property ofand its second-norm can be defined as
Note that the normalized multivariate Gegenbauer polynomials (for ) will be used as the basis functions in subsequent sparse approximations, and the normalized polynomial will be denoted as for the sake of simplicity.
Besides the normalization issue, a linear transformation of the parameter is required to match the definition domain of input random variables:and parameters and are empirically defined aswherein represents the inverse cumulative distribution function (iCDF) of the random variable associated with the mean and the standard deviation .
The parameter is expected to connect with the truncated probability in equation (14). If is a Gaussian variable, the truncated probability will be less than and , respectively, as and 4. Therefore, the linear transformation in equation (13) with the parameter will be utilized in numerical examples to tackle nonuniformly distributed and skewed random variables.
3. The Gegenbauer Polynomial-Based Sparse Surrogate Model
Structural reliability simulations are always realized based on a multivariate performance function and its surrogate model. Once the surrogate model is numerically available, the subsequent reliability analysis can be alternatively realized by the brutal Monte Carlo simulation. Note that the total number of functional evaluations is limited to the development of the surrogate model , rather than all random samples. As a comparison, the original random simulation based on will require samples [33]. The large number of model repetitions implies a huge amount of computational cost. Therefore, the paper proposes utilizing the chaotic Gegenbauer polynomials to build the surrogate model at first, and the total number of mechanistic model reevaluations will be only limited to the number of numerical operations to develop the surrogate model, rather than directly carrying out the physical model-based random simulations.
Conventional response surface models usually contain hundreds or even thousands of polynomial terms for a robust estimation result. The global sensitivity analysis result has shown that a relatively small number of component functions is capable of accurately approximating a complex performance function ; that is, a sparse approximation is available for the structural reliability analysis. Based on the consideration, the section presents utilizing the chaotic Gegenbauer polynomial-based sparse model to mimic the true performance function. The principle polynomials among all potential explanatory elements are detected based on a small number of model simulation results. To begin with, a standard procedure for the surrogate model developed based on the statistical regression method is briefly summarized as follows.
3.1. The Regression-Based Surrogate Model
To begin with, the set consists of realizations of input random vector generated based on a low-discrepancy sequence, for example, the Sobol’, the Helton, or the Hammersley algorithm, whereas the corresponding realizations of the chaotic Gegenbauer polynomial set are expressed in a matrix form as
Following the theory of multivariate “linear” regression [34], a response surface model for the performance function can be obtained as
Note that the vector consists of all regression coefficients that are attached to explanatory variables . In addition, the residual error is expressed asHere, the model response vector comprises total realizations of the mechanistic model by continuously feeding each element of the training dataset . And the unknown regression coefficients can be expressed in terms of the training dataset as
Therefore, once the training matrix defined in equation (15) and corresponding structural response samples are numerically available, the least-square minimization based on the 2-norm of the residual error vector allows deriving an empirical surrogate model noted in equation (16) for reliability analysis of the structural system.
3.2. A Sparse Regression Model Based on the OMP Algorithm
A fundamental problem in structural reliability analysis signal processing is to develop a reliable sparse surrogate model for multivariate input uncertainties. The orthogonal matching pursuit (OMP) algorithm for the signal processing is presented with the basis dictionary defined by the chaotic Gegenbauer polynomials.
Given a realization of the polynomial parameter , the full dictionary set can be initialized as , together with principle components . Based totally on training samples in generated based on the Halton or the Sobol’ low-discrepancy scheme, the corresponding model responses and the polynomial matrix in equation (15) are prepared for subsequent model evaluations. Specifically, the residual error is initialized as , whereas the significant elements in will be .
For a realization of the iterative counter , one could implement the OMP-based sparse regression analysis as follows:(1)Detect the most principle polynomial based on the following criterion: wherein the symbol denotes an th column of the polynomial matrix that is evaluated based on samples of the input random vector . Therefore, the polynomial index parameter for an th significant polynomial will be updated as and , respectively. Meanwhile, the explanatory dictionary is revised by excluding the element , that is, for subsequent iterations.(2)Update the model result by calculating the coefficients , which determines the model after a th iteration as Note that the residual error will be , and , and the symbol denotes an identity matrix.(3)Evaluate the sparse regression model in equation (20) based on stopping criteria listed in Table 1. Or else, set and repeat Steps 1–3 till the largest realization of the iterative counter .
It is observed that the OMP algorithm is a stepwise forward greedy algorithm to select principle components. In this regard, a main issue for the sparse regression can stop the greedy selection process at “a right time.” However, stopping rules of the greedy algorithm are mainly defined by mathematical characteristics of the residual error. Typical realizations include the and measures, that is, and , wherein, parameters (as ) denote the predefined error thresholds. In addition, a relative-error-based measure can also be used:which is evaluated by two successive residual errors and (as ) in model iterations. Note that the -norm used here is much more strict than criteria based on the norm.
Rather than the hard stopping criteria that are evaluated directly based on the residual error term , an -value-based soft stopping criterion can be further used to exclude incorrectly identified component functions. This is because the multivariate Gegenbauer polynomials might not be exactly orthogonally defined with each other due to the truncated simulation domain as shown in equation (14). It is highly possible that the residual error cannot be exactly represented by the remaining polynomial set after a th iteration. This is the motivation to utilize an -test-based procedure to exclude spurious basis functions and maintain the high sparsity of the response surrogate model.
Recall that the covariance matrix for regression coefficients is estimated aswherein denotes the variance of the error term , and its unbiased estimator isHerein, the integers and represent the total number of training samples and polynomial elements in , respectively. Then, an -statistics for the significant test of regression coefficients are defined aswhich will be used to detect the most insignificant coefficient with the minimum -value:
Together with the significant level in numerical examples, the -value-based “soft” stopping criterion will be numerically implemented depending on the following cases: Case 1: and . This implies that the null hypothesis cannot be rejected based on the test for the th regression coefficient, and the surrogate model is the sparse regression result for the performance function . Case 2: but . One needs to update the significant polynomial set by excluding the spurious principle component and recalculate regression coefficients for afterward significant tests.
Otherwise, go to Step 3 till the counting variable reaches its upper bound .
In summary, all considered hard and soft stopping criteria are listed in Table 1. In numerical simulations, a total of eight cases about realizations of parameters and will be considered, that is, (Cases 1 and 2), (Cases 3 and 4), (Cases 5 and 6), and (Cases 7 and 8) given a realization of and . Together with several examples in the literature, potential applications of the Gegenbauer polynomial-based surrogate model for structural reliability analysis will be demonstrated below.
4. Numerical Assessments of the Gegenbauer Polynomial-Based Surrogate Model
This section examines the performance of the chaotic Gegenbauer polynomial-based sparse regression method for structural reliability analysis based on several examples in the literature. In this regard, an accuracy measure of the surrogate model is defined aswherein the mean value of the model response is estimated as based on realizations of the input random vector . In addition, a model sparsity indicator is defined asand a small value of the ratio of the sparsity (RoS) (e.g., ) is expected in engineering realities to reduce the total number of training samples for structural reliability analysis.
4.1. The Ishigami function
The mathematical example investigates the numerical performance of the chaotic Gegenbauer polynomial in sparse regressions analysis by considering the Ishigami function:
All input variables are i.i.d. uniform random variables within the interval , and constants and are assumed as 7.0 and 0.1, respectively, in the literature. Note that the full model consists of 286 terms for the highest polynomial order .
Since the analytical result of the Ishigami function is available, a numerical experiment is carried out to examine the effect of various stopping criteria for uncertainty analysis of the Ishigami function. With 200 random realizations of the input vector , the proposed algorithm is followed to determine dominant basis functions and the corresponding sparse regression model.
Figure 3 presents simulation results of the regression error (Re) and the ratio of sparsity (RoS) for various realizations of the polynomial parameter for and , whereas investigated stopping criteria in Table 1 will be investigated. This allows one to determine a better stopping criterion for small realizations of the structural performance function. Results have justified that the polynomial parameter can determine divergent surrogate models in terms of the approximation error and the sparsity ratio. Specifically, realizations and determine the minimal error and the sparsity ratio results for the uniform random variables, respectively. Besides, an increase in the absolute value of will generally increase the global errors and the sparsity ratio. In this regard, the Chebyshev polynomial determined by the parameter will be further examined for various structural reliability problems.
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Specifically, the effect of the parameter on the numerical accuracy of the surrogate model is further investigated by considering varying from to 0.2 with an incremental Step 0.05. Results in Figure 4 show that there are no significant differences among prediction error results, which are less than for all realizations cases, yet the cases and can determine relatively smaller sparsity, that is, . Therefore, the case will be used to check various stopping criteria in Table 1 as follows.
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The polynomial parameter is further fixed as , numerical results of the regression error and the sparsity ratio are estimated for various stopping criteria in Figure 5. It is observed that the global regression error and sparse ratio are much higher than the average result for the hard stop criterion with the threshold parameter (Case 4) and the -value-based soft rule with the parameter (Case 7). Specifically, the largest regression error and the sparse ratio results are observed for Case 2, that is, . In general, the optimum result is determined by the hard rule (Case 4) and the soft criterion (Case 7) for the Ishigami example. The criteria will be further examined by a high-dimensional uncertain model as follows.
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4.2. The High-Dimensional Function
To further examine the performance of the proposed Gegenbauer polynomial-based regression method for uncertainty analysis of high-dimensional models, the example considers the multivariate function:Herein, are uniform i.i.d. variables within the region , and various realizations of the dimensionality parameter are considered.
To examine the performance of the proposed sparse regression approach for a problem with various dimensions of the input random vector , the dimensionality parameter is generally assumed as 10 and 20. Combining the polynomial parameter and various stopping criteria, Figures 6 and 7 summarize simulation results for the global regression error and the sparsity ratio associated with the dimensionality parameters and 20.
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Results have shown that the stop criterion (Case 4) and the statistic-based criteria in Cases 7 and 8 are able to determine relatively small results of the regression error and the sparsity ratio. Specifically, the criterion (Case 2), that is, , cannot determine the surrogate model as accurate as other rules if . Consider a better balance of the global error and the sparsity ratio achieved by the hard criteria and the soft rule ; they will be used to build the surrogate model for structural reliability analysis in the following simulations.
To further examine the utility of various polynomials in the model approximation, the Chebyshev polynomial of the first kind, that is, the polynomial parameter , is further used to build a surrogate model of the high-dimensional function. Results in Figure 8 have shown that the proposed regression method is rather effective for the high-dimensional problem, for example, the dimensionality parameter . The corresponding sparsity ratio is less than , and the regression errors are in the magnitude of . The high accuracy ensures the numerical effectiveness of the surrogate model for structural reliability simulations.
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Figure 9 finally presents numerical results for the response probability density function of the high-dimensional function. With 500 Sobol’ sequences to build the sparse model at first, the brutal-force MCS is fully realized based on the surrogate model . The close agreement of simulation results between the surrogate and the true models has confirmed the high accuracy of the proposed approach.
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4.3. Reliability Analysis of a Steel Frame Structure
The example further demonstrates potential applications of the proposed Gegenbauer polynomial-based sparse regression method for reliability analysis of a two-bay six-story steel frame. As depicted in Figure 10, random variables of the frame structure include the modulus of elasticity , the moment of inertia , and structural external loads . The probabilistic characteristics of input random variables are summarized in Table 2, whereas the performance function is defined as the structural maximal interstorey drift over the limit cm:Herein, denotes the maximal interstorey drift, whereas represents the nodal lateral displacements that are explicitly evaluated based on a finite element (FE) model of the steel frame structure.
To determine the failure probability of the steel frame structure, the Gegenbauer polynomial with the parameter is used to develop a sparse surrogate model with the polynomial order and the sample size . Note that a similar result based on the parameter is determined, yet omitted here for the sake of brevity.
Figure 11(a) depicts numerical results for the empirical PDF of the structural maximum roof drift, and 500 samples are sufficient to determine a surrogate model to mimic the true performance function. The POE result pictured in Figure 11(b) provides the estimation of the structural failure probability as , which is fairly close to the benchmark result in [35].
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To further evaluate the robustness of the proposed approach, 100 repetitions of the sparse regression analysis have been implemented, and the corresponding results for the structural failure probability are summarized in Figure 12. Since each round of the sparse regression analysis only requires 500 nonintrusive model runs. It has exhibited the high efficiency of the proposed approach for the structural reliability analysis, whereas the small standard deviation result () confirms the robustness of the proposed method as well.
4.4. Reliability Analysis of a Bar Structure with Spatially Varying Stochastic Material Properties
This section illustrates an application of the proposed approach by considering a bar structure with spatially varying Young’s modulus. As depicted in Figure 13, a bar structure with unit length and cross-sectional area (i.e., and ) is subjecting axially distributed load . Specifically, the modulus of elasticity is model via a homogeneous Gaussian random field:which is the exponential covariance function and symbols and represent any two positions along the bar. The parameters unit and denote the standard deviation and correlation length of the Gaussian random field, respectively.
Following the theory of the Karhunen–Loève expansion, the material random field is numerically parameterized based on its first- eigenpairs {} [36]:wherein denote independent standard Gaussian random variables, whereas the mean value of the random field is constantly assumed as .
Failure events for reliability analysis of the bar structure are defined by the maximum axial displacement over a threshold value of 0.40:Here, random vector comprises all Gaussian random variables in equation (32) for numerical discretization of the random field.
With various realizations of the correlation length parameter, that is, , and 100, numerical realizations of the elasticity random field are presented in Figure 14. Note that ten random variables will be enough to represent more than 95% of the original variability. This implies that the random vector consists of ten standard Gaussian random variables. In addition, a large value of the correlation length parameter (e.g., ) increases the statistical dependency between positions of the random modulus elasticity . This allows one to examine the numerical performance of the proposed approach in dealing with dependent input uncertainties.
To implement, the sparse regression method with the polynomial order and 500 Sobol’ sequences is employed to develop a surrogate model for the structural maximal axial displacement . Results for PDFs of are presented in Figure 15 for various realizations of the correlation length parameter . It is observed that the response variability is directly related to the parameter in the random field model, and the case determines the largest variation of the structural response quantity.
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Based on the structural performance function defined in equation (33), Table 3 summarizes the structural failure probability for various realizations of the correlation length parameter . The close agreement between the estimation and benchmark results has verified the high accuracy and numerical efficiency of the sparse regression method.
Figure 16 further summarizes simulation results for the sparsity ratio and the global regression error of the surrogate model. The sparsity ratio result is dramatically decreased with an increase of the correlation parameter . A larger value of the parameter implies numerical realizations of the Young’s Modulus at two different locations are strongly correlated. The statistical dependency will increase the structural failure probability as presented in Figure 16(b). Note that a small variation result for the structural failure probability further verifies the robustness of the proposed Gegenbauer polynomial-based regression method for reliability analysis of a structural model with dependent input uncertainties.
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5. Conclusion
Structural reliability analysis is typically evaluated based on a multivariate performance function that defines small failure probabilities. It is significant to develop a surrogate model to mimic the true performance function as the brutal-force MCS based on the realistic model might be computationally intensive. This paper presents utilizing the Gegenbauer polynomials to constitute the explanatory dictionary, whereas principle component functions are adaptively selected via the OMP-based sparse regression algorithm. Due to the regression bias introduced by utilizing random samples, an excluding procedure to detect spuriously component functions is proposed based on the statistics. Simulation results have shown that the Gegenbauer polynomial-based regression method can determine reliable estimation results for the investigated performance function. Small regression errors and the high sparsity of the surrogate model have demonstrated potential applications of the adaptive sparse regression algorithm for structural reliability analysis.
Data Availability
The simulation data within this submission are available based on the request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.