Structural reliability analysis based on polynomial chaos, Voronoi cells and dimension reduction technique

doi:10.1016/j.ress.2019.01.001

Reliability Engineering & System Safety

Volume 185, May 2019, Pages 329-340

https://doi.org/10.1016/j.ress.2019.01.001 Get rights and content

Highlights

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An unequal weighted sampling strategy is adopted to construct the sparse PCE.
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Slice inverse regression is employed for dimension reduction when the dimension is very large.
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A hybrid method combing polynomial chaos, Voronoi cells and dimension reduction is proposed to construct the surrogate models.
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Numerical examples show the proposed method is fairly efficient and accurate for structural reliability analysis.

Abstract

Polynomial chaos expansions (PCEs) has been widely used to construct meta-models for structural reliability analysis. The computational effort of classical PCEs is unaffordable as the required number of deterministic model analyses grows exponentially with the dimension. Alternatively, the sparse PCEs are always built to alleviate this problem. This paper proposes an efficient method, which combines the sparse PCE with a novel unequal-weighted sampling strategy, i.e. Voronoi cells and the dimension reduction technique for structural reliability analysis. The unequal-weighted sampling strategy could converge fast to the ultimate goal of sequentially building a sparse PCE. Besides, when the dimension is high, the sliced inverse regression technique is employed to convert the original high-dimensional problem to a low-dimensional one. Then, a stepwise weighted regression method is involved to automatically determine the significant terms of the PCE and discard the insignificant ones for the reduced model. In this regard, the sparsity of the basis, the dimension reduction technique and the fast convergence of unequal-weighted sampling strategy lead to a considerably reduced computational cost. Four numerical examples with a large number of random variables are presented to validate the proposed method. The computational results show that the proposed method can establish fairly accurate meta-models for structural reliability assessment with low computational effort.

Introduction

Surrogate models (also known as meta-models) are widely found to approximate the real models for uncertainty quantification. There are a number of commonly used meta-models, such as the polynomial chaos expansions (PCEs) [1], [2], [3], [4], [5], [6], [7], [8], high-dimensional model representation [9], [10], Kriging models [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], support vector machines [21], [22], [23], [24], [25], artificial neural networks [26], [27], [28], [29], [30], etc. Among these techniques, the PCE, which is non-intrusive, could be one of the most promising methods since it can determine the effect of input uncertainties on output response [31]. The key idea of PCE is to decompose the output response onto multivariate polynomial bases, which are orthogonal with respect to the joint probability density function (PDF) of input variables. In this regard, once the coefficients of polynomials are available, the output response can be reconstructed by PCE and the reliability can be straightforwardly computed based on PCE by using Monte Carlo simulations (MCS)

It is well-known that the original PCEs suffer from the “curse of dimensionality”, which means the number of coefficients for determination exponentially grows with the dimension. It is also known that the coefficients are evaluated based on a set of points in the input random-variate space and their corresponding output responses of the original model. In this regard, the number of required model analyses could be prohibitively large when the number of input random variables is large [4], [32]. To circumvent this problem, several sparse PCEs have been developed, where a small number of significant terms are retained. The sparse PCEs can be constructed based on the following representative methods: (1) stepwise regression method [4], [32]; (2) least angle regression method [33]; (3) weighted L1-minimization method [34]; (4) adaptive-sparse expansion scheme [35] and so forth. The sparse PCEs have been proven to be able to provide a considerable computational effort saving compared to the classic full PCE for structural reliability analysis.

In the construction of a sparse PCE, extensive simulations are required at a finite number of samples to build the sparse PCE with acceptable prediction accuracy. A challenging issue is how to achieve an accurate sparse PCE by using samples as few as possible. The number of samples is actually closely related to the total computational effort for building a sparse PCE. In this regard, the sampling strategies, also known as the experiments design, are of paramount importance to achieve the tradeoff of accuracy and efficiency. The Latin hypercube sampling (LHS) is widely used for constructing the sparse PCE, where an over-sampling of 2–3 times of unknown coefficients of a sparse PCE may be always recommended to get satisfactory results. This feature could still result in large computational effort in high-dimensional cases because the equal-weighted samples are used to construct the sparse PCE, which may converge slowly to the ultimate goal. For a complex function, one would intuitively expect that the weights of samples are higher in the core region and lower in the less important region to capture the behavior of this function with a small sample size. Besides, the samples with unequal weights always have a faster convergence rate compared to that of equal-weighted ones. In this regard, an unequal-weighted sampling strategy, should be used to build the sparse PCE within a small amount of computational effort.

Further, when the dimension is high, e.g. larger than 50, it is still difficult to build the sparse PCE with fair accuracy even if the unequal-weighted samples are employed. To mitigate this difficulty, the dimension reduction techniques should be incorporated to convert the original high-dimensional input vector to be a low-dimensional one. Since the output response is often caused by only a few of latent parameters varying within a low-dimensional subspace [36], many high-dimensional problems are intrinsically low-dimensional. This low-dimensional subspace is always referred to as the effective dimension reduction (EDR) subspace. Once the low-dimensional input vector is available, building a sparse PCE over the EDR subspace becomes computationally viable.

The objective of the paper is to build a sparse PCE based on the unequal-weighted sampling strategy and dimension reduction technique for efficient structural reliability analysis. First, the fundamentals of PCE and its measures of accuracy are introduced in Section 2. Then, Section 3 is devoted to presenting the unequal-weighted sampling strategy, where the Voronoi cells are employed to fulfil the aim. Section 4 introduces a specific dimension reduction technique, say the sliced inverse regression. In Section 5, the sparse PCE is constructed accordingly. Four numerical examples are given in Section 6 to validate the proposed method. This paper ends up with some concluding remarks in the final Section.

Section snippets

Polynomial chaos expansion and its measures of accuracy

In this section, the polynomial chaos expansion (PCE) is first revisited.

The unequal weighted sampling strategy: Voronoi cells

As a matter of fact, the first step to build a sparse PCE is to generate the unequal-weighted samples. One of the appropriate ways to generate the unequal weighted samples is to use the Voronoi diagram approach [42], [43], [44], [45], where a set of Voronoi cells $C = {C_{1}, C_{2}, \dots, C_{N}}$ are employed to partition the distribution domain. First, a set of points need to be generated in the interested random-variate space, which is denoted as $P = {θ_{i} = (θ_{1, i}, θ_{2, i}, \dots, θ_{d, i}), i = 1, 2, \dots, N}$ . This can be implemented by

Dimension reduction technique for high-dimensional case

When the dimension is high, e.g. d > 50, it is always difficult to build the sparse PCE even by using the unequal-weighted samples due to the “curse of dimensionality”. In this regard, the dimension reduction technique can be employed to seek the central subspace and mitigate this issue. Several dimension reduction techniques can be used, e.g. principal component analysis (PCA) [49], partial least square (PLS) regression [50] and active subspace [51], etc. However, the knowledge of output

Construction of sparse PCE

As mentioned, the high-order interaction terms in a full PCE (Eq. (4)) could be negligible, where the main effects and low-order interactions of random variables should be retained. In this regard, a hyperbolic truncation scheme can be employed to fulfill the aim [33]. This can be performed by defining a new set A_β of multi-indices as follows $A_{β} = {α \in N^{d}, {∥ α ∥}_{β} = {(\sum_{j = 1}^{d} {α_{j}}^{β})}^{1 / β} \leq p}$ where 0 < β < 1 is called the β-quasi-norm and Card(A_β) < P.

It is noted that only the terms whose β-quasi-norms are smaller

Numerical examples

In this section, four numerical examples are presented to demonstrate the efficacy of the proposed method. They are selected to show that the proposed method is of accuracy and efficiency for both explicit and implicit functions when a large number of random variables are involved. The first example considers a linear analytical function with 20, 50 and 100 random variables, respectively, while the second example involves a nonlinear analytical function with 25 random variables. In Examples 3

Concluding remarks

In this paper, an efficient method is developed to build the sparse PCE based on the unequal-weighted sequential sampling strategy and dimension reduction technique. The unequal-weighted sampling strategy, which can be obtained by partitioning the input random-variate space via Voronoi cells, has a much faster convergent rate compared to the equal ones, e.g. Latin hyper cube sampling. For high-dimensional cases, the sliced inverse regression (SIR) method is employed to conduct the dimension

Acknowledgments

The research reported in this paper is supported by the National Natural Science Foundation of China (Grant No. 51608186) the Natural Science Foundation of Hunan Province (No. 2017JJ3016) and the Fundamental Research Funds for the Central Universities (No. 531107040890). The support is gratefully appreciated. The anonymous reviewers are highly appreciated for their constructive comments to improve the original manuscript.

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