Bivariate Dimension Reduction Method Research Articles

Efficient method for updating the failure probability of a deteriorating structure without repeated reliability analyses

Bayesian updating is widely applied to evaluate the failure probability of a deteriorating structure; however, it requires considerable computational effort. This is because reliability analysis based on posterior distributions must be repeated each time the inspection data are updated. To overcome this problem, this study proposes an efficient method to directly update the failure probability of a deteriorating structure. In the proposed method, an updating factor is proposed to quantify the updated information by separating the components to be updated from the reliability analysis with the direct prior distribution. Thus, both the need to repeat the structural reliability analysis and reacquire posterior distributions/samples each time the inspection data are updated can be avoided. The point-estimate method combined with the bivariate dimension-reduction method is introduced to directly update the failure probability, from which only the updating factor at the fixed estimation points, that is, the components that have no relationship with the structural analysis, needs to be updated, thus considerably reducing the computation time. The efficacy and accuracy of the proposed method were verified through four numerical examples, wherein commonly used methods were employed for comparison.

Efficient method for fully quantifying the uncertainty of failure probability

It has been recognized that the uncertainty of distribution parameters has a significant effect on the outcome of structural reliability analysis. In this study, an efficient and accurate method is proposed to fully quantify failure probability, considering the uncertainty of the distribution parameters. This method obtains predictive failure probability by integrating the probability space of the conditional reliability index instead of directly utilizing the point-estimate method on the conditional failure probability as in previous studies. In the proposed method, a three-parameter lognormal distribution is suggested to approach the distribution of the conditional reliability index. The first three central moments of the conditional reliability index are estimated utilizing the point-estimate method combined with the bivariate dimension-reduction method. Based on this, the analytical solutions for the quantiles and probability distribution of the conditional failure probability can be readily derived without redundant calculations. Four illustrative examples were investigated to verify the efficiency and accuracy of the proposed method. The results show that the proposed method produces sufficiently accurate results in an efficient and simple way. It also presents a complete picture of the structural reliability evaluation results considering distribution parameter uncertainty in a wide range of applications.

An adaptive dimension-reduction method-based sparse polynomial chaos expansion via sparse Bayesian learning and Bayesian model averaging

Polynomial chaos expansion (PCE) is a popular surrogate method for stochastic uncertainty quantification (UQ). Nevertheless, when dealing with high-dimensional problems, the well-known curse of dimensionality and overfitting pose great challenges to the PCE. This study proposes an adaptive sparse learning method based on the automatic relevance determination (ARD) and the Bayesian model averaging (BMA) to solve this problem. Firstly, we construct the sparse PCE model in the framework of sparse Bayesian learning. Secondly, the dimension-reduction method (DRM) is used to reduce the size of candidate PCE bases. Different from the common univariate and bivariate DRMs, this study considers the high-dimensional components in an adaptive way. Thirdly, we propose a novel ARD method based on the analytical B-LASSO to prune the candidate PCE bases. The pruned PCE is regarded as an initial model, and then the BMA is used to refine the high-fidelity sparse PCE. Compared with the direct use of the noninformation prior, the intervention of the pruned PCE model can enhance the rationality of the prior, and improve the accuracy of the posterior estimation. We illustrate the validity of the proposed method from different angles using five classical numerical examples and solve one practical engineering problem. The results indicate that the proposed method is a good choice for UQ under limited design samples.

Estimation of statistical moments of performance functions using efficient importance measure

ABSTRACT This paper introduces an efficient methodology for evaluating the first four central moments of performance functions in structural reliability assessment. The procedure of the proposed method is divided into three major steps. First, a performance function is approximated as the sum of univariate and bivariate functions by the bivariate dimension-reduction method. Next, a new criterion based on importance measure analysis is proposed for dividing the bivariate functions into significant bivariate functions and non-significant bivariate functions, and then the new approximation of the performance function is obtained. Finally, formulas are derived for evaluating the first four central moments of the proposed performance function, in which the first four raw moments of the univariate and bivariate functions can be achieved by a point estimating method in a standard normal space. The effectiveness and efficiency of the proposed method are demonstrated through three numerical examples.

A novel framework for reliability assessment of payload fairing separation considering multi-source uncertainties and multiple failure modes

Payload fairing separation is a critical process for various aerospace structures. To further guarantee the safety of separation, multi-source uncertainties in the processes of manufacturing, fabrication and assembly need to be considered in the stage of structural design. Reliability assessment is a powerful tool to deal with multi-source uncertainties, however, the high computational cost of structural analyses is the main challenge of the numerical simulation of payload fairing separation. In this paper, a novel framework for reliability assessment of payload fairing separation is proposed. In general, since not all potential failure modes will be happened for a certain design, sampling method is utilized to identify the active failure mode firstly. Then, an efficient global sensitivity analysis method is used for the dimension reduction to simply the high dimensional problem. And then, the MPP-based simplified bivariate dimension reduction method (MPP-based SBDRM) is proposed to improve the accuracy of MPP-based DRM for calculating the failure probability. In the proposed method, the augmented step size adjustment (ASSA) method is employed to ensure the robustness and efficiency of the most probable point (MPP) searching process. To further improve the efficiency, an adaptive surrogate model is constructed to replace the time-consuming finite element analysis (FEA). Two mathematical benchmarks are used to validate the performance of the proposed method. Finally, a typical practical payload fairing demonstrates that the proposed framework is able to assess the reliability of fairing separation in an efficient and accurate manner.

Robust concurrent topology optimization of multiscale structure under load position uncertainty

Concurrent topology optimization of macrostructure and microstructure has attracted significant interest due to its high structural performance. However, most of the existing works are carried out under deterministic conditions, the obtained design may be vulnerable or even cause catastrophic failure when the load position exists uncertainty. Therefore, it is necessary to take load position uncertainty into consideration in structural design. This paper presents a computational method for robust concurrent topology optimization with consideration of load position uncertainty. The weighted sum of the mean and standard deviation of the structural compliance is defined as the objective function with constraints are imposed to both macro- and micro-scale structure volume fractions. The Bivariate Dimension Reduction method and Gauss-type quadrature (BDRGQ) are used to quantify and propagate load uncertainty to calculate the objective function. The effective properties of microstructure are evaluated by the numerical homogenization method. To release the computation burden, the decoupled sensitivity analysis method is proposed for microscale design variables. The bi-directional evolutionary structural optimization (BESO) method is used to obtain the black-and-white designs. Several 2D and 3D examples are presented to validate the effectiveness of the proposed robust concurrent topology optimization method.

An improved adaptive bivariate dimension-reduction method for efficient statistical moment and reliability evaluations

Statistical moment estimations of the performance function with the aim of balancing accuracy and efficiency still remains a challenge for moment-based structural reliability analysis. In this paper, an improved adaptive bivariate dimension-reduction method in terms of vectors (i-VBDRM) is proposed for efficient statistical moments and reliability evaluations. In the proposed method, the delineation of cross terms of two-dimensional functions involved in the bivariate dimension-reduction method (BDRM) is first implemented, where the random variables are classified into several sub-vectors after the non-normal to normal transformation. Then, the explicit expressions of the moments of multiple component sub-vector functions are formulated, where a novel cubature formula is proposed and the Gauss-Hermite quadrature is employed to evaluate the involved two- and one-dimensional Gaussian-weighted integrals. In that regard, the first-four central moments can be calculated with accuracy and efficiency. Then, the probability density function of the performance function is rebuilt by a flexible distribution model called the shifted generalized lognormal distribution based on the first-four central moments evaluated by the proposed method. To demonstrate the efficacy of the proposed method, four numerical examples are presented, where some other forms of BDRM, univariate dimension-reduction method and the crude Monte Carlo simulation are performed for comparisons. The results show that the proposed method can keep the trade-off of precision and efficiency for both the statistical moment assessments and structural reliability analysis.

Probabilistic surrogate models for uncertainty analysis: Dimension reduction‐based polynomial chaos expansion

SummaryThis paper presents an approach for efficient uncertainty analysis (UA) using an intrusive generalized polynomial chaos (gPC) expansion. The key step of the gPC‐based uncertainty quantification (UQ) is the stochastic Galerkin (SG) projection, which can convert a stochastic model into a set of coupled deterministic models. The SG projection generally yields a high‐dimensional integration problem with respect to the number of random variables used to describe the parametric uncertainties in a model. However, when the number of uncertainties is large and when the governing equation of the system is highly nonlinear, the SG approach‐based gPC can be challenging to derive explicit expressions for the gPC coefficients because of the low convergence in the SG projection. To tackle this challenge, we propose to use a bivariate dimension reduction method (BiDRM) in this work to approximate a high‐dimensional integral in SG projection with a few one‐ and two‐dimensional integrations. The efficiency of the proposed method is demonstrated with three different examples, including chemical reactions and cell signaling. As compared to other UA methods, such as the Monte Carlo simulations and nonintrusive stochastic collocation (SC), the proposed method shows its superior performance in terms of computational efficiency and UA accuracy.

Maximum Entropy Method-Based Reliability Analysis With Correlated Input Variables via Hybrid Dimension-Reduction Method

The estimation of the statistical moments is widely involved in the industrial application, whose accuracy affects the reliability analysis result considerably. In this study, a novel hybrid dimension-reduction method based on the Nataf transformation is proposed to calculate the statistical moments of the performance function with correlated input variables. Nataf transformation is intrinsically the Gaussian copula, which is commonly used to transform the correlated input variables into independent ones. To calculate the numerical integration of the univariate component function in the proposed method, a normalized moment-based quadrature rule is employed. According to the statistical moments obtained by the proposed method, the probability density function of the performance function can be recovered accurately via maximum entropy method. Six examples are tested to illustrate the accuracy and efficiency of the proposed method, compared with that of Monte Carlo simulation, the conventional univariate dimension-reduction method, and the bivariate dimension-reduction method. It is confirmed that the proposed method achieves a good tradeoff between accuracy and efficiency for structural reliability analysis with correlated input variables.

Uncertainty propagation method for probabilistic fatigue crack growth life prediction

Predicting the probability distribution of fatigue crack growth (FCG) life is very important for probabilistic damage tolerance analysis and structural integrity assessment. This paper proposes a novel uncertainty propagation method for probabilistic FCG life prediction, which allows efficiently estimating the probability density function (PDF), and cumulative distribution function (CDF) of the FCG life considering the randomness of the loads, material properties of crack propagation rate, and crack geometries. First, the relationship between the PDF of FCG life and the one of its logarithm is established by a logarithmic analysis. Then, the random variables are uniformly transformed into independent normal variables that obey distributions of N(0,1/2) using variable transformation technique. Thus, the bivariate dimension reduction method can be conveniently applied to calculate the first four moments of the logarithmic FCG life, and the corresponding PDF is further estimated by the maximum entropy principle. Finally, the PDF and CDF of the FCG life are analytically derived from the ones of the logarithmic FCG life. The validity of the proposed method is verified by numerical, experimental and engineering examples. It is demonstrated that the PDF and CDF of FCG life are well predicted by the proposed method, regardless of the strong nonlinearity of FCG life function and the large uncertainties of input random variables. This method outperforms the Monte Carlo simulation (MCS) in terms of computational efficiency, and in the meanwhile it has an acceptable computational accuracy.

Efficient seismic reliability analysis of large-scale coupled systems including epistemic and aleatory uncertainties

Reliability analysis of large coupled systems is a challenging task in dam safety. Not only the number of random variables is increased, but also a complex numerical model makes the simulations computationally expensive. To overcome this problem, an efficient analytical model should be developed which does not affect the accuracy of reliability analysis. In this paper, a gravity dam-foundation-reservoir coupled system is taken as a vehicle for the numerical simulations. Epistemic (i.e. material) uncertainties, as well as the ground motion record-to-record variability (i.e. aleatory) are incorporated in the simulations. A new reliability technique is proposed and its efficiency is compared to the Latin Hypercube Sampling (LHS). In this technique, a continuous probability density function is replaced by a finite number of fractional moments. Furthermore, an improved bivariate dimension reduction method is developed for dynamic systems. Several parametric simulations are performed to investigate the impact of the reservoir water level and the dam size. Results of the proposed method are in good agreement with the LHS. Finally, a response surface meta-model is proposed to estimate the dam behavior as a function of ground motion intensity measures.

A new bivariate dimension reduction method for efficient structural reliability analysis

This paper presents a new bivariate dimension reduction method (BDRM) for statistical moments evaluation and structural reliability analysis with accuracy and efficiency. A high-order unscented transformation (HUT) is introduced to evaluate the two-dimensional integrals involved in BDRM, and the free parameter involved in HUT is suggested. In this regard, the proposed BDRM can be formulated accordingly for statistical moments assessment. Then, the performance function’s probability density function (PDF) is reconstructed by the shifted generalized lognormal distribution (SGLD) with the available statistical moments as constraints. Thus, the failure probability can be straightforwardly obtained by a simple integral over the PDF. The proposed method is verified by five numerical examples, including linear and non-linear, explicit and implicit performance functions. Besides, some other existing methods are also employed to demonstrate the advantages of the proposed method. It is found that the proposed method can keep the trade-off of accuracy and efficiency for structural reliability analysis.

An efficient method for statistical moments and reliability assessment of structures

The probability density function (PDF) of a performance function can be constructed from the perspective of first four statistical moments, and the failure probability can be evaluated accordingly. Since the shifted generalized lognormal distribution (SGLD) model will be fitted to recover the PDF based on the first four statistical moments, the evaluation of statistical moments of the performance function is of critical significance for the structural reliability analysis. This paper presents a new method for statistical moments and reliability assessment of structures with efficiency and accuracy, especially when large variabilities in the input random vector and nonlinearities are considered . First, a numerical method is established based on rotating the points in the quasi-symmetric point method (Q-SPM), which is very efficient for evaluating the statistical moments. This numerical method is called the rotational quasi-symmetric point method (RQ-SPM). The optimal angles of rotation in RQ-SPM can be determined via an optimization problem, where the objective function is adopted as minimizing the differences between the marginal moments of input random variables estimated by the points after rotation and their exact values. By doing so, the information of marginal distributions and their tail distributions could be better reproduced, which is of paramount importance to the statistical moments assessment of the performance function, especially for the high-order moments. Once the statistical moments are available, the PDF of the performance function can be recovered by the SGLD model. Finally, the failure probability can be evaluated by a simple integral over the PDF of the performance function. Several numerical examples are given to demonstrate the efficacy of the proposed method. Comparisons of the new method, the original Q-SPM, the univariate dimension reduction method (UDRM) and the bivariate dimension reduction method (BDRM) are also made on the statistical moments assessment. The results manifest the accuracy and efficiency of the proposed method for both the statistical moments and reliability assessment of structures.

Improved Neumann Expansion Method for Stochastic Finite Element Analysis

This paper presents an improved Neumann expansion method for calculating statistical moments of either a structural response or system inverse matrix under stochastic uncertainties in system properties and loads. The essentials of the proposed method are the successive matrix inversion and partial bivariate dimension reduction method that enable efficient multidimensional probability integration. The successive matrix inversion computes an exact realization of a stochastic system response, or an inverse matrix, by using the concept of inverse Neumann expansion. In characterizing the stochastic response, multidimensional probability integration is solved for statistical moments and covariance matrices. The existing univariate decomposition method is highly efficient to approximate the multidimensional integration. However, due to the missing interaction effects, the univariate decomposition method could lead to a huge error. In this study, a partial bivariate decomposition method is proposed to capture bivariate interaction effects approximately with a marginal increase in computational cost. Several numerical examples, including representative structural problems, are discussed to demonstrate the accuracy and efficiency of the proposed method compared with other existing methods. It is found that the proposed method provides better accuracy with significant computational cost savings when random fluctuations make locally limited changes in a system.

A new method for reliability assessment of structural dynamic systems with random parameters

The reliability of structural dynamic systems involving random parameters can be assessed based on the equivalent extreme value distribution of response. To derive such probability density function, a new method is proposed in the present paper, in which an iterative scheme is involved. In each iterative step, an estimated solution is easily obtained firstly by solving a linear system of equations and the more accurate result is then searched around the estimated solution with high efficiency. The derivation is based on the principle of maximum entropy, in which an un-constrained optimization problem with fractional moments is involved. Three different methods for numerical integration of fractional moments are compared, which indicates that the bivariate dimension reduction method ensures the accuracy and efficiency simultaneously. Two examples, of which one deals with a linear random structure subjected to seismic excitation, the other deals with a nonlinear structure with random parameters subjected to random ground motions, are illustrated to validate the proposed method. The investigations show that the proposed method is of satisfactory accuracy and applicable to the reliability assessment of practical structural dynamic systems.

Reliability and Sensitivity Analysis of Mechanical Structures Based on Dimension Reduction Method

This paper presents a method for probabilistic sensitivity analysis of mechanical components or structural systems subject to random uncertainties in loads, material properties and geometry. The bi-variate dimension reduction method is applied to compute the response moments and their sensitivities with respect to the distribution parameters of basic random variables. Saddlepoint approximations with truncated cumulant generating functions are employed to estimate the probability density functions and cumulative distribution functions of the random responses. The rigorous analytic derivation of the sensitivities of the probability of failure of the systems under consideration with respect to the distribution parameters of basic random variables is derived. Finally, the practicality and efficiency of the proposed method are demonstrated by an application example.

Hybrid Dimension-Reduction Method for Robust Design Optimization

Robust design optimization has gained increasing concern in the engineering design process because it can provide an economical design that is insensitive to variations in the input variables without eliminating their causes. Robustness assessment, which estimates the mean values and standard deviations of the objective function and constraint functions, is an important and inevitable component in robust design optimization. A hybrid dimension-reduction method is proposed for efficient and accurate robustness assessment in this paper. It is a combination of the univariate dimension-reduction method and the bivariate dimension-reduction method for the balance of efficiency and accuracy. The significant random variables are identified by their variation contributions to the output variation of the objective function and constraints using the univariate dimension-reduction method. The approximate part of the significant random variables in the approximate function is then extended to use the bivariate dimension-reduction method, while the rest of the approximate part is still kept using the univariate dimension-reduction method. It then couples with a numerical optimization algorithm, formulating an iterative robust design-optimization procedure. Two engineering examples are used to illustrate its advantages. The proposed method is found to be competitive in estimating the mean values and standard deviations of the objective function and constraint functions in robust design optimization.

Uncertainty Analysis by Dimension Reduction Integration and Saddlepoint Approximations

Uncertainty analysis, which assesses the impact of the uncertainty of input variables on responses, is an indispensable component in engineering design under uncertainty, such as reliability-based design and robust design. However, uncertainty analysis is an unaffordable computational burden in many engineering problems. In this paper, an uncertainty analysis method is proposed with the purpose of accurately and efficiently estimating the cumulative distribution function (CDF), probability density function (PDF), and statistical moments of a response given the distributions of input variables. The bivariate dimension reduction method and numerical integration are used to calculate the moments of the response; then saddlepoint approximations are employed to estimate the CDF and PDF of the response. The proposed method requires neither the derivatives of the response nor the search of the most probable point, which is needed in the commonly used first and second order reliability methods (FORM and SORM) and the recently developed first order saddlepoint approximation. The efficiency and accuracy of the proposed method is illustrated with three example problems. With the same computational cost, this method is more accurate for reliability assessment and much more efficient for estimating the full range of the distribution of a response than FORM and SORM. This method provides results as accurate as Monte Carlo simulation, with significantly reduced computational effort.