Directional Importance Sampling Research Articles

Metamodel-Based Directional Importance Sampling for Structural Reliability Analysis

Metamodel-Based Directional Importance Sampling for Structural Reliability Analysis

Single-loop cross-entropy-based directional importance sampling for efficient estimation of failure probability function

Failure probability function (FPF) can not only reflect the effect of random input distribution parameter (DP) on structural safety degree, but also decouple reliability based design optimization. Nowadays, it still desires the efficient algorithms for FPF. Thus, this paper proposes an efficient and novel single-loop cross-entropy-based directional importance sampling method (SL-CE-DIS). The proposed SL-CE-DIS has two innovations. The first is constructing the transformation relationship (Tr-Re) of the samples on the limit state surface (LSS) in the standard normal space corresponding to different DP realizations. This Tr-Re is established by the property of the LSS in original physical space independent of DP, and it makes the LSS sample explored by single CE-DIS at a given DP realization can be shared by the CE-DIS at other DP realizations. The second is designing an inverse updating rule to reconstruct the quasi-optimal DIS density (DIS-D) corresponding to other DP realization, which is in the adjacent region of the given DP realization, based on CE principle, and this designed inverse updating rule can easily estimate the FPF values corresponding to other DP realization by matching the transformed LSS samples. Based on the designed LSS sample sharing strategy and inverse updating rule of DIS-D, the proposed SL-CE-DIS can use the LSS information explored by single CE-DIS at the given DP realization to estimate the FPF corresponding to the DP over the adjacent of the given DP realization, and the presented examples fully verify the superiority of the SL-CE-DIS over the other existing single-loop sampling methods for FPF.

Reliability-based design tool for gas storage in lined rock caverns

ABSTRACT The transition to a fossil-free energy matrix may require large quantities of hydrogen gas, which could be stored efficiently in underground lined rock caverns (LRCs). Since the consequences of failure can be catastrophic, the LRC design needs to have a small failure probability. However, the current deterministic design practice for LRCs limits the possibility to stringently address geotechnical uncertainties. In this paper, a reliability-based design tool is presented for LRCs. The adaptive directional importance sampling (ADIS) method, which requires a relatively small number of samples, is used with a 3D finite element (FE) model to evaluate small probabilities of failure. An illustrative example based on the LRC in Skallen, southwestern Sweden, demonstrates the implementation and applicability of the developed design tool. The considered uncertainties are related to the geological conditions and the steel lining. The results show that the reliability of this LRC design meets the expected safety requirements. Considering different geological conditions with correlations, at least “good” quality rock mass is needed for the LRC design. An additional sensitivity analysis is performed to study the potential influence of corrosion and hydrogen embrittlement, showing that the LRC design could meet safety requirements for a lower category of the weld quality.

Rare event estimation with sequential directional importance sampling

In this paper, we propose a sequential directional importance sampling (SDIS) method for rare event estimation. SDIS expresses a small failure probability in terms of a sequence of auxiliary failure probabilities, defined by magnifying the input variability. The first probability in the sequence is estimated with Monte Carlo simulation in Cartesian coordinates, and all the subsequent ones are computed with directional importance sampling in polar coordinates. Samples from the directional importance sampling densities used to estimate the intermediate probabilities are drawn in a sequential manner through a resample-move scheme. The latter is conveniently performed in Cartesian coordinates and directional samples are obtained through a suitable transformation. For the move step, we discuss two Markov Chain Monte Carlo (MCMC) algorithms for application in low and high-dimensional problems. Finally, an adaptive choice of the parameters defining the intermediate failure probabilities is proposed and the resulting coefficient of variation of the failure probability estimate is analyzed. The proposed SDIS method is tested on five examples in various problem settings, which demonstrate that the method outperforms existing sequential sampling reliability methods.

On the use of Directional Importance Sampling for reliability-based design and optimum design sensitivity of linear stochastic structures

This contribution focuses on reliability-based design and optimum design sensitivity of linear dynamical structural systems subject to Gaussian excitation. Directional Importance Sampling (DIS) is implemented for reliability assessment, which allows to obtain first-order derivatives of the failure probabilities as a byproduct of the sampling process. Thus, gradient-based solution schemes can be adopted by virtue of this feature. In particular, a class of feasible-direction interior point algorithms are implemented to obtain optimum designs, while a direction-finding approach is considered to obtain optimum design sensitivity measures as a post-processing step of the optimization results. To show the usefulness of the approach, an example involving a building structure is studied. Overall, the reliability sensitivity analysis framework enabled by DIS provides a potentially useful tool to address a practical class of design optimization problems.

Vector-angle geometric mapping-based directional importance sampling method for reliability analysis

Vector-angle geometric mapping-based directional importance sampling method for reliability analysis

Cross-entropy-based directional importance sampling with von Mises-Fisher mixture model for reliability analysis

In structural reliability analysis, the estimation of rare failure probability is a significant challenge. Directional importance sampling may address this challenge when a quasi-optimal sampling density is chosen. Directional importance sampling involves sampling on the unit hypersphere and one-dimensional reliability analysis. This paper develops a cross-entropy-based directional importance sampling (CE-DIS) for reliability analysis. The von Mises-Fisher mixture (vMFM) model is selected as sampling density to generate direction vectors on the unit hypersphere. The updating rules based on the cross-entropy method are derived for the vMFM model to find a quasi-optimal sampling density. In addition, the DBSCAN algorithm is used to choose the number of distributions in the vMFM model in case of no prior information of failure domain, and the intermediate failure event is introduced in case of not enough samples to update the distribution parameters. Several benchmark numerical examples and engineering applications are used to test the performance of the proposed CE-DIS method. The results demonstrate that the CE-DIS method significantly improves the computational efficiency compared to original directional sampling and other CE-based importance sampling methods. In addition, the CE-DIS method can be applied to different types of reliability problems, e.g., multiple design points, multiple failure modes and rare failure probability.

Augmented reliability analysis for estimating imprecise first excursion probabilities in stochastic linear dynamics

Imprecise probability allows quantifying the level of safety of a system taking into account the effect of both aleatory and epistemic uncertainty. The practical estimation of an imprecise probability is usually quite demanding from a numerical viewpoint, as it is necessary to propagate separately both types of uncertainty, leading in practical cases to a nested implementation in the so-called double loop approach. In view of this issue, this contribution presents an alternative approach that avoids the double loop by replacing the imprecise probability problem by an augmented, purely aleatory reliability analysis. Then, with the help of Bayes’ theorem, it is possible to recover an expression for the failure probability as an explicit function of the imprecise parameters from the augmented reliability problem, which ultimately allows calculating the imprecise probability. The implementation of the proposed framework is investigated within the context of imprecise first excursion probability estimation of uncertain linear structures subject to imprecisely defined stochastic quantities and crisp stochastic loads. The associated augmented reliability problem is solved within the context of Directional Importance Sampling, leading to an improved accuracy at reduced numerical costs. The application of the proposed approach is investigated by means of two examples. The results obtained indicate that the proposed approach can be highly efficient and accurate.

Sensitivity estimation of first excursion probabilities of linear structures subject to stochastic Gaussian loading

This contribution focuses on evaluating the sensitivity associated with first excursion probabilities of linear structural systems subject to stochastic Gaussian loading. The sensitivity measure considered is the partial derivative of the probability with respect to parameters that affect the structural response, such as dimensions of structural elements. The actual calculation of the sensitivity demands solving high dimensional integrals over hypersurfaces, which can be challenging from a numerical viewpoint. Hence, sensitivity evaluation is cast within the context of a reliability analysis that is conducted with Directional Importance Sampling. In this way, the sought sensitivity is obtained as a byproduct of the calculation of the failure probability, where the post-processing step demands performing a sensitivity analysis of the unit impulse response functions of the structure. Thus, the sensitivity is calculated using sampling by means of an estimator, whose precision can be quantified in terms of its standard deviation. Numerical examples involving both small- and large-scale structural models illustrate the procedure for probability sensitivity estimation.

Bounding the first excursion probability of linear structures subjected to imprecise stochastic loading

This paper presents a highly efficient and accurate approach to determine the bounds on the first excursion probability of a linear structure that is subjected to an imprecise stochastic load. Traditionally, determining these bounds involves solving a double loop problem, where the aleatory uncertainty has to be fully propagated for each realization of the epistemic uncertainty or vice versa. When considering realistic structures such as buildings, whose numerical models often contain thousands of degrees of freedom, such approach becomes quickly computationally intractable. In this paper, we introduce an approach to decouple this propagation by applying operator norm theory. In practice, the method determines those epistemic parameter values that yield the bounds on the probability of failure, given the epistemic uncertainty. The probability of failure, conditional on those epistemic parameters, is then computed using the recently introduced framework of Directional Importance Sampling. Two case studies involving a modulated Clough-Penzien spectrum are included to illustrate the efficiency and exactness of the proposed approach.

An active learning Kriging model combined with directional importance sampling method for efficient reliability analysis

In this paper, the active learning Kriging model (ALK), which has been studied extensively in recent years, has been expanded by combining with the directional importance sampling (DIS) method. The directional sampling method can reduce the dimensionality of the variable space by random sampling or interpolation in the direction of vector diameter, which can improve the efficiency of reliability analysis. It is especially suitable for the surfaces whose limit state is spherical or near-spherical. By introducing the control coefficient and constructing the directional importance sampling density function, the sampling efficiency can be further improved in the design point domain. A novel reliability analysis method called ALK-DIS method is proposed. The greatest advantage of the proposed method is its ability on great computational efficiency and dealing with small failure probability problem In addition, due to the excellent performance of directional sampling method in dealing with multi-failure model reliability problems, the ALK-DIS method has the advantage of being applied to system reliability analysis in this paper successfully. The applicability, feasibility and efficiency of the proposed method are proved on examples which contain linearity equation, non-linear numerical example, non-linear oscillator and system reliability engineering problems.

Application of directional importance sampling for estimation of first excursion probabilities of linear structural systems subject to stochastic Gaussian loading

This contribution addresses the estimation of first excursion probabilities of linear structural systems subject to stochastic Gaussian loading by means of simulation. This probability is estimated by combining existing knowledge on the geometry of the associated failure domain with Directional Importance Sampling. In this way, the space associated with the stochastic loading is explored by generating some random directions according to a prescribed importance sampling distribution; then, each random direction is analyzed taking advantage of the linearity of the response with respect to the stochastic loading. Such an approach allows estimating small failure probabilities with high accuracy and precision while requiring a reduced number of samples. The application of Directional Importance Sampling is illustrated by means of a series of examples, indicating that failure probabilities in the order of 10-3 or less can be estimated reliably with a reduced number of samples, even in problems comprising involved structural models.

Directional importance sampling simulation for structural reliability analysis using a procedure to update structural probability estimator through a repetitive simulation method

Directional importance sampling simulation for structural reliability analysis using a procedure to update structural probability estimator through a repetitive simulation method

An adaptive directional importance sampling method for structural reliability analysis

The importance sampling is merged with directional simulation in this paper. A sampling function is defined on the unit hyper sphere which samples random directions. The directions are sampled around a direction that aims to the design point. The sampling function uses spherical coordinates to generate random directions. The method is made adaptive by a closed form updating rule to renew the sampling parameters. To reduce the number of calls on the limit state function, a root finding procedure is put forward. The proposed method is tested with well-known test problems and its performance is compared with the conventional directional simulation. The results demonstrate the accuracy and efficiency of the proposed method for rare event estimation.

A novel hybrid scattering order-dependent variance reduction method for Monte Carlo simulations of radiative transfer in cloudy atmosphere

We present a novel hybrid scattering order-dependent variance reduction method to accelerate the convergence rate in both forward and backward Monte Carlo radiative transfer simulations involving highly forward-peaked scattering phase function. This method is built upon a newly developed theoretical framework that not only unifies both forward and backward radiative transfer in scattering-order-dependent integral equation, but also generalizes the variance reduction formalism in a wide range of simulation scenarios. In previous studies, variance reduction is achieved either by using the scattering phase function forward truncation technique or the target directional importance sampling technique. Our method combines both of them. A novel feature of our method is that all the tuning parameters used for phase function truncation and importance sampling techniques at each order of scattering are automatically optimized by the scattering order-dependent numerical evaluation experiments. To make such experiments feasible, we present a new scattering order sampling algorithm by remodeling integral radiative transfer kernel for the phase function truncation method. The presented method has been implemented in our Multiple-Scaling-based Cloudy Atmospheric Radiative Transfer (MSCART) model for validation and evaluation. The main advantage of the method is that it greatly improves the trade-off between numerical efficiency and accuracy order by order.

305 方向重点サンプリング・シミュレーションに基づく構造破損確率の推定法 : 動径変数のサンプリングをβ超球領域外に限定する場合(GS3 破壊解析・評価,研究討論セッション)

305 方向重点サンプリング・シミュレーションに基づく構造破損確率の推定法 : 動径変数のサンプリングをβ超球領域外に限定する場合(GS3 破壊解析・評価,研究討論セッション)

FULLY PROBABILISTIC DUNE SAFETY ASSESSMENT USING AN ADVANCED PROBABILISTIC METHOD

In this paper, an advanced probabilistic method - Adaptive Directional Importance Sampling (ADIS) - is used in concert with a process-based dune erosion model - XBeach - to perform a dune safety assessment. The efficiency of the probabilistic method removes the restriction of only using simple empirical dune erosion models for probabilistic calculations. The proposed combination of ADIS and XBeach is able to handle complex cross-shore profiles that empirical dune erosion models cannot, as shown in several case studies.

G0310307 限定方向重点サンプリング・シミュレーションに基づく構造システムの信頼性解析(材料力学部門一般セッション:解析・シミュレーション)

This study describes an efficient directional importance sampling method for the simulation-based estimation of the structural failure probability. The proposed method is composed of two-stage procedures. In the first stage, executing specified numbers of directional simulations, the conditional failure probabilities on various random directions are determined by calculating the radial distances from the origin to the failure surfaces. When the basic random variables are subject to the normal distributions, the conditional failure probabilities on the respective random directions are evaluated by using the upper probabilities of the chi-square distribution. Random directional vectors with the radial distances smaller than the minimum radial distance+3 are considered to have effective contributions to the structural failure probability, while random directions with the radial distances larger than minimum radial distance+3 are considered to be ineffective to the structural failure probability. It is proposed that by using random directions with the effective radial distances only, a directional importance sampling probability density is constructed on the basis of the respective upper probability of the chi-square distribution per unit hypersurrace area. In the second stage, a directional importance sampling simulation is executed to estimate the structural failure probability by using the directional vector samples generated from the directional importance sampling probability density constructed in the first stage. Numerical emples to estimate the failure probability of structure with multiple failure surfaces and structure with nonlinear limit state function are presented to illustrate that the proposed method gives accurate estimations effectively.

An Efficient Method of Structural Reliability Analysis Based on a Directional Importance Sampling Simulation (A Determination of Equivalent Directional Importance Sampling Probability Densities Weighted in the Direction of the Design Points)

An Efficient Method of Structural Reliability Analysis Based on a Directional Importance Sampling Simulation (A Determination of Equivalent Directional Importance Sampling Probability Densities Weighted in the Direction of the Design Points)

J045082 設計点方向重点サンプリング・シミュレーションを結合した効率的数値積分による構造システムの信頼性解析法(〔J045-08〕知的材料・構造システム(8))

A new method for estimating the structural failure probability based on the directional importance sampling simulation combined with a numerical radial integration procedure to determine the conditional failure probability in the sampled direction is proposed. The standardized orthogonal space is divided with several concentric hyperspheres and the conditional failure probability in the sampled direction is evaluated by a radial piecewise integration of the failure probability volume contained in the respective concentric hyperspheres. Samples are generated from the importance sampling p.d.f.s centered at the design points on the respective limit state surfaces and directions from the origin to the respective sampled points are proposed to utilize as directional vector samples equivalently. Numerical examples show that the proposed method gives accurate estimations of the structural failure probability efficiently.