Sensitivity estimation of first excursion probabilities of linear structures subject to stochastic Gaussian loading
Introduction
A widespread means for quantifying the level of safety of a structure subject to dynamic uncertain loading is the so-called first excursion probability, which measures the chances that one or more responses of interest exceed a prescribed threshold level within the duration of the stochastic excitation [1]. Clearly, the properties of the structure such as dimensions of structural members, stiffnesses or masses affect this probability and hence, it is of paramount interest to calculate the associated sensitivity. That is, to quantify how this probability changes due to modifications of those properties, see e.g. [2], [3], [4], [5], [6]. Such information is of utmost importance, for example, within the context of decision making and risk optimization [7], [8], as it allows determining optimal design solutions which balance construction, operation and eventual failure consequences of engineering systems. In particular, the gradient of the failure probability provides a practical sensitivity measure. Nonetheless, the calculation of both first excursion probability and its sensitivity in terms of a gradient is a challenging task, as the estimation of the probability itself usually demands solving a high dimensional integral that does not possess a closed-form solution.
The estimation of the gradient of a failure probability is a topic which has been addressed in several contributions considering different approaches, see e.g. [9], [10], [11], [12], [13]. However, the number of contributions addressing the specific issue of estimating the gradient of first excursion probabilities of dynamical systems subject to stochastic loading is more restricted and can be classified into two major groups. In the first of these groups, the gradient of the failure probability is calculated with respect to distribution parameters of some structural properties characterized as random variables. In such case, by means of the so-called score function [10], [14], the probability sensitivity of complex nonlinear structural systems has been estimated by means of Subset Simulation, as reported in [15], [16]. The second group of approaches focuses on estimating the gradient of the first excursion probability with respect to deterministic parameters of a structural system. Within this second group, one class of approaches employs stochastic simulation in conjunction with Bayes’ theorem in order to calculate the sought sensitivity measure [17], [18]. Another class of approaches employs stochastic simulation as well, but in combination with local approximations of the responses of interest [19], [20], [21]. Yet another class of approaches considers some approximations of second-order moments [22].
This work focuses on estimating the gradient of the first excursion probability with respect to deterministic parameters (such as dimensions of structural members, stiffnesses or masses) of linear structures subjected to dynamic Gaussian loading. These last considerations are focused on applications that involve mainly structural systems where serviceability conditions are of interest, see e.g. [23], [24], [25]. The theoretical framework for assessing such gradient is based on Directional Sampling [26], [27], [28]. The reason for choosing Directional Sampling over other techniques for reliability analysis lies in the fact that it allows identifying explicitly the limit state hypersurface that separates safe and failure domains. This is most useful for reliability sensitivity estimation [13]. The novel contribution of this work comes into the integration and application of Directional Sampling for performing sensitivity analysis for the specific case of first excursion probabilities of linear structures. Specifically, an Importance Sampling density function developed in [29], [30] is injected within the framework of Directional Sampling, allowing the development of a Directional Importance Sampling (DIS) scheme for probability and probability sensitivity estimation. It is shown that the sought sensitivity becomes a byproduct of the failure probability estimation carried out by means of DIS and that its practical implementation demands performing a sensitivity analysis of the unit impulse response functions of the structure under consideration.
The remaining part of this contribution is organized as follows. Section 2 presents the problem considered in this contribution, namely the calculation of the gradient of a first excursion probability. Then, Section 3 discusses the framework for estimating the aforementioned gradient, which combines concepts of Directional Sampling and Directional Importance Sampling. The application of that framework is illustrated in Section 4, including both small- and large-scale structural models. Finally, Section 5 closes with some conclusions and outlook.
Section snippets
Preliminary remarks
This section states the class of problems considered in this work. In particular, Section 2.2 describes the type of stochastic loading considered and the evaluation of the structural response. Then, Section 2.3 presents formal definitions of both the first excursion probability and its gradient.
Gaussian loading and structural dynamic response
Consider a dynamic load p acting over a structural system, which is represented as a discrete Gaussian process at time instants , where is the time discretization, the number of
General remarks
Approaches for estimating the gradient of the failure probability are usually cast as a post-processing step of a reliability analysis, see e.g. [10], [13], [15], [40], [41]. This makes sense from a practical viewpoint, as one usually requires the information on probability and its sensitivity simultaneously. Furthermore, the information provided by structural analyses performed at the reliability analysis stage can be considered as well for sensitivity analysis. Therefore, this work follows
Overview of the examples
This Section presents two examples that illustrate the application of the framework for the estimation of the probability and its gradient as discussed above. The first example comprises a test model while the second one involves the finite element model of a structural system. In both examples, comparisons with reference results are included, in order to provide a quantitative assessment of the advantages of the framework.
Example 1: Two-degree-of-freedom oscillator subject to discrete white noise excitation
The first example involves a test problem which consists of the
Conclusions and outlook
This contribution has explored the application of Directional Importance Sampling for the estimation of the sensitivity of the first excursion probability of a linear structure subject to stochastic Gaussian loading. The sensitivity measure considered herein is the partial derivative of the failure probability with respect to parameters that affect the structural response. The results obtained in this contribution suggest that:
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It is possible to estimate the sought derivatives in a
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research is partially supported by ANID— (National Agency for Research and Development, Chile) under its program FONDECYT, Grant Nos. 1180271 and 1200087, and Universidad Tecnica Federico Santa Maria – Dirección de Postgrado y Programas under its programs PAC (Programa Asistente Científico 2017) and PIIC (Programa de Incentivo a la Iniciación Científica). This support is gratefully acknowledged by the authors.
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