System reliability analysis of spatial variance frames based on random field and stochastic elastic modulus reduction method

This paper presents the stochastic elastic modulus reduction method for system reliability analysis of spatial variance frames based on the perturbation stochastic finite element method (PSFEM) and the local average of a random field. The stochastic responses and reliability index of each element of a structural frame are characterized by the PSFEM and the first-order second-moment method, to properly handle the correlation structures and scale of fluctuation of random fields. A strategy of elastic modulus adjustment for the estimation of system reliability is developed to determine the range and magnitude of elastic modulus reduction, by taking the element reliability index as a governing parameter. The collapse mechanism and system reliability index of a stochastic framed structure are determined through iterative computations of the PSFEM. Compared with the failure mode approaches in traditional system reliability analysis, the proposed method avoids two major difficulties, namely the identification of significant failure modes and estimation of the joint probability of failure modes. The influences of the correlation structure and scale of fluctuation of the random field upon system reliability are investigated to demonstrate the accuracy and computational efficiency of the proposed methodology in system reliability analysis of spatial variance frames.     

Generalized F-discrepancy-based point selection strategy for dependent random variables in uncertainty quantification of nonlinear structures

In the performance evaluation of structures under disastrous actions, for example, earthquakes, it is important to take into account the randomness of structural parameters. Generally, these random parameters are treated either as independent or perfectly dependent, but practically they are partly dependent. This article aims at developing a point selection strategy for uncertainty quantification of nonlinear structures involving probabilistically dependent random parameters characterized by copula function. For this purpose, the point selection strategy for structures involving independent basic variables is first revisited. As an improvement, a generalized F-discrepancy diminishing oriented iterative screening algorithm is proposed. Then, combining with the conditional sampling method, a conditional point set rearrangement method and a conditional iterative screening-rearrangement method are proposed for probabilistically dependent variables. These new point selection strategies are readily incorporated into the probability density evolution method for uncertainty quantification of nonlinear structures involving dependent random parameters, which is characterized by copula function. The proposed methods are illustrated by two examples including a shear frame with hysteretic restoring forces and a reinforced concrete frame structure with the damage constitutive model of concrete, where the material parameters are probabilistically dependent. The results demonstrate the effectiveness of the proposed method. Problems to be studied are discussed.     

钢筋砼结构非线性随机地震响应分析

考虑地震作用和结构参数的随机性,建立了钢筋砼结构药非线性随机动力学模型。文中导出了随机结构动力分析的非线性随机有限元法的增量列式,并据此对多层钢筋砼结构进行了弹塑性随机地震响应分析。计算结果与该建筑物的实际震害作了对比,效果良好。还讨论了动力模型中随机变量对响应量的影响。     

Performance optimization of multidisciplinary mechanical systems subject to uncertainties

This paper presents a methodology to solve a new class of stochastic optimization problems for multidisciplinary systems (multidisciplinary stochastic optimization or MSO) wherein the objective is to maximize system mechanical performance (e.g. aerodynamic efficiency) while satisfying reliability-based constraints (e.g. structural safety). Multidisciplinary problems require a different solution approach than those solved in earlier research in reliability-based structural optimization (single discipline) wherein the goal is usually to minimize weight (or cost) for a structural configuration subject to a limiting probability of failure or to minimize probability of failure subject to a limiting weight (or cost). For the problems solved herein, the objective is to maximize performance over the range of operating conditions, while satisfying constraints that ensure safe and reliable operation. Because the objective is performance based and because the constraints are reliability based, the random variables used in the objective must model variability in operating conditions, while the random variables used in the constraints must model uncertainty in extreme values (to ensure safety). Thus, the problem must be formulated to treat these two different types of variables at the same time, including the case when the same physical quantity (e.g. a particular load) appears in both the objective function and the constraints. In addition, the problem must be formulated to treat multiple load cases, which can again require modeling the same physical quantity with different random variables. Deterministic multidisciplinary optimization (MDO) problems have advanced to the stage where they are now commonly formulated with multiple load cases and multiple disciplines governing the objective and constraints. This advancement has enabled MDO to solve more realistic problems of much more practical interest. The formulation used herein solves stochastic optimization problems that are posed in this same way, enabling similar practical benefits but, in addition, producing optimum designs that are more robust than the deterministic optimum designs (since uncertainties are accounted for during the optimization process). The methodology has been implemented in the form of a baseline MSO shell that executes on both a massively parallel computer and a network of workstations. The MSO shell is demonstrated herein by a stochastic shape optimization of an axial compressor blade involving fully coupled aero-structural analysis.     

Identification of probabilistic distribution parameters for the mesoscopic stochastic fracture model

As a kind of multiphase composite material, the basic mechanical behaviors of concrete are randomness and nonlinearity. The mesoscopic stochastic fracture model (MSFM) which can reflect the coupling effect of randomness and nonlinearity, has been widely used for the nonlinear analysis of concrete structures. In this paper, we proposed a new stochastic modeling principle to identify the probabilistic distribution parameters of MSFM. In order to reduce the modeling works, a dimension-reduced algorithm is proposed as well. In this paper, an overview of MSFM is firstly presented to introduce the background of the research. Then the stochastic harmonic function (SHF) representation is introduced to express the random field mentioned in the MSFM, and the probability density evolution method (PDEM) is applied to obtain the theoretical probability density function (PDF) of the stress–strain relationships. Furthermore, a stochastic modeling principle is proposed, in which minimizing the Kullback–Leibler divergence (KLD) is taken as the optimization object. Based on the framework of genetic algorithm, a dimension-reduced algorithm is proposed to identify the parameters with reference to the data from tested complete curves of uniaxial compressive and uniaxial tensile stress–strain relationship of concrete. The results indicate that the proposed principle and algorithm can be used to identify the parameters of MSFM accurately and efficiently.     

Constraint effects on probabilistic analysis of cracks in ductile solids

This paper presents a method for evaluating constraint effects on probabilistic elastic–plastic analysis of cracks in ductile solids. It is based on fracture parameters J and Q , correlation between Q and J– resistance curve of the material, and J -tearing theory for predicting fracture initiation and instability in cracked structures. Based on experimental data from small-scale fracture specimens, correlation equations were developed for fracture toughness at crack initiation and the slope of the J– resistance curve as a function of constraint condition. The random parameters may involve crack geometry, tensile and fracture toughness properties of the material, and applied loads. Standard reliability methods were applied to predict probabilistic fracture response and reliability of cracked structures. The results suggest that crack-tip constraints have little effect on the probability of crack initiation. However, the probability of fracture instability can be significantly reduced when constraint effects are taken into account. Hence, for a structure where some amount of stable crack-growth can be tolerated, crack-tip constraints should be considered for probabilistic fracture-mechanics analysis.     

既有钢筋混凝土桥梁构件抗力模糊随机时变概率模型研究

构件的抗力概率模型是进行桥梁结构时变可靠性研究的基础之一。既有钢筋混凝土桥梁材料的老化与损伤情况复杂使其抗力同时具有随机性、模糊性和时变性是一个模糊随机过程。在分析影响既有钢筋混凝土桥梁构件抗力不确定性因素的基础上,考虑桥梁在服役过程中的耐久性损伤对构件抗力的影响,在常规方法只能考虑抗力随机时变性基础上,进一步考虑模糊性,结合实测数据和现有资料建立了既有钢筋混凝土桥梁中混凝土强度、钢筋截面积和钢筋强度模糊随机时变模型,进而研究了在不修复情况下桥梁构件抗力模糊时变概率模型,分析了抗力平均值和标准差随时间和阈值变化的规律,并以受弯构件为例给出了具体分析过程和结果。     

Data-driven identification for approximate analytical solution of first-passage problem

The first-passage problem plays a significant role in engineering performance evaluation and design optimization. To address general stochastic dynamical systems, a data-driven method is proposed to identify approximate analytical solutions for the first-passage problem which explicitly includes parameters of the system, excitation, and those related to the initial and boundary conditions. The method consists of two successive processes. First, the probability density of the first-passage time is assumed to satisfy the modified Weibull distribution and its expansion expression is constructed by using the rule of dimensional consistency. Second, by comparing the expansion with the probability density of the first-passage time estimated from random state data, the coefficients are determined by solving a set of overdetermined linear algebraic equations. Two representative examples, including the Duffing oscillator and a 2-DOF nonlinear dynamical system, are discussed in detail to illustrate the application and efficiency of the data-driven method. The efficacies of the approximate analytical solutions for the external parameters are also verified.     

Optimal structural design under stochastic uncertainty by stochastic linear programming methods

In the optimal plastic design of mechanical structures one has to minimize a certain cost function under the equilibrium equation, the yield condition and some additional simple constraints, like box constraints. A basic problem is that the model parameters and the external loads are random variables with a certain probability distribution. In order to get reliable/robust optimal designs with respect to random parameter variations, by using stochastic optimization methods, the original random structural optimization problem must be replaced by an appropriate deterministic substitute problem. Starting from the equilibrium equation and the yield condition, the problem can be described in the framework of stochastic (linear) programming problems with ‘complete fixed recourse’. The main properties of this class of substitute problems are discussed, especially the ‘dual decomposition’ data structure which enables the use of very efficient special purpose LP-solvers.     

随机结构模态频率与物理参数的相关分析

本文应用Monte-Carlo随机有限元方法进行了随机结构系统的模态频率与随机物理参数的相关分析。数值试验表明，在一定的变异范围内（8=1％-30％），随着物理参数变异性增大，模态密集程度降低，模态频率与随机物理参数互相关系数发生变化。     

Probabilistic criteria of structural stochastic optimal controls

A family of probabilistic criteria for stochastic optimal controls is developed in the context of physical stochastic optimal control scheme of structures. A physical form of the control policy is firstly conducted in conjunction with the classical optimal control theory, specifically, an LQR control. In order to obtain the optimal weighting matrices included in the control gain, two classes of probabilistic criteria, in accordance with the objective structural performance, are then proposed, including single-objective criteria, of which the statistics and tail details of probability density of equivalent extreme-value vectors of interest are involved, and multi-objective criteria, of which the ensemble-expectation and exceedance probability of equivalent extreme-value processes in the sense of performance and energy trade-off are involved. A linear single-degree-of-freedom structural system subjected to random ground motion is investigated for illustrative purpose. The results indicate that the effectiveness of response control hinges on the physical meanings of the probabilistic criteria. The implementation of control criteria related to the exceedance probability with multiple constraints results in a more economic and more accurate control effectiveness than that of control criteria related to statistics with single constraint. The exceedance probability criterion in energy trade-off sense accommodates system performance to a better trade-off between response reductions and control requirements, which is also included in the comparative study against other control criteria currently in use. The former, meanwhile, provides accurate reliabilities of system quantities simultaneously, while other control criteria fail to do so. It is thus the primary criterion of structural performance controls. Following that, a randomly base-excited eight-storey shear frame controlled by active tendons is investigated as a numerical example. The numerical results reveal that using the advocated probabilistic criterion, the structural stochastic optimal control operates efficiently and achieves a desirable objective performance.     

Feedback minimization of the first-passage failure of a hysteretic system under random excitations

The stochastic optimal bounded control of a hysteretic system for minimizing its first-passage failure is presented. The hysteretic system subjected to random excitation is firstly replaced by an equivalent nonlinear non-hysteretic system. The controlled non-hysteretic system is reduced to a one-dimensional controlled diffusion process by using the stochastic averaging of the energy envelope method. The dynamical programming equations and their associated boundary and final-time conditions for the problems of maximization of reliability and mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The dynamical programming equations for the maximum reliability problem and the mean first-passage time problem are finalized and solved numerically. Finally, numerical results are worked out to illustrate the application and effectiveness of the proposed method.     

Probabilistic nonconvex constrained optimization with fixed number of function evaluations

A methodology is proposed for the efficient solution of probabilistic nonconvex constrained optimization problems with uncertain. Statistical properties of the underlying stochastic generator are characterized from an initial statistical sample of function evaluations. A diffusion manifold over the initial set of data points is first identified and an associated basis computed. The joint probability density function of this initial set is estimated using a kernel density model and an Itô stochastic differential equation (ISDE) constructed with this model as its invariant measure. This ISDE is adapted to fluctuate around the manifold yielding additional joint realizations of the uncertain parameters, design variables, and function values, which are obtained as solutions of the ISDE. The expectations in the objective function and constraints are then accurately evaluated without performing additional function evaluations. The methodology brings together novel ideas from manifold learning and stochastic Hamiltonian dynamics to tackle an outstanding challenge in stochastic optimization. Three examples are presented to highlight different aspects of the proposed methodology.     

基于Copula理论的序列型地震动随机模型

通过考虑序列型地震动中主震和余震间的空间相关性,将已有地震动物理随机模型推广至序列型地震动随机模型。依据序列型地震动的产生与传播机制,将其表示为包含16个随机变量的Fourier模型。考虑主震与余震的空间相关性,基于Copula理论,将模型更准确地表示为6个二维随机变量和2个一维随机变量;为描述地震动在局部场地上传播的过程,建立了序列型地震动的随机场模型;根据1038组实测序列型地震动及5组地震动台阵数据,对模型中的物理随机变量进行参数识别和统计分析,结合Copula函数,给出了各参数的概率分布。与实测地震动对比表明:该文提出的序列型地震动随机模型,能够较为真实地再现序列型地震动的空间相关性以及地震动在局部场地的行波效应。     

Data-driven reliability assessment with scarce samples considering multidimensional dependence

This study proposes a data-driven method for assessing reliability, based on the scarce input dataset with multidimensional correlation. Since considering the distribution parameters estimated from the scarce dataset as those of the population may lead to epistemic uncertainty, the bootstrap resampling algorithm is adopted to infer the distribution parameters as interval parameters. To account for the variable dependence, vine copula theory is utilized to construct the joint probability density function (PDF) of input variables, and maximum likelihood estimation (MLE) and Akaike information criterion (AIC) analysis are employed to select optimal copulas based on the samples for the vine structure. Subsequently, the failure probability bounds of a response function are calculated based on the constructed joint PDF with interval distribution parameters by the active learning Kriging (AK) method combining the sparse grid integration (SGI) method. Finally, several examples are provided to demonstrate the feasibility and efficiency of the proposed method.     

Uncertainty quantification in non-linear dynamic response of functionally graded materials plate

This study deals with the stochastic non-linear dynamic response of functionally graded materials (FGMs) plate with uncertain system properties subjected to time-dependent uniformly distributed transverse load in thermal environments. System properties, such as material properties of each constituent's material, volume fraction index, and transverse load, are taken as uncorrelated random input variables. Material properties are assumed as temperature dependent (TD). The formulation is based on higher-order shear deformation theory (HSDT) with von-Karman non-linear strain kinematics using modified C° continuity. A Newton–Raphson-based non-linear finite element method along with a first-order perturbation technique (FOPT) and Monte Carlo sampling (MCS) is outlined to examine the second-order statistics (mean, standard deviation (SD), and probability density function (PDF)) of the non-linear dynamic response of the FGM plate. The governing dynamic equation is solved by Newmark's time integration scheme. The effects of volume fraction index, load parameters, plate thickness ratios, and temperature changes with random system properties are examined through parametric studies. The present outlined approach is validated with the results available in the literature and by MCS.     

Variability response functions for beams with nonlinear constitutive laws

The ability to determine probabilistic information of response quantities in structural mechanics (e.g. displacements, stresses) is restricted due to lack of information on the probabilistic characteristics of uncertain system parameters. The concept of the Variability Response Function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous stochastic fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). In this paper, the existence, the uniqueness, and the SDF- and PDF-independence of a variability response function is formally proven for the first time for statically determinate beam structures following a specific class of nonlinear constitutive laws (power laws). For statically indeterminate nonlinear structures, the generalized variability response function (GVRF) methodology is shown to produce GVRFs for statically indeterminate nonlinear beams with a square-root constitutive law that are almost SDF-independent and only mildly dependent on the marginal PDF. This PDF-dependence is not significant and all GVRFs computed in this study have very similar shapes. This is important as it implies that conclusions related to the effect of correlation length scales on the response uncertainty can be inferred in general. However, the GVRF methodology for nonlinear statically indeterminate structures is only suitable when a closed-form expression is known to exist for the VRF of statically determinate structures having the same constitutive law.