A two-stage Subset Simulation-based approach for calculating the reliability of inelastic structural systems subjected to Gaussian random excitations

This paper presents a methodology for calculating the reliability of inelastic structural systems subjected to Gaussian random excitations. The method adopts a two-stage approach, involving separate calculation of the failure probabilities associated with linear elastic and inelastic structural response. The method exploits the fact that the calculation of failure probabilities associated with a linear problem can be performed extremely efficiently, using minimal computational effort compared to the effort required for solving a corresponding nonlinear problem. The calculation of failure probability associated with inelastic response is performed using a modified Subset Simulation procedure where the first step involves the direct simulation of samples in the inelastic domain rather than standard Monte Carlo simulations as in Standard Subset Simulation. It is demonstrated with a numerical example that the proposed two-stage approach offers significant computational savings over the Standard Subset Simulation approach.     

Stochastic computation based on orthogonal expansion of random fields

In solving stochastic differential equations, recently a random variable based Polynomial Chaos (rv-PC) method has been developed as a major numerical solver. For many realistic random media problems the rv-PC method however confronts a critical challenge of curse-of-dimensionality. Since a random field is represented by random variables, the use of various optimal sampling techniques and conventional high dimensional methods still faces the curse-of-dimensionality. Distinguished from all the random variable based methods, in this study a novel Random Field based Orthogonal Expansion (RF-OE) method is proposed in aim to circumvent the curse-of-dimensionality for many physical systems whereas the input information is represented as random fields or stochastic processes, e.g. seismic/ocean wave, wind load, shock wave, and geophysical media. Multiscale modeling of random media problems is selected as the benchmark problem to test the RF-OE method. Especially, the RF-OE method provides a perfect matching with the higher-order Mehler’s formula. By replacing high dimensional random variable representations with a series of orthogonal expansion terms about an underlying random field/process, the RF-OE method reduces the number of dimensions of a stochastic differential equation exponentially. In the first example the RF-OE method is verified with Monte Carlo simulation on a lognormal random media flow transport problem. In the second example the RF-OE method is applied to a time domain problem involving orthogonal expansion of random excitations. In the conclusion the items for further development of the RF-OE method are identified.     

不确定性转子系统的随机有限元建模及响应分析

随机特性和随机载荷会引起转子系统动力响应的不确定性,是转子动力学分析中的重要影响因素.本文基于Timosheke梁理论,把转轴的材料和几何随机特性表示为一维随机场函数,推导出随机转轴有限元列式,建立转子系统随机动力学模型,并给出随机载荷作用下随机转子系统动力响应统计量的分析方法.分别对线性和非线性涡轮泵转子系统进行了随机动力响应分析,并同Monte Carlo仿真结果进行对比,结果表明所建立的随机有限元动力学模型和给出的随机响应分析方法是合理可行的,可以有效应用于实际转子系统随机动力学分析和设计中.     

Neumann dynamic stochastic finite element method of vibration for structures with stochastic parameters to random excitation

This paper presents a method for the dynamic analysis of structures with stochastic parameters to random excitation. A procedure to derive the statistical characteristics of the dynamic response for structure is proposed by using dynamic Neumann stochastic finite element method presented herein. Random equation of motion for structure is transformed into a quasi-static equilibrium equation for the solution of displacement in time domain. Neumann expansion method is developed and applied to the equation for deriving the statistical solution of the dynamic response of such a random structure system, within the framework of Monte Carlo simulation. Then, the results from Neumann dynamic stochastic finite element method are compared with those from the first- and second-order perturbation stochastic finite element methods and the direct Monte Carlo simulation with respect to accuracy, convergence and computational efficiency. Numerical examples are examined to show that the approach proposed in this paper has a very high accuracy and efficiency in the analysis of compound random vibration.     

Reliability analysis of structures with complex limit state functions using probability density evolution method

In reliability-based structural analysis and design optimization, there exist some limit state functions exhibiting disjoint failure domains, multiple design points and discontinuous responses. This study addresses this type of challenging problem of reliability assessment of structures with complex limit state functions based on the probability density evolution method (PDEM). Probability density function (PDF) of stochastic structures under static and dynamic loads can be acquired, which is independent of the specific form of limit state functions. Numerical results of several typical examples illustrate that, the time-invariant and instantaneous PDF curves and failure probabilities of stochastic structures with disjoint failure domains, multiple design points and discontinuous responses are calculated effectively and accurately. Moreover, the PDEM is validated to be more efficient than the Monte Carlo simulation and the subset simulation, and is a feasible and general approach to tackle the reliability analysis of complicated problems. In addition, the influence of random design parameters of structures on uncertainty propagation is also scrutinized.     

Neural networks in stochastic mechanics

Summary  A state of art on the application of neural networks in Stochastic Mechanics is presented. The use of these Artificial Intelligence numerical devices is almost exclusively carried out in combination with Monte Carlo simulation for calculating the probability distributions of response variables, specific failure probabilities or statistical quantities. To that purpose the neural networks are trained with a few samples obtained by conventional Monte Carlo techniques and used henceforth to obtain the responses for the rest of samples. The advantage of this approach over standard Monte Carlo techniques lies in the fast computation of the output samples which is characteristic of neural networks in comparison to the lengthy calculation required by finite element solvers. The paper considers this combined method as applied to three categories of stochastic mechanics problems, namely those modelled with random variables, random fields and random processes. While the first class is suitable to the analysis of static problems under the effect of values of loads and resistances independent from time and space, the second is useful for describing the spatial variability of material properties and the third for dynamic loads producing random vibration. The applicability of some classical and special neural network types are discussed from the points of view of the type of input/output mapping, the accuracy and the numerical efficiency.     

基于混合蒙特卡洛算法的容差分配研究

针对航空制造业中,当容差分配问题中含有装配成功率等随机约束时,常用的数值算法往往难以处理。为提高产品制造精度,提出了混合蒙特卡洛(Hybrid Monte Carlo,HMC)算法,即把动态蒙特卡洛(Dynamic Monte Carlo,DMC)算法和静态蒙特卡罗(SMC)算法结合起来,将DMC用于容差分配的优化仿真运算,把SMC用来处理装配成功率约束。通过仿真验证了该方案的可行性,混合蒙特卡洛法既合理地处理了随机约束,证明装配准确度计算和容差分配的一致性。结果说明求解这类问题是最佳算法。     

An efficient and robust integration technique for applied random vibration analysis

The mode-based finite element formulation of the equations of motion is usually adopted for linear random vibration analysis (RVA). In general, the RVA of large systems requires a large number of numerical integrations which is very time-consuming for a reasonable level of desired accuracy. Moreover, conventional numerical integration methods may fail to converge when the integrands are highly oscillatory due to slow propagation velocities. In this paper, a robust general-purpose RVA integration technique which can overcome these drawbacks is presented. Multi-point base and nodal excitations including wave passage effect and frequency-independent spatial correlation can be taken into account in the analysis. The proposed technique is based on the closed-form solutions for polynomial-type power spectral density functions and has been verified to be efficient and accurate for many engineering problems. This paper describes the implementation details, presents two examples taken from engineering applications and demonstrates the dramatic time-saving in the computation compared to numerical integration solutions. Response statistics, such as standard deviation of structural responses, are computed and displayed over the entire structures for these examples.     

A fast approximate method for parameter sensitivity estimation in Monte Carlo structural reliability

In estimating the effect of a change in a random variable parameter on the (time-invariant) probability of structural failure estimated through Monte Carlo methods the usual approach is to carry out a duplicate simulation run for each parameter being varied. The associated computational cost may become prohibitive when many random variables are involved. Herein a procedure is proposed in which the numerical results from a Monte Carlo reliability estimation procedure are converted to a form that will allow the basic ideas of the first order reliability method to be employed. Using these allows sensitivity estimates of low computational cost to be made. Illustrative examples with sensitivities computed both by conventional Monte Carlo and the proposed procedure show good agreement over a range of probability distributions for the input random variables and for various complexities of the limit state function.     

Optimal digital simulations for random linear systems with integration constraints

A generalized approach involving concepts from optimization theory is developed for realizing optimal digital simulations for linear, time-varying, continuous dynamical systems having random inputs by modifying discrete input signal variances. The minimization of a cost functional based on the state covariance matrices of the continuous system and its discrete model leads to a two-point boundary value problem which can be solved by known numerical techniques. The result is a systematic procedure for determining optimal digital simulations under the constraints that the numerical integration formula and integration step size have been specified in advance. An example is presented to illustrate the procedure, including a verification using Monte Carlo simulation runs.     

Analytical evaluation of modal covariance matrices

Dynamic excitations in the form of stationary random processes with normal distribution are completely defined by their power spectral and cross spectral density functions. The stationary response of a linear structure to such excitations will also consist of random processes with normal distribution. In a modal formulation the statistical quantities of all output processes are obtained from modal covariance matrices. The elements of these matrices represent integrals which are usually evaluated numerically. In lightly damped structures, however, the integrand shows pronounced peaks. Thus small integration steps may be necessary for accurate results. In the applications the spectral density functions are conveniently described by discrete values and piecewise polynomial interpolation. The elements of the modal covariance matrices can then be evaluated analytically. For lightly damped structures this method is much more effective than numerical integration and maintains full accuracy in the modal properties of the structural model. The accuracy and efficiency of the method is illustrated by a numerical example.     

Large-scale reliability-based structural optimization

A robust and efficient methodology is presented for treating large-scale reliability-based structural optimization problems. The optimization is performed with evolution strategies, while the reliability analysis is carried out with the Monte Carlo simulation method incorporating the importance sampling technique to reduce the sample size. Efficient hybrid methods are implemented to solve the reanalysis-type problems that arise in the optimization phase with evolution strategies and in the reliability analysis with Monte Carlo simulations. These hybrid solution methods are based on the preconditioned conjugate gradient algorithm using efficient preconditioning schemes. The numerical tests presented demonstrate the computational advantages of the proposed methods, which become more pronounced for large-scale optimization problems.     

Discrete analysis method for random vibration of structures subjected to spatially correlated filtered white noises

Random vibration analysis of large-span space structures or high-rise structures which are subjected to spatially correlated filtered white noise excitations such as wind load and earthquake motion, has been a difficult problem in engineering computation. Based on the idea of the discrete analysis method of random vibration, this paper attempts to solve this problem. The formulae of calculating structural mean and mean square responses are given. As an example, the wind-induced vibration of a cable roof structure is analysed by using these formulae.     

基于混合法的监控系统可靠性分析

针对复杂监控系统规模庞大及关键设备为双机冗余结构的特点,提出以动态故障树(DFT)为基础并结合蒙特卡罗方法对监控系统进行可靠性分析的混合方法。利用DFT建立系统可靠性模型,通过蒙特卡罗仿真算法对模型进行仿真计算,得到系统的可靠性指标。通过对地铁车站级监控系统的可靠性分析,证明了该模型的可行性和算法的有效性。     

Realistic and efficient reliability estimation for aerospace structures

In this paper an efficient procedure for calculating non-exceedance probabilities of the structural response is presented, with emphasis on structures modeled by large finite element systems with many uncertain parameters. This is a problem which receives considerable attention in numerous applications of engineering mechanics, such as space and aerospace engineering. For this purpose, a novel sampling procedure is introduced, which allows a significant reduction of the variance of the estimator of the probability of failure when compared to that of direct Monte Carlo simulation. This improvement in the computational efficiency is most important, as the computational efforts are much higher when uncertainties are considered.The only prerequisite for the application of this sampling procedure is an estimate of the gradient of the performance function of the structure. The calculation of the gradient is carried out efficiently, by exploiting the correlation between a randomly chosen input and the corresponding output of the system. The proposed concept is especially suited for high-dimensional problems in reliability engineering, e.g. for a rather large number n of random variables, say n > 100.To demonstrate the practical value of the methodology a reliability analysis of the INTEGRAL-satellite of the European Space Agency (ESA) has been performed. The results show that both for the frequency response analysis and the structural reliability analysis a substantial number of parameters of the finite element model play an important role.     

基于RMC的蒙特卡罗程序性能优化

蒙特卡罗MC方法是核反应堆设计和分析中重要的粒子输运模拟方法.MC方法能够模拟复杂几何形状且计算结果精度高,缺点是需要耗费大量时间进行上亿规模粒子模拟.如何提高蒙特卡罗程序的性能成为大规模蒙特卡罗数值模拟的挑战.基于堆用蒙特卡罗分析程序RM C,先后开展了基于TCMalloc动态内存分配优化、OpenMP线程调度策略优...     

Testing random number generators for microcomputer applications of Monte Carlo simulation

Monte Carlo simulation is a venerable method of solving probabilistic modeling problems, but has historically been thought of as a mainframe technique because of its computational burden. With the great increases in the speed and power of microcomputers, there is increasing interest in microcomputer-based Monte Carlo techniques applied to probabilistic environmental modeling. Effective Monte Carlo simulation requires fast, accurate random number generators and generator performance is often software and hardware specific. This paper describes a package of diagnostic programs titled RANDALIZE that has been developed to interrogate proposed random number generators to determine if they yield sufficiently random variates with the desired probabilistic properties. The methods used to interrogate a random number sequence are defined and illustrated. Methods for interpreting the results are discussed. Example results are presented to illustrate program functions.     

Robust optimization of structural dynamic characteristics based on adaptive Kriging model and CNSGA

Dynamic characteristics greatly influence the comprehensive performance of a structure. But they are rarely included as objectives in traditional robust optimization of structures. In this study, a robust optimization model including both means and standard deviations of dynamic characteristic indices in the objective and constraint functions is constructed for improving the structural dynamic characteristics and reducing their fluctuations under uncertainty. Adaptive Kriging models are employed for the efficient computation of dynamic characteristics. An intelligent resampling technology is proposed to save computational costs and accelerate convergence of Kriging models, which takes full advantage of the test points for precision verification, the sample points within the local region of the biggest relative maximum absolute error and the near-optimal point to improve the global and local precision of Krigings. The high efficiency of proposed intelligent resampling technology is demonstrated by a numerical example. Finally, an efficient algorithm integrating adaptive Kriging models, Monte Carlo (MC) method, constrained non-dominated sorting genetic algorithm (CNSGA) is proposed to solve the robust optimization model of structural dynamic characteristics. Kriging models are interfaced with MC method to efficiently compute the fitness of individuals during CNSGA. The implementation of proposed methodology is explained in detail and highlighted by the robust optimization of a cone ring fixture with lots of circumferentially distributed holes in a large turbo generator aimed at moving its natural frequencies away from the exciting one. The comparison of the optimized design with the initial one demonstrates that the proposed methodology is feasible and applicable in engineering practice.     

基于Monte Carlo method的均衡度确定模型

蒙特卡罗方法(Monte Carlo method),也称统计模拟方法,是一种以概率统计理论为指导的一类非常重要的数值计算方法,是指使用随机数(或更常见的伪随机数)来解决很多计算问题的方法,本文尝试建立警察服务平台的均衡度模型并用蒙特卡罗方法求解,实验结果可以满足一般的应用需求。     

Robust topology optimization accounting for misplacement of material

The use of topology optimization for structural design often leads to slender structures. Slender structures are sensitive to geometric imperfections such as the misplacement or misalignment of material. The present paper therefore proposes a robust approach to topology optimization taking into account this type of geometric imperfections. A density filter based approach is followed, and translations of material are obtained by adding a small perturbation to the center of the filter kernel. The spatial variation of the geometric imperfections is modeled by means of a vector valued random field. The random field is conditioned in order to incorporate supports in the design where no misplacement of material occurs. In the robust optimization problem, the objective function is defined as a weighted sum of the mean value and the standard deviation of the performance of the structure under uncertainty. A sampling method is used to estimate these statistics during the optimization process. The proposed method is successfully applied to three example problems: the minimum compliance design of a slender column-like structure and a cantilever beam and a compliant mechanism design. An extensive Monte Carlo simulation is used to show that the obtained topologies are more robust with respect to geometric imperfections.