Numerical solution of first passage problems using an approximate Chapman-Kolmogorov relation

A new deterministic numerical method for solving first passage time problem is described, analyzed and computationally tested. The method is based on recursively solving an integral equation for the reliability function. The integral equation is derived from the Chapman-Kolmogorov relation and involves an approximation to the Green's function for the forward Kolmogorov equation. An error analysis yields estimates of convergence rates. Numerical experiments indicate that the method is stable and can accurately approximate the reliability function and first passage times.     

First passage failure of quasi integrable-Hamiltonian systems under combined harmonic and white noise excitations

The first passage failure of multi-degree-of-freedom (MDOF) quasi integrable-Hamiltonian systems under combined harmonic and white noise excitations in the case of external resonance is studied. First, a stochastic averaging method for quasi integrable-Hamiltonian systems under combined harmonic and white noise excitations using generalized harmonic functions is reviewed briefly. Then, a backward Kolmogorov equation governing the conditional reliability function and a Pontryagin equation governing the conditional mean of the first passage time are established from the averaged Itô equations, respectively. The conditional reliability function, and the conditional probability density and conditional mean of the first passage time are obtained from solving these equations together with suitable initial condition and boundary conditions. The comparison between the analytical results and those from Monte Carlo simulation for an example shows that the proposed method works very well. It is also shown by using Monte Carlo simulation that the reliability of the system in the case of external resonance is much lower than that in the non-resonant case.     

复合随机振动系统的动力可靠度分析

建议了一类新的复合随机振动系统动力可靠度分析方法.基于复合随机振动系统反应分析的密度演化方法,根据首次超越破坏准则,对密度演化方程施加相应的边界条件,进而求解密度演化方程,在安全域内积分给出结构的动力可靠度.结合精细时程积分方法与具有TVD性质的差分格式,研究了基于密度演化方法求解结构动力可靠度问题的数值方法.以受到随机地震作用、具有随机参数的八层层间剪切型结构为例,进行了结构动力可靠度分析并与随机模拟结果进行了比较.研究表明,建议的方法具有较高的精度和效率.     

First passage failure of MDOF quasi-integrable Hamiltonian systems with fractional derivative damping

The first passage failure of multi-degree-of-freedom (MDOF) quasi-integrable Hamiltonian systems with damping described by a fractional derivative is studied. The stochastic averaging procedure is applied to derive the averaged equations for first integrals. The conditional reliability function and the conditional mean of first passage failure time are obtained by solving the associated backward Kolmogorv equation and Pontryagin equation together with suitable boundary conditions and initial condition, respectively. One example of two coupled nonlinear oscillators with fractional derivative damping is given to illustrate the proposed procedure. The accuracy of the method is substantiated by comparing the analytical results with those from Monte Carlo simulation. Effects of some parameters of fractional order, damping coefficients and nonlinear strength on the system??s reliability are examined.     

随机结构动力可靠度分析的概率密度演化方法

基于随机结构动力反应分析的概率密度演化方法，提出了一类新的随机结构动力可靠度分析方法。在随机结构动力反应概率密度演化方程的基础上，对于首次超越问题，根据所给的首次超越破坏准则施加相应的吸收壁边界条件，求解具有吸收壁边界条件的概率密度演化方程并在安全域内积分．给出结构的动力可靠度。结合精细时程积分方法和具有TVD性质的差分格式，讨论了计算结构动力可靠度的数值方法。以八层框架结构为例进行了动力可靠度分析并与随机模拟分析结果进行了比较。     

A bounded optimal control for maximizing the reliability of randomly excited nonlinear oscillators with fractional derivative damping

In this paper, a bounded optimal control for maximizing the reliability of randomly excited nonlinear oscillators with fractional derivative damping is proposed. First, the partially averaged It? equations for the energy processes of individual degree of freedom are derived by using the stochastic averaging method. Second, the dynamical programming equations for the control problems of maximizing the reliability function and maximizing the mean first passage time are established from the partially averaged It? equations by using the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraints. Third, the conditional reliability function and mean first passage time of the optimally controlled system are obtained by solving the backward Kolmogorov equation and Pontryagin equation associated with the fully averaged It? equation, respectively. The application of the proposed procedure and effectiveness of the control strategy are illustrated by using two examples. Besides, the effect of fractional derivative order on the reliability of the optimally controlled system is examined.     

On the estimation of failure probability having prescribed statistical moments of first passage time

A method of estimating the probability density function and cumulative distribution function when only the ordinary or central moments of the distribution are known is examined. The technique is used in conjunction with previous work which yields the ordinary moments of time to first passage failure to obtain accurate estimates of the failure probability for two representative oscillators. The results are then compared to those obtained by a nearly exact numerical scheme.     

First passage failure of SDOF nonlinear oscillator with lightly fractional derivative damping under real noise excitations

The first passage failure of single-degree-of-freedom (SDOF) nonlinear oscillator with lightly fractional derivative damping under real noise excitations is investigated in this paper. First, the system state is approximately represented by one-dimensional time-homogeneous diffusive Markov process of amplitude through stochastic averaging. Then, the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of first passage time are established from the averaged Itô equation for Hamiltonian. The conditional reliability function, the conditional probability density and mean of the first passage time are obtained by solving these equations together with suitable initial condition and boundary conditions. Finally, two examples are worked out in detail and the analytical solutions are checked by those from the Monte Carlo simulation of original systems.     

Numerical solution of the forward one-dimensional problem of critical flow in nozzles

The principles of an algorithm are formulated for a numerical solution of problems of one-dimensional flow in nozzles with passage through a singularity. The results of calculations are compared with experimental data.     

响应谱MC模拟的桥梁抖振首次超越研究

在桥梁耦合抖振谱分析的基础上,采用谐波合成技术,依据Monte Carlo思想大量模拟位移谱时程,由此便可以方便、快捷地计算桥梁抖振响应首次超越失效概率,克服传统Monte Carlo方法时域内再现桥梁抖振首次超越小失效概率事件效率偏低的问题。采用江阴长江大桥和上海杨浦大桥作为算例,比较了抖振首次超越响应谱Monte Carlo模拟结果与目前广泛使用的基于Poisson、Poisson包络、Markov过程三种假定的近似解析算法结果差异,表明由于考虑大气紊流环境下桥梁抖振背景分量,在较高阈值情况下,近似解析算法不再适用,而本文方法具有更高的精度。     

Control of population and atomic coherence by adiabatic rapid passage and optimization of coherent anti-Stokes Raman scattering signal by maximal coherence

Abstract

Robust control of atomic coherence and population transfer among Zeeman sublevels in the ground states of rubidium atom is investigated using adiabatic rapid passage in a nanosecond time scale, which is smaller than the lifetime of first excited Rb. It is shown that a slight change in the pump pulse time delay relative to the Stokes pulse leads to a significant modification of atomic coherence and population transfer, consequently having remarkable impacts on the generation of coherent anti-Stokes Raman scattering (CARS) signal and probe pulse absorption. This coherent control of quantum state and population is presented by numerical simulations based on self-consistent set of density matrix equations and Maxwell equations as well as experimental demonstration in rubidium atom with different atomic densities. Experimental observations are in good agreement with numerical calculations.     

三塔两跨悬索桥行波效应振动台试验及数值研究

为了研究行波效应对大跨多塔连跨悬索结构抗震性能的影响,以泰州长江公路大桥为背景,设计并制作1/40缩尺比例模型,进行了全桥振动台模型试验。采用中塔与主梁间弹性索连接的纵向约束结构体系,试验分别测试了不同视波速下行波效应对全桥位移响应的影响。振动台试验表明,考虑行波效应时,中塔顶位移、北塔梁纵向相对位移会有明显增大,最大增幅在50%以上;主桥梁与引桥梁纵向相对位移也会有明显增大,增大幅值在1倍以上。因此,仅考虑一致地震激励不能保证大跨度多塔悬索桥的结构安全。通过比较试验结果和数值计算结果可以发现,试验结果与有限元数值模拟结果较为接近、吻合较好。数值计算所采用的绝对位移法分析行波效应方法操作简单,力学概念清晰,可方便的应用到大跨度桥梁行波效应分析中。     

On reliability of systems with moving material subjected to fracture and instability

The reliability of systems with moving cracked elastic and isotropic material is considered. The material is modeled as a moving plate which continually has a crack on the edge. The plate is subjected to homogeneous tension acting in the traveling direction and the tension varies temporally around a constant value, the set tension. The tension and the length of the crack are modeled by an Ornstein–Uhlenbeck process and an exponential Ornstein–Uhlenbeck process, respectively. Failure is regarded as the state at which the plate becomes unstable or fractures (or both) and a lower bound for the reliability of the system is derived. Considering reliability of the system leads to first passage time problems and, in solving them, a known explicit result for the first passage time of an Ornstein–Uhlenbeck process to a constant boundary is exploited. A change in the set tension has opposite effects on the probabilities of instability and fracture, and a safe range of set tension is studied. Numerical examples are computed for material and machine parameters typical of paper and printing presses. The results suggest that tension variations may significantly affect the pressroom runnability.     

Reliability Bounds based on Universal Generating Function and Discrete Stress-strength Interference Model

A method for estimating the component reliability is proposed when the probability density functions of stress and strength can not be exactly determined. For two groups of finite experimental data about the stress and strength,an interval statistics method is introduced. The processed results are formulated as two interval-valued random variables and are graphically represented by using two histograms. The lower and upper bounds of component reliability are proposed based on the universal generating function method and are calculated by solving two discrete stress-strength interference models. The graphical calculations of the proposed reliability bounds are presented through a numerical example and the confidence of the proposed reliability bounds is discussed to demonstrate the validity of the proposed method. It is showed that the proposed reliability bounds can undoubtedly bracket the real reliability value. The proposed method extends the exciting universal generating function method and can give an interval estimation of component reliability in the case of lake of sufficient experimental data. An application example is given to illustrate the proposed method.     

On suitable formulations of the method of modal analysis for numerical calculations

The method of modal analysis is a method widely used in structural analysis of linear systems. After a short introduction two step by step formulations for modal calculations are discussed. One of them is of more theoretical interest, the other is recommended for practical calculations for multi-degree-of-freedom systems because it is efficient in computing time. The first is formulated for the whole system the latter for each mode. Likewise a combination of the method of modal analysis and the LAPLACE transformation is established. The numerical calculation of the LAPLACE transformation is done by using the algorithm of the Fast FOURIER transformation. The advantages of the different formulations for numerical calculations are discussed.     

Excilamps: Promising sources for exciting spontaneous fluorescence of organic molecules

We present the results of real and numerical experiments devoted to investigation of the spatial gradient of the intensity of chlorophyll fluorescence in the volume of a plant leaf. It is demonstrated for the first time that potential effective sources capable of exciting spontaneous fluorescence from the photosystem of a leaf are the exciplex lamps (exilamps) that generate continuous UV radiation. Estimations using the Monte Carlo method show that the main factor responsible for a transformation of the fluorescence spectrum during the passage of radiation through the volume of the leaf is the reabsorption process.     

On the application of the “Russian Roulette and Splitting” simulation technique for the reliability assessment of mechanical systems

A selective Monte Carlo simulation (MCS) technique, namely, the “Russian Roulette and Splitting” technique, is applied for the assessment of the reliability of structures under stochastic excitations. The basic features of the algorithm are described and discussed. As a numerical example the technique is used for the numerical solution of the first passage problem. The results are compared with available solutions obtained by another variance reduction technique—the “Double & Clump” procedure. Translated from Problemy Prochnosti, No. 3, pp. 131–138, May–June, 1997.     

Boundary Conditions Generated by Dynamic Particles in SPH Methods

Smoothed Particle Hydrodynamics is a purely Lagrangian method that can be applied to a wide variety of fields. The foundation and properties of the so called dynamic boundary particles (DBPs) are described in this paper. These boundary particles share the same equations of continuity and state as the moving particles placed inside the domain, although their positions and velocities remain unaltered in time or are externally prescribed. Theoretical and numerical calculations were carried out to study the collision between a moving particle and a boundary particle. The boundaries were observed to behave in an elastic manner in absence of viscosity. They allow the fluid particles to approach till a critical distance depending on the energy of the incident particle. In addition, a dam break confined in a box was used to check the validity of the approach. The good agreement between experiments and numerical results shows the reliability of DBPs.     

Efficient solution of the first passage problem by Path Integration for normal and Poissonian white noise

In this paper the first passage problem is examined for linear and nonlinear systems driven by Poissonian and normal white noise input. The problem is handled step-by-step accounting for the Markov properties of the response process and then by Chapman–Kolmogorov equation. The final formulation consists just of a sequence of matrix–vector multiplications giving the reliability density function at any time instant. Comparison with Monte Carlo simulation reveals the excellent accuracy of the proposed method.     

Augmenting approximate solutions for consistent reliability analysis

Approximate solutions provide a great source of insight for understanding system behavior. Due to their approximate nature, however, their use in reliability analysis does not lead to ‘consistent’ reliability estimates, in the sense that the resulting estimates do not converge to the reliability based on the original target response as the computational effort spent in the reliability procedure increases. This paper intends to develop a method that incorporates the information from approximate solutions to yield efficient and consistent reliability estimates. The governing formula is developed using the Theorem of Total Probability to relate information provided by approximate solutions to the target reliability. The method is applied to studying first passage reliability of structures subjected to stochastic loadings.