Stochastic collocation method for secondary bifurcation of a nonlinear aeroelastic system

A stochastic collocation method is proposed to investigate the secondary bifurcation of a two-dimensional aeroelastic system with structural nonlinearity represented by cubic restoring forces, and uncertainties expressed by random parameters in the cubic stiffness coefficient and in the initial pitch angle. The accuracy of the stochastic collocation method is improved by incorporating higher order schemes, such as piecewise cubic interpolation and piecewise cubic spline interpolation, instead of a piecewise linear interpolation formula. For an aeroelastic problem with the uncertainty expressed by a time dependent combination of five random variables, an efficient collocation method is developed using a sparse grid approach with a dimension adaptive strategy. Numerical simulations are carried out to demonstrate the effectiveness of the proposed method for long term computation and discontinuous problems.     

A dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos – capturing all scales of random modes on independent grids

In stochastic computations, or uncertainty quantification methods, the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion. The size of this system increases drastically when the number of independent random variables and/or order of polynomial chaos expansions increases. This is invariably the case for large scale simulations and/or problems involving steep gradients and other multiscale features; such features are variously reflected on each solution component or random/uncertainty mode requiring the development of adaptive methods for their accurate resolution. In this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space-refinement on physical space that allows all scales of each solution component to be refined independently of the rest. We exemplify this using the convection–diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method. Thus we establish a new, elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions.     

An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations

A computational methodology is developed to address the solution of high-dimensional stochastic problems. It utilizes high-dimensional model representation (HDMR) technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower-order to higher-order component functions. HDMR is efficient at capturing the high-dimensional input–output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower-order terms. An adaptive version of HDMR is also developed to automatically detect the important dimensions and construct higher-order terms using only the important dimensions. The newly developed adaptive sparse grid collocation (ASGC) method is incorporated into HDMR to solve the resulting sub-problems. By integrating HDMR and ASGC, it is computationally possible to construct a low-dimensional stochastic reduced-order model of the high-dimensional stochastic problem and easily perform various statistic analysis on the output. Several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method. The cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large-input variability. The efficiency of the proposed method is examined by comparing with Monte Carlo (MC) simulation.     

A stochastic mixed finite element heterogeneous multiscale method for flow in porous media

A computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales. In order to capture the small scale heterogeneity, a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method (HMM) in the spatial domain. This new method ensures both local and global mass conservation. Starting from a specified covariance function, the stochastic log-permeability is discretized in the stochastic space using a truncated Karhunen–Loève expansion with several random variables. Due to the small correlation length of the covariance function, this often results in a high stochastic dimensionality. Therefore, a newly developed adaptive high dimensional stochastic model representation technique (HDMR) is used in the stochastic space. This results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method (ASGC). Numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method.     

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method.     

Uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids

Quantitative predictions of the behavior of many deterministic systems are uncertain due to ubiquitous heterogeneity and insufficient characterization by data. We present a computational approach to quantify predictive uncertainty in complex phenomena, which is modeled by (partial) differential equations with uncertain parameters exhibiting multi-scale variability. The approach is motivated by flow in random composites whose internal architecture (spatial arrangement of constitutive materials) and spatial variability of properties of each material are both uncertain. The proposed two-scale framework combines a random domain decomposition (RDD) and a probabilistic collocation method (PCM) on sparse grids to quantify these two sources of uncertainty, respectively. The use of sparse grid points significantly reduces the overall computational cost, especially for random processes with small correlation lengths. A series of one-, two-, and three-dimensional computational examples demonstrate that the combined RDD–PCM approach yields efficient, robust and non-intrusive approximations for the statistics of diffusion in random composites.     

Bulk-SQUID effect in a discrete superconductor as a consequence of generation frequency locking in constituent josephson junctions

The theory of the bulk-SQUID effect in discrete superconductors is constructed for the first time. It is shown that the bulk-SQUID effect emerges in the system of 2D intrinsically stochastic multijunction SQUID (i.e., with random values of critical currents of the junctions, injection currents, and the coupling coefficients between the junction) due to generation frequency locking in all junctions. It is demonstrated that the bulk-SQUID effect occurs in a wide range of random parameters of the system. This domain of variation of the system parameters can be divided into three subdomains. The first one is the subdomain of coherent dynamics of phases at the junctions, the second is the subdomain of incoherent dynamics, in which the phases of the junctions are not locked, but the bulk-SQUID effect persists, and the third is the subdomain of transient dynamics, in which coherent dynamics and the bulk-SQUID effect are observed in parts. A simple mathematical model of noninteracting junctions, which correctly describes basic features of the dynamics of the initial system and makes it possible to calculate some of its statistical characteristics, is proposed and analyzed.     

一种改进的MEMS陀螺信号去噪方法研究

 从工程实用的角度出发,探讨了MEMS陀螺仪随机漂移误差的有效补偿方法。根据小波阈值去噪原理,结合多项式函数插值法提出了一种MEMS陀螺仪输出信号的有效去噪补偿方法,克服了传统软、硬阈值去噪方法的缺陷,通过对MEMS陀螺数据分析研究,验证了该方法对于MEMS陀螺输出信号滤波消噪的优越性。     

Stochastic model for dielectric breakdown

We discuss a model for the development of discharge patterns in dielectric breakdown based on the Laplace equation associated with a probability field. The model gives rise to random fractals with well-defined Hausdorff dimensions. The relations of this model with the diffusion-limited aggregation are discussed in detail. The possibility of application to other stochastic phenomena like fracture propagation is proposed.     

水声时变线谱建模与双稳态随机共振分析

建立了水声时变线谱的模型,将双稳态随机共振系统应用于水声时变线谱信号的检测,考查了随机共振系统对水声环境的适应能力,提出了设计水声线谱检测系统可以利用的外在参数。在信噪比变化、声压起伏、线谱漂移等情况下,仿真结果表明双稳态随机共振系统都能较好地工作。为水声领域新型线谱检测系统的设计提供了依据。     

Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

This paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws. We consider problems with parametric uncertainty in initial conditions and model coefficients, whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables. The stochastic spectral method relies on multi-resolution schemes where the stochastic domain is discretized using tensor-product stochastic elements supporting local polynomial bases. A Galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution. Hyperbolicity of the resulting Galerkin system is analyzed. A finite volume scheme with a Roe-type solver is used for discretization of the spatial and time variables. An original technique is introduced for the fast evaluation of approximate upwind matrices, which is particularly well adapted to local polynomial bases. Efficiency and robustness of the overall method are assessed on the Burgers and Euler equations with shocks.     

Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty

We estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations, within the context of dynamically orthogonal (DO) field equations. Transient nonlinear dynamics, irregular data and non-stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation. To efficiently quantify uncertainties in such systems, it is essential to vary the dimensionality of the stochastic subspace with time. An objective here is to provide criteria to do so, working directly with the original equations of the dynamical system under study and its DO representation. We first analyze the scaling of the computational cost of these DO equations with the stochastic dimensionality and show that unlike many other stochastic methods the DO equations do not suffer from the curse of dimensionality. Subsequently, we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous (i) stability arguments and (ii) Bayesian data updates. We then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2D stochastic fluid flows, specifically a double-gyre wind-driven circulation and a lid-driven cavity flow in a basin. In these two applications, we focus on the growth of uncertainty due to internal instabilities in deterministic flows. We consider a range of flow conditions described by varied Reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions.     

Noise-induced chaos in the electrostatically actuated MEMS resonators

In this Letter, nonlinear dynamic and chaotic behaviors of electrostatically actuated MEMS resonators subjected to random disturbance are investigated analytically and numerically. A reduced-order model, which includes nonlinear geometric and electrostatic effects as well as random disturbance, for the resonator is developed. The necessary conditions for the rising of chaos in the stochastic system are obtained using random Melnikov approach. The results indicate that very rich random quasi-periodic and chaotic motions occur in the resonator system. The threshold of bounded noise amplitude for the onset of chaos in the resonator system increases as the noise intensity increases.     

静电排斥型微机电系统变形镜驱动器

为了克服常规静电微机械驱动器所存在的静电吸合现象,基于非均匀分布的静电场可以产生排斥力的原理,设计了一种新型的微机电系统变形镜驱动器。驱动器由5组电极构成,最大的一组由中心质量块和位于正下方的下电极组成,其它4组分别位于各条边上。中心质量块由4根L形弹簧支撑,每根弹簧分别固定于驱动器的4个锚点。利用有限元软件对驱动器的频率响应和暂态响应特性进行了仿真,结果表明,谐振频率高达4kHz,暂态响应时间小于0.05s。利用表面硅工艺加工出了驱动器,并利用白光扫描干涉仪对驱动器的静态位移电压特性进行了测试,测得驱动器在70V激励电压下的形变量为1.4μm。     

Passage Time Statistics in a Stochastic Verhulst Model

Using the stochastic paths perturbation approach analytic individual realizations of a stochastic Verhulst model are introduced. The escape of the unstable state is studied for any kind of noise from these individual realizations. We infer from these paths the statistics of the first passage time distribution invoking the solution of an explicit equation with a random coefficient. A stochastic population Verhulst’s dynamics with small perturbations of the Wiener class is explicitly worked out. The method can also be implemented for other type of stochastic perturbations like Poisson-noise (shot white pulses), etc.     

On causal stochastic equations for log-stable multiplicative cascades

We reformulate various versions of infinitely divisible cascades proposed in the literature using stochastic equations. This approach sheds a new light on the differences and common points of several formulations that have been recently provided by several teams. In particular, we focus on the simplification occurring when the infinitely divisible noise at the heart of such model is stable: an independently scattered random measure becomes a stable stochastic integral. In the last section we discuss the D-dimensional generalization.     

基于Allan方差的MEMS陀螺随机误差建模

针对由于MEMS陀螺随机误差较大而影响MEMS惯性测量系统测量精度的问题,提出一种利用Allan方差分析随机误差并建模的方法。在分析Allan方差原理的基础上,通过Allan方差分析法分离和辨识了MEMS陀螺仪的各项随机误差以及误差系数,并利用随机误差系数进行了数学建模。通过与ARMA模型比较,表明利用Allan方差建立的模型更加精确。该方法为MEMS惯性导航系统中姿态测量的误差补偿和滤波提供了新的思路,对提高MEMS惯性测量系统的测量精度具有一定的实际应用价值。     

STOCHASTIC DYNAMIC SENSITIVITY OF UNCERTAIN STRUCTURES SUBJECTED TO RANDOM EARTHQUAKE LOADING

The present study involves computation of stochastic sensitivity of structures with uncertain structural parameters subjected to random earthquake loading. The formulations are provided in frequency domain. A strong earthquake-induced ground motion is considered as a random process defined by respective power spectral density function. The uncertain structural parameters are modelled as homogeneous Gaussian stochastic field and discretized by the local averaging method. The discretized stochastic field is simulated by the Cholesky decomposition of respective co-variance matrix. By expanding the dynamic stiffness matrix about its reference value, the advantage of Neumann Expansion technique is explored within the framework of Monte Carlo simulation, to compute responses as well as sensitivity of response quantities. This approach involves only a single decomposition of the dynamic stiffness matrix for the entire simulated structure and the facility that several stochastic fields can be tackled simultaneously are basic advantages of the Neumann Expansion. The proposed algorithm is explained by an example problem.     

First passage time of nonlinear ship rolling in narrow band non-stationary random seas

The generalized extended stochastic central difference (GESCD) method is applied to study the response statistics and first passage time of nonlinear ship rolling in narrow band stationary and non-stationary random seas. The GESCD method is based on a combination of the extended stochastic central difference method with a statistical linearization technique, modified adaptive time scheme, and time coordinate transformation. The extended stochastic central difference method is, however, an extension of the stochastic central difference method for the determination of the recursive mean square or covariance of responses of systems under narrow band stationary and non-stationary random disturbances. Approximate first passage probabilities of nonlinear systems based on the modified mean rate of various crossings proposed earlier by the first author were determined. It is concluded that the GESCD method is very accurate, simple and efficient to apply compared with Monte Carlo simulation. The proposed method is applicable to cases with large nonlinearities and intensive random excitations. The approximate first passage probabilities of the nonlinear system determined by the proposed approach are very accurate as they are in excellent agreement with those evaluated by the Monte Carlo simulation. It is believed that the model considered in this paper is a closer representation to reality than those reported earlier in the literature.     

随机过程动态自适应小波独立网格多尺度模拟

在随机过程数值仿真中,由多项式混沌展开谱方法得到求解展开系数的确定性偶合方程组。该方程组比相应的确定性仿真时增大许多。并且当多项式展开阶数和随机空间维数提高时,方程维数急剧增加。由于待求未知分量为表征不同尺度波动的混沌展开模,形成节点意义下的的多尺度问题,传统的网格细分自适应逼近不再适用。为此我们采用了小波的多尺度离散,并建立基于空间细化的动态自适应系统,让每个求解点上的多个未知分量有各自独立的小波网格。本文以随机对流扩散方程为例,进行了二个算例的数值实验,论证了此方法的优点。