Stochastic modal interaction in linear and nonlinear aeroelastic structures

The linear and autoparametric modal interactions in a three defree-of-freedom structure under wide band random excitation are examined. For a structure with constant parameters the linear response is obtained in a closed form. When the structure stiffness matrix involves random fluctuations, the governing equations of motion, in terms of the normal coordinates, are found to be coupled through parametric terms. The structural response is mainly governed by the condition of mean square stability. The boundary of stable-unstable responses is obtained as a function of the internal detuning parameter. The results of the linear system with constant parameters are used as a reference to measure the deviation of the system response when the nonlinear inertia coupling is included. In the neighbourhood of combination internal resonance the system random response is determined by using the Fokker Planck equation approach together with the Gaussian closure scheme. This approach results in 27 coupled first order differential equations in the first and second response moments. These equations are solved numerically. The response is found to deviate significantly from the linear solution when the system internal detuning is close to the exact internal resonance. The autoparametric interaction is found to depend significantly on the system damping ratios and a nonlinear coupling parameter. In the vicinity of combination internal resonance, the second normal mode mean square exhibits an increase associated with a corresponding decrease in the first and third normal modes.     

联肢剪力墙的刚度、稳定性以及二阶效应

该文采用连续化模型,对双肢剪力墙结构平面内稳定性进行了研究,求得了顶部作用集中压力时临界荷载的精确显式表达式和显式屈曲波形。这个临界荷载公式表明,联肢剪力墙是一种双重抗侧力结构,并且可以采用串并联电路模型来表示两者之间的相互作用。串并联模型推广到线性分析的情况,得到顶部抗侧刚度的显式表达式,与精确解进行了比较。推导了顶部作用竖向集中荷载时,在不同水平荷载作用下结构的侧移、墙肢弯矩、墙肢轴力和连梁弯矩放大系数,并提供了近似计算公式。     

Small noise expansion of moment lyapunov exponents for two-dimensional systems

We construct an approximation for the moment Lyapunov exponent, the asymptotic growth rate of the moments of the response of a two-dimensional linear system driven by real or white noise. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noise. As an example, we study the moment stability of the stationary solution of nonlinear structural and mechanical systems subjected to real noise excitation. The usefulness of the moment Lyapunov exponent in predicting parameter values at which qualitative changes in the probability density function occur (stochastic bifurcation) is also illustrated.     

Sensitivity of expected exceedance rate of SDOF-system response to statistical uncertainties of loading and system parameters

The effects of the statistical uncertainties of the loading and system parameters on the mean exceedance rate of a particular threshold are investigated for a linear SDOF-system with viscous damping. For this purpose the structural loading is described by the well-known stationary Kanai-Tajimi-earthquake model. The analysis is simplified by utilizing an approximate solution for the threshold-crossing rate, for which the error with respect to the exact solution is shown to be small. Each of the parameters involved in the expression for the mean exceedance rate of the stationary response of the structure is considered a random variable. The respective effects of the statistical uncertainties of the parameters on the threshold-crossing rate, as expressed by the first- and second moments, are shown explicitely in the numerical examples.     

Application of the generalized perturbation-based stochastic boundary element method to the elastostatics

This paper is entirely devoted to the demonstration of a solution for some boundary value problems of isotropic linear elastostatics with random parameters using the boundary element method. The stochastic perturbation technique in its general nth-order Taylor series expansion version is used to express all the random parameters and the state functions of the problem. These expansions inserted in the classical deterministic equilibrium statement return up to the nth-order (both PDEs and matrix) equations. Contrary to the previous implementations of the stochastic perturbation technique, any order partial derivatives with respect to the random input are derived from the deterministic structural response function (SRF) at a given point. This function is approximated using polynomials by the least-squares method from the multiple solution of the initial deterministic problem solved for the expectations of random structural parameters. First two probabilistic moments have been computed symbolically here using the computational MAPLE environment, also as the polynomial expressions including perturbation parameter ε. It should be mentioned that such a generalized perturbation approach makes it possible to analyze all types of random variables (not only Gaussian) and to compute even higher probabilistic moments with a priori given accuracy. The entire methodology can be implemented after minor modifications to analyze nonlinear phenomena for both statics and dynamics of even heterogeneous domains.     

Probabilistic analysis of truss structures with uncertain parameters (virtual distortion method approach)

A new approach for probabilistic characterization of linear elastic redundant trusses with uncertainty on the various members subjected to deterministic loads acting on the nodes of the structure is presented. The method is based on the simple observation that variations of structural parameters are equivalent to superimposed strains on a reference structure depending on the axial forces on the elastic modulus of the original structure as well as on the uncertainty (virtual distortion method approach). Superposition principle may be applied to separate contribution to mechanical response due to external loads and parameter variations. Statically determinate trusses dealt with the proposed method yields explicit analytic solution in terms of displacements while redundant trusses have been studied by means of an asymptotic expansion exhibiting explicit dependence on parameter fluctuations. Probabilistic characterization of the response may then be obtained both for statically determinate and statically indeterminate stochastic trusses.     

Non-stationary statistical solutions of a class of random diffusion equations: Analytical and numerical considerations

Non-stationary, behaviour of statistical moments up to the second order, of solutions of linear one-dimensional diffusion equations having random initial conditions and random external excitations the latter of which is represented by non-stationary Gaussian white noises, is considered. Three approaches are presented: the first is concerned with the analytical solution procedure based on separation of variables together with the superposition principle; the second deals with a semi-analytical approach by the use of finite element and finite difference approximations in space; and the third is related to numerical analysis using the simplest explicit and implicit finite differences. Comparison is made for the results obtained by these three solution procedures. Convergence behaviour of the analytical solutions is investigated, and the consideration of stability of the finite difference solutions is also given.     

有阻尼体系动力分析的一种显式差分法

本文利用拉格朗日型的二次插值函数近似位移反应，建立了速度、加速度的差商近似公式，进而推得集中质量阻尼体系动力分析的一种显式差分法。该显式差分格式具有二阶计算精度，与目前常用的有阻尼体系动力求解的几种格式相比，具有明显减少计算工作量的优点。     

Mechanistic procedure for parameter determination of multiplicative decomposition based constitutive models

By exploiting the Clausius–Planck local energy dissipation inequality, a large strain, three-dimensional constitutive model has been developed for the monotonic and cyclic response prediction of various asphaltic materials. The model consists of a Zener non-linear, visco-elastic component acting in series with a stress dependent viscous component. A novel computational scheme has been developed for solution of the coupled system of equations expressing the interdependent response of these two in series components. An explicit, mechanistic, parameter determination procedure is presented for the laboratory determination of all necessary model parameters. Examples of model parameter determination and utilisation for prediction of the response of a recycled asphalt mix and a stone mastic asphalt mix are presented.     

Structural optimization of non-linear systems under stochastic excitation

This work concentrates on the structural optimization of a class of non-linear systems with deterministic structural parameters subject to stochastic excitation. The optimization problem is formulated as the minimization of an objective function subject to constraints on the response level. The stochastic response is characterized by its first two statistical moments, which are computed by a statistical equivalent linearization technique. The implicit structural optimization problem is replaced by a sequence of explicit sub-optimization problems. The sub-problems are constructed by using a conservative first-order approximation of the objective and constraint functions. The applicability of the proposed design process is demonstrated in three numerical examples where the methodology is applied to systems with nonlinearity of hardening and hysteretic type. The effects of the nonlinearity on the general performance of the final designs are discussed. At the same time, some engineering implications of the results obtained in this work are addressed.     

Least squares stochastic Boundary Element Method

The main issue in this paper is mathematical formulation and computational implementation of the stochastic Boundary Element Method based on the generalized stochastic perturbation technique. The key feature is a replacement of the given order polynomial response function with the least squares method leading to a numerical determination of this response function. This new approach minimizes the approximation error during the recovery of the structural response indexed with the random input parameter, which is a decisive factor for the entire stochastic method accuracy; contrary to some lower order techniques, numerical implementation of up to the fourth order probabilistic moments is displayed. Computational experiments obey both analyses for the homogeneous and heterogeneous structures with Gaussian random material parameters and also some comparison against the Monte-Carlo simulation and analytical results.     

Global format for energy–momentum based time integration in nonlinear dynamics

A global format is developed for momentum and energy consistent time integration of second‐order dynamic systems with general nonlinear stiffness. The algorithm is formulated by integrating the state‐space equations of motion over the time increment. The internal force is first represented in fourth‐order form consisting of the end‐point mean value plus a term containing the stiffness matrix increment. This form gives energy conservation for systems with internal energy as a quartic function of the displacement components. This representation is then extended to general energy conservation via a discrete gradient representation. The present procedure works directly with the internal force and the stiffness matrix at the time integration interval end‐points, and in contrast to previous energy‐conserving algorithms, it does not require any special form of the energy function nor use of mean value products at the element level or explicit use of a geometric stiffness matrix. An optional monotonic algorithmic damping, increasing with response frequency, is developed in terms of a single damping parameter. In the solution procedure, the velocity is eliminated and the nonlinear iterations are based on the displacement components alone. The procedure represents an energy consistent alternative to available collocation methods, with an equally simple implementation. Copyright © 2014 John Wiley & Sons, Ltd.     

A new method for selection of population distribution and parameter estimation

An optimization method for parameter estimation is presented with the Kolmogorov-Smirnov distance used as the objective. A step-by-step implementation procedure is given. The method is demonstrated by estimating the parameters for three-parameter Weibull distributions from three different samples (with different sample sizes). A comparison of the proposed method and the usual methods such as the least-squares method, the matching moments method and the maximum likelihood method shows that more reasonable estimates of the parameters are given by the proposed optimization method. Then, the proposed method is successfully extended to estimate the parameters for the sum of two three-parameter Weibull distributions. Based on these findings, a new procedure for selection of population distribution and parameter estimation is presented.     

Second order sensitivity analysis of lumped mechanical systems in the frequency domain

This paper presents a method for determining the second order sensitivity matrix and logarithmic sensitivity functions in the frequency domain. The procedure described is applicable to any discrete and viscously damped mechanical system. A definition of the second order logarithmic sensitivity function has been proposed. The examples shown indicate that logarithmic sensitivity functions of the second order preserve the qualitative character of first order sensitivity functions and are more sensitive to changes of system parameters. The influence of large parameter changes on second order sensitivity functions has also been investigated.     

Robust and provably second‐order explicit–explicit and implicit–explicit staggered time‐integrators for highly non‐linear compressible fluid–structure interaction problems

An explicit–explicit staggered time‐integration algorithm and an implicit–explicit counterpart are presented for the solution of non‐linear transient fluid–structure interaction problems in the Arbitrary Lagrangian–Eulerian (ALE) setting. In the explicit–explicit case where the usually desirable simultaneous updating of the fluid and structural states is both natural and trivial, staggering is shown to improve numerical stability. Using rigorous ALE extensions of the two‐stage explicit Runge–Kutta and three‐point backward difference methods for the fluid, and in both cases the explicit central difference scheme for the structure, second‐order time‐accuracy is achieved for the coupled explicit–explicit and implicit–explicit fluid–structure time‐integration methods, respectively, via suitable predictors and careful stagings of the computational steps. The robustness of both methods and their proven second‐order time‐accuracy are verified for sample application problems. Their potential for the solution of highly non‐linear fluid–structure interaction problems is demonstrated and validated with the simulation of the dynamic collapse of a cylindrical shell submerged in water. The obtained numerical results demonstrate that, even for fluid–structure applications with strong added mass effects, a carefully designed staggered and subiteration‐free time‐integrator can achieve numerical stability and robustness with respect to the slenderness of the structure, as long as the fluid is justifiably modeled as a compressible medium. Copyright © 2010 John Wiley & Sons, Ltd.     

混凝土结构材料非线性本构参数识别算法研究

针对混凝土材料特性离散性较大、本构参数不确定性较强的问题,本研究提出了基于RSM-CMA的钢筋混凝土结构材料非线性本构参数的识别算法。首先,讨论分析了ABAQUS混凝土损伤塑性模型与规范提供的混凝土本构模型的统一方法,开发了混凝土材料子程序,获取了影响混凝土单轴应力-应变关系的敏感待识别参数,给出了损伤塑性模型屈服面函数与塑性势函数等非识别参数的建议取值。其次,基于响应面法(RSM),实现了结构宏观响应与细观本构参数间隐性关系的显式表达,基于带约束的最小二乘原理,构建目标函数,利用混沌猴群算法(CMA),实现本构参数的优化识别。最后,以钢筋混凝土拱构件的数值仿真为算例,验证了本研究所提算法的准确性。结果表明:抗压强度、峰值压应变是混凝土本构敏感性参数;抗压强度为影响结构极限承载力的最显著本构参数,弹性模量次之,峰值压应变影响最小;本研究所提算法能够较为准确地识别材料的本构参数,有助于获取精确的结构层次宏观响应,可为有限信息下材料本构参数的辨识、获取提供参考。     

Stochastic analysis of structures with uncertain-but-bounded parameters via improved interval analysis

The stochastic analysis of linear structures, with slight variations of the structural parameters, subjected to zero-mean Gaussian random excitations is addressed. To this aim, the fluctuating properties, represented as uncertain-but-bounded parameters, are modeled via interval analysis. In the paper, a novel procedure for estimating the lower and upper bounds of the second-order statistics of the response is proposed. The key idea of the method is to adopt a first-order approximation of the random response derived by properly improving the ordinary interval analysis, based on the philosophy of the so-called affine arithmetic. Specifically, the random response is split as sum of two aliquots: the midpoint or nominal solution and a deviation. The latter is approximated by superimposing the responses obtained considering one uncertain-but-bounded parameter at a time. After some algebra, the sets of first-order ordinary differential equations ruling the midpoint covariance vector and the deviations due to the uncertain parameters separately taken are obtained. Once such equations are solved, the region of the response covariance vector is determined by handy formulas.To validate the procedure, two structures with uncertain stiffness properties under uniformly modulated white noise excitation are analyzed.     

A novel non‐linearly explicit second‐order accurate L‐stable methodology for finite deformation: hypoelastic/hypoelasto‐plastic structural dynamics problems

A novel non‐linearly explicit second‐order accurate L‐stable computational methodology for integrating the non‐linear equations of motion without non‐linear iterations during each time step, and the underlying implementation procedure is described. Emphasis is placed on illustrative non‐linear structural dynamics problems employing both total/updated Lagrangian formulations to handle finite deformation hypoelasticity/hypoelasto‐plasticity models in conjunction with a new explicit exact integration procedure for a particular rate form constitutive equation. Illustrative numerical examples are shown to demonstrate the robustness of the overall developments for non‐linear structural dynamics applications. Copyright © 2004 John Wiley & Sons, Ltd.     

Time response of structures with uncertain parameters under stochastic inputs based on coupled polynomial chaos expansion and component mode synthesis methods

This article presents a numerical procedure to compute the stochastic dynamic response of large finite element models with uncertain parameters based on polynomial chaos and component mode synthesis methods. Polynomial chaos expansions at higher orders are used to derive the statistical solution of the dynamic response as well as the Monte Carlo simulation procedure. Based on various component mode synthesis methods, the size of the model is reduced. These methods are coupled with polynomial chaos expansion and the explicit mathematical formulations are given. Numerical results illustrating the accuracy and efficiency of the proposed coupled methodological procedures are presented.     

Approximate analysis of response variability of uncertain linear systems

A probabilistic methodology is presented for obtaining the variability and statistics of the dynamic response of multi-degree-of-freedom linear structures with uncertain properties. Complex mode analysis is employed and the variability of each contributing mode is analyzed separately. Low-order polynomial approximations are first used to express modal frequencies, damping ratios and participation factors with respect to the uncertain structural parameters. Each modal response is then expanded in a series of orthogonal polynomials in these parameters. Using the weighted residual method, a system of linear ordinary differential equations for the coefficients of each series expansion is derived. A procedure is then presented to calculate the variability and statistics of the uncertain response. The technique is extended to the stochastic excitation case for obtaining the variability of the response moments due to the variability of the system parameters. The methodology can treat a variety of probability distributions assumed for the structural parameters. Compared to existing analytical techniques, the proposed method drastically reduces the computational effort and computer storage required to solve for the response variability and statistics. The performance and accuracy of the method are illustrated by examples.