Higher order weighted integral stochastic finite element method and simplified first-order application

It is well known that expansion-based stochastic methods are approximate schemes, as they are based on a first or, at most, second order series expansion on the basic variable, e.g. displacement. Therefore, expansion-based stochastic analysis schemes are bound to show small response variability when compared with Monte Carlo simulation (MCS) results, and application of these schemes is limited to stochastic problems with relatively small variability. In order to overcome these general drawbacks of the expansion methods, we suggest a higher-order stochastic field function that can be employed in the expansion-based stochastic analysis scheme of the weighted integral method. We then propose a new weighted integral formulation using the higher-order stochastic field function. The new formulation is not only applicable to stochastic problems with a high degree of uncertainty but also can reproduce the phenomenon of accelerated increase in the response variability when the coefficient of variation of the stochastic field increases, as observed in the MCS. In order to show the validity of the proposed formulation, we provide two numerical examples and the results are discussed in detail.     

STOCHASTIC OPTIMAL CONTROL OF HYSTERETIC SYSTEMS UNDER EXTERNALLY AND PARAMETRICALLY RANDOM EXCITATIONS

A stochastic optimal control method for nonlinear hysteretic systems under exter-nally and/or parametrically random excitations is presented and illustrated with an example ofhysteretic column system.A hysteretic system subject to random excitation is first replaced bya nonlinear non-hysteretic stochastic system.An It stochastic differential equation for the to-tal energy of the system as a one-dimensional controlled diffusion process is derived by usingthe stochastic averaging method of energy envelope.A dynamical programming equation is thenestablished based on the stochastic dynamical programming principle and solved to yield the op-timal control force.Finally,the responses of uncontrolled and controlled systems are evaluatedto determine the control efficacy.It is shown by numerical results that the proposed stochasticoptimal control method is more effective and efficient than other optimal control methods.     

PATH INTEGRAL SOLUTION OF NONLINEAR DYNAMIC BEHAVIOR OF STRUCTURE UNDER WIND EXCITATION

IntroductionThe dynamic behavior of the nonlinear structure under wind excitation has beenobserved very complicated.Taking guyed masts as an example,only a few collapsingaccidents occurred under extreme atmospheric conditions[1],many took place under mild…     

Dynamic Analysis of a Flexible Vehicle Structure with Nonlinear Force Elements Using Substructure Technique∗

ABSTRACT

The substnicturing technique is used to analyze the motion of a flexible vehicle structure with nonlinear force elements. The number of effective degrees of freedom of the combined system model is greatly reduced by synthesizing the individually modeled substructures, using constraints of geometric compatibility at the interfaces. When combining the substructures, recursive-type nonlinear integral equations (NIE) are introduced instead of the conventional nonlinear differential equations (NDE). It is shown that the NIE formulation is computationally more efficient than the NDE formulation in simulating steady-state and transient responses of a flexible vehicle with nonlinear dampers. The NIE formulation is further applied to the nonlinear damping optimization problem, and it is found that this method is efficient for optimization.     

Solute Transport Analysis Through Heterogeneous Media in Nonuniform in the Average Flow by a Stochastic Approach

The stochastic approach has been shown to be an excellent tool for the characterisation and analysis of velocity fields and transport processes through heterogeneous porous formations. The main results (linear theory) have been obtained for problems with simplified flow conditions, usually in the assumption of uniform in the average flow, but a great effort is spent to reach theoretical results for more complex situations.This paper deals with 2D heterogeneous aquifers subject to uniform recharge; the stochastic approach is adopted to characterise, as ensemble behaviour, the velocity field and transport processes of a nonreactive solute. The impact of transmissivity conditioning on solute particles trajectories is analysed and an application is carried out. The analytical formulations, obtained by a first order analysis, are compared to the one resulting from constant in the average hydraulic gradient, and their reliability is investigated with numerical tests performed by a Monte Carlo method.The result of this study is that strong non-stationarities are present in the flow and transport process. A detailed analysis shows that the theoretical results cannot be extended to cases with high heterogeneity level, unlike the uniform in the average flow fields.     

A non-intrusive methodology for the representation of crack growth stochastic processes

In this paper, we present a methodology to pursue the uncertainty quantification of the stochastic process that represents the crack growth problem. The main idea of this methodology is to discretize the crack growth process in a sequence of random variables and then, approximate each of them using a stochastic polynomial approach. This methodology is non-intrusive, i.e. it is based on the representation of random variables using stochastic polynomials, whose coefficients are evaluated using a least squares method and only a few realizations of the stochastic process. The Paris–Erdogan law was used as crack growth model in order to focus the reader's attention on the uncertainty quantification methodology. We modeled the parameters of the Paris–Erdogan law as random variables, i.e. the initial crack length and the coefficients of the Paris–Erdogan model are treated as random variables. Two numerical examples are presented in order to shown the effectiveness and accuracy of the proposed methodology. From the results of these examples, it is shown that the proposed methodology is able to successfully approximate the stochastic process that represents the crack growth for the Paris–Erdogan model, with a much lower computational cost than the MCS. The main limitation of the proposed approach is that, in the form it was presented, it is not able to handle random processes as input parameters.     

An inverse for the Jaumann derivative and some applications to the rheology of viscoelastic fluids

Summary By using a generalization of the matrizant of matrix calculus, it is shown how one can construct formally an inverse, or integral, for the well-knownJaumann derivative of continuum mechanics. Some applications to fluid rheology are then considered. First, it is shown that this integral provides, via theBoltzmann super-position principle, a generalization of Oldroyd's quasi-linear fluid model, which is related to the molecular model ofBueche. Explicit expressions for the stresses arising in a general laminar shear flow are then derived for this model. Secondly, it is indicated how the operation can be used with rheological equations which are nonlinear in the deformation-rate, but quasi-linear in stress, to solve explicitly for the stress in terms of kinematic quantities. As an example, a rheological equation for suspensions of viscoelastic spheres in aNewtonian fluid is treated.     

Stochastic response of an axially moving viscoelastic beam with fractional order constitutive relation and random excitations

A convenient and universal residue calculus method is proposed to study the stochastic response behaviors of an axially moving viscoelastic beam with random noise excitations and fractional order constitutive relationship, where the random excitation can be decomposed as a nonstationary stochastic process, Mittag-Leffler internal noise, and external stationary noise excitation. Then, based on the Laplace transform approach, we derived the mean value function, variance function and covariance function through the Green's function technique and the residue calculus method, and obtained theoretical results. In some special case of fractional order derivative α , the Monte Carlo approach and error function results were applied to check the effectiveness of the analytical results, and good agreement was found. Finally in a general-purpose case, we also confirmed the analytical conclusion via the direct Monte Carlo simulation.     

T-stress evaluation for slightly curved crack using perturbation method

A general formulation for evaluating the T-stress at crack tips in a curved crack is introduced. In the formulation, a singular integral equation with the distribution of dislocation along the curve is suggested. For a slightly curved crack, a small parameter is generally assumed for the crack configuration. By using the assumption for the small parameter, the perturbation method is suggested and it reduces the singular integral equation into many successive singular integral equations. If the cracked plate has a remote loading and the curve configuration is a quadratic function, the mentioned successive singular integral equations can be solved in a closed form. Therefore, the solution for the T-stress in a closed form is obtained. The obtained results for T-stress are shown by figures. It is found that if the involved parameter is not too small, the influence of the curve configuration is significant. Comparison for T-stresses obtained from a quadratic-shaped curved crack and an arc crack is presented.     

Dynamical problems of continuous media with random boundary data

A spatial correlation method is formulated for linear dynamical problems in continuum mechanics with random boundary data. The essential feature of the method is the formulation of a nonstochastic mixed initial-boundary value problem for the (matrix) spatial correlation function of the (vector) state variable. Whenever the Green's function of the (stochastic) problem can not be obtained in terms of known functions, a numerical solution of the meansquare response and other second order response statistics by the spatial correlation method is several hundred folds more efficient than any other available method. Further improvements in the computational efficiency of the method for a steady state stationary response process are also noted.     

Boundary element method for thermoelastic analysis of three-dimensional transversely isotropic solids

Thermal effects are well known to manifest themselves as additional volume integral terms in the direct formulation of the boundary integral equation (BIE) for linear elastic solids when using the boundary element method (BEM). This domain integral has been successfully transformed in an exact manner to surface ones only in isotropy and in 2D anisotropy, thereby restoring the BEM as a truly boundary solution technique. The difficulties with extending it to 3D general anisotropic solids lie in the mathematical complexity of the Green’s function and its derivatives for such materials. These quantities are required items in the BEM formulation. In this paper, the exact, analytical transformation of the volume integral associated with thermal effects to surface ones is achieved for a transversely isotropic material using a similar approach which the authors have previously employed for the same task in BEM for 2D general anisotropy. A numerical scheme, however, needs to be employed to evaluate some of the new terms introduced in the surface integrals that arise from this process here. The mathematical soundness of the formulation is demonstrated by a few examples; the numerical results obtained are checked by alternative means, including those obtained from the commercial FEM code, ANSYS.     

Stochastic filtering of multiphase transport phenomena

Whether or not a multiphase mixture is to be considered heterogeneous is a function of scale and hence also a function of the measurement process. Heterogeneities manifested through the measurement process and sampling frequency (extrinsic heterogeneity) are discussed in relationship to the intrinsic heterogeneity associated with the true physical environment. The question of whether or not a multiphase environment may be scaled is partially a function of the scale of the extrinsic heterogeneity of interest. Thus, it is also a function of the instrumental apparatus, which implies that the instrument must also be scaled. We review restrictive scaling constraints in frequency space that an instrument must satisfy for proper scaling of an experiment. Cushman (1984 Water Resour. Res. and 1985 Acta Appl. Math) developed a theory that incorporates the scale and measurement process directly into the multiphase transport equations and which in turn allows for correlation of field properties over scales of motion. We discuss a generalization of this theory which accounts for integral representations of distributional stochastic processes as a function of the scale of the instrument used in the measurement of the field variables. This allows us to solve the stochastic transport equations in an operational setting, i.e., in terms of measurable quantities. Throughout the article examples are presented to illustrate the concepts.     

A subdomain boundary element method for high‐Reynolds laminar flow using stream function‐vorticity formulation

The paper presents a new formulation of the integral boundary element method (BEM) using subdomain technique. A continuous approximation of the function and the function derivative in the direction normal to the boundary element (further ‘normal flux’) is introduced for solving the general form of a parabolic diffusion‐convective equation. Double nodes for normal flux approximation are used. The gradient continuity is required at the interior subdomain corners where compatibility and equilibrium interface conditions are prescribed. The obtained system matrix with more equations than unknowns is solved using the fast iterative linear least squares based solver. The robustness and stability of the developed formulation is shown on the cases of a backward‐facing step flow and a square‐driven cavity flow up to the Reynolds number value 50 000. Copyright © 2004 John Wiley & Sons, Ltd.     

Fokker–Planck Equations for a Free Energy Functional or Markov Process on a Graph

The classical Fokker–Planck equation is a linear parabolic equation which describes the time evolution of the probability distribution of a stochastic process defined on a Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy. In recent years, it has been shown that the Fokker–Planck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this paper, we consider analogous matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If N ≧ 2 is the number of vertices of the graph, we show that the corresponding Fokker–Planck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to stochastic processes defined on Euclidean spaces, the situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker–Planck equations for the same process. It is shown that there is a strong connection but there are also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker–Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples are also discussed.     

一类Markov过程的最大绝对值过程概率密度求解的新方法

随机过程或随机系统响应的最大绝对值概率分布往往是科学与工程中关心的重要挑战性问题.本文从理论与数值上进行了Markov过程的时变最大绝对值过程及其概率分布研究.文中,通过引入扩展状态向量,构造了最大绝对值$\!$-$\!$-$\!$状态量联合向量过程,由此将不具有Markov性的最大值过程转化为具有Markov性的向量随机过程.在此基础上,通过最大绝对值$\!$-$\!$-$\!$状态量之间的关系,建立了联合向量过程的转移概率密度函数.进而,结合Chapman-Kolmogorov方程和路径积分方法,提出了最大绝对值概率密度函数求解的数值方法.由此,可以得到Markov过程最大绝对值过程的时变概率密度函数,可进一步用于结构动力可靠度分析等.通过数值算例,验证了本文所提方法的有效性. 该方法有望推广到更一般随机系统的极值分布估计之中.      

Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media

This study is concerned with developing a two-dimensional multiphase model that simulates the movement of NAPL in heterogeneous aquifers. Heterogeneity is dealt with in a probabilistic sense by modeling the intrinsic permeability of the porous medium as a stochastic process. The deterministic finite element method is used to spatially discretize the multiphase flow equations. The intrinsic permeability is represented in the model via its Karhunen–Loeve expansion. This is a computationally expedient representation of stochastic processes by means of a discrete set of random variables. Further, the nodal unknowns, water phase saturations and water phase pressures, are represented by their stochastic spectral expansions. This representation involves an orthogonal basis in the space of random variables. The basis consists of orthogonal polynomial chaoses of consecutive orders. The relative permeabilities of water and oil phases, and the capillary pressure are expanded in the same manner, as well. For these variables, the set of deterministic coefficients multiplying the basis in their expansions is evaluated based on constitutive relationships expressing the relative permeabilities and the capillary pressure as functions of the water phase saturations. The implementation of the various expansions into the multiphase flow equations results in the formulation of discretized stochastic differential equations that can be solved for the deterministic coefficients appearing in the expansions representing the unknowns. This method allows the computation of the probability distribution functions of the unknowns for any point in the spatial domain of the problem at any instant in time. The spectral formulation of the stochastic finite element method used herein has received wide acceptance as a comprehensive framework for problems involving random media. This paper provides the application of this formalism to the problem of two-phase flow in a random porous medium.     

Dynamic Models of Simple Judgments: I. Properties of a Self-Regulating Accumulator Module

This is the first of two papers comparing connectionist and traditional stochastic latency mechanisms with respect to their ability to account for simple judgments. In this paper, we show how the need to account for additional features of judgment has led to the formulation of progressively more sophisticated models. One of these, a self-regulating, generalized accumulator process, is treated in detail, and its simulated performance across a sample of tasks is described. Since an adaptive decision module of this kind possesses all the ingredients of intelligent behavior, it is eminently suited as a basic computing element in more complex networks.     

Optimal calculation steps for the evaluation of residual stress by the incremental hole-drilling method

The integral method is a suitable calculation procedure for the determination of nonuniform residual stresses by semidestructive mechanical methods such as the hole-drilling method and the ring-core method. However, the high sensitivity to strain measurement errors due to the ill conditioning of the equations has hindered its practical use. the analysis of the influence of the strain measurment error on the computed stresses carried out in the present work has showed that, given both maximum hole depth and number of total steps, the error sensitivity depends on the particular depth increment distribution used. By means of the matrix formulation, the depth increment distribution that optimizes the numerical conditioning is investigated. Numerical simulations and an experimental test have corroborated the best performance of the proposed step distribution with respect to the constant or increasing distributions commonly used.     

Application of Fractional Derivatives in Thermal Analysis of Disk Brakes

This paper presents a Fractional Derivative Approach for thermal analysis of disk brakes. In this research, the problem is idealized as one-dimensional. The formulation developed contains fractional semi integral and derivative expressions, which provide an easy approach to compute friction surface temperature and heat flux as functions of time. Given the heat flux, the formulation provides a means to compute the surface temperature, and given the surface temperature, it provides a means to compute surface heat flux. A least square method is presented to smooth out the temperature curve and eliminate/reduce the effect of statistical variations in temperature due to measurement errors. It is shown that the integer power series approach to consider simple polynomials for least square purposes can lead to significant error. In contrast, the polynomials considered here contain fractional power terms. The formulation is extended to account for convective heat loss from the side surfaces. Using a simulated experiment, it is also shown that the present formulation predicts accurate values for the surface heat flux. Results of this study compare well with analytical and experimental results.     

Application of Fractional Derivatives in Thermal Analysis of Disk Brakes

This paper presents a Fractional Derivative Approach for thermal analysis of disk brakes. In this research, the problem is idealized as one-dimensional. The formulation developed contains fractional semi integral and derivative expressions, which provide an easy approach to compute friction surface temperature and heat flux as functions of time. Given the heat flux, the formulation provides a means to compute the surface temperature, and given the surface temperature, it provides a means to compute surface heat flux. A least square method is presented to smooth out the temperature curve and eliminate/reduce the effect of statistical variations in temperature due to measurement errors. It is shown that the integer power series approach to consider simple polynomials for least square purposes can lead to significant error. In contrast, the polynomials considered here contain fractional power terms. The formulation is extended to account for convective heat loss from the side surfaces. Using a simulated experiment, it is also shown that the present formulation predicts accurate values for the surface heat flux. Results of this study compare well with analytical and experimental results.