The Thermodynamic Significance of the Local Volume Averaged Temperature

Since the temperature is not an additive function, the traditional thermodynamic point of view suggests that the volume integral of the temperature has no precise physical meaning. This observation conflicts with the customary analysis of non-isothermal catalytic reactors, heat pipes, driers, geothermal processes, etc., in which the volume averaged temperature plays a crucial role. In this paper we identify the thermodynamic significance of the volume averaged temperature in terms of a simple two-phase heat transfer process. Given the internal energy as a function of the point temperature and the density

Averaged Momentum Equation for Flow Through a Nonhomogenenous Porous Structure

This paper addresses the derivation of the macroscopic momentum equation for flow through a nonhomogeneous porous matrix, with reference to dendritic structures characterized by evolving heterogeneities. A weighted averaging procedure, applied to the local Stokes' equations, shows that the heterogeneous form of the Darcy's law explicitly involves the porosity gradients. These extra terms have to be considered under particular conditions, depending on the rate of geometry variations. In these cases, the local closure problem becomes extremely complex and the full solution is still out of reach. Using a simplified two-phase system with continuous porosity variations, we numerically analyze the limits where the usual closure problem can be retained to estimate the permeability of the structure.     

Random Vibration of Strongly Nonlinear Systems

Procedures developed recently to extend the quasi-conservativeaveraging method to the case of non-white broad-band excitations arefurther extended and presented in a unified framework. The excitationscan be stationary or nonstationary with known evolutionary spectra, andthey can be additive, or multiplicative, or both. Relative merits ofthese procedures are compared, and their accuracies are substantiated byMonte Carlo simulations.     

非线性粘弹性板的失稳条件

研究了给定面内周期激励作用下简支各向同性均匀粘弹性板平衡失稀问题,板的材料特性由Leaderman非线性本构关系描述,将板的动力学方程进行（Galerkin截断得到简化数学模型为弱非线性系统,采用平均法得到系统的平均化方程,对平均化方程进行稳定性分析得到了板平衡失稳的解析条件,对原系统用数值仿真进行研究,数值结果表明,随着激励幅值的增加或粘弹性材料系数的减少,系统平衡点推失稳,激励幅值和粘弹性材料系数的临界值均与解析结果接近。     

Nonlinear Normal Modes in a System with Nonholonomic Constraints

We investigate the dynamics of a system involving the planar motionof a rigid body which is restrained by linear springs and whichpossesses a skate-like nonholonomic constraint known as aplygin'ssleigh. It is shown that the system can be reduced to one with 2 degrees of freedom. The resulting phase flow is shownto involve a curve of nonisolated equilibria. Using second-orderaveraging, the system is shown to possess two families of nonlinearnormal modes (NNMs). Each NNM involves two amplitude parameters.The structure of the NNMs is shown to depart from the generic formin the neighborhood of a 1:1 internal resonance.     

On the closure problem for Darcy's law

In a previous derivation of Darcy's law, the closure problem was presented in terms of an integro-differential equation for a second-order tensor. In this paper, we show that the closure problem can be transformed to a set of Stokes-like equations and we compare solutions of these equations with experimental data. The computational advantages of the transformed closure problem are considerable.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the averaging volume, m² - B second-order tensor used to respresent the velocity deviation - b vector used to represent the pressure deviation, m^–1 - C second-order tensor related to the permeability tensor, m^–2 - D second-order tensor used to represent the velocity deviation, m² - d vector used to represent the pressure deviation, m - g gravity vector, m/s² - I unit tensor - K C ^–1,–D, Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - l _i i=1, 2, 3, lattice vectors, m - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - p pressure in the-phase, N/m ² - p intrinsic phase average pressure, N/m² - p –p , spatial deviation of the pressure in the-phase, N/m² - r position vector locating points in the-phase, m - r ₀ radius of the averaging volume, m - t time, s - v velocity vector in the-phase, m/s - v intrinsic phase average velocity in the-phase, m/s - v phase average or Darcy velocity in the \-phase, m/s - v –v , spatial deviation of the velocity in the-phase m/s - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ Greek Letters V /V volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m²     

Flow of Maxwell fluids in porous media

In this paper we analyze the flow of a Maxwell fluid in a rigid porous medium using the method of volume averaging. We first present the local volume averaged momentum equation which contains Darcy-scale elastic effects and undetermined integrals of the spatial deviations of the pressure and velocity. A closure problem is developed in order to determine the spatial deviations and thus obtain a closed form of the momentum equation that contains a time-dependent permeability tensor. To gain some insight into the effects of elasticity on the dynamics of flow in porous media, the entire problem is transformed to the frequency domain through a temporal Fourier transform. This leads to a dynamic generalization of Darcy's law. Analytical results are provided for the case in which the porous medium is modeled as a bundle of capillary tubes, and a scheme is presented to solve the transformed closure problem for a general microstructure.     

非线性粘弹性拟静态问题与非线性弹性静力问题对应原理

本文应用多重单边拉氏变换导出了非线性粘弹性拟静态问题与非线性弹性静力问题的对应关系,从而把过去认为只有线性条件下存在的粘弹性——弹性对应原理拓展到了非线性范畴     

Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment

In this paper we examine the closure problem associated with the volume averaged form of the Stokes equations presented in Part II. For both ordered and disordered porous media, we make use of a spatially periodic model of a porous medium. Under these circumstances the closure problem, in terms of theclosure variables, is independent of the weighting functions used in the spatial smoothing process. Comparison between theory and experiment suggests that the geometrical characteristics of the unit cell dominate the calculated value of the Darcy's law permeability tensor, whereas the periodic conditions required for thelocal form of the closure problem play only a minor role.Roman Letters A interfacial area of the- interface contained within the macroscopic region, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A interfacial area of the- interface associated with the local closure problem, m² - A _p surface area of a particle, m² - b vector used to represent the pressure deviation, m^–1 - B ⁰ B+I, a second order tensor that maps v _m ontov - B second-order tensor used to represent the velocity deviation - d _p 6V _p/A_p, effective particle diameter, m - d a vector related to the pressure, m - D a second-order tensor related to the velocity, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor calculated on the basis of a spatially periodic model, m² - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p characteristic length for the volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - characteristic length (pore scale) for the-phase - _i i=1, 2, 3 lattice vectors, m - weighting function - m(-y) , convolution product weighting function - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - n unit normal vector pointing from the-phase toward the -phase - p pressure in the-phase, N/m² - p _m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p – p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function - r position vector, m - r position vector locating points in the-phase, m. - V averaging volume, m³ - B volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - v velocity vector in the-phase, m/s - v _m superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - v traditional superficial volume averaged velocity, m/s - v – v _m , spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the -phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * , weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

An Optimal Nonlinear Feedback Control Strategy for Randomly Excited Structural Systems

A strategy for optimal nonlinear feedback control of randomlyexcited structural systems is proposed based on the stochastic averagingmethod for quasi-Hamiltonian systems and the stochastic dynamicprogramming principle. A randomly excited structural system isformulated as a quasi-Hamiltonian system and the control forces aredivided into conservative and dissipative parts. The conservative partsare designed to change the integrability and resonance of the associatedHamiltonian system and the energy distribution among the controlledsystem. After the conservative parts are determined, the system responseis reduced to a controlled diffusion process by using the stochasticaveraging method. The dissipative parts of control forces are thenobtained from solving the stochastic dynamic programming equation. Boththe responses of uncontrolled and controlled structural systems can bepredicted analytically. Numerical results for a controlled andstochastically excited Duffing oscillator and a two-degree-of-freedomsystem with linear springs and linear and nonlinear dampings, show thatthe proposed control strategy is very effective and efficient.     

Upscaling of Nonwetting Phase Residual Transport in Porous Media: A Network Approach

Based on the volume averaging method, a macroscopic model is developed for the upscaling of NAPL transport in a porous medium idealised by a network model. Under the assumption of local mass non-equilibrium, a macroscopic equation involving a dispersion tensor, additional convective terms and a linear form for the interfacial mass flux is obtained. The resolution of the two local closure problems obtained allow the determination of the local properties without adjustable parmeters. These problems are solved in a semi-analytical, semi-numerical manner on the network. The originality of this work is the association of the upscaling by volume averaging method with the network approach. The local properties, including the dispersion tensor and the mass exchange coefficient, can therefore be calculated over a large number of pore-bodies and pore-throats in a computationaly tractable manner, thus leading to more significant results. Results are presented for 3D, spatially periodic models of porous media.     

Maxwell’s Equations in Two-Phase Systems I: Local Electrodynamic Equilibrium

The method of volume averaging is used to examine the form of Maxwell's equations in two-phase systems. The volume averaged equations for the individual phases are developed and used to identify the condition of local electrodynamic equilibrium. When this condition is satisfied, the electric and magnetic fields in the two phases are close enough so that single equations can be used to describe these fields. The one-equation model for the electric and magnetic fields is obtained without the need for closure problems; however, the constraints associated with the principle of local electrodynamic equilibrium are very severe and are not satisfied for many processes.     

Nonlinear implicit iterative method for solving nonlinear ill-posed problems

In the paper, we extend the implicit iterative method for linear ill-posed operator equations to solve nonlinear ill-posed problems. We show that under some conditions the error sequence of solutions of the nonlinear implicit iterative method is monotonically decreasing and, with this monotonicity, prove convergence of the new method for both the exact and perturbed equations.     

Forcing Function Effects on Nonlinear Trajectories: Identifying Very Local Brain Dynamics

Effects of imposing a sinusoidal acoustic and visual forcing function at various frequencies onto an EEG process are examined in terms of various indices of the nonlinear dynamics. Conjoint use of four methods of data analysis; Lyapunov exponents, the entropic analogue of the Schwarzian derivative, surrogate distributions, and higher-order kernel analyses in the time domain, is illustrated. Local epochs with unstable dynamics are identifed on very short series.     

Versal Deformation and Local Bifurcation Analysis of Time-Periodic Nonlinear Systems

In this study a local semi-analytical method of quantitativebifurcation analysis for time-periodic nonlinear systems is presented.In the neighborhood of a local bifurcation point the system equationsare simplified via Lyapunov–Floquet transformation whichtransforms the linear part of the equation into a dynamically equivalenttime-invariant form. Then the time-periodic center manifoldreduction is used to separate the `critical' states and reduce thedimension of the system to a possible minimum. The center manifoldequations can be simplified further via time-dependent normal formtheory. For most codimension one cases these nonlinear normal forms arecompletely time-invariant. Versal deformation of thesetime-invariant normal forms can be found and the bifurcation phenomenoncan be studied in the neighborhood of the critical point. However, ingeneral, it is not a trivial task to find a quantitatively correctversal deformation for time-periodic systems. In order to do so, onemust find a relationship between the bifurcation parameter of theoriginal time-periodic system and the versal deformation parameter ofthe time-invariant normal form. Essentially one needs to find theeigenvalues of the fundamental solution matrix of the time-periodicproblem in terms of the system parameters, which, in general, cannot bedone due to computational difficulties. In this work two ideas areproposed to achieve this goal. The eigenvalues of the fundamentalsolution matrix can be related to the versal deformation parameter bysensitivity analysis and an approximation of any desired order can beobtained. This idea requires a symbolic computational procedure whichcan be very time consuming in some cases. An alternative method issuggested for faster results in which a second or higher order curvefitting technique is used to find the relationship. Once thisrelationship is established, closed form post-bifurcation steady-statesolutions can be obtained for flip, symmetry breaking, transcritical andsecondary Hopf bifurcations. Unlike averaging and perturbation methods,the proposed technique is applicable at any bifurcation point in theparameter space. As physical examples, a simple and a double pendulumsubjected to periodic parametric excitation are considered. A simple twodegrees of freedom model is also studied and the results are comparedwith those obtained from the traditional averaging method. All resultsare verified by numerical integration. It is observed that the proposedtechnique yields results which are very close to the numericalsolutions, unlike the averaging method.     

用无网格局部Petrov-Galerkin法分析非线性地基梁

利用无网格局部Petrov-Galerkin法求解了非线性地基梁。在Petrov-Galerkin方法中，采用移动最小二乘（MLS）近似函数作为场主量挠度的试函数并取移动最小二乘近似函数中的体验函数作为近似场函数的加权函数，采用罚因子法施加本质边界条件。文末给出了两个计算实例，算例的结果表明，Petrov-galerkin法不仅能成功地分析线性地基梁，而且也适用于解非线性地基梁，在分析非线性地基梁时具有收敛快，稳定性好的优点。     

Crossing Rate Analysis with a Non-Gaussian Closure Method for Nonlinear Stochastic Systems

The mean upcrossing rate of the stationary responses of nonlinear stochastic system excited by white noise is analyzed based on the assumption that the probability density function (PDF) of the responses is a linear superposition of basic functions. The Gaussian PDFs are used as the basic functions of which the coefficients are the reciprocal of the number of the basic functions. The Gaussian closure method is a special case of the proposed method. Based upon the approximate PDF, the explicit expression for the mean upcrossing rate (MCR) is given. Numerical results show that the approximate MCRs approach the exact ones in the tails of the MCR curves as the number of the basic functions increases.     

中心有刚体质量的环形薄板的非线性强迫振动

本文研究中心带有刚体质量外部固定铰支或活动铰支的环形薄板的非线性强迫振动。考虑板的弯曲变形、面内位移和几何非线性，用哈米尔顿原理建立板的运动方程，用Ｋａｎｔｏｒｏｖｉｃｈ平均法消去时间变量，然后用数值积分求得非线性振动的振幅随激振力的大小及激振力的频率而变化的关系。求解过程中用打靶法逐步改进未知参量，以保证边界条件的满足。最后讨论薄膜力、激振力的分布、板的内外半径比等因素对响应的影响