Study on application of random walk method to calculate water exchange in large-scale bay

The water exchange matrix is an efficient tool to study the water exchange among the sub-areas in large-scale bays. The application of the random walk method to calculate the water exchange matrix is studied. Compared with the advection-diffusion model, the random walk model is more flexible to calculate the water exchange matrix. The forecast matrix suggested by Thompson et al. is used to evaluate the water exchange characteristics among the sub-areas fast. According to the theoretic analysis, it is found that the precision of the predicted results is mainly affected by three factors, namely, the particle number, the generated time of the forecast matrix, and the number of the sub-areas. The impact of the above factors is analyzed based on the results of a series of numerical tests. The results show that the precision of the forecast matrix increases with the increase of the generated time of the forecast matrix and the number of the particles. If there are enough particles in each sub-area, the precision of the forecast matrix will increase with the number of the sub-areas. Moreover, if the particles in each sub-area are not enough, the excessive number of the sub-areas can result in the decrease of the precision of the forecast matrix.     

A method to study interactions between narrow-banded random waves and multi-chamber perforated structures

A time-domain method, based on linear velocity potential theory, is presented to study the interaction between narrow-banded random waves and perforated structures. A simple relation is derived to estimate the jet length of flows through the perforated wall. The reflection coefficient of narrow banded random waves from perforated structures is calculated by assuming a Rayleigh distribution of the heights of incident random waves. For reflection of narrow-banded waves from a single-chamber perforated breakwater, a comparison of the predicted and measured reflection coefficients shows that the method presented in this paper can provide a prediction better than that of regular waves. Numerical results are also reported on the reflection of narrow-banded waves from multi-chamber perforated breakwaters.The project partially supported by the Hong Kong Research Grant Council (DAG03/04.EG39, DAG04/05.EG32).     

Distribution and dispersion in the Euler-Lagrange random walk

Summary In a new kind of random walk, displaying both Lagrangian and Eulerian statistical properties¹⁾, calculations were made previously of the Eulerian and Lagrangian velocity auto-correlations. Now the characteristic function of particle displacement has been calculated, and possible continuum limit forms for the probability density equation have been deduced. One of these turns out to be the telegraph equation, given by Goldstein²⁾ as the limit of a different kind of random walk. For this case the auto-correlation functions have been determined.This work was supported by the Mechanics Branch, U.S. Office of Naval Research, under contract Nonr 248 (38) and was presented in Session F of the 1959 Annual Meeting of the American Physical Society at New York under the title A Random Walk with both Lagrangian and Eulerian Statistics.Associate Professor of Engineering Research, Ordnance Research Laboratory, The Pennsylvania State University. This work was done as Research Associate, Post-Doctoral Fellow, Mechanical Engineering Department, The Johns Hopkins University.     

Random walk on graphs: An application to the double diffusivity model

Preliminary but interesting and definite results are given on the application of graph theory concepts (random walk on graphs) to the double diffusivity theory proposed by Aifantis in the late 70s to model transport in media with high diffusivity paths such as metal polycrystals with a continuous distribution of grain boundaries possessing much higher diffusivity than the bulk, as well as in nanopolycrystals for which it has been shown recently that the double diffusivity model fits experimental observations. The new information provided by employing the graph theory tool is concerned with certain restrictions and relations that the phenomenological coefficients, entering in the coupled partial differential equations of double diffusivity, should satisfy depending on the topology and related details of the graph model adopted.     

Thermodynamics-based gradient plasticity theories with an application to interface models

In the framework of small deformations, the so-called residual-based gradient plasticity theory is reconsidered and improved. Using the notion of moving geometrically necessary dislocations (GNDs), suitable micromechanics interpretations are heuristically given for the higher order boundary conditions and the long distance particle interactions. Also, a comparison is made between this theory and the analogous virtual work principle (VWP)-based one, whereby their respective conceptual and methodological features are pointed out. The conditions under which the two theories lead to a same constitutive model are investigated, showing that, correspondingly, a certain indeterminacy exhibited by the VWP-based theory disappears. A phenomenological interface model is used to better point out differences and analogies of the two theories.     

Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential

We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high-degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weak Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten Laplacian, with detailed applications.     

The interrelation between void fraction fluctuations and flow patterns in two-phase flow

A fast response, linearized X-ray void measurement system has been used to obtain statistical measurements in normally fluctuating air-water flow in a rectangular channel. It is demonstrated that the probability density function (PDF) of the fluctuations in void fraction may be used as an objective and quantitative flow pattern discriminator for the three dominant patterns of bubbly, slug, and annular flow. This concept is applied to data over the range of 0.0 to 37 m/sec mixture velocities to show that slug flow is simply a transitional, periodic time combination of bubbly and annular flows. Film thicknesses calculated from the PDF data are similar in magnitude in both slug and annular flows. Calculation of slug length and residence time ratios along with bubble lengths in slug flow are also readily obtainable from the statistical measurements. Spectral density measurements showed bubbly flow to be stochastic while slug and annular flows showed periodicities correlatable in terms of the liquid volume flux.     

Evolutionary solutions to the brachistochrone problem with Coulomb friction

The classical brachistochrone problem was originally proposed by Johann Bernoulli more than 300 years ago, and is still one of the most elegant problems in modern mathematics. This paper examines a generalisation of the original brachistochrone problem by the inclusion of a non-conservative force in the form of Coulomb friction. Solutions to this problem are found using an evolutionary computational technique based on Darwinian natural selection and survival of the fittest. These evolutionary solutions tend to indicate that as friction is introduced the optimum curves develop from cycloidal (the analytical solution for the conservative problem) toward straight lines. The flexibility of the evolutionary method to optimise for other objectives is then discussed.     

NMR for equilateral triangular geometry under conditions of surface relaxivity—analytical and random walk solution

We consider analytical and numerical solution of NMR relaxation under the condition of surface relaxation in an equilateral triangular geometry. We present an analytical expression for the Green’s function in this geometry. We calculate the transverse magnetic relaxation without magnetic gradients present, single-phase, both analytically and numerically. There is a very good match between the analytical and numerical results. We also show that the magnetic signal from an equilateral triangular geometry is qualitatively different from the known solution: plate, cylinder, and sphere, in the case of a nonuniform initial magnetization. Nonuniform magnetization close to the sharp corners makes the magnetic signal very fast multiexponential. This type of initial configuration fits qualitatively with the experimental results by Song (Phys. Rev. Lett. 85, 3878 (2000)), Song et al. (Nature 406, 178 (2000)), Song (Mag. Reson. Imag. 19, 417 (2001)) and Lisitza and Song (Phys. Rev. B 65, 172406 (2002)). It should also be noted that the solution presented here can be used to describe absorption of a chemical substance in an equilateral triangular geometry (for a stationary fluid).     

3D macro and micro-block models for limit analysis of out-of-plane loaded masonry walls with non-associative Coulomb friction

In this paper, a masonry system composed of a façade wall connected with two sidewalls and subjected to out-of-plane loading is investigated within the framework of three-dimensional limit analysis. Two different modeling approaches, namely macro and micro-block models, are adopted. A rigid-perfectly plastic model with dry contact interfaces governed by Coulomb failure criterion is assumed for masonry walls with regular units and staggering (non-standard limit analysis). Three classes of failure modes are investigated, involving rocking, sliding, twisting failure and combinations of them. The macro-block model is based on the assumption that the failure involves a number of cracks which separate the structure into a few macro-blocks and all the possible relative motions among micro-blocks are concentrated along the cracks. Two limiting conditions for the ultimate load factor are kinematically computed by use of minimization routines. The micro-block model is based on a concave contact formulation in which contact points are located at the corners of interfaces, allowing failure modes involving opening and sliding to be simulated. An iterative solution procedure is used to solve the non-associative friction problem, with second order cone programming (SOCP) used to allow the conic yield function to be solved directly. Both models are validated against experimental outcomes from the literature. A parametric analysis is carried out in order to highlight the influence of each geometrical and mechanical parameter on the prevalence of a mechanism over the other. The presence of an unrestrained horizontal floor system with different orientations is also analyzed.     

Experimental and computational models for three-dimensional crack-fiber interactions

This study sheds light on the small-scale interaction of a three-dimensional matrix crack with a fiber. The experiments with a model brittle-matrix/brittle-fiber system record the three-dimensional growth history of an initially penny-shaped fracture which quasi-statically propagates toward and around a cylindrical inclusion. Crack growth histories are obtained by hydraulically fracturing a cement matrix with embedded glass rods. These experimentally determined crack patterns support micromechanical computational simulations which were conducted using a three-dimensional surface integral method. The implications for tailoring interfacial friction to increase the crack resistance of brittle materials (e.g., ceramic matrix/ceramic fiber composites) are discussed.     

Comparison and analysis of two Coulomb friction models on the dynamic behavior of slider-crank mechanism with a revolute clearance joint

The objective of this study is to investigate the effects of the Coulomb dry friction model versus the modified Coulomb friction model on the dynamic behavior of the slider-crank mechanism with a revolute clearance joint. The normal and tangential forces acting on the contact points between the journal and the bearing are described by using a Hertzian-based contact force model and the Coulomb friction models, respectively. The dynamic equations of the mechanism are derived based on the Lagrange equations of the first kind and the Baumgarte stabilization method. The frictional force is solved via the linear complementarity problem (LCP) algorithm and the trial-and-error algorithm. Finally, three numerical examples are given to show the influence of the two Coulomb friction models on the dynamic behavior of the mechanism. Numerical results show that due to the stick friction, the slider-crank mechanism may exhibit stick-slip motion and can balance at some special positions, while the mechanism with ideal joints cannot.     

Waves with random coupling and random propagation constants

We study the mathematical treatment of transmission statistics of coupled wave devices with random imperfections. Such devices include multi-mode waveguides or optical fibers, directional couplers at waveguide or optical frequencies, dominant-wave transmission lines (with the reflected wave the spurious mode), lumped filters composed of circuit or of waveguide elements, multi-layer optical coatings to achieve high or low reflectivity.If the random parameters have very rapid spatial variations, we assume they have white spectra. Exact results for transmission statistics are obtained in such cases. These exact results can be extended to random parameters with almost-white spectra, and narrow-band spectra that are far from white.These calculations are carried out in a simple way by using Kronecker matrix products.     

Prediction models of fatigue crack closure and growth under random loading

The effects of plastic zones both in front of and behind crack tip on crack closure have been analysed. The total residual deformations of crack surfaces involve two parts, that is, the amount of plastic blunting of crack tip and the residual deformation in the wake of the tip. This paper presents a fatigue crack closure model in which the influences of compressive load on closure stress are discussed. The model is applied to random loading conditions by the assumption of limited memory properties. The fatigue lives are predicted using the proposed crack growth model for CCT plane stress specimen cut from 2219-T851 aluminum alloy under flight spectrum loadings, and the prediction values agree with the test results.The project was supported by the natural science foundation of China.     

Discrete and continuous models for in plane loaded random elastic brickwork

The aim of this paper is to study non-periodic masonries – typical of historical buildings – by means of a perturbation approach and to evaluate the effect of a random perturbation on the elastic response of a periodic masonry wall. The random masonry is obtained starting from a periodic running bond pattern. A random perturbation on the horizontal positions of the vertical interfaces between the blocks which form the masonry wall is introduced. In this way, the height of the blocks is uniform, while their width in the horizontal direction is random. The perturbation is limited such as each block has still exactly 6 neighboring blocks. In a first discrete model, the blocks are modeled as rigid bodies connected by elastic interfaces (mortar thin joints). In other words, masonry is seen as a “skeleton” in which the interactions between the rigid blocks are represented by forces and moments which depend on their relative displacements and rotations. A second continuous model is based on the homogenization of the discrete model. Explicit upper and lower bounds on the effective elastic moduli of the homogenized continuous model are obtained and compared to the well-known effective elastic moduli of the regular periodic masonry. It is found that the effective moduli are not very sensitive to the random perturbation (less than 10%). At the end, the Monte Carlo simulation method is used to compare the discrete random model and the continuous model at the structural level (a panel undergoing in plane actions). The randomness of the geometry requires the generation of several samples of size L of the discrete masonry. For a sample of size L, the structural discrete problem is solved using the same numerical procedure adopted in [Cecchi, A., Sab, K., 2004. A comparison between a 3D discrete model and two homogenized plate models for periodic elastic brickwork, International Journal of Solids Structures 41 (9–10), 2259–2276] and the average solution over the samples gives an estimation which depends on L. As L increases, an asymptotic limit is reached. One issue is to find the minimum size for L and to compare the asymptotic average solution to the one obtained from the continuous homogenized model.     

Noise-to-state finite-time practical stability for random nonlinear systems and its application in random Lagrange systems

The noise-to-state finite-time practical stability for random nonlinear systems and its application is studied in this paper. The definition of noise-to-state finite-time practical stability is firstly introduced in probability sense for random nonlinear systems. Next, the related stability criterion is also given by Lyapunov approach. For random benchmark system, the finite-time adaptive tracking control problem is investigated by the vectorial backstepping method and the obtained stability theorem. Simulation example illustrates that the constructed controller design scheme is effective and feasible.

     

一类铁电模型的参变量变分原理及其应用

将参变量变分原理引入铁电问题。对一类借用了经典弹塑性理论中的概念和方法的多轴铁电模型建立基于Helmholtz自由能的参变量变分原理,可以有效处理传统变分原理中由非关联流动法则或屈服面不考虑材料系数变化所引起的切线模量非对称困难。相应于参变量变分原理,引入参数二次规划算法,可获得具有可靠数值稳定性的一套铁电算法。将该算法应用于一个具体的铁电模型,数值计算结果表明本文方法的有效性。     

Planar multiple-contact problems subject to unilateral and bilateral kinetic constraints with static Coulomb friction

This paper is concerned with contact problems. A planar multiple-contact problem subject to unilateral and bilateral kinetic constraints with static friction is studied using the complementarity method. First, this paper discusses the one-to-one correspondence of solutions of the contact problems of concern and of the corresponding complementarity models. An enhanced complementarity model is proposed by adding missed tangential acceleration constraints into previous complementarity models. Solutions of the proposed complementarity model and solutions of the contact problem are proven to exhibit one-to-one correspondence, which may not be guaranteed in the previous complementarity models. Then, this paper applies linear complementarity theory to investigate the properties of the solutions of the proposed complementarity model. For both unilaterally constrained contact problems and bilaterally constrained contact problems, the existence of solutions and boundedness of solutions are proven. Sufficient conditions for the uniqueness of solutions and finiteness of the number of solutions are also provided. Several numerical examples are given to show the non-uniqueness of solutions or the infiniteness of the number of solutions. Such phenomena demonstrate the non-smoothness of the contact problems discussed herein.