Rhythmic precipitate patterns and fractal structure

Liesegang patterns of parallel precipitate bands are obtained when solutions containing co-precipitate ions interdiffuse in a 1D gel matrix.The sparingly soluble salt formed,displays a beautiful stratification of discs of precipitate perpendicular to the 1D tube axis.The Liesegang structures are analyzed from the viewpoint of their fractal nature.Geometric Liesegang patterns are constructed in conformity with the well-known empirical laws such as the time,band spacing and band width laws.The dependence of the band spacing on the initial concentrations of diffusing(outer)and immobile(inner)electrolytes(A0 and B0,respectively)is taken to follow the Matalon-Packter law.Both mathematical fractal dimensions and box-count dimensions are calculated.The fractal dimension is found to increase with increasing A0 and decreasing B0.We also analyze mosaic patterns with random distribution of crystallites,grown under different conditions than the classical Liesegang gel method,and report on their fractal properties.Finally,complex Liesegang patterns wherein the bands are grouped in multiplets are studied,and it is shown that the fractal nature increases with the multiplicity.     

Representation of robotic fractional dynamics in the pseudo phase plane

This paper analyses robotic signals in the perspective of fractional dynamics and the pseudo phase plane (PPP).It is shown that the spectra of several experimental signals can be approximated by trend lines whose slope characterizes their fractional behavior.For the PPP reconstruction of each signal,the time lags are calculated through the fractal dimension.Moreover,to obtain a smooth PPP,the noisy signals are filtered through wavelets.The behavior of the spectra reveals a relationship with the fractal dimension of the PPP and the corresponding time delay.     

Fractal geometry and topology abstracted from hair fibers

Based on the concepts of fractal super fibers, the （3, 9＋2）-circle and （9＋2, 3）-circle binary fractal sets are abstracted form such prototypes as wool fibers and human hairs, with the （3）-circle and the （9＋2）-circle fractal sets as subsets. As far as the （9＋2） topological patterns are concerned, the following propositions are proved： The （9＋2） topological patterns accurately exist, but are not unique. Their total number is 9. Among them, only two are allotropes. In other words, among the nine topological patterns, only two are independent （or fundamental）. Besides, we demonstrate that the （3, 9＋2）-circle and （9＋2, 3）-circle fractal sets are golden ones with symmetry breaking.     

Peristaltic motion of third grade fluid in curved channel

＂ Analysis is performed to study the slip effects on the peristaltic flow of non-Newtonian fluid in a curved channel with wall properties. The resulting nonlinear partial differential equations are transformed to a single ordinary differential equation in a stream function by using the assumptions of long wavelength and low Reynolds number. This differential equation is solved numerically by employing the built-in routine for solving nonlinear boundary value problems （BVPs） through the software Mathematica. In addition, the analytic solutions for small Deborah number are computed with a regular perturbation technique. It is noticed that the symmetry of bolus is destroyed in a curved channel. An intensification in the slip effect results in a larger magnitude of axial velocity. Further, the size and circulation of the trapped boluses increase with an increase in the slip parameter. Different from the case of planar channel, the axial velocity profiles are tilted towards the lower part of the channel. A comparative study between analytic and numerical solutions shows excellent agreement.     

Extended multiscale finite element method for mechanical analysis of heterogeneous materials

An extended multiscale finite element method （EMsFEM） is developed for solving the mechanical problems of heterogeneous materials in elasticity.The underlying idea of the method is to construct numerically the multiscale base functions to capture the small-scale features of the coarse elements in the multiscale finite element analysis.On the basis of our existing work for periodic truss materials, the construction methods of the base functions for continuum heterogeneous materials are systematically introduced. Numerical experiments show that the choice of boundary conditions for the construction of the base functions has a big influence on the accuracy of the multiscale solutions, thus,different kinds of boundary conditions are proposed. The efficiency and accuracy of the developed method are validated and the results with different boundary conditions are verified through extensive numerical examples with both periodic and random heterogeneous micro-structures.Also, a consistency test of the method is performed numerically. The results show that the EMsFEM can effectively obtain the macro response of the heterogeneous structures as well as the response in micro-scale,especially under the periodic boundary conditions.     

A numerical approach for pressure transient analysis of a vertical well with complex fractures

A new well test model for a vertical fractured well is developed based on a discrete-fracture model in which the fractures are discretized as one dimensional (1-D) entities. The model overcomes the weakness of complex meshing, a large number of grids, and instability in conventional stripe-fracture models. Then, the discrete-fracture model is implemented using a hybrid element finite-element method. Triangular elements are used for matrix and line elements for the fractures. The finite element formulation is validated by comparing with the semi-analytical solution of a single ver-tical fractured well. The accuracy of the approach is shown through several examples with different fracture apertures, fracture conductivity, and fracture amount. Results from the discrete-fracture model agree reasonably well with the stripe-fracture model and the analytic solutions. The advantages of the discrete-fracture model are presented in mesh gen-eration, computational improvement, and abilities to handle complex fractures like wedge-shaped fractures and fractures with branches. Analytical results show that the number of grids in the discrete-fracture model is 10%less than stripe-fracture model, and computational efficiency increases by about 50%. The more fractures there are, the more the com-putational efficiency increases.     

Orientation distribution of fibers and rheological property in fiber suspensions flowing in a turbulent boundary layer

A model relating the translational and rotational transport of orientation distribution function （ODF） of fibers to the gradient of mean ODF and the dispersion coefficients is proposed to derive the mean equation for the ODE Then the ODF of fibers is predicted by numerically solving the mean equation for the ODF together with the equations of turbulent boundary layer flow. Finally the shear stress and first normal stress difference of fiber suspensions are obtained. The results, some of which agree with the available relevant experimental data, show that the most fibers tend to orient to the flow direction. The fiber aspect ratio and Reynolds number have significant and negligible effects on the orientation dis- tribution of fibers, respectively. The additional normal stress due to the presence of fibers is anisotropic. The shear stress of fiber suspension is larger than that of Newtonian solvent, and the first normal stress difference is much less than the shear stress. Both the additional shear stress and the first normal stress difference increase with increasing the fiber concentration and decreasing fiber aspect ratio.     

On transition of type V interaction in double-wedge flow with non-equilibrium effects

The transition between regular reflection(RR) and Mach reflection(MR) of type V shock–shock interaction on a double-wedge geometry with high temperature non-equilibrium effects is investigated by extended shock-polar method and numerical simulation. First, the critical angles of transition from detachment criterion and von Neumann criterion are determined by the extended shock-polar method considering the non-equilibrium effects. Then wave patterns and the transition process are numerically obtained. Results of the critical transition angles from shock-polar calculation and numerical simulation show evident disagreement, indicating transition mechanism between RR and MR of type V interaction is changed. By comparing with the frozen counterpart, it is also found that non-equilibrium effects lead to a larger critical wedge angle and a larger hysteresis interval.     

Mode decomposition of nonlinear eigenvalue problems and application in flow stability

Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.     

THE EVALUATION OF STRESS INTENSITY FACTORS OF PLANE CRACK FOR ORTHOTROPIC PLATE WITH EQUAL PARAMETER BY F2LFEM

In this paper, the evaluation of stress intensity factor of plane crack problems for orthotropic plate of equal-parameter is investigated using a fractal two-level finite element method (F2LFEM). The general solution of an orthotropic crack problem is obtained by assimilating the problem with isotropic crack problem, and is employed as the global interpolation function in F2LFEM. In the neighborhood of crack tip of the crack plate, the fractal geometry concept is introduced to achieve the similar meshes having similarity ratio less than one and generate an infinitesimal mesh so that the relationship between the stiffness matrices of two adjacent layers is equal. A large number of degrees of freedom around the crack tip are transformed to a small set of generalized coordinates. Numerical examples show that this method is efficient and accurate in evaluating the stress intensity factor (SIF).     

分形理论在煤层气非平衡吸附模型中的应用

传统的煤层气动力学模型均是建立在欧几里得几何基础上的,难以描述煤层孔隙结构的复杂性及形状的不规则性。本文以分形理论为基础,通过引入分形维数来刻画煤层孔隙结构的复杂性并考虑煤层的吸附特性、双重介质特征及介质的变形,建立基于Fick第二定律的分形介质煤层气非稳态渗流数学模型。由于流动方程的强非线性,结合各类边界条件用正则摄动法和Laplace变换得到模型在拉氏空间上的近似解析解,再利用Laplace数值反演求得实空间上的数值解。对参数进行敏感性分析并绘制了典型压力曲线,这些结果为煤层气开采提供了理论依据和试井方法。     

Low Mach number prediction of the acoustic signature of fractal-generated turbulence

In this work, we compare the acoustic properties of a fractal square grid with those of a regular grid by means of a hybrid approach based on Lighthill’s analogy and Direct Numerical Simulation (DNS). Our results show that the sound levels corresponding to our fractal square grid of three fractal iterations are significantly reduced by comparison to a regular grid of same porosity and mesh-based Reynolds number. We also find a well-defined peak at a Strouhal number between 0.2 and 0.3 in the acoustic spectrum of our fractal square grid which is absent in the case of our regular grid. We explain this effect in terms of a new criterion for quasi-periodic vortex shedding from a regular or fractal grid.     

Properties of Chaotic and Regular Boundary Crisis in Dissipative Driven Nonlinear Oscillators

The phenomenon of the chaotic boundary crisis and the related concept of the chaotic destroyer saddle has become recently a new problem in the studies of the destruction of chaotic attractors in nonlinear oscillators. As it is known, in the case of regular boundary crisis, the homoclinic bifurcation of the destroyer saddle defines the parameters of the annihilation of the chaotic attractor. In contrast, at the chaotic boundary crisis, the outset of the destroyer saddle which branches away from the chaotic attractor is tangled prior to the crisis. In our paper, the main point of interest is the problem of a relation, if any, between the homoclinic tangling of the destroyer saddle and the other properties of the system which may accompany the chaotic as well as the regular boundary crisis. In particular, the question if the phenomena of fractal basin boundary, indeterminate outcome, and a period of the destroyer saddle, are directly implied by the structure of the destroyer saddle invariant manifolds, is examined for some examples of the boundary crisis that occur in the mathematical models of the twin-well and the single-well potential nonlinear oscillators.     

Solution set on the natural satellite formation orbits under first-order earth＇s non-spherical perturbation

Using the reference orbital element approach, the precise governing equations for the relative motion of formation flight are formulated. A number of ideal formations with respect to an elliptic orbit can be designed based on the relative motion analysis from the equations. The features of the oscillating reference orbital elements are studied by using the perturbation theory. The changes in the relative orbit under perturbation are divided into three categories, termed scale enlargement, drift and distortion respectively. By properly choosing the initial mean orbital elements for the leader and follower satellites, the deviations from originally regular formation orbit caused by the perturbation can be suppressed. Thereby the natural formation is set up. It behaves either like non-disturbed or need little control to maintain. The presented reference orbital element approach highlights the kinematics properties of the relative motion and is convenient to incorporate the results of perturbation analysis on orbital elements. This method of formation design has advantages over other methods in seeking natural formation and in initializing formation.     

考虑变形耦合的几何非线性空间梁单元

以"精确几何模型梁单元"为代表的很多几何非线性梁单元,在构造过程中分别对描述截面转动的转角和描述截面形心位置的位移进行了独立插值,由此引起了诸如运动学描述冗余和剪切闭锁等困难。其根本原因在于单元形函数没能体现细长梁中的变形耦合关系。本文对这类传统单元进行了改造,通过深入研究单元变形之间的内在联系,提出了一种变形场完全满足Bernoulli梁变形耦合关系的新单元,避免了构造过程中对转动矢量的插值,并通过数值算例检验了单元的有效性。     

基于改进遗传算法主动柔性结构压电元件位置优化

基于改进的遗传算法提出了一种解决多压电片布置在柔性结构上以实现振动主动控制的有效方法.本文采用D优化设计准则,即把Fisher信息矩阵行列式最大值作为目标函数的一种优化方法.通过对结构模态振型和动力特性的研究,使用一种简单的方法将所选择的低阶模态振型转化为归一化形式,最后通过结构模态振型可确定压电片最优位置.为了达到更好的振动控制效果,压电片布置在结构模态应变最大位置处.当在结构不同位置上布置压电片时,结构各阶模态振型也将随之发生不同程度的变化.本文建立了ANSYS软件和遗传算法的接口来实现对结构的模态重分析,从而提取各阶模态振型.     

Finite Element Analysis of Elasto-plastic Plate Bending Problems using Transition Rectangular Plate Elements

In this work, the finite element analysis of the elasto-plastic plate bending problems is carried out using transition rectangular plate elements. The shape functions of the transition plate elements are derived based on a practical rule. The transition plate elements are all quadrilateral and can be used to obtain efficient finite element models using minimum number of elements. The mesh convergence rates of the models including the transition elements are compared with the regular element models. To verify the developed elements, simple tests are demonstrated and various elasto-plastic problems are solved. Their results are compared with ANSYS results.The English text was polished by Keren Wang.     

岩体结构要素分形几何研究

本文研究了岩体结构诸要素, 包括结构面形态、结构面厚度、结构面分布、产状分布、孔隙分布颗粒表面形态等的分形表征, 目的是提高岩体结构的定量描述程度, 为岩体工程的定量评价提供新的途径。     

Two-dimensional lattice gases for regular binary mixtures

This paper reports on the development of two-dimensional lattice gas models for regular binary mixtures. In particular, results are reproduced concerning equilibrium solutions and an expression for the diffusion coefficient. In our model, a volume force is incorporated, and the system of macroscopic evolution equations resembles the Boussinesq approximation in convection theory. As an example, a lattice gas Rayleigh-Bénard system is considered. We conclude with a few remarks on implementation and optimisation of the program using a SIMD parallel computer.