Words that almost commute

The Hamming distance $\mathrm{ham}(u,v)$ between two equal-length words u, v is the number of positions where u and v differ. The words u and v are said to be conjugates if there exist non-empty words $x,y$ such that $u=xy$ and $v=yx$. The smallest value $\mathrm{ham}(xy,yx)$ can take on is 0, when x and y commute. But, interestingly, the next smallest value $\mathrm{ham}(xy,yx)$ can take on is 2 and not 1. In this paper, we consider conjugates $u=xy$ and $v=yx$ where $\mathrm{ham}(xy,yx)=2$. More specifically, we provide an efficient formula to count the number $h(n)$ of length-n words $u=xy$ over a k-letter alphabet that have a conjugate $v=yx$ such that $\mathrm{ham}(xy,yx)=2$. We also provide efficient formulae for other quantities closely related to $h(n)$. Finally, we show that $h(n)$ grows erratically: cubically for n prime, but exponentially for n even.     

Interface‐fitted moving mesh method for axisymmetric two‐phase flow in microchannels

A boundary‐fitted moving mesh scheme is presented for the simulation of two‐phase flow in two‐dimensional and axisymmetric geometries. The incompressible Navier‐Stokes equations are solved using the finite element method, and the mini element is used to satisfy the inf‐sup condition. The interface between the phases is represented explicitly by an interface adapted mesh, thus allowing a sharp transition of the fluid properties. Surface tension is modelled as a volume force and is discretized in a consistent manner, thus allowing to obtain exact equilibrium (up to rounding errors) with the pressure gradient. This is demonstrated for a spherical droplet moving in a constant flow field. The curvature of the interface, required for the surface tension term, is efficiently computed with simple but very accurate geometric formulas. An adaptive moving mesh technique, where smoothing mesh velocities and remeshing are used to preserve the mesh quality, is developed and presented. Mesh refinement strategies, allowing tailoring of the refinement of the computational mesh, are also discussed. Accuracy and robustness of the present method are demonstrated on several validation test cases. The method is developed with the prospect of being applied to microfluidic flows and the simulation of microchannel evaporators used for electronics cooling. Therefore, the simulation results for the flow of a bubble in a microchannel are presented and compared to experimental data.     

Invariant discrete flows

In this paper, we investigate the evolution of joint invariants under invariant geometric flows using the theory of equivariant moving frames and the induced invariant discrete variational complex. For certain arc length preserving planar curve flows invariant under the special Euclidean group , the special linear group , and the semidirect group , we find that the induced evolution of the discrete curvature satisfies the differential‐difference mKdV, KdV, and Burgers' equations, respectively. These three equations are completely integrable, and we show that a recursion operator can be constructed by precomposing the characteristic operator of the curvature by a certain invariant difference operator. Finally, we derive the constraint for the integrability of the discrete curvature evolution to lift to the evolution of the discrete curve itself.     

Dynamic and buckling analysis of polymer hybrid composite beam with variable thickness

This work deals with a study of the dynamic and buckling analysis of polymer hybrid composite(PHC) beam. The beam has variable thickness and is reinforced by carbon nanotubes(CNTs) and nanoclay(NC) simultaneously. The governing equations are derived based on the first shear deformation theory(FSDT). A three-phase HalpinTsai approach is used to predict the mechanical properties of the PHC. We focus our attention on the effect of the simultaneous addition of NC and CNT on the vibration and buckling analysis of the PHC beam with variable thickness. Also a comparison study is done on the sensation of three impressive parameters including CNT, NC weight fractions, and the shape factor of fillers on the mechanical properties of PHC beams,as well as fundamental frequencies of free vibrations and critical buckling load. The results show that the increase of shape factor value, NC, and CNT weight fractions leads to considerable reinforcement in mechanical properties as well as increase of the dimensionless fundamental frequency and buckling load. The variation of CNT weight fraction on elastic modulus is more sensitive rather than shear modulus but the effect of NC weight fraction on elastic and shear moduli is fairly the same. The shape factor values more than the medium level do not affect the mechanical properties.     

A Runge-Kutta–based WENO reconstruction on moving mesh for compressible fluids

In this paper, we construct a high-order moving mesh method based on total variation diminishing Runge-Kutta and weighted essential nonoscillatory reconstruction for compressible fluid system. Beginning with the integral form of fluid system, we get the semidiscrete system with an arbitrary mesh velocity. We use weighted essential nonoscillatory reconstruction to get the space accuracy on moving meshes, and the time accuracy is obtained by modified Runge-Kutta method; the mesh velocity is determined by moving mesh method. One- and two-dimensional numerical examples are presented to demonstrate the efficient and accurate performance of the scheme.     

A proper orthogonal decomposition variational multiscale meshless interpolating element-free Galerkin method for incompressible magnetohydrodynamics flow

In the recent decade, the meshless methods have been handled for solving most of PDEs due to easiness of the meshless methods. One of the popular meshless methods is the element-free Galerkin (EFG) method that was first proposed for solving some problems in the solid mechanics. The test and trial functions of the EFG are based on the special basis. Recently, some modifications have been developed to improve the EFG method. One of these improvements is the variational multiscale EFG procedure. In the current article, the shape functions of interpolation moving least squares approximation have been applied to the variational multiscale EFG technique for solving the incompressible magnetohydrodynamics flow. In order to reduce the elapsed CPU time of simulation, we employ a reduced-order model based on the proper orthogonal decomposition technique. The current combination can be referred to as the reduced-order variational multiscale EFG technique. To illustrate the reduction in CPU time used as well as the efficiency of the proposed method, we applied it for the two-dimensional cases.     

无轴承旋翼柔性梁载荷特性研究

应用Hamilton原理建立了双路传力的无轴承旋翼运动方程。采用均匀入流模型,基于直升机飞行平衡条件,建立了无轴承旋翼柔性梁载荷的计算模型,并通过算例验证了模型的精度。利用该模型,研究了全机重心位置、机身气动阻力以及平尾安装角对柔性梁载荷特性的影响,给出了各因素对柔性梁载荷的影响趋势,得出了降低柔性梁载荷的方法。数值结果表明:2cm左右重心位置的变化能够引起9%~11%的柔性梁载荷变化量,15%气动阻力的增加会导致约9%的柔性梁载荷的增大;2°平尾安装角的变化引起约10%柔性梁载荷的变化量,3°平尾安装角的变化引起约26%柔性梁载荷的变化。     

A new Monte Carlo model for predicting the mechanical properties of fiber yarns

Understanding the complicated failure mechanisms of hierarchical composites such as fiber yarns is essential for advanced materials design. In this study, we developed a new Monte Carlo model for predicting the mechanical properties of fiber yarns that includes statistical variation in fiber strength. Furthermore, a statistical shear load transfer law based on the shear lag analysis was derived and implemented to simulate the interactions between adjacent fibers and provide a more accurate tensile stress distribution along the overlap distance. Simulations on two types of yarns, made from different raw materials and based on distinct processing approaches, predict yarn strength values that compare favorably with experimental measurements. Furthermore, the model identified very distinct dominant failure mechanisms for the two materials, providing important insights into design features that can improve yarn strength.     

Structural response of stiffened plates in similitude under a turbulent boundary layer excitation

In structural dynamics, similitude laws usually deal with simple configurations as thin flat plates with point forces. Only recently, few papers have analyzed stiffened shells or stochastic pressure loads.This research activity extends the applicability of some similitude laws, developed for thin flat plates under a turbulent boundary layer load, to ribbed plates forced by the same wall pressure fluctuations.The work addresses the problem of designing a scaled experimental test-article and, successively, of re-modulating the measured data in order to get the structural response of an original (unscaled) configuration.Due to the complexity of the structural domain, the design of a scaled configuration leads to a distorted similitude. Then, a simple approach, to circumvent the distortion effects, is proposed.     

基于粒子滤波的运动目标光电定位仿真研究

运动目标的光电定位不能像静止目标那样简单做均值滤波，鉴于此，引入粒子滤波算法，它不仅可以应用于线性系统，而且还适用于非线性系统。结合光电定位需求，详细推导了计算公式及初值和参数选取公式，对只含测量噪声以及含有测量和运动噪声等的海面运动目标光电无源定位算法进行了仿真计算，验证了算法的有效性，讨论了噪声强度对滤波效果的影响，滤波参数选择对滤波效果的影响，目标运动方式对滤波跟随性的影响，重采样算法对滤波效果的影响等。所得结论为:粒子滤波可用于运动目标光电定位过程，可有效降低定位误差；粒子滤波算法具有较强鲁棒性，适用于噪声较大、目标运动形态变化大等情况。