Solution of Two-Dimensional Stokes Flow Problems Using Improved Singular Boundary Method

In this paper, an improved singular boundary method (SBM), viewed as one kind of modified method of fundamental solution (MFS), is firstly applied for the numerical analysis of two-dimensional (2D) Stokes flow problems. The key issue of the SBM is the determination of the origin intensity factor used to remove the singularity of the fundamental solution and its derivatives. The new contribution of this study is that the origin intensity factors for the velocity, traction and pressure are derived, and based on that, the SBM formulations for 2D Stokes flow problems are presented. Several examples are provided to verify the correctness and robustness of the presented method. The numerical results clearly demonstrate the potentials of the present SBM for solving 2D Stokes flow problems.     

头部电阻抗成像正问题的解析解研究

建立头颅的球型仿真模型.用头皮、颅骨、脑脊髓和脑组织四层同心球结构模拟头颅.对各层组织采用复数电导率,用分离变量法分析头颅球模型在最外层(头皮层)表面施加点电流激励的情况下,内层的电位分布.根据电位分布的表达式,绘制出颅内的电位等位线图.分析电流注入角度对电位分布的影响.仿真结果表明,当颅内脑组织电导率变化时,头皮表面电位的实部变化明显大于虚部,激励电流频率变化对头皮表面电位虚部的影响较为明显.研究结果可用于分析头部电阻抗成像问题.     

Ergodicity of the Finite Dimensional Approximation of the 3D Navier–Stokes Equations Forced by a Degenerate Noise

We prove ergodicity of the finite dimensional approximations of the three dimensional Navier–Stokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential.     

用拉普拉斯变换差分法求解试井分析中的一维渗流问题

结合拉普拉斯变换和有限差分法给出求解试井分析中一维渗流问题的拉普拉斯变换差分法:首先对渗流方程采用拉普拉斯变换消去时间变量得到拉普拉斯空间数学模型,采用有限差分法求解拉普拉斯空间数学模型,最后通过拉普拉斯反演算法得到井底压力或产量.通过与有限差分法结果和解析解对比,拉普拉斯变换差分法比有限差分法计算误差小.虽然单步计算耗时长,但计算任意时刻结果时对空间网格的适应性和不依赖其它时刻计算结果的特性使得拉普拉斯变换差分法在试井分析中有非常好的应用前景.     

The Stochastic Traveling Salesman Problem: Finite Size Scaling and the Cavity Prediction

We study the random link traveling salesman problem, where lengths l _ij between city i and city j are taken to be independent, identically distributed random variables. We discuss a theoretical approach, the cavity method, that has been proposed for finding the optimum tour length over this random ensemble, given the assumption of replica symmetry. Using finite size scaling and a renormalized model, we test the cavity predictions against the results of simulations, and find excellent agreement over a range of distributions. We thus provide numerical evidence that the replica symmetric solution to this problem is the correct one. Finally, we note a surprising result concerning the distribution of k th-nearest neighbor links in optimal tours, and invite a theoretical understanding of this phenomenon.     

Multigrid Solution of the Incompressible Navier–Stokes Equations for Density-Stratified Flow past Three-Dimensional Obstacles

The steady incompressible Navier–Stokes equations in three dimensions are solved for neutral and stably stratified flow past three-dimensional obstacles of increasing spanwise width. The continuous equations are approximated using a finite volume discretisation on staggered grids with a flux-limited monotonic scheme for the advective terms. The discrete equations which arise are solved using a nonlinear multigrid algorithm with up to four grid levels using the SIMPLE pressure correction method as smoother. When at its most effective the multigrid algorithm is demonstrated to yield convergence rates which are independent of the grid density. However, it is found that the asymptotic convergence rate depends on the choice of the limiter used for the advective terms of the density equation, and some commonly used schemes are investigated. The variation with obstacle width of the influence of the stratification on the flow field is described and the results of the three-dimensional computations are compared with those of the corresponding computation of flow over a two-dimensional obstacle (of effectively infinite width). Also given are the results of time-dependent computations for three-dimensional flows under conditions of strong static stability when lee-wave propagation is present and the multigrid algorithm is used to compute the flow at each time step.     

An Iterative Two-Grid Method of a Finite Element PML Approximation for the Two Dimensional Maxwell Problem

In this paper, we propose an iterative two-grid method for the edge finite element discretizations (a saddle-point system) of Perfectly Matched Layer (PML) equations to the Maxwell scattering problem in two dimensions. Firstly, we use a fine space to solve a discrete saddle-point system of $H(grad)$ variational problems, denoted by auxiliary system 1. Secondly, we use a coarse space to solve the original saddle-point system. Then, we use a fine space again to solve a discrete$\boldsymbol{H}(curl)$-elliptic variational problems, denoted by auxiliary system 2. Furthermore, we develop a regularization diagonal block preconditioner for auxiliary system 1 and use $H$-$X$ preconditioner for auxiliary system 2. Hence we essentially transform the original problem in a fine space to a corresponding (but much smaller) problem on a coarse space, due to the fact that the above two preconditioners are efficient and stable. Compared with some existing iterative methods for solving saddle-point systems, such as PMinres, numerical experiments show the competitive performance of our iterative two-grid method.     

The Three Dimensional Non-conforming Finite Element Solution of the Chapman–Ferraro Problem

We demonstrate the feasibility of using a non-conforming, piecewise harmonic finite element method on an unstructured grid in solving a magnetospheric physics problem. We use this approach to construct a global discrete model of the magnetic field of the magnetosphere that includes the effects of shielding currents at the outer boundary (the magnetopause). As in the approach of F. R. Toffolettoet al.(1994,Geophys. Res. Lett.21, 7) the internal magnetospheric field model is that of R. V. Hilmer and G.-H. Voigt (1995,J. Geophys. Res.) while the magnetopause shape is based on an empirically determined approximation (1997, J. Shueet al.,J. Geophys. Res.102, 9497). The results is a magnetic field model whose field lines are completely confined within the magnetosphere. The presented numerical results indicate that the discrete non-conforming finite element model is well-suited for magnetospheric field modeling.     

Solution of the Electrodiffusion Equations in the Quasi-Stationary Approximation

The electrodiffusion equations are solved together with the equation for the current running through the boundary of the examined system with the environment. The complete system of equations allows the volt-ampere characteristic of the system to be obtained for an arbitrary time dependence of the external potential. The procedure of finding the first-order approximation is developed to solve the system of electrodiffusion equations.     

On Recent Analytical Results for Solution of the Scattering Problem for the Sharply Screened Coulomb Potential

Exact representations for the wave function and for the Green function of the Hamiltonian with the sharply screened Coulomb potential are given. The representations are obtained by summing up the partial wave series. The final form of the wave function and the Green function in the region of the coordinate space where the potential is not zero are given in terms of the Coulomb wave function and the Coulomb Green function, respectively. The exact representation has been obtained for the transition operator in the configuration space.     

Analytical Solution for an Ultracold Trapped Ion at the Node of the Standing Wave Laser Without Rotating Wave Approximation

A method of solving an ultracold trapped ion at the node of the standing wave laser without rotating wave approximation is proposed and the analytical forms of the eigenfunctions and eigenenergies of the system are presented.     

Analytical Solution for an Ultracold Trapped Ion at the Node of the Standing Wave Laser Without Rotating Wave Approximation

A method of solving an ultracold trapped ion at the node of the standing wave laser without rotating wave approximation is proposed and the analytical forms of the eigenfunctions and eigenenergies of the system are presented.     

头部分层球模型磁感应成像正问题的解析解

建立适用于磁感应成像正问题研究的头颅四层同心球模型,分别代表大脑、脑脊髓层、颅骨层和头皮层.以矢量磁位为变量,建立球坐标系下的亥姆赫兹方程,作为磁感应成像正问题的控制方程,用分离变量法求解亥姆赫兹方程,得到模型内矢量磁位的分布,进而推导出球内涡流场的分布特性,绘制出其等位线图.分析激励电流频率和幅值对感应电压的影响.仿真结果表明该解析方法可以计算磁感应成像正问题,并可作为生成逆问题灵敏度矩阵的一种快速算法.     

A Finite Volume Method Based on the Constrained Nonconforming Rotated Q1-Constant Element for the Stokes Problem

We construct a finite volume element method based on the constrained nonconforming rotated Q₁-constant element (CNRQ₁-P₀) for the Stokes problem. Two meshes are needed, which are the primal mesh and the dual mesh. We approximate the velocity by CNRQ₁ elements and the pressure by piecewise constants. The errors for the velocity in the H¹ norm and for the pressure in the L² norm are O(h) and the error for the velocity in the L² norm is O(h²). Numerical experiments are presented to support our theoretical results.     

Numerical and Asymptotic Solution of the Problem of Oscillations of an Inhomogeneous Waveguide with an Annular Crack of Finite Width

Acoustical Physics - The problem of waves in an inhomogeneous cylindrical waveguide with an annular crack is considered. A system of integral equations with hypersingular kernels is obtained for...     

Approximated Analytical Solution of a Problem on Indentation of an Electro-Elastic Half-Space with Inhomogeneous Coating by a Conductive Punch

The paper addresses to the development of the bilateral asymptotic method of solution of contact problems taking into account electro-mechanical fields coupling. Approximated analytical solutions are provided for contact problems on indentation of an electroelastic piezoelectric half-space with inhomogeneous in depth piezoelectric coating by a rigid conductive circular flat, spherical or conical punch.