Solution of Dirichlet problem for a rectangular region in terms of elliptic functions

In this study, the Green function of the (interior) Dirichlet problem for the Laplace (also Poisson) differential equation in a rectangular domain is expressed in terms of elliptic functions and the solution of the problem is based on the Green function and therefore on the elliptic functions. The method of solution for the Dirichlet problem by the Green function is presented; the Green function and transformation required for the solution of the Dirichlet problem in the rectangular region is found and the problem is solved in the rectangular region. An example for the problem in the rectangular region is given in order to present an application of the solution of Dirichlet problem. The equation is solved first by the known method of separation of variables and then in terms of elliptic functions; the results of both methods are compared. The results are found to be consistent but the advantage of this method is that the solution is obtained in terms of elementary functions.     

FIDISOL: A ‘black box’ solver for partial differential equations

FIDISOL (finite difference solver) is a program package for the solution of nonlinear systems of 2-D and 3-D elliptic and parabolic partial differential equations subjected to arbitrary nonlinear boundary conditions on a rectangular domain. The solution method is a variable step size/variable order finite difference method. This paper is a survey of this project in which the principles of developing a fully vectorized program package designed as a ‘data flow’ algorithm for vector computers are explicated.     

Continuous Collision Detection for Two Moving Elliptic Disks

Collision detection and avoidance are important in robotics. Compared with commonly used circular disks, elliptic disks provide a more compact shape representation for robots or other vehicles confined to move in the plane. Furthermore, elliptic disks allow a simpler analytic representation than rectangular boxes, which makes it easier to perform continuous collision detection (CCD). We shall present a fast and accurate method for CCD between two moving elliptic disks, which avoids any need to sample the time domain of the motion, thus avoiding the possibility of missing collisions between time samples. Based on some new algebraic conditions on the separation of two ellipses, we reduce collision detection for two moving ellipses to the problem of detecting real roots of a univariate equation, which is the discriminant of the characteristic polynomial of the two ellipses. Several techniques are investigated for robust and accurate processing of this univariate equation for two classes of commonly used motions: planar cycloidal motions and planar rational motions. Experimental results demonstrate the efficiency, accuracy, and robustness of our method.     

Finite difference solution to the flow between two rotating spheres

This paper describes a numerical method for calculating incompressible viscous flows between two concentric rotating spheres. The dependent variables describing the axisymmetric flow field are the azimuthal components of the vorticity, of the velocity vector potential and of the velocity. The coupled set of governing partial differential equations is written as a system of strictly second-order equations by introducing vorticity conditions of an integral character in a meridional plane. Such conditions generalize the one-dimensional integral conditions employed by Dennis and Singh to calculate steady-state solutions of the same problem using Gegenbauer polynomials and finite differences. The basic equations are discretized in space and in time by means of the finite-difference method. A fourth-order accurate centred-difference approximation of the advection terms is employed and a nonlinearly implicit scheme for the discrete time integration is here considered. A general finite-difference algorithm for steady-state and time-dependent problems is obtained which has no relaxation parameter and makes extensive use of fast elliptic solvers. The numerical results obtained by the present method are found to be in good agreement with the literature and confirm the nonuniqueness of the steady-state solution in a narrow spherical gap at certain regimes.     

Spectral domain decomposition techniques for viscous incompressible flows

Spectral domain decomposition methods are described for solving the equations governing the flow of viscous incompressible fluids in rectangularly decomposable domains. The domain of interest is divided into a number of rectangular subdomains on each of which a spectral approximation of the flow variables is sought. For Newtonian flows a stream function formulation is used whereas for nonNewtonian flows the components of the extra-stress tensor are also used. Efficient direct methods for the solution of the algebraic systems are discussed. Numerical results are presented for laminar flow through a channel contraction and for the stick-slip problem.     

基于与中间块比较的快速分形图像编码

针对基本分形图像编码算法时间过长的问题,提出了一种基于与中间块比较的快速分形编码算法。该算法是对基于形态特征的快速分形图像编码算法的改进,将形态特征意义下的最近邻匹配改进为与中间块的误差意义下的最近邻匹配,并且在最近邻的邻域内进一步搜索最优匹配父块时,对各父块进行8种等距变换,而不是在找到最优匹配父块后进行,同时引进误差阈值来控制子块搜索的邻域范围。实验结果表明,该算法编码速度大大提高,并且在相近编码时间的前提下该算法的解码图像质量比基于形态特征的算法好。     

Blind separation of audio signals using trigonometric transforms and wavelet denoising

This paper deals with the problem of blind separation of audio signals from noisy mixtures. It proposes the application of a blind separation algorithm on the discrete cosine transform (DCT) or the discrete sine transform (DST) of the mixed signals, instead of performing the separation on the mixtures in the time domain. Wavelet denoising of the noisy mixtures is recommended in this paper as a preprocessing step for noise reduction. Both the DCT and the DST have an energy compaction property, which concentrates most of the signal energy in a few coefficients in the transform domain, leaving most of the transform domain coefficients close to zero. As a result, the separation is performed on a few coefficients in the transform domain. Another advantage of signal separation in transform domains is that the effect of noise on the signals in the transform domains is smaller than that in the time domain due to the averaging effect of the transform equations, especially when the separation algorithm is preceded by a wavelet denoising step. The simulation results confirm the superiority of transform domain separation to time domain separation and the importance of the wavelet denoising step.     

基于HVS分类及邻域搜索的快速分形图像编码

针对基本分形图像编码算法时间过长的问题,提出了一种基于HVS分类及邻域搜索的快速算法。根据HVS特性将子块分为平滑类子块和非平滑类子块,对于平滑类子块直接存储其均值,以减少需要搜索匹配父块的子块数;对于非平滑类子块,从离其最近的父块开始搜索,在搜索父块时,剔除与当前子块的近似度不满足要求的父块,并引入误差阂值和搜索父块的最大次数来控制子块的搜索过程。实验结果证明,该算法大大提高了编码速度。     

A multi-level boundary element method for Stokes flows in irregular two-dimensional domains

Recently, we have developed multi-level boundary element methods (MLBEM) for the solution of the Laplace and Helmholtz equations that involve asymptotically decaying non-oscillatory and oscillatory singular kernels, respectively. The accuracy and efficiency of the fast boundary element methods for steady-state heat diffusion and acoustics problems have been investigated for square domains. The current work extends the MLBEM methodology to the solution of Stokes equation in more complex two-dimensional domains. The performance of the fast boundary element method for the Stokes flows is first investigated for a model problem in a unit square. Then, we consider an example problem possessing an analytical solution in a rectangular domain with 5:1 aspect ratio, and finally, we study the performance of the MLBEM algorithm in a C-shaped domain.     

Non-Reflecting Boundary Conditions for Maxwell’s Equations

A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in the Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equation.     

System decomposition technique for spray modelling in CFD codes

A new decomposition technique for a system of ordinary differential equations is suggested, based on the geometrical version of the integral manifold method. This is based on comparing the values of the right hand sides of these equations, leading to the separation of the equations into ‘fast’ and ‘slow’ variables. The hierarchy of the decomposition is allowed to vary with time. Equations for fast variables are solved by a stiff ODE system solver with the slow variables taken at the beginning of the time step. The solution of the equations for the slow variables is presented in a simplified form, assuming linearised variation of these variables for the known time evolution of the fast variables. This can be considered as the first order approximation for the fast manifold. This technique is applied to analyse the explosion of a polydisperse spray of diesel fuel. Clear advantages are demonstrated from the point of view of accuracy and CPU efficiency when compared with the conventional approach widely used in CFD codes. The difference between the solution of the full system of equations and the solution of the decomposed system of equations is shown to be negligibly small for practical applications. It is shown that in some cases the system of fast equations is reduced to a single equation.     

Electric potential in the earth’s ionosphere: a numerical model

Modeling the global distribution of the electric potential in the Earth’s ionosphere is based on the solution of a 2D continuity equation in the ionospheric-magnetospheric current circuit. The potential distribution is described by the boundary value problem for an elliptic system of partial differential equations on the spherical shell approximating the ionosphere, which is divided into three subregions with nonlocal boundary conditions. Implementation of the boundary conditions, which reflect the continuity of the common current circuit and potential equalization at the boundaries of the polar caps, is leading to the mutual dependence of the potential distribution within the northern and southern caps and their influence on the potential distribution in the midlatitude region. The problem is solved by an iterative method with a regularizing operator which is inverted using the separation of the variables and the fast Fourier transform with respect to the azimuthal variable and the sweep method with respect to the latitudinal one.     

Preconditioning of the coarse problem in the method of balanced domain decomposition by constraints

The method of balanced domain decomposition by constraints is an iterative algorithm for numerical solution of partial differential equations which exploits a non-overlapping partition of a domain. As an essential part of each step, restricted problems are solved on every subdomain and a certain coarse grid solution is found. In this paper we present a new strategy of preconditioning of the coarse problem. This is based on the algebraic multilevel preconditioning technique. We present numerical estimates of constants defining the condition numbers of the preconditioned coarse problems for several two- and three-dimensional elliptic equations.     

A Fast Spectral Subtractional Solver for Elliptic Equations

The paper presents a fast subtractional spectral algorithm for the solution of the Poisson equation and the Helmholtz equation which does not require an extension of the original domain. It takes O(N ² log N) operations, where N is the number of collocation points in each direction. The method is based on the eigenfunction expansion of the right hand side with integration and the successive solution of the corresponding homogeneous equation using Modified Fourier Method. Both the right hand side and the boundary conditions are not assumed to have any periodicity properties. This algorithm is used as a preconditioner for the iterative solution of elliptic equations with non-constant coefficients. The procedure enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points. We also apply the basic solver to the solution of the Poisson equation in complex geometries.     

Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method

We have studied previously a generalized conjugate gradient method for solving sparse positive-definite systems of linear equations arising from the discretization of elliptic partial-differential boundary-value problems. Here, extensions to the nonlinear case are considered. We split the original discretized operator into the sum of two operators, one of which corresponds to a more easily solvable system of equations, and accelerate the associated iteration based on this splitting by (nonlinear) conjugate gradients. The behavior of the method is illustrated for the minimal surface equation with splittings corresponding to nonlinear SSOR, to approximate factorization of the Jacobian matrix, and to elliptic operators suitable for use with fast direct methods. The results of numerical experiments are given as well for a mildy nonlinear example, for which, in the corresponding linear case, the finite termination property of the conjugate gradient algorithm is crucial.     

Thermo-elastic stresses within a rectangular conductor carrying an alternating current

A finite element algorithm is presented for the solution of the basic equations describing thermomechanical problems in a conductor carrying an alternating current. The analyzed algorithm is based on a combination of the Bubnov-Galerkin method and the method of separation of variables. Calculations are carried out for a copper conductor with strong skin effect.     

A multidomain spectral method for solving elliptic equations

We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three distinct features. First, the combined problem of solving the PDE, satisfying the boundary conditions, and matching between different subdomains is cast into one set of equations readily accessible to standard linear and nonlinear solvers. Second, touching as well as overlapping subdomains are supported; both rectangular blocks with Chebyshev basis functions as well as spherical shells with an expansion in spherical harmonics are implemented. Third, the code is very flexible: The domain decomposition as well as the distribution of collocation points in each domain can be chosen at run time, and the solver is easily adaptable to new PDEs. The code has been used to solve the equations of the initial value problem of general relativity and should be useful in many other problems. We compare the new method to finite difference codes and find it superior in both runtime and accuracy, at least for the smooth problems considered here.     

Blind separation of audio signals using trigonometric transforms and Kalman filtering

This paper deals with the problem of blind separation of audio signals from noisy mixtures. It proposes the application of a blind separation algorithm on the Discrete Cosine Transform (DCT) or the Discrete Sine Transform (DST) of the mixed signals, instead of performing the separation on the mixtures in the time domain. Kalman Filtering of the noisy separated signals is recommended in this paper as a post-processing step for noise reduction. Both the DCT and the DST have an energy compaction property, which concentrates most of the signal energy in a few coefficients in the transform domain, leaving the rest of the transform-domain coefficients close to zero. As a result, the separation is performed on a few coefficients in the transform domain. Another advantage of signal separation in transform domains is that the effect of noise on the signals in the transform domains is smaller than that in the time domain due to the averaging effect of the transform equations. The simulation results confirm the effectiveness of transform-domain signal separation and the feasibility of the post-processing Kalman filtering step.     

Simple and efficient motion estimation algorithm by continuum search

Motion estimation plays a vital role in reducing temporal correlation in video codecs but it requires high computational complexity. Different algorithms have tried to reduce this complexity. However these reduced-complexity routines are not as regular as the full search algorithm (FSA). Also, regularity of an algorithm is very important in order to have a hardware implementation of that algorithm even if it leads to more complexity burden. The goal of this paper is to develop an efficient and regular algorithm which mimics FSA by searching a small area exhaustively. Our proposed algorithm is designed based on two observations. The first observation is that the motion vector of a block falls within a specific rectangular area designated by the prediction vectors. The second observation is that in most cases, this rectangular area is smaller than one fourth of the FSA’s search area. Therefore, the search area of the proposed method is adaptively found for each block of a frame. To find the search area, the temporal and spatial correlations among motion vectors of blocks are exploited. Based on these correlations, a rectangular search area is determined and the best matching block in this area is selected. The proposed algorithm is similar to FSA in terms of regularity but requires less computational complexity due to its smaller search area. Also, the suggested algorithm is as simple as FSA in terms of implementation and is comparable with many of the existing fast search algorithms. Simulation results show the claimed performance and efficiency of the algorithm.     

基于正四棱锥结构的机器人声源定位系统研究

声源定位成为机器人智能研究的重要方向。针对当前声源定位精度不理想、实时性不佳等问题,提出了一种正四棱锥麦克风阵列声源定位结构。采用时间延迟估计的声源定位方法,并提出时延值的快速搜索策略;推导了该结构的基于信号时延的时空映射关系,建立了声源目标位置的几何计算模型,并依据正四棱锥结构特点及冗余的时延值对值域划分,缩小求解范围,运用迭代算法得到声源的位置坐标,并通过双重筛选机制剔除错误的定位结果。实验结果证明了该结构及定位算法在提高系统定位精度和实时性能的有效性,能满足机器人应用中对声源定位的需求。