ORTHOVEC: A REDUCE program for 3-D vector analysis in orthogonal curvilinear coordinates

ORTHOVEC is a collection of REDUCE procedures and operations which provide a simple to use environment for the manipulation of scalars and vectors. Operations include addition, subtraction, dot and cross products, division, modulus, div, grad, curl, Laplacian, differentiation, integration, a · ▽ and Taylor expansion.     

A finite difference method for 3D incompressible flows in cylindrical coordinates

In this work, a finite difference method to solve the incompressible Navier-Stokes equations in cylindrical geometries is presented. It is based upon the use of mimetic discrete first-order operators (divergence, gradient, curl), i.e. operators which satisfy in a discrete sense most of the usual properties of vector analysis in the continuum case. In particular the discrete divergence and gradient operators are negative adjoint with respect to suitable inner products. The axis r = 0 is dealt with within this framework and is therefore no longer considered as a singularity. Results concerning the stability with respect to 3D perturbations of steady axisymmetric flows in cylindrical cavities with one rotating lid, are presented.     

New subgrid artificial viscosity Galerkin methods for the Navier–Stokes equations

We study subgrid artificial viscosity methods for approximating solutions to the Navier–Stokes equations. Two methods are introduced that add viscous stabilization via an artificial viscosity, then remove it only on a coarse mesh. These methods can be considered as conforming, mixed methods, the first for velocity and vorticity, and the second for velocity and its gradient, the former incorporating a naturally arising grad–div stabilization term. In this paper, we rigorously study the first scheme analytically, showing that it is unconditionally stable and optimally convergent, as well as both schemes computationally. Numerical experiments demonstrate the advantages of both of these methods.     

Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions

We derive $\mathit{H}$(curl)-error estimates and improved $\mathit{L}$²-error estimates for the Maxwell equations approximated using edge finite elements. These estimates only invoke the expected regularity pickup of the exact solution in the scale of the Sobolev spaces, which is typically lower than $\frac{1}{2}$ and can be arbitrarily close to $0$ when the material properties are heterogeneous. The key tools for the analysis are commuting quasi-interpolation operators in $\mathit{H}$(curl)- and $\mathit{H}$(div)-conforming finite element spaces and, most crucially, newly-devised quasi-interpolation operators delivering optimal estimates on the decay rate of the best-approximation error for functions with Sobolev smoothness index arbitrarily close to $0$. The proposed analysis entirely bypasses the technique known in the literature as the discrete compactness argument.     

A Weak Galerkin Finite Element Method for the Maxwell Equations

This paper introduces a numerical scheme for the time-harmonic Maxwell equations by using weak Galerkin (WG) finite element methods. The WG finite element method is based on two operators: discrete weak curl and discrete weak gradient, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on arbitrary shape of polyhedra and, at the same time, is parameter free. Optimal-order of convergence is established for the WG approximations in various discrete norms which are either \(H^1\)-like or \(L^2\) and \(L^2\)-like. An effective implementation of the WG method is developed through variable reduction by following a Schur-complement approach, yielding a system of linear equations involving unknowns associated with element boundaries only. Numerical results are presented to confirm the theory of convergence.     

Galerkin method as a tool to investigate the planar and non-planar behavior of curved beams

The equations of equilibrium which describe the three-dimensional behavior of curved beams are derived. These equations include first order geometric nonlinear influences. The equations are solved by the Galerkin method. The method is very general and allows general variation of the geometric and structural properties along the beam. Any combination of boundary conditions is possible and the most general distribution of loads along the beam can be treated. Numerical results for a few examples are presented and compared to other theoretical and experimental results. The agreement between the results is generally good. It is shown that Galerkin method is an efficient method which enables one to solve the problem using small number of unknowns compared to other methods which are in use.     

Parallel asynchronous iterations for the solution of a 3D continuous flow electrophoresis problem

This paper deals with the parallel numerical solution of a 3D continuous flow electrophoresis problem governed by Navier-Stokes equations coupled with transport and potential equations. For this problem, using the properties of the discrete operators, the convergence of synchronous and asynchronous parallel Schwarz alternating methods is analysed. Finally, parallel solvers are implemented and the results of simulations are given.     

The structure of pan-addition operator with pre-determined pan-multiplication

Based on the structure of general pan-multiplication operators and the theory of first order linear partial differential equations, we study the structure of pan-addition operators with pre-determined pan-multiplication operators. At first, we describe the structure of pan-addition operators under the ordinary multiplication. Then, with this structure in place, we derive the general representation for the structure of pan-addition operators.     

On smoothness of a vector field-application to optical flow

Two measures of smoothness of a vector field over a domain are suggested, together with associated smoothing operators that satisfy given constrains. The technique is illustrated using operators that smooth the invariants of optical flow curl and divergence, estimated from intensity profiles of moving nondeformable objects. The operators have been applied with satisfactory results to derived flow estimates with high noise levels     

An algorithm to mine general association rules from tabular data

Most methods for mining association rules from tabular data mine simple rules which only use the equality operator “=” in their items. For quantitative attributes, approaches tend to discretize domain values by partitioning them into intervals. Limiting the operator only to “=” results in many interesting frequent patterns that may not be identified. It is obvious that where there is an order between objects, operators such as greater than or less than a given value are as important as the equality operator. This motivates us to extend association rules, from the simple equality operator, to a more general set of operators. We address the problem of mining general association rules in tabular data where rules can have all operators {?, >, ≠, =} in their antecedent part. The proposed algorithm, mining general rules (MGR), is applicable to datasets with discrete-ordered attributes and on quantitative discretized attributes. The proposed algorithm stores candidate general itemsets in a tree structure in such a way that supports of complex itemsets can be recursively computed from supports of simpler itemsets. The algorithm is shown to have benefits in terms of time complexity, memory management and has good potential for parallelization.     

ℋ2-matrices – Multilevel methods for the approximation of integral operators

Multigrid methods are typically used to solve partial differential equations, i.e., they approximate the inverse of the corresponding partial differential operators. At least for elliptic PDEs, this inverse can be expressed in the form of an integral operator by Greens theorem.This implies that multigrid methods approximate certain integral operators, so it is straightforward to look for variants of multigrid methods that can be used to approximate more general integral operators.²-matrices combine a multigrid-like structure with ideas from panel clustering algorithms in order to provide a very efficient method for discretizing and evaluating the integral operators found, e.g., in boundary element applications.     

结合加权局部旋度模式的3维人脸表情识别

目的 3维人脸的表情信息不均匀地分布在五官及脸颊附近,对表情进行充分的描述和合理的权重分配是提升识别效果的重要途径。为提高3维人脸表情识别的准确率,提出了一种基于带权重局部旋度模式的3维人脸表情识别算法。方法 首先,为了提取具有较强表情分辨能力的特征,提出对3维人脸的旋度向量进行编码,获取局部旋度模式作为表情特征;然后,提出将ICNP（interactive closest normal points）算法与最小投影偏差算法结合,前者实现3维人脸子区域的不规则划分,划分得到的11个子区域保留了表情变化下面部五官和肌肉的完整性,后者根据各区域对表情识别的贡献大小为各区域的局部旋度模式特征分配权重;最后,带有权重的局部旋度模式特征被输入到分类器中实现表情识别。结果 基于BU-3DFE 3维人脸表情库对本文提出的局部旋度模式特征进行评估,结果表明其分辨能力较其他表情特征更强;基于BU-3DFE库进行表情识别实验,与其他3维人脸表情识别算法相比,本文算法取得了最高的平均识别率,达到89.67%,同时对易混淆的“悲伤”、“愤怒”和“厌恶”等表情的误判率也较低。结论 局部旋度模式特征对3维人脸的表情有较强的表征能力; ICNP算法与最小投影偏差算法的结合,能够实现区域的有效划分和权重的准确计算,有效提高特征对表情的识别能力。试验结果表明本文算法对3维人脸表情具有较高的识别率,并对易混淆的相似表情仍具有较好的识别效果。     

Monotonie- und Randmaximumsätze bei Diskretisierungen des Dirichletproblems allgemeiner nichtlinearer elliptischer Differentialgleichungen

In this paper we consider finite difference operators which approximate the dirichlet problem of general nonlinear elliptic equations. For such nonlinear operators we give several criteria which insure the validity of an discrete monotonicity and maximum principle. In the case of linear difference operators one of these criteria is known as a necessary and sufficient condition.     

Generalized variational principles with relaxed continuity requirements for certain nonlinear problems,with an application to nonlinear elasticity

We consider two systems of differential equations (statics and kinematics) and two systems of algebraic equations (constitutive relations and loading). The differential operators (including the boundary terms) are connected in such a manner that (1) certain potential properties are guaranteed and (2) the boundary terms are consistent with the differential operators. Besides the external boundaries, internal boundaries are taken into account — a necessary step if a division of the whole spatial region into subregions (finite elements) is made. Assuming further that the constitutive and loading operators are potential, various variational principles are constructed systematically using some concepts of functional analysis. In the most general case one gets a four-field principle. It is important to note that the admissible functions — besides the usual continuity requirements inside the subregions — are subject to no restrictions there. The use of Lagrangian multipliers is entirely dispensable.     

Direct simulation of the infinitesimal dynamics of semi-discrete approximations for convection-diffusion-reaction problems

In this paper a scheme for approximating solutions of convection-diffusion-reaction equations by Markov jump processes is studied. The general principle of the method of lines reduces evolution partial differential equations to semi-discrete approximations consisting of systems of ordinary differential equations. Our approach is to use for this resulting system a stochastic scheme which is essentially a direct simulation of the corresponding infinitesimal dynamics. This implies automatically the time adaptivity and, in one space dimension, stable approximations of diffusion operators on non-uniform grids and the possibility of using moving cells for the transport part, all within the framework of an explicit method. We present several results in one space dimension including free boundary problems, but the general algorithm is simple, flexible and on uniform grids it can be formulated for general evolution partial differential equations in arbitrary space dimensions.     

New oscillation criteria for second order linear difference equations with positive and negative coefficients

The oscillation of second order neutral difference equations with positive and negative coefficients of the form

is investigated. We obtain many new results using the comparison between both first order and second order difference equations. An example is given to show the strength of the obtained results.     

Parallel adaptive solution for two dimensional 3-T energy equation on UG

Two-dimensional (2D) energy equation coupled with three temperatures such as electron, ion and photon is widely used to approximately describe the evolution of radiation energy across multiple materials and to study the exchange of energy among electrons, ions and photons for numerical research on laser-driven implosion of a fuel capsule in inertial confinement fusion experiments. The numerical solution of such equations is always fascinating because of its strongly nonlinear phenomena and strongly discontinuous interfaces. Using the UG framework, this paper successfully solves such equations on 2D unstructured grids with a fully implicit finite volume discretization scheme and parallel adaptive multigrid. Significant numerical results using 32 processors are given and analyzed.     

Iterative methods for solving extended general mixed variational inequalities

In this paper, we introduce and consider a new class of mixed variational inequalities involving four operators, which are called extended general mixed variational inequalities. Using the resolvent operator technique, we establish the equivalence between the extended general mixed variational inequalities and fixed point problems as well as resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving general mixed variational inequalities. We study the convergence criteria for the suggested iterative methods under suitable conditions. Our methods of proof are very simple as compared with other techniques. The results proved in this paper may be viewed as refinements and important generalizations of the previous known results.     

A synthesis of Lyapunov's first and second methods

This paper considers the relation which exists between the first and second methods of Lyapunov. As a result of this study a useful nth order generating technique for Lyapunov functions is developed which applies to linear and nonlinear systems. The function obtained has a negative definite time derivative so that regions of asymptotic stability or instability can be determined. The function is derived from the equations of the system and the curl relations. The final result requires the calculation of single and inverse coefficients using recursive formulae. The basic method for generating Lyapunov functions makes use of the first method of Lyapunov. It is then extended to systems where the first method fails, to variations which have proved useful, and to more general and, hence, more complex forms. Two theorems are proved and examples are given in each section to illustrate the method presented.     

Algebra manipulating programs for structural analogies

Two computer programs are described for solving sets of simultaneous equations with coefficients in the form of algebraic expressions.Both programs retain the algebraic nature of these coefficients throughout, although the methods of storage and solution are entirely different. The problems which the programs were originally designed to solve relate to regular structures, for which modular stiffness matrix equations can be written. The coefficients of these matrices are expressions of simple algebraic quantities and finite difference operators. These operators are transformed into differential operators by the programs, the output giving continuum approximations to the structures. Other applications of the programs are discussed.