A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.

A FINITE DIMENSIONAL METHOD FOR SOLVINGNONLINEAR ILL-POSED PROBLEMS

We propose a finite dimensional method to compute the solution of nonlinear ill-posed problems approximately and show that under certain conditions, the convergence can be guaranteed. Moreover, we obtain the rate of convergence of our method provided that the true solution satisfies suitable smoothness condition. Finally, we present two examples from the parameter estimation problems of differential equations and illustrate the applicability of our method.

The dimension split element-free Galerkin method for three-dimensional potential problems

This paper presents the dimension split element-free Galerkin （DSEFG） method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares （IMLS） approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method.

Parametric study on single shot peening by dimensional analysis method incorporated with finite element method

Shot peening is a widely used surface treatment method by generating compressive residual stress near the surface of metallic materials to increase fatigue life and re- sistance to corrosion fatigue, cracking, etc. Compressive re- sidual stress and dent profile are important factors to eval- uate the effectiveness of shot peening process. In this pa- per, the influence of dimensionless parameters on maximum compressive residual stress and maximum depth of the dent were investigated. Firstly, dimensionless relations of pro- cessing parameters that affect the maximum compressive residual stress and the maximum depth of the dent were de- duced by dimensional analysis method. Secondly, the in- fluence of each dimensionless parameter on dimensionless variables was investigated by the finite element method. Fur- thermore, related empirical formulas were given for each di- mensionless parameter based on the simulation results. Fi- nally, comparison was made and good agreement was found between the simulation results and the empirical formula, which shows that a useful approach is provided in this pa- per for analyzing the influence of each individual parameter.

Three Dimensional Simulation of Upset Forging by Using Variational Upper Bound Method

The variation principle is discussed and Rayleigh-Ritz method is proposed for construction of veloci ty field. A kinematically admissible velocity field based on polynomials was appIied to the determina tion of forging load and deformed buIge profile during upset forging of blocks. Simulation of upsetforging of rectangular blocks under various friction condjtions was performed. Comparison of the computed results with experiments and FEM shows good agreement. It is shown that this techniquecan be used for 3D simulation of metal forming process.

Iterative Dichotomiser Posteriori Method Based Service Attack Detection in Cloud Computing

Cloud computing(CC)is an advanced technology that provides access to predictive resources and data sharing.The cloud environment represents the right type regarding cloud usage model ownership,size,and rights to access.It introduces the scope and nature of cloud computing.In recent times,all processes are fed into the system for which consumer data and cache size are required.One of the most security issues in the cloud environment is Distributed Denial of Ser-vice(DDoS)attacks,responsible for cloud server overloading.This proposed sys-tem ID3(Iterative Dichotomiser 3)Maximum Multifactor Dimensionality Posteriori Method(ID3-MMDP)is used to overcome the drawback and a rela-tively simple way to execute and for the detection of(DDoS)attack.First,the pro-posed ID3-MMDP method calls for the resources of the cloud platform and then implements the attack detection technology based on information entropy to detect DDoS attacks.Since because the entropy value can show the discrete or aggregated characteristics of the current data set,it can be used for the detection of abnormal dataflow,User-uploaded data,ID3-MMDP system checks and read risk measurement and processing,bug ratingfile size changes,orfile name changes and changes in the format design of the data size entropy value.Unique properties can be used whenever the program approaches any data error to detect abnormal data services.Finally,the experiment also verifies the DDoS attack detection capability algorithm.

A Fast Element-Free Galerkin Method for 3D Elasticity Problems

In this paper,a fast element-free Galerkin(FEFG)method for three-dimensional(3D)elasticity problems is established.The FEFG method is a combination of the improved element-free Galerkin(IEFG)method and the dimension splitting method(DSM).By using the DSM,a 3D problem is converted to a series of 2D ones,and the IEFG method with a weighted orthogonal function as the basis function and the cubic spline function as the weight function is applied to simulate these 2D problems.The essential boundary conditions are treated by the penalty method.The splitting direction uses the finite difference method(FDM),which can combine these 2D problems into a discrete system.Finally,the system equation of the 3D elasticity problem is obtained.Some specific numerical problems are provided to illustrate the effectiveness and advantages of the FEFG method for 3D elasticity by comparing the results of the FEFG method with those of the IEFG method.The convergence and relative error norm of the FEFG method for elasticity are also studied.

Dimension Splitting Method for the Three Dimensional Rotating Navier-Stokes Equations

In this paper, we propose a dimensional splitting method for the three dimensional （3D） rotating Navier-Stokes equations. Assume that the domain is a channel bounded by two surfaces and is decomposed by a series of surfaces i into several sub-domains, which are called the layers of the flow. Every interface i between two sub-domains shares the same geometry. After establishing a semi-geodesic coordinate （S-coordinate） system based on i, Navier-Stoke equations in this coordinate can be expressed as the sum of two operators, of which one is called the membrane operator defined on the tangent space on i, another one is called the bending operator taking value in the normal space on i. Then the derivatives of velocity with respect to the normal direction of the surface are approximated by the Euler central difference, and an approximate form of Navier-Stokes equations on the surface i is obtained, which is called the two-dimensional three-component （2D-3C） Navier-Stokes equations on a two dimensional manifold. Solving these equations by alternate iteration, an approximate solution to the original 3D Navier-Stokes equations is obtained. In addition, the proof of the existence of solutions to 2D-3C Navier-Stokes equations is provided, and some approximate methods for solving 2D-3C Navier-Stot4es equations are presented.

Settlement Analysis for An Embankment Considering Visco-Elasto-Plastic Characteristics

In the paper. a visco-elasto plastic constitutive model and a method for determining model parameters for soft clay are presented. In this model total strain of soft clay is assumed to be divided into three parts: instantaneous elastic, visco-elastic and visco-plastic. The characteristics of instantaneous and visco-elastic deformation of soft clay are simulated by Merchant's model, the plastic is by a model with two yield surfaces. And related constitutive equation is conducted. A number of stress-controlled triaxial tests are performed to calculated the model parameters. The visco-elasto-plastic model is used for analysis of the displacement of an embankment on soft ground by use of three-dimensional finite element method. The predicted settlements agree well with the measured data. It is indicated that the viscous characteristics should be taken into account in deformation analysis for soft clay.

THE PERTURBATION PROBLEM OF AN ELLIPTIC SYSTEM WITH SOBOLEV CRITICAL GROWTH

In this paper,we study the following perturbation problem with Sobolev critical exponen t:{-Δu=(1+εK(x))u^2*-1+α/2*u^a-1v^3+εh(x)u^pP,x∈R^N,-Δu=(1+εQ(x))v^2*-1+β/2*u^B-1+εl(x)u^q,x∈R^N,u>0,v>0,x∈R^N where 0<p,q<1,α+β=2*:=2N/N-2,α,β≥3,4.Using a perturbation argument and a finite dimensional reduc tion met hod,we get the exis tence of positive solutions to problem(0.1)and the asymptotic property of the solutions.

ON THE FINITE DIFFERENCE METHODS FOR TWO DIMENSIONAL TURBULENCE BOUNDARY LAYER

In this paper,the author first establishes the general finite difference formula for the governing equations of the turbulent average velocities in a steady two dimensional incompressible fluid boundary layer-inner layer.Next, three key parameters of the difference scheme are determined respectively by several simple flow models with known analytical solutions.Finally a special five points difference system is given and its application value is showed by a numerical example for the vertical velocity distribution in an Ekman's layer.

EFFICIENT NUMERICAL ALGORITHMS FOR THREE-DIMENSIONAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional dif- fusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.

Approximate contact solutions for non-axisymmetric homogeneous and power-law graded elastic bodies:A practical tool for design engineers and tribologists

In two recent papers,approximate solutions for compact non-axisymmetric contact problems of homogeneous and power-law graded elastic bodies have been suggested,which provide explicit analytical relations for the force–approach relation,the size and the shape of the contact area,as well as for the pressure distribution therein.These solutions were derived for profiles,which only slightly deviate from the axisymmetric shape.In the present paper,they undergo an extensive testing and validation by comparison of solutions with a great variety of profile shapes with numerical solutions obtained by the fast Fourier transform(FFT)-assisted boundary element method(BEM).Examples are given with quite significant deviations from axial symmetry and show surprisingly good agreement with numerical solutions.

Evaluation of dimension of fractal time series with the least square method

Properties of fractional Brownian motions(f Bms)have been investigated by researchers in different fields,e.g.statistics,hydrology,biology,finance,and public transportation,which has helped us better understand many complex time series observed in nature[1-4].The Hurst exponent H(0<H<1)is the most important parameter characterizing any given time series F(t),where t represents the time steps,and the

Comparison of dimension reduction methods for DEA under big data via Monte Carlo simulation

Data with large dimensions will bring various problems to the application of data envelopment analysis(DEA).In this study,we focus on a“big data”problem related to the considerably large dimensions of the input-output data.The four most widely used approaches to guide dimension reduction in DEA are compared via Monte Carlo simulation,including principal component analysis(PCA-DEA),which is based on the idea of aggregating input and output,efficiency contribution measurement(ECM),average efficiency measure(AEC),and regression-based detection(RB),which is based on the idea of variable selection.We compare the performance of these methods under different scenarios and a brand-new comparison benchmark for the simulation test.In addition,we discuss the effect of initial variable selection in RB for the first time.Based on the results,we offer guidelines that are more reliable on how to choose an appropriate method.

A Finite Element Method Solver for Time-Dependent and Stationary Schrodinger Equations with a Generic Potential

A general finite element solution of the Schrodinger equation for a onedimensional problem is presented.The solver is applicable to both stationary and time-dependent cases with a general user-selected potential term.Furthermore,it is possible to include external magnetic or electric fields,as well as spin-orbital and spinmagnetic interactions.We use analytically soluble problems to validate the solver.The predicted numerical auto-states are compared with the analytical ones,and selected mean values are used to validate the auto-functions.In order to analyze the performance of the time-dependent Schrodinger equation,a traveling wave package benchmark was reproduced.In addition,a problem involving the scattering of a wave packet over a double potential barrier shows the performance of the solver in cases of transmission and reflection of packages.Other general problems,related to periodic potentials,are treated with the same general solver and a Lagrange multiplier method to introduce periodic boundary conditions.Some simple cases of known periodic potential solutions are reported.

Motion Planning for Robots with Topological Dimension Reduction Method

This paper explores the realization of robotic motion planning, especially Findpath problem, which is a basic motion planning problem that arises in the development of robotics. Findpath means: Give the initial and desired final configurations of a robotic arm in 3-dimensionnl space, and give descriptions of the obstacles in the space, determine whether there is a continuous collision-free motion of the robotic arm from one configure- tion to the other and find such a motion if it exists. There are several branches of approach in motion planning area, but in reality the important things are feasibility, efficiency and accuracy of the method. In this paper ac- cording to the concepts of Configuration Space (C-Space) and Rotation Mapping Graph (RMG) discussed in [1], a topological method named Dimension Reduction Method (DRM) for investigating the connectivity of the RMG (or the topologic structure of the RMG )is presented by using topologic technique. Based on this ap- proach the Findpath problem is thus transformed to that of finding a connected way in a finite Characteristic Network (CN). The method has shown great potentiality in practice. Here a simulation system is designed to embody DRM and it is in sight that DRM can he adopted in the first overall planning of real robot sys- tem in the near future.

A Topological Implementation for Motion Planning of a Robotic Arm

An efficient path planning algorithm based on topologic method is presented in this paper.The colli- sion free path planning for three-joint robotic arm consists of three parts:partition of C-space,construc- tion of CN and search for a path in CN.We mainly solved the problems of partitioning the C-space and judging the connectivity between connected blocks,etc.For the motion planning of a robotic arm with a gripper,we developed the concepts of global planning and local planning,and discussed the basic fac- tors for constructing the planning system.In the paper,some evaluation and analysis of the complexity and reliability of the algorithm are given,together with some ideas to improve the efficiency and increase the reliability.At last,some experimental results are presented to show the efficiency and accuracy of the nigorithm.

Fractal dimension study of polaron effects in cylindrical GaAs/AlxGa1-xAs core-shell nanowires

Polaron effects in cylindrical GaAs/AlxGa1-xAs core-shell nanowires are studied by applying the fractal dimension method. In this paper, the polaron properties of GaAs/AlxGa1-xAs core-shell nanowires with different core radii and aluminum concentrations are discussed. The polaron binding energy, polaron mass shift, and fractal dimension parameter are numerically determined as functions of shell width. The calculation results reveal that the binding energy and mass shift of the polaron first increase and then decrease as the shell width increases. A maximum value appears at a certain shell width for different aluminum concentrations and a given core radius. By using the fractal dimension method, polaron problems in cylindrical GaAs/AlxGa1-xAs core-shell nanowires are solved in a simple manner that avoids complex and lengthy calculations.

Catenoidal Layers for the Allen-Cahn Equation in Bounded Domains

This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α~2Δu + u(1- u^2) = 0 in a smooth bounded domain Ω R^3, with Neumann boundary condition and α > 0 a small parameter. These solutions have the property that as α→ 0, their level sets collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature intersecting ?Ω orthogonally and that is non-degenerate respect to ?Ω. The authors provide explicit examples of surfaces to which the result applies.