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.

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.

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.

Fractal Dimension Analysis of the Fine Aggregate Gradation of Interlocking Skeleton Asphalt Mixture

By analyzing the fine aggregate gradation scales from standards,fine aggregate in the asphalt mixture is regarded as a whole research object and fractal dimensions X of the samples were obtained by linear regression and（AC-13）is 2.43-2.56,Sup-13 is 2.28-2.54,SMA-13 is 2.66-2.88 and SAC-13 is 2.54-2.73.In the dense gradation mixture,there are little different between fractal dimensions of coarse and fine aggregates but it makes sense for skeleton asphalt mixture.For a given coarse aggregate gradation and the same percentage of coarse aggregates,the compressive strengths and splitting strengths of the asphalt mixture are studied when the fractal dimensions are selected as 2.60,2.65 and 2.70,respectively.When asphalt-stone ratio is less than optimal asphalt-stone ratio,the higher compressive strength is,the bigger X can be gotten.When asphalt-stone ratio is larger than optimal asphalt-stone ratio,little difference of compressive strength can be observed under these three conditions.The largest splitting strength can be got when X is 2.65,and larger splitting strength can be observed with the ascending of the asphalt-stone ratio.

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.

Focusing microwaves into subwavelength dimensions with a half-cylindrical hyperlens based on split ring resonators

Hyperlenses based on metamaterials can be applied to subwavelength imaging in the lightwave band. In this letter, we demonstrate both through simulations and experimentally verified results that our proposed half- cylindrical shaped hyperlens can be used for super-resolution microwave focusing in a TE mode. Based on split ring resonators, the hyperlens satisfies a hyperbolic dispersion relationship. Simulations demonstrate that the focused spot size and position are insensitive to the rotation angle of the hyperlens around its geometric center. Experimental results show that a focused spot size 1/3 of the vacuum wavelength is achieved in the microwave band.