Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes,which can provide a good balance between the numerical accuracy and computational cost.The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions.The multiscale basis functions have abilities to capture originally perturbed information in the local problem,as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes,where the layer-adapted meshes are generated by a given parameter.Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L^(2)norm and first order convergence in the energy norm on graded meshes,which is independent ofε.In contrast with the conventional methods,our method is much more accurate and effective.

SUPERCONVERGENCE OF GRADIENT RECOVERY SCHEMES ON GRADED MESHES FOR CORNER SINGULARITIES

For the linear finite element solution to the Poisson equation, we show that supercon- vergence exists for a type of graded meshes for corner singularities in polygonal domains. In particular, we prove that the L^2-projection from the piecewise constant field △↓UN to the continuous and piecewise linear finite element space gives a better approximation of △↓U in the Hi-norm. In contrast to the existing superconvergence results, we do not assume high regularity of the exact solution.

FINITE ELEMENT APPROXIMATION USING A GRADED MESH ON DOMAINS WITH REENTRANT CORNERS

Finite element approximation of the elliptic operator on a non-convexdomain composed of rectangles is considered using a graded mesh.Some errorestimates and error expansion are presented.

A Compact Difference Scheme on Graded Meshes for the Nonlinear Fractional Integro-differential Equation with Non-smooth Solutions

In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M denote the numbers of grids in temporal and spatial direction,α ∈(0,1)is the fractional order.To recover the full accuracy based on the regularity requirement of the solution,we adopt the L1 method and the trapezoidal product integration(PI)rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral,respectively,further handle the nonlinear term carefully by the Newton linearized method.Based on the discrete fractional Gr¨onwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral,the stability and convergence of the proposed scheme are analyzed by the energy method.Theoretical results are also confirmed by a numerical example.

L1 scheme on graded mesh for the linearized time fractional KdV equation with initial singularity

Numerical approximation for a linearized time fractional KdV equation with initial singularity using L1 scheme on graded mesh is considered.It is proved that the L1 scheme can attain order 2−αconvergence rate with appropriate choice of the grading parameter,whereα(0<α<1)is the order of temporal Caputo fractional derivative.A fully discrete spectral scheme is constructed combing a Petrov-Galerkin spectral method for the spatial discretization,and its stability and convergence are theoretically proved.Some numerical results are provided to verify the theoretical analysis and demonstrated the sharpness of the error analysis.

Explicit Isogeometric Topology Optimization Method with Suitably Graded Truncated Hierarchical B-Spline

This work puts forward an explicit isogeometric topology optimization(ITO)method using moving morphable components(MMC),which takes the suitably graded truncated hierarchical B-Spline based isogeometric analysis as the solver of physical unknown(SGTHB-ITO-MMC).By applying properly basis graded constraints to the hierarchical mesh of truncated hierarchical B-splines(THB),the convergence and robustness of the SGTHB-ITOMMC are simultaneously improved and the tiny holes occurred in optimized structure are eliminated,due to the improved accuracy around the explicit structural boundaries.Moreover,an efficient computational method is developed for the topological description functions(TDF)ofMMC under the admissible hierarchicalmesh,which consists of reducing the dimensionality strategy for design space and the locally computing strategy for hierarchical mesh.We apply the above SGTHB-ITO-MMC with improved efficiency to a series of 2D and 3Dcompliance design problems.The numerical results show that the proposed SGTHB-ITO-MMC method outperforms the traditional THB-ITO-MMCmethod in terms of convergence rate and efficiency.Therefore,the proposed SGTHB-ITO-MMC is an effective way of solving topology optimization(TO)problems.

GRADE/MESH及GRADE/SAP应用简介

本文对GRADE/MESH的基本功能、GRADE/SAP的原始数据生成及其后处理做了扼要介绍。认为应用该软件可进行大桥方案计算、高桩梁板式码头的空间计算及对多种复杂结构的静力、动力进行分析,是一种实用、高效、准确方便的软件。

A Modified Paving Approach and Automatic Mesh Grading Method to Quadrilateral Mesh Generation

A modified paving technique for automatic generation of all-quadrilateral mesh fromarbitrary 2-D geometry is presented. The generated mesh elementS are nearly square andperpendicular to boundaries. Aner the nodes and elementS formation is completed. a fully automaticgrading method is applied to increase the accuracy and reliability of engineering analysis. In thispaper, we mainly describe the theory of mathematical algorithm and present some examples ofautomatically generated mesh.

Longitudinal Compression Behavior of Functionally Graded Ti–6Al–4V Meshes

The compressive deformation behavior in the longitudinal direction of graded Ti–6Al–4V meshes fabricated by electron beam melting was investigated using experiments and finite element methods（FEM）.The results indicate that the overall strain along the longitudinal direction is the sum of the net strain carried by each uniform mesh constituent and the deformation behavior fits the Reuss model well. The layer thickness and the sectional area have no effect on the elastic modulus, whereas the strength increases with the sectional area due to the edge effect of each uniform mesh constituent. By optimizing3 D graded/gradient design, meshes with balanced superior properties, such as high strength, energy absorption and low elastic modulus, can be fabricated by electron beam melting.

ERROR ANALYSIS FOR A FAST NUMERICAL METHOD TO A BOUNDARY INTEGRAL EQUATION OF THE FIRST KIND

For two-dimensional boundary integral equations of the first kind with logarithmic kernels, the use of the conventional boundary element methods gives linear systems with dense matrix. In a recent work [J. Comput. Math., 22 （2004）, pp. 287-298], it is demonstrated that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. The numerical experiments also indicate that the proposed numerical methods require less computational time than the conventional ones while the formal rate of convergence can be preserved. The purpose of this work is to establish a stability and convergence theory for this fast numerical method. The stability analysis depends on a decomposition of the coefficient matrix for the collocation equation. The formal orders of convergence observed in the numerical experiments are proved rigorously.

A fast finite difference/finite element method for the two-dimensional distributed-order time-space fractional reaction-diffusion equation

In this work,we develop a finite difference/finite element method for the two-dimensional distributed-order time-space fractional reaction-diffusion equation(2D-DOTSFRDE)with low regularity solution at the initial time.A fast evaluation of the distributedorder time fractional derivative based on graded time mesh is obtained by substituting the weak singular kernel for the sum-of-exponentials.The stability and convergence of the developed semi-discrete scheme to 2D-DOTSFRDE are discussed.For the spatial approximation,the finite element method is employed.The convergence of the corresponding fully discrete scheme is investigated.Finally,some numerical tests are given to verify the obtained theoretical results and to demonstrate the effectiveness of the method.

Discontinuous Galerkin Methods for Multi-Pantograph Delay Differential Equations

In this paper,the discontinuous Galerkin method is applied to solve the multi-pantograph delay differential equations.We analyze the optimal global convergence and local superconvergence for smooth solutions under uniform meshes.Due to the initial singularity of the forcing term f,solutions of multi-pantograph delay differential equations are singular.We obtain the relevant global convergence and local superconvergence for weakly singular solutions under graded meshes.The numerical examples are provided to illustrate our theoretical results.