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 mu...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.展开更多
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-pr...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.展开更多
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 den...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.展开更多
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 th...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.展开更多
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 indi...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.展开更多
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...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.展开更多
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.
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 ap...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.展开更多
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 uni...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.展开更多
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. Mat...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.展开更多
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 tim...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.展开更多
基金National Natural Science Foundation of China(Grant No.11301462)University Science Research Project of Jiangsu Province(Grant No.13KJB110030)Yangzhou University Overseas Study Program and New Century Talent Project to Shan Jiang。
文摘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.
基金supported in part by NSF Grant DMS-0811272in part by NIH Grant P50GM76516 and R01GM75309supported in part by NSF Grant DMS 0555831, and DMS 0713743
文摘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.
基金supported by the National Natural Science Foundation of China(No.11701103,11801095)Young Top-notch Talent Program of Guangdong Province(No.2017GC010379)+2 种基金Natural Science Foundation of Guangdong Province(No.2022A1515012147,2019A1515010876,2017A030310538)the Project of Science and Technology of Guangzhou(No.201904010341,202102020704)the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University(2021023)。
文摘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.
基金supported by the National Key R&D Program of China (2020YFB1708300)the Project funded by the China Postdoctoral Science Foundation (2021M701310).
文摘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.
基金supported by 863 Project(No.2015AA033702)the National Basic Research Program of China(Nos.2012CB619103,2012CB933901 and 2012CB933902)+1 种基金the National Natural Science Foundation of China(Nos.51271182 and 51271180)the Shandong Provincial Natural Science Foundation,China(No.ZR2014JL031)
文摘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.
文摘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.
基金supported by the Deutsche Forschungsgemeinschaft(DFG),SFB 123."Stochatistische Mathematische Modelle",Universitat Heidelberg
文摘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.
基金The work of Hu Chen is supported in part by NSF of China(No.11801026)and China Postdoctoral Science Foundation Under No.2018M631316the work of Xiaohan Hu and Yifa Tang is supported in part by NSF of China(No.11771438)+3 种基金the work of Jincheng Ren is supported in part by NSF of China(No.11601119)sponsored by Program for HASTIT(No.18HASTIT027)Young talents Fund of HUELthe work of Tao Sun is supported in part by NSF of China(No.11401380).
文摘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.
基金supported by the Natural Science Foundation of China(No.11571027),the International Research Cooperation Seed of Beijing University of Technology(No.2018B32)Science and Technology Projects of Beijing Education Commission Foundatio(No.KM201510005032),and the 16th graduate science and technology fund of Beijing university of technology(No.ykj-2017-00127).
文摘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.
基金supported by an CERG grant of Hong Kong Research Grant Council and by FRG grants of Hong Kong Baptist University
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.11671343,11601460)the Natural Science Foundation of Hunan Province of China(No.2018JJ3491)the Project of Scientific Research Fund of Hunan Provincial Science and Technology Department,China(No.2018WK4006).
文摘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.
