The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin （EFG） method which is based on the moving least-square approximation for the tes...The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin （EFG） method which is based on the moving least-square approximation for the test and trial functions. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for hyperbolic problems needs only the scattered nodes instead of meshing the domain of the problem. It neither requires any element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for two-dimensional hyperbolic problems is investigated by two numerical examples in this paper.展开更多
A generalized Fisher equation （GFE） relates the time derivative of the average of the intrinsic rate of growth to its variance. The exact mathematical result of the GFE has been widely used in population dynamics an...A generalized Fisher equation （GFE） relates the time derivative of the average of the intrinsic rate of growth to its variance. The exact mathematical result of the GFE has been widely used in population dynamics and genetics, where it originated. Many researchers have studied the numerical solutions of the GFE, up to now. In this paper, we introduce an element-free Galerkin （EFG） method based on the moving least-square approximation to approximate positive solutions of the GFE from population dynamics. Compared with other numerical methods, the EFG method for the GFE needs only scattered nodes instead of meshing the domain of the problem. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. In comparison with the traditional method, numerical solutions show that the new method has higher accuracy and better convergence. Several numerical examples are presented to demonstrate the effectiveness of the method.展开更多
The present paper deals with the numerical solution of the coupled Schrodinger-KdV equations using the elementfree Galerkin （EFG） method which is based on the moving least-square approximation. Instead of traditiona...The present paper deals with the numerical solution of the coupled Schrodinger-KdV equations using the elementfree Galerkin （EFG） method which is based on the moving least-square approximation. Instead of traditional mesh oriented methods such as the finite difference method （FDM） and the finite element method （FEM）, this method needs only scattered nodes in the domain. For this scheme, a variational method is used to obtain discrete equations and the essential boundary conditions are enforced by the penalty method. In numerical experiments, the results are presented and compared with the findings of the finite element method, the radial basis functions method, and an analytical solution to confirm the good accuracy of the presented scheme.展开更多
The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin （EFG） method, which is based on the moving least-square approximation. Compared w...The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin （EFG） method, which is based on the moving least-square approximation. Compared with numerical methods based on meshes, the EFG method for time-fractional partial differential equations needs only scattered nodes instead of meshing the domain of the problem. It neither requires element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. In this method, the first-order time derivative is replaced by the Caputo fractional derivative of order α（0 〈 α≤ 1）. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Several numerical examples are presented and the results we obtained are in good agreement with the exact solutions.展开更多
We proposed an improved graphics processing unit(GPU)acceleration approach for three-dimensional structural topology optimization using the element-free Galerkin(EFG)method.This method can effectively eliminate the ra...We proposed an improved graphics processing unit(GPU)acceleration approach for three-dimensional structural topology optimization using the element-free Galerkin(EFG)method.This method can effectively eliminate the race condition under parallelization.We established a structural topology optimization model by combining the EFG method and the solid isotropic microstructures with penalization model.We explored the GPU parallel algorithm of assembling stiffness matrix,solving discrete equation,analyzing sensitivity,and updating design variables in detail.We also proposed a node pair-wise method for assembling the stiffnessmatrix and a node-wise method for sensitivity analysis to eliminate race conditions during the parallelization.Furthermore,we investigated the effects of the thread block size,the number of degrees of freedom,and the convergence error of preconditioned conjugate gradient(PCG)on GPU computing performance.Finally,the results of the three numerical examples demonstrated the validity of the proposed approach and showed the significant acceleration of structural topology optimization.To save the cost of optimization calculation,we proposed the appropriate thread block size and the convergence error of the PCG method.展开更多
基金Project supported by the Natural Science Foundation of Ningbo, China (Grant Nos 2009A610014, 2009A610154, 2008A610020 and 2007A610050)
文摘The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin （EFG） method which is based on the moving least-square approximation for the test and trial functions. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for hyperbolic problems needs only the scattered nodes instead of meshing the domain of the problem. It neither requires any element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for two-dimensional hyperbolic problems is investigated by two numerical examples in this paper.
基金supported by the National Natural Science Foundation of China (Grant No. 11072117)the Natural Science Foundation of Ningbo City (Grant Nos. 2012A610038 and 2012A610152)+1 种基金the Scientific Research Fund of Education Department of Zhejiang Province,China (Grant No. Z201119278)K.C. Wong Magna Fund in Ningbo University
文摘A generalized Fisher equation （GFE） relates the time derivative of the average of the intrinsic rate of growth to its variance. The exact mathematical result of the GFE has been widely used in population dynamics and genetics, where it originated. Many researchers have studied the numerical solutions of the GFE, up to now. In this paper, we introduce an element-free Galerkin （EFG） method based on the moving least-square approximation to approximate positive solutions of the GFE from population dynamics. Compared with other numerical methods, the EFG method for the GFE needs only scattered nodes instead of meshing the domain of the problem. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. In comparison with the traditional method, numerical solutions show that the new method has higher accuracy and better convergence. Several numerical examples are presented to demonstrate the effectiveness of the method.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11072117 and 61074142)the Natural Science Foundation of Zhejiang Province,China(Grant No.Y6110007)+3 种基金Scientific Research Fund of Zhejiang Provincial Education Department,China(Grant No.Z201119278)the Natural Science Foundation of Ningbo City(Grant Nos.2012A610152 and 2012A610038)the Disciplinary Project of Ningbo City,China(Grant No.SZXL1067)K.C.Wong Magna Fund in Ningbo University
文摘The present paper deals with the numerical solution of the coupled Schrodinger-KdV equations using the elementfree Galerkin （EFG） method which is based on the moving least-square approximation. Instead of traditional mesh oriented methods such as the finite difference method （FDM） and the finite element method （FEM）, this method needs only scattered nodes in the domain. For this scheme, a variational method is used to obtain discrete equations and the essential boundary conditions are enforced by the penalty method. In numerical experiments, the results are presented and compared with the findings of the finite element method, the radial basis functions method, and an analytical solution to confirm the good accuracy of the presented scheme.
基金Project supported by the National Natural Science Foundation of China(Grant No.11072117)the Natural Science Foundationof Zhejiang Province,China(Grant Nos.Y6110007and Y6110502)the K.C.Wong Magna Fund in Ningbo University,China
文摘The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin （EFG） method, which is based on the moving least-square approximation. Compared with numerical methods based on meshes, the EFG method for time-fractional partial differential equations needs only scattered nodes instead of meshing the domain of the problem. It neither requires element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. In this method, the first-order time derivative is replaced by the Caputo fractional derivative of order α（0 〈 α≤ 1）. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Several numerical examples are presented and the results we obtained are in good agreement with the exact solutions.
基金This work is supported by the National Natural Science Foundation of China(Nos.51875493,51975503,11802261)The financial support to the first author is gratefully acknowledged.
文摘We proposed an improved graphics processing unit(GPU)acceleration approach for three-dimensional structural topology optimization using the element-free Galerkin(EFG)method.This method can effectively eliminate the race condition under parallelization.We established a structural topology optimization model by combining the EFG method and the solid isotropic microstructures with penalization model.We explored the GPU parallel algorithm of assembling stiffness matrix,solving discrete equation,analyzing sensitivity,and updating design variables in detail.We also proposed a node pair-wise method for assembling the stiffnessmatrix and a node-wise method for sensitivity analysis to eliminate race conditions during the parallelization.Furthermore,we investigated the effects of the thread block size,the number of degrees of freedom,and the convergence error of preconditioned conjugate gradient(PCG)on GPU computing performance.Finally,the results of the three numerical examples demonstrated the validity of the proposed approach and showed the significant acceleration of structural topology optimization.To save the cost of optimization calculation,we proposed the appropriate thread block size and the convergence error of the PCG method.
