Many problems in engineering shape design involve eigenvalue optimizations.The relevant difficulty is that the eigenvalues are not continuously differentiable with respect to the density.In this paper,we are intereste...Many problems in engineering shape design involve eigenvalue optimizations.The relevant difficulty is that the eigenvalues are not continuously differentiable with respect to the density.In this paper,we are interested in the case of multi-density inhomogeneous materials which minimizes the least eigenvalue.With the finite element discretization,we propose a monotonically decreasing algorithm to solve the minimization problem.Some numerical examples are provided to illustrate the efficiency of the present algorithm as well as to demonstrate its availability for the case of more than two densities.As the computations are sensitive to the choice of the discretization mesh sizes,we adopt the refined mesh strategy,whose mesh grids are 25-times of the amount used in[S.Osher and F.Santosa,J.Comput.Phys.,171(2001),pp.272-288].We also show the significant reduction in computational cost with the fast convergence of this algorithm.展开更多
Some block iterative methods for solving variational inequalities with nonlinear operators are proposed. Monotone convergence of the algorithms is obtained. Some comparison theorems are also established. Compared with...Some block iterative methods for solving variational inequalities with nonlinear operators are proposed. Monotone convergence of the algorithms is obtained. Some comparison theorems are also established. Compared with the research work in given by Pao in 1995 for nonlinear equations and research work in given by Zeng and Zhou in 2002 for elliptic variational inequalities, the algorithms proposed in this paper are independent of the boundedness of the derivatives of the nonlinear operator.展开更多
This paper presents and analyzes a monotone domain decomposition algorithm for solving nonlinear singularly perturbed reaction-diffusion problems of parabolic type. To solve the nonlinear weighted average finite diffe...This paper presents and analyzes a monotone domain decomposition algorithm for solving nonlinear singularly perturbed reaction-diffusion problems of parabolic type. To solve the nonlinear weighted average finite difference scheme for the partial differential equation, we construct a monotone domain decomposition algorithm based on a Schwarz alternating method and a box-domain decomposition. This algorithm needs only to solve linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear discrete problem. domain decomposition algorithm is estimated The rate of convergence of the monotone Numerical experiments are presented.展开更多
基金supported by the Chinese National Science Foundation(No.10871179)the National Basic Research Programme of China(No.2008CB717806)Specialized Research Fund for the Doctoral Program of Higher Education of China(SRFDP No.20070335201).
文摘Many problems in engineering shape design involve eigenvalue optimizations.The relevant difficulty is that the eigenvalues are not continuously differentiable with respect to the density.In this paper,we are interested in the case of multi-density inhomogeneous materials which minimizes the least eigenvalue.With the finite element discretization,we propose a monotonically decreasing algorithm to solve the minimization problem.Some numerical examples are provided to illustrate the efficiency of the present algorithm as well as to demonstrate its availability for the case of more than two densities.As the computations are sensitive to the choice of the discretization mesh sizes,we adopt the refined mesh strategy,whose mesh grids are 25-times of the amount used in[S.Osher and F.Santosa,J.Comput.Phys.,171(2001),pp.272-288].We also show the significant reduction in computational cost with the fast convergence of this algorithm.
基金the National Natural Science Foundation of China(No.10671060)the Doctoral Fund of Ministry of Education of China granted[2003]0532006
文摘Some block iterative methods for solving variational inequalities with nonlinear operators are proposed. Monotone convergence of the algorithms is obtained. Some comparison theorems are also established. Compared with the research work in given by Pao in 1995 for nonlinear equations and research work in given by Zeng and Zhou in 2002 for elliptic variational inequalities, the algorithms proposed in this paper are independent of the boundedness of the derivatives of the nonlinear operator.
文摘This paper presents and analyzes a monotone domain decomposition algorithm for solving nonlinear singularly perturbed reaction-diffusion problems of parabolic type. To solve the nonlinear weighted average finite difference scheme for the partial differential equation, we construct a monotone domain decomposition algorithm based on a Schwarz alternating method and a box-domain decomposition. This algorithm needs only to solve linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear discrete problem. domain decomposition algorithm is estimated The rate of convergence of the monotone Numerical experiments are presented.
