The preconditioning method is used to solve the low Mach number flow. The space discritisation scheme is the Roe scheme and the DES turbulence model is used. Then, the low Mach number turbulence flow around the NACA00...The preconditioning method is used to solve the low Mach number flow. The space discritisation scheme is the Roe scheme and the DES turbulence model is used. Then, the low Mach number turbulence flow around the NACA0012 airfoil is used to verify the efficiency of the proposed method. Two cases of the low Mach number flows around the multi-element airfoil and the circular cylinder are also used to test the proposed method. Numerical results show that the methods combined the preconditioning method and compressible Navier-Stokes equations are efficient to solve low Mach number flows.展开更多
To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and invest...To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and investigated. The time derivatives of three-dimensional Navier-Stokes equations are preconditioned by Choi-Merkle preconditioning matrix that is originally designed for two-dimensional low Mach number viscous flows. An extension to three-dimensional viscous flow is implemented, and a method improving the convergence for transonic flow is proposed. The space discretizaition is performed by employing a finite-volume cell-centered scheme and using a central difference. The time marching is based on an explicit Rtmge-Kutta scheme proposed by Jameson. An efficient FAS multigrid method is used to accelerate the convergence to steady-state solutions. Viscous flows over ONERA M6 wing and M100 wing are numerically simulated with Mach numbers ranging from 0.010 to 0.839. The inviscid flow over the DLR-F4 wing-body configuration is also calculated to preliminarily examine the performance of the presented method for complex configuration. The computed results are compared with the experimental data and good agreement is achieved. It is shown that the presented method is efficient and robust for both compressible and incompressible flows and is very attractive for aerodynamic optimization designs of wing and complex configuration.展开更多
The preconditioned Gauss-Seidel type iterative method for solving linear systems, with the proper choice of the preconditioner, is presented. Convergence of the preconditioned method applied to Z-matrices is discussed...The preconditioned Gauss-Seidel type iterative method for solving linear systems, with the proper choice of the preconditioner, is presented. Convergence of the preconditioned method applied to Z-matrices is discussed. Also the optimal parameter is presented. Numerical results show that the proper choice of the preconditioner can lead to effective by the preconditioned Gauss-Seidel type iterative methods for solving linear systems.展开更多
Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image...Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.展开更多
Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering （CAE） and computer aided manufacturing ...Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering （CAE） and computer aided manufacturing （CAM）. This paper presents a high-efficiency improved symmetric successive over-relaxation （ISSOR） preconditioned conjugate gradient （PCG） method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.展开更多
The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value...The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value methods.In this paper,we propose a new circulant preconditioner to speed up the convergence rate of the GMRES method, which is a convex linear combination of P-circulant and Strang-type circulant preconditioners. Theoretical and practical arguments are given to show that this preconditioner is feasible and effective in some cases.展开更多
Several preconditioners are proposed for improving the convergence rate of the iterative method derived from splitting. In this paper, the comparison theorem of preconditioned iterative method for regular splitting is...Several preconditioners are proposed for improving the convergence rate of the iterative method derived from splitting. In this paper, the comparison theorem of preconditioned iterative method for regular splitting is proved. And the convergence and comparison theorem for any preconditioner are indicated. This comparison theorem indicates the possibility of finding new preconditioner and splitting. The purpose of this paper is to show that the preconditioned iterative method yields a new splitting satisfying the regular or weak regular splitting. And new combination preconditioners are proposed. In order to denote the validity of the comparison theorem, some numerical examples are shown.展开更多
A class of preconditioned iterative methods, i.e., preconditioned generalized accelerated overrelaxation （GAOR） methods, is proposed to solve linear systems based on a class of weighted linear least squares problems...A class of preconditioned iterative methods, i.e., preconditioned generalized accelerated overrelaxation （GAOR） methods, is proposed to solve linear systems based on a class of weighted linear least squares problems. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned iterative methods is better than that of the original methods. Furthermore, the effectiveness of the proposed methods is shown in the numerical experiment.展开更多
Let the linear system Ax=b where the coefficient matrix A=(a<sub>ij</sub>)∈R<sup>m,n</sup> is an L-ma-trix(that is,a<sub>ij</sub>】0 （?） i and a<sub>ij</sub>≤0 （?...Let the linear system Ax=b where the coefficient matrix A=(a<sub>ij</sub>)∈R<sup>m,n</sup> is an L-ma-trix(that is,a<sub>ij</sub>】0 （?） i and a<sub>ij</sub>≤0 （?） i≠j),A=I-L-U,I is the identity matrix,-L and-U are,respectively,strictly lower and strictly upper triangular parts of A.In[1]theauthors considered two preconditioned linear systems?x=（?） and ?x=（?）展开更多
In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method (PCCG). The theorems discuss respectively the qualitative property of the ite...In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method (PCCG). The theorems discuss respectively the qualitative property of the iterative solution and the construction principle of the iterative matrix. The authors put forward a new incompletely LU factorizing technique for non-M-matrix and the method of constructing the iterative matrix. This improved PCCG is used to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems, and simultaneously contrasted with other methods. The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations, ft is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.展开更多
Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential ...Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.展开更多
Bai, Golub and Pan presented a preconditioned Hermitian and skew-Hermitian splitting(PHSS) method [Numerische Mathematik, 2004, 32: 1-32] for non-Hermitian positive semidefinite linear systems. We improve the method t...Bai, Golub and Pan presented a preconditioned Hermitian and skew-Hermitian splitting(PHSS) method [Numerische Mathematik, 2004, 32: 1-32] for non-Hermitian positive semidefinite linear systems. We improve the method to solve saddle point systems whose(1,1) block is a symmetric positive definite M-matrix with a new choice of the preconditioner and compare it with other preconditioners. The results show that the new preconditioner outperforms the previous ones.展开更多
In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the...In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.展开更多
Explicit Exact and Approximate Inverse Preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is ...Explicit Exact and Approximate Inverse Preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is discussed. The second order iterative methods behave quite similar to first order methods and the development of efficient preconditioners for solving the original linear system is a decisive factor for making the second order iterative methods superior to the first order iterative methods. Adaptive preconditioned Conjugate Gradient methods using explicit approximate preconditioners for solving efficiently large sparse systems of algebraic equations are also presented. The generalized Approximate Inverse Matrix techniques can be efficiently used in conjunction with explicit iterative schemes leading to effective composite semi-direct solution methods for solving large linear systems of algebraic equations.展开更多
The preconditioned methods for solving linear system are discussed. The convergence rate of accelerated overrelaxation (AOR) method can be enlarged by using the preconditioned method when the classical AOR method conv...The preconditioned methods for solving linear system are discussed. The convergence rate of accelerated overrelaxation (AOR) method can be enlarged by using the preconditioned method when the classical AOR method converges, and the preconditioned method is invalid when the classical iterative method does not converge. The results in corresponding references are improved and perfected.展开更多
This paper proposes a new accelerating technique for simulating low speed flows,termed as p Seudo High Speed method(SHS),which uses governing equations and numerical methods of compressible flows.SHS method has advant...This paper proposes a new accelerating technique for simulating low speed flows,termed as p Seudo High Speed method(SHS),which uses governing equations and numerical methods of compressible flows.SHS method has advantages of simple formula,easy manipulation,and only need to modify flux of Euler equations.It can directly employ the existing well-developed schemes of hyperbolic conservation laws.To verify the technique,several numerical experiments are performed,such as:flow past airfoils and flow past a cylinder.Analysis of SHS method and comparisons with some precondition methods are made numerically.All tests show that SHS method can greatly improve the efficiency of compressible method simulating low speed flow fields,which exhibits in accelerating the convergence rate and increasing the accuracy of the numerical results.展开更多
A vorticity-velocity method was used to study the incompressible viscous fluid flow around a circular cylinder with surface suction or blowing. The resulted high order implicit difference equations were effeciently so...A vorticity-velocity method was used to study the incompressible viscous fluid flow around a circular cylinder with surface suction or blowing. The resulted high order implicit difference equations were effeciently solved by the modified incomplete LU decomposition conjugate gradient scheme ( MILU-CG). The effects of surface suction or blowing' s position and strength on the vortex structures in the cylinder wake, as well as on the drag and lift forces at Reynoldes number Re = 100 were investigated numerically. The results show that the suction on the shoulder of the cylinder or the blowing on the rear of the cylinder can effeciently suppress the asymmetry of the vortex wake in the transverse direction and greatly reduce the lift force; the suction on the shoulder of the cylinder, when its strength is properly chosen, can reduce the drag force significantly, too.展开更多
We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections.We provide upper bounds and in a ...We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections.We provide upper bounds and in a special case,a lower bound for preconditioners defined via the method of successive subspace corrections.展开更多
A hybrid finite difference method and vortex method (HDV), which is based on domain decomposition and proposed by the authors (1992), is improved by using a modified incomplete LU decomposition conjugate gradient meth...A hybrid finite difference method and vortex method (HDV), which is based on domain decomposition and proposed by the authors (1992), is improved by using a modified incomplete LU decomposition conjugate gradient method (MILU-CG), and a high order implicit difference algorithm. The flow around a rotating circular cylinder at Reynolds number R-e = 1000, 200 and the angular to rectilinear speed ratio alpha is an element of (0.5, 3.25) is studied numerically. The long-time full developed features about the variations of the vortex patterns in the wake, and drag, lift forces on the cylinder are given. The calculated streamline contours agreed well with the experimental visualized flow pictures. The existence of critical states and the vortex patterns at the states are given for the first time. The maximum lift to drag force ratio can be obtained nearby the critical states.展开更多
An inexact Halley＇s method-Halley-PCG（preconditioned conjugate gradient） method is proposed for solving the systems of linear equations for improved Halley method either by Cholesky factorization exactly or by prec...An inexact Halley＇s method-Halley-PCG（preconditioned conjugate gradient） method is proposed for solving the systems of linear equations for improved Halley method either by Cholesky factorization exactly or by preconditioned conjugate gradient method approximately. The convergence result is given and the efficiency of the method compared to the improved Halley＇s method is shown.展开更多
文摘The preconditioning method is used to solve the low Mach number flow. The space discritisation scheme is the Roe scheme and the DES turbulence model is used. Then, the low Mach number turbulence flow around the NACA0012 airfoil is used to verify the efficiency of the proposed method. Two cases of the low Mach number flows around the multi-element airfoil and the circular cylinder are also used to test the proposed method. Numerical results show that the methods combined the preconditioning method and compressible Navier-Stokes equations are efficient to solve low Mach number flows.
文摘To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and investigated. The time derivatives of three-dimensional Navier-Stokes equations are preconditioned by Choi-Merkle preconditioning matrix that is originally designed for two-dimensional low Mach number viscous flows. An extension to three-dimensional viscous flow is implemented, and a method improving the convergence for transonic flow is proposed. The space discretizaition is performed by employing a finite-volume cell-centered scheme and using a central difference. The time marching is based on an explicit Rtmge-Kutta scheme proposed by Jameson. An efficient FAS multigrid method is used to accelerate the convergence to steady-state solutions. Viscous flows over ONERA M6 wing and M100 wing are numerically simulated with Mach numbers ranging from 0.010 to 0.839. The inviscid flow over the DLR-F4 wing-body configuration is also calculated to preliminarily examine the performance of the presented method for complex configuration. The computed results are compared with the experimental data and good agreement is achieved. It is shown that the presented method is efficient and robust for both compressible and incompressible flows and is very attractive for aerodynamic optimization designs of wing and complex configuration.
基金Project supported by MOE's 2004 New Century Excellent Talent Program (NCET)the Applied Basic Research Foundations of Sichuan Province (No.05JY029-068-2)
文摘The preconditioned Gauss-Seidel type iterative method for solving linear systems, with the proper choice of the preconditioner, is presented. Convergence of the preconditioned method applied to Z-matrices is discussed. Also the optimal parameter is presented. Numerical results show that the proper choice of the preconditioner can lead to effective by the preconditioned Gauss-Seidel type iterative methods for solving linear systems.
基金supported by the National Basic Research Program (No.2005CB321702)the National Outstanding Young Scientist Foundation(No. 10525102)the Specialized Research Grant for High Educational Doctoral Program(Nos. 20090211120011 and LZULL200909),Hong Kong RGC grants and HKBU FRGs
文摘Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.
基金Project supported by the National Natural Science Foundation of China(Nos.5130926141030747+3 种基金41102181and 51121005)the National Basic Research Program of China(973 Program)(No.2011CB013503)the Young Teachers’ Initial Funding Scheme of Sun Yat-sen University(No.39000-1188140)
文摘Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering （CAE） and computer aided manufacturing （CAM）. This paper presents a high-efficiency improved symmetric successive over-relaxation （ISSOR） preconditioned conjugate gradient （PCG） method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.
基金Supported by the Scientific Research Foundation for Advisor Program of Higher Education of Gansu Province(1009-6)Supported by the Scientific Research Foundation for Youth Scholars of Hexi University(qn201015)
文摘The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value methods.In this paper,we propose a new circulant preconditioner to speed up the convergence rate of the GMRES method, which is a convex linear combination of P-circulant and Strang-type circulant preconditioners. Theoretical and practical arguments are given to show that this preconditioner is feasible and effective in some cases.
文摘Several preconditioners are proposed for improving the convergence rate of the iterative method derived from splitting. In this paper, the comparison theorem of preconditioned iterative method for regular splitting is proved. And the convergence and comparison theorem for any preconditioner are indicated. This comparison theorem indicates the possibility of finding new preconditioner and splitting. The purpose of this paper is to show that the preconditioned iterative method yields a new splitting satisfying the regular or weak regular splitting. And new combination preconditioners are proposed. In order to denote the validity of the comparison theorem, some numerical examples are shown.
基金supported by the National Natural Science Foundation of China (No. 11071033)the Fundamental Research Funds for the Central Universities (No. 090405013)
文摘A class of preconditioned iterative methods, i.e., preconditioned generalized accelerated overrelaxation （GAOR） methods, is proposed to solve linear systems based on a class of weighted linear least squares problems. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned iterative methods is better than that of the original methods. Furthermore, the effectiveness of the proposed methods is shown in the numerical experiment.
文摘Let the linear system Ax=b where the coefficient matrix A=(a<sub>ij</sub>)∈R<sup>m,n</sup> is an L-ma-trix(that is,a<sub>ij</sub>】0 （?） i and a<sub>ij</sub>≤0 （?） i≠j),A=I-L-U,I is the identity matrix,-L and-U are,respectively,strictly lower and strictly upper triangular parts of A.In[1]theauthors considered two preconditioned linear systems?x=（?） and ?x=（?）
文摘In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method (PCCG). The theorems discuss respectively the qualitative property of the iterative solution and the construction principle of the iterative matrix. The authors put forward a new incompletely LU factorizing technique for non-M-matrix and the method of constructing the iterative matrix. This improved PCCG is used to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems, and simultaneously contrasted with other methods. The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations, ft is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.
文摘Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.
基金Supported by the National Natural Science Foundation of China(11301330)Supported by the Shanghai College Teachers Visiting Abroad for Advanced Study Program(B.60-A101-12-010)Supported by the First-class Discipline of Universities in Shanghai
文摘Bai, Golub and Pan presented a preconditioned Hermitian and skew-Hermitian splitting(PHSS) method [Numerische Mathematik, 2004, 32: 1-32] for non-Hermitian positive semidefinite linear systems. We improve the method to solve saddle point systems whose(1,1) block is a symmetric positive definite M-matrix with a new choice of the preconditioner and compare it with other preconditioners. The results show that the new preconditioner outperforms the previous ones.
文摘In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.
文摘Explicit Exact and Approximate Inverse Preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is discussed. The second order iterative methods behave quite similar to first order methods and the development of efficient preconditioners for solving the original linear system is a decisive factor for making the second order iterative methods superior to the first order iterative methods. Adaptive preconditioned Conjugate Gradient methods using explicit approximate preconditioners for solving efficiently large sparse systems of algebraic equations are also presented. The generalized Approximate Inverse Matrix techniques can be efficiently used in conjunction with explicit iterative schemes leading to effective composite semi-direct solution methods for solving large linear systems of algebraic equations.
文摘The preconditioned methods for solving linear system are discussed. The convergence rate of accelerated overrelaxation (AOR) method can be enlarged by using the preconditioned method when the classical AOR method converges, and the preconditioned method is invalid when the classical iterative method does not converge. The results in corresponding references are improved and perfected.
文摘This paper proposes a new accelerating technique for simulating low speed flows,termed as p Seudo High Speed method(SHS),which uses governing equations and numerical methods of compressible flows.SHS method has advantages of simple formula,easy manipulation,and only need to modify flux of Euler equations.It can directly employ the existing well-developed schemes of hyperbolic conservation laws.To verify the technique,several numerical experiments are performed,such as:flow past airfoils and flow past a cylinder.Analysis of SHS method and comparisons with some precondition methods are made numerically.All tests show that SHS method can greatly improve the efficiency of compressible method simulating low speed flow fields,which exhibits in accelerating the convergence rate and increasing the accuracy of the numerical results.
基金Foundation item:the Natural Science Foundation of Jiangsu Province(BK97056109)
文摘A vorticity-velocity method was used to study the incompressible viscous fluid flow around a circular cylinder with surface suction or blowing. The resulted high order implicit difference equations were effeciently solved by the modified incomplete LU decomposition conjugate gradient scheme ( MILU-CG). The effects of surface suction or blowing' s position and strength on the vortex structures in the cylinder wake, as well as on the drag and lift forces at Reynoldes number Re = 100 were investigated numerically. The results show that the suction on the shoulder of the cylinder or the blowing on the rear of the cylinder can effeciently suppress the asymmetry of the vortex wake in the transverse direction and greatly reduce the lift force; the suction on the shoulder of the cylinder, when its strength is properly chosen, can reduce the drag force significantly, too.
文摘We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections.We provide upper bounds and in a special case,a lower bound for preconditioners defined via the method of successive subspace corrections.
文摘A hybrid finite difference method and vortex method (HDV), which is based on domain decomposition and proposed by the authors (1992), is improved by using a modified incomplete LU decomposition conjugate gradient method (MILU-CG), and a high order implicit difference algorithm. The flow around a rotating circular cylinder at Reynolds number R-e = 1000, 200 and the angular to rectilinear speed ratio alpha is an element of (0.5, 3.25) is studied numerically. The long-time full developed features about the variations of the vortex patterns in the wake, and drag, lift forces on the cylinder are given. The calculated streamline contours agreed well with the experimental visualized flow pictures. The existence of critical states and the vortex patterns at the states are given for the first time. The maximum lift to drag force ratio can be obtained nearby the critical states.
文摘An inexact Halley＇s method-Halley-PCG（preconditioned conjugate gradient） method is proposed for solving the systems of linear equations for improved Halley method either by Cholesky factorization exactly or by preconditioned conjugate gradient method approximately. The convergence result is given and the efficiency of the method compared to the improved Halley＇s method is shown.
