Residual a posteriori error estimate two-grid methods for the steady Navier-Stokes equation with stream function form

Residual based on a posteriori error estimates for conforming element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.     

Solution of spline function of elastic plates

In this paper.from four and three-order differential equations defined by cubic andquadratic splines of generaized beam.The beam functions with many boundary conditionsand under various loads are reduced.The approximate solution of deformation surface andstress of elastic thin plate is very accurate.     

The dynamic plastic response characteristics of circular beam

In this paper the problem of a circular beam subjected to radial impact by a rigid mass at its tip in its own plane is investigated on the basis of rigid-perfectly plastic assumption. The analytical solution of the particle velocities is obtained as the function of travelling plastic hinge location. By analysing the solution, some special properties of circular beam problem are found.     

Perturbation method in the problem of large deflections of circular plates with nonuniform thickness

In this paper the perturbation method about two parameters is applied to the problem of large deflection of a cricular plate with exponentially varying thickness under uniform pressure. An asymptotic solution up to the third-order is derived. In comparison with the exact solutions in special cases, the asymptotic solution shows a precise accuracy.     

Efficient computation of mean drag for the subcritical flow past a circular cylinder using general Galerkin G2

General Galerkin (G2) is a new computational method for turbulent flow, where a stabilized Galerkin finite element method is used to compute approximate weak solutions to the Navier–Stokes equations directly, without any filtering of the equations as in a standard approach to turbulence simulation, such as large eddy simulation, and thus no Reynolds stresses are introduced, which need modelling. In this paper, G2 is used to compute the drag coefficient c_D for the flow past a circular cylinder at Reynolds number Re=3900, for which the flow is turbulent. It is found that it is possible to approximate c_D to an accuracy of a few percent, corresponding to the accuracy in experimental results for this problem, using less than 10⁵ mesh points, which makes the simulations possible using a standard PC. The mesh is adaptively refined until a stopping criterion is reached with respect to the error in a chosen output of interest, which in this paper is c_D. Both the stopping criterion and the mesh‐refinement strategy are based on a posteriori error estimates, in the form of a space–time integral of residuals times derivatives of the solution of a dual problem, linearized at the approximate solution, and with data coupling to the output of interest. Copyright © 2008 John Wiley & Sons, Ltd.     

A NONLINEAR GALERKIN MIXED ELEMENT METHOD AND A POSTERIORI ERROR ESTIMATOR FOR THE STATIONARY NAVIER-STOKES EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｎｏｎｌｉｎｅａｒＧａｌｅｒｋｉｎｍｅｔｈｏｄｉｓａｍｕｌｔｉ＿ｌｅｖｅｌｓｃｈｅｍｅｔｏｆｉｎｄｔｈｅａｐｐｒｏｘｉｍａｔｅｓｏｌｕｔｉｏｎｆｏｒｔｈｅｄｉｓｓｉｐａｔｉｖｅＰＤＥ (ｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ) .Ｔｈｉｓｍｅｔｈｏｄｃｏｎｓｉｓｔｓｉｎｓｐｌｉｔｔｉｎｇｔｈｅｕｎｋｎｏｗｎｉｎｔｏｔｗｏ (ｏｒｍｏｒｅ)ｔｅｒｍｓ ,ｗｈｉｃｈｂｅｌｏｎｇｔｏｔｈｅｄｉｓｃｒｅｔｅｓｐａｃｅｓｗｉｔｈｄｉｆｆｅｒｅｎｔｍｅｓｈｓｉｚｅ .Ｔｈｅ…     

A mixed finite element method for the generalized Stokes problem

We present and analyse a new mixed finite element method for the generalized Stokes problem. The approach, which is a natural extension of a previous procedure applied to quasi‐Newtonian Stokes flows, is based on the introduction of the flux and the tensor gradient of the velocity as further unknowns. This yields a two‐fold saddle point operator equation as the resulting variational formulation. Then, applying a slight generalization of the well known Babu?ka–Brezzi theory, we prove that the continuous and discrete formulations are well posed, and derive the associated a priori error analysis. In particular, the finite element subspaces providing stability coincide with those employed for the usual Stokes flows except for one of them that needs to be suitably enriched. We also develop an a posteriori error estimate (based on local problems) and propose the associated adaptive algorithm to compute the finite element solutions. Several numerical results illustrate the performance of the method and its capability to localize boundary layers, inner layers, and singularities. Copyright © 2005 John Wiley & Sons, Ltd.     

A least square extrapolation method for the a posteriori error estimate of the incompressible Navier Stokes problem

A posteriori error estimators are fundamental tools for providing confidence in the numerical computation of PDEs. To date, the main theories of a posteriori estimators have been developed largely in the finite element framework, for either linear elliptic operators or non‐linear PDEs in the absence of disparate length scales. On the other hand, there is a strong interest in using grid refinement combined with Richardson extrapolation to produce CFD solutions with improved accuracy and, therefore, a posteriori error estimates. But in practice, the effective order of a numerical method often depends on space location and is not uniform, rendering the Richardson extrapolation method unreliable. We have recently introduced (Garbey, 13th International Conference on Domain Decomposition, Barcelona, 2002; 379–386; Garbey and Shyy, J. Comput. Phys. 2003; 186 :1–23) a new method which estimates the order of convergence of a computation as the solution of a least square minimization problem on the residual. This method, called least square extrapolation, introduces a framework facilitating multi‐level extrapolation, improves accuracy and provides a posteriori error estimate. This method can accommodate different grid arrangements. The goal of this paper is to investigate the power and limits of this method via incompressible Navier Stokes flow computations. Copyright © 2005 John Wiley & Sons, Ltd.     

Adaptive finite element method for an optimal control problem of Stokes flow with L2‐norm state constraint

An adaptive finite element approximation for an optimal control problem of the Stokes flow with an L²‐norm state constraint is proposed. To produce good adaptive meshes, the a posteriori error estimates are discussed. The equivalent residual‐type a posteriori error estimators of the H ¹‐error of state and L²‐error of control are given, which are suitable to carry out the adaptive multi‐mesh finite element approximation. Some numerical experiments are performed to illustrate the efficiency of the a posteriori estimators. Copyright © 2011 John Wiley & Sons, Ltd.     

不可压Navier-Stokes方程基于误差估算的网格自适应解法

首先导出了广义Stokes方程Petrov—Galerkin有限元数值解的当地事后误差估算公式;以非连续二阶鼓包（bump）函数空间为速度、压强误差的近似空间,该估算基于求解当地单元上的广义Stokes问题。然后,证明了误差估算值与精确误差之间的等价性。最后,将误差估算方法应用于Navier—Stokes环境,以进行不可压粘流计算中的网格自适应处理。数值实验中成功地捕获了多强度物理现象,验证了本文所发展的方法。     

TWO-GRID ERROR ESTIMATES FOR THE STREAM FUNCTION FORM OF NAVIER-STOKES EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｃｏｎｓｉｄｅｒｔｗｏ＿ｇｒｉｄｍｅｔｈｏｄｆｏｒｔｈｅｓｔｒｅａｍｆｕｎｃｔｉｏｎｆｏｒｍｏｆｔｈｅｓｔａｔｉｏｎａｒｙＮａｖｉｅｒ＿Ｓｔｏｋｅｓｅｑｕａｔｉｏｎｓ.Ｔｈｅａｄｖａｎｔａｇｅｓｏｆｔｈｅｓｔｒｅａｍｆｕｎｃｔｉｏｎｆｏｒｍａｒｅｔｈａｔｔｈｅｉｎｃｏｍｐｒｅｓｓｉｂｉｌｉｔｙｃｏｎｄｉｔｉｏｎｉｓｓａｔｉｓｆｉｅｄａｕｔｏｍａｔｉｃａｌｌｙａｎｄｔｈｅｐｒｅｓｓｕｒｅｉｓｎｏｔｐｒｅｓｅｎｔｉｎｔｈｅｗｅａｋｆｏｒｍ .Ｔｈｅｍｅｔｈｏｄｉｓｂａｓｅｄ…     

A nonlinear Galerkin/Petrov-least squares mixed element method for the stationary Navier-Stokes equations

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｎｏｎｌｉｎｅａｒＧａｌｅｒｋｉｎｍｅｔｈｏｄｉｓａｍｕｌｔｉ＿ｌｅｖｅｌｓｃｈｅｍｅｔｏｆｉｎｄｔｈｅａｐｐｒｏｘｉｍａｔｅｓｏｌｕｔｉｏｎｆｏｒｔｈｅｄｉｓｓｉｐａｔｉｖｅＰＤＥ .ＴｈｉｓｍｅｔｈｏｄｈａｓｆｉｒｓｔｍａｉｎｌｙｂｅｅｎａｄｄｒｅｓｓｅｄｂｙＦｏｉａｓ＿Ｍａｎｌｅｙ＿Ｔｅｍａｍ[1],Ｍａｒｉｏｎ＿Ｔｅｍａｍ[2 ],Ｆｏｉａｓ＿Ｊｏｌｌｙ＿Ｋｅｖｒｅｋｉｄｉｓ＿Ｔｉｔｉ[3]ａｎｄＤｅｖｕｌｄｅｒ＿Ｍａｒｉｏｎ＿Ｔｉｔｉ[4 ]ｉｎｔｈｅｃａｓｅｏｆｓｐｅｃｔ…     

A Nonlinear Galerkin/Petrov-Least Squares Mixed Element Method for the Stationary Navier-Stokes Equation

A nonlinear Galerkin/ Petrov- least squares mixed element (NGPLSME) method for the stationary Navier-Stokes equations is presented and analyzed. The scheme is that Petrov-least squares forms of residuals are added to the nonlinear Galerkin mixed element method so that it is stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. The existence, uniqueness and convergence ( at optimal rate ) of the NGPLSME solution is proved in the case of sufficient viscosity ( or small data).     

High accuracy eigensolution and its extrapolation for potential equations

From the potential theorem, the fundamental boundary eigenproblems can be converted into boundary integral equations （BIEs） with the logarithmic singularity. In this paper, mechanical quadrature methods （MQMs） are presented to obtain the eigensolutions that are used to solve Laplace＇s equations. The MQMs possess high accuracy and low computation complexity. The convergence and the stability are proved based on Anselone＇s collective and asymptotical compact theory. An asymptotic expansion with odd powers of the errors is presented. By the h3-Richardson extrapolation algorithm （EA）, the accuracy order of the approximation can be greatly improved, and an a posteriori error estimate can be obtained as the self-adaptive algorithms. The efficiency of the algorithm is illustrated by examples.     

The effect of the hydrodynamic interaction on the rheological properties of Hookean dumbbell suspensions in steady state shear flow

The diffusion equation for the configurational distribution function of Hookean dumbbell suspensions with the hydrodynamic interaction (HI) was solved, in terms of Galerkin’s method, in steady state shear flow; and viscosity,first and second normal-stress coefficients and molecular stretching were then calculated. The results indicate that the HI included in a microscopic model of molecules gives rise to a significant effect on the macroscopic properties of Hookean dumbbell suspensions. For example, the viscosity and the first normal stress coefficient, decreasing as shear rate increases, are no longer constant; the second normal-stress coefficient, being negative with small absolute value and shear-rate dependent, is no longer zero; and an additional stretching of dumbbells is yielded by the HI. The viscosity function and the first normal-stress coefficient calculated from this method are in agreement with those predicted from the self-consistent average method qualitatively, while the negative second normal-stress coefficient from the former seems to be more reasonable than the positive one from the latter.     

An approximate solution considering flow inertia between spherical surfaces

In this paper, the analytical expressions of the pressure distribution, velocity distribution and discharge of the flow between spherical surfaces are found by using the method of iterative approximate solution when the inertia terms of Navier-Stokes equations in spherical coordinates are taken into consideration. Furthermore, using these expressions, we can directly obtain the corresponding analytical expressions of the laminar radial flow between parallel disks, which are fully identical with corresponding results presented by refs. [3,4].     

A posteriori error estimate of the DSD method for first-order hyperbolic equations

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔ’ｓｋｎｏｗｎｔｈａｔｉｎｎｕｍｅｒｉｃａｌａｐｐｒｏｘｉｍａｔｉｏｎｏｆｆｉｒｓｔ＿ｏｒｄｅｒｈｙｐｅｒｂｏｌｉｃｅｑｕａｔｉｏｎｓ,ｔｈｅｕｓｅｏｆａｄａｐｔｉｖｅｆｉｎｉｔｅｅｌｅｍｅｎｔｍｅｔｈｏｄｓ (ｓｅｅ [1 ] )ｈａｓｂｅｅｎｅｘｐａｎｄｅｄｔｏｍａｎｙｆｉｅｌｄｓｓｕｃｈａｓｃｏｍｐｕｔａｔｉｏｎａｌｆｌｏｗｍｅｃｈａｎｉｃｓ,ｔｈｅｒｍａｌａｎａｌｙｓｅｓ,ｅｌｅｃｔｒｉｃａｌｅｎｇｉｎｅｅｒｉｎｇ ,ｅｔｃ.Ｔｈｅｈ＿ｖｅｒｓｉｏｎａｄａｐｔｉｖｅｆｉｎｉ…     

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.     

Geotechnical yield criteria and constitutive relations in strain space

Based on Hyushins postulate this paper deals with the necessity and features ofresearching the geotechnical elasto-plastie theory in strain space.In the paper,weestablished the relations between stress in variants and elastic strain invariants.broughtabout the transformation from the stress yield surfaces into the strain yield surfaces,derived and discussed the strain expressions from 12 yield criteria expressed by stress.Bynormality rule.we also derived 12 constitutive relations for ideal plastic materialsassociated with the above expressions.The results presented here can be applied to practiceand are helpful to the study of the plastic theory in strain space.     

Anisotropic adaptive finite element method for magnetohydrodynamic flow at high Hartmann numbers

This paper presents an anisotropic adaptive finite element method (FEM) to solve the governing equations of steady magnetohydrodynamic (MHD) duct flow. A residual error estimator is presented for the standard FEM, and two-sided bounds on the error independent of the aspect ratio of meshes are provided. Based on the Zienkiewicz-Zhu estimates, a computable anisotropic error indicator and an implement anisotropic adaptive refinement for the MHD problem are derived at different values of the Hartmann number. The most distinguishing feature of the method is that the layer information from some directions is captured well such that the number of mesh vertices is dramatically reduced for a given level of accuracy. Thus, this approach is more suitable for approximating the layer problem at high Hartmann numbers. Numerical results show efficiency of the algorithm.