Newton's method and Schwarzian derivatives

We discuss Newton's method with respect to obtaining convergence to a fixed point with orders of convergence greater than 2. We identify the role played by the Schwarzian derivative in controlling the convergence of Newton's map to a super stable fixed point.     

Analytical formulas of solutions of geometrically nonlinear equations of axisymmetric plates and shallow shells

Starting from the step-by-step iterative method, the analytical formulas of solutions of the geometrically nonlinear equations of the axisymmetric plates and shallow shells, have been obtained. The uniform convergence of the iterative method, is used to prove the convergence of the analytical formulas of the exact solutions of the equations.     

Accuracy and convergence of element-by-element iterative solvers for incompressible fluid flows using penalty finite element model

The ability of two types of Conjugate Gradient like iterative solvers (GMRES and ORTHOMIN) to resolve large-scale phenomena as a function of mesh density and convergence tolerance limit is investigated. The flow of an incompressible fluid inside a sudden expansion channel is analysed using three meshes of 400, 1600 and 6400 bilinear elements. The iterative solvers utilize the element-by-element data structure of the finite element technique to store and maintain the data at the element level. Both the mesh density and the penalty parameter are found to influence the choice of the convergence tolerance limit needed to obtain accurate results. An empirical relationship between the element size, the penalty parameter, and the convergence tolerance is presented. This relationship can be used to predict the proper choice of the convergence tolerance for a given penalty parameter and element size.     

Numerical instabilities and convergence control for convex approximation methods

Convex approximation methods could produce iterative oscillation of solutions for solving some problems in structural optimization. This paper firstly analyzes the reason for numerical instabilities of iterative oscillation of the popular convex approximation methods, such as CONLIN (Convex Linearization), MMA (Method of Moving Asymptotes), GCMMA (Global Convergence of MMA) and SQP (Sequential Quadratic Programming), from the perspective of chaotic dynamics of a discrete dynamical system. Then, the usual four methods to improve the convergence of optimization algorithms are reviewed, namely, the relaxation method, move limits, moving asymptotes and trust region management. Furthermore, the stability transformation method (STM) based on the chaos control principle is suggested, which is a general, simple and effective method for convergence control of iterative algorithms. Moreover, the relationships among the former four methods and STM are exposed. The connection between convergence control of iterative algorithms and chaotic dynamics is established. Finally, the STM is applied to the convergence control of convex approximation methods for optimizing several highly nonlinear examples. Numerical tests of convergence comparison and control of convex approximation methods illustrate that STM can stabilize the oscillating solutions for CONLIN and accelerate the slow convergence for MMA and SQP.     

Comparison of using Cartesian and covariant velocity components on non-orthogonal collocated grids

In this paper, the Cartesian velocity components and the covariant velocity components are adopted respectively as the main variables in solving the momentum equations in the SIMPLE-like method to calculate a lid-driven cavity flow on non-orthogonal collocated grids. In total, more than 400 computer runs are carried out for a two-dimensional problem. The accuracy and convergence performance of using Cartesian and covariant velocity components are compared in detail. Comparisons show that both the Cartesian and covariant velocity methods have the same numerical accuracy. The convergence rate of the covariant velocity method can be faster than that of the Cartesian velocity method if the relaxation factor for pressure is small enough. However, the convergence range of the relaxation factor for pressure in the covariant velocity method is quite narrow. When the cross-derivatives in the pressure-correction equation are retained approximately, its convergence performance can be greatly improved. Copyright © 1999 John Wiley & Sons, Ltd.     

同位网格上SIMPLE算法收敛特性的Fourier分析

应用Fourier方法研究了同位网格上SIMPLE算法求解浅水方程的收敛特性,并就松弛因子组合及阻力项的影响进行了分析.结果表明,采用合适的松弛因子组合可以很快地消除高频区域的误差,同时也可逐步消减迭代中低频区域的误差以获得收敛解.在保证收敛的前提下,低频误差分量决定了迭代速度,而且浅水方程中阻力项越大越利于SIMPLE算法收敛.     

A Remark on the Convergence of Global and Bounded Solutions for a Semilinear Wave Equation on ∝ N

We give a new and easier proof for the result of Feireisl [Fe1] concerning the convergence of global and bounded solutions of the wave equation with small initial energy. We also prove that the convergence takes place with an exponential decay.     

On uniform convergence of linear operators on the probabilistic normed space

In this paper, we introduced the notion of uniform convergence of the linear operators on the probabilistic normed space, and the notion of probabilistic distance between the operators, which describes the above convergence completely. In terms of these notions, we obtained the essential features of the continuity of operators, and of the uniform convergence of operator sequences, and we also obtained the closure of continuity and complete continuity under the operation of the limit of uniform convergence.     

全机绕流Euler方程多重网格分区计算方法

全机三维复杂形状绕流数值求解只能采用分区求解的方法,本文采用可压缩Euler方程有限体积方法以及多重网格分区方法对流场进行分区计算。数值方法采用改进的van Leer迎风型矢通量分裂格式和MUSCL方法,基于有限体积方法和迎风型矢通量分裂方法,建立一套处理子区域内分界面的耦合条件。各个子区域之间采用显式耦合条件,区域内部采用隐式格式和局部时间步长等,以加快收敛速度。计算结果飞机表面压力分布等气动力特性与实验值进行了比较,二者基本吻合。计算结果表明采用分析“V”型多重网格方法,能提高计算效率,加快收敛速度达到接近一个量级。根据全机数值计算结果和可视化结果讨论了流场背风区域旋涡的形成过程。     

最优化算法的收敛准则

收敛准则是最优化算法的重要组成部分，其选择得好与坏将直接影响到算法的成功与否以及收敛得快与慢。现有常用的收敛准则基本上是建立在前后迭代点的逼近和它们相应函数值的逼近是否达到一定的精度要求以及迭代点处函数梯度是否接近于零的基础上的。它们各自有自己的适用范围。但它们的共同特点是对迭代终止点的性质不能做出判断。本文在总结和分析现有算法收敛准则的基础上，借助于正定矩阵、一维优化方法中对分法和黄金分割法，提出了新的算法收敛准则。算例结果表明，这些收敛准则是有效实用的。     

Rates of strong uniform convergence of nearest neighbor density estimates on any compact set

In this paper, we propose the concept of rates of strong uniform convergence of nearest neighbor density estimates on any compact set and obtain some better convergence rates. Hence the problem of the strong uniform convergence rates predetermined is its special example. The applied region of the estimate is extended.     

Multigrid solution of the incompressible Navier–Stokes equations for three‐dimensional recirculating flow: coupled and decoupled smoothers compared

A comparison of multigrid methods for solving the incompressible Navier–Stokes equations in three dimensions is presented. The continuous equations are discretised on staggered grids using a second‐order monotonic scheme for the convective terms and implemented in defect correction form. The convergence characteristics of a decoupled method (SIMPLE) are compared with those of the cellwise coupled method (SCGS). The convergence rates obtained for computations of the three‐dimensional lid‐driven cavity problem are found to be very similar to those obtained for computations of the corresponding two‐dimensional problem with comparable grid density. Although the convergence rate of SCGS is thus superior to that of SIMPLE, the decoupled method is found to be more efficient computationally and requires less computing time for a given level of convergence. The linewise implementation of the coupled method (CLGS) is also investigated and shown to be more efficient than SCGS, although the convergence rate and computing time required per cycle are both found to depend on the direction of sweep. The optimal implementation of CLGS is found to be only marginally more effective than SIMPLE, but a change to the structure of the data storage would increase the advantage. Copyright © 1999 John Wiley & Sons, Ltd.     

重载滑动轴承非牛顿流体的三维弹性流体动力润滑分析的加权平均法

本文采用加权平均法成功地求解了由广义Ｒｅｙｎｏｌｄｓ方程（幂次型）和三维弹性方程组成的非牛顿体弹流模型，该方法较好的解决了大偏心率下的弹流迭代收敛困难的问题。该方法具有收敛快，对初值要求松且计算精度高等优点，是一种解决弹流问题较好的迭代模式。     

The pansystems view of prediction and blow-up of fluid

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Convergence and exact solutions of spline finite strip method using unitary transformation approach

The spline finite strip method(PSM) is one of the most popular numerical methods for analyzing prismatic structures.Efficacy and convergence of the method have been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems.To date,no exact solutions of the method or its explicit forms of error terms have been derived to show its convergence analytically. As such,in this paper,the mathematical exact solutions of spline finite strips in the plat...     

Convergence predictions for aeroelastic calculations of tuned and mistuned bladed disks

Mistuning changes the dynamics of bladed disks significantly. Frequency domain methods for predicting the dynamics of mistuned bladed disks are typically based on iterative aeroelastic calculations. Converged aerodynamic stiffness matrices are required for accurate aeroelastic results of eigenvalue and forced response problems. The tremendous computation time needed for each aerodynamic iteration would greatly benefit from a fast method of predicting the number of iterations needed for converged results. A new hybrid technique is proposed to predict the convergence history based on several critical ratios and by approximating as linear the relation between the aerodynamic force and the complex frequencies (eigenvalues) of the system. The new technique is hybrid in that it uses a combined theoretical and stochastic/computational approach. The dynamics of an industrial bladed disk is investigated, and the predicted convergence histories are shown to match the actual results very well. Monte Carlo simulations using the new hybrid technique show that the aerodynamic ratio and the aerodynamic gradient ratio are the two most important factors affecting the convergence history.     

Convergence towards equilibrium for a gas of Maxwellian Pseudomolecules

We give a new proof of convergence towards equilibrium for a gas of Maxwellian pseudo-molecules with finite mass, momentum and energy. No entropy is needed for weak * convergence. Strong convergence follows in the case of planar velocities by a suitable use of theJ-functional introduced by McKean for the Kac's caricature of the Maxwellian gas.     

可变体系的几何稳定平衡状态及受力分析

索和撑杆组成的张拉集成体系，膜结构及悬索体系，一般都要施加预应力使其成为能承受荷载的几何不变体系，整体受力表现为几何非线性。本文将其受力状态分解为两部分，一是几何稳定平衡状态，二是弹性稳定平衡状态。文中提出几何稳定平衡状态的分析计算方法，证明增量迭代法的收敛性。结果表明，所提出的方法收敛快，精度高。     

Projection-algebraic approximation of linear and nonlinear operator differential equations in Banach spaces

We develop a Calogero-type projection-algebraic method of discrete approximations for linear differential equations in Banach spaces and analyze the convergence of finite-dimensional approximations based on the functional-analytic approach to discrete approximations and methods of operator theory in Banach spaces. Applications of the obtained results to the functional-interpolation scheme of the projection-algebraic method of discrete approximations are considered. Based on a generalized Leray–Schauder-type theorem, we consider the projection-algebraic scheme of discrete approximations and analyze its solvability and convergence for a special class of nonlinear operator equations.     

On use of the Dirac delta function in the vibration analysis of elastic structures

The Dirac delta function has often been employed to represent the amplitude of concentrated harmonic forces in the analysis of vibration of elastic structures such as beams and plates. It is known that this function, as represented by a truncated Fourier series, does not provide a true representation of a concentrated force, nevertheless, it is frequently employed and good convergence is usually, though not always, encountered in solutions thereby obtained. In this paper, the nature of the function is discussed and for illustrative purposes it is used to obtain series solutions for some selected beam and plate free vibration problems. In some cases problems are chosen for which exact solutions are already obtainable by analytical means. This permits powerful checks to be made on rates of convergence experienced when the series solutions are investigated. Rates of convergence are discussed in detail and it is explained why convergence is to be expected when analyzing certain families of problems when employing this function and a lack of convergence is to be expected in others.