A Newton's method scheme for solving free-surface flow problems

A Newton's method scheme is described for solving the system of non-linear algebraic equations arising when finite difference approximations are applied to the Navier–Stokes equations and their associated boundary conditions. The problem studied here is the steady, buoyancy-driven motion of a deformable bubble, assumed to consist of an inviscid, incompressible gas. The linear Newton system is solved using both direct and iterative equation solvers. The numerical results are in excellent agreement with previous work, and the method achieves quadratic convergence.     

有限长轴承非稳态油膜力自由-移动边界问题的有限条解法

讨论了不可压缩流体润滑的动载径向滑动轴承油膜压力分布的自由移动边界问题的有限条解法．将自由边界问题转化为全域(矩形域)的具有不等式约束的微分方程的边值问题,进一步化为具有不等式约束的泛函极值问题。借助有限条法在矩形域上离散这个泛函,得到了一个特殊的二次泛函的规化问题。通过变量平移变换,使其化为标准的二次规划问题。然后借助于牛顿非光滑算法,迭代求解非线性的互补方程。给出了有限长轴承真实的油膜应力分布。对于所求解方程的系数矩阵的高度稀疏性。给出了紧缩存储算法。节省了存储空间和减少了计算量。算例表明该方法是有效的。     

SOLVING THE FREE BOUNDARY PROBLEM IN CONTINUOUS CASTING BY USING BOUNDARY ELEMENT METHOD

ＳＯＬＶＩＮＧＴＨＥＦＲＥＥＢＯＵＮＤＡＲＹＰＲＯＢＬＥＭＩＮＣＯＮＴＩＮＵＯＵＳＣＡＳＴＩＮＧＢＹＵＳＩＮＧＢＯＵＮＤＡＲＹＥＬＥＭＥＮＴＭＥＴＨＯＤＬｉＹａｏｙｏｎｇ（李耀勇）；ＺｈａｎｇＺｈｉｌｉ（张自立）（ＲｅｃｅｉｖｅｄＪｕｎｅ，１８，１９...     

A boundary element method for a non-linear free surface problem

In this paper a numerical method to compute the wave resistance of a body submerged in a free stream of finite and infinite depth is presented. Non-linear effects on the free surface are taken into account by an iterative procedure; the solution is in the form of a single-layer potential. For the 2D problem, results are shown for both the cases of finite and infinite depth of the fluid domain, with special emphasis on the supercritical flow in which the consistency of the scheme is pointed out. The method is also extended to the 3D case of a spheroid submerged in deep water. All the results presented are compared with experimental data and analytical solutions available in the literature.     

Newton's method for solving heat transfer,fluid flow and interface shapes in a floating molten zone

Newton's method is applied to the finite volume approximation for the steady state heat transfer, fluid flow and unknown interfaces in a floating molten zone. The streamfunction/vorticity and temperature formulation of the Navier–Stokes and energy equations and their associated boundary conditions are written in generalized curvilinear co-ordinates and conservative law form with the Boussinesq approximation. During Newton iteration the ILU(0) preconditioned GMRES matrix solver is applied for solving the linear system, where the sparse Jacobian matrix is estimated by finite differences. Nearly quadratic convergence of the method is observed. Sample calculations are reported for sodium nitrate, a high-Prandtl-number material (Pr = 9.12). Both natural convection and thermocapillary flow as well as an overall mass balance constraint in the molten zone are considered. The effects of convection and heat input on the flow patterns, zone position and interface shapes are illustrated. After the lens effect due to the molten zone is considered, the calculated flow patterns and interface shapes are compared with the observed ones and are found to be in good agreement.     

A NEW EFFICIENT METHOD TO BOUNDARY VALUE PROBLEM FOR BALLISTIC ROCKET GUIDANCE

IntroductionThe problem of motion control for the mass center of ballistic rocket pertained typicallyto the boundary value problem for two points with the non-linear and multi-variables,namely,given the initial states of rocket,in order to satisfy the res…     

A fully implicit direct Newton's method for the steady-state Navier–Stokes equations

Newton's method and banded Gaussian elimination can be a CPU efficient method for steady-state solutions to two-dimensional Navier–Stokes equations. In this paper we look at techniques that increase the radius of convergence of Newton's method, reduce the number of times the Jacobian must be factored, and simplify evaluation of the Jacobian. The driven cavity and natural convection problems are used as test problems, and finite volume discretization is employed.     

An effective boundary method for the analysis of elastoplastic problems

In this paper, a series of effective formulae of the boundary element method is presented. In these formulae, by using a new variable two kernels are only of the weaker singularity of Lnr (where r is the distance between a source point and a field point). Hence, the singularities in the conventional displacement formulation and stress formulation at internal points are reduced respectively so that the "boundary-layer" effect which strongly degenerates the accuracy of stress calculation by using original formulae is eliminated. Also the direct evaluation of coefficients C (boundary tensor), which are difficult to calculate, is avoided. This method is used in elastoplastic analysis. The results of the numerical investigation demonstrate the potential advantages of this method.     

Numerical method for unsteady compressible boundary layers driven by compression or expansion wave

A numerical analysis is presented for the unsteady compressible laminar boundary layer driven by a compression or expansion wave. Approximate or series expansion methods have been used for the problems because of the characteristics of the governing equations, such as non-linearity, coupling with the thermal boundary layer equation and initial conditions. Here a transformation of the governing equations and the numerical linearization technique are introduced to deal with the difficulties. First, the governing equations are transformed for the initial conditions by Howarth and semisimilarity variables. These transformations reduce the number of independent variables from three to two and the governing equations from partial to ordinary differential equations at the initial point. Next, the numerical linearization technique is introduced for the non-linearity and the coupling with the thermal boundary layer equation. Because the non-linear terms are linearized without sacrifice of numerical accuracy, the solutions can be obtained without numerical iterations. Therefore the exact numerical solution, not approximate or series expansion, can be obtained. Compared with the approximate or series expansion method, this method is much improved. Results are compared with the series expansion solutions.     

不同材料契合弹性力学问题的虚边界元最小二乘法

本文针对不同材料契合弹性力学问题,由虚边界元方法出发,建立了引入拉氏乘子的最小二科解法,对该类问题避免了采用Ｈｅｔｅｎｙｉ’ｓ基本解的麻烦和限制。数值结果表明本文算法的有效性和计算精度高。     

Effects of the Jacobian evaluation on Newton's solution of the Euler equations

Newton's method is developed for solving the 2‐D Euler equations. The Euler equations are discretized using a finite‐volume method with upwind flux splitting schemes. Both analytical and numerical methods are used for Jacobian calculations. Although the numerical method has the advantage of keeping the Jacobian consistent with the numerical residual vector and avoiding extremely complex analytical differentiations, it may have accuracy problems and need longer execution time. In order to improve the accuracy of numerical Jacobians, detailed error analyses are performed. Results show that the finite‐difference perturbation magnitude and computer precision are the most important parameters that affect the accuracy of numerical Jacobians. A method is developed for calculating an optimal perturbation magnitude that can minimize the error in numerical Jacobians. The accuracy of the numerical Jacobians is improved significantly by using the optimal perturbation magnitude. The effects of the accuracy of numerical Jacobians on the convergence of the flow solver are also investigated. In order to reduce the execution time for numerical Jacobian evaluation, flux vectors with perturbed flow variables are calculated only for neighbouring cells. A sparse matrix solver that is based on LU factorization is used. Effects of different flux splitting methods and higher‐order discretizations on the performance of the solver are analysed. Copyright © 2005 John Wiley & Sons, Ltd.     

An analytical method for a mixed boundary value problem of circular plates under arbitrary lateral loads

In this paper by using the concept of mixed boundary functions,an analytical method is proposed for a mixed boundary value problem of circular plates.The trial functions are constructed by using the series of particular solutions of the biharmonic equations in the polar coordinate system.Three examples are presented to show the stability and high convergence rate of the method.     

Fixed domain methods for free and moving boundary flows in porous media

A survey is presented concerning fixed domain methods used to solve mathematical models of free and moving boundary flow problems in porous media. These include the following: variational inequality or quasi-variational inequality formulations; general inequality formulations which have been set and solved in fixed domains; and the residual flow procedure. Finally, some parallel computing methods and mesh adaptation methods are discussed to demonstrate how these fixed domain formulations can be solved with current technology.The fixed domain methods that are referenced herein can be classified into two groups: the variational inequality method and the extended pressure head method. Baiocchi was the first to apply the variational inequality method to free boundary problems of flows through porous media. This method in general also uses an extension of the pressure head but adds an application of an integral transformation (a Baiocchi transformation) to the problem. The method possesses a beautiful mathematical structure for its theory and yields simple numerical solution algorithms. However, application of the method is difficult if not impossible in some cases depending upon the regularity of the seepage domain.The extended pressure head method is based on the concept that the pressure is extended smoothly across the free or moving boundary into the unsaturated region from the flow domain. The extension of the pressure head to the entire porous medium yields an extended coefficient of permeability of the medium which is equal to the saturated coefficient in the seepage region and is equal to zero or some small value (for computational purposes) in the unsaturated region.     

一种投影浸入边界法的快速直接求解方法

根据投影浸入边界法分步投影求解的特点,同时针对压力泊松方程离散后的大型稀疏线性方程组是非奇异非对称的特点,结合开源函数库UMFPACK,在传递线性方程组的系数矩阵和右端向量时,采用函数库Eigen将系数矩阵的数据结构改写优化,大大降低了存储空间,实现对高维大型稀疏线性方程组的快速求解,同时求解保持良好的稳定性。本文首先利用一具有解析解的数值算例验证了求解泊松方程数值方法的准确性和网格依赖性,进而利用VC++编写投影浸入边界法的数值计算程序,以单圆柱绕流为基准数值算例,通过与其他文献和实验结果的对比,验证了投影浸入边界法数值计算结果的可靠性,并进一步分析了不同雷诺数下圆柱绕流的流场结构特征和尾涡结构的动态演化过程。     

A SOLVER USING NEWTON‘S METHOD FOR UNSTEADY VISCOUS FLOWS

A solver is developed for time-accurate computations of viscous flows based on the conception of Newton‘s method. A set of pseudo-time derivatives are added into governing equations and the discretized system is solved using GMRES algorithm. Due to some special properties of GMRES algorithm, the solution procedure for unsteady flows could be regarded as a kind of Newton iteration. The physical-time derivatives of governing equations are discretized using two different approaches, I.e., 3-point Euler backward, and Crank-Nicolson formulas, both with 2nd-order accuracy in time but with different truncation errors. The turbulent eddy viscosity is calculated by using a version of Spalart~Allmaras one-equation model modified by authors for turbulent flows. Two cases of unsteady viscous flow are investigated to validate and assess the solver, I.e., low Reynolds number flow around a row of cylinders and transonic bi-circular-arc airfoil flow featuring the vortex shedding and shock buffeting problems, respectively. Meanwhile, comparisons between the two schemes of timederivative discretizations are carefully made. It is illustrated that the developed unsteady flow solver shows a considerable efficiency and the Crank-Nicolson scheme gives better results compared with Euler method.     

A Newton/upwind method and numerical study of shock wave/boundary layer interactions

The objective of the paper is twofold. First we describe an upwind/central differencing method for solving the steady Navier–Stokes equations. The symmetric line relaxation method is used to solve the resulting algebraic system to achieve high computational efficiency. The grid spacings used in the calculations are determined from the triple-deck theory, in terms of Mach and Reynolds numbers and other flow parameters. Thus the accuracy of the numerical solutions is improved by comparing them with experimental, analytical and other computational results. Secondly we proceed to study numerically the shock wave/boundary layer interactions in detail, with special attention given to the flow separation. The concept of free interaction is confirmed. Although the separated region varies with Mach and Reynolds numbers, we find that the transverse velocity component behind the incident shock, which has not been identified heretofore, is also an important parameter. A small change of this quantity is sufficient to eliminate the flow separation entirely.     

A free boundary problem arising in oil production

For a stable two-phase free boundary problem arising in oil production, we prove the existence of a weak solution and the continuity of the free boundary.

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Sommario Si prova un teorema di esistenza di una soluzione debole per un problema stazionario a contorno libero a due fasi che interessa la produzione di petrolio. Si dimostra inoltre la continuità del contorno libero.

     

三维非稳定渗流自由面边界积分项的精确数值计算

利用固定网格法分析三维非稳定渗流问题时,将要面对两项积分难题:以自由面及单元表面为边界的空间积分及以自由面为边界的曲面积分。针对常用的任意8结点6平面三维普通单元,提出采用坐标变换及等参变换技术求取空间积分项的精确数值解;至于曲面积分项,建议改用单元非饱和区部分表面作为积分边界,经过坐标变换及等参变换处理积分边界后,利用高斯数值积分可求出曲面积分项的精确数值解。通过一个普通单元及一项均质半无限边界堤坝的实例分析,表明此方法的精确性和稳定性良好。     

INTERPOLATION PERTURBATION METHOD FOR SOLVING THE BOUNDARY LAYER TYPE PROBLEMS

ＹｕａｎＹｉｗｕ（袁镒吾）（ＲｅｃｅｉｖｅｄＯｃｔ．２，１９９４；ＣｏｍｍｕｎｉｃａｔｅｄｂｙＣｈｉｅｎＷｅｉｚａｎｇ）ＩＮＴＥＲＰＯＬＡＴＩＯＮＰＥＲＴＵＲＢＡＴＩＯＮＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＴＨＥＢＯＵＮＤＡＲＹＬＡＹＥＲＴＹＰＥＰＲＯＢ...     

A total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast.