A Newton type iterative method for heat-conduction inverse problems

An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterative methods are considered. The implicit iterative method is applied to the linearized Newton equation, and the key step in the process is that a new reasonable a posteriori stopping rule for the inner iteration is presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given, and these results show the obvious advantages of the new method over the other ones.     

A Newton type iterative inverse method for heat-conduction problems

An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterative methods are considered. The implicit iterative method is applied to the linearized Newton equation, and the key step in the process is that a new reasonable a posteriori stopping rule for the inner iteration is presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given, and these results show the obvious advantages of the new method over the other ones.     

The energy-integral method: application to linear hyperbolic heat-conduction problems

This paper utilizes the energy-integral method to obtain approximate analytic solutions to a linear hyperbolic heat-conduction problem for a semi-infinite one-dimensional medium. As for the mathematical formulation of the problem, a time-dependent relaxation model for the energy flux is assumed, leading to a hyperbolic differential equation which is solved under suitable initial and boundary conditions. In fact, analytical expressions are derived for uniform as well as varying initial conditions along with (a) prescribed surface temperature, or (b) prescribed heat flux at the surface boundary. The case when a heat source (or sink) of certain type takes place has also been discussed. Comparison of the approximate analytic solutions obtained by the energy-integral method with the corresponding available or obtainable exact analytic solutions are made; and the accuracy of the approximate solutions is generally acceptable.Nomenclature A,C constants - a ₀(t),a ₁(t),...,a _n (t) arbitrary time-dependent coefficients, equation (3.2) - b thermal propagation speed - C _p specific heat of solid at constant pressure - g(x) given function, equation (5.1) - h(t) specified function of time - I _n modified Bessel function of the first kind - K thermal conductivity - j,n positive constants - P _n (x,t) polynomial of degreen - q(x,t) heat flux - Q(t),R(t),H(t),E(t) see equations (3.9), (II.d), (4.10), (4.12), respectively - (t) thermal penetration depth - (t,) approximate thermal penetration depth - T(x,t) temperature distribution - t time - y dimensionless time, equation (3.17) - V(y) dimensionless surface heat flux - W(y) dimensionless surface temperature - U-(t) unit-step function - G(x;t,) Green's function - x spatial variable - ()₀ surface value (atx=0) Greek symbols thermal diffusivity - density of solid - parameter, see equations (3.11) and (3.13) - parameter depending onn and - specified parameter, equations (4.5a) and (5.12b) - (t),(t) given functions of time, equations (4.6) and (5.5b) - , dummy variables - relaxation time - energy integral - (y),(y) specified functions ofy; equations (3.22) and (4.19)     

不规则区域平面问题的条形传递函数方法

对条形传递函数方法进行了改进,提出了映射条形传递函数方法,用于处理非正规形状区域的平面问题。在本文方法中,一个非正规区域被映射成为若干矩形子区域的组合,在这些矩形子区域内划分条形单元,进而建立起位移离散模型。利用变分关系对模型处理,可以得到问题的动态控制方程。应用改进后得到的数值传递函数求解,就可以得到系统的动力、静力响应。文后应用上述方法建立了应用模型并给出了数值算法,结果表明本方法继承了原方法精度高、处理规范、便于求解动态问题等,并成功地应用到了非规则区域的平面问题中。     

An adaptive moving finite element method for steady low Mach number compressible combustion problems

This work surveys an r-adaptive moving mesh finite element method for the numerical solution of premixed laminar flame problems. Since the model of chemically reacting flow involves many different modes with diverse length scales, the computation of such a problem is often extremely time-consuming. Importantly, to capture the significant characteristics of the flame structure when using detailed chemistry, a much more stringent requirement on the spatial resolution of the interior layers of some intermediate species is necessary. Here, we propose a moving mesh method in which the mesh is obtained from the solution of so-called moving mesh partial differential equations. Such equations result from the variational formulation of a minimization problem for a given target functional that characterizes the inherent difficulty in the numerical approximation of the underlying physical equations. Adaptive mesh movement has emerged as an area of intense research in mesh adaptation in the last decade. With this approach, points are only allowed to be shifted in space leaving the topology of the grid unchanged. In contrast to methods with local refinement, data structure hence is unchanged and load balancing is not an issue as grid points remain on the processor where they are. We will demonstrate the high potential of moving mesh methods for effectively optimizing the distribution of grid points to reach the required resolution for chemically reacting flows with extremely thin boundary layers.     

Solution of nonlinear heat-conduction problems in a medium with phase transitions

A new method is proposed for solving one-dimensional nonlinear heat-conduction problems in a medium having any number of phase transitions (nonlinear Stefan problems). The method consists of a direct calculation of the isotherms and reduces to the Cauchy problem for a system of ordinary differential equations.The author thanks V. M. Kartvelishvili, a student at the Moscow Physicotechnical Institute, for writing the program and carrying out the calculations on a BÉSM-3M computer. The author thanks L. A. Chudov for useful comments.     

On the steady solution of linear advection problems in steady recirculating flows

Numerical modelling of non-Newtonian flows typically involves the coupling between equations of motion characterized by an elliptic behaviour, and the fluid constitutive equation, which is an advection equation linked to the fluid history. In this paper we prove that linear steady advection problems in steady recirculating flows have only one solution when the kinematics differs from a rigid motion. We also give a numerical procedure to determine this steady solution. We will describe this numerical procedure for two linear models the first will be the SFRT flow model and the second will be a simplified linear formulation of the Pom–Pom viscoelastic model.     

A multigrid pseudospectral method for steady flow computation

In this work two‐dimensional steady flow problems are cast into a fixed‐point formulation, Q = F(Q). The non‐linear operator, F, is an approximate pseudospectral solver to the Navier–Stokes equations. To search the solution we employ Picard iteration together with a one‐dimensional error minimization and a random perturbation in case of getting stuck. A monotone convergence is brought out, and is greatly improved by using a multigrid strategy. The efficacy of this approach is demonstrated by computing flow between eccentric rotating cylinders, and the regularized lid‐driven cavity flow with Reynolds number up to 1000. Copyright © 2003 John Wiley & Sons, Ltd.     

A flux-vector splitting method for steady Navier-Stokes equations

The flux-vector splitting method is applied to the convective part of the steady Navier-Stokes equations for incompressible flow. By the use of partial upwind differences in the split first-order part and central differences in the second-order part, a set of discrete equations is obtained which can be solved by vector variants of classical relaxation schemes. It is shown that accurate results can be obtained on one of the GAMM backward-facing step test problems.     

Local pressure preconditioning method for steady incompressible flows

The convergence and accuracy characteristics of the preconditioned incompressible Euler and Navier–Stokes equations are studied. An object-oriented C++ numerical code has been developed for solving the inviscid and viscous, steady, incompressible flows problems. The code is based on the cell-centred finite volume method. In this scheme, two-dimensional incompressible Euler and Navier–Stokes equations are modified by a robust artificial compressibility (AC) and a local preconditioning matrix of pressure-sensor type. The preconditioned equations are solved with the Jameson's numerical approach, i.e. artificial dissipation and artificial viscosity terms under the form of a fourth- and second-order derivative, respectively. An explicit four-stage Runge–Kutta integration algorithm is applied to obtain the steady-state condition. The computed results include the steady-state solution of flow past the NACA-hydrofoils and a circular cylinder in free stream, for which the numerical results are compared with numerical works of other researchers. Good agreement is observed. The effects of AC parameter, artificial viscosity and dissipation factor, and local preconditioning coefficient on convergence rate and solution accuracy are tested by computing flow over the NACA0012 hydrofoil. In addition, some important design criteria of a preconditioner, such as stiffness reduction, hyperbolicity, symmetrisability, accuracy preservation for M → 0, and M-property have been examined analytically.     

共轭梯度法求解非线性多宗量稳态传热反问题

应用共轭梯度法求解非线性多宗量稳态热传导反问题。采用八节点的等参单元在空间上进行离散,建立了便于敏度分析的非线性正演和反演的有限元模型,可直接求导进行敏度分析。给出了相关的数值验证,对测量误差及测点数目的影响作了初步探讨,结果表明,采用的算法能够对非线性稳态热传导中导热系数和边界条件联合反问题进行有效的求解,并具有较高精度。     

Extremum problems of boundary control for steady equations of thermal convection

An inverse extremum problem of boundary control for steady equations of thermal convection is considered. The cost functional in this problem is chosen to be the root-mean-square deviation of flow velocity or vorticity from the velocity or vorticity field given in a certain part of the flow domain; the control parameter is the heat flux through a part of the boundary. A theorem on sufficient conditions on initial data providing the existence, uniqueness, and stability of the solution is given. A numerical algorithm of solving this problem, based on Newton’s method and on the finite element method of discretization of linear boundary-value problems, is proposed. Results of computational experiments on solving extremum problems, which confirm the efficiency of the method developed, are discussed.     

Boundary element method for steady 2D high-Reynolds-number flow

This paper deals with the numerical simulation of fluid dynamics using the boundary–domain integral technique (BEM). The steady 2D diffusion–convection equations are discussed and applied to solve the plane Navier-Stokes equations. A vorticity–velocity formulation has been used. The numerical scheme was tested on the well-known ‘driven cavity’ problem. Results for Re = 1000 and 10,000 are compared with benchmark solutions. There are also results for Re = 15,000 but they have only qualitative value. The purpose was to show the stability and robustness of the method even when the grid is relatively coarse.     

Winkler partial slip solution for harmonic oscillations in steady rolling contact problems

A Winkler model (Kalker’s simplified theory) is adopted for solving analytically partial slip rolling contact problem in the first order perturbation form of small periodic oscillations of generally both normal and tangential load about a steady state. At present, only numerical investigations exist for this problem, with various approximations to deal with the transient effects (often, simply neglected), and particularly the effect of varying normal load and hence contact area, has not been investigated in detail, despite the problem of corrugation is essentially driven by the change of normal load.The linear perturbation analysis is used to obtain closed form expressions for the receptances of the tangential load. Also, similar expressions are obtained for the energy dissipation, which is correlated with the local wear.     

Anisotropic Cartesian grid method for steady inviscid shocked flow computation

The anisotropic Cartesian grid method, initially developed by Z.N. Wu (ICNMFD 15, 1996; CFD Review 1998, pp. 93–113) several years ago for efficiently capturing the anisotropic nature of a viscous boundary layer, is applied here to steady shocked flow computation. A finite‐difference method is proposed for treating the slip wall conditions. Copyright © 2003 John Wiley & Sons, Ltd.     

An efficient quasi-Newton method for two-dimensional steady free surface flow

Steady free surface flows are of interest in the fields of marine and hydraulic engineering. Fitting methods are generally used to represent the free surface position with a deforming grid. Existing fitting methods tend to use time-stepping schemes, which is inefficient for steady flows. There also exists a steady iterative method, but that one needs to be implemented with a dedicated solver. Therefore a new method is proposed to efficiently simulate two-dimensional (2D) steady free surface flows, suitable for use in conjunction with black-box flow solvers. The free surface position is calculated with a quasi-Newton method, where the approximate Jacobian is constructed in a novel way by combining data from past iterations with an analytical model based on a perturbation analysis of a potential flow. The method is tested on two 2D cases: the flow over a bottom topography and the flow over a hydrofoil. For all simulations the new method converges exponentially and in few iterations. Furthermore, convergence is independent of the free surface mesh size for all tests.     

Variational iteration method for two-dimensional steady slip flow in micro-channels

In this paper, the two-dimensional steady slip flow in microchannels is investigated. Research on micro flow, especially on micro slip flow, is very important for designing and optimizing the micro electromechanical system (MEMS). The Navier-Stokes equations for two-dimensional steady slip flow in microchannels are reduced to a nonlinear third-order differential equation by using similarity solution. The variational iteration method (VIM) is used to solve this nonlinear equation analytically. Comparison of the result obtained by the present method with numerical solution reveals that the accuracy and fast convergence of the new method.     

A boundary-value problem for a nonlinear heat-conduction equation

Methods for the numerical solution of the boundary-value problem are comidered. One method is related to the method of successive approximations and the other employs the collocation method [1]. A relationship between the latter method and the Ritz and Galerkin methods [2] is shown. An application of the collocation method to the nonstationary problem is given. The approximate solution is represented in analytic form. A way of finding the absolute error of the approximate solution is given.     

A 3D direct coupling method for steady ship hydroelastic analysis

Asides from the influence of incoming waves, ships can experience steady motions, such as rigid-body sinkage and trim motions, and flexible-body vertical bending motions, due to a constant forward speed even under calm water conditions. In this paper, a novel approach to analyze steady-ship hydroelasticity, particularly for the steady-ship motions and surrounding steady-wave disturbances, is proposed using a three-dimensional (3D) direct coupling method, based on a higher-order boundary element method (HOBEM) and a higher-order shell finite element method (FEM). Within the linearized framework, a solution method is proposed based on a two-step procedure, using two types of Neumann–Kelvin (NK) linear flow models for the fluid part and a virtual work equilibrium equation for the structural part. The first step is to compute a mean position wave-resistance problem using the modified NK equation, the second step is to solve a perturbed position wave-resistance problem, by employing a classical NK model and a virtual work equation based on the first step’s solution. Detailed mathematical formulation and numerical procedures are described, and a few numerical results are illustrated. These include both rigid and flexible steady-ship motions, Von-Mises stress distributions, and wave-resistance coefficients for Froude numbers ranging from 0.15 to 0.5. Furthermore, the numerical results obtained using the present direct coupling method and a modal-based one are compared.