A Nodal Method for Phase Change Moving Boundary Problems

A semi-analytic numerical scheme has been developed to solve the one-dimensional, moving boundary phase change problem with time-dependent boundary conditions. Locally analytic, approximate solutions are developed for the position of the moving boundary, and for temperature distribution. Set of discrete equations are obtained by applying these solutions over space-time nodes, and by imposing continuity of temperature and heat flux. Application of this so-called nodal integral approach to the nonlinear Stefan problem shows that the scheme is O(Δx ²), and that it predicts the position of the moving boundary and the temperature distribution within the domain very accurately. For example, with as little as two nodes in the spatial domain, the location of the moving boundary for the case of an exponentially increasing surface temperature on the boundary, after one dimensionless time unit, is found with an error of less than 1%. In addition to large size nodes in space, this scheme also allows the use of very large size time steps. Comparison of numerical results with reference solutions is presented.     

浸入边界法及其应用

浸入边界法主要用于模拟存在复杂外形结构的流场的运动情况和处理各种动边界问题,目前已广泛应用于计算流体力学领域.浸入边界法既是数学建模方法又是数值离散方法,它将物体边界与流体的相互作用通过在流体运动方程中加体积力项来体现,并在数值计算中采用简单的笛卡尔网格,避免了按照物体边界形状生成贴体网格时所遇到的各种问题.浸入边界法分为连续力法和离散力法:连续力法主要用于处理弹性边界问题,它的力源项满足特定的力学关系式;离散力法主要用于处理固体界面问题,它的力源项由边界条件推导得到.着重阐述了浸入边界法的基本原理和数学构造,对目前已有的几种不同的浸入边界法做了简单地介绍,并给出了一些应用实例,最后提出了浸入边界法未来的发展方向.     

Boundary Layers in Weak Solutions of Hyperbolic Conservation Laws

. This paper is concerned with the initial‐boundary‐value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing‐viscosity method and finite‐difference schemes (Lax‐Friedrichs‐type schemes and the Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform bounds on the approximate solutions and so dealing with solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. From the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based on a set of admissible boundary values, following the terminology of Dubois & LeFloch[15]. The local structure of these sets and the well‐posedness of the corresponding initial‐boundary‐value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics. (Accepted July 2, 1998)     

Boundary element method for arbitrary elastic thin shell

In this paper, a boundary element scheme for arbitrary elastic thin shells is elaborated, Based on BEM of 3D linear elasticity and Kirchhoff's hypothesis, boundary integral equations for shells are deduced. As a result, only Kelvin's solution is used, the difficulty in finding a fundamental solution of arbitrary shells is successfully avoided. The project supported by National Natural Science Foundation of China     

Adaptive Boundary Conditions for Exterior Flow Problems

We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain in two dimensions. This corresponds to studying the stationary fluid flow past a body. The necessity to truncate for numerical purposes the infinite exterior domain to a finite domain leads to the problem of finding appropriate boundary conditions on the surface of the truncated domain. We solve this problem by providing a vector field describing the leading asymptotic behavior of the solution. This vector field is given in the form of an explicit expression depending on a real parameter. We show that this parameter can be determined from the total drag exerted on the body. Using this fact we set up a self-consistent numerical scheme that determines the parameter, and hence the boundary conditions and the drag, as part of the solution process. We compare the values of the drag obtained with our adaptive scheme with the results from using traditional constant boundary conditions. Computational times are typically reduced by several orders of magnitude.     

非连续边界元积分的精确表达式及相关问题

以二维位势问题边界元分析为例,给出了利用线性非连续边界元离散边界积分方程时系数矩阵积分计算的精确表达式,通过和利用Gauss积分方法计算系数矩阵所得数值结果的比较表明：配位点选择不同对数值计算结果精度影响的主要原因是积分计算的精度,尤其当配位因子选择较大时,存在的准奇异积分（Nearly Singular Integrals)很难利用常规Gauss积分方法准确求得。     

Boundary Integral Procedures for Unsaturated Flow Problems

Abstract. A novel numerical scheme based on the singular integral theory of the boundary element method. (BEM) is presented for the solution of transient unsaturated flow in porous media. The effort in the present paper is directed in facilitating the application of the boundary integral theory to the solution of the highly non-linear equations that govern unsaturated flow. The resulting algorithm known as the Green element method (GEM) presents a robust attractive method in the state-of -the-art application of the boundary element methodology. Three GEM models based on their different methods of handling the non-linear diffusivity, illustrate the suitability and robustness of this approach for solving highly non-linear 1-D and 2-D flows which would have proved cumbersome or too difficult to implement with the classical BEM approach.     

A Conservative Difference Scheme for Conservative Differential Equation with Periodic Boundary

1　ＤｉｆｆｅｒｅｎｔｉａｌＥｑｕａｔｉｏｎａｎｄＤｉｆｆｅｒｅｎｔｉａｂｉｌｉｔｙＰｒｏｐｅｒｔｉｅｓｏｆｔｈｅＳｏｌｕｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｃｏｎｓｅｒｖａｔｉｖｅｆｏｒｍａｎｄｓｉｎｇｕｌａｒｐｅｒｔｕｒｂｅｄｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｗｉｔｈｐｅｒｉｏｄｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ :Ｌｕ(ｘ) ≡ε(ｐ(ｘ)ｕ′(ｘ) )′ (ｑ(ｘ)ｕ(ｘ) )′-ｒ(ｘ)ｕ(ｘ) =ｆ(ｘ)　　( 0 <ｘ<1 ) ,( 1 )ｕ( 0 ) ≡ｕ( 1 ) ,ｌｕ≡ｕ′( 1 )…     

Boundary Singularity for Thermal Transpiration Problem of the Linearized Boltzmann Equation

We study the boundary singularity of the fluid velocity for the thermal transpiration problem in the kinetic theory. Logarithmic singularity has been demonstrated through the asymptotic and computational analysis. The goal of this paper is to confirm this logarithmic singularity through exact analysis. We use an iterated scheme, with the “gain” part of the collision operator as a source. The iterated scheme is appropriate for large Knudsen numbers considered here and yields an explicit leading term.     

On the Numerical-Analytic Investigation of Parametrized Problems with Nonlinear Boundary Conditions

We consider a parametrized boundary-value problem containing an unknown parameter both in nonlinear ordinary differential equations and in nonlinear boundary conditions. By using a suitable change of variables, we reduce the original problem to a family of problems with linear boundary conditions plus certain nonlinear algebraic determining equations. We construct a numerical-analytic scheme suitable for studying the solutions of the transformed boundary-value problem.     

Boundary layer of a body of revolution in a drag-reducing polymer solution

A possible method is presented for calculating an axisymmetric boundary layer when a body moves in weak polymer solutions with constant concentration. The method is based on the use of a velocity profile and system of integral equations which take into account most fully the effects of the transverse curvature of the body's surface. The computational scheme makes it possible to take into account the change in the flow regimes in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–48, May–June, 1981.     

弹性力学平面问题中一类无奇异边界积分方程

从理论上提出一种新的方法，归化出间接变量无奇异边界积分方 程. 采用Lagrange二次单元，建立一个数值求解框架系统. 此外，基于问题的计算区域的 特殊性，给出一种边界近似方法. 数值算例表明该方法所取得的数值结果与精确解相当接 近，特别是边界量的数值结果. 此外，该方法容易被推广到三维问题. 和已有的直接变量的情形相比较，具有优点：1)无需处理HFP积分. 大大降低处理问 题的复杂性，并提高了计算效率和解的精度；2)摆脱了问题的具体形式，进入纯代数操作. 这样做的好处是从理论上建立一种普遍适用的方法，不仅适用于弹性力学问题，同样可应用 于其它问题，如位势问题, Stokes问题等. 3)提供了一种计算CPV积分的方法.     

Boundary element method for moving and rolling contact of 2D elastic bodies with defects

A scheme of boundary element method for moving contact of two-dimensional elastic bodies using conforming discretization is presented. Both the displacement and the traction boundary conditions are satisfied on the contacting region in the sense of discretization. An algorithm to deal with the moving of the contact boundary on a larger possible contact region is presented. The algorithm is generalized to rolling contact problem as well. Some numerical examples of moving and rolling contact of 2D elastic bodies with or without friction, including the bodies with a hole-type defect, are given to show the effectiveness and the accuracy of the presented schemes. The project supported by the National Natural Science Foundation of China (19772025)     

Optimal Control of Nonmodal Disturbances in Boundary Layers

The optimal control of infinitesimal flow disturbances experiencing the largest transient gain over a fixed time span, commonly termed “optimal perturbations,” is undertaken using a variational technique in two- and three-dimensional boundary layer flows. The cost function employed includes various energy metrics which can be weighted according to their perceived importance, simplifying the task of determining which terms are essential for a “good” control scheme. In the accelerated boundary layers investigated, disturbance kinetic energy can be typically reduced by about one order of magnitude. However, it seems impossible to suppress completely over the entire control interval; “good” control strategies still permit approximately an order magnitude growth over the initial energy at some point in the interval. It is shown that the control effort efficiently targets the physical mechanisms behind transient growth. Received 5 February 2001 and accepted 15 June 2001     

Nonlinear Boundary Control of the Generalized Burgers Equation

In this paper, the adaptive and non-adaptive stabilization of the generalized Burgers equation by nonlinear boundary control are analyzed. For the non-adaptive case, we show that the controlled system is exponentially stable in L². As for the adaptive case, we present a novel and elegant approach to show the L² regulation of the solution of the generalized Burgers system. Numerical results supporting and reinforcing the analytical ones of both the controlled and uncontrolled system for the non-adaptive and adaptive cases are presented using the Chebychev collocation method with backward Euler method as a temporal scheme.     

一类平衡大气边界层边界条件研究

基于标准k-ε湍流模型,首先利用湍流粘度方程和剪切应力在整个边界层内恒定的假设,推导出一类耗散率表达式,并根据常用的湍动能入口剖面方程以及平均风速剖面方程,计算获得相应的耗散率方程;然后在输运方程中添加自定义源项,通过已经确定的平均速度方程、湍动能方程、耗散率方程计算得到相应输运方程的自定义源项表达式,并进行空风洞数值模拟,从而得到了一类满足平衡大气边界层的来流边界条件．通过将这种边界条件与由湍流平衡条件得到的边界条件进行比较,表明本方法获得的边界条件更适用．并且,本方法无需考虑修正壁面函数和修正湍流模型常数,因而计算更为简单,可为平衡大气边界层的研究提供一种新的思路．     

Boundary element method for thermoelastic analysis of three-dimensional transversely isotropic solids

Thermal effects are well known to manifest themselves as additional volume integral terms in the direct formulation of the boundary integral equation (BIE) for linear elastic solids when using the boundary element method (BEM). This domain integral has been successfully transformed in an exact manner to surface ones only in isotropy and in 2D anisotropy, thereby restoring the BEM as a truly boundary solution technique. The difficulties with extending it to 3D general anisotropic solids lie in the mathematical complexity of the Green’s function and its derivatives for such materials. These quantities are required items in the BEM formulation. In this paper, the exact, analytical transformation of the volume integral associated with thermal effects to surface ones is achieved for a transversely isotropic material using a similar approach which the authors have previously employed for the same task in BEM for 2D general anisotropy. A numerical scheme, however, needs to be employed to evaluate some of the new terms introduced in the surface integrals that arise from this process here. The mathematical soundness of the formulation is demonstrated by a few examples; the numerical results obtained are checked by alternative means, including those obtained from the commercial FEM code, ANSYS.     

Solution of Dynamic Problems for Thin-Walled Structural Elements with Nonuniform Dynamic Boundary Conditions

The characteristics of the stress–strain state of thin-walled structural elements are determined in the case where dynamic boundary loads or displacements described by pulse functions are specified. A general scheme for realization of the method of natural-mode expansion is stated as applied to differential equations with unknown functions of one spatial coordinate and time. Theoretical relations for rods, plates, and shells are given. The potential of the approach developed is illustrated by solving specific problems     

Non-Darcy Mixed Convection in a Vertical Porous Channel with Boundary Conditions of Third Kind

Combined free and forced convection flow in a parallel plate vertical channel filled with porous matrix is analyzed in the fully developed region with boundary conditions of third kind. The flow is modeled using the Brinkman?CForchheimer-extended Darcy equations. The plates exchange heat with an external fluid. Both conditions of equal and different reference temperatures of the external fluid are considered. Governing equations are solved numerically by shooting technique that uses classical explicit Runge?CKutta scheme and Newton?CRaphson method as a correction scheme and analytically using perturbation series method for Darcy model. The velocity field, the temperature field and Nusselt numbers are obtained for governing parameters such as porous parameter, inertia term and perturbation parameter for equal and unequal Biot numbers and are displayed graphically. The dimensionless mean velocity and bulk temperature are also determined. It is found that the numerical solutions agree for small values of the perturbation parameter in the absence of the inertial forces.     

水闸闸室结构三维样条边界元分析

用三维样边界元法分析水闸闸室结构。底板，闸墩和载水墙等为其子结构，交通桥，工作桥和胸墙等处理为内部支撑。地基和边载可以是任意的，只要能给定地表位移面力关系。在各种工况下，不论是设置平板门还是弧形门，是平底板是反拱底板，即使在稀疏剖分下也能给出高精度的位移场，应力场和地基反力场。