横向剪变形对具有一小圆孔的浅壳应力集中系数的影响

本文导出了分析计及横向剪变形的浅壳小孔应力集中问题的简化方程.对浅球壳和圆柱壳带一小圆孔的情况,获得了方程的级数形式通解.对均匀内压作用下的圆柱壳,求得了小圆孔边上应力集中系数的近似显式解,并计算了数值结果.     

当绝热指数很大时气体的一种非定常平面平行运动的渐近解

文献[1]在气体的速度分量只与极角θ及时间t有关,而与极距r无关的条件下求解理想气体的非定常平面平行具势运动方程组(1.2)～(1.3).文献[1]指出,在一般的情况下,不能得到解的显式表示式,只是对于二种特殊情况得到了显式解.本文研究了文献[1]的同样问题.第一部分,对音速作了一些补充限制,从而得到了方程组的显式解.第二部分,假设气体的绝热指数γ>>1,求得了方程组的一级近似解.     

一类双参数非线性高阶反应扩散方程的摄动解法

研究了一类两参数非线性反应扩散奇摄动问题的模型.利用奇摄动方法,对该问题解的结构在两个小参数相互关联的情形下作了讨论.首先,构造问题的外部解; 之后在区域的边界邻域构造局部坐标系,再在该邻域中引入多尺度变量,得到问题解的边界层校正项; 然后引入伸长变量,构造初始层校正项,并得到问题解的形式渐近展开式;最后建立了微分不等式理论,并由此证明了问题的解的一致有效的渐近展开式.用上述方法得到的各次近似解,具有便于求解、精度高等特点.     

两种坐标系中旋转体的体积计算

分别在直角坐标系和极坐标系情形下,计算笛卡尔叶形线、心形线、四叶玫瑰线和伯努利双纽线所围区域绕轴旋转所得旋转体体积。     

用"残缺"的坐标系解题

建立直角坐标系来解立体几何问题是非常有效的常用方法,但是往往要建立确定的坐标系,即X轴、Y轴、Z轴和某一确定直线是重合的,其实有时并不需要每个坐标轴都要确定.     

一类证券市场中投资组合及消费选择的最优控制问题

研究一类证券市场中投资组合及消费选择的最优控制问题．在随机干扰源相互关联情形下,运用动态规划方法,对一类典型的效用函数CRRA(Constant Relative Risk Aversion,常数相对风险厌恶)情形,得到了最优投资组合及消费选择的显式解,并给出了最优解的经济解释和关于部分参数的灵敏度分析．     

直线爆炸所形成的柱面波的近似分析解

由直线爆炸所形成的柱面波,文献[1]得到了数值解。本文对于绝热指数γ=C_p/C_v较大的情形,取ε=1/γ2为小参数,用伸缩坐标法得到了问题的一级近似分析解。算例(γ=3)表明,近似解与准确解(数值解)十分符合。     

三级四阶显式辛R-K-N格式的完全构造

s级p阶辛Runge-Kutta-Nystr\"om(R-K-N)方法的一种充要条件是用关于参数的非线性方程组来表示的,辛R-K-N格式的构造问题因而转化为该方程组的求解问题. 在一些特殊的限定条件下, 已有该方程组在s=3,p=4时的两组解,即得到了两个三级四阶显式辛格式. 对于s=3,p=4情形,基于吴方法,利用计算机代数系统Maple及软件包wsolve给出了对应的非线性方程组的全部解, 这样就构造了所有的三级四阶显式辛R-K-N格式, 并证明了三级四阶显式辛R-K-N方法所满足的条件方程有冗余. 数值实验结果显示出新的辛格式在一定的条件下有着较好的误差精度.     

薄板弯曲自由振动问题的高精度近似解析解及改进研究

对于薄板弯曲自由振动问题,已有如下方法:在Hamilton(哈密顿)体系下基于分离变量法得到挠度的解析形式,并建立自振频率联立方程组,给出求解振动频率和振型函数的方法.笔者指出该方法中所用挠度函数的解析式实际上是一种满足位移边界条件的高精度近似解,基于Rayleigh-Ritz(瑞利-里茨)法再次求近似频率后发现,原方法的近似解的精度很高.另外,对于含有固支、简支等不同的边界形式,恰当地选取不同位置作为坐标系的原点,得到含有频率的方程组的统一形式,且较为简洁.这些形式可基于四边固支、四边简支等边界条件的矩形板研究,依照板变形的对称性可验证频率方程组形式的正确性,并得到不同边界条件下频率方程形式之间的联系与转化.     

Jaulent-Miodek方程组和长水波近似方程组的新精确解

首先,利用直接代数法给出了一类非线性方程的四组显式精确解的公式.进而,很方便地得到了Jaulent-Miodek方程组和长水波近似方程组的若干新精确解.     

AN EMBEDDED BOUNDARY METHOD FOR ELLIPTIC AND PARABOLIC PROBLEMS WITH INTERFACES AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS

The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella （1998, J. Comput. Phys. 147, 60） has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems.     

A symmetric integrated radial basis function method for solving differential equations

In this article, integrated radial basis functions (IRBFs) are used for Hermite interpolation in the solution of differential equations, resulting in a new meshless symmetric RBF method. Both global and local approximation‐based schemes are derived. For the latter, the focus is on the construction of compact approximation stencils, where a sparse system matrix and a high‐order accuracy can be achieved together. Cartesian‐grid‐based stencils are possible for problems defined on nonrectangular domains. Furthermore, the effects of the RBF width on the solution accuracy for a given grid size are fully explored with a reasonable computational cost. The proposed schemes are numerically verified in some elliptic boundary‐value problems governed by the Poisson and convection‐diffusion equations. High levels of the solution accuracy are obtained using relatively coarse discretisations.     

分数外微分在三维空间中的应用

首先介绍了分数微积分和分数微分形式。讨论了在原点处对曲线坐标的分数外微分变换,并且获得了从三维卡氏坐标到球面坐标和柱面坐标的两个分数微分变换。特别地,当v=m=1时,这两个分数微分变换约化的结果与通过外微积分获得的结果是一致的。     

A regularization of a reaction–diffusion system approximation to the two-phase Stefan problem

Reaction–diffusion system approximations to the classical two-phase Stefan problem are considered in the present study. A reaction–diffusion system approximation to the Stefan problem has been proposed by Hilhorst et al. from an ecological point of view, and they have given convergence results for the system. In the present study, a new reaction–diffusion system approximation to the Stefan problem is proposed based on regularization of the enthalpy–temperature constitutive relation. For a deeper understanding of the approximation mechanism by means of reaction–diffusion systems, the rates of convergence for both the solutions and the free boundaries are investigated.     

Kriging surrogate model with coordinate transformation based on likelihood and gradient

The Kriging surrogate model, which is frequently employed to apply evolutionary computation to real-world problems, with a coordinate transformation of the design space is proposed to improve the approximation accuracy of objective functions with correlated design variables. The coordinate transformation is conducted to extract significant trends in the objective function and identify the suitable coordinate system based on either one of two criteria: likelihood function or estimated gradients of the objective function to each design variable. Compared with the ordinary Kriging model, the proposed methods show higher accuracy in the approximation of various test functions. The proposed method based on likelihood shows higher accuracy than that based on gradients when the number of design variables is less than six. The latter method achieves higher accuracy than the ordinary Kriging model even for high-dimensional functions and is applied to an airfoil design problem with spline curves as an example with correlated design variables. This method achieves better performances not only in the approximation accuracy but also in the capability to explore the optimal solution.     

A hybrid coupling interface method for elliptic complex interface problems

We propose a coupling interface method (CIM) under Cartesian grid for solving elliptic complex interface problems in arbitrary d dimensions, where the coefficients, the source terms, and the solutions may be discontinuous or singular across the interfaces. It consists of a first-order version (CIM1) and a second-order version (CIM2). In one dimension, this finite difference method at a grid point adjacent to the interface is derived based on piecewise linear (CIM1) or quadratic (CIM2) approximation of the solution and two jump conditions. The method is extended to high dimensions through a dimensionby-dimension approach. To connect information from each dimension, a coupled equation for the principal derivatives is derived through the jump conditions in each coordinate direction. For CIM2, one-side interpolation for cross derivatives is need. This coupling approach reduces number of grid point in the finite difference stencil. The hybrid method uses CIM1 or CIM2 adaptly for complex interface. Numerical tests demonstrate that CIM1 and CIM2 are respectively first order and second order in the maximal norm with less error as compared with other methods. In addition, the hybrid CIM can solve complex interface problems in two and three dimensions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

(Robust) Edge-based semidefinite programming relaxation of sensor network localization

Recently Wang, Zheng, Boyd, and Ye (SIAM J Optim 19:655–673, 2008) proposed a further relaxation of the semidefinite programming (SDP) relaxation of the sensor network localization problem, named edge-based SDP (ESDP). In simulation, the ESDP is solved much faster by interior-point method than SDP relaxation, and the solutions found are comparable or better in approximation accuracy. We study some key properties of the ESDP relaxation, showing that, when distances are exact, zero individual trace is not only sufficient, but also necessary for a sensor to be correctly positioned by an interior solution. We also show via an example that, when distances are inexact, zero individual trace is insufficient for a sensor to be accurately positioned by an interior solution. We then propose a noise-aware robust version of ESDP relaxation for which small individual trace is necessary and sufficient for a sensor to be accurately positioned by a certain analytic center solution, assuming the noise level is sufficiently small. For this analytic center solution, the position error for each sensor is shown to be in the order of the square root of its trace. Lastly, we propose a log-barrier penalty coordinate gradient descent method to find such an analytic center solution. In simulation, this method is much faster than interior-point method for solving ESDP, and the solutions found are comparable in approximation accuracy. Moreover, the method can distribute its computation over the sensors via local communication, making it practical for positioning and tracking in real time.     

Addendum to the paper “on certain exact solutions of the fourier equation for regions varying with time” : PMM vol. 37, n6, 1973, pp. 1145–1146

The exact solutions given in [1] are generalized to the case of cylindrical and spherical sectors rotating about the azimuth relative to the coordinate origin either at a uniform rate or with uniform acceleration (or deceleration). The class of equations of motion of the boundaries of the half-space (in the Cartesian coordinates) which lead to exact solutions of the Fourier equation defined in these domains, is enlarged.     

An approximation technique for the numerical solution of a Stefan problem

Summary A nonlinear approximation technique for the numerical solution of certain free boundary problems is proposed. The method is shown for a degenerate one-dimensional Stefan problem. For this problem, an error estimate, which is independent of the used algorithm, is derived. Numerical examples are discussed.This paper was written when the author held a 1 1/2 year postdoctoral position at the Department of Mathematics and Applied Mathematics Institute of the University of Delaware     

Solving a set of global optimization problems by the parallel technique with uniform convergence

In this paper, we consider solving a set of global optimization problems in parallel. The proposed novel algorithm provides uniform convergence to the set of solutions for all problems treated simultaneously. The current accuracy for each particular solution is estimated by the difference in each coordinate from the point of global decision. The main statement is given in the corresponding theorem. For the sake of illustration some computational results with hundreds of multidimensional global problems are provided.