Numerical solution of 2D elastostatic problems formulated by potential functions

2D linear elastostatic problems formulated in Cartesian coordinates by potential functions are numerically solved by network simulation method which allows an easy implementation of the complex boundary conditions inherent to this type of formulation. Four potential solutions are studied as governing equations: the general Papkovich–Neuber formulation, which is defined by a scalar potential plus a vector potential of two components, and the three simplified derived formulations obtained by deleting one of the three original functions (the scalar or one of the vector components). Application of this method to a rectangular plate subjected to mixed boundary conditions is presented. To prove the reliability and accurate of the proposed numerical method, as well as to demonstrate the suitability of the different potential formulations, numerical solutions are compared with those coming from the classical Navier formulation.     

具组合型非线性项与调和位势的非线性Schr(o)dinger方程

讨论了一类带有组合型非线性项与调和位势的非线性Schr(o)dinger方程.通过构造变分问题,引入位势井方法.给出了位势井的结构和位势井深度函数的性质.得到了问题的相关集合在流之下的不变性.揭示了只要问题的初值属于位势井内或位势井外,则问题在今后所有时间内的解都存在于位势井内或井外.结合凹性方法,给出了解的整体存在性的最佳条件.     

Potential Renormalization Method for a Model of the Hartree–Fock–Slater Type

We propose a new potential renormalization method for a semiclassical model of the Hartree–Fock–Slater type. The method allows easily constructing the wave functions and, in contrast to most of the similar methods, does not require an explicit form of the potential.     

The WKB method and differential consequences of the Riccati equation

A generalized WKB method based on the use of the differential consequences of the Riccati equation is presented. The method combines the simplicity of the traditional WKB method and the universality of the Maslov method: in the case of a smooth potential with classical turning points in a bounded space interval, the leading term of the expansion is found as a root of an algebraic equation and provides a regular approximate solution in the whole domain of the potential; we can increase the accuracy of this solution by taking new differential consequences into account.     

A MIXED FINITE ELEMENT METHOD FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE

The transient behavior of a semiconductor device is described by a system of three quasilinear partial differential equations. One is elliptic in form for the electric potential and the other two are parabolic in form for the conservation of electron and hole concentrations. The electric potential equation is discretized by a mixed finite element method. The electron and hole density equations are treated by a Galerkin method that applies a variant of the method of characteristics to the transport terms. Optimal order convergence analysis in L2 is given for the proposed method.     

Effective potentials and Bogoliubov’s quasi-averages

The effective potential method, used in quantum field theory to study spontaneous symmetry violation, is discussed from the point of view of Bogoliubov’s quasi-averaging procedure. It is shown that the effective potential method is a disguised type of this procedure. The catastrophe theory approach to the study of phase transitions is discussed and the existence of the potentials used in that approach is proved from the statistical point of view. It is shown that in the case of broken symmetry, the nonconvex effective potential is not a Legendre transform of the generating functional for connected Green’s functions. Instead, it is a part of the potential used in catastrophe theory. The relationship between the effective potential and the Legendre transform of the generating functional for connected Green’s functions is given by Maxwell’s rule. A rigorous rule for evaluating quasi-averaged quantities within the framework of the effective potential method is established. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 149–161, October, 1997.     

Potential method applied to Boussinesq equation

In this paper a solution of the nonlinear Boussinesq equation is presented using the potential similarity transformation method. The equation is first written in a conserved form, a potential function is then assumed reducing it to a system of equations which is further solved through the group transformation method. New transformations are found.     

On Direct and Inverse Problems of Logarithmic Potential With Finite Number of Parameters

We describe possibilities of realizing V. K. Ivanov’s method for solving inverse problems of logarithmic potential and that of A. V. Tsirul’skii’s method for solving direct problems of logarithmic potential. We solve two direct problems for logarithmic potential and for simple layer potential in the case of polygonal contour.     

Reproducing-kernel-based splines for the regularization of the inverse ellipsoidal gravimetric problem

To solve the inverse gravimetric problem, i.e. to reconstruct the Earth's mass density distribution by using the gravitational potential, we introduce a spline interpolation method for the ellipsoidal Earth model, where the ellipsoid has a rotational symmetry. This problem is ill-posed in the sense of Hadamard as the solution may not exist, it is not unique and it is not stable. Since the anharmonic part (orthogonal complement) of the density function produces a zero potential, we restrict our attention only to reconstruct the harmonic part of the density function by using the gravitational potential. This spline interpolation method gives the existence and uniqueness of the unknown solution. Moreover, this method represents a regularization, i.e. every spline continuously depends on the given gravitational potential. These splines are also combined with a multiresolution concept, i.e. we get closer and closer to the unknown solution by increasing the scale and adding more and more data at each step.     

Boundary element method for simulating the coupled motion of a fluid and a three-dimensional body

A boundary element method for potential flow problem coupled with the dynamics of rigid body was developed to determine numerically the resultant force and moment of force acting on an arbitrarily three-dimensional solid body and its motion in a current of an infinite fluid. An accurate integration method for singular integrands occurring in the boundary integral equations, a computational method for the tangential gradient of a velocity potential on a surface, and a method to properly treat the singularities appearing in the system of the dynamic equations of a rigid body, were proposed to complete the numerical solution of the problem. Several numerical examples were given to show the validity of the method.     

企业的潜显信息资源探讨

企业在发展过程中,会碰到各式各样的信息资源矛盾问题.解决此类问题的首要条件是企业能够全面认识自己的信息资源.可拓学提供了共轭分析方法来帮助企业从物质性、系统性、动态性、对立性全面分析信息资源.重点研究潜显信息,抽象出潜显信息的概念模型,提出真伪显信息的判别法以及在挖掘潜信息过程中生成策略的方法.     

The potential method for conditional maxima in the locally compact metric spaces

Summary In the paper there is presented a method of reducing the constrained maximization problem to an unconditional one. The method consists of introducing a function, called potential, which at an unconditional maximum converges to the constrained maximum when a positive parameter of the potential tends to zero. A proof of such convergence for the locally compact metric spaces is given.     

二维流面上的流动问题的速度图方法

对于一些特殊的流动,尤其是平面上的位势流动,速度图方法有其显著的优点．对于理想流体来说,流面总是存在的,在流面上,流动的速度向量总是在其切空间里．通过引入流函数和势函数,采用张量分析作为工具,给出了二维曲流面上位势流动的速度图方法,得到了流函数满足的速度图方程,为一些特殊的流动问题提供了一类分析方法．并且,对于得到的二维速度图方程,得到了相应的特征方程和特征根,从而可以对方程的类型进行分类．最后,给出了一些特殊流动的实例．     

一类非线性波动方程的势对称分类

先给出了含有一个任意函数的线性波动方程的古典和势对称的完全分类.然后,在此基础上给出了含有两个任意函数的一类非线性波动方程的两种情形势对称分类,得到了该方程的新势对称.在处理对称群分类问题的难点-求解确定方程组时我们提出了微分形式吴方法算法,克服了以往难于处理的困难.在整个计算过程中反复使用了吴方法,吴方法起到了关键的作用.     

A Fourier method for computing the perturbation to a two‐dimensional electric potential by a conductor of arbitrary shape

Introducing an electric conductor into a region pervaded by an initial electric potential perturbs that potential by inducing a charge distribution on the conductor's surface, necessary to guarantee that the surface is an equipotential of the total potential. Some numerical method is required to compute the perturbation potential, when the conductor's shape does not admit a standard analytic solution. For two‐dimensional situations, a method is proposed for solving for the perturbation potential that involves expansion of the boundary perturbation potential and its normal derivative as truncated Fourier series. This boundary potential is known to within an additive constant from the requirement that its sum with the initial potential must be a constant. The standard representation theorem for the Dirichlet problem gives a consistency relation between the boundary function and its normal derivative, which here becomes a set of linear algebraic relations between Fourier series coefficients, with matrix entries found by appropriate applications of the fast Fourier transform. These are solved for the boundary derivative coefficients; at any point exterior to the conductor, the perturbation potential can then be evaluated from the two sets of Fourier coefficients, using further application of the fast Fourier transform. Examples are shown for two conductor shapes, with several initial potentials. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 673–683, 2001     

A Kind of Discretization of the KdV Hierarchy

We propose a method for introducing higher-order terms in the potential expansion in order to study the continuum limits of the Toda hierarchy. These higher-order terms are differential polynomials in the lower-order terms. This type of potential expansion allows using fewer equations in the Toda hierarchy to recover the KdV hierarchy by the so-called recombination method. We show that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend toward the corresponding objects of the KdV hierarchy in the continuum limit. This method gives a kind of discretization of the KdV hierarchy.     

A potential function method for the Cauchy problem of elliptic operators

In this paper, we deal with the Cauchy problem of elliptic operators. Through the use of a single-layer potential function, we devise a numerical method for approximating the solution of the Cauchy problem of elliptic operators, which are well known to be highly ill-posed in nature. The method is based on the denseness of single-layer potential functions. Convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method. It has been observed that the use of more Cauchy data will greatly improve the accuracy of the approximate solutions.     

Radial Schrödinger equation: The spectral problem

Using the integral transformation method involving the investigation of the Laplace transforms of wave functions, we find the discrete spectra of the radial Schrödinger equation with a confining power-growth potential and with the generalized nuclear Coulomb attracting potential. The problem is reduced to solving a system of linear algebraic equations approximately. We give the results of calculating the discrete spectra of the S-states for the Schrödinger equation with a linearly growing confining potential and the nuclear Yukawa potential.