复超球上Poisson-华积分边界性质的一个新证明

把复超球 Bn看作多复变典型域 RI(m,n)当 m=1时的特例 ,本文给出复超球上 Poisson-华积分边界性质的不同于文献 [3 ]的一个新证明 ,并研究了 Cauchy积分的边界性质及 Bn上的 Dirichlet问题     

A new boundary integral equation formulation for plane orthotropic elastic media

A new boundary integral equation formulation for solving plane elasticity problems involving orthotropic media is presented in this paper. Based on the real variable fundamental solutions of the considered problems, a limit theorem for the transformation from domain integral equations into boundary integral equations (BIEs) and a novel decomposition technique to the fundamental solutions, the regularized BIEs with indirect unknowns, which do not involve the direct calculation of CPV and HFP integrals, are established. The limiting process is done in global coordinates and no separate numerical treatment for strong and weak singular integrals was necessary. The current method does not need to transform the considered problems into isotropic ones as is normally done in the existing literature, so no inverse transform is required. The numerical implementation is carried out using both discontinuous quadratic elements and exact elements, which is developed to model its boundary with negligible error. The validity of the proposed scheme is demonstrated by three numerical examples. Excellent agreement between the numerical results and exact solutions was obtained even with using small amounts of element.     

时标上的推广的Pachpatte型不等式及其在边值问题中的应用

该文利用单调化技巧研究了时标上的推广的Pachpatte型不等式, 该不等式右端有一个非常数项和三个包含未知函数与没有假设单调性的非线性函数的复合函数的积分项, 不等式左端是未知函数与非线性函数的复合函数. 所得不等式不仅把Pachpatte型不等式的离散形式和连续形式统一起来, 而且推广了已有的时标上的相应不等式. 最后, 用得到的结果研究时标上边值问题解的估计.     

Evaluation of regular and singular domain integrals with boundary-only discretization—theory and Fortran code

In this paper, a set of boundary integrals are derived based on a radial integration technique to accurately evaluate two dimensional (2D) and three dimensional (3D), regular and singular domain integrals. A self-contained Fortran code is listed and described for numerical implementation of these boundary integrals. The main feature of the theory is that only the boundary of the integration domain needs to be discretized into elements. This feature cannot only save considerable efforts in discretizing the integration domain into internal cells (as in the conventional method), but also make computational results for singular domain integrals more accurate since the integrals have been regularized. Some examples are provided to verify the correctness of the presented formulations and the included code.     

A new boundary element technique for solving plane problems of linear elasticity: 1. Theory

A new boundary elements technique for solving plane problems of linear elasticity theory is described. The method is based upon the Muskhelishvili complex variable representation of the displacement and stress fields involving two independent complex functions. These functions are represented by complex Cauchy integrals where the path of integration is taken around the external boundary of the solid. Two complex density functions appearing in the integrands of the Cauchy integrals are represented by spline functions and these are determined by the application of appropriate boundary conditions. The theory presented is suitable only for bounded simply-connected regions.     

POSITIVE SOLUTIONS TO A NONHOMOGENEOUS NONAUTONOMOUS SECONDORDER BOUNDARY VALUE PROBLEM

The existence of multiple positive solutions is studied for a nonlinear nonautonomous second-order boundary value problem with nonhomogeneous boundary conditions. In order to describe the growth behaviors of nonlinearity on some bounded sets, two height functions are introduced. By considering the integrals of the height functions and applying the Krasnosel'skii fixed point theorems on a cone, several new results are proved.     

改进的无奇异局部边界积分方程方法

在局部边界积分方程方法中,当源节点位于分析域的整体边界上时,局部边界积分将出现奇异积分问题,这些奇异积分需要做特别的处理．为此,提出了对域内节点采用局部积分方程,而对边界节点直接采用移动最小二乘近似函数引入边界条件来解决奇异积分问题,这同时也解决了对积分边界进行插值引入近似误差的问题．作为应用和数值实验,对Laplace方程和Helmholtz方程问题进行了分析,取得了很好的数值结果．进而,在Helmholtz方程求解中,采用了含波解信息的修正基函数来代替单项式基函数进行近似．数值结果显示,这样处理是简单高效的,在高波数声传播问题的求解中非常具有前景．     

A stream function-vorticity formulation coupled with boundary integrals for the two-dimensional exterior Stokes problem

In this work, we derive a stream function-vorticity variational formulation coupled with boundary integrals for the exterior Stokes problem in two dimensions, when the right-hand side has a bounded support. The stream function-vorticity formulation is expressed in a bounded region containing the support of the right-hand side, and the boundary conditions on the artificial boundary are obtained by an integral representation. We prove that this coupled formulation is equivalent to the original Stokes problem.     

A stable boundary elements method for magnetohydrodynamic channel flows at high Hartmann numbers

The article is devoted to extension of boundary element method (BEM) for solving coupled equations in velocity and induced magnetic field for time dependent magnetohydrodynamic (MHD) flows through a rectangular pipe. The BEM is equipped with finite difference approach to solve MHD problem at high Hartmann numbers up to 10⁶. In fact, the finite difference approach is used to approximate partial derivatives of unknown functions at boundary points respect to outward normal vector. It yields a numerical method with no singular boundary integrals. Besides, a new approach is suggested in this article where transforms 2D singular BEM's integrals to 1D nonsingular ones. The new approach reduces computational cost, significantly. Note that the stability of the numerical scheme is proved mathematically when computational domain is discretized uniformly and Hartmann number is 40 times bigger than length of boundary elements. Numerical examples show behavior of velocity and induced magnetic field across the sections.     

A boundary value problem for hypermonogenic functions in Clifford analysis

This paper deals with a boundary value problem for hypermonogenic functions in Clifford analysis. Firstly we discuss integrals of quasi-Cauchy's type and get the Plemelj formula for hypermonogenic functions in Clifford analysis, and then we address Riemman boundary value problem for hypermonogenic functions.     

On Taylor coefficients of Cauchy-type integrals of finite complex measures

Boundary values of Cauchy-type integrals of finite complex measures given on a unit circle, generally speaking, are not Lebesgue integrable, and therefore at expansion of Cauchy-type integrals in Taylor series, the expansion coefficients cannot be expressed by boundary values using the Lebesgue integral. In this paper, using the notion of A-integration and N-integration, we get a formula for calculating the Taylor expansion coefficients of Cauchy-type integrals of finite complex measures.     

A Burton-Miller boundary element-free method for Helmholtz problems

A Burton-Miller boundary element-free method is developed by using the Burton-Miller formulation for meshless and boundary-only analysis of Helmholtz problems. The method can produce a unique solution at all wavenumbers and is valid for Dirichlet, Neumann and mixed problems simultaneously. An efficient numerical integration procedure is presented to handle both strongly singular and hypersingular boundary integrals directly and uniformly. Numerical results reveal that this direct meshless method only involves boundary nodes and can deal with Helmholtz problems at extremely large wavenumbers.     

Semi-analytic integration of hypersingular Galerkin BIEs for three-dimensional potential problems

An accurate and efficient semi-analytic integration technique is developed for three-dimensional hypersingular boundary integral equations of potential theory. Investigated in the context of a Galerkin approach, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and field variables on the boundary. In the inner integration procedure, all singular and non-singular integrals over a triangular boundary element are expressed exactly as analytic formulae over the edges of the integration triangle. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent terms are identified explicitly and are shown to cancel with corresponding terms from the edge-adjacent case. The remaining surface integrals, containing only weak singularities, are carried out successfully by use of standard numerical cubatures. Sample problems are included to illustrate the performance and validity of the proposed algorithm.     

Transformation of hypersingular integrals and black-box cubature

In this paper, we will consider hypersingular integrals as they arise by transforming elliptic boundary value problems into boundary integral equations. First, local representations of these integrals will be derived. These representations contain so-called finite-part integrals. In the second step, these integrals are reformulated as improper integrals. We will show that these integrals can be treated by cubature methods for weakly singular integrals as they exist in the literature.

     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.     

Boundary Integrals and Approximations of Harmonic Functions

Steklov expansions for a harmonic function on a rectangle are derived and studied with a view to determining an analog of the mean value theorem for harmonic functions. It is found that the value of a harmonic function at the center of a rectangle is well approximated by the mean value of the function on the boundary plus a very small number (often 3 or fewer) of specific further boundary integrals. These integrals are coefficients in the Steklov representation of the function. Similar approximations are found for the central values of solutions of Robin and Neumann boundary value problems. The results follow from analyses of the explicit expressions for the Steklov eigenvalues and eigenfunctions.     

A spectral projection method for transmission eigenvalues

We consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides whether the region contains eigenvalue(s) or not. It is particularly suitable to test whether zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples.     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.     

A boundary value problem for hypermonogenic functions in Clifford analysis

This paper deals with a boundary value problem for hypermonogenic functions in Clifford analysis. Firstly we discuss integrals of quasi-Cauchy’s type and get the Plemelj formula for hypermonogenic functions in Clifford analysis, and then we address Riemman boundary value problem for hypermonogenic functions.

     

A general backwards calculus of variations via duality

We prove Euler–Lagrange and natural boundary necessary optimality conditions for problems of the calculus of variations which are given by a composition of nabla integrals on an arbitrary time scale. As an application, we get optimality conditions for the product and the quotient of nabla variational functionals.