Elasticity problems involving coupled half-planes

A general method is proposed for reducing problems concerning cracks, cuts, inclusions and interacting blocks in coupled half-planes to complex integral equations, both singular and hyper-singular. The method is based on the fact that if the Kolosov-Muskhelishvili functions are known for a whole plane, then the corresponding functions for coupled half-planes are obtained from them by simple transformations. Boundary integral equations (BIE) are presented, as well as fundamental solutions for isolated forces and periodic systems of forces, which may be used to construct new complex BIEs.     

Elastic-plastic stress analysis of cracked structures using the boundary element method

The application of the boundary element method (BEM) for the 3D-stress analysis of cracked structures considering elastic-plastic material behavior is presented. For separating the coincident crack surfaces the DUAL-BEM is utilized. The relevant boundary integral equations (BIE) – the strongly singular displacement BIE and the hypersingular traction BIE – are evaluated in the framework of a collocation procedure. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

This paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two-dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free-space Green function but in turn entails evaluation of integrals over the unit-cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite-rank operator correction—used to properly impose the Rayleigh radiation condition—yield a robust second-kind BIE that produces superalgebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh–Wood anomalies. The corrected windowed BIE can be discretized by means of off-the-shelf Nyström and boundary element methods, and it leads to linear systems suitable for iterative linear algebra solvers as well as standard fast matrix–vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodology.     

The Mechanical Quadrature Methods and Their Extrapolation for Solving BIE of Steklov Eigenvalue Problems

By means of the potential theory Steklov eigenvalue problems are transformed into general eigenvalue problems of boundary integral equations (BIE) with the logarithmic singularity.Using the quadrature rules, the paper presents quadrature methods for BIE of Steklov eigenvalue problem, which possess high accuracies O(h^3) and low computing complexities. Moreover, an asymptotic expansion of the errors with odd powers is shown. Using h^3-Richardson extrapolation, we can not only improve the accuracy order of approximations, but also derive a posterior estimate as adaptive algorithms. The efficiency of the algorithm is illustrated by some examples.     

An effective boundary-integral approach for the mixed boundary-value problems of linear elastostatics

A recent BIE formulation is applied to the mixed problem of two-dimensional elastostatics. The discretized form of the equations is presented in detail for easy numerical implementation. Some numerical examples are presented from which it is seen that this method is a significant improvement over a previous indirect approach.     

The Coupling of Boundary Integral and Finite Element Methods for the Navier-Stokes Equations in an Exterior Domain

In this paper, a technique of coupling variational formulation of FEM and BIE (boundary integral equation) is used to deal with stationary Navier-Stokes equations in an unbounded domain. We discuss well-posedness for the coupling variational problem, the regularization method and FEM-BEM approximation. Finally, operator splitting and optimal control techniques are used to treat the difficulty of nonlinearity and constraints in computer implementation.     

Numerical solution for degenerate scale problem arising from multiple rigid lines in plane elasticity

This paper studies the degenerate scale problem arising from multiple rigid lines in plane elasticity. In the first step, the problem should be formulated on a degenerate scale by distribution of body force densities along rigid lines. The condition of vanishing displacement along lines is also assumed. The coordinate transform with a reduced factor “h” is performed in the next step. The new obtained BIE is a particular non-homogenous BIE defined in the transformed coordinates with normal scale. In the normal scale, the integral operator is invertible. By using two fundamental solutions that are formulated in the normal scale, the new obtained BIE can be reduced to an equation for finding the factor “h”. Finally, the degenerate scale is obtained. It is proved from computed results that the degenerate scale only depends on the configuration of rigid lines, and does not depend on the initial normal scale used. In addition, the degenerate scale is invariant with respect to the rotation of rigid lines. Many examples are carried out.     

On the boundary integral equations for the crack opening displacement of flat cracks

The boundary integral equations for the crack opening displacement in acoustic and elastic scattering problems are discussed in the case of flat cracks by means of the Fourier analysis technique. The pseudo-differential nature of the hypersingular integral operators is shown and their symbols explicited. It is then proved that the variational problems assocaited with these BIE are well-posed in a Sobolev functional framework which is closely linked with the elastic energy. A decomposition of the vector integral equation in the elastic case into scalar integral equations is obtained as a by-product of the variational formulation.     

Evaluation of the degenerate scale in antiplane elasticity using null field BIE

This paper provides a numerical solution for the degenerate scale in antiplane elasticity using the null field boundary integral equation (BIE). With coordinate transformation, the BIE can be formulated in a normal scale, and a basic solution for the BIE in the normal scale is illustrated. The degenerate scale is easily obtained from the basic solution. Several numerical examples are given.     

Boundary Integral Equation Solution of Non-linear Plane Potential Problems

This paper presents the boundary integral equation (BIE) formulation,and numerical solution procedure for two-dimensional problemsgoverned by Laplac's equation and subject to non-linear boundaryconditions. The introduction of non-linear terms constitutesa fundamental extension of the BIE method, as previous applicationshave been restricted entirely to linear problems. Furthermore,non-linearities necessitate the use of iterative solution techniqueswhich present the conceptual disadvantage that a solution isnot guaranteed. However, such difficulties were not encounteredwith the Newton—Raphson method employed in this study.The various features of the BIE technique are illustrated bythe application to a physical problem which is of significancein heat exchanger design.     

Second kind boundary integral equation for multi-subdomain diffusion problems

We consider isotropic scalar diffusion boundary value problems whose diffusion coefficients are piecewise constant with respect to a partition of space into Lipschitz subdomains. We allow so-called material junctions where three or more subdomains may abut. We derive a boundary integral equation of the second kind posed on the skeleton of the subdomain partition that involves, as unknown, only one trace function at each point of each interface. We prove the well-posedness of the corresponding boundary integral equations. We also report numerical tests for Galerkin boundary element discretisations, in which the new approach proves to be highly competitive compared to the well-established first kind direct single-trace boundary integral formulation. In particular, GMRES seems to enjoy fast convergence independent of the mesh resolution for the discrete second kind BIE.     

平面定常Stokes问题的无奇异第一类边界积分方程

对无奇异边界积分方程归化法的研究,已有的结果都是针对直接变量的,其核心思想是利用刚体位移(包括刚体的转动和平移)或均匀场．然而,对第一类边界积分方程的无奇异边界归化法的研究,至今还未涉足．本文提交一种新方法,归化出平面定常Stokes问题的第一类无奇异边界积分方程,并建立完整的数值求解体系．一个简单的算例表明本文方法可获得理想的数值结果,特别是边界量的数值结果。     

Monotone and bounded interval equilibria in a coordination game with information aggregation

We analyze how private learning in a class of games with common stochastic payoffs affects the form of equilibria, and how properties such as player welfare and the extent of strategic miscoordination relate across monotone and non-monotone equilibria. Researchers typically focus on monotone equilibria. We provide conditions under which non-monotone equilibria also exist, where players attempt to coordinate to obtain the stochastic payoff whenever signals are in a bounded interval. In bounded interval equilibria (BIE), an endogenous fear of miscoordination discourages players from coordinating to obtain the stochastic payoff when their signals suggest coordination is most beneficial. In contrast to monotone equilibria, expected payoffs from successful coordination in BIE are lower than the ex-ante expected payoff from ignoring signals and always trying to coordinate to obtain the stochastic payoff. We show that BIE only exist when, absent private information, the game would be a coordination game.     

A consistency analysis for the numerical solution of boundary integral equations

A method is presented for assessing the nature of the error incurred in the boundary integral equation (BIE) solution of both harmonic and biharmonic boundary value problems (BVPs). It is shown to what order of accuracy the governing partial differential equation is actually represented by the approximating numerical scheme, and how raising the order of the boundary ‘shape functions’ affects this representation. The effect of varying both the magnitude and the aspect ratio of the solution domain is investigated; it is found that the present technique may suggest an optimum nondimensional scaling for the BIE solution of a particular harmonic or biharmonic BVP.     

A separation between the boundary shape and the boundary functions in the parametric integral equation system for the 3D Stokes equation

Numerical Algorithms - The paper introduces the analytical modification of the classic boundary integral equation (BIE) for Stokes equation in 3D. The performed modification allows us to obtain...     

Integration of singular Galerkin-type boundary element integrals for 3D elasticity problems

Summary. A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional integrals. Received June 25, 1995 / Revised version received January 29, 1996     

Coupling of boundary integral equation and finite element methods for transmission problems in acoustics

Numerical Algorithms - In this paper, we propose a coupling of finite element method (FEM) and boundary integral equation (BIE) method for solving acoustic transmission problems in two dimensions....     

Prolongation of solutions of measure differential equations and dynamic equations on time scales

In this paper, we prove the results on existence and uniqueness of the maximal solutions for measure differential equations, considering more general conditions on functions f and g by using the correspondence between the solutions of these equations and the solutions of generalized ODEs. Moreover, we prove these results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations.     

On the existence of multiple steady-state solutions in the theory of electrodiffusion. Part I: the nonelectroneutral case. Part II: a constructive method for the electroneutral case

We give a constructive method for giving examples of doping functions and geometry of the device for which the nonelectroneutral voltage driven equations have multiple solutions. We show in particular, by performing a singular perturbation analysis of the current driven equations that if the electroneutral voltage driven equations have multiple solutions then the nonelectroneutral voltage driven equations have multiple solutions for sufficiently small normed Debye length. We then give a constructive method for giving examples of data for which the electroneutral voltage driven equations have multiple solutions.

     

Nondegeneracy and uniqueness of positive solutions for Robin problem of second order ordinary differential equations and its applications

This article studies positive solutions of Robin problem for semi-linear second order ordinary differential equations. Nondegeneracy and uniqueness results are proven for homogeneous differential equations. Necessary and sufficient conditions for the existence of one or two positive solutions for inhomogeneous differential equations or differential equations with concave-convex nonlinearities are obtained by making use of the nondegeneracy and uniqueness results for positive solutions of homogeneous differential equations.