A boundary integral method applied to plates of arbitrary plan form and arbitrary boundary conditions

An integral equation method which has been applied to thin elastic clamped plates of arbitrary plan form[1] has been extended to include arbitrary boundary conditions. Numerical difficulties which arise in the case of free boundaries have been avoided by defining an integration contour which differs from the actual plate boundary. Thus second order singularities which arise in the integrand of the equations are avoided.The method is applied to two rectangular plates with mixed boundary conditions (i.e. clamped, simply-supported and free edges). Average errors are 1.5% for displacements and 3% for bending moments.     

A boundary integral equation method for axisymmetric elastic contact problems

A numerical method for the solution of axisymmetric contact problems has been developed using the Boundary Integral Equation (BIE) technique. An automatic load incrementation technique is implemented in a BIE axisymmetric computer program using isoparametric quadratic elements. The method is successfully applied to some frictionless contact problems and the results are compared to other numerical and analytical solutions to demonstrate the accuracy of the BIE method.     

A boundary integral method applied to plates of arbitrary plan form

An integral equation method for the solution of thin elastic plates of arbitrary plan form has been presented. The method involves embedding the real plate in a fictitious plate for which the Green's function is known. An unknown load vector is then introduced on the boundary of the real plate (line load and line normal moment). The deflection field due to both known transverse and unknown boundary loads can then be found everywhere by superposition. Satisfaction of the boundary conditions on the real plate results in a vector integral equation in the unknown boundary vector.In concept, any consistent set of boundary conditions will yield a solution. Practically, boundary conditions requiring higher derivatives of the deflection are both very cumbersome and yield singularities in the integral equations which cause numerical difficulties. For these reasons only clamped boundary conditions are treated numerically in the present paper.For interior bending moments and deflections (greater than distances of the order of one boundary subdivision from the boundary) the method is both highly accurate and inexpensive. Errors right on the boundary, e.g. the clamping moment in the clamped boundary condition case, can be appreciable, however. While this can be improved by a more sophisticated treatment of the unknown boundary vector in the numerical solution (increased expense) it is shown in the paper that a simple boundary extrapolation procedure gives excellent accuracy there.     

Combination of the boundary integral equation method and the extrapolation method

Many numerical methods including the boundary integral equation method start with division of the domain of calculation into intervals. The accuracy of their results can be improved considerably by extrapolation. To be able to apply the extrapolation method it is necessary to know the asymptotic expansion of the error.In this paper the principle of the extrapolation method and subjects important for its application are described. Above all it is shown how to determine the asymptotic expansions numerically by trial and error. In the first sections the matter is explained in a general manner to encourage users of various numerical methods—among them users of the finite element method—to try to extrapolate their results. Then the investigations are exemplified in detail by the boundary integral equation method. The accuracy of approximate solutions of integral equations for plane elastostatic problems with prescribed boundary tractions and displacements is improved by extrapolation. Particular attention is paid to boundary tractions and displacements with discontinuous derivatives.To induce also practically orientated readers without specialized mathematical knowledge to think about applying the extrapolation method the basic topics are represented in an extensive manner and illustrated by simple examples. (For a survey of this paper see end of Section 1.)     

The p-adaptive boundary integral equation method

This work is concerned with the development and application of the p-adaptive boundary integral equation method (BIEM) in practical elastostatics engineering situations. Some basic concepts inherent to the p-adaptive technique are summarized and discussed. A pseudocode which illustrates the way for generating the p-adaptive system of equations in microcomputers is also provided.

Two numerical examples, which show the accuracy of the method discussed herein are included.     

A boundary integral equation method for a class of crack problems in anisotropic elasticity

A boundary integral procedure for the solution of an important class of crack problems in anisotropic elasticity is outlined. A specific numerical example is considered in order to assess the effectiveness of the procedure.     

A compression technique for the boundary integral equation reduced from Helmholtz equation

Applying the trigonometric wavelets and the multiscale Galerkin method, we investigate the numerical solution of the boundary integral equation reduced from the exterior Dirichlet problem of Helmholtz equation by the potential theory. Consequently, we obtain a matrix compression strategy, which leads us to a fast algorithm. Our truncated treatment is simple, the computational complexity and the condition number of the truncated coefficient matrix are bounded by a constant. Furthermore, the entries of the stiffness matrix can be evaluated from the Fourier coefficients of the kernel of the boundary integral equation. Examples given for demonstrating our numerical method shorten the runtime obviously.     

A modified boundary integral evolution formulation for the wave equation

We apply a modified boundary integral formulation otherwise known as the Green element method (GEM) to the solution of the two-dimensional scalar wave equation.GEM essentially combines three techniques namely: (a) finite difference approximation of the time term (b) finite element discretization of the problem domain and (c) boundary integral replication of the governing equation. These unique and advantageous characteristics of GEM facilitates a direct numerical approximation of the governing equation and obviate the need for converting the governing partial differential equation to a Helmholtz-type Laplace operator equation for an easier boundary element manipulation. C₁ continuity of the computed solutions is established by using Overhauser elements. Numerical tests show a reasonably close agreement with analytical results. Though in the case of the Overhauser GEM solutions, the level of accuracy obtained does not in all cases justify the extra numerical rigor.     

A natural stress boundary integral equation for calculating the near boundary stress field

The stress computational accuracy of internal points by conventional boundary element method becomes more and more deteriorate as the points approach to the boundary due to the nearly singular integrals including nearly strong singularity and hyper-singularity. For calculating the boundary stress, a natural boundary integral equation in which the boundary variables are the displacements, tractions and natural boundary variables was established in the authors’ previous work. Herein, a natural stress boundary integral equation (NSBIE) is further proposed by introducing the natural variables to analyze the stress field of interior points. There are only nearly strong singular integrals in the NSBIE, i.e., the singularity is reduced by one order. The regularization algorithm proposed by the authors is taken over to deal with these singular integrals. Consequently, the NSBIE can analyze the stress field closer to the boundary. Numerical examples demonstrated that two orders of magnitude improvement in reducing the approaching degree can be achieved by NSBIE compared to the conventional one when the near boundary stress field is evaluated. Furthermore, this new way is extended to the multi-domain elasticity problem to calculate the stress field near the boundary and interface.     

Direct boundary integral equation method for electromagnetic scattering by partly coated dielectric objects

We present a new variational direct boundary integral equation approach for solving the scattering and transmission problem for dielectric objects partially coated with a PEC layer. The main idea is to use the electromagnetic Calderón projector along with transmission conditions for the electromagnetic fields. This leads to a symmetric variational formulation which lends itself to Galerkin discretization by means of divergence-conforming discrete surface currents. A wide array of numerical experiments confirms the efficacy of the new method. Dedicated to George C. Hsiao on the occasion of his 70th birthday. Communicated by: W. L. Wendland     

Asymptotic expansions of results of the boundary integral equation method for plane elastostatic problems

The accuracy of numerical results can be improved by extrapolation if the asymptotic expansion of the error for step sizes (or element lengths) h tending to zero is well-known. In this paper expansions are determined for results of an integral equation for the plane elastostatic problem with prescribed boundary tractions. Special care is paid to discontinuous derivatives of the boundary values and of the boundary itself. Furthermore, the influence of the degree of interpolation of the sought function of the integral equation and the influence of non-equidistant division of the boundary on the structure of the expansion is investigated.The paper represents a continuation and partly a completion of [1].An extensive survey of the paper is given at the end of Section 1.     

Meshless local boundary integral equation method for simply supported and clamped plates resting on elastic foundation

Simply supported and clamped thin elastic plates resting on a two-parameter foundation are analyzed in the paper. The governing partial differential equation of fourth order for a plate is decomposed into two coupled partial differential equations of second order. One of them is Poisson’s equation whereas the other one is Helmholtz’s equation. The local boundary integral equation method is used with meshless approximation for both the Poisson and the Helmholtz equation. The moving least square method is employed as the meshless approximation. Independent of the boundary conditions fictitious nodal unknowns used for the approximation of bending moments and deflections are always coupled in the resulting system of algebraic equations. The Winkler foundation model follows from the Pasternak model if the second parameter is equal to zero. Numerical results for a square plate with simply and/or clamped edges are presented to prove the efficiency of the proposed formulation.     

边界积分方程再认识及其软件研发——纪念杜庆华院士诞辰100周年

值杜庆华院士诞辰一百周年之际，感念与杜先生交往的点滴及杜先生对边界元法研究的贡献，对比有限差分法、有限元法和边界元法各自适用问题的特点，呼吁学者们勇敢面对边界元法及其自主CAD/CAE软件研发劳动强度大、研究经费难的现状，直面边界元法被日渐边缘化的困境，遇山钻洞、遇堑搭桥，继承和发扬杜先生开创的边界元法，并以此纪念杜先生！     

A boundary integral method for the simulation of two-dimensional particle coarsening

A boundary integral method for the solution of a time-dependent free-boundary problem in a two-dimensional, multiply-connected, exterior domain is described. The method is based on an iterative solution of the resulting integral equations at each time step, with the initial guesses provided by extrapolation from previous time steps. The method is related to a technique discussed by Baker for the study of water waves. The discretization is chosen so that the solvability conditions required for the exterior Dirichlet problem do not degrade the convergence rate of the iterative solution procedure. Consideration is given to the question of vectorizing the computation. The method is applied to the problem of the coarsening of two-dimensional particles by volume diffusion.     

A numerical method for transmission problems for the Helmholtz equation

This method consists in decoupling the transmission problem into two boundary value problems, which can be solved separately by well known procedures. A convergence proof is given with the help of the integral equation method and convergence results on projection methods.     

A meshless Galerkin method for Stokes problems using boundary integral equations

A meshless Galerkin scheme for the simulation of two-dimensional incompressible viscous fluid flows in primitive variables is described in this paper. This method combines a boundary integral formulation for the Stokes equation with the moving least-squares (MLS) approximations for construction of trial and test functions for Galerkin approximations. Unlike the domain-type method, this scheme requires only a nodal structure on the bounding surface of a body for approximation of boundary unknowns, thus it is especially suitable for the exterior problems. Compared to other meshless methods such as the boundary node method and the element free Galerkin method, in which the MLS is also introduced, boundary conditions do not present any difficulty in using this meshless method. The convergence and error estimates of this approach are presented. Numerical examples are also given to show the efficiency of the method.     

A direct boundary integral equation method for the numerical construction of harmonic functions in three-dimensional layered domains containing a cavity

We consider boundary value problems for the Laplace equation in three-dimensional multilayer domains composed of an infinite strip layer of finite height and a half-space containing a bounded cavity. The unknown (harmonic) function satisfies the Neumann boundary condition on the exterior boundary of the strip layer (i.e. at the bottom of the first layer), the Dirichlet, Neumann or Robin boundary condition on the boundary surface of the cavity and the corresponding transmission (matching) conditions on the interface layer boundary. We reduce this boundary value problem to a boundary integral equation over the boundary surface of the cavity by constructing Green's matrix for the corresponding transmission problem in the domain consisting of the infinite layer and the half-space (not with the cavity). This direct integral equation approach leads, for any of the above boundary conditions, to boundary integral equations with a weak singularity on the cavity. The numerical solution of this equation is realized by Wienert's [Die Numerische approximation von Randintegraloperatoren für die Helmholtzgleichung im R ³, Ph.D. thesis, University of Göttingen, Germany, 1990] method. The reduction of the problem, originally set in an unbounded three-dimensional region, to a boundary integral equation over the boundary of a bounded domain, is computationally advantageous. Numerical results are included for various boundary conditions on the boundary of the cavity, and compared against a recent indirect approach [R. Chapko, B.T. Johansson, and O. Protsyuk, On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R ³ with cavities, J. Numer. Appl. Math. (Kyiv) 105 (2011), pp. 4–18], and the results obtained show the efficiency and accuracy of the proposed method. In particular, exponential convergence is obtained for smooth cavities.     

A fast singular boundary method for 3D Helmholtz equation

This paper presents a fast singular boundary method (SBM) for three-dimensional (3D) Helmholtz equation. The SBM is a boundary-type meshless method which incorporates the advantages of the boundary element method (BEM) and the method of fundamental solutions (MFS). It is easy-to-program, and attractive to the problems with complex geometries. However, the SBM is usually limited to small-scale problems, because of the operation count of $O\left(N^3\right)$ with direct solvers or $O\left(N^2\right)$ with iterative solvers, as well as the memory requirement of $O\left(N^2\right)$. To overcome this drawback, this study makes the first attempt to employ the precorrected-FFT (PFFT) to accelerate the SBM matrix–vector multiplication at each iteration step of the GMRES for 3D Helmholtz equation. Consequently, the computational complexity can be reduced from $O\left(N^2\right)$ to $O\left(NlogN\right)$ or $O\left(N\right)$. Three numerical examples are successfully tested on a desktop computer. The results clearly demonstrate the accuracy and efficiency of the developed fast PFFT-SBM strategy.