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 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.     

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.     

Richardson extrapolation applied to boundary element method results in a Dirichlet problem for the Laplace equation

Richardson extrapolation is used to improve the accuracy of the numerical solutions for the normal boundary flux and for the interior potential resulting from the boundary element method. The boundary integral equations arise from a direct boundary integral formulation for solving a Dirichlet problem for the Laplace equation. The Richardson extrapolation is used in two different applications: (i) to improve the accuracy of the collocation solution for the normal boundary flux and, separately, (ii) to improve the solution for the potential in the domain interior. The main innovative aspects of this work are that the orders of dominant error terms are estimated numerically, and that these estimates are then used to develop an a posteriori technique that predicts if the Richardson extrapolation results for applications (i) and (ii) are reliable. Numerical results from test problems are presented to demonstrate the technique.     

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 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.     

The plane stress/strain analysis of non-homogeneous continua by the boundary integral equation method

The numerical analysis of non-homogeneous bodies by the Boundary Integral Equation method is discussed. The body is divided into subregions where the material properties are constant, and each subregion is enclosed within its own boundary. Quadratic shape functions are used to interpolate the variables defined on these boundaries. At corners, either on the external boundaries or on the subregion interfaces, the tractions are discontinuous which can cause numerical difficulties. Particular attention is given to overcoming this problem by using either a set of auxiliary equations, or a number of auxiliary nodes at these corner points. In addition, the elegant method of calculating the body force terms by boundary integration, instead of the more conventional domain integration, is discussed. Finally, two examples are presented which examine the performance of the BIEM for problems of this type.     

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     

A hybrid-mode boundary integral equation method for normal- and superconducting transmission lines of arbitrary cross-section

A space domain Boundary Integral Equation (BIE) method for full-wave analysis of general waveguide is presented. The method is demonstrated to be applicable to arbitrary shielded or unshielded waveguide cross-sections ranging from optical waveguide to MMIC transmission lines. It allows for arbitrary isotropic complex media including normal (imperfect) conductors and superconductors as is demonstrated with full-wave loss analysis of coplanar stripline and several thin-film microstrip line configurations employing Au and YBCO conductors. An outline of the theory of the BIE method is given. The implications of nonsmooth boundary curves are considered, and special attention is given to the reliability of the method in being free of spurious solutions. © 1992 John Wiley & Sons, Inc.     

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 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.     

On the problem of elastic torsion of variable diameter circular shafts by the boundary integral equation method

The paper presents a formulation of the boundary integral equation for elastic torsion of variable diameter circular shafts. Stress function approach is used. This form of BIE has an advantage in that the resultant shearing stresses at the lateral boundary of the shaft can be computed directly from the numerical solution of the equation. Two numerical examples are given at the end of the paper.     

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.     

Application of the dual reciprocity boundary integral equation technique to solve the nonlinear Klein-Gordon equation

This paper aims to obtain approximate solutions of the Nonlinear Klein-Gordon (NLKG) equation by employing the Boundary Integral Equation (BIE) method and the Dual Reciprocity Boundary Element Method (DRBEM). This method is improved by using a predictor-corrector scheme to the nonlinearity which appears in the problem. We employ the time stepping scheme to approximate the time derivative, and the Linear Radial Basis Functions (LRBFs), are used in the Dual Reciprocity (DR) technique. To confirm the accuracy of the new approach, the numerical results of a Double-Soliton and a problem with inhomogeneous terms are compared with analytical solutions and for the examples possessing single and periodic waves, two conserved quantities associated to the (NLKG) equation, the energy and the momentum are investigated.     

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.     

The Trefftz method as an integral equation

The Trefftz method may be described in terms of an integral equation formulation based on eigenfunction expansions rather than the usual element (BEM) solution method. Using an ‘Embedding Integral Equation’ approach, the Trefftz method may not only be seen to be related to integral equation methods but it is also seen to be amenable to partitioning, allowing more rapid convergence.     

A sixth-order extrapolation method for special nonlinear fourth-order boundary value problems

A sixth-order convergent finite difference method is developed for the numerical solution of the special nonlinear fourth-order boundary value problem y^(iv)(x) = f(x, y), a < x < b, y(a) = A₀, y″(a) = B₀, y(b) = A₁ y′(b) = B₁, the simple-simple beam problem.The method is based on a second-order convergent method which is used on three grids, sixth-order convergence being obtained by taking a linear combination of the (second-order) numerical results calculated using the three individual grids.Special formulas are proposed for application to points of the discretization adjacent to the boundaries x = a and x= b, the first two terms of the local truncation errors of these formulas being the same as those of the second-order method used at the other points of each grid.Modifications to these two formulas are obtained for problems with boundary conditions of the form y(a) = A₀, y′(a) = C₀, y(b) = A₁, y′(b) = C₁, the clamped-clamped beam problem.The general boundary value problem, for which the differential equation is y^(iv)(x) = f(x, y, y′, y″, y‴), is also considered.     

Load sensitivity analyses of elastic structures by differential and boundary integral equation formulations

A direct unification of the boundary integral equations (BIE) of elasticity and load sensitivity analysis procedures is proposed in a novel approach. In the so-called BIE formulation, it is found that more than one type of adjoint functions are necessary, which are defined by different integral equations at complementary points of the structural boundary. The presence of double volume/surface integrals and the switching of the source and field points in the fundamental solutions are further interesting characteristics of the proposed formulation. An analytical example is solved by using the standard differential equation formulation, as well as the new BIE formulation of load sensitivity analysis.