NOVEL REGULARIZED BOUNDARY INTEGRAL EQUATIONS FOR POTENTIAL PLANE PROBLEMS

The universal practices have been centralizing on the research of regularization to the direct boundary integal equations （DBIEs）. The character is elimination of singularities by using the simple solutions. However, up to now the research of regularization to the first kind integral equations for plane potential problems has never been found in previous literatures. The presentation is mainly devoted to the research on the regularization of the singular boundary integral equations with indirect unknowns. A novel view and idea is presented herein, in which the regularized boundary integral equations with indirect unknowns without including the Cauchy principal value （CPV） and Hadamard-finite-part （HFP） integrals are established for the plane potential problems. With some numerical results, it is shown that the better accuracy and higher efficiency, especially on the boundary, can be achieved by the present system.

EQUIVALENT BOUNDARY INTEGRAL EQUATIONS WITH INDIRECT UNKNOWNS FOR THIN ELASTIC PLATE BENDING THEORY

Equivalent Boundary Integral Equations (EBIE) with indirect unknowns for thin elastic plate bending theory, which is equivalent to the original boundary value problem, is established rigorously by mathematical technique of non-analytic continuation and is fully proved by means of the variational principle. The previous three kinds of boundary integral equations with indirect unknowns are discussed thoroughly and it is shown that all previous results are not EBIE.

RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS

This paper studies several problems , which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1] are chosen as a starting point for characterizations of functions in Besom spaces B(?)(0,1) with 0<σ<∞ and (1+σ)-1<γ<∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.

Computational model for short-fiber composites with eigenstrain formulation of boundary integral equations

A computational model is proposed for short-fiber reinforced materials with the eigenstrain formulation of the boundary integral equations （BIE） and solved with the newly developed boundary point method （BPM）. The model is closely derived from the concept of the equivalent inclusion Of Eshelby tensors. Eigenstrains are iteratively determined for each short-fiber embedded in the matrix with various properties via the Eshelby tensors, which can be readily obtained beforehand either through analytical or numerical means. As unknown variables appear only on the boundary of the solution domain, the solution scale of the inhomogeneity problem with the model is greatly reduced. This feature is considered significant because such a traditionally time-consuming problem with inhomogeneity can be solved most cost-effectively compared with existing numerical models of the FEM or the BEM. The numerical examples are presented to compute the overall elastic properties for various short-fiber reinforced composites over a representative volume element （RVE）, showing the validity and the effectiveness of the proposed computational modal and the solution procedure.

Boundary integral equations for dynamic rupture propagation on vertical complex fault system in half-space:Theory

The boundary integral equation method （BIEM） is now widely used in numerical studies on earthquake rupture dynamics, and is proved to be a powerful tool to deal with problems on complex fault system. However, since this method heavily lies on the specific forms of Green＇s function and only the Green＇s function in full-space has a closed analytic expression, it is usually limited to a full-space medium. In this study, as a first step to extend this method to an arbitrary complex fault system in half-space, the boundary integral equations （BIEs） for dynamic strike-slip on vertical complex fault system in half-space are derived based on exact Green＇s function for isotropic and homogeneous half-space. Effect of the geometry of the complex fault system are dealt with carefully. Final BIEs is composed of two parts： contribution from full-space, which has been thoroughly investigated by Aochi and his co-workers by using the Green＇s function for full-space, and that from free surface, which is studied in detail in this study.

BOUNDARY INTEGRAL EQUATIONS FOR BENDING PROBLEM OF REISSNER'S PLATES ONTWO-PARAMETER FOUNDATION

Two fundamental solutions for bending problem of Reissner's plates on twoparameter foundation are derived by means of Fouier integral transformation of generalized function in this paper.On the basis of virtual work principles, three boundary integral equations which fit for arbitrary shapes, loads and boundary conditions of thick plates are presented according to Hu Haichang's theory about Reissner's plates. It provides the fundamental theories for the application of BEM. A numerical example is given for clamped, simply supported and free boundary conditions. The results obtained are satisfactory as compared with the analytical methods.

Approximate Formulation and Numerical Solution for Hypersingular Boundary Integral Equations in Plane Elasticity

Based on the fact that the singular boundary integrals in the sense of Cauchy principal value can be represented approximately by the mean values of two companion nearly singular boundary integrals, a vary general approach was developed in the paper. In the approach, the approximate formulation before discretization was constructed to cope with the difficulties encountered in the corner treatment in the formulations of hypersingular boundary integral equations. This makes it possible to solve the hypersingular boundary integral equation numerically in a non regularized form and in a local manner by using conforming C 0 quadratic boundary elements and standard Gaussian quadratures similar to those employed in the conventional displacement BIE formulations. The approximate formulation is very convenient to use because the corner information is comprised naturally in the representations of those approximate integrals. Numerical examples in plane elasticity show that with the present approach, the compatible or better results can be achieved in comparison with those of the conventional BIE formulations.

EQUIVALENT BOUNDARY INTEGRAL EQUATIONS WITH INDIRECT VARIABLES FOR PLANE ELASTICITY PROBLEMS

The exact form of the exterior problem for plane elasticity problems was produced and fully proved by the variational principle.Based on this,the equivalent boundary integral equations(EBIE) with direct variables,which are equivalent to the original boundary value problem,were deduced rigorously.The conventionally prevailing boundary integral equation with direct variables was discussed thoroughly by some examples and it is shown that the previous results are not EBIE.

Desingularized boundary integral equations and their applications in wave dynamics and wave-body interaction problems

Over the past 30 years or so,desingularized boundary integral equations(DBIEs)have been used to study water wave dynamics and body motion dynamics.Within the potential flow modeling,unlike conventional boundary integral methods,a DBIE separates the integration surface and the control(collocation)surface,resulting in a BIE with non-singular kernels.The desingularization allows simpler and faster numerical evaluation of the boundary integrals,and consequently faster numerical solutions.In this paper,derivations of different forms of DBIEs are given and the fundamental aspects and advantages of the DBIEs are reviewed and discussed.Numerical examples of applications of DBIEs in wave dynamics and body motion dynamics are given and the outlook of future development of the desingularized methods is discussed.

On the Boundary Integral Equations for a Two-Dimensional Slowly Rotating Highly Viscous Fluid Flow

In this paper,the two-dimensional slowly rotating highly viscous fluid flow in small cavities is modelled by the triharmonic equation for the streamfunction.The Dirichlet problem for this triharmonic equation is recast as a set of three boundary integral equations which however,do not have a unique solution for three exceptional geometries of the boundary curve surrounding the planar solution domain.This defect can be removed either by using modified fundamental solutions or by adding two supplementary boundary integral conditions which the solution of the boundary integral equations must satisfy.The analysis is further generalized to polyharmonic equations.

Second-Kind Boundary Integral Equations for Scattering at Composite Partly Impenetrable Objects

We consider acoustic scattering of time-harmonic waves at objects composed of several homogeneous parts.Some of those may be impenetrable,giving rise to Dirichlet boundary conditions on their surfaces.We start from the recent secondkind boundary integral approach of[X.Claeys,and R.Hiptmair,and E.Spindler.A second-kind Galerkin boundary element method for scattering at composite objects.BIT Numerical Mathematics,55(1):33-57,2015]for pure transmission problems and extend it to settings with essential boundary conditions.Based on so-called global multipotentials,we derive variational second-kind boundary integral equations posed in L^(2)(S),where S denotes the union of material interfaces.To suppress spurious resonances,we introduce a combined-field version(CFIE)of our new method.Thorough numerical tests highlight the low andmesh-independent condition numbers of Galerkin matrices obtained with discontinuous piecewise polynomial boundary element spaces.They also confirm competitive accuracy of the numerical solution in comparison with the widely used first-kind single-trace approach.

Weakly-Singular Traction and Displacement Boundary Integral Equations and Their Meshless Local Petrov-Galerkin Approaches

The general meshless local Petrov-Galerkin (MLPG) weak forms of the displacement and trac- tion boundary integral equations (BIEs) are presented for solids undergoing small deformations. Using the directly derived non-hyper-singular integral equations for displacement gradients, simple and straight- forward derivations of weakly singular traction BIEs for solids undergoing small deformations are also pre- sented. As a framework for meshless approaches, the MLPG weak forms provide the most general basis for the numerical solution of the non-hyper-singular displacement and traction BIEs. By employing the various types of test functions, several types of MLPG/BIEs are formulated. Numerical examples show that the pre- sent methods are very promising, especially for solving the elastic problems in which the singularities in dis- placements, strains, and stresses are of primary concern.

Boundary Integral Equations and A Posteriori Error Estimates

Adaptive methods have been rapidly developed and applied in many fields of scientific and engi- neering computing. Reliable and efficient a posteriori error estimates play key roles for both adaptive finite element and boundary element methods. The aim of this paper is to develop a posteriori error estimates for boundary element methods. The standard a posteriori error estimates for boundary element methods are obtained from the classical boundary integral equations. This paper presents hyper-singular a posteriori er- ror estimates based on the hyper-singular integral equations. Three kinds of residuals are used as the esti- mates for boundary element errors. The theoretical analysis and numerical examples show that the hyper- singular residuals are good a posteriori error indicators in many adaptive boundary element computations.

BOUNDARY INTEGRAL EQUATIONS FOR ISOTROPIC LINEAR ELASTICITY

This articles first investigates boundary integral operators for the three-dimensional isotropic linear elasticity of a biphasic model with piecewise constant Lam´e coefficients in the form of a bounded domain of arbitrary shape surrounded by a background material.In the simple case of a spherical inclusion,the vector spherical harmonics consist of eigenfunctions of the single and double layer boundary operators and we provide their spectra.Further,in the case of many spherical inclusions with isotropic materials,each with its own set of Lam´e parameters,we propose an integral equation and a subsequent Galerkin discretization using the vector spherical harmonics and apply the discretization to several numerical test cases.

A study on boundary integral equations for dynamic elastoplastic analysis for the plane problem by TD-BEM

The equivalent stress fundamental solution for the elastoplastic dynamic plane strain problem is proposed to transform the virtual work in the third direction to the plane.Subsequently,based on Betti reciprocal theorem,by adopting the time dependent fundamental solutions in terms of displacement,traction and equivalent stress,the boundary integral equations for dynamic elastoplastic analysis for the plane strain problem are established.The establishment procedures for the displacement and the stress boundary integral equations,together with the stress equation at boundary points,are presented in details,while the standard discretization both in time and space under the frame of time domain boundary element method(TD-BEM)and the solution of the algebraic equations are also briefly stated.Two verification examples are presented from different viewpoints,for elastic and elastoplastic analysis,for 1-D and 2-D geometries,and for finite and infinite domains.The TD-BEM formulation for dynamic elastoplastic analysis is presented for the plane strain problem as an example,where the formulation is also applicable for the plane stress problem by properly transforming the elastic constants and adopting the corresponding fundamental solutions.

SPLITTING EXTRAPOLATIONS FOR SOLVING BOUNDARY INTEGRAL EQUATIONS OF LINEAR ELASTICITY DIRICHLET PROBLEMS ON POLYGONS BY MECHANICAL QUADRATURE METHODS

Taking hm as the mesh width of a curved edge Гm （m = 1, ..., d ） of polygons and using quadrature rules for weakly singular integrals, this paper presents mechanical quadrature methods for solving BIES of the first kind of plane elasticity Dirichlet problems on curved polygons, which possess high accuracy O（h0^3） and low computing complexities. Since multivariate asymptotic expansions of approximate errors with power hi^3 （i = 1, 2, ..., d） are shown, by means of the splitting extrapolations high precision approximations and a posteriori estimate are obtained.

BOUNDARY INTEGRAL EQUATIONS OF UNIQUE SOLUTIONS IN ELASTICITY

The properties of the fundamental solution are derived in linear elastostatics. These properties are used to show that the conventional displacement and traction boundary integral equations yield non-unique displacement solutions in a traction boundary value problem. The condition for the existence of unique displacement solutions is proposed for the traction boundary value problem. The degrees of freedom of the displacement solution are removed by the condition to obtain the boundary integral equations of unique solutions for the traction boundary value problems. Numerical example is presented to demonstrate the accuracy and efficiency of the present equations.

The Collocation Method and the Splitting Extrapolation for the First Kind of Boundary Integral Equations on Polygonal Regions

In this paper,the collocation methods are used to solve the boundary integral equations of the first kind on the polygon.By means of Sidi’s periodic transformation and domain decomposition,the errors are proved to possess the multi-parameter asymptotic expansion at the interior point with the powers h^(3)/_(i)(i=1,...,d),which means that the approximations of higher accuracy and a posteriori estimation of the errors can be obtained by splitting extrapolations.Numerical experiments are carried out to show that the methods are very efficient.

A STUDY ON THE WEIGHT FUNCTION OF THE MOVING LEAST SQUARE APPROXIMATION IN THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.

THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN

We consider a kind of scattering problem by a crack F that is buried in a bounded domain D, and we put a point source inside the domain D. This leads to a mixed boundary value problem to the Helmholtz equation in the domain D with a crack Г. Both sides of the crack F are given Dirichlet-impedance boundary conditions, and different boundary condition （Dirichlet, Neumann or Impedance boundary condition） is set on the boundary of D. Applying potential theory, the problem can be reformulated as a system of boundary integral equations. We establish the existence and uniqueness of the solution to the system by using the Fredholm theory.