Geometric Conversion Approach for the Numerical Evaluation of Hypersin gular and Nearly Hypersingular Boundary Integrals over Curved Surface Boundary Elements

With the aid of the properties of the hypersingular kernels, a geometric conversion approach was presented in this paper. The conversion leads to a general approach for the accurate and reliable numerical evaluation of the hypersingular surface boundary integrals encountered in a variety of applications with boundary element method. Based on the conversion, the hypersingularity in the boundary integrals could be lowered by one order, resulting in the simplification of the computer code. Moreover, an integral transformation was introduced to damp out the nearly singular behavior of the kernels by the distance function defined in the local polar coordinate system for the nearly hypersingular case. The approach is simple to use, which can be inserted readily to computer code, thus getting rid of the dull routine deduction of formulae before the numerical implementations, as the expressions of these kernels are in general complicated. The numerical examples were given in three dimensional elasticity, verifying the effectiveness of the proposed approach, which makes it possible to observe numerically the behavior of the boundary integral values with hypersingular kernels across the boundary.

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.

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.

Solution of stress intensity factors of multiple cracks in plane elasticity with eigen COD formulation of boundary integral equation

The concept of eigen crack opening displacement （COD） can be defined as the COD of a crack in infinite plate under the tractions acting on the crack surface. By introducing this concept, the eigen COD formulation of boundary integral equation is proposed in this paper, together with the solution procedures for multiple crack problems in plane elasticity. With the proposed approach, the multiple crack problems can be solved with the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix as that in the numerical Green’s function （NGF） approach but without the trouble to determine the complementary solutions since the standard boundary element discretization on the crack surface is no longer required with the proposed approach. Some numerical examples computing the stress intensity factors are presented and compared with those in literature to show the accuracy and the effectiveness of the proposed approach.

ON THE UNIQUENESS OF BOUNDARY INTEGRAL EQUATION FOR THE EXTERIOR HELMHOLTZ PROBLEM

From the point of view of energy analysis, the cause that the uniqueness of the boundary integral equation induced from the exterior Helmholtz problem does not hold is investigated in this paper. It is proved that the Sommerfeld's condition at the infinity is changed so that it is suitable not only for the radiative wave but also for the absorptive wave when we use the boundary integral equation to describe the exterior Helmholtz problem. There fore, the total energy of the system is conservative. The mathematical dealings to guarantee the uniqueness are discussed based upon this explanation.

RESEARCH ON THE COMPANION SOLUTION FOR A THIN PLATE IN THE MESHLESS LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless local boundary integral equation method is a currently developed numerical method, which combines the advantageous features of Galerkin finite element method(GFEM), boundary element method(BEM) and element free Galerkin method(EFGM), and is a truly meshless method possessing wide prospects in engineering applications. The companion solution and all the other formulas required in the meshless local boundary integral equation for a thin plate were presented, in order to make this method apply to solve the thin plate problem.

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.

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.

Analytical Solution of Boundary Integral Equations for 2-D Steady Linear Wave Problems

Based on the Fourier transform, the analytical solution of boundary integral equations formulated for the complex velocity of a 2-D steady linear surface flow is derived. It has been found that before the radiation condition is imposed,free waves appear both far upstream and downstream. In order to cancel the free waves in far upstream regions, the eigensolution of a specific eigenvalue, which satisfies the homogeneous boundary integral equation, is found and superposed to the analytical solution. An example, a submerged vortex, is used to demonstrate the derived analytical solution. Furthermore,an analytical approach to imposing the radiation condition in the numerical solution of boundary integral equations for 2-D steady linear wave problems is proposed.

BOUNDARY INTEGRAL FORMULAS FOR ELASTIC PLANE PROBLEM OF EXTERIOR CIRCULAR DOMAIN

After the stress function and the normal derivative on the boundary for the plane problem of exterior circular domain are expanded into Laurent series, comparing them with the Laurent series of the complex stress function and making use of some formulas in Fourier series and the convolutions, the boundary integral formula of the stress function is derived further. Then the stress function can be obtained directly by the integration of the stress function and its normal derivative on the boundary. Some examples are given. It shows that the boundary integral formula of the stress function is convenient to be used for solving the elastic plane problem of exterior circular domain.

Two-dimensional polynomial eigenstrain formulation of boundary integral equation with numerical verification

The low-order polynomial-distributed eigenstrain formulation of the boundary integral equation （BIE） and the corresponding definition of the Eshelby tensors are proposed for the elliptical inhomogeneities in two-dimensional elastic media. Taking the results of the traditional subdomain boundary element method （BEM） as the control, the effectiveness of the present algorithm is verified for the elastic media with a single elliptical inhomogeneity. With the present computational model and algorithm, significant improvements are achieved in terms of the efficiency as compared with the traditional BEM and the accuracy as compared with the constant eigenstrain formulation of the BIE.

BOUNDARY INTEGRAL FORMULA OF ELASTICPROBLEMS IN CIRCLE PLANE

By bianalytic functions, the boundary integral formula of the stress function for the elastic problem in a circle plane is developed. But this integral formula includes a strongly singular integral and can not be directly calculated. After the stress function is expounded to Fourier series, making use of some formulas in generalized functions to the convolutions, the boundary integral formula which does not include strongly singular integral is derived further. Then the stress function can be got simply by the integration of the values of the stress function and its derivative on the boundary. Some examples are given. It shows that the boundary integral formula of the stress function for the elastic problem is convenient.

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.

Improved non-singular local boundary integral equation method

When the source nodes are on the global boundary in the implementation of local boundary integral equation method （LBIEM）, singularities in the local boundary integrals need to be treated specially. In the current paper, local integral equations are adopted for the nodes inside the domain and moving least square approximation （MLSA） for the nodes on the global boundary, thus singularities will not occur in the new al- gorithm. At the same time, approximation errors of boundary integrals are reduced significantly. As applications and numerical tests, Laplace equation and Helmholtz equation problems are considered and excellent numerical results are obtained. Furthermore, when solving the Helmholtz problems, the modified basis functions with wave solutions are adapted to replace the usually-used monomial basis functions. Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.

Numerical Evaluation of CPV Boundary Integrals with Symmetrical Quadrature Schemes

Stemming from the definition of the Cauchy principal values (CPV) integrals, a newly developed symmetrical quadrature scheme was proposed in the paper for the accurate numerical evaluation of the singular boundary integrals in the sense of CPV encountered in the boundary element method. In the case of inner element singularities, the CPV integrals could be evaluated in a straightforward way by dividing the element into the symmetrical part and the remainder(s). And in the case of end singularities, the CPV integrals could be evaluated simply by taking a tangential distance transformation of the integrand after cutting out a symmetrical tiny zone around the singular point. In both cases, the operations are no longer necessary before the numerical implementation, which involves the dull routine work to separate out singularities from the integral kernels. Numerical examples were presented for both the two and the three dimensional boundary integrals in elasticity. Comparing the numerical results with those by other approaches demonstrates the feasibility and the effectiveness of the proposed scheme.

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.

Numerical solution of multiple hole problem by using boundary integral equation

This paper studies a numerical solution of multiple hole problem by using a boundary integral equation.The studied problem can be considered as a supposition of many single hole problems.After considering the interaction among holes,an algebraic equation is formulated,which is then solved by using an iteration technique.The hoop stress around holes can be finally determined. One numerical example is provided to check its accuracy.