A semi-analytical algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods

The nearly singular integrals occur in the boundary integral equations when the source point is close to an integration element (as compared to its size) but not on the element. In this paper, the concept of a relative distance from a source point to the boundary element is introduced to describe possible influence of the singularity of the integrals. Then a semi-analytical algorithm is proposed for evaluating the nearly strongly singular and hypersingular integrals in the three-dimensional BEM. By using integration by parts, the nearly singular surface integrals on the elements are transformed to a series of line integrals along the contour of the element. The singular behavior, which appears as factor, is separated from remaining regular integrals. Consequently standard numerical quadrature can provide very accurate evaluation of the resulting line integrals. The semi-analytical algorithm is applied to analyzing the three-dimensional elasticity problems, such as very thin-walled structures. Meanwhile, the displacements and stresses at the interior points very close to its bounding surface are also determined efficiently. The results of the numerical investigation demonstrate the accuracy and effectiveness of the algorithm.     

Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems

A new completely analytical integral algorithm is proposed and applied to the evaluation of nearly singular integrals in boundary element method (BEM) for two-dimensional anisotropic potential problems. The boundary layer effect and thin body effect are dealt with. The completely analytical integral formulas are suitable for the linear and non-isoparametric quadratic elements. The present algorithm applies the analytical formulas to treat nearly singular integrals. The potentials and fluxes at the interior points very close to boundary are evaluated. The unknown potentials and fluxes at boundary nodes for thin body problems with the thickness-to-length ratios from 1E−1 to 1E−8 are accurately calculated by the present algorithm. Numerical examples on heat conduction demonstrate that the present algorithm can effectively handle nearly singular integrals occurring in boundary layer effect and thin body effect in BEM. Furthermore, the present linear BEM is especially accurate and efficient for the numerical analysis of thin body problems.     

Analysis of thin piezoelectric solids by the boundary element method

The piezoelectric boundary integral equation (BIE) formulation is applied to analyze thin piezoelectric solids, such as thin piezoelectric films and coatings, using the boundary element method (BEM). The nearly singular integrals existing in the piezoelectric BIE as applied to thin piezoelectric solids are addressed for the 2-D case. An efficient analytical method to deal with the nearly singular integrals in the piezoelectric BIE is developed to accurately compute these integrals in the piezoelectric BEM, no matter how close the source point is to the element of integration. Promising BEM results with only a small number of elements are obtained for thin films and coatings with the thickness-to-length ratio as small as 10⁻⁶, which is sufficient for modeling many thin piezoelectric films as used in smart materials and micro-electro-mechanical systems.     

Global and local Trefftz boundary integral formulations for sound vibration

The sound-pressure field harmonically varying in time is governed by the Helmholtz equation. The Trefftz boundary integral equation method is presented to solve two-dimensional boundary value problems. Both direct and indirect BIE formulations are given. Non-singular Trefftz formulations lead to regular integrals counterpart to the conventional BIE with the singular fundamental solution. The paper presents also the local boundary integral equations with Trefftz functions as a test function. Physical fields are approximated by the moving least-square in the meshless implementation. Numerical results are given for a square patch test and a circular disc.     

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.     

基于边界面法的完整实体应力分析理论与应用

提出基于边界面法（Boundary Face Method,BFM）的完整实体应力分析方法.在该分析中,避免对结构作几何上的简化,结构的所有局部细节都按实际形状尺寸作为三维实体处理.以边界积分方程为理论基础的BFM是完整实体应力分析的自然选择.在该方法中,边界积分和场变量插值都在实体边界曲面的参数空间里实现.高斯积分点的几何数据,如坐标、雅可比和外法向量都直接由曲面算得,而不是通过单元插值近似获得,从而避免几何误差.该方法的实现直接基于边界表征的CAD模型,可做到与CAD软件的无缝连接.线弹性问题的应用实例表明,该方法可以简单有效地模拟具有细小特征的复杂结构,并且计算结果的应力精度比边界元法（Boundary Element Method,BEM）和有限元法（Finite Element Method,FEM）高.     

Treatment of singular integral caused by employing linear boundary elements for two-dimensional elastostatic problems

The boundary element method employing linear elements has the advantage of high precision in calculations. However, there exists a problem of singularity when the diagonal terms of the boundary influence coefficients are calculated. A method by which the singular terms are cancelled is proposed in this paper. A computer program for the solution of two-dimensional elastostatic problems using linear boundary elements is developed and verified through two examples. The technique dealing with the nodes at corners is discussed. The results show that the present method of solving singular integrals is credible.     

Non-equivalence of the conventional boundary integral formulation and its elimination for plane elasticity problems

Compared with a given boundary value problem of plane elasticity, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions which are dependent upon Poisson's ratio and geometry. Such a non-equivalence of solutions of boundary integral equations can be eliminated by using a necessary and sufficient boundary integral formulation proposed by He [Necessary and sufficient BIE-BEM: its theory and practice. Ph.D. Dissertation, Zhejiang University, Hangzhou, China (1993)]. Numerical analysis shows that the conventional boundary integral equation yields incorrect non-equivalent results when the scale in the fundamental solution is near its degenerate scale value. Also, this non-equivalence can be remedied by using the necessary and sufficient boundary integral equation.     

The boundary element-free method for 2D interior and exterior Helmholtz problems

The boundary element-free method (BEFM) is developed in this paper for numerical solutions of 2D interior and exterior Helmholtz problems with mixed boundary conditions of Dirichlet and Neumann types. A unified boundary integral equation is established for both interior and exterior problems. By using the improved interpolating moving least squares method to form meshless shape functions, mixed boundary conditions in the BEFM can be satisfied directly and easily. Detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells. Three numerical integration procedures are developed for the computation of strongly singular integrals. Numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method.     

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.     

Dynamic elastoplastic analysis of 3-D structures by the domain/boundary element method

The main objective of this paper is to present a general three-dimensional boundary element methodology for solving transient dynamic elastoplastic problems. The elastostatic fundamental solution is used in writing the integral representation and this creates in addition to the surface integrals, volume integrals due to inertia and inelasticity. Thus, an interior discretization in addition to the usual surface discretization is necessary. Isoparametric linear quadrilateral elements are used for the surface discretization and isoparametric linear hexahedra for the interior discretization. Advanced numerical integration techniques for singular and nearly singular integrals are employed. Houbolt's step-by-step numerical time integration algorithm is used to provide the dynamic response. Numerical examples are presented to illustrate the method and demonstrate its accuracy.     

Preconditioners for Solving Stochastic Boundary Integral Equations with Weakly Singular Kernels

A stochastic linear heat conduction problem is reduced to a special weakly singular integral equation of the second kind. The smoothness of the solution to a multidimensional weakly singular integral equation is investigated. It is also indicated that the derivatives of solutions may have singularities of certain order near the boundary of domain. The solution in the form of a multidimensional cubic spline is studied using circulant integral operators and a special mesh near the boundary with respect to all variables. Furthermore, stable numerical algorithms are given. Received: June 22, 1998; revised November 11, 1998     

Transient analysis of the dynamic stress intensity factors using SGBEM for frequency-domain elastodynamics

In this paper, a two-dimensional symmetric-Galerkin boundary integral formulation for elastodynamic fracture analysis in the frequency domain is described. The numerical implementation is carried out with quadratic elements, allowing the use of an improved quarter-point element for accurately determining frequency responses of the dynamic stress intensity factors (DSIFs). To deal with singular and hypersingular integrals, the formulation is decomposed into two parts: the first part is identical to that for elastostatics while the second part contains at most logarithmic singularities. The treatment of the elastostatic singular and hypersingular singular integrals employs an exterior limit to the boundary, while the weakly singular integrals in the second part are handled by Gauss quadrature. Time histories (transient responses) of the DSIFs can be obtained in a post-processing step by applying the standard fast Fourier transform (FFT) and algorithm to the frequency responses of these DSIFs. Several test examples are presented for the calculation of the DSIFs due to two types of impact loading: Heaviside step loading and blast loading. The results suggest that the combination of the symmetric-Galerkin boundary element method and standard FFT algorithms in determining transient responses of the DSIFs is a robust and effective technique.     

Evaluation of singular integrals in the symmetric Galerkin boundary element method

The implementation of the symmetric Galerkin boundary element method (SGBEM) involves extensive work on the evaluation of various integrals, ranging from regular integrals to hypersingular integrals. In this paper, the treatments of weak singular integrals in the time domain are reviewed, and analytical evaluations for the spatial double integrals which contain weak singular terms are derived. A special scheme on the allocation of Gaussian integration points for regular double integrals in the SGBEM is developed to improve the efficiency of the Gauss–Legendre rule. The proposed approach is implemented for the two-dimensional elastodynamic problems, and two numerical examples are presented to verify the accuracy of the numerical implementation.     

Weibull renewal equation solution

The existing solution methods for the Weibull Renewal Equation suffer from a lack of sufficient accuracy due to the singularity at the origin for some parameter values of the weibull density. The proposed method of solution provides accuracy to any desired degree of precision for all parameter values particularly in the singular range. The method utilizes a cubic spline approximation of the unknown renewal function and applies the Galerkin technique of integral equation solution. Gaussian quadratures are used to evaluate integrals. The singular nature of the integrand is handled by the Gauss-Jacobi quadrature. Results are compared with those obtained by simulation.     

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Dual boundary integral equations at a corner using contour approach around singularity

A dual integral formulation for the Laplace equation problem with a corner is derived by using the contour approach surrounding the singularity. It is found that using the contour approach the jump term comes half and half from the free terms in the L and M kernel integrations, which is different from the limiting process from an interior point to a boundary point where the jump term comes from the L kernel only. Thus, the definition of the Hadamard principal value for hypersingular integration at the collocation point of a corner is extended to a generalized sense for both the tangent and normal derivative of double layer potentials in comparison with the conventional definition. Two regularized versions of dual boundary integral equations with corners are proposed to avoid the boundary effect and are tested by an example. The numerical implementation is incorporated in the BEPO2D program. Also, a numerical example with a Dirichlet boundary condition on the corner is verified to determine the validity of the dual integral formulation.     

A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics

We introduce a novel enriched Boundary Element Method (BEM) and Dual Boundary Element Method (DBEM) approach for accurate evaluation of Stress Intensity Factors (SIFs) in crack problems. The formulation makes use of the Partition of Unity Method (PUM) such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation. An enrichment strategy is described, in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns. In addition, an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. Finally, results are shown for mixed mode crack problems; these illustrate that the introduction of PUM enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched DBEM.     

High-accuracy quadrature methods for solving boundary integral equations of steady-state anisotropic heat conduction problems with Dirichlet conditions

This paper presents the mechanical quadrature methods (MQMs) for solving the boundary integral equations of steady-state anisotropic heat conduction equation on the smooth domains and polygons, respectively. The costless and high-accurate Sidi–Israeli quadrature formula are applied to deal with the integrals in which the kernels have a logarithmic singularity. Especially, the Sidi transformation is used for the polygon cases in order to obtain a rapid convergence by degrading the singularity at the corners on the boundary. The convergence and stability of the MQMs solution are proved based on Anselone's collective compact theory. In addition, asymptotic error expansion of the MQMs shows that the approximation order is of O(h³), where h is the partition size of the boundary. Finally, numerical examples are tested and results verify the theoretical analysis.