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.     

An algorithm for solving a dual cosine series

Solutions to dual trigonometric series are frequently given as double singular integrals, which are not conducive to automatic numerical evaluation. It is suggested that by performing a preliminary Fourier analysis on the data, the double singular integrals can be eliminated, and the solutions reduced to a finite number of precisely specified elementary arithmetic operations. This technique is used to reduce to an algorithm the analytic solution to a dual cosine series which had earlier been derived by the method of orthogonality relations. Some computational results are also presented.     

An Implementation of Fast Wavelet Galerkin Methods for Integral Equations of the Second Kind

We present a numerical implementation of the fast Galerkin method for Fredholm integral equations of the second kind using the piecewise polynomial wavelets. We focus on addressing critical issues for the numerical implementation of such a method. They include a choice of practical truncation strategy, numerical integration of weakly singular integrals and the error control of the numerical quadrature. We also implement a multiscale iteration method for solving the resulting compressed linear system. Numerical examples are given to demonstrate the proposed ideas and methods.     

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.     

Numerical quadratures for singular and hypersingular integrals

We present a procedure for the design of high-order quadrature rules for the numerical evaluation of singular and hypersingular integrals; such integrals are frequently encountered in solution of integral equations of potential theory in two dimensions. Unlike integrals of both smooth and weakly singular functions, hypersingular integrals are pseudo-differential operators, being limits of certain integrals; as a result, standard quadrature formulae fail for hypersingular integrals. On the other hand, such expressions are often encountered in mathematical physics (see, for example, [1]), and it is desirable to have simple and efficient “quadrature” formulae for them. The algorithm we present constructs high-order “quadratures” for the evaluation of hypersingular integrals. The additional advantage of the scheme is the fact that each of the quadratures it produces can be used simultaneously for the efficient evaluation of hypersingular integrals, Hilbert transforms, and integrals involving both smooth and logarithmically singular functions; this results in significantly simplified implementations. The performance of the procedure is illustrated with several numerical examples.     

弱非线性耦合二维各向异性谐振子的动力学行为

研究了弱非线性耦合二维各向异性谐振子的奇点稳定性及其在相空间中的轨迹.首先,求得弱非线性耦合二维各向异性谐振子的奇点;其次,分别利用Lyapunov间接法和梯度系统方法讨论该系统的平衡点稳定性;最后,用Matlab方法对系统进行数值模拟,并运用庞加赖截面观察系统在相空间的运动轨迹,发现随着能量的增加系统经历规则运动、规则运动与混沌并存等阶段,最后出现了混沌现象.     

Numerical evaluation of singular and finite-part integrals on curved surfaces using symbolic manipulation

The numerical integration of all singular surface integrals arising in 3-d boundary element methods is analyzed theoretically and computationally. For all weakly singular integrals arising in BEM, Duffy's triangular or local polar coordinates in conjunction with tensor product Gaussian quadrature are efficient and reliable for bothh-andp-boundary elements. Cauchy- and hypersingular surface integrals are reduced to weakly singular ones by analytic regularization which is done automatically by symbolic manipulation.     

SINGINT: Automatic numerical integration of singular integrands

We explore the combination of deterministic and Monte Carlo methods to facilitate efficient automatic numerical computation of multidimensional integrals with singular integrands. Two adaptive algorithms are presented that employ recursion and are runtime and memory optimized, respectively. SINGINT, a C implementation of the algorithms, is introduced and its utilization in the calculation of particle scattering amplitudes is exemplified.     

Quadrature for parabolic Galerkin BEM with moving surfaces

The successful implementation of the Galerkin Boundary Element Method hinges on the accurate and effective quadrature of the influence coefficients. For parabolic boundary integral operators quadrature must be performed in space and time where integrals have singularities when source- and evaluation points coincide. For problems where the surface is fixed, the time integration can be performed analytically, but for moving geometries numerical quadrature in space and time must be used. For this case a set of transformations is derived that render the singular space–time integrals into smooth integrals that can be treated with standard tensor product Gauss quadrature rules. This methodology can be applied to the heat equation and to transient Stokes flow.     

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.     

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.     

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.     

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.     

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.     

A finite element based level-set method for multiphase flow applications

A numerical method for simulating incompressible two-dimensional multiphase flow is presented. The method is based on a level-set formulation discretized by a finite-element technique. The treatment of the specific features of this problem, such as surface tension forces acting at the interfaces separating two immiscible fluids, as well as the density and viscosity jumps that in general occur across such interfaces, have been integrated into the finite-element framework. Using a method based on the weak formulation of the Navier-Stokes equations has its advantages. In this formulation, the singular surface tension forces are included through line integrals along the interfaces, which are easily approximated quantities. In addition, differentiation of the discontinuous viscosity is avoided. The discontinuous density and viscosity are included in the finite element integrals. A strategy for the evaluation of integrals with discontinuous integrands has been developed based on a rigorous analysis of the errors associated with the evaluation of such integrals. Numerical tests have been performed. For the case of a rising buoyant bubble the results are in good agreement with results from a front-tracking method. The run presented here is a run including topology changes, where initially separated areas of one fluid merge in different stages due to buoyancy effects. Received: 1 March 1999 / Accepted: 17 June 1999     

An extrapolation procedure for the evaluation of singular integrals

An extrapolation procedure for the evaluation of finite range singular integrals is suggested which is based on the application of the e-algorithm to accelerate sequences of quadrature approximations. These sequences are produced by integrating over increasingly small sub-intervals using the powerful pseudo-Gaussian quadrature formulae of Patterson. Extensive numerical tests are carried out on a large number of test integrals and critical comparisons made with existing methods.     

On the numerical evaluation of derivatives of Cauchy principal value integrals

Quadrature rules for the approximate evaluation of derivatives of Cauchy principal value integrals (with respect to the free variable inside the integral) can be obtained by formal differentiations of the right sides of the corresponding quadrature rules (without derivatives). The justification of this method, under appropriate conditions, is presented in this short communication. The results of this short communication are also applicable to the numerical evaluation of a class of finite-part integrals and to the numerical solution of singular integrodifferential equations.     

Dielectric‐coated conducting curvilinear surface electrostatic model for spacecraft charging applications

In this article, free space capacitance of dielectric‐coated conducting curvilinear surfaces is obtained using method of moments (MOM). The surfaces are discretized using triangular subsections. The curvilinear surfaces are represented by quadratic parametric elements. The MOM based on the pulse basis function and point matching is used. The integrand is divided into a regular and a singular part. The regular part of the integral is numerically evaluated by using the Gaussian quadrature algorithm for triangles. The Duffy transformation is used to evaluate the singular part of the integrals. Computed results are given for the capacitance of the dielectric‐coated conducting cylinder of finite axial length and truncated cone, and compared with the previously published results. © 2010 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2010.     

A node-centric indirect boundary element method: three-dimensional displacement discontinuities

An indirect boundary element formulation based on unknown physical values, defined only at the nodes (vertices) of a boundary discretization of a linear elastic continuum, is introduced. As an adaptation of this general framework, a linear displacement discontinuity density distribution using a flat triangular boundary discretization is considered. A unified element integration methodology based on the continuation principle is introduced to handle regular as well as near-singular and singular integrals. The boundary functions that form the basis of the integration methodology are derived and tabulated in the appendix for linear displacement discontinuity densities. The integration of the boundary functions is performed numerically using an adaptive algorithm which ensures a specified numerical accuracy. The applications include verification examples which have closed-form analytical solutions as well as practical problems arising in rock engineering. The node-centric displacement discontinuity method is shown to be numerically efficient and robust for such problems.     

Efficient and Accurate Computation of Electric Field Dyadic Green’s Function in Layered Media

Concise and explicit formulas for dyadic Green’s functions, representing the electric and magnetic fields due to a dipole source placed in layered media, are derived in this paper. First, the electric and magnetic fields in the spectral domain for the half space are expressed using Fresnel reflection and transmission coefficients. Each component of electric field in the spectral domain constitutes the spectral Green’s function in layered media. The Green’s function in the spatial domain is then recovered involving Sommerfeld integrals for each component in the spectral domain. By using Bessel identities, the number of Sommerfeld integrals are reduced, resulting in much simpler and more efficient formulas for numerical implementation compared with previous results. This approach is extended to the three-layer Green’s function. In addition, the singular part of the Green’s function is naturally separated out so that integral equation methods developed for free space Green’s functions can be used with minimal modification. Numerical results are included to show efficiency and accuracy of the derived formulas.