Finite-part Integrals Research Articles

Three-dimensional interfacial Green’s functions in anisotropic bimaterials

In this paper, we derive, for the first time, the complete set of three-dimensional interfacial elastostatic Green’s functions in anisotropic bimaterials, including displacements, stresses, and their derivatives with respect to the source coordinates. We make use of the extended Stroh formalism and the Mindlin’s superposition method, and express these Green’s functions in terms of one-dimensional finite-part integrals with variable θ over [0, π]. Denoting by r the distance between the field and source points on the interfacial planes, we show that the interfacial displacements, stresses and derivatives of displacements, and derivatives of stresses are proportional, respectively, to 1/ r, 1/ r 2, and 1/ r 3, while their finite-part integrals are, respectively, in the orders of 1/cos θ, 1/cos 2 θ, and 1/cos 3 θ. Because of the special dependence upon the distance r, the interfacial Green’s functions on the whole interfacial plane are completely determined by their values on the unit circle on the interfacial plane. An efficient and accurate method is also proposed for the evaluation of the involved finite-part integrals, and some typical numerical examples are given to show the general features of the interfacial Green’s functions. In particular, it is remarked that some of them are discontinuous across the interface. These interfacial Green’s functions are essential to various integral-equation methods in solving inclusion and interfacial crack problems in anisotropic bimaterials. Furthermore, they are also required in the study of strained quantum dot semiconductor devices should the Green’s function method be applied.

On the use of interpolative quadratures for hypersingular integrals in fracture mechanics

The implementation of finitepart integration of hypersingular boundary integrals is discussed in the context of the applications in engineering fracture mechanics. The approach uses a formulation o...

Solving strongly singular integral equations by Lp approximation methods

It has previously been shown that integral equations can be solved effectively by means of an L p approximation technique (and particularly by means of an L 1 approximation). This method has also successfully been applied to singular integral equations. In the present discussion attention is paid to solving strongly singular integral equations via L p approximation techniques, and in particular to the case of integral equations containing Hadamard (or finite-part) integrals.

Transformations for evaluating singular boundary element integrals

Accurate numerical integration of line integrals is of fundamental importance for the reliable implementation of the boundary element method. Usually, the regular integrals arising from a boundary element method implementation are evaluated using standard Gaussian quadrature. However, the singular integrals which arise are often evaluated in another way, sometimes using a different integration method with different nodes and weights. This paper presents a straightforward transformation to improve the accuracy of evaluating singular integrals. The transformation is, in a sense, a generalisation of the popular method of Telles with the underlying idea being to utilise the same Gaussian quadrature points as used for evaluating nonsingular integrals in a typical boundary element method implementation. The new transformation is also shown to be equivalent to other existing transformations in certain situations. Comparison of the new method with existing coordinate transformation techniques shows that a more accurate evaluation of weakly singular integrals can be obtained. The technique can also be extended to evaluate certain Hadamard finite-part integrals. Based on the observation of several integrals considered, guidelines are suggested for the best transformation order to use (i.e. the degree to which nodes should be clustered near the singular point).

An appropriate quadrature rule for the analysis of plane crack problems in the boundary‐element method

AbstractAn hypersingular integral equation of a three‐dimensional elastic solid with an embedded planar crack subjected to a uniform stress field at infinity is derived. The solution of the boundary‐integral equation is succeeded taking into consideration an appropriate Gauss quadrature rule for finite part integrals which is suitable for the numerical treatment of any plane crack with a smooth‐contour shape and permit the fast convergence for the results. The problem of a circular and of an elliptical crack in an infinite body subjected to a uniform stress field at infinity is confronted; and the stress intensity factors are calculated. Copyright © 2001 John Wiley & Sons, Ltd.

A Chebyshev Quadrature Rule for One Sided Finite Part Integrals

This paper is concerned with a Chebyshev quadrature rule for approximating one sided finite part integrals with smooth density functions. Our quadrature rule is based on the Chebyshev interpolation polynomial with the zeros of the Chebyshev polynomial T N +1 ( τ )− T N −1 ( t ). We analyze the stability and the convergence for the quadrature rule with a differentiable function. Also we show that the quadrature rule has an exponential convergence when the density function is analytic.

Evaluations of certain hypersingular integrals on interval

AbstractIn this paper, we investigate a hypersingular integral on an interval. The definition of Hadamard's finite‐part integrals and some of its properties are given. Some numerical methods on approximate computation of the finite‐part integrals are constructed. The new method is very simple, easy to implement, reliable, and above all, not affected by the location of singular point. Some numerical experiments are carried out using the current formulae, and numerical results show that the current methods are feasible and effective. Copyright © 2001 John Wiley & Sons, Ltd.

Transformation of hypersingular integrals and black-box cubature

In this paper, we will consider hypersingular integrals as they arise by transforming elliptic boundary value problems into boundary integral equations. First, local representations of these integrals will be derived. These representations contain so-called finite-part integrals. In the second step, these integrals are reformulated as improper integrals. We will show that these integrals can be treated by cubature methods for weakly singular integrals as they exist in the literature.

Differentiation of finite element solutions to non‐linear problems

Two extended numerical differentiation methods based on Green's second identity are presented. These may be used for postprocessing approximate solutions in general material distributions, including inhomogeneous and discontinuous material characteristics. The first method uses a general formulation with Green's functions and extended Poisson kernels for standard domains, while the second applies Green's functions to certain restricted, analytically known configurations. The singularities encountered in the necessary integral kernels for second derivatives are evaluated using finite part integration techniques. Both methods are illustrated by numerical experiments, and results are shown for differentiation of quasi-harmonic functions in inhomogeneous domains. Copyright © 2000 John Wiley & Sons, Ltd.

Numerical solution of integral equations with finite part integrals

We obtain convergence rates for several algorithms that solve a class of Hadamard singular integral equations using the general theory of approximations for unbounded operators.

The complex one-sided integrals of Cauchy and Hadamard and application to boundary element method

The definitions of complex integrals of Cauchy and Hadamard with the singular point coinciding with the end point of the integration curve are proposed. It is shown that the new integrals satisfy most of the properties of the regular ones, including the change of variables. It is also shown that the Cauchy principal value (CPV) and Hadamard finite-part (HFP) integrals can be considered as a sum of the new type integrals. The application to numerical solution by the boundary element method (BEM) and the complex hypersingular integral equation (CHSIE) for the multiregions of interacting elastic bodies and bodies with cracks and holes is discussed. The different ways to place the collocation points are considered. The numerical results for the problems of circular hole and circular elastic inclusion in infinite plate indicated that the appropriate choice of the approximating functions leads to a high accuracy of the calculation. Applications of the new technique to geomechanics problems are discussed. © 1998 John Wiley & Sons, Ltd.

The Euler-Maclaurin expansion and finite-part integrals

In this paper we compare G(p), the Mellin transform (together with its analytic continuation), and $\overline{\overline{G}}(p)$ , the related Hadamard finite-part integral of a function g(x), which decays exponentially at infinity and has specified singular behavior at the origin. Except when p is a nonpositive integer, these coincide. When p is a nonpositive integer, $\overline{\overline{G}}(p)$ is well defined, but G(p) has a pole. We show that the terms in the Laurent expansion about this pole can be simply expressed in terms of the Hadamard finite-part integral of a related function. This circumstance is exploited to provide a conceptually uniform proof of the various generalizations of the Euler-Maclaurin expansion for the quadrature error functional.

The Euler-Maclaurin formula revisited

The Euler-Maclaurin summation formula for the approximate evaluation of I = \int 0 1 f ( x ) d x comprises a sum of the form (1/ m )\sum j =0 m -1 f (( j + t ? )/ m ), where 0? t ? ? 1, a second sum whose terms involve the difference between the derivatives of f at the end-points 0 and 1 and a truncation error term expressed as an integral. By introducing an appropriate change of variable of integration using a sigmoidal transformation of order r ?1, (other authors call it a periodizing transformation) it is possible to express I as a sum of m terms involving the new integrand with the second sum being zero. We show that for all functions in a certain weighted Sobolev space, the truncation error is of order O (1/ m n 1 ) , for some integer n 1 which depends on r . In principle we may choose n 1 to be arbitrarily large thereby giving a good rate of convergence to zero of the truncation error. This analysis is then extended to Cauchy principal value and certain Hadamard finite-part integrals over (0,1). In each case, the truncation error is O (1/ m n 1 ). This result should prove particularly useful in the context of the approximate solution of integral equations although such discussion is beyond the scope of this paper.

Calculation of strongly singular and hypersingular surface integrals

Efficient numerical computation of integrals defined on closed surfaces in ℝ3 with non-integrable point singularities that arise in physical geodesy is discussed. The method is based on the use of polar coordinates and the definition of integrals with non-integrable point singularities as Hadamard finite part integrals. First the behavior of singular integrals under smooth parameter transformations is studied, and then it is shown how they can be reduced to absolutely integrable functions over domains in ℝ2. The correction terms that usually arise if the substitution rule is formally applied, in contrast to absolutely integrable functions, are calculated. It is shown how to compute the regularized integrals efficiently, and, numerical efforts for various orders of singularity are compared. Finally, efficient numerical integration methods are discussed for integrals of functions that are defined as singular integrals, a task that typically arises in Galerkin boundary element methods.

Derivation of exact expressions for two-dimensional singular and finite-part integrals applicable in solid mechanics

The purpose of this paper is to develop exact expressions for the evaluation of singular and finite-part integrals appearing in the stress analysis of elastic solids. A numerical solution is presented for the solution of problems in three dimensional elastostatics. The kernels of the integral equation solution are given in the case where the surfaces of the solid are discretized with flat triangular elements. Exact expressions are derived for the singular and finite-part integrals over a triangular domain. The availability of exact expressions for these integrals will increase the accuracy of the numerical results without the need of any cubature formulae. Finally, the example considered in the paper indicates the excellent approximation obtained by the method even when using only a few classical elements.

Integration of singular integrals for the Galerkin-type boundary element method in 3D elasticity

The mixed boundary value problem in three-dimensional linear elasticity is solved via a system of boundary integral equations. The Galerkin approximation of the singular and hypersingular integral equations leads to (hyper)singular and regular double integrals. The numerical cubature of the singular integrals is discussed in the case of domains which have piecewise smooth surfaces with a Riemann metric structure. We give a method for reducing finite part integrals to at most Cauchy singular integrals. The presented integration method applied to Cauchy singular integrals leads to explicitly given regular integrand functions which can be integrated by standard Gaussian product quadrature rules. The number of necessary Gaussian knots depends on the shape of the boundary elements and on the curvature of the surface. We give estimates of the quadrature error including constants, which mark the properties of the boundary elements. The error estimates can be taken to implement adaptive cubature methods. The numerical example is a mixed boundary value problem with corner and edge singularities.

Theoretical analysis of three-dimensional interface crack

Using the fundamental solution of interface crack and the method of finite-part integral, the problem of three-dimensional interface crack is reduced to solve a set of two-dimensional hypersingular integrodifferential equations with unknown displacement discontinuities of crack surface. Then a systematically theoretical analysis for solving these equations is presented.

Error bounds for spline-based quadrature methods for strongly singular integrals

For the numerical evaluation of finite-part integrals with singularities of order p ⩾ 1, we give error bounds for quadrature methods based on spline approximation. These bounds behave in the same way as the optimal ones. The ideas of the proof are also useful for methods based on other approximation processes.

A method for the numerical evaluation of a hadamard finite partintegral

An accurate numerical method for the evaluation of a particular Hadamard finite part integral using the IMT quadrature is indicated. The proposed method exploits a standard relation between the Cauchy principal value integral and the Hadamard finite part integral. The paper also provides an asymptotic estimate of the discretization error, starting from a contour integral representation of the error term.

Generalized compound quadrature formulae for finite-part integrals

We investigate the error term of the dth degree compound quadrature formulae for finite-part integrals of the form f(x -p f(x)dx where p ∈ R and p ≥ 1. We are mainly interested in error bounds of the form (R[f]( ≤ c∥f (s) ∥∞ with best possible constants c. It is shown that, for p ∈ N and n uniformly distributed nodes, the error behaves as O(n p-s-1 ) for f ∈ C 5 [0,1], p - 1 < s ≤ d + 1. In a previous paper we have shown that this is not true for p ∈ N. As an improvement, we consider the case of non-uniformly distributed nodes. Here, we show that for all p ≥ 1 and f ∈ C 5 [0,1], an O(n -s ) error estimate can be obtained in theory by a suitable choice of the nodes. A set of nodes with this property is stated explicitly. In practice, this graded mesh causes stability problems which are computationally expensive to overcome.