Hypersingular Integrals Research Articles

A unified approach with spectral convergence for the evaluation of hypersingular and supersingular integrals with a periodic kernel

In this paper, a general class of methods is proposed for the evaluation of hypesingular/supersingular integrals with a periodic integrand, of singularity higher than or equal to 2. The method is based on regularizing the singular integrals into proper regular integrals when we can apply the classical trapezoidal rule. Error analysis is presented for hypesingular/supersingular integrals with a singularity of order p+1, where p is a positive integer. We prove that the convergence rate of the method is of O(lnn/nk+α) if the integrand u satisfies u∈C̃k+p,α[0,2π] (for any positive integer k), with exponential convergence rate if the integrand u is analytic. Several numerical examples are provided to support the theoretical error analysis.

Plane problem of elasticity theory for anisotropic bodies with thin elastic inclusions

Based on the coupling principle for continua of different dimensions, we constructed integral equations for determining the plane stress of an anisotropic elastic body with thin inhomogeneities of the material’s structure. For the numerical solution of these equations, we used the boundary element method and proposed new basic functions for the description of the near-end area of an inhomogeneity. The interpolation quadratures and the polynomial transformations are adopted for the efficient numerical evaluation of the corresponding singular and hypersingular integrals. Numerical examples show a high efficiency of the approach developed for the determination of the stressed state of anisotropic bodies that contain cracks and thin elastic and rigid inclusions.

The boundedness for commutators of maximal hypersingular integrals with rough kernels

In this paper, we prove that the commutators of maximal hypersingular integrals with rough kernels are bounded from the Sobolev space \(\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)\) to the Lebesgue space L p (ℝ n ), which is a substantial improvement and an extension of some known results.

Iterative method for the determination of a diffraction field in the interaction of a longitudinal shear wave with a system of cracks

UDC 539.3 We consider the problem of determination of a diffraction field that is formed due to the interaction of longitudinal shear waves with a system of cracks arbitrarily located in the infinite body. The initial problem is reduced to a system of singular integro-differential equations. We propose an iterative method of solution of this system, where the zero approximation is the solutions of integral equations for separate cracks. Modern methods of the mechanics of deformable bodies (method of potentials, method of discontinuous solutions, etc.) allow one to easily reduce the problem of determination of a wave field in the body with an arbitrary system of cracks to a system of integral equations. As a matter of fact, the interaction of longitudinal-shear waves (SH-waves) with an arbitrary system of cracks in the infinite elastic body was studied within the method of integral equations in [7, 11]. In [4], an analogous problem for radially positioned cracks was solved within the method of discontinuous solutions. These studies were continued in [3], where an elastic body under conditions of a plane deformation was considered. In these works, it is proposed to solve the systems of integrodifferential equations for the opening displacements of cracks and their derivatives in order to avoid hypersingular integrals. For such systems, an efficient procedure of numerical solution is available. To determine the wave fields in bodies with systems of cracks, the method of boundary elements is used more and more frequently in recent times. For example, this method was applied to the solution of two- and three-dimensional problems in [6, 8] and [9, 10], respectively. However, an analysis of these works shows that the reduction of initial boundary-value problems to boundary integral equations encounters serious difficulties, which consist of the necessity to numerically solve the systems of integral equations of high dimensions. The number of equations of such systems is proportional to the number of cracks, and the kernels of the equations themselves have singularities of high orders. For this reason, all authors restricted themselves to the case of several, most often two, cracks, while performing the numerical calculations of wave fields and stress intensity factors. Therefore, the problem of determination of a stressed state and diffraction fields in bodies with systems of cracks is not completely solved. In the present work, we propose an iterative approach to the determination of a diffraction field and the stress intensity factor in the case of the interaction of an SH-wave with a system of N cracks. As a result, there is no necessity to solve a system of N integro-differential equations, and, at each iteration, N separate equations corresponding to isolated cracks are solved. Statement of the Problem Let N through cracks be located in an elastic body in the state of plane deformation. In the plane Oxy , these cracks occupy intervals 2d k in length, whose middle points have coordinates (a k , b k ) , k = 1, 2, … , N

3D acoustic wave simulation using BEM formulations: Closed form integration of singular and hypersingular integrals

Abstract Among the obstacles to applying boundary element techniques to three-dimensional wave propagation problems is the difficulty of accurately representing the singular and hypersingular terms at the points of application of the virtual loads. This paper presents the analytical evaluation of the singular and hypersingular integrals for constant boundary elements. First, the singular integral results are compared with those evaluated by means of a Gaussian quadrature scheme, which uses an enormous amount of sampling points. In the case of hypersingular integrals the comparison makes use of the results provided by the method presented by Terai [T. Terai, On calculation of sound fields around three dimensional objects by integral equation methods, J Sound Vib 69 (1980) 71–100.]. An additional verification is performed by comparing the boundary element method (BEM) results with known analytical solutions for cylindrical inclusions.

Stabilization of time-domain acoustic boundary element method for sound radiation problems

Time-domain acoustic solution from the Kirchhoff integral equation for the exterior problem is not unique due to the presence of fictitious internal modes and also suffers from the instability that stems from the time marching scheme. In this work, methods to stabilize the time-domain acoustic boundary element calculation were suggested. Low-order fictitious internal modes within the effective frequency range of boundary element calculation were suppressed by the newly formulated time-domain CHIEF (Combined Helmholtz Integral Equation Formulation) method. Additional interior points were included, similar to frequency-domain problems, to satisfy the zero pressure constraint considering the shortest retarded time between boundary nodes and interior points. However, the calculation was yet unstable due to remaining unstable high-order modes beyond the effective frequency limit. To further stabilize the computation, unstable high-order internal modes were nullified using the wave vector filtering method. In comparison with the time-domain Burton-Miller formulation, the proposed method has no hyper-singular integral and the monotonically increasing instability was not observed. As a test example, sound radiation from a pulsating sphere was used and a good stabilization performance was shown. Average relative-difference norm of the stabilized time response from the analytic solution was 2.7%. (This work was partially supported by BK21 project)

3D ACOUSTIC SHAPE SENSITIVITY ANALYSIS USING FAST MULTIPOLE BOUNDARY ELEMENT METHOD

This paper presents a fast multipole boundary element approach to the 3D acoustic shape sensitivity analysis based on the discrete adjoint variable method. The Burton–Miller formula is used to conquer the fictitious eigenfrequency problem of the original boundary integral equation method in solving exterior acoustic problems. Constant elements are used to discretize the boundary, so that the hypersingular integrals contained in the formulae can be evaluated explicitly and directly. Numerical examples are presented to demonstrate the accuracy and efficiency of the present approach.

Taming hypersingular integrals using dimensional continuation

We use the method of dimensional continuation to isolate singularities in integrals containing products of Green's functions or their derivatives. Rules for the extraction of the finite part of so-called hypersingular integrals are developed, which should be useful in methods based on boundary integral techniques in science and engineering. In applications to potential theory, electromagnetic scattering, and crack dynamics in continuum mechanics, boundary integrals now can be readily evaluated using computational techniques without recourse to complex analysis or contour distortions since the hypersingularities occurring in intermediate steps of the computations can be isolated and ignored while taking the finite parts of the integrals into account in a consistent manner. We have also identified new forms of the Dirac δ function in D dimensions, which are useful and convenient in the calculations. A summary of the integrable singular integrals is given in tabular form. We extend the considerations to a wider class of Green's functions and present a theorem, with additional results arising from it, that shows that hypersingular integrals associated with three-dimensional potential problems can be reduced to one-dimensional finite integrals rather than two-dimensional integrals, again leading to direct evaluations in such cases. These calculations are compared with existing results to show the efficacy of the approach.

Closed Form Integration of Singular and Hypersingular Integrals in 3D BEM Formulations for Heat Conduction

The evaluation of the singular and hypersingular integrals that appear in three‐dimensional boundary element formulations for heat diffusion, in the frequency domain, is presented in analytical form. This improves computational efficiency and accuracy. Numerical integrations using existing techniques based on standard Gaussian integration schemes that incorporate an enormous amount of sampling points are used to verify the solutions of singular integrals. For the hypersingular integrals the comparison is evaluated by making use of an analytical solution that is valid for circular domains, combined with a standard Gaussian integration scheme for the remaining boundary element domain. Closed form solutions for cylindrical inclusions (with null temperatures and null heat fluxes prescribed on the boundary) are then derived and used to validate the three‐dimensional boundary element formulations.

Hp-DGFEM FOR KOLMOGOROV–FOKKER–PLANCK EQUATIONS OF MULTIVARIATE LÉVY PROCESSES

We analyze the discretization of nonlocal degenerate integrodifferential equations arising as so-called forward equations for jump-diffusion processes. Such equations arise in option pricing problems when the stochastic dynamics of the markets is modeled by Lévy driven stochastic volatility models. Well-posedness of the arising equations is addressed. We develop and analyze stable discretization schemes, in particular the discontinuous Galerkin Finite Element Methods (DG-FEM). In the DG-FEM, a new regularization of hypersingular integrals in the Dirichlet form of the pure jump part of infinite variation processes is proposed, allowing in particular a stable DG discretization of hypersingular integral operators. Robustness of the stabilized discretization with respect to various degeneracies in the characteristic triple of the stochastic process is proved. We provide in particular an hp-error analysis of the DG-FEM. Numerical experiments for model equations confirm the theoretical results.

AUTOMATIC QUADRATURE SCHEME FOR EVALUATING HYPERSINGULAR INTEGRALS

Hasegawa constructed the automatic quadrature scheme (AQS), of Cauchy principle value integrals for smooth functions. There is a close connection between Hadamard and Cauchy principle value integral. In this paper, we modify AQS for hypersingular integrals with second-order singularities, using hasegawa's formula and based on the relations between Hadamard finite part integral and Cauchy principle value integral. Numerical experiments are also given, to validate the modified AQS.

Asymptotic error expansions for hypersingular integrals

This paper presents quadrature formulae for hypersingular integrals $\int_a^b\frac{g(x)}{|x-t|^{1+\alpha }}\mathrm{d}x$ , where a?<?t?<?b and 0?<?????1. The asymptotic error estimates obtained by Euler---Maclaurin expansions show that, if g(x) is 2m times differentiable on [a,b], the order of convergence is O(h 2μ ) for ??=?1 and O(h 2μ???? ) for 0?<???<?1, where μ is a positive integer determined by the integrand. The advantages of these formulae are as follows: (1) using the formulae to evaluate hypersingular integrals is straightforward without need of calculating any weight; (2) the quadratures only involve g(x), but not its derivatives, which implies these formulae can be easily applied for solving corresponding hypersingular boundary integral equations in that g(x) is unknown; (3) more precise quadratures can be obtained by the Richardson extrapolation. Numerical experiments in this paper verify the theoretical analysis.

On near-trapped modes and fictitious frequencies for water wave problems containing an array of circular cylinders using a null-field boundary integral equation

To avoid using the addition theorem to translate the Bessel function, scattering of water waves by an array of circular cylinders is solved by using the null-field boundary integral equations, in conjunction with the adaptive observer system. Both the near-trapped modes (physics) and fictitious frequencies (mathematics) are observed. To alleviate the resonance problem of fictitious frequency for multiple cylinders, Combined Helmholtz Interior integral Equation Formulation (CHIEF) approach and Burton and Miller formulation are both considered. Regarding the Burton and Miller approach, hypersingular integrals can be easily calculated by using series summability without any difficulty owing to the introduction of degenerate kernels. The highly rank-deficient matrices for equal radius of cylinders are numerically examined and the rank is improved by adding valid CHIEF constraints. Besides, the selection of location and number for CHIEF points is addressed instead of trial and error. Parameter study of the incident angle on the resultant force is investigated. The effect of spacing and radius of cylinders on the near-trapped mode and fictitious frequency is also discussed. Several examples of water wave interaction by circular cylinders were demonstrated to see the validity of the present formulation.

A numerical solution for the equation of the lifting surface in subsonic flow including tunnel effects

We employ the images method to establish the lifting surface equation including tunnel effects. We obtain a 2d hypersingular integral and we utilize Gauss-type quadrature formulas to discretize it. Numerical calculations are performed for the elliptical and rectangular wings in order to calculate the jump of the pressure.

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.

Divergent Integrals in Elastostatics: General Considerations

This article considers weakly singular, singular, and hypersingular integrals, which arise when the boundary integral equation methods are used to solve problems in elastostatics. The main equations related to formulation of the boundary integral equation and the boundary element methods in 2D and 3D elastostatics are discussed in details. For their regularization, an approach based on the theory of distribution and the application of the Green theorem has been used. The expressions, which allow an easy calculation of the weakly singular, singular, and hypersingular integrals, have been constructed.

Explicit evaluation of hypersingular boundary integral equations for acoustic sensitivity analysis based on direct differentiation method

This paper presents a new set of boundary integral equations for three dimensional acoustic shape sensitivity analysis based on the direct differentiation method. A linear combination of the derived equations is used to avoid the fictitious eigenfrequency problem associated with the conventional boundary integral equation method when solving exterior acoustic problems. The strongly singular and hypersingular boundary integrals contained in the equations are evaluated as the Cauchy principal values and Hadamard finite parts for constant element discretization without using any regularization technique in this study. The present boundary integral equations are more efficient to use than the usual ones based on any other singularity subtraction technique and can be applied to the fast multipole boundary element method more readily and efficiently. The effectiveness and accuracy of the present equations are demonstrated through some numerical examples.

Algorithms for approximating finite Hilbert transform with end-point singularities and its derivatives

Algorithms are proposed for the numerical evaluation of Cauchy principal value integrals ⨍ − 1 1 w ( t ) f ( t ) / ( t − x ) d t , − 1 < x < 1 , with weight functions of Jacobi type singularities w ( t ) = ( 1 − t ) α ( 1 + t ) β , where α = ± 1 / 2 and β = ± 1 / 2 , for a given function f ( t ) and Hadamard finite-part integrals ⨎ − 1 1 w ( t ) f ( t ) / ( t − x ) 2 d t . The function f is interpolated by using a finite sum of Chebyshev polynomials. The present algorithms require O ( N log N ) arithmetic operations, where N is the order of the interpolation polynomial. It is shown that the present scheme gives uniform approximations, namely the errors are bounded independently of x , and is very efficient for smooth f . Further, we discuss approximations of hyper-singular integrals ∫ − 1 1 w ( t ) f ( t ) / ( t − x ) n d t , n ≥ 3 , and show their uniform convergences. Numerical examples are given to demonstrate the performance of the present schemes.

Fast and Accurate Computation of Hypersingular Integrals in Galerkin Surface Integral Equation Formulations via the Direct Evaluation Method

Hypersingular 4-D integrals, arising in the Galerkin discretization of surface integral equation formulations, are computed by means of the direct evaluation method. The proposed scheme extends the basic idea of the singularity cancellation methods, usually employed for the regularization of the singular integral kernel, by utilizing a series of coordinate transformations combined with a reordering of the integrations. The overall algebraic manipulation results in smooth 2-D integrals that can be easily evaluated via standard quadrature rules. Finally, the reduction of the dimensionality of the original integrals together with the smooth behavior of the associated integrands lead up to unmatched accuracy and efficiency.

Evaluation of free terms in hypersingular boundary integral equations

Summary The accurate numerical solution of hypersingular boundary integral equations necessitates the precise evaluation of free terms, which are required to counter discontinuous and often unbounded behaviour of hypersingular integrals at a boundary. The common approach for the evaluation of free terms involves integration over a portion of a spherical shaped surface centred at a singularity and allowing the radius of the sphere to tend to zero. In this paper two alternative methods, which are shape invariant, are proposed and investigated for the determination of free terms. One approach, the point-limiting method, involves moving a singularity towards a shrinking integration domain at a faster rate than the domain shrinks. Issues surrounding the choice of approach and shrinkage rates, and path dependency are examined. A related approach, the boundary-limiting method, involves moving an invariant but shrinking boundary toward the singularity again at a faster rate than the shrinkage of the domain. The latter method can be viewed as a vanishing exclusion zone approach but the actual boundary shape is used for the boundary of the exclusion zone. Both these methods are shown to provide consistent answers and can be shown to be directly related to the result obtained by moving a singularity towards a boundary, i.e. by comparison with the direct method. Unlike the spherical approach the two methods involve integration over the actual boundary shape and consequently shape dependency is not a concern. A particular highlight of the point limiting approach, as a result of field approximations being restricted to the boundary, is the ability to obtain free terms in a mixed formulation without reference to the underpinning constitutive equations, which is not available to the spherical method. Focus in the paper is on the 2-D potential equation as this is shown to be sufficient to demonstrate the concepts involved.