Direct Integration of the Newton Potential over Cubes

In boundary element methods, the evaluation of the weakly singular integrals can be performed either a) numerically, b) symbolically, i.e., by explicit expressions, or c) in a combined manner. The explicit integration is of particular interest, when the integrals contain the singularity or if the singularity is rather close to the integration domain. In this paper we describe the explicit expressions for the sixfold volume integrals arising for the Newton potential, i.e., for a 1/r integrand. The volume elements are axi-parallel bricks. The sixfold integrals are typical for the Galerkin method. However, the threefold integral arising from collocation methods can be derived in the same way. Received April 18, 2001; revised September 17, 2001 Published online April 25, 2002     

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.     

May the singular integrals in BEM be replaced by zero?

When approximating the singular integrals arising in the boundary element method by quadrature techniques, it is important to keep the quadrature error consistent with the discretization error in order to reach the optimal order of convergence. In classical approaches, this means that the order of the quadrature grows logarithmically in the number of degrees of freedom. We present a quadrature scheme based on alternative representations of the singular integrands that allows us to use a constant quadrature order without giving up consistency.     

Computational Aspects Of Numerical Integration Based On Optimal Nodal Splines

The numerical construction of nodal spline integration rules is considered for the evaluation both of weakly singular integrals and of certain integrals defined in the Hadamard finite part sense. A bound for the quadrature sum condition number is also given.     

An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals

In this paper, a robust method is presented for numerical evaluation of weakly, strongly, hyper- and super-singular boundary integrals, which exist in the Cauchy principal value sense in two- and three-dimensional problems. In this method, the singularities involved in integration kernels are analytically removed by expressing the non-singular parts of the integration kernels as power series in the local distance ρ of the intrinsic coordinate system. For three-dimensional boundary integrals, the radial integration method [1] is applied to transform the surface integral into a line integral over the contour of the surface and to remove various orders of singularities within the radial integrals. Some examples are provided to verify the correctness and robustness of the presented method.     

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.     

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.     

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.     

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.     

Adaptive quadratures over volumes

We consider the numerical approximation of volume integrals over bounded domainsD:={D∈R ³:H(x>≤0}, whereH:R ³→R is a suitable decidability function. The integrands may be smooth maps or singular maps such as those arising in the volume potentials for boundary integral methods. An adaptive extrapolation method is described which is based on some simple quadrature rules. It utilizes an automatic simplicial subdivision of the domain. The method offers improvements over recently given approaches. A special version is offered for the important application of the numerical computation of volume potentials in boundary integral methods. Several examples illustrate the performance of the method.     

An adaptive Huber method for weakly singular second kind Volterra integral equations with non-linear dependencies between unknowns and their integrals

Numerical methods for weakly singular Volterra integral equations with non-linear dependencies between unknowns and their integrals, are almost non-existent in the literature. In the present work an adaptive Huber method for such equations is proposed, by extending the method previously formulated for the first kind Abel equations. The method is tested on example integral equations involving integrals with kernels K(t, τ) = (t ? τ)^?1/2, K(t, τ) = exp[?λ(t ? τ)](t ? τ)^?1/2 (where λ > 0), and K(t, τ) = 1. By controlling estimated local discretisation errors, the integral equation can be solved adaptively on a discrete grid of nodes in the independent variable domain, in a step-by-step fashion. The practical accuracy order is close to 2. The accuracy can be varied by varying the prescribed local error tolerance parameter tol, although the actual errors tend to be larger than tol. Approximations to off-nodal solution values can also be computed, with a comparable accuracy. The method appears numerically stable when partial derivatives, of the non-linear function representing the equation, with respect to the unknown and its integral(s), are of the same sign. The stability of the method in the opposite case may be debated.     

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 functional integrals to stochastic equations

Representing a probability density function (PDF) and other quantities describing a solution of stochastic differential equations by a functional integral is considered in this paper. Methods for the approximate evaluation of the arising functional integrals are presented. Onsager–Machlup functionals are used to represent PDF by a functional integral. Using these functionals the expression for PDF on a small time interval Δt can be written. This expression is true up to terms having an order higher than one relative to Δt. A method for the approximate evaluation of the arising functional integrals is considered. This method is based on expanding the action along the classical path. As an example the application of the proposed method to evaluate some quantities to solve the equation for the Cox–Ingersol–Ross type model is considered.     

On numerical cubatures of nearly singular surface integrals arising in BEM collocation

In this paper we present efficient methods to approximate nearly singular surface integrals arising massively when discretizing boundary integral equations via the collocation method. The idea is to introduce local polar coordinates centred at a corner of the triangle. Thus it is possible to perform the inner integration analytically, where either the corresponding formulae can be evaluated numerically stable or can be replaced by simple (rational) approximation quite efficiently. We show that the outer integration can be performed by simple Gauß-Legendre quadrature and how to adapt the order of the Gauß formulae to a required order of consistency. Numerical tests will emphasize the efficiency of our method.     

Hierarchical Quadrature for Singular Integrals

We introduce a method for the computation of singular integrals arising in the discretization of integral equations. The basic method is based on the concept of admissible subdomains, known, e.g., from panel clustering techniques and -matrices: We split the domain of integration into a hierarchy of subdomains and perform standard quadrature on those subdomains that are amenable to it. By using additional properties of the integrand, we can significantly reduce the algorithmic complexity of our approach. The method works also well for hypersingular integrals.     

Numerical solution of the singular Ginzburg-Landau equations by multiple shooting

A useful method for the numerical solution of singular two-point boundary value problems by multiple shooting techniques is treated. In a small interval Taylor series expansions are used in orderto kill the numerical singularity. The reliability of the method is demonstrated by solving the Ginzburg-Landau equations arising in the theory of superconductivity.     

Application of global optimization and radial basis functions to numerical solutions of weakly singular volterra integral equations

A novel approach to the numerical solution of weakly singular Volterra integral equations is presented using the C^∞ multiquadric (MQ) radial basis function (RBF) expansion rather than the more traditional finite difference, finite element, or polynomial spline schemes. To avoid the collocation procedure that is usually ill-conditioned, we used a global minimization procedure combined with the method of successive approximations that utilized a small, finite set of MQ basis functions. Accurate solutions of weakly singular Volterra integral equations are obtained with the minimal number of MQ basis functions. The expansion and optimization procedure was terminated whenever the global errors were less than 5 · 10⁻⁷.     

Singularity crossing phenomena in DAEs: A two-phase fluid flow application case study

This paper analyzes the existence of smooth trajectories through singular points of differential algebraic equations, or DAEs, arising from traveling wave solutions of a degenerate convection-diffusion model. The DAE system can be written in the quasilinear form A(x)x′ = b(x). In this setting, singularities are displayed when the matrix A(x) undergoes a rank change. The singular hypersurface may be smoothly crossed by trajectories in a finite time if x* is a geometric singularity satisfying certain directional conditions. The basis of our analysis is a two-phase fluid flow model in one spatial dimension with dissipative mechanism involved.     

Numerical analysis of singular weighted integrals

In this article we investigate the numerical aspects of integrals of the form (1) $$\int_a^b {f(x)\psi (x)dx} $$ wheref is an unobjectionable function and ψ is singular, i.e. ψ is oscillating with high frequency, is discontinuous or unbounded. Suitable integration algorithms are presented.     

Boundary element methods for transient convective diffusion. Part II: 2D implementation

A boundary element method for transient convective diffusion phenomena presented in Part I of the paper is extended to two dimensional problems. We introduce a series representation for the transient convective kernel and perform a time integration for the double integrals to evaluate coefficients of the time-discrete boundary integral equation. The time-integrated kernels are evaluated for the linear, quadratic and quartic time interpolation functions utilized in the paper. Then, linear, quadratic and quartic boundary elements as well as bi-linear, bi-quadratic and bi-quartic volume cells are introduced to ensure proper resolution in space for the two-dimensional formulation. Due to the singular nature of the transient convective diffusion kernels, integration of the kernels over the boundary elements and volume cells requires a considerable effort to maintain a desired level of accuracy. We define influence domains due to time-integrated and time-delayed kernels arising for the surface and volume integrals, respectively. Note that the kernel influences are extremely localized due to the convective nature of the kernels, thus, the surface and volume integrations are performed only within these domains of influence. The localization of the kernels becomes more prominent as the Peclet number of the flow increases. Due to increasing sparsity of the global matrix, iterative solvers become the primary choice for the convective diffusion problems.