Duffy Transformation Research Articles

A Family of Augmented Duffy Transformations for Near-Singularity Cancellation Quadrature

A new family of systematically constructed near-singularity cancellation transformations is presented, yielding quadrature rules for integrating near-singular kernels over triangular surfaces. This family results from a structured augmentation of the well-known Duffy transformation. The benefits of near-singularity cancellation quadrature are that no analytical integral evaluations are required and applicability in higher-order basis function and curvilinear settings. Six specific transformations are constructed for near-singularities of orders one, two and three. Two of these transformations are found to be equivalent to existing ones. The performance of the new schemes is thoroughly assessed and compared with that of existing schemes. Results for the gradient of the scalar Green function are also presented. For simplicity, static kernel results are shown. The new schemes are competitive with and in some cases superior to the existing schemes considered.

Computation of free space capacitance of cylinder and truncated cone for spacecraft circuit modelling

Absolute free space capacitance of cylinder and truncated cone is computed for spacecraft equivalent circuit modelling. The method of moment with triangular patch modelling based on the pulse basis function is employed. The curved triangular surfaces are represented by parametric geometric transformation. The moment integral is evaluated numerically in a parameter space using Gaussian quadrature algorithm and Duffy transformation method. The results are good matching with the previously published results. The derived expressions are simple and also fit for the spacecraft charging analysis software. Since this approach is more general, any curved surfaces that are encountered during spacecraft modelling, free space capacitance of that surface can be easily evaluated and hence, spacecraft potential can be predicted accurately.

Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary-domain integral and integro-differential equation methods

This paper presents a formulation of the boundary-domain integral equation (BDIE) and the boundary-domain integro-differential equation (BDIDE) methods for the numerical solution of two-dimensional mixed boundary-value problems (BVP) for a second-order linear elliptic partial differential equation (PDE) with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the BVP to a BDIE or BDIDE. The numerical formulation of the BDIDE employs an approximation for the boundary fluxes in terms of the potential function within the domain cells; therefore, the solution is fully described in terms of the variation of the potential function along the boundary and domain. Linear basis functions localised on triangular elements and standard quadrature rules are used for the calculation of boundary integrals. For the domain integrals, we have implemented Gaussian quadrature rules for two dimensions with Duffy transformation, by mapping the triangles into squares and eliminating the weak singularity in the process. Numerical examples are presented for several simple problems with square and circular domains, for which exact solutions are available. It is shown that the present method produces accurate results even with coarse meshes. The numerical results also show that high rates of convergence are obtained with mesh refinement.

Full numerical quadrature of weakly singular double surface integrals in Galerkin boundary element methods

When a Galerkin discretization of a boundary integral equation with a weakly singular kernel is performed over triangles, a double surface integral must be evaluated for each pair of them. If these pairs are not contiguous or not coincident, the kernel is regular and a Gauss–Legendre quadrature can be employed. When the pairs have a common edge or a common vertex, then edge and vertex weak singularities appear. If the pairs have both facets coincident, the whole integration domain is weakly singular. Taylor (IEEE Trans. Antenn. Propag. 2003; 51(7):1630–1637) proposed a systematic evaluation based on a reordering and partitioning of the integration domain, together with a use of the Duffy transformations in order to remove the singularities, in such a way that a Gauss–Legendre quadrature was performed on three coordinates with an analytic integration in the fourth coordinate. Since this scheme is a bit restrictive because it was designed for electromagnetic kernels, a full numerical quadrature is proposed in order to handle kernels with a weak singularity with a general framework. Numerical tests based on modifications of the one proposed by Wang and Atalla (Commun. Numer. Meth. Engng 1997; 13(11):885–890) are included. Copyright © 2009 John Wiley & Sons, Ltd.

Semi-analytic treatment of nearly-singular Galerkin surface integrals

On utilizing piecewise linear test and interpolation functions over flat triangles, three semi-analytic techniques for evaluating nearly-singular surface integrals of potential theory are developed and investigated in the context of a three-dimensional Galerkin approximation. In all procedures, the inner surface integral is calculated exactly via recursive expressions defined over the edges of the integration triangle. In addition, the proposed methods respectively employ a Duffy transformation, sinh transformations, and polynomial transformations to weaken or smooth out near singularities, followed by a standard product Gauss–Legendre quadrature rule to compute the outer boundary integral. Numerical experiments and comparisons of the proposed approaches are included to provide more insight into the performance of these techniques. Computational tests have demonstrated that the algorithm based on a Duffy transformation is the most efficient and effective method for dealing with nearly-singular as well as well-separated integrals.

Generalized Duffy transformation for integrating vertex singularities

For an integrand with a 1/r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evaluate the integral. In this paper, we generalize the Duffy transformation to power singularities of the form p(x)/r α , where p is a trivariate polynomial and α > 0 is the strength of the singularity. We use the map (u, v, w) → (x, y, z) : x = u β , y = x v, z = x w, and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation (β = 1) is optimal, whereas if α ≠ 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transformation. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation.

Numerical evaluation of singular and near-singular potential Integrals

A simple and efficient numerical procedure using a singularity cancellation scheme is presented for evaluating singular and near-singular potential integrals with 1/R singularities. The procedure not only has several advantages over singularity subtraction methods, but also improves on some aspects of other singularity cancellation methods such as polar and Duffy transformations. A theoretical analysis is presented for triangles, quadrilaterals, tetrahedrons, bricks, and prisms, and numerical results are presented for triangles and prisms.

Computational cost estimations and comparisons for three methods of applied electromagnetics (MoM, MAS, MMAS)

The computational costs of three numerical techniques used in electromagnetics, namely the moment method (MoM), the method of auxiliary sources (MAS), and its modified version (MMAS), are estimated for various calculation schemes and configurations. Both surface and volumetric problems are considered. The number of multiplications required for the system-matrix fill is calculated and added to the algorithmic cost of the matrix inversion. The Green's function singularity extraction is also taken into account, particularly for the MoM. The original integrals are transformed into the local (area or volume) coordinate systems, and are subsequently evaluated on the basis of standard numerical quadrature schemes. For the surface integral equation (SIE), some calculations using either the well-known Duffy transformations or some analytical-numerical integration schemes are also presented (expressions are available only for the scalar potential integral case). For the MAS and MMAS, the matrix fill is shown to be much faster, since no time-consuming integrations are involved. The analysis is applied to various objects, such as a perfectly conducting (PEC) parallelpiped, a PEC sphere, and a microstrip patch antenna, and useful conclusions are drawn on the relative efficiency of the three methods.

A higher-order Nystro/spl uml/m scheme for electromagnetic scattering by arbitrarily shaped surfaces

A higher-order Nystro/spl uml/m scheme is developed for electromagnetic scattering by arbitrary conducting scatterers. In our implementation, we employ a superparametric geometry mapping for arbitrary curvilinear surfaces to minimize the geometry error. The local correction for singular integral kernels is manipulated efficiently with the Lagrange interpolation of the unknown functions followed by singularity extraction and Duffy's transformation. Since this local correction approach removes the local method of moments (MoM) procedure, the scheme is easier to implement and more efficient in controlling errors compared with other higher-order Nystro/spl uml/m schemes. Two numerical examples for scattering by sharp-corner scatterers are conducted to demonstrate the robustness of this method.

Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra

AbstractA numerical solution of integral equations typically requires calculation of integrals with singular kernels. The integration of singular terms can be considered either by purely numerical techniques, e.g. Duffy's method, polar co‐ordinate transformation, or by singularity extraction. In the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically. This method has been well established for linear and constant shape functions. In this paper we extend the method for polynomial shape functions of arbitrary order. We present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the Helmholtz equation, or its gradient, and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra. The presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high‐order basis functions. Numerical experiments show that the developed method leads to a more accurate and robust integration scheme, and in many cases also a faster method than, for example, Duffy's transformation. Copyright © 2003 John Wiley & Sons, Ltd.

Spectrally accurate numerical solution of the single-particle Schrödinger equation

We have formulated a three-dimensional fully numerical (i.e., chemical basis-set free) method and applied it to the solution of the single-particle Schr\odinger equation. The numerical method combines the rapid ``exponential'' convergence rates of spectral methods with the geometric flexibility of finite-element methods and can be viewed as an extension of the spectral element method. Singularities associated with multicenter systems are efficiently integrated by a Duffy transformation and the discrete operator is formulated by a variational statement. The method is applicable to molecular modeling for quantum chemical calculations on polyatomic systems. The complete system is shown to be efficiently inverted by the preconditioned conjugate gradient method and exponential convergence rates in numerical approximations are demonstrated for suitable benchmark problems including the hydrogenlike orbitals of nitrogen.