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

accurate and efficient numerical integration of weakly singular integrals in galerkin efie solutions a galerkin descretization of the electric field integral equation for perfectly conducting surfaces using rao wilton glisson 1982 basis functions requires the numerical evaluation of integrals with singular kernels over triangular regions these singularities have been traditionally handled by utilizing a singularity extraction procedure to produce a regular integral and an analytic function to replace the original singular integral a new approach is presented here in which the four dimensional 4 d weakly singular integrals unique to the galerkin rao wilton glisson electric field integral equation solution for perfectly conducting surfaces are transformed into integrals with regular integrands the transformations allow some of the integrations to be performed analytically in some cases reducing the original 4 d integral into a 1 d numerical integration the accuracy and convergence properties of the new method are demonstrated by evaluating the scalar potential function over a unit triangle

on the calculation of potential integrals for linear source distributions on triangular domains present analytical formulas for the singular field contributions of linear source distributions on triangular domains the formulas can be used for near field calculations of electric and magnetic source currents especially for the calculation of method of moments mom matrix elements due to the solution of surface integral equations for mom self coupling terms involving the scalar green function of free space the authors present expressions comprising analytical solutions of both the source integrals and the testing integrals

direct evaluation of hypersingular galerkin surface integrals a direct algorithm for evaluating hypersingular integrals arising in a three dimensional galerkin boundary integral analysis is presented the singular integrals are defined as limits to the boundary and by integrating two of the four dimensions analytically the coincident integral is shown to be divergent however the divergent terms can be explicitly calculated and shown to cancel with corresponding singularities in the adjacent edge integrals a single analytic integration is employed for the edge and vertex singular integrals this is sufficient to display the divergent term in the edge adjacent integral and to show that the vertex integral is finite by explicitly identifying the divergent quantities we can compute the hypersingular integral without recourse to stokes s theorem or the hadamard finite part the algorithms are developed in the context of a linear element approximation for the laplace equation but are expected to be generally applicable as an example the algorithms are applied to solve a thermal problem in an exponentially graded material

quadrature formulas obtained by variable transformation and the de rule abstract this paper gives a survey of the results known to date about quadrature formulas obtained by variable transformation followed by an application of the trapezoidal rule with an equal mesh size it has been shown that a formula obtained by an appropriate transformation is in general efficient and also robust against the end point singularity various kinds of useful transformations together with the asymptotic error behaviors of the resulting quadrature formulas are summarized in particular special emphasis is placed on an asymptotically optimal formula called the double exponential formula abbreviated as the de rule which is characterized by the double exponential decrease of its weights in the neighborhood of the end points of the transformed interval of integration

h_ infty unbiased filtering for linear descriptor systems via lmi this note concerns the optimal h infin unbiased filtering problem for linear descriptor systems it unifies the design of reduced order minimal order and full order filters for continuous and discrete time systems necessary and sufficient conditions for the solvability of the problem are obtained in terms of a set of linear matrix inequalities lmis the parametrization of all h infin unbiased filters is presented application to standard systems with or without unknown inputs is given a numerical example is given to show the applicability of the presented approach

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

fast and accurate mom analysis of periodic arrays of multilayered stacked rectangular patches with application to the design of reflectarray antennas the scattering of plane waves by periodic arrays of stacked rectangular patches in multilayered substrates is a problem that has to be solved many times when designing reflectarray antennas made of those patches under the local periodicity assumption the solution to the periodic multilayered problem has been traditionally carried out by means of the galerkin s version of the method of moments mom in the spectral domain this approach involves the computation of double infinite summations and whereas some of these summations converge very fast some other converge very slowly in this paper the slowly convergent summations are computed by making use of an enhanced mixed potential integral equation mpie formulation of the mom in the spatial domain this enhanced formulation is based on the interpolation of the multilayered periodic green s functions and on the efficient computation of the four dimensional 4 d integrals leading to the mom matrix entries both the novel hybrid spectral spatial mom code and the standard spectral domain mom code have been used for the design of a contoured beam reflectarray antenna it has been verified that the spectral spatial mom code requires cpu times that are typically 30 times smaller than those required by the pure spectral domain mom code

on the direct evaluation of weakly singular integrals in galerkin mixed potential integral equation formulations weakly singular integrals over coincident triangles arising in the galerkin discretization of mixed potential integral equation formulations are calculated using a direct evaluation method the proposed method utilizes a series of coordinate transformations together with a re ordering of the integrations in order to reduce the dimensionality of the original four dimensional 4d weakly singular integrals into 1d numerical integrations of smooth functions the final formulas can be easily evaluated with a standard gaussian quadrature rule resulting in a scheme with great accuracy and efficiency properties numerical results for the comparison of the proposed method with both singularity subtraction and singularity cancellation methods often used for the evaluation of multidimensional singular integrals are presented indicating the superior overall performance of the direct evaluation scheme

convergence and gibbs phenomenon in cubic spline interpolation of discontinuous functions convergence of cubic spline interpolation for discontinuous functions are investigated it is shown that the complete cubic spline interpolation of the heaviside step function converges in the lp norm at rate o h1p for quasi uniform meshes when 1 p and diverges in the l norm when the uniform meshes are used no matter how small the uniform mesh size is the complete cubic spline interpolation always oscillates near the discontinuity although this oscillation decays exponentially away from the discontinuous point the maximum overshoot is not decreasing especially we obtain the asymptotic maximum overshoot when the uniform mesh size goes to zero the knowledge on the heaviside function is utilized to discuss convergence properties of cubic spline interpolation for functions with isolated discontinuous points

singularity subtraction technique for high order polynomial vector basis functions on planar triangles in this paper a singularity subtraction technique is developed for computing the impedance matrix elements of various electromagnetic surface integral equation formulations with the galerkin method and high order basis functions analytical closed form formulas for computing surface integrals with r r sup n n spl ges 3 singularities times polynomial nodal shape functions of arbitrary order on a planar triangle are presented

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

evaluation of weakly singular integrals via generalized cartesian product rules based on the double exponential formula various weakly singular integrals over triangular and quadrangular domains arising in the mixed potential integral equation formulations are computed with the help of novel generalized cartesian product rules the proposed integration schemes utilize the so called double exponential quadrature rule originally developed for the integration of functions with singularities at the endpoints of the associated integration interval the final formulas can easily be incorporated in the context of singularity subtraction singularity cancellation and fully numerical methods often used for the evaluation of multidimensional singular integrals the performed numerical experiments clearly reveal the superior overall performance of the proposed method over the existing numerical integration methods

calculation of mom interaction integrals in highly conductive media the construction of the impedance matrix in the method of moments requires the calculation of interaction integrals between the expansion functions through the green s function and its derivatives the singular behavior of the green s function poses considerable problems for an accurate numerical evaluation of these integrals requiring techniques such as singularity extraction or cancellation in this contribution we will show why these methods fail when the medium is highly conductive a novel technique is proposed to handle these highly challenging integrals the complexity of the new method is independent of the conductivity

potential integrals for uniform and linear source distributions on polygonal and polyhedral domains formulas for the potentials due to uniform and linearly varying source distributions defined on simply shaped domains are systematically developed and presented domains considered are infinite planar strips infinite cylinders of polygonal cross sections planar surfaces with polygonal boundaries and volumetric regions with polyhedral boundaries the expressions obtained are compact in form and their application in the numerical solution of electromagnetics problems by the method of moments is illustrated

the double exponential transformation in numerical analysis the double exponential transformation was first proposed by takahasi and mori in 1974 for the efficient evaluation of integrals of an analytic function with end point singularity afterwards this transformation was improved for the evaluation of oscillatory functions like fourier integrals recently it turned out that the double exponential transformation is useful not only for numerical integration but also for various kinds of sinc numerical methods the purpose of the present paper is to review the double exponential transformation in numerical integration and in a variety of sinc numerical methods

hypersingular kernel integration in 3d galerkin boundary element method we consider cauchy singular and hypersingular boundary integral equations associated with 3d potential problems defined on polygonal domains whose solutions are approximated with a galerkin boundary element method related to a given triangulation of the boundary in particular for constant and linear shape functions the most frequently used basis functions we give explicit results of the analytical inner integrations and suggest suitable quadrature schemes to evaluate the outer integrals required to form the galerkin matrix elements these numerical indications are given after an analysis of the singularities arising in the whole integration process which is valid also for shape functions of higher degrees

numerical integration schemes for the bem solution of hypersingular integral equations in this paper we consider singular and hypersingular integral equations associated with 2d boundary value problems defined on domains whose boundaries have piecewise smooth parametric representations in particular given any polynomial local basis we show how to compute efficiently all integrals required by the galerkin method the proposed numerical schemes require the user to specify only the local polynomial degrees therefore they are quite suitable for the construction of p and h p versions of galerkin bem copyright 1999 john wiley sons ltd

on the numerical integration of the linear shape functions times the 3 d green s function or its gradient on a plane triangle formulas for the static and dynamic potentials due to linearly varying source distributions defined on a planar triangle are presented the general linear variation of the source is represented in terms of three linear basis functions each one associated with a different vertex of the triangular domain the singular kernels considered are given by the 3 d green s function and its gradient both for the static and dynamic case in the static case the evaluation of potentials is performed analytically and compact form results are given in the dynamic case the results are given in a form suited for numerical integration

a comparison of the error function and the tanh transformation as progressive rules for double and triple singular integrals abstract transformations of the form x tanh g t for g t n and for g t c sinh t as well as transformations of the error function type are employed on double and triple singular integrals extensions of the one dimensional approach allow variable limit multiple integrals to be attempted successfully some shortcomings of the double exponential or de rule are exposed in the comparisons finally comparison is made with the method of good lattice points sugihara 1987

adaptive singularity cancellation for efficient treatment of near singular and near hypersingular integrals in surface integral equation formulations a proposed singularity cancellation technique for fully numerical evaluation of method of moments integrals in surface integral equation solutions produces reasonably accurate results with few quadrature points for singular and hypersingular integrals however for near singular and near hypersingular integrals time consuming computations need to be repeatedly performed over unnecessary regions outside the actual integration domain for a more efficient treatment of these integrals an adaptive singularity cancellation technique is proposed as such the source triangular domain is subdivided in a way that all sample points remain inside the desired integration domain and unnecessary computations are avoided second the accuracy of results in existing singularity cancellation transformations is greatly affected by variations in height of observation point above the plane of source domain this drawback has been removed in the adaptive singularity cancellation transformations additionally an optimum selection criterion for the distribution of quadrature samples is presented the criterion enables run time selection of optimum number of samples in different directions by consideration of the instantaneous geometry of the transformed integration domain

machine precision evaluation of singular and nearly singular potential integrals by use of gauss quadrature formulas for rational functions a new technique for machine precision evaluation of singular and nearly singular potential integrals with 1 r singularities is presented the numerical quadrature scheme is based on a new rational expression for the integrands obtained by a cancellation procedure in particular by using library routines for gauss quadrature of rational functions readily available in the literature this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions the rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed and several numerical examples are provided