Evaluation Of Near-singular Integrals Research Articles

Evaluation of near-singular integrals with application to vortex sheet flow

Evaluation of near-singular integrals with application to vortex sheet flow

High order isoparametric elements in boundary element method—Smooth spheroidal element

The smooth spheroidal element, a high order isoparametric element, is constructed and applied in the boundary element method (BEM) for modeling closed surface of spheroid as well as the field variables using a single element. The smoothness of the element is realized by repeated use of real nodes as auxiliary nodes of interpolation along both the latitude and meridian directions to remove the end node/line effects existing in the conventional closure element. The special techniques of shape function manipulations are suggested and presented to deal with the integrals with hyper- or near hyper-singularities in a uniform and simple way encountered in the BEM. A combined technique, the distance transformation with possible subdivision is proposed to improve the evaluation of near singular integrals. Through a number of numerical examples, the accuracy and efficiency of using high order smooth elements are demonstrated by checking the fitting accuracy of geometrical parameters, the computation costs and some bench mark tests in elasticity.

High order isoparametric elements in boundary element method—Smooth elliptical element

The smooth elliptic element, a high order isoparametric element, is constructed and applied in the boundary element method for modeling closed geometries of ellipses as well as the field variables using a single element. The smoothness of the element is realized by repeated use of real nodes as auxiliary nodes along the circumferential direction of ellipse to remove the end node effect. By writing Lagrange polynomials into the summation of monomials in descending order then into the nested products, the coefficients of shape functions can be generated automatically to get rid of manual work encountered in the construction of a variety of high order elements with different node numbers and distributions. Special techniques of shape function manipulations are suggested and presented in dealing with the integrals with hyper- or near hyper-singularities in a uniform and simple way. A combined technique, the distance transformation with possible element subdivision is proposed to improve the evaluation of near singular integrals. The accuracy and efficiency of using high order elements are demonstrated through a number of numerical examples, by checking the fitting accuracy of geometrical parameters, the computation costs and some simple bench mark tests in elasticity.

A New Concept in Near-Singular Integral Evaluation: The Continuation Approach

The functional homogeneity of Green functions is exploited in the derivation of continuation formulae for the accurate evaluation of near singular integrals. These formulae provide a means of systematically continuing such unconventional integrals as the principal value and finite part integrals of singular functions to conventional integrals of nonsingular functions. A new concept, the notion of continuum integral, which is applicable to singular as well as to nonsingular integrals, is introduced as a generalization of principal value and finite part integrals. Numerical examples are provided that illustrate the computational advantages of the continuation method and give a new perspective on the subtleties of singular and near-singular integrals.