Singular vector expansion functions for finite methods

This paper describes the fundamental properties of new singular vector bases that incorporate the edge conditions in curved triangular elements. The bases are fully compatible with the interpolatory or hierarchical high-order regular vector bases used in adjacent elements. Several numerical results confirm the faster convergence of these bases on wedge problems and the capability to model regular fields when the singularity is not excite

Evaluation of hierarchical vector basis functions for quadrilateral cells

New hierarchical vector basis functions for quadrilateral cells are introduced, and the matrix condition numbers associated with their use are compared to those of existing vector basis families to assess the relative linear independence of the functions. Scale factors are employed to improve the condition numbers. In addition, the proper use of subsets of these families to transition from one order to another (as needed for adaptive -refinement) without exciting spurious modes is considere

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 provide

Singular Higher-Order Complete Vector Bases for Finite Methods

This paper presents new singular curl- and divergence- conforming vector bases that incorporate the edge conditions. Singular bases complete to arbitrarily high order are described in a unified and consistent manner for curved triangular and quadrilateral elements. The higher order basis functions are obtained as the product of lowest order functions and Silvester-Lagrange interpolatory polynomials with specially arranged arrays of interpolation points. The completeness properties are discussed and these bases are proved to be fully compatible with the standard, high-order regular vector bases used in adjacent elements. The curl (divergence) conforming singular bases guarantee tangential (normal) continuity along the edges of the elements allowing for the discontinuity of normal (tangential) components, adequate modeling of the curl (divergence), and removal of spurious modes (solutions). These singular high-order bases should provide more accurate and efficient numerical solutions of both surface integral and differential problems. Sample numerical results confirm the faster convergence of these bases on wedge problems