Evaluation and Integration of the Thin Wire Kernel

Abstract

New approaches for numerically computing the thin wire kernel and wire potential integrals are presented. The singular behavior of the kernel integral is removed by transforming the integration variable to produce a smooth integrand. Subsequent integration of the kernel to obtain potential integrals uses quadrature schemes catering to its behavior. This technique allows standard algorithms for numerical quadrature to be used with updated integration weights that account for the transformed behavior, obviating the need for singularity subtraction techniques. The result is a procedure for evaluating the potential integrals that is independent of the basis functions.

Evaluating the gradient of the thin wire kernel

“…Results are compared with previous formulations to demonstrate the accuracy of the approach. This formulation is readily applied to integrations over the wire using approaches similar to those in [1].…”

“…Recently, a formulation for evaluating the thin wire kernel was developed that employed a change of variable to smooth the kernel integrand, canceling the singularity in the integrand [1]. Hence, the typical expansion of the wire kernel in a series for use in the potential integrals is avoided.…”

The Analysis of Thin Wires using Higher-Order Elements and Basis Functions

“…The right-hand side of (4) now may be merely interpreted as an integration over a linear segment of length 0 J with a modified integrand. This integral form is similar to that used in [4] for singularity analysis on linear wire segments. To see this, consider a Taylor series expansion of ′ r about 0 r , …”

Transient behavior of wind towers grounding systems under lightning strikes

“…Following the procedure presented in [37,38], since the sources and the potentials about a linear tubular section present a rotational symmetry, assuming, without loss of generality, sources distributed uniformly on a cylindrical tube of constant radius r 0 centered along the z-axis and an observation point in cylindrical coordinates r 0 ; 0; z ð Þ, the wire kernel K r; r 0 ð Þ is defined as …”