# Calculation of currents around a triangular indentation by the hodograph method

## TL;DR

This paper develops a hodograph series method to analyze current distributions and discontinuity lines around a triangular indentation in non-linear conductors, revealing how different creep regimes influence current behavior and d-line shapes.

## Contribution

It introduces a novel hodograph series approach for triangular indentations, extending understanding of current disturbances in non-linear conductors with different creep exponents.

## Key findings

- Currents near the indentation are significantly disturbed only in the mixed creep exponent case.
- For uniform creep exponent, the d-line shapes resemble those of planar indentations, with asymptotic behaviors matching critical state models.
- The shape of d-lines depends on the creep exponent, showing parabolic-like features.

## Abstract

Border indentations in non-linear conductors, such as superconducting thin films in the creep regime, alter the distribution of currents and magnetic fields near and far from the indentation. One of such disturbances are the discontinuity lines, or \textit{d}-lines, a parabolic-like line originating from the indentation where the current density direction changes abruptly. Hodograph series results are obtained for the currents around a triangular indentation and its corresponding $d$-lines in a conducting stripe of finite width and in an infinite half plane, considering two cases: uniform creep exponent and mixed infinite and ohmic exponents. The mixed creep exponent case presents currents distributions resembling the purely ohmic case, with significant current disturbances only near the indentation. For uniform creep exponent, results similar to a planar indentation are obtained, with far ranged currents features and parabolic-like $d$-lines with shapes depending on the creep exponent. In particular, the same $d$-line asymptotic behaviour is obtained for the triangle indentation as that of the planar defect in the critical state, a result obtained here just on continuity considerations of the hodograph expansions. This equivalence is due to identical contributions to the Fourier series of the current stream-function in the hodograph space, obtained from an images method expansion.