# $T$-$\Omega$ formulation with higher order hierarchical basis functions for non simply connected conductors

## TL;DR

This paper enhances the $T$-$
Omega$ formulation for eddy current problems by incorporating higher order hierarchical basis functions and non-local basis functions to handle conductors with complex topologies.

## Contribution

It introduces a novel extension of the $T$-$
Omega$ formulation using non-local basis functions derived from the first de Rham cohomology group, enabling analysis of arbitrary conductor topologies.

## Key findings

- Effective handling of arbitrary topology conductors
- Efficient computation of non-local basis functions with DS algorithm
- Extension of $T$-$
Omega$ formulation for complex geometries

## Abstract

This paper extends the $T$-$\Omega$ formulation for eddy currents based on higher order hierarchical basis functions so that it can deal with conductors of arbitrary topology. To this aim we supplement the classical hierarchical basis functions with non-local basis functions spanning the first de Rham cohomology group of the insulating region. Such non-local basis functions may be efficiently found in negligible time with the recently introduced D{\l}otko--Specogna (DS) algorithm.