Resolving the sign conflict problem for hp-hexahedral Nédélec elements with application to eddy current problems

- A procedure for addressing the sign conflict issue for conforming hexahedral meshes.
- An efficient implementation of hierarchic hp-Nedelec elements in deal.II.
- Solution of a series of challenging 3D eddy current benchmark problems.
- Description of the extension to non-conforming hexahedral meshes.

The eddy current approximation of Maxwell's equations is relevant for Magnetic Induction Tomography (MIT), which is a practical system for the detection of conducting inclusions from measurements of mutual inductance with both industrial and clinical applications. An MIT system produces a conductivity image from the measured fields by solving an inverse problem computationally. This is typically an iterative process, which requires the forward solution of a Maxwell's equations for the electromagnetic fields in and around conducting bodies at each iteration. As the (conductivity) images are typically described by voxels, a hexahedral finite element grid is preferable for the forward solver. Low order Nédélec (edge element) discretisations are generally applied, but these require dense meshes to ensure that skin effects are properly captured. On the other hand, hp-Nédélec finite elements can ensure the skin effects in conducting components are accurately captured, without the need for dense meshes and, therefore, offer possible advantages for MIT. Unfortunately, the hierarchic nature of hp-Nédélec basis functions introduces edge and face parameterisations leading to sign conflict issues when enforcing tangential continuity between elements. This work describes a procedure for addressing this issue on general conforming hexahedral meshes and an implementation of a hierarchic hp-Nédélec finite element basis within the deal.II finite element library. The resulting software is used to simulate Maxwell forward problems, including those set on multiply connected domains, to demonstrate its potential as an MIT forward solver.