Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems

Abstract: The eddy current problem has many relevant practical applications in science, ranging from non-destructive testing to magnetic confinement of plasma in fusion reactors. It arises when electrical conductors are immersed in an external time-varying magnetic field operating at frequencies for which electromagnetic wave propagation effects can be neglected. Popular formulations of the eddy current problem either use the magnetic vector potential or the magnetic scalar potential. They have individual advantages and disadvantages. One challenge is related to differential geometry: Scalar potential based formulations run into trouble when conductors are present in non-trivial topology, as approximation spaces must be then augmented with generators of the first cohomology group of the non-conducting domain. For all existing algorithms based on lowest order methods it is assumed that the extension of the graph-based algorithms to high-order approximations requires hierarchical bases for the curl-conforming discrete spaces. However, building on insight on de Rham complexes approximation with splines, we will show in the present submission that the hierarchical basis condition is not necessary. Algorithms based on spanning tree techniques can instead be adapted to work on an underlying hexahedral mesh arising from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.