Optimized cycle basis in volume integral formulations for large scale eddy-current problems

We present a Volume Integral formulation for the solution of large scale eddy-current problems coupled with low-rank approximation techniques. Two alternative approaches are introduced to map the problem unknowns into a subset of grid elements forming a base of global or mixed (global and local) cycles, respectively, and guarantee the well-posedness of the problem both in simply and multiply connected domains. The paper shows that the adoption of mixed cycles is computationally more efficient than global ones. In particular, integral formulations based on global cycles cannot be safely coupled with low-rank approximation techniques, which, however, are crucial to increase the size of the largest solvable problem, like the ones involving conducting structures in magnetic confinement fusion devices. The aim of this paper is to demonstrate how such bottleneck can be overcome by considering local and global cycles differently, on the basis of the cohomology theory. An improved, efficient, and robust algorithm for computing a base of global cycles is described in detail. In particular, the presented algorithm is able to almost minimize the cohomology basis length, i.e. the number of mesh edges forming such a basis, in order to allow an efficient solution of large scale problems. Furthermore, a novel and general method to handle global and local cycles together, in the context of low-rank approximated matrices, is shown to be efficient for the solution of large scale eddy-current problems in multiply connected domains. Along the manuscript, pseudo-codes are given, which clarify the proposed methods and help to implement them by Volume Integral Equation practitioners.