Generators of H₁(Γₕ,) for Triangulated Surfaces: Construction and Classification

We consider a bounded Lipschitz-polyhedron Ω³ of general topology equipped with a tetrahedral triangulation that induces a mesh Γₕ of the surface ∂Ω. We seek a maximal set of surface edge cycles that are independent in H₁(Γₕ,Z) and bounding with respect to the exterior of Ω. We present an algorithm for constructing suitable 1-cycles in Γₕ: First, representatives of a basis of the homology group H₁(Γₕ,Z) are constructed, merely using the combinatorial description of the surface mesh Γₕ. Then, a duality pairing based on linking numbers is used to determine those combinations that are bounding with respect to R³Ω. This is the key to circumventing a triangulation of the exterior region R³Ω in the computations. For shape-regular, quasi-uniform families of meshes, the asymptotic complexity of the algorithm is shown to be O(N 2 ), where N is the number of edges of Γₕ. The scheme provides an essential preprocessing step for all boundary element methods for eddy current simulation, which rely on discrete divergence-free vectorfields and their description through stream functions.