A Challenge for Magnetic Scalar Potential Formulations of 3-D Eddy Current Problems: Multiply Connected Cuts in Multiply Connected Regions which Necessarily Leave the Cut Complement Multiply Connected

Finite Element-based Algorithms to Make Cuts for Magnetic Scalar Potentials: Topological Constraints and Computational Complexity

“…Noting that Ampère's law is a statement about closed loops in R which link nonzero current [15], we define the first homology group of R with integer coefficients, denoted by H 1 (R; Z), as the group of equivalence classes of closed loops in R which link closed paths in R c which may be current paths [15]. Two closed loops in R lie in the same equivalence class if together they comprise the boundary of a surface in R (Figure 2).…”

“…The obstruction to ideal complexity is tied up in the structure of the fundamental group of the region. One implementation was designed to have ideal complexity [29], however, the implementation has heuristic assumptions which make it generally inadequate for making cuts in regions for which the fundamental group of R has a nontrivial commutator subgroup [13,15]. This paper details an algorithm that can be implemented with O(m 2 0 ) time complexity and O(m 4/3 0 ) storage complexity given no more topological data than that contained in the finite element connection matrix.…”

“…Such algorithms are limited to problems reducible to a planar problem and their success for this limited class of problems hinges on the fact that the topology of 2-d surfaces is "perfectly understood." In three dimensions, intuition from Ampère's law leads to a formal definition of cuts in terms of a basis of the (abelian) homology groups of the region [20,21,15], even though such cuts do not necessarily render the region simply-connected [15]. The abelian nature of homology groups makes the problem computationally tractable while avoiding the subtle aspects of 3-dimensional topology which are trivial in two dimensions.…”

Geometrical and Topological Methods in Time Domain Antenna Synthesis

“…This isomorphism plays a key role in understanding counterintuitive aspects of making "cuts" for magnetic scalar potentials in three dimensions [40]. Our immediate concern, is to find commutative algebraic gadgets, more complicated than H 1 , which help articulate t' -omplexity of irl.…”

Data Structures for Geometric and Topological Aspects of Finite Element Algorithms

“…The rank of H p (K) is the number of independent equivalence classes in the group and is known as the pth Betti number of K, denoted by β p (K); intuitively, β 0 (K) counts the number of connected components of K, and β 1 (K) counts the "number of holes in K" [20,12,11]. Calling Z p (X; Z) = ker∂ T p the group of p-cocycles, and B p (X; Z) = Im ∂ T p−1 the group of p-coboundaries, the pth cohomology group is:…”