Critical Analysis of the Spanning Tree Techniques

Abstract

Abstract.Two algorithms based upon a tree-cotree decomposition, called in this paper spanning tree technique (STT) and generalized spanning tree technique (GSTT), have been shown to be useful in computational electromagnetics. The aim of this paper is to give a rigorous description of the GSTT in terms of homology and cohomology theories, together with an analysis of its termination. In particular, the authors aim to show, by concrete counterexamples, that various problems related with both STT and GSTT algorithms exist. The counterexamples clearly demonstrate that the failure of STT and GSTT is not an exceptional event, but something that routinely occurs in practical applications.Key words. algebraic topology, scalar potential in multiply connected regions, tree-cotree decomposition, belted tree, computational topology, homology theory, cohomology theory, homology and cohomology generators, homology-cohomology duality AMS subject classifications. 65N30, 78M10, 78M25, 55N99, 55M05, 55N33 DOI. 10.1137/0907663341. Introduction. The tree-cotree decomposition arises from graph theory and consists in partitioning the edges of a graph into a spanning tree and its complement, referred to as the cotree. The idea of taking advantage of the tree-cotree decomposition is at the root of electric network theory [1], [2]. It has been used, for example, to generate a maximal set of independent Kirchhoff's equations for the network analysis (see, for example, [3] [10]. The tree-cotree decomposition became popular in computational electromagnetics after [11]. It had been widely used as a gauging technique to set well-posed magnetostatic and magneto-quasi-static boundary value problems (BVP). Nowadays, such gauging techniques have lost their importance, since the ungauged formulations were shown to be more effective, improving the condition number of the matrix for the linear system of equations; see, for example, [12].More recently, two algorithmic techniques based upon the tree-cotree decomposition have been shown to be quite useful in computational electromagnetics. The first one, introduced in [13] (see also [14], [15], [16] and referred to as spanning tree technique (STT) in this paper, is commonly employed in order to compute the socalled generalized source magnetic fields, needed to enforce the source currents when solving magnetostatic and magneto-quasi-static BVP formulated by using a magnetic scalar potential. In this application, the STT is used to compute a 1-cochain when its coboundary 2-cochain is given as input.

Construction of a Finite Element Basis of the First de Rham Cohomology Group and Numerical Solution of 3D Magnetostatic Problems

“…Let us note, however, that in three-dimensions the construction of a belted tree is not straightforward (see Rapetti et al [53], D lotko et al [26], D lotko and Specogna [21]). …”

“…A careful analysis of the termination properties of this algorithm can be found in D lotko and Specogna [21], where it is referred to as GSTT (generalized spanning tree technique) if the initialization set is K = L ∪ E , or STT (spanning tree technique), if the initialization set is K = L. Clearly, if the domain Ω has a simple topological shape (namely, g = 0) the two algorithms coincide.…”

“…We use a spanning tree (similar but different techniques, based on the so-called belted tree, have been proposed by several authors, but they do not work for all topological situations: see Ren and Razek [55], Kettunen et al [38], Bossavit [13], Rapetti et al [53], Henrotte and Hameyer [34], D lotko et al [26]). Another fundamental tool is the direct algorithm of Webb and Forghani [62]: however, since it is known to fail in certain topological situations (see D lotko and Specogna [21]), we modify it in a suitable way, in order to be able to construct the finite element loop fields for every domain Ω. A basic point here is the fact that we provide an explicit formula for expressing the discrete loop fields in terms of linking numbers.…”

“…This problem has been widely considered, mainly for simple topological domains (see, e.g., Webb and Forghani [62], Preis et al [52], Dular et al [28], Le Ménach et al [43], Rapetti et al [53], Dular [27], Badics and Cendes [6], D lotko and Specogna [21]). We show that a discrete source field can be determined by adopting a similar procedure to that employed for the construction of the finite element loop fields: again, the main point is the use of the Webb-Forghani algorithm, followed by the introduction of a graph for the edges whose degree of freedom has not been yet determined when the algorithm stops and by a simple algebraic direct solver (for a similar approach, see also D lotko and Specogna [24]).…”

LDRD 149045 final report distinguishing documents.

“…Homology can model the dynamics of adding simplices, but homological tools for removing them are limited to zigzag complexes and are more difficult. Homology has ties to electromagnetic PDE solutions, in particular the kernel of operators related to the phenomena of orthogonality between electric and magnetic fields and eddy currents [4]. Homology is related to Morse Theory, and the Sandia-CA combustion group (Pebay, Bennett, et al) has an ongoing project with Sandia-NM's visualization department (Shepherd et al) on using a topological tool called Reeb graphs together with statistics to study ignition and extinguishment in combustion simulations.…”