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A Novel Technique for Cohomology Computations in Engineering Practice Comput. Methods Appl. Mech. Engrg. 253 (2013) 530–542 Contents lists available at SciVerse ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma A novel technique for cohomology computations in engineering practice ⇑ Paweł Dłotko a,1, Ruben Specogna b, a Jagiellonian University, Institute of Computer Science, Lojasiewicza 4, 30348 Kraków, Poland b Università di Udine, Dipartimento di Ingegneria Elettrica, Gestionale e Meccanica, Via delle Scienze 208, 33100 Udine, Italy article info abstract Article history: The problem of computing cohomology generators of a cell complex is gaining more and more interest in Received 14 February 2012 various branches of science ranging from computational physics to biology. Focusing on engineering Received in revised form 13 June 2012 applications, cohomology generators are currently used in computer aided design (CAD) and in potential Accepted 9 August 2012 definition for computational electromagnetics and fluid dynamics. Available online 24 August 2012 The aim of this paper is to introduce a novel technique to effectively compute cohomology generators focusing on the application involving the potential definition for h-oriented eddy-current formulations. Keywords: This technique, which has been called Thinned Current Technique (TCT), is completely automatic, compu- (Co) homology computation tationally efficient and general. The TCT runs in most cases in linear time and exhibits a speed up of (Co) homology generators Definition of potentials orders of magnitude with respect to the best alternative documented implementation. Eddy-currents Ó 2012 Elsevier B.V. All rights reserved. Thick cuts 1. Introduction mathematics where cohomology group has been computed (with cup and cap products [32,5]) is group theory [33,34]. As for the The problem of computing cohomology generators [1–5] of a case of computing cohomology of cell complexes the only ap- cell complex is gaining more and more interest in various branches proach known to the Authors has been by using the concept of of science ranging from computational physics [6–23] to biology graph pyramids in [29]. However, this approach is limited to [24] and quantum chemistry [25]. Focusing on engineering, coho- two-dimensional cubical complexes only. The lack of results con- mology generators are currently used in computer aided design cerning cohomology computations—in contrast with the great (CAD) for feature detection [26,27], shape analysis [28,29], param- number of publications concerning homology computations— etrization and mesh generation [30,31] and in definition of poten- could be due to several reasons. One may be that cohomology the- tials for magnetostatic, eddy-current problems [6–23] and for ory is less intuitive and render finding novel applications more Navier–Stokes equations [16]. In particular, in the last 25 years, difficult. cohomology computation draw sensible attention in the computa- A general algorithm for the computation of a basis of the coho- tional electromagnetics community. In fact, generators of the first mology group over integers is well known since many decades cohomology group—usually called thick cuts [10–23] in this con- ago and it is based on the Smith Normal Form [1–5] computation. text—are expressly needed to solve eddy-current problems with The problem of this algorithm is that its computational complexity the efficient magnetic scalar potential-based T-X formulation is hyper-cubical with the best implementation available [35] and [11,17,18,20,23]. Even though a considerable effort has been in- consequently it cannot be used in practice even on extremely coarse vested by the computational electromagnetics community to de- meshes. The same problem in case of homology computations has velop fast and general algorithms to produce cohomology been solved by using a sparse matrix data structure [36] and reduc- generators, the required computational complexity and memory tions techniques [37]. Sparse matrices are used to effectively store consumption render them still far from being attractive. boundary matrices and reduction techniques are used to reduce In mathematics and computer science communities, computa- the complex before the Smith Normal Form computation is run. A tions of cohomology groups and generators has not been exten- survey of these reduction techniques in the context of computa- sively explored so far. As far as we know, the only field of tional electromagnetics is addressed in the review paper [38]. One can in fact use Smith Normal Form and some of the reductions ⇑ designed for homology computations in order to obtain cohomology Corresponding author. Tel.: +39 0432 558285; fax: +39 0432 558251. groups and their generators. Once the cohomology generators are E-mail addresses: [email protected] (P. Dłotko), [email protected] (R. Specogna). found on the reduced complex via the standard Smith Normal Form 1 P.D. was partially supported by grant IP 2010 046370. algorithm, they are restored into the original complex by the 0045-7825/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cma.2012.08.009 P. Dłotko, R. Specogna / Comput. Methods Appl. Mech. Engrg. 253 (2013) 530–542 531 so-called pull-back operation [37]. Pulling back the generators rep- (TCT) which is easy to implement in a parallel or distributed com- resents, however, a considerable extra computational cost. There- puting environment. fore, a so-called shaving for cohomology is desired, which enables The paper is structured as follows. In Section 2 the need of a first a reduction of the complex without the need of pulling-back the cohomology group basis for electromagnetic potentials definition generators. This is due to the property that the cohomology gener- is recalled. In Section 3, we present a novel algorithm for cohomol- ators of the shaved complex are generators also of the original com- ogy computation called Thinned Current Technique (TCT), which plex. The concept of shaving for cohomology has been introduced in stems from an alternative approach with respect to the ones pre- [20]. In the same paper, the authors introduced an automatic, gen- sented in the literature survey. Section 4 describes the modifica- eral, and efficient algorithm to compute cohomology generators in tions that need to be applied when one wants to take advantage computational electromagnetics. The [20] algorithm uses as shav- of some symmetry in the eddy-current problem. In Section 5, ings the reduction procedures presented in [39,40]. real-life examples are provided to benchmark the efficiency and An attempt of producing thick cuts avoiding (co) homology robustness of the presented method with respect to the ones doc- computations has been reported in [11]. First, the first cohomology umented in literature. Finally, in Section 6, the conclusions are group generators are found on the surface of conductors. However, drawn. only half of them have to be selected, namely the ones that are non-bounding in the conducting region. The question of how to se- 2. (Co) homology in electromagnetic modeling lect them automatically and efficiently is left unaddressed in the paper. Second, it grows an acyclic sub-complex inside the insulat- For the sake of brevity, the relevant concepts from algebraic ing region and, at the end of this process, the complement of the topology as (co) chains, (co) boundaries and (co) homology are acyclic sub-complex is considered as the support of thick cuts. not recalled in this paper. For an informal introduction to the sub- Again, how to construct the acyclic sub-complex is not described ject the reader is invited to consult [13,18,21]. A more formal pre- at all in the paper. The lack of these and other necessary details sentation can be found in any algebraic topology textbook as [5]. does not allow a serious analysis of this algorithm. Let us assume that the cell complex K provided by the input In [12], a different algorithm—which the Authors call General- mesh of the computational domain is homeomorphic to the ized Spanning Tree Technique (GSTT)—has been introduced. This three-dimensional ball, which is a standard assumption in practical algorithm attempts to generate a basis for the first cohomology meaningful problems. Elements of K belonging to the conducting group once a basis for the first homology group is given as input. region are stored in the sub-complex Kc, whereas the elements In [12], homology generators have been constructed ‘‘by hand’’ of K belonging to the insulating region form the sub-complex Ka. and no discussion about the termination of the algorithm has been Moreover, we assume, that K; Ka and Kc are combinatorial mani- addressed. In [18], an efficient algorithm to automatically produce folds (see [42]). the homology generators has been described and in [21] a detailed In this section, we intuitively explain why cohomology theory is analysis of the GSTT has been presented by the Authors. The con- expressly needed in the context of computational electromagnet- clusion is that the GSTT algorithm may present many termination ics. A more comprehensive and detailed treatment on this topic failures that are very difficult to solve in practice. is presented in [23]. To this aim, let us concentrate on the defini- We would like also to point out that an algorithm similar to the tion of potentials in the insulating region Ka for h-oriented eddy- GSTT has been introduced by Kotiuga in the pioneering works [6– current formulations.2 The discrete Ampère’s law in the insulating 8,13]. More precisely, a GSTT-like algorithm is applied on the dual region can be written as complex to obtain the so-called thin cuts, that are generators of the first cohomology group of the insulating domain on the dual com- dF I 0; 1 plex. This technique is not attractive from the computational com- ¼ ¼ ð Þ plexity point of view because a non-physical Poisson problem have to be solved for each generator to avoid cut self-intersections (i.e.
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