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Enumeration of Architectures with Perfect Matchings Proceedings of the ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC/CIE 2016 August 21-24, 2016, Charlotte, North Carolina DETC2016-60212 ENUMERATION OF ARCHITECTURES WITH PERFECT MATCHINGS Daniel R. Herber, Tinghao Guo, James T. Allison University of Illinois at Urbana-Champaign Industrial & Enterprise Systems Engineering Urbana, IL 61801 Email: {herber1,guo32,jtalliso}@illinois.edu ABSTRACT further evaluation (subjective or quantitative), helping to over- In this article a class of architecture design problems is ex- come design fixation. A popular class of methods for generating plored with perfect matchings. A perfect matching in a graph architecture candidates is generative representations [6–13]. This is a set of edges such that every vertex is present in exactly one class covers a range of candidate architectures in an implicit form edge. The perfect matching approach has many desirable prop- based on repeated application of rules that modify the graph. It erties such as complete design space coverage. Improving on the has been recognized that generative approaches generate topo- pure perfect matching approach, a tree search algorithm is de- logically simple designs, not covering the entire design space veloped that more efficiently covers the same design space. The [14]. Furthermore, the design space is sensitive to design knowl- effect of specific network structure constraints and colored graph edge [4, 9] and rules [10, 11]. While these designs may satisfy isomorphisms on the desired design space is demonstrated. This functional requirements elegantly, generation of more elaborate is accomplished by determining all unique feasible graphs for a architectures is needed in some cases. select number of architecture problems, explicitly demonstrating It can be challenging to describe the design space of an ar- the specific challenges of architecture design. Additional appli- chitecture generation method, partially due to the combinatorial cations of this work to the larger architecture design process is nature of architecture design problems. A better understanding also discussed. of how certain rules restrict the design space can lead to better generative approaches but this requires a complete design space to compare against. Furthermore, the ultimate goal is a set of all 1 Introduction architectures that are feasible with respect to constraints [15] and System architecture is defined as the elements or compo- that are unique [7]. Arriving at such a design space efficiently is nents contained within a system and their relationships [1–3]. a considerable challenge. Designing breakthrough engineering systems with new capabil- In this article, the design space is completely captured by ities and new levels of performance requires innovations in sys- a perfect matchings approach for a certain class of architec- tem architecture. Engineers often rely on heuristics such as de- ture design problems, more specifically, problems that are rep- sign by analogy [4] and intuition when considering system ar- resented by undirected colored graphs under the component/port chitecture, but this may result in fixation on example designs and paradigm [2, 14, 16]. The proposed approach generates truly stifle innovation [5]. novel architectures (in fact all of them) but still leverages some Many studies have concentrated on effective representation of the natural constraints found in architecture design problems and generation methods, primarily based on graph representa- to reduce the number of graphs generated. This approach leads tions of system architecture (see Fig. 1 for some common en- to a number of interesting insights into the fundamental nature of gineering systems represented as graphs). The value of these architecture design problems. methods often is to present new valid topologies to engineers for 1 Copyright © 2016 by ASME M P F P P V T F K B L G M R L P P L M R K R E P M (a) Suspension. (b) Hybrid powertrain. (c) Mechanism. FIGURE 1: Architectures represented as graphs. The remainder of the paper is organized as follows. The Definition 2 (Colored Graph). A colored graph G is a three- next section outlines the some of the basic theory behind candi- tuple (V; E; P) where (V; E) specifies an undirected graph and P = f gk \ date architectures with perfect matchings. Next network struc- Vi i=1 is a partition of the vertices into color sets (Vi V j = Φ, ture constraints and the colored graph isomorphism problem are [k i , j, and ( i=1Vi) = V). For convenience, define color(v) = i if discussed to achieve feasible unique architectures. Using the in- v 2 Vi. sights from the previous two sections, a tree search algorithm is The graphs in Fig. 1 are colored graphs where each vertex developed that more efficiently covers the same design space. A represents a component. The colored labels indicate different number of case studies are then presented. Finally, a discussion component types. For example, in Fig. 1a, K represents a ver- is given of the results and how the proposed approaches can be tex with a coloring K indicating that it is a spring and that B used in current architecture design research. represents a damper. These are termed 2-port components since they can have up to 2 unique edges (this port notion is analogous to bond graph modeling [20]). However there is a fundamental 2 Candidate Architectures with Perfect Matchings limitation with this representation: if the order that the ports of First some relevant graph theory background is given. a component are connected to edges prescribed in A is impor- Definition 1 (Graph). A graph is a pair G = (V; E) of sets sat- tant, then pure component graph representation is not sufficient 2 isfying E ⊂ [V] where the elements of V are the vertices and the for determining a unique architecture. elements of E are its edges. Consider the planetary gear P in Fig. 1b. Since the plane- A simple graph is an unweighted, undirected graph contain- tary gear is represented by a single vertex, it is unclear which of ing no graph loops or multiple edges. The adjacency matrix of G the connected components { E , G , M } is connected to the sun, is the n × n matrix A = A(G) whose entries ai j are given by: ring, and carrier (names for the planetary gear ports). Permuta- 8 tions of this decision would result in a different architectures but <>1 if the set (vi;v j) 2 E ai j = the same adjacency matrix. A better representation would deter- >0 otherwise : mine unique graphs motivating a pure ports graph representation For a simple graph, the adjacency matrix must have 0s on the of architectures. diagonal. For an undirected graph, the adjacency matrix is sym- metric and if only a subset of the edges are present in E, the cor- rect A(G) can be constructed with sign(A+ AT ). The connectivity 2.1 Ports Graph P matrix of simple graph G can be found with: A port graph G is constructed from a three-tuple (C;R; P): n • C is the colored label set representing distinct component AC(n) = A (1) types, whose size is denoted by nC where the interpretation of AC(n) is for every nonzero entry, there • R is a column vector indicting the number of replicates for exists at most n undirected walks required to go from vi to v j, each component i.e., the pair of vertices are connected in some sense [17, p. 165]. • P is a column vector indicating the number of ports for each We will assume that n that same as the length of A giving all component walks. Using (C;R; P) we will create the three-tuple (V; E; L) that The degree of a vertex is the number of edges at the vertex defines a proper colored graph (see Definition 2). The definition and the average degree is d (G) = 2jEj=jVj [18, p. 5]. A matching of an n-port component in this context is all n ports are com- in a graph is a set of edges such that no two have a vertex in pletely connected to each other. Therefore each component can common and a perfect matching is a matching that covers every be considered a complete graph of its ports (see Fig. 2 for some vertex [19, p. 255]. complete graphs). The vertex and edge set for GP is then defined 2 Copyright © 2016 by ASME 1 1 1 1 1 1 2 5 2 4 2 3 3 4 2 3 2 K K K K K 1 2 3 4 5 (a) 1!! perfect matchings for K2. 1 1 1 FIGURE 2: Complete graphs on n vertices between 1 and 5. 2 4 2 4 2 4 3 3 3 as the union of these complete subgraphs: (b) 3!! perfect matchings for K4. n R [C [k 1 1 1 1 1 P 2 6 2 6 2 6 2 6 2 6 (V; E) = KPk (2) k=1 j=1 3 5 3 5 3 5 3 5 3 5 4 4 4 4 4 where KPk is a complete graph of size Pk. The complete label for each vertex is constructed from a 1 1 1 1 1 naming scheme where the base is the colored label from C, the 2 6 2 6 2 6 2 6 2 6 subscript is the replicate number, and the superscript is the port 3 5 3 5 3 5 3 5 3 5 number. Then the set of colored labels for GP can be constructed 4 4 4 4 4 as: 1 1 1 1 1 nC Rk Pk 2 6 2 6 2 6 2 6 2 6 P [[[n i o L = (Ck) j (3) 3 5 3 5 3 5 3 5 3 5 k=1 j=1 i=1 4 4 4 4 4 where each label is unique at this point.
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