Combinatorial Laws for Physically Meaningful Design

A typical computer representation of a design includes geometric and physical information organized in a suitable combinatorial data structure. Queries and transformations of these design representations are used to formulate most algorithms in computational design, including analysis, optimization, evolution, generation, and synthesis. Formal properties, and in particular existence and validity of the computed solutions, must be assured and preserved by all such algorithms. Using tools from algebraic topology, we show that a small set of the usual combinatorial operators: boundary (∂), coboundary (δ), and dualization ( ) – are sufficient to represent a variety of physical laws and invariants. Specific examples include geometric integrity, balance and equilibrium, and surface smoothing. Our findings point a way toward systematic development of data structures and algorithms for design in a common formal computational framework. 1 Combinatorial representations in design Given a rich variety of representations and algorithms for computational design, there appears to be little reason to believe that most of them may be formulated within a single combinatorial framework. We show that not only is such a framework possible [15], but it is formal [3] because it is based on the tools from algebraic topology [7], practical [5,14] because it arises naturally from computational considerations, and ubiquitous [25] because it spans most of the geometric and physical laws. Please direct all correspondence to this author at Mechanical Engineering, 1513 University Avenue, Madison, WI 53706 USA 1.1 Design as an oriented cell complex Computational design of shapes in most cases involves working with both geometry as well as physics. It is common that shaping and analysis of artifacts rely on two separate representations. For example, in the evolutionary design of tables in [8] the assumed geometric primitives are voxels, but evaluating the structural integrity of the table requires that the voxel model be converted into another representation suitable for performing analysis. Furthermore, many different design procedures appear to rely on a variety of seemingly incompatible representations and techniques. Thus, shape-grammar rules may be implemented as transformations of truss structures [17], while subdivision and fairing rules are commonly applied to curve and surface representations [24] – both are quite distinct from the space-aligned three-dimensional voxel-based representation of tables. It is well known that these and most other representations of designs are simple instances of an oriented cell complex – a standard combinatorial structure widely used in topology and more recently in geometric modeling. Perhaps less obvious is that most geometric, physical, and design quantities can be formally defined as algebraic objects called chains and cochains intuitively vectors whose components are associated with individual cells. The immediate consequence of this fact is that most spatial, physical and design laws may be expressed algebraically and applied uniformly across a variety of design representations, tasks, and algorithms. Furthermore, most such laws are compositions of only a few primitive laws that are generic in nature. The purpose of this paper is to elucidate this algebraic topological view of computational design and to demonstrate its ubiquity. We substantially extend the earlier ideas proposed in [15] while drawing 1 Copyright c 2003 by ASME heavily on efforts of others who have used algebraic topological tools for explaining analogies and classifying physical theories [3, 25] and improving numerical solution techniques [13]. Much of the present research in the network formulation of many physical systems is based on the earlier work by Kron and Roth. Kron modelled various physical systems including elastic and plastic structures as electric circuits and solved them piecewise by the method of tearing [11]. Roth related the various quantities in the physical systems as homology and cochomology sequences using his now well-known Roth diagrams [20]. Branin extended the algebraic-topological formulation of networks by linking it with standard operations in vector calculus [3]. Tonti classified most physical theories into a framework of topological and constitutive relationships using his own diagrams [25]. Acceptance of the proposed computational framework should streamline and simplify development of computational procedures in terms of standard computational primitives, lead to more efficient search algorithms that generate fewer invalid candidate designs, and encourage cross-fertilization of ideas and techniques across a wide range of application domains. Combinatorially, cell complexes are compositions of oriented k-cells with k = 0(vertex), 1(edge), 2(surface), 3(volume), typically embedded in 3-D space without intersections and subject to the additional constraint that the closure of every cell is a union of other cells in the complex. For the purposes of this paper, we will draw on three substantially distinct but fairly common examples used in computational design, as illustrated in Figure 1. Voxel-based representation, such as used in evolutionary design of tables [8], are essentially 3-dimensional cubical cell complexes (Figure 1a), where 3-cells correspond to individual voxels, and lower-dimensional cells implicitly correspond to adjacency relationships (2-cells for face adjacency, 1cells for edge adjacency, and 0-cells for corner adjacency). Truss and frame structures that are commonly used in shape-grammar based design [17] are 1-dimensional complexes composed of 0cells (joints) and 1-cells (members) (Figure 1c). Boundary representations of solid models [18] and subdivision surfaces [24] (Figure 1b) are 2-dimensional geometric cell complexes with faces, edges, and vertices corresponding to 2-cells, 1-cells, and 0-cells respectively. More generally, different types of cell complexes arise frequently in computational science and engineering. Finite elements, finite differences, and finite volume models in numerical analysis are defined on collections of k-cells forming a mesh or a grid [13]. Many graphical computations are performed on polygonal meshes, while manifolds are usually represented by underlying triangulations. In each case, the cellular decompositions of shapes and spaces are supported by graph-based data structures and algorithms to perform all relevant computations cell by cell. Furthermore, all spatial properties and functions of the represented spaces may now be treated algebraically as vectors over the decomposed spaces. 1.2 Properties as chains and cochains The simplest operation that is performed on a cell complex is the identification and logical selection of certain cells for specific computational purposes. For example, the convex hull area computation in [8] requires isolating the corners (0-cells) and edges (1-cells) of faces of voxels at a certain height, while application of boundary conditions (tractions and displacements) requires isolation of 0-cells in a truss. Application of a smoothing procedure on a certain portion of a surface requires isolation of a subset of faces from the whole domain. In each case, a group of p-dimensional cells is selected among all cells of the complex for further processing. Such a group is represented algebraically as an elementary p-chain defined as a formal sum: Cp np ∑ i 1 fi cell i p (1) where Cp is the p-chain, cell p denotes the i th cell of dimension p, np is the number of p-cells in the cell complex and fi 1 1 0 is a coefficient that determines whether a particular oriented cell is selected with positive or negative orientation ( fi 1) or not selected at all ( fi 0). The top row in Figure 2 illustrates the selection of three 2-cells (with coefficient f i 1 and shaded grey) on a surface as a chain C2. More general coefficients may be associated with the cells in the cell complex in order to capture other uses and properties of the cell complex algebraically. Thus, integers may correspond to how many times a particular p-cell is visited in a path, reals may represent a total mass associated with a cell, and vectors may be used to specify forces acting on certain cells, etc. The generalization of coefficients is accompanied by a subtle transition from elementary p-chains to functions on p-chains called p-cochains. A p-cochain can be defined as a formal sum: C np ∑ i 1 gi cell i p (2) which is similar to Equation (1) with the difference that C p is the p-cochain, and gi denote the generic coefficients that captures specific properties of the associated cell, cell p. Cochains in our examples include geometric properties (for instance, geometric locations of 0-cells of trusses, voxel-based table designs, and subdivision surface patches). Instances of physical and nonphysical attributes include forces and displacements on a truss structure or voxel-based models of tables, and weights associated with 1-cells of a subdivision surface in [24]. Figure 3 shows a cell complex composed of 0, 1 and 2-cells (subscripts of the alphabets denote the dimension of the cell) and coefficients associated with all 1and 2-cells. The corresponding 1-cochain C 1 and a 2-cochain C2 are: C1 5 a1 10 b1 23 c1 16 d1 16 e1 28 f1 29 g1 53 h1 13 i1 3 j1 19 k1 14 l1 and C2 80 a2 67 b2 76 c2 50 d2. The coefficients in C1 and C2 typically represent discrete distributions of physical 2 Copyright c 2003 by ASME