Combinatorial Laws for Physically Meaningful Design

[+] Author and Article Information
Vasu Ramaswamy, Vadim Shapiro

Spatial Automation Laboratory, University of Wisconsin-Madison, 1513 University Avenue, Madison, WI 53706

J. Comput. Inf. Sci. Eng 4(1), 3-10 (Mar 23, 2004) (8 pages) doi:10.1115/1.1645863 History: Received April 01, 2003; Revised December 01, 2003; Online March 23, 2004
Copyright © 2004 by ASME
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Grahic Jump Location
Common examples of combinatorial structures used in computational synthesis; each is an instance of an oriented cell complex
Grahic Jump Location
Dimensionally homogeneous portions of the cell-complex may be represented by chains. Here a 2-chain identifies three cells with non-zero coefficients; Boundary operation on this 2-chain produces an oriented 1-chain of edges bounding the selected area. Note the change in orientation of some 1-cells as ∂ reorients them based on their relative orientation with the 2-cells.
Grahic Jump Location
Cochains are collections of oriented cells along with their associated coefficients. These coefficients represent the integral quantities associated with cells. For example, vector-valued coefficients on 1-cells could denote relative displacements and scalar coefficients on 2-cells could represent their areas.
Grahic Jump Location
The force balance law can be applied via coboundary operation to any cellular structure, in this case to two adjacent voxels
Grahic Jump Location
The force balance law may be more naturally specified using the boundary operation for some cellular structures. For trusses, the boundary of the 1-cochain of member forces is balanced to the 0-chain of forces acting on joints (center). The coboundary formulation of the equilibrium of forces requires the association of the various cochains in their respective dual space (right).
Grahic Jump Location
In the 2-D cell complex (n=2), for every primal p-cell, there exists an (n-p)-dual cell. For example, A0 is the dual 0-cell of the primal 2-cell a2,B0 is the dual to b2 and so on. Similarly A1 is the dual 1-cell of the primal 1-cell d1,B1 is dual to f1 and so on. The 2-cell A2 (shaded) is dual to the primal 0-cell e0.
Grahic Jump Location
The boundary ∂ operation on the primal cell complex is equivalent to the three-operation sequence  * δ * . The operation ∂ (top row) transfers the coefficients from 1-cells onto the 0-cell e0. This procedure is equivalent to applying a  *  operation (left column), which transfers the quantities from the primal cells to their dual, followed by a coboundary operation δ (bottom row), which transfers the coefficients from the dual 1-cells to the dual 2-cell A2. Application of another  *  operation (right column) then transfers the coefficient on A2 on to its dual, which is the primal 0-cell e0.
Grahic Jump Location
Roth diagram (left) shows some possible relationships between cochains on a three-dimensional cell complex. An example of composite law is shown on the right.
Grahic Jump Location
The incidence relationships between the 0 and 1-cells via boundary and coboundary can be represented as matrices that are transpose of each other. Orientation of a 1-cell toward an incident 0-cell is considered positive and negative otherwise.



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