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TECHNICAL PAPERS

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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Figures

Grahic Jump Location
Common examples of combinatorial structures used in computational synthesis; each is an instance of an oriented cell complex
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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.
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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.
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The force balance law can be applied via coboundary operation to any cellular structure, in this case to two adjacent voxels
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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).
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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.
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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.
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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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