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
Your Session has timed out. Please sign back in to continue.


Palmer,  R., and Shapiro,  V., 1993, “Chain Models of Physical Behavior for Engineering Analysis and Design,” Res. Eng. Des., 5, pp. 161–184.
Brain, F., H., 1966, “The Algebraic-Topological Basis for Network Analogies and the Vector Calculus,” In Proceedings of the Symposium on Generalized Networks, 16 , Polytechnic Institute of Brooklyn, pp. 453–491.
Hocking, J., G., and Young, G., S., 1988, Topology, Dover, pp. 298–299.
Palmer,  R. S., 1995, “Chain Models and Finite Element Analysis: An Executable CHAINS Formulation of Plane Stress,” Computer Aided Geometric Design, 12, pp. 733–770.
Egli,  R., and Stewart,  N., 1999, “A Framework for System Specification Using Chains on Cell Complexes,” Comput.-Aided Des., 31(11), pp. 669–681.
Tonti, E., 1975, On the Formal Structure of Physical Theories, Instituto Di Matematica Del Politecnico Di Milano, Milan.
Hornby, G., S., and Pollack, J., B., 2001, “The Advantages of Generative Grammatical Encodings for Physical Design,” Congress on Evolutionary Computation, pp. 600–607.
Reddy,  G., and Cagan,  J., 1995, “An Improved Shape Annealing Algorithm for Truss Topology Generation,” ASME J. Mech. Des., 117(2(A)), pp. 315–321.
Taubin, G., 1995, “A Signal Processing Approach to Fair Surface Design,” Siggraph ’95 Conference Proceedings, August, pp. 351–358.
Mattiussi,  C., 2000, “The Finite Volume, Finite Element, and Finite Difference Methods as Numerical Methods for Physical Field Problems,” Adv. Hydrosci., 113, pp. 1–146.
Kron, G., 1957–1959, Diakoptics—The Piecewise Solution of Large-Scale Systems, The Electrical Journal, London. A series of 20 articles beginning June 7, 1957.
Roth, J., P., 1971, “Existence and Uniqueness of Solution to Electrical Network Problem via Homology Sequences,” In SIAM-AMS Proceedings, Vol. III , American Mathematical Society, pp. 113–118.
Requicha, A., A. G., 1977, Mathematical Models of Rigid Solid Objects, Tech. Memo 28, Production Automation Project, University of Rochester, Rochester, NY.
Tonti, E., 1977, “The Reason of the Analogies in Physics,” In Problem Analysis in Science and Engineering, H. Branin Jr., F., and K. Huseyin, Eds., Academic Press, pp. 463–514.
Samuelsson, A., 1962, “Linear Analysis of Frame Structures by use of Algebraic Topology,” PhD thesis, Chalmer Tekniska Hogskola, Goteborg.
Raghothama,  S., and Shapiro,  V., 1998, “Boundary Representation Deformation in Parametric Solid Modeling,” ACM Transactions on Computer Graphics, 17(4), October, pp. 259–286.
Mantyla, M., 1988, An Introduction to Solid Modeling, Computer Science Press, Maryland, USA, pp. 139–160.
Heisserman, J., A., 1991, “Generative Geometric Design and Boundary Solid Grammars,” PhD thesis, Carnegie Mellon University, Pittsburgh, PA, May.
Roth,  J. P., 1955, “An Application of Algebraic Topology to Numerical Analysis: On the Existence of a Solution to the Network Problem,” Proc. Natl. Acad. Sci. U.S.A., 41, pp. 518–521.
Whitney, H., 1957, Geometric Integration Theory, Princeton University Press, Princeton, New Jersey.
Chard,  J. A., and Shapiro,  V., 2000, “A Multivector Data Structure for Differential Forms and Equations,” IMACS Transactions Journal, Mathematics and Computers in Simulation, 54, pp. 33–64.
Tonti, E., 1999, Finite Formulation of Field Laws, unpublished.
Bamberg, P., and Sternberg, S., 1988–1990, A Course in Mathematics for Students of Physics, Cambridge University Press, Cambridge, England, pp. 420.
Strang,  G., 1988, “A Framework for Equilibrium Equations,” SIAM Rev., 30(2), June, pp. 283–297.
Blackett, D., W., 1967, Elementary Topology: A Combinatorial and Algebraic Approach, Academic Press Text-books in Mathematics, Academic Press, New York London, pp. 162–164.
Bjorke, O., 1978, “The Finite Element Method as Multi-Terminal Networks,” In System Structures in Engineering (Oyvind Bjorke and Ole Immanuel Franksen, Ed.), Tapir, Trondheim, Norway, pp. 179–218.
Schwalm,  W. M. B., and Giona,  M., 1999, “Vector Difference Calculus for Physical Lattice Models,” Phys. Rev. E, 59(1), January, pp. 1217–1233.
Hyman,  J. M., and Shashkov,  M., 1997, “Natural Discretizations for the Divergence, Gradient and Curl on Logically Rectangular Grids,” International Journal of Computers and Mathematics with Applications, 33(4), pp. 81–104.
Ilies, H., and Shapiro, V., 2003, “On the Synthesis of Functionally Equivalent Mechanical Designs,” In 2003 AAAI Spring Symposium on Computational Synthesis, March 24–26, 2003, Stanford, AAAI.


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.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In