Skip to main content
SpringerLink
Log in

Chain models of physical behavior for engineering analysis and design

  • Published:
Research in Engineering Design Aims and scope Submit manuscript

Abstract

The relationship between geometry (form) and physical behavior (function) dominates many engineering activities. The lack of uniform and rigorous computational models for this relationship has resulted in a plethora of inconsistent (and thus usually incompatible) computer-aided design (CAD) tools and systems, causing unreasonable overhead in time, effort, and cost, and limiting the extent to which CAD tools are used in practice. It seems clear that formalization of the relationship between form and function is a prerequisite to taking full advantage of computers in automating design and analysis of engineering systems.

We present a unified computational model of physical behavior that explicitly links geometric and physical representations. The proposed approach characterizes physical systems in terms of their algebraic-topological properties:cell complexes, chains, and operations on them.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Arbab. “Set Models and Boolean Operations for Solids and Assemblies”.IEEE Computer Graphics and Applications, pp. 76–86, November 1990.

  2. Richard L. Bishop and Samuel I. Goldberg.Tensor Analysis on Manifolds. Dover, New York, 1980.

    Google Scholar 

  3. Franklin H. Branin. “The Algebraic-Topological Basis for Network Analogies and the Vector Calculus”. InProceedings of the Symposium on Generalized Networks, Vol. 16, pp. 453–491, Brooklyn, New York, 1966. Polytechnic Institute of Brooklyn.

    Google Scholar 

  4. Paul Bamberg and Schlomo Sternberg.A Course in Mathematics for Students of Physics. Cambridge University Press, Cambridge, 1988, 1990.

    Google Scholar 

  5. William L. Burke.Applied Differential Geometry. Cambridge University Press, Cambridge, 1985.

    Google Scholar 

  6. J. J. Cox and D. C. Anderson. “Single Model Formulation that Link Engineering Analysis with Geometric Modeling”. In J. Turner, J. Pegna, and M. Wozny, editors,Product Modeling for Computer-Aided Design and Manufacturing. Elsevier Science Publishers, North-Holland, 1991.

    Google Scholar 

  7. Richard Courant and David Hilbert.Methods of Mathematical Physics. John Wiley, New York, 1989.

    Google Scholar 

  8. R. D. Cook, D. S. Malkus, and M. E. Plesha.Concepts and Applications of Finite Element Analysis, 3rd edn. John Wiley, New York, 1989.

    Google Scholar 

  9. M. R. Duffey and J. R. Dixon, “Automating Extrusion Design: A Case Study in Geometric and Topological Reasoning for Mechanical Design”.Computer-Aided Design, 20(10): 589–596, 1988.

    Google Scholar 

  10. J. R. Dixon, E. C. Libardi, and E. H. Nielson. “Unresolved Research Issues in Development of Design-with-Features Systems”. In M. Wozny, J. Turner, and K. Preiss, editors,Geometric Modeling for Product Engineering, pp. 183–196. Elsevier Science Publishers B.V. (North Holland), 1990.

    Google Scholar 

  11. F. J. Evans and J. J. van Dixhoorn. “Towards More Physical Structure in Systems Theory”. In F. J. Evans and J. J. van Dixhoorn, editors,Physical Structure in Systems Theory, pp. 1–15. Academic Press, London, 1974.

    Google Scholar 

  12. S. Finger and J. A. Dixon. “A Review of Research in Mechanical Engineering Design”.Research in Engineering Design, 1(1 and 2), 1989. Parts I and II.

  13. O. I. Franksen. “The Nature of Data — from Measurements to Systems”.Bit, 25: 25–50, 1985.

    Google Scholar 

  14. E. L. Gursoz, Y. Choi, and F. B. Prinz. “Vertex-based Representation of Non-Manifold Boundaries”. In M. Wozny, J. Turner, and K. Preiss, editors,Geometric Modeling for Product Engineering, pp. 107–130. North-Holland, 1990.

  15. Jin-Kang Gui and Marti Mantyla. “New Concepts for Complete Product Assembly Modelling”. In J. Rossignac, J. Turner, and G. Allen, editors,Second Symposium on Solid Modeling and Applications, pp. 397–406, Montreal, Canada, May 19–21, 1993.

  16. M. R. Henderson. “Representing Functionality and Design Intent in Product Models”. In J. Rossignac, J. Turner, and G. Allen, editors,Second Symposium on Solid Modeling and Applications, pp. 387–396, Montreal, Canada, May 19–21, 1993.

  17. C. H. Hoffmann.Geometric and Solid Modeling. Morgan Kaufmann, San Mateo, CA, 1989.

    Google Scholar 

  18. Gabriel Kron.Tensor Analysis of Networks. John Wiley, New York, 1939.

    Google Scholar 

  19. Gabriel Kron. “Numerical Solution of Ordinary and Partial Differential Equations by Means of Equivalent Circuits”.Journal of Applied Physics, 126: 172–186, March 1945.

    Google Scholar 

  20. Gabriel Kron.Diakoptics — The Piecewise Solution of Large-Scale Systems. The Electrical Journal, London, 1957–59. A series of 20 articles beginning 7 June 1957.

  21. A. E. H. Love.A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, New York, 1927.

    Google Scholar 

  22. Y. T. Lee and A. A. G. Requicha. “Algorithms for Computing the Volume and Other Integral Properties of Solids: Parts I and II”.Communications of the ACM, 25(9):635–650, 1982.

    Google Scholar 

  23. H. Margenau.The Nature of Physical Reality. Ox Bow Press, Woodbridge, Connectiout, 1977.

    Google Scholar 

  24. M. Bremicker, M. Chirehdast, N. Kikuchi, and P. Y. Papalambros. “Integrated Topology and Shape Optimization in Structural Design”.Mechanical Structures and Machines, 19(4): 551–587, 1991.

    Google Scholar 

  25. James R. Munkres.Elements of Algebraic Topology. Addison-Wesley, New York, 1984.

    Google Scholar 

  26. C. A. Nickle. “Osciilographic Solution of Electro-Mechanical Systems”.Transactions of the American Institute of Electrical Engineers, 44: 844–856, 1925.

    Google Scholar 

  27. H. F. Olson.Dynamical Analogies. Van Nostrand, Princeton, 1943.

    Google Scholar 

  28. Henry M. Payneter.Analysis and Design of Engineering Systems. MIT Press, Cambridge, Massachusetts, 1961.

    Google Scholar 

  29. Richard S. Palmer and James F. Cremer. “Simlab: Automatically Creating Physical Systems Simulators”. Computer Science Technical Report TR91-1246, Cornell University, November 1991.

  30. A. A. G. Requicha. “Mathematical Models of Rigid Solid Objects”. Tech. Memo 28, Production Automation Project, University of Rochester, Rochester, NY, November 1977.

    Google Scholar 

  31. J. R. Rossignac and M. A. O'Connor. “SGC: A Dimension-independent Model for Pointsets with Internal Structures and Incomplete Boundaries”. In M. Wozny, J. Turner, and K. Preiss, editors,Geometric Modeling for Product Engineering, pp. 145–180. Elsevier Science Publishers B.V. (North Holland), 1990.

    Google Scholar 

  32. A. A. G. Requicha and H. B. Voelcker. “An Introduction to Geometric Modeling and its Applications in Mechanical Design and Production”. In J. T. Tou, editor,Advances in Information Systems Science, Vol. 8. Plenum Publishing, New York, 1981.

    Google Scholar 

  33. A. A. G. Requicha and H. B. Voelcker. “Solid Modeling: A Historical Summary and Contemporary Assessment.IEEE Computer Graphics and Applications, pp. 9–24, March 1982.

  34. K. Shimada and D. Gossard. “Automated Shape Generation of Components in Mechanical Assemblies”. InASME Design Automation Conference, Tempe, AZ, September 1992.

  35. Jami J. Shah. “Conceptual Development of Form Features and Feature Modelers”.Research in Engineering Design, 2: 93–108, 1991.

    Google Scholar 

  36. V. Shapiro.Representations of Semi-Algebraic Sets in Finite Algebras Generated by Space Decompositions. PhD thesis, Cornell University, Cornell Programmable Automation, Ithaca NY 14853, February 1991.

    Google Scholar 

  37. V. Shapiro and H. B. Voelcker. “On Role of Geometry in Mechanical Design”.Research in Engineering Design, 1(1):69–73, 1989.

    Google Scholar 

  38. W. W. Tworzydlo and J. T. Oden. “Towards an Automated Environment in Computational Mechanics”.Computer Methods in Applied Mechanics and Engineering, 104: 87–143, 1993.

    Google Scholar 

  39. Enzo Tonti.On the Formal Structure of Physical Theories. Istituto di Matematica del Politecnico di Milano, Milan, 1975.

    Google Scholar 

  40. K. T. Ulrich and W. P. Seering. “Synthesis of Schematic Descriptions in Mechanical Design”.Research in Engineering Design, 1(1): 3–18, 1989.

    Google Scholar 

  41. H. B. Voelcker. “Modeling in the Design Process”. In W. Dale Compton, editor,Design and Analysis of Integrated Manufacturing Systems. National Academy Press, Washington, DC, 1988.

    Google Scholar 

  42. K. J. Weiler.Topological Structures for Geometric Modeling. PhD thesis, Rensselaer Polytechnic Institute, 1986.

  43. T. C. Wilson, J. A. Talbert, and J. J. Cox. “Modeling Primitives: An Object Oriented Formulation of Boundary Value Problems in a Solid Geometric Modeling Context”. In J. Rossignac, J. Turner, and G. Allen, editors,Second Symposium on Solid Modeling and Applications, pp. 441–448, Montreal, Canada, May 19–21, 1993.

  44. O. C. Zienkiewicz and R. L. Taylor.The Finite Element Method, 4th edn. McGraw-Hill, London, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Cornell University, 14853, Ithaca, NY, USA

    Richard S. Palmer

  2. Analytic Process Department, General Motors R&D Center, 48090-9055, Warren, MI, USA

    Vadim Shapiro

Authors
  1. Richard S. Palmer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vadim Shapiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Richard S. Palmer or Vadim Shapiro.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Palmer, R.S., Shapiro, V. Chain models of physical behavior for engineering analysis and design. Research in Engineering Design 5, 161–184 (1993). https://doi.org/10.1007/BF01608361

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01608361

Keywords

Search

Navigation