Skip to main content
SpringerLink
Log in

Optimization of Continuous Heterogeneous Models

  • Chapter
Heterogeneous Objects Modelling and Applications

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 4889))

Abstract

A heterogeneous model consists of a solid model and a number of spatially distributed material attributes. Much progress has been made in developing methods for construction, design, and editing of such models. We consider the problem of optimization of a heterogeneous model, and show that its representation by a continuous function defined over a constructively represented domain naturally leads to simple and effective optimization procedures. Using minimum compliance optimization problem as an example, we show that the design sensitivities are directly obtainable in terms of material and geometric parameters, which can be used in any standard gradient-based optimization procedures. The proposed approach allows both local control of the material properties and global control of geometric variations, and can be used with many existing techniques for material modeling. Numerical experiments are given to demonstrate these representational advantages.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adzhiev, V., Kartasheva, E., Kunii, T., Pasko, A., Schmitt, B.: Hybrid Cellular-functional Modeling of Heterogeneous Objects. Journal of Computing and Information Science in Engineering 2, 312 (2003)

    Article  Google Scholar 

  2. Allaire, G., Jouve, F., Toader, A.M.: Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics 194, 363–393 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Belytschko, T., Xiao, S.P., Parimi, C.: Topology optimization with implicit functions and regularization. International Journal for Numerical Methods in Engineering 57, 1177–1196 (2003)

    Article  MATH  Google Scholar 

  4. Bendsøe, M.P.: Optimal shape design as a material distribution problem. Structural Optimization 1, 193–202 (1989)

    Article  Google Scholar 

  5. Bendsøe, M.P., Sigmund, O.: Material interpolations in topology optimization. Archive of Applied Mechanics 69, 635–654 (1999)

    Google Scholar 

  6. Bendsøe, M.P., Sigmund, O.: Topology Optimization: Theory, Methods and Applications. Springer, Heidelberg (2003)

    Google Scholar 

  7. Biswas, A., Fenves, S., Shapiro, V., Sriram, R.: Representation of Heterogeneous Material Properties in Core Product Model. Engineering with Computers (2007) (in press)

    Google Scholar 

  8. Biswas, A., Shapiro, V.: Approximate distance fields with non-vanishing gradients. Graphical Models 66(3), 133–159 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bloomenthal, J.: Introduction to Implicit Surfaces. Morgan Kaufmann Publishers, San Francisco (1997)

    MATH  Google Scholar 

  10. Chen, J., Freytag, M., Shapiro, V.: Shape sensitivity of constructive representations. In: Proceedings of the 2007 ACM Symposium on Solid and Physical Modeling, pp. 85–95. ACM Press, New York (2007)

    Chapter  Google Scholar 

  11. Chen, J., Shapiro, V., Suresh, K., Tsukanov, I.: Shape optimization with topological changes and parametric control. International Journal of Numerical Methods in Engineering 71(3), 313–346 (2007)

    Article  MathSciNet  Google Scholar 

  12. de Boor, C.: A Practical Guide to Splines. Springer, New York (2001)

    MATH  Google Scholar 

  13. Eschenauer, H.A., Olhoff, N.: Topology optimization of continuum structures: A review. Applied Mechanics Review 54, 331–390 (2001)

    Article  Google Scholar 

  14. Frisken, S.F., Perry, R.N., Rockwood, A.P., Jones, T.R.: Adaptively sampled distance fields: A general representation of shape for computer graphics. In: Proceedings of the ACM SIGGRAPH Conference on Computer Graphics, pp. 249–254 (2000)

    Google Scholar 

  15. Haug, E.J., Choi, K.K., Komkov, V.: Design Sensitivity Analysis of Structural Systems. Academic Press, New York, NY (1986)

    MATH  Google Scholar 

  16. Kojekine, N., Hagiwara, I., Savchenko, V.: Software tools using csrbf’s for processing scattered data. Computers and Graphics 27(2) (2003)

    Google Scholar 

  17. Kou, X.Y., Tan, S.T.: Heterogeneous object modeling: A review. Computer-Aided Design 39(4), 284–301 (2007)

    Article  Google Scholar 

  18. Kumar, V., Burns, D., Dutta, D., Hoffmann, C.: A framework for object modeling. Computer-Aided Design 31, 541–556 (1999)

    Article  MATH  Google Scholar 

  19. Lim, C., Turkiyyah, G.M., Ganter, M.A., Storti, D.W.: Implicit reconstruction of solids from cloud point sets. In: Proceedings of the Third Symposium on Solid Modeling and Applications, pp. 393–402. ACM Press, New York (1995)

    Chapter  Google Scholar 

  20. Liu, H., Cho, W., Jackson, T.R., Patrikalakis, N.M., Sachs, E.M.: Algorithms for design and interrogation of functionally gradient material objects. In: Proceedings of ASME 2000 IDETC/CIE 2000 ASME Design Automation Conference, Baltimore, MD (2000)

    Google Scholar 

  21. Matsui, K., Terada, K.: Continuous approximation of material distribution for topology optimization. International Journal of Numerical Methods in Engineering 59, 1925–1944 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Muraki, S.: Volumetric shape description of range data using “Blobby Model. In: Proceedings of the ACM SIGGRAPH Conference on Computer Graphics, vol. 25(4), pp. 227–235 (July 1991)

    Google Scholar 

  23. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Heidelberg (1999)

    MATH  Google Scholar 

  24. Ohtake, Y., Belyaev, A., Alexa, M., Turk, G., Seidel, H.-P.: Multi-level partition of unity implicits. ACM Transactions on Graphics (TOG) 22(3), 463–470 (2003)

    Article  Google Scholar 

  25. Pegna, J., Safi, A.: CAD modeling of multi-model structures for freeform fabrication. In: In Proceedings of the 1998 Solid Freeform Fabrication Symposium, Austin, TX (August 1998)

    Google Scholar 

  26. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C, 2nd edn. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  27. Qian, X., Dutta, D.: Design of heterogeneous turbine blade. Computer-Aided Design 35, 319–329 (2003)

    Google Scholar 

  28. Raviv, A., Elber, G.: Three-dimensional freeform sculpting via zero sets of scalar trivariate functions. Computer-Aided Design 32, 513–526 (2000)

    Article  Google Scholar 

  29. Ricci, A.: A constructive geometry for computer graphics. Computer Journal 16(3), 157–160 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rvachev, V.L.: Geometric Applications of Logic Algebra (in Russian), Naukova Dumka (1967)

    Google Scholar 

  31. Rvachev, V.L.: Theory of R-functions and Some Applications (in Russian), Naukova Dumka (1982)

    Google Scholar 

  32. Rvachev, V.L., Sheiko, T.I.: R-functions in boundary value problems in mechanics. Applied Mechanics Reviews 48(4), 151–188 (1996)

    Article  Google Scholar 

  33. Rvachev, V.L., Sheiko, T.I., Shapiro, V., Tsukanov, I.: On completeness of RFM solution structures. Computational Mechanics 25, 305–316 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Savchenko, V.V., Pasko, A.A., Okunev, O.G., Kunii, T.L.: Function Representation of Solids Reconstructed from Scattered Surface Points and Contours. Computer Graphics Forum 14(4), 181–188 (1995)

    Article  Google Scholar 

  35. Schmitt, B., Pasko, A., Schlick, C.: Constructive sculpting of heterogeneous volumetric objects using trivariate b-splines. The Visual Computer 20(2), 130–148 (2004)

    Article  Google Scholar 

  36. Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  37. Sethian, J.A., Wiegmann, A.: Structural boundary design via level set and immersed interface methods. Journal of Computational Physics 163(2), 489–528 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Shapiro, V.: Real functions for representation of rigid solids. Computer-Aided Geometric Design 11(2), 153–175 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. Shapiro, V.: Well-formed set representations of solids. International Journal on Computational Geometry and Applications 9(2), 125–150 (1999)

    Article  MATH  Google Scholar 

  40. Shapiro, V.: A convex deficiency tree algorithm for curved polygons. International Journal of Computational Geometry and Applications 11(2), 215–238 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Shapiro, V.: Semi-analytic geometry with R-functions. Acta Numerica 16, 239–303 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  42. Shapiro, V., Tsukanov, I.: Implicit functions with guaranteed differential properties. In: Fifth ACM Symposium on Solid Modeling and Applications, Ann Arbor, MI, pp. 258–269 (1999)

    Google Scholar 

  43. Shapiro, V., Tsukanov, I.: Meshfree simulation of deforming domains. Computer Aided Design 31, 459–471 (1999)

    Article  MATH  Google Scholar 

  44. Sigmund, O., Petersson, J.: Numerical instabilities in topology optimization: A survey on precedures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization 16, 68–75 (1998)

    Article  Google Scholar 

  45. Tsukanov, I., Shapiro, V.: The architecture of SAGE – a meshfree system based on RFM. Engineering with Computers 18(4), 295–311 (2002)

    Article  Google Scholar 

  46. Turk, G., O’Brien, J.: Modeling with implicit surfaces that interpolate. ACM Transactions on Graphics 21(4), 855–873 (1999)

    Article  Google Scholar 

  47. Velho, L., Gomes, J., de Figueiredo, L.H.: Implicit Objects in Computer Graphics. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  48. Wang, M.Y., Wang, X.M., Guo, D.M.: A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering 192(1-2), 227–246 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wang, S.Y., Wang, M.Y.: Radial basis functions and level set method for structural topology optimization. International Journal for Numerical Methods in Engineering 65(12), 2060–2090 (2005)

    Article  Google Scholar 

  50. Washizu, K.: Variational methods in elasticity and plasticity, 3rd edn. Pergamon Press, Oxford (1982)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Spatial Automation Laboratory, University of Wisconsin, Madison,  

    Jiaqin Chen & Vadim Shapiro

Authors
  1. Jiaqin Chen
    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

Editor information

Alexander Pasko Valery Adzhiev Peter Comninos

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Chen, J., Shapiro, V. (2008). Optimization of Continuous Heterogeneous Models. In: Pasko, A., Adzhiev, V., Comninos, P. (eds) Heterogeneous Objects Modelling and Applications. Lecture Notes in Computer Science, vol 4889. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68443-5_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-68443-5_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-68441-1

  • Online ISBN: 978-3-540-68443-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation