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Precise Construction of Micro-structures and Porous Geometry via Functional Composition

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Mathematical Methods for Curves and Surfaces (MMCS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10521))

Abstract

We introduce a modeling constructor for micro-structures and porous geometry via curve-trivariate, surface-trivariate and trivariate-trivariate function (symbolic) compositions. By using 1-, 2- and 3-manifold based tiles and paving them multiple times inside the domain of a 3-manifold deforming trivariate function, smooth, precise and watertight, yet general, porous/micro-structure geometry might be constructed, via composition. The tiles are demonstrated to be either polygonal meshes, (a set of) Bézier or B-spline curves, (a set of) Bézier or B-spline (trimmed) surfaces, (a set of) Bézier or B-spline (trimmed) trivariates or any combination thereof, whereas the 3-manifold deforming function is either a Bézier or a B-spline trivariate.

We briefly lay down the theoretical foundations, only to demonstrate the power of this modeling constructor in practice, and also present a few 3D printed tangible examples. We then discuss these results and conclude with some future directions and limitations.

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Notes

  1. 1.

    http://www.rhino3d.com.

  2. 2.

    https://en.wikipedia.org/wiki/Borromean_rings.

References

  1. Armillotta, A., Pelzer, R.: Modeling of porous structures for rapid prototyping of tissue engineering scaffolds. Int. J. Adv. Manuf. Technol. 39, 501–511 (2008)

    Article  Google Scholar 

  2. Carmo, M.P.D.: Differntial Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)

    Google Scholar 

  3. Chen, J., Shapiro, V.: Optimization of continuous heterogeneous models. In: Pasko, A., Adzhiev, V., Comninos, P. (eds.) Heterogeneous Objects Modelling and Applications. LNCS, vol. 4889, pp. 193–213. Springer, Heidelberg (2008). doi:10.1007/978-3-540-68443-5_8

    Chapter  Google Scholar 

  4. Coquillart, S.: Extended free-form deformation: a sculpturing tool for 3D geometric modeling. In: Proceedings of the 17th Annual Conference on Computer Graphics and Interactive Techniques, vol. 24, pp. 187–196. ACM Press, August 1990

    Google Scholar 

  5. Coquillart, S., Jancne, P.: Animated free-form deformation: an interactive animation technique. In: Proceedings of the 18th Annual Conference on Computer Graphics and Interactive Techniques, vol. 25, pp. 23–26. ACM Press, July 1991

    Google Scholar 

  6. DeRose, T.D., Goldman, R.N., Hagen, H., Mann, S.: Functional composition algorithms via blossoming. ACM Trans. Graph. 12(2), 113–135 (1993)

    Article  MATH  Google Scholar 

  7. Elber, G.: Free form surface analysis using a hybrid of symbolic and numerical computation. Ph.D. thesis, University of Utah (1992)

    Google Scholar 

  8. Elber, G.: Geometric deformation-displacement maps. In: The Tenth Pacific Graphics, pp. 156–165, October 2002

    Google Scholar 

  9. Elber, G.: Geometric texture modeling. IEEE Comput. Graph. Appl. 25(4), 66–76 (2005)

    Google Scholar 

  10. Elber, G.: Constructing porous geometry. In: FASE 2016, June 2016

    Google Scholar 

  11. Ellens, M.S., Cohen, E.: An approach to \(C^{-1}\) and \(C^0\) feature lines. In: Mathematical Methods for Curves and Surfaces, pp. 121–132 (1995)

    Google Scholar 

  12. Farin, G.: Curves and Surfaces for Computer Aided Geometric Design. Academic Press Professional, Boston (1993)

    MATH  Google Scholar 

  13. Farouki, R.T., Rajan, V.T.: Algorithms for polynomials in Bernstein form. Comput. Aided Geom. Des. 5(1), 1–26 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. Griessmair, J., Purgathofer, W.: Deformation of solids with trivariate B-splines. Eurograpghic 89, 137–148 (1989)

    Google Scholar 

  15. MacCracken, R., Joy, K.I.: Free-form deformations with lattices of arbitrary topology. In: Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, pp. 181–188. ACM Press (1996)

    Google Scholar 

  16. Massarwi, F., Elber, G.: A B-spline based framework for volumetric object modeling. Comput. Aided Des. 78, 36–47 (2016)

    Article  MathSciNet  Google Scholar 

  17. Morken, K.: Some identities for products and degree raising of splines. Constr. Approx. 7(1), 195–208 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Paik, K.L.: Trivariate B-splines. MSc. Department of Computer Science, University of Utah (1992)

    Google Scholar 

  19. Pasko, A., Fryazinov, O., Vilbrandt, T., Fayolle, P.-A., Adzhiev, V.: Procedural function-based modelling of volumetric microstructures. Graph. Models 73(5), 165–181 (2011)

    Article  Google Scholar 

  20. Pauletti, M.S., Martinelli, M., Cavallini, N., Antolin, P.: Igatools: an isogeometric analysis library. SIAM J. Sci. Comput. 37(4), 465–496 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Piegl, L., Tiller, W.: The NURBS Book. Springer, Heidelberg (1997)

    Book  MATH  Google Scholar 

  22. Porumbescu, S.D., Budge, B., Feng, L., Joy, K.I.: Shell maps. ACM Trans. Graph. 24(3), 626–633 (2005)

    Article  Google Scholar 

  23. Schein, S., Elber, G.: Discontinuous free form deformations. In: Proceedings of Pacific Graphics 2004, pp. 227–236 (2004)

    Google Scholar 

  24. Schroeder, C., Regli, W.C., Shokoufandeh, A., Sun, W.: Computer-aided design of porous artifacts. Comput. Aided Des. 37, 339–353 (2005)

    Article  Google Scholar 

  25. Sederberg, T.W., Parry, S.R.: Free-form deformation of solid geometric models. Comput. Graph. 20, 151–160 (1986)

    Article  Google Scholar 

  26. Xiao, F., Yin, X.: Geometry models of porous media based on Voronoi tessellations and their porosity-permeability relations. Comput. Math. Appl. 72, 328–348 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This research was supported in part by the ISRAEL SCIENCE FOUNDATION (grant No. 278/13). I also like to thank Boris van Sosin for his help in implementing the trivariate-trivariate composition operator.

The IGA of the model in Fig. 12 has been performed with the help of Pablo Antolin (EPFL Lausanne), Annalisa Buffa (EPFL Lausanne and IMATI-CNR Pavia), Massimiliano Martinelli (IMATI-CNR Pavia); Giancarlo Sangalli (University of Pavia and IMATI-CNR Pavia)

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Authors and Affiliations

  1. Department of Computer Science, Technion – IIT, 32000, Haifa, Israel

    Gershon Elber

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  1. Gershon Elber
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Correspondence to Gershon Elber .

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Editors and Affiliations

  1. University of Oslo, Oslo, Norway

    Michael Floater

  2. University of Oslo, Oslo, Norway

    Tom Lyche

  3. Université Joseph Fourier, Grenoble, France

    Marie-Laurence Mazure

  4. University of Oslo, Oslo, Norway

    Knut Mørken

  5. Vanderbilt University, Nashville, Tennessee, USA

    Larry L. Schumaker

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Elber, G. (2017). Precise Construction of Micro-structures and Porous Geometry via Functional Composition. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2016. Lecture Notes in Computer Science(), vol 10521. Springer, Cham. https://doi.org/10.1007/978-3-319-67885-6_6

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  • DOI: https://doi.org/10.1007/978-3-319-67885-6_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67884-9

  • Online ISBN: 978-3-319-67885-6

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