“Parametric self-supporting surfaces via direct computation of airy stress functions”

  • ©Masaaki Miki, Takeo Igarashi, and Philippe Block



Session Title:

    Fabricating Fabulous Forms


    Parametric self-supporting surfaces via direct computation of airy stress functions



    This paper presents a method that employs parametric surfaces as surface geometry representations at any stage of a computational process to compute self-supporting surfaces. This approach can be differentiated from existing relevant methods because such methods represent surfaces by a triangulated mesh surface or a network consisting of lines. The proposed method is based on the theory of Airy stress functions. Although some existing methods are also based on this theory, they apply its discrete version to discrete geometries. The proposed method simultaneously applies the theory to parametric surfaces directly and the discrete theory to the edges of parametric patches. The discontinuous boundary between continuous patches naturally corresponds to ribs seen in traditional vault masonry buildings. We use nonuniform rational B-spline surfaces in this study; however, the basic idea can be applied to other parametric surfaces. A variety of self-supporting surfaces obtained by the proposed computational scheme is presented.


    1. Aris, R. 1962. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover, New York.Google Scholar
    2. Arnold, D. N., Falk, R. S., and Winther, R. 2006. Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15 (May), 1.Google ScholarCross Ref
    3. Arnold, D. N., Falk, R. S., and Winther, R. 2010. Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47, 2 (Jan.), 281–354.Google ScholarCross Ref
    4. Bazilevs, Y., Calo, V., Cottrell, J., Evans, J., Hughes, T., Lipton, S., Scott, M., and Sederberg, T. 2010. Isogeometric analysis using T-splines. Computer Methods in Applied Mechanics and Engineering 199, 5-8 (Jan.), 229–263.Google ScholarCross Ref
    5. Block, P., and Ochsendorf, J. 2007. Thrust Network Analysis: A new methodology for three-dimensional equilibrium. Journal of the International Association for Shell and Spatial Structures 48, 3, 167–173.Google Scholar
    6. Buffa, A., Sangalli, G., and Vázquez, R. 2014. Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations. Journal of Computational Physics 257 (Jan.), 1291–1320. Google ScholarDigital Library
    7. Ciarlet, P. G. 2006. An Introduction to Differential Geometry with Applications to Elasticity. Springer, New York.Google Scholar
    8. Cottrell, J. A., Hughes, T. J. R., and Bazilevs, Y. 2009. Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley & Sons. Google ScholarCross Ref
    9. Darling, R. W. R. 1994. Differential forms and connections. Cambridge University Press.Google Scholar
    10. de Goes, F., Alliez, P., Owhadi, H., and Desbrun, M. 2013. On the Equilibrium of Simplicial Masonry Structures. ACM Transactions on Graphics (TOG) 32, 4. Google ScholarDigital Library
    11. de Goes, F., Liu, B., Budninskiy, M., Tong, Y., and Desbrun, M. 2014. Discrete 2-Tensor Fields on Triangulations. Computer Graphics Forum 33, 5 (Aug.), 13–24. Google ScholarDigital Library
    12. Dörfel, M. R., Jüttler, B., and Simeon, B. 2010. Adaptive isogeometric analysis by local h-refinement with T-splines. Computer Methods in Applied Mechanics and Engineering 199, 5-8 (Jan.), 264–275.Google ScholarCross Ref
    13. Eisenhart, L. 1947. An introduction to differential geometry: with use of the tensor calculus. Princeton University Press.Google Scholar
    14. Eisenhart, L. 1997. Riemannian geometry. Princeton University Press, London.Google Scholar
    15. Fraternali, F., Angelillo, M., and Fortunato, A. 2002. A lumped stress method for plane elastic problems and the discrete-continuum approximation. International Journal of Solids and Structures 39, 25 (Dec.), 6211–6240.Google ScholarCross Ref
    16. Fraternali, F., Farina, I., and Carpentieri, G. 2014. A discrete-to-continuum approach to the curvatures of membrane networks and parametric surfaces. Mechanics Research Communications 56, 18–25.Google ScholarCross Ref
    17. Fraternali, F. 2010. A thrust network approach to the equilibrium problem of unreinforced masonry vaults via polyhedral stress functions. Mechanics Research Communications 37, 2, 198–204.Google ScholarCross Ref
    18. Gurtin, M. E. 1963. A generalization of the Beltrami stress functions in continuum mechanics. Archive for Rational Mechanics and Analysis 13, 1 (Dec.), 321–329.Google ScholarCross Ref
    19. Hardy, R. L. 1971. Multiquadric equations of topography and other irregular surfaces. Journal of Geophysical Research 76, 8 (Mar.), 1905–1915.Google ScholarCross Ref
    20. Hughes, T., Cottrell, J., and Bazilevs, Y. 2005. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194, 39-41 (Oct.), 4135–4195.Google ScholarCross Ref
    21. Johannessen, K. A., Kvamsdal, T., and Dokken, T. 2014. Isogeometric analysis using LR B-splines. Computer Methods in Applied Mechanics and Engineering 269 (Feb.), 471–514.Google ScholarCross Ref
    22. Li, X., Deng, J., and Chen, F. 2007. Surface modeling with polynomial splines over hierarchical T-meshes. The Visual Computer 23, 12 (Aug.), 1027–1033. Google ScholarDigital Library
    23. Liu, Y., Pan, H., Snyder, J., Wang, W., and Guo, B. 2013. Computing self-supporting surfaces by regular triangulation. ACM Transactions on Graphics (TOG) 32, 4. Google ScholarDigital Library
    24. Panozzo, D., Block, P., and Sorkine-Hornung, O. 2013. Designing unreinforced masonry models. ACM Transactions on Graphics (TOG) 32, 4 (July). Google ScholarDigital Library
    25. Scott, M., Simpson, R., Evans, J., Lipton, S., Bordas, S., Hughes, T., and Sederberg, T. 2013. Isogeometric boundary element analysis using unstructured T-splines. Computer Methods in Applied Mechanics and Engineering 254 (Feb.), 197–221.Google ScholarCross Ref
    26. Sederberg, T. W., Zheng, J., Bakenov, A., and Nasri, A. 2003. T-splines and T-NURCCs. ACM Transactions on Graphics 22, 3 (July), 477. Google ScholarDigital Library
    27. Sederberg, T. W., Finnigan, G. T., Li, X., Lin, H., and Ipson, H. 2008. Watertight trimmed NURBS. In ACM SIGGRAPH 2008 papers on – SIGGRAPH ’08, ACM Press, New York, New York, USA, vol. 27, 1. Google ScholarDigital Library
    28. Simpson, R., Bordas, S., Trevelyan, J., and Rabczuk, T. 2012. A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis. Computer Methods in Applied Mechanics and Engineering 209-212 (Feb.), 87–100.Google Scholar
    29. Tang, C., Sun, X., Gomes, A., Wallner, J., and Pottmann, H. 2014. Form-finding with polyhedral meshes made simple. ACM Transactions on Graphics (TOG) 33, 4 (July). Google ScholarDigital Library
    30. Vouga, E., Höbinger, M., Wallner, J., and Pottmann, H. 2012. Design of self-supporting surfaces. ACM Transactions on Graphics (TOG) 31, 4. Google ScholarDigital Library
    31. Williams, C. 1990. The Generation of a Class of Structural Forms for Vaults and Sails. The Structural Engineer 68, 12, 231–235.Google Scholar

ACM Digital Library Publication: