“Computing self-supporting surfaces by regular triangulation” by Liu, Pan, Snyder, Wang and Guo

  • ©Yang Liu, Hao Pan, John M. Snyder, Wenping Wang, and Baining Guo

Conference:


Type:


Title:

    Computing self-supporting surfaces by regular triangulation

Session/Category Title: Building Structures & Layouts


Presenter(s)/Author(s):


Moderator(s):



Abstract:


    Masonry structures must be compressively self-supporting; designing such surfaces forms an important topic in architecture as well as a challenging problem in geometric modeling. Under certain conditions, a surjective mapping exists between a power diagram, defined by a set of 2D vertices and associated weights, and the reciprocal diagram that characterizes the force diagram of a discrete self-supporting network. This observation lets us define a new and convenient parameterization for the space of self-supporting networks. Based on it and the discrete geometry of this design space, we present novel geometry processing methods including surface smoothing and remeshing which significantly reduce the magnitude of force densities and homogenize their distribution.

References:


    1. Alliez, P., Cohen-Steiner, D., Yvinec, M., and Desbrun, M. 2005. Variational tetrahedral meshing. ACM Trans. Graph. (SIGGRAPH) 24, 3, 617–625. Google ScholarDigital Library
    2. Aurenhammer, F. 1987. Power diagrams: properties, algorithms and applications. SIAM J. Comput. 16, 1, 78–96. Google ScholarDigital Library
    3. Aurenhammer, F. 1987. A criterion for the affine equivalence of cell complexes in Rd and convex polyhedra in Rd+1. Discrete Comp. Geom. 2, 49–64.Google ScholarDigital Library
    4. Block, P., and Lachauer, L. 2011. Closest-Fit, Compression-only solutions for freeform shells. In Proc. of the IABSE-IASS Symp.Google Scholar
    5. Block, P., and Ochsendorf, J. 2007. Thrust Network Analysis: a new methodology for three-dimensional equilibrium. J. Int. Assoc. Shell and Spatial Structures 48, 167–173.Google Scholar
    6. Block, P. 2009. Thrust Network Analysis: Exploring Three-dimensional Equilibrium. PhD thesis, Massachusetts Institute of Technology.Google Scholar
    7. CGAL, Computational Geometry Algorithms Library. http://www.cgal.org.Google Scholar
    8. Chen, L., and Xu, J. 2004. Optimal Delaunay triangulations. J. Comput. Math. 22, 299–308.Google Scholar
    9. Cheng, S.-W., and Dey, T. K. 2004. Quality meshing with weighted Delaunay refinement. SIAM J. Comput. 33, 1, 69–93. Google ScholarDigital Library
    10. Cremona, L. 1890. Graphical Statics. Oxford University Press.Google Scholar
    11. de Goes, F., Breeden, K., Ostromoukhov, V., and Desbrun, M. 2012. Blue noise through optimal transport. ACM Trans. Graph. (SIGGRAPH ASIA) 31, 6, 171:1–171:11. Google ScholarDigital Library
    12. Du, Q., Faber, V., and Gunzburger, M. 1999. Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev. 41, 637–676. Google ScholarDigital Library
    13. Fraternali, F., Angelillo, M., and Fortunato, A. 2002. A lumped stress method for plane elastic problems and the discrete-continuum approximation. J. Int. Solids and Structures 39, 6211–6240.Google ScholarCross Ref
    14. Fraternali, F. 2010. A thrust network approach to the equilibrium problem of unreinforced masonry vaults via polyhedral stress functions. Mechanics Research Communications 37, 198–204.Google ScholarCross Ref
    15. Gertz, E. M., and Wright, S. J. 2003. Object-oriented software for quadratic programming. ACM Trans. Math. Softw. 29, 58–81. Google ScholarDigital Library
    16. Glickenstein, D. 2005. Geometric triangulations and discrete Laplacians on manifolds. arXiv:0508188 {math.MG}.Google Scholar
    17. Heyman, J. 1966. The stone skeleton. J. Int. Solids and Structures 2, 249–279.Google ScholarCross Ref
    18. Heyman, J. 1997. The Stone Skeleton: Structural Engineering of Masonry Architecture. Cambridge University Press.Google Scholar
    19. Kilian, A., and Ochsendorf, J. 2005. Paticle-spring systems for structural form finding. J. Int. Assoc. Shell and Spatial Structures 46, 77–84.Google Scholar
    20. Kilian, A. 2006. Design Exploration through Bidirectional Modeling of Constraints. PhD thesis, Massachusetts Institute of Technology. Google ScholarDigital Library
    21. Lawson, C. 1977. Software for C1 surface interpolation. In Mathematical Software III, Academic Press, New York, 161–194.Google Scholar
    22. Liu, Y., Wang, W., Lévy, B., Sun, F., Yan, D.-M., Lu, L., and Yang, C. 2009. On centroidal Voronoi tessellation — energy smoothness and fast computation. ACM Trans. Graph. 28, 101:1–101:17. Google ScholarDigital Library
    23. Maxwell, J. C. 1869. On reciprocal diagrams, frames and diagrams of forces. Transactions of the Royal Society of Edinburgh 26, 1–40.Google ScholarCross Ref
    24. Mullen, P., Memari, P., de Goes, F., and Desbrun, M. 2011. HOT: Hodge-optimized triangulations. ACM Trans. Graph. (SIGGRAPH) 30, 4, 103:1–103:12. Google ScholarDigital Library
    25. ODwyer, D. 1999. Funicular analysis of masonry vaults. Computers & Structures 73, 187–197.Google ScholarCross Ref
    26. Rippmann, M., Lachauer, L., and Block, P. 2012. Interactive vault design. J. Int. Space Structures 27, 219–230.Google ScholarCross Ref
    27. Schek, H.-J. 1974. The force density method for form finding and computation of general networks. Comput. Methods in Applied Mech. and Eng. 3, 115–134.Google ScholarCross Ref
    28. Schiftner, A., and Balzer, J. 2010. Statics-sensitive layout of planar quadrilateral meshes. In Advances in Architectural Geometry, 221–236.Google Scholar
    29. Smith, J., Hodgins, J., Oppenheim, I., and Witkin, A. 2002. Creating models of truss structures with optimization. ACM Trans. Graph. (SIGGRAPH) 21, 3, 295–301. Google ScholarDigital Library
    30. Umetani, N., Igarashi, T., and Mitra, N. J. 2012. Guided exploration of physically valid shapes for furniture design. ACM Trans. Graph. (SIGGRAPH) 31, 4, 86:1–86:11. Google ScholarDigital Library
    31. Vouga, E., Höbinger, M., Wallner, J., and Pottmann, H. 2012. Design of self-supporting surfaces. ACM Trans. Graph. (SIGGRAPH) 31, 4, 87:1–87:11. Google ScholarDigital Library
    32. Wardetzky, M., Mathur, S., Kälberer, F., and Grinspun, E. 2007. Discrete Laplace operators: no free lunch. In Symp. on Geom. Proc. Google ScholarDigital Library
    33. Whiting, E., Ochsendorf, J., and Durand, F. 2009. Procedural modeling of structurally-sound masonry buildings. ACM Trans. Graph. (SIGGRAPH ASIA) 28, 112:1–112:9. Google ScholarDigital Library
    34. Whiting, E., Shin, H., Wang, R., Ochsendorf, J., and Durand, F. 2012. Structural optimization of 3D masonry buildings. ACM Trans. Graph. (SIGGRAPH ASIA) 31, 159:1–159:11. Google ScholarDigital Library
    35. Zorin, D., Schröder, P., and Sweldens, W. 1996. Interpolating subdivision for meshes with arbitrary topology. In SIGGRAPH ’96, 189–192. Google ScholarDigital Library


ACM Digital Library Publication:



Overview Page: