Computing self-supporting surfaces by regular triangulation

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.
2. Aurenhammer, F. 1987. Power diagrams: properties, algorithms and applications. SIAM J. Comput. 16, 1, 78--96.
3. Aurenhammer, F. 1987. A criterion for the affine equivalence of cell complexes in R^d and convex polyhedra in R^d+1. Discrete Comp. Geom. 2, 49--64.
4. Block, P., and Lachauer, L. 2011. Closest-Fit, Compression-only solutions for freeform shells. In Proc. of the IABSE-IASS Symp.
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.
6. Block, P. 2009. Thrust Network Analysis: Exploring Three-dimensional Equilibrium. PhD thesis, Massachusetts Institute of Technology.
7. CGAL, Computational Geometry Algorithms Library. http://www.cgal.org.
8. Chen, L., and Xu, J. 2004. Optimal Delaunay triangulations. J. Comput. Math. 22, 299--308.
9. Cheng, S.-W., and Dey, T. K. 2004. Quality meshing with weighted Delaunay refinement. SIAM J. Comput. 33, 1, 69--93.
10. Cremona, L. 1890. Graphical Statics. Oxford University Press.
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.
12. Du, Q., Faber, V., and Gunzburger, M. 1999. Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev. 41, 637--676.
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.
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.
15. Gertz, E. M., and Wright, S. J. 2003. Object-oriented software for quadratic programming. ACM Trans. Math. Softw. 29, 58--81.
16. Glickenstein, D. 2005. Geometric triangulations and discrete Laplacians on manifolds. arXiv:0508188 {math.MG}.
17. Heyman, J. 1966. The stone skeleton. J. Int. Solids and Structures 2, 249--279.
18. Heyman, J. 1997. The Stone Skeleton: Structural Engineering of Masonry Architecture. Cambridge University Press.
19. Kilian, A., and Ochsendorf, J. 2005. Paticle-spring systems for structural form finding. J. Int. Assoc. Shell and Spatial Structures 46, 77--84.
20. Kilian, A. 2006. Design Exploration through Bidirectional Modeling of Constraints. PhD thesis, Massachusetts Institute of Technology.
21. Lawson, C. 1977. Software for C¹ surface interpolation. In Mathematical Software III, Academic Press, New York, 161--194.
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.
23. Maxwell, J. C. 1869. On reciprocal diagrams, frames and diagrams of forces. Transactions of the Royal Society of Edinburgh 26, 1--40.
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.
25. ODwyer, D. 1999. Funicular analysis of masonry vaults. Computers & Structures 73, 187--197.
26. Rippmann, M., Lachauer, L., and Block, P. 2012. Interactive vault design. J. Int. Space Structures 27, 219--230.
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.
28. Schiftner, A., and Balzer, J. 2010. Statics-sensitive layout of planar quadrilateral meshes. In Advances in Architectural Geometry, 221--236.
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.
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.
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.
32. Wardetzky, M., Mathur, S., Kälberer, F., and Grinspun, E. 2007. Discrete Laplace operators: no free lunch. In Symp. on Geom. Proc.
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.
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.
35. Zorin, D., Schröder, P., and Sweldens, W. 1996. Interpolating subdivision for meshes with arbitrary topology. In SIGGRAPH '96, 189--192.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

      • Parametric curve and surface models

      • Volumetric models