“On the equilibrium of simplicial masonry structures” by Goes, Alliez, Owhadi and Desbrun

  • ©Fernando de Goes, Pierre Alliez, Houman Owhadi, and Mathieu Desbrun

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Title:

    On the equilibrium of simplicial masonry structures

Session/Category Title: Building Structures & Layouts

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Abstract:


    We present a novel approach for the analysis and design of self-supporting simplicial masonry structures. A finite-dimensional formulation of their compressive stress field is derived, offering a new interpretation of thrust networks through numerical homogenization theory. We further leverage geometric properties of the resulting force diagram to identify a set of reduced coordinates characterizing the equilibrium of simplicial masonry. We finally derive computational form-finding tools that improve over previous work in efficiency, accuracy, and scalability.

References:


    1. Alliez, P., Cohen-Steiner, D., Yvinec, M., and Desbrun, M. 2005. Variational tetrahedral meshing. ACM Trans. Graph. 24, 3 (July), 617–625. Google ScholarDigital Library
    2. Angelillo, M., Babilio, E., and Fortunato, A. 2012. Singular stress fields for masonry-like vaults. Continuum Mechanics and Thermodynamics, 1–19.Google Scholar
    3. Aurenhammer, F., Hoffmann, F., and Aronov, B. 1998. Minkowski-type theorems and least-squares clustering. Algorithmica 20, 1, 61–76.Google ScholarCross Ref
    4. Block, P., and Lachauer, L. 2011. Closest-fit, compression-only solutions for free form shells. In IABSE/IASS London Symposium, Int. Assoc. Shell Spatial Structures.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, 3, 167–173.Google Scholar
    6. Block, P. 2009. Thrust Network Analysis: Exploring Three-dimensional Equilibrium. PhD thesis, Department of Architecture, Massachusetts Institute of Technology.Google Scholar
    7. CGAL, 2012. Computational Geometry Algorithms Library (release 4.1). http://www.cgal.org.Google Scholar
    8. Desbrun, M., Kanso, E., and Tong, Y. 2007. Discrete differential forms for computational modeling. In Discrete Differential Geometry, A. Bobenko and P. Schröder, Eds. Springer.Google Scholar
    9. Desbrun, M., Donaldson, R., and Owhadi, H. 2013. Modeling across scales: Discrete geometric structures in homogenization and inverse homogenization. In Multiscale analysis and nonlinear dynamics, M. Z. Pesenson, Ed., vol. 8 of Reviews of Nonlinear Dynamics and Complexity. Wiley.Google Scholar
    10. Fisher, M., Schröder, P., Desbrun, M., and Hoppe, H. 2007. Design of tangent vector fields. In Proceedings of ACM SIGGRAPH. Google ScholarDigital Library
    11. Fosdick, R., and Schuler, K. 2003. Generalized Airy stress functions. Meccanica 38, 5, 571–578.Google ScholarCross Ref
    12. 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, 6211–6240.Google ScholarCross Ref
    13. 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
    14. Fraternali, F. 2011. A mixed lumped stress–displacement approach to the elastic problem of masonry walls. Mechanics Research Communications 38, 176–180.Google ScholarCross Ref
    15. Giaquinta, M., and Giusti, E. 1985. Researches on the equilibrium of masonry structures. Archive for Rational Mechanics and Analysis 88, 359–392.Google ScholarCross Ref
    16. Glickenstein, D., 2005. Geometric triangulations and discrete Laplacians on manifolds. arXiv.org:math/0508188.Google Scholar
    17. Grady, L. J., and Polimeni, J. R. 2010. Discrete Calculus: Applied Analysis on Graphs for Computational Science. Springer. Google ScholarCross Ref
    18. Green, A., and Zerna, W. 2002. Theoretical Elasticity. Dover.Google Scholar
    19. Heyman, J. 1966. The stone skeleton. International Journal of Solids and Structures 2, 2, 249–279.Google ScholarCross Ref
    20. Mercat, C. 2001. Discrete Riemann surfaces and the Ising model. Comm. Math. Phys. 218, 177–216.Google ScholarCross Ref
    21. Mérigot, Q. 2011. A multiscale approach to optimal transport. Computer Graphics Forum 30, 5, 1583–1592.Google ScholarCross Ref
    22. Mullen, P., Memari, P., de Goes, F., and Desbrun, M. 2011. HOT: Hodge-optimized triangulations. ACM Trans. Graph. 30, 4 (July), 103:1–103:12. Google ScholarDigital Library
    23. O’Dwyer, D. 1999. Funicular analysis of masonry vaults. Computers & Structures 73, 1–5, 187–197.Google Scholar
    24. Preparata, F. P., and Shamos, M. I. 1985. Computational Geometry: An Introduction. Springer-Verlag. Google ScholarCross Ref
    25. Tong, Y., Alliez, P., Cohen-Steiner, D., and Desbrun, M. 2006. Designing quadrangulations with discrete harmonic forms. In Symposium on Geometry Processing, 201–210. Google ScholarDigital Library
    26. Vouga, E., Höbinger, M., Wallner, J., and Pottmann, H. 2012. Design of self-supporting surfaces. ACM Trans. Graph. 31, 4, 87:1–87:11. Google ScholarDigital Library
    27. Wächter, A., and Biegler, L. T. 2006. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106, 1, 25–57. Google ScholarDigital Library
    28. Wardetzky, M., Mathur, S., Kälberer, F., and Grinspun, E. 2007. Discrete Laplace operators: no free lunch. In Symposium on Geometry Processing, 33–37. Google ScholarDigital Library
    29. Whiting, E., Ochsendorf, J., and Durand, F. 2009. Procedural modeling of structurally-sound masonry buildings. ACM Transactions on Graphics 28, 5, 112:1–112:10. Google ScholarDigital Library
    30. Whiting, E., Shin, H., Wang, R., Ochsendorf, J., and Durand, F. 2012. Structural optimization of 3D masonry buildings. ACM Transactions on Graphics 31, 6, 159:1–159:11. Google ScholarDigital Library
    31. Wong, Y. W., and Pellegrino, S. 2006. Wrinkled membranes Part II: analytical models. Journal of Mechanics of Materials and Structures 1, 25–59.Google ScholarCross Ref
    32. Zayer, R., Rossl, C., and Seidel, H.-P. 2005. Discrete tensorial quasi-harmonic maps. In Proceedings of Shape Modeling and Applications, 278–287. Google ScholarDigital Library

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