Structural and Multidisciplinary Optimization

, Volume 57, Issue 3, pp 1187–1211 | Cite as

Form-finding of grid-shells using the ground structure and potential energy methods: a comparative study and assessment

  • Yang Jiang
  • Tomás Zegard
  • William F. Baker
  • Glaucio H. PaulinoEmail author


The structural performance of a grid-shell depends directly on the geometry of the design. Form-finding methods, which are typically based on the search for bending-free configurations, aid in achieving structurally efficient geometries. This manuscript proposes two form-finding methods for grid-shells: one method is the potential energy method, which finds the form in equilibrium by minimizing the total potential energy in the system; the second method is based on an augmented version of the ground structure method, in which the load application points become variables of the topology optimization problem. The proposed methods, together with the well-known force density method, are evaluated and compared using numerical examples. The advantages and drawbacks of the methods are reviewed, compared and highlighted.


Form-finding Grid-shells Potential energy method Ground structure method 



The authors acknowledge the financial support from the US National Science Foundation (NSF) under projects #1559594 (formerly #1335160), which is a GOALI (Grant Opportunities for Academic Liaison with Industry) project with SOM (Skidmore, Owings & Merrill LLP), and project #1321661. Ms Haley Simms provided valuable comments that contributed to improve the manuscript. We are also grateful for the endowment provided by the Raymond Allen Jones Chair at the Georgia Institute of Technology. The information provided in this paper is the sole opinion of the authors and does not necessarily reflect the views of the sponsoring agencies.


  1. ACI Committee 318, American Concrete Institute (2014) Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI 318R-14)Google Scholar
  2. Addis B (2007) Building: 3000 years of design, engineering and construction. Phaidon Press, LondonGoogle Scholar
  3. Akbarzadeh M, Van Mele T, Block P (2015) Spatial compression-only form finding through subdivision of external force polyhedron. In: Proceedings of the international association for shell and spatial structures (IASS). Symposium, AmsterdamGoogle Scholar
  4. American Institute of Steel Construction (2011) Steel Construction Manual. 14th EdnGoogle Scholar
  5. Argyris JH, Angelopoulos T, Bichat B (1974) A general method for the shape finding of lightweight tension structures. Comput Methods Appl Mech Eng 3:135–149CrossRefGoogle Scholar
  6. Baker WF (1992) Energy-based design of lateral systems. Struct Eng Int 2(2):99–102CrossRefGoogle Scholar
  7. Baker WF, Beghini LL, Mazurek A, Carrion J, Beghini A (2013) Maxwell’s reciprocal diagrams and discrete Michell frames. Struct Multidiscip Optim 48:267–277MathSciNetCrossRefGoogle Scholar
  8. Barnes MR (1977) Form-finding and analysis of tension space structures by dynamic relaxation. PhD Thesis, City University LondonGoogle Scholar
  9. Barnes MR (1988) Form-finding and analysis of prestressed nets and membranes. Comput Struct 30(3):685–695MathSciNetCrossRefGoogle Scholar
  10. Barnes MR, Topping BHV, Wakefield DS (1977) Aspects of form-finding by dynamic relaxation. In: International conference on the behaviour of slender structuresGoogle Scholar
  11. Bendsøe MP, Ben-Tal A, Zowe J (1994) Optimization methods for truss geometry and topology design. Struct Multidiscip Optim 7(3):141–159CrossRefGoogle Scholar
  12. Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications. Springer, BerlinzbMATHGoogle Scholar
  13. Bergós J, Llimargas M (1999) Gaudí: The Man and His Work. Taschen, KölnGoogle Scholar
  14. Bletzinger K-U, Ramm E (1993) Form finding of shells by structural optimization. Eng Comput 9(1):27–35CrossRefGoogle Scholar
  15. Bletzinger K-U, Ramm E (1999) A general finite element approach to the form finding of tensile structures by the upyeard reference strategy. Int J Space Struct 14(2):131–145CrossRefGoogle Scholar
  16. Bletzinger K-U, Ramm E (2001) Structural optimization and form finding of light weight structures. Comput Struct 79(22):2053–2062CrossRefGoogle Scholar
  17. Bletzinger K-U, Wüchner R, Daoud F, Camprubí N (2005) Computational methods for form finding and optimization of shells and membranes. Comput Methods Appl Mech Eng 194(30):3438–3452MathSciNetCrossRefzbMATHGoogle Scholar
  18. Block P (2009) Thrust network analysis: exploring three-dimensional equilibrium. PhD thesis, Massachusetts Institute of Technology, USAGoogle Scholar
  19. Block P, Ochsendorf J (2007) Thrust network analysis: a new methodology for three-dimensional equilibrium. J Intern Assoc Shell Spatial Struct 48(3):167–173Google Scholar
  20. Chiandussi G, Codegone M, Ferrero S (2009) Topology optimization with optimality criteria and transmissible loads. Comput Math Appl 57(5):772–788MathSciNetCrossRefzbMATHGoogle Scholar
  21. Chilton J (2010) Heinz Isler’s infinite spectrum form-finding in design. Archit Des 80(4):64–71Google Scholar
  22. Christensen PW, Klarbring A (2009) An introduction to structural optimization, vol 153 of solid mechanics and its applications. Springer Netherlands, DordrechtGoogle Scholar
  23. Coelho RF, Tysmans T, Verwimp E (2014) Form finding & structural optimization. Struct Multidiscip Optim 49(6):1037–1046CrossRefGoogle Scholar
  24. Collins GR (1977) The Drawings of Antonio gaudí. The drawing center, New YorkGoogle Scholar
  25. Darwich W, Gilbert M, Tyas A (2010) Optimum structure to carry a uniform load between pinned supports. Struct Multidiscip Optim 42(1):33–42CrossRefGoogle Scholar
  26. Dorn W, Gomory R, Greenberg H (1964) Automatic design of optimal structures. J de mécanique 3 (1):25–52Google Scholar
  27. Fuchs MB, Moses E (2000) Optimal structural topologies with transmissible loads. Struct Multidiscip Optim 19(4):263– 273CrossRefGoogle Scholar
  28. Gallagher RH, Zienkiewicz OC (1973) Optimum structural design: theory and applications. Wiley, LondonzbMATHGoogle Scholar
  29. Gilbert M, Darwich W, Tyas A, Shepherd P (2005) Application of large-scale layout optimization techniques in structural engineering practice. In: 6th world congress of structural and multidisciplinary optimization, June 1–10Google Scholar
  30. Haber R, Abel J (1982) Initial equilibrium solution methods for cable reinforced membranes part I-formulations. Comput Methods Appl Mech Eng 30(3):263–284CrossRefzbMATHGoogle Scholar
  31. Hemp W (1973) Optimum structures, 1st edn. Oxford University Press, OxfordGoogle Scholar
  32. International Code Council (2015) 2015 International Building CodeGoogle Scholar
  33. Jiang Y (2015) Free form finding of grid shell structures. Master’s thesis, University of Illinois at Urbana-Champaign, USAGoogle Scholar
  34. Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4):373–395MathSciNetCrossRefzbMATHGoogle Scholar
  35. Kilian A, Ochsendorf J (2006) Particle spring systems for structural form finding. J Intern Assoc Shell Spatial Struct 46(2):77– 84Google Scholar
  36. Leon SE, Paulino GH, Pereira A, Menezes IFM, Lages EN (2011) A unified library of nonlinear solution schemes. Appl Mech Rev 64(4):040803CrossRefGoogle Scholar
  37. Lewiński T, Rozvany GIN, Sokół T, Bołbotowski K (2013) Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains revisited. Struct Multidiscip Optim 47 (6):937–942MathSciNetCrossRefzbMATHGoogle Scholar
  38. Lewiński T, Zhou M, Rozvany GIN (1994) Extended exact solutions for least-weight truss layouts—Part I: cantilever with a horizontal axis of symmetry. Int J Mech Sci 36(5):375–398CrossRefzbMATHGoogle Scholar
  39. Lewis WJ (2003) Tension structures: form and behaviour. Thomas Telford Publishing, LondonCrossRefGoogle Scholar
  40. Linkwitz K, Schek H (1971) Einige Bemerkungen zur Berechnung von vorgespannten Seilnetzkonstruktionen. Ingenieur-Archiv 40(3):145–158CrossRefGoogle Scholar
  41. Martinell C, Collins GR, Rohrer J (1975) Gaudí: his life, his theories, his work. MIT Press, CambridgeGoogle Scholar
  42. Maxwell JC (1890) In: Niven WD (ed) The scientific papers of James Clerk Maxwell. Library Collection, CambridgeGoogle Scholar
  43. Michell A (1904) The limits of economy of material in frame structures. Phil Mag 8(47):589–597CrossRefzbMATHGoogle Scholar
  44. Miki M, Adriaenssens S, Igarashi T, Kawaguchi K (2014) The geodesic dynamic relaxation method for problems of equilibrium with equality constraint conditions. Int J Numer Methods Eng 99:682–710MathSciNetCrossRefzbMATHGoogle Scholar
  45. Mitchell T (2013) A limit of eonomy of material in shell structures. PhD thesis, University of California Berkeley, USAGoogle Scholar
  46. Nouri-Baranger T (2004) Computational methods for tension-loaded structures. Arch Comput Meth Eng 11 (2003):143– 186CrossRefzbMATHGoogle Scholar
  47. Pauletti RMO, Pimenta PM (2008) The natural force density method for the shape finding of taut structures. Comput Methods Appl Mech Eng 197(49):4419–4428CrossRefzbMATHGoogle Scholar
  48. Pichugin A, Tyas A, Gilbert M (2012) On the optimality of Hemp’s arch with vertical hangers. Struct Multidiscip Optim 46(1):17– 25MathSciNetCrossRefzbMATHGoogle Scholar
  49. Ramos AS, Paulino GH (2015) Convex topology optimization for hyperelastic trusses based on the ground-structure approach. Struct Multidiscip Optim 51(2):287–304MathSciNetCrossRefGoogle Scholar
  50. Richardson JN, Adriaenssens S, Coelho RF, Bouillard P (2013) Coupled form-finding and grid optimization approach for single layer grid shells. Eng Struct 52:230–239CrossRefGoogle Scholar
  51. Rozvany GIN (2001) On design-dependent constraints and singular topologies. Struct Multidiscip Optim 21 (2):164–172MathSciNetCrossRefGoogle Scholar
  52. Rozvany GIN, Gollub W, Zhou M (1997) Exact Michell layouts for various combinations of line supports-Part II. Struct Optim 14(2-3):138–149CrossRefGoogle Scholar
  53. Rozvany GIN, Prager W (1979) A new class of structural optimization problems: optimal archgrids. Comput Methods Appl Mech Eng 19(1):127–150MathSciNetCrossRefzbMATHGoogle Scholar
  54. Rozvany GIN, Sokół T (2013) Validation of numerical methods by analytical benchmarks, and verification of exact solutions by numerical methods. Topology Optimization in Structural and Continuum MechanicsGoogle Scholar
  55. Rozvany GIN, Wang CM (1983) On plane Prager-structures—I. Int J Mech Sci 25(7):519–527CrossRefzbMATHGoogle Scholar
  56. Rozvany GIN, Wang CM, Dow M (1982) Prager-structures: Archgrids and cable networks of optimal layout. Comput Methods Appl Mech Eng 31(1):91–113MathSciNetCrossRefzbMATHGoogle Scholar
  57. Sánchez J, Serna MÁ, Morer P (2007) A multi-step force–density method and surface-fitting approach for the preliminary shape design of tensile structures. Eng Struct 29(8):1966– 1976CrossRefGoogle Scholar
  58. Schek HJ (1974) The force density method for form finding and computation of general networks. Comput Methods Appl Mech Eng 3(1):115–134MathSciNetCrossRefGoogle Scholar
  59. Siev A, Eidelman J (1964) Stress analysis of prestressed suspended roofs. J Struct Div 90(4):103–122Google Scholar
  60. Sokół T (2011) A 99 line code for discretized Michell truss optimization written in Mathematica. Struct Multidiscip Optim 43(2):181–190CrossRefzbMATHGoogle Scholar
  61. Sokół T (2014) Multi-load truss topology optimization using the adaptive ground structure approach. In: Łodygowski T, Rakowski J, Litewka P (eds) Recent advances in computational mechanics. CRC Press, Boca Raton, pp 9–16Google Scholar
  62. Sokół T, Rozvany GIN (2013) On the adaptive ground structure approach for the multi-load truss topology optimization. In: 10th world congress on structural and multidisciplinary optimization, pp 1–10Google Scholar
  63. Tabarrok B, Qin Z (1992) Nonlinear analysis of tension structures. Comput Struct 45(5):973–984CrossRefzbMATHGoogle Scholar
  64. Talischi C, Paulino GH, Pereira A, Menezes IFM (2012) Polymesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45:309–328MathSciNetCrossRefzbMATHGoogle Scholar
  65. Thrall AP, Billington DP, Bréa KL (2012) The Maria Pia Bridge: A major work of structural art. Eng Struct 40:479–486CrossRefGoogle Scholar
  66. Tyas A, Pichugin AV, Gilbert M (2010) Optimum structure to carry a uniform load between pinned supports: exact analytical solution. Proc R Soc A Math Phys Eng sciences 467(2128):1101–1120MathSciNetCrossRefzbMATHGoogle Scholar
  67. Veenendaal D, Block P (2012) An overview and comparison of structural form finding methods for general networks. Int J Solids Struct 49(26):3741–3753CrossRefGoogle Scholar
  68. Wang CM, Rozvany GIN (1983) On plane prager-structures—II: non-parallel external loads and allowances for selfweight. Int J Mech Sci 25(7):529–541CrossRefzbMATHGoogle Scholar
  69. Wright M (2004) The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull Am Math Soc 42(01):39–56MathSciNetCrossRefzbMATHGoogle Scholar
  70. Yang XY, Xie YM, Steven GP (2005) Evolutionary methods for topology optimisation of continuous structures with design dependent loads. Comput Struct 83(12-13):956–963CrossRefGoogle Scholar
  71. Zalewski W, Allen E (1997) Shaping structures: statics. Wiley, New YorkGoogle Scholar
  72. Zegard T (2014) Structural optimization: from continuum and ground structures to additive manufacturing. PhD thesis, University of Illinois at Urbana-Champaign, USAGoogle Scholar
  73. Zegard T, Paulino GH (2014) GRAND — Ground Structure based topology optimization for arbitrary 2D domains using MATLAB. Struct Multidiscip Optim 50(5):861–882MathSciNetCrossRefGoogle Scholar
  74. Zegard T, Paulino GH (2015) GRAND3 — Ground Structure based topology optimization for arbitrary 3D domains using MATLAB. Struct Multidiscip Optim 52(6):1161–1184CrossRefGoogle Scholar
  75. Zhang Y (1996a) Solving large-scale linear programs by interior-point methods under the MATLAB Environment. Technical report, Department of Mathematics and Statistics University of Maryland, Baltimore, USAGoogle Scholar
  76. Zhang Y (1996b) Solving large-scale linear programs by interior-point methods under the Matlab Environment. Optim Methods Softw 10(1):1–31Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of Civil & Environmental EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Pontificia Universidad Católica de ChileSantiagoChile
  3. 3.Skidmore, Owings & Merrill LLPChicagoUSA

Personalised recommendations