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

## Abstract

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.

## Keywords

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

### Acknowledgments

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.

## References

- ACI Committee 318, American Concrete Institute (2014) Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI 318R-14)Google Scholar
- Addis B (2007) Building: 3000 years of design, engineering and construction. Phaidon Press, LondonGoogle Scholar
- 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
- American Institute of Steel Construction (2011) Steel Construction Manual. 14th EdnGoogle Scholar
- 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
- Baker WF (1992) Energy-based design of lateral systems. Struct Eng Int 2(2):99–102CrossRefGoogle Scholar
- 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
- Barnes MR (1977) Form-finding and analysis of tension space structures by dynamic relaxation. PhD Thesis, City University LondonGoogle Scholar
- Barnes MR (1988) Form-finding and analysis of prestressed nets and membranes. Comput Struct 30(3):685–695MathSciNetCrossRefGoogle Scholar
- Barnes MR, Topping BHV, Wakefield DS (1977) Aspects of form-finding by dynamic relaxation. In: International conference on the behaviour of slender structuresGoogle Scholar
- Bendsøe MP, Ben-Tal A, Zowe J (1994) Optimization methods for truss geometry and topology design. Struct Multidiscip Optim 7(3):141–159CrossRefGoogle Scholar
- Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications. Springer, BerlinzbMATHGoogle Scholar
- Bergós J, Llimargas M (1999) Gaudí: The Man and His Work. Taschen, KölnGoogle Scholar
- Bletzinger K-U, Ramm E (1993) Form finding of shells by structural optimization. Eng Comput 9(1):27–35CrossRefGoogle Scholar
- 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
- Bletzinger K-U, Ramm E (2001) Structural optimization and form finding of light weight structures. Comput Struct 79(22):2053–2062CrossRefGoogle Scholar
- 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
- Block P (2009) Thrust network analysis: exploring three-dimensional equilibrium. PhD thesis, Massachusetts Institute of Technology, USAGoogle Scholar
- 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
- Chiandussi G, Codegone M, Ferrero S (2009) Topology optimization with optimality criteria and transmissible loads. Comput Math Appl 57(5):772–788MathSciNetCrossRefzbMATHGoogle Scholar
- Chilton J (2010) Heinz Isler’s infinite spectrum form-finding in design. Archit Des 80(4):64–71Google Scholar
- Christensen PW, Klarbring A (2009) An introduction to structural optimization, vol 153 of solid mechanics and its applications. Springer Netherlands, DordrechtGoogle Scholar
- Coelho RF, Tysmans T, Verwimp E (2014) Form finding & structural optimization. Struct Multidiscip Optim 49(6):1037–1046CrossRefGoogle Scholar
- Collins GR (1977) The Drawings of Antonio gaudí. The drawing center, New YorkGoogle Scholar
- 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
- Dorn W, Gomory R, Greenberg H (1964) Automatic design of optimal structures. J de mécanique 3 (1):25–52Google Scholar
- Fuchs MB, Moses E (2000) Optimal structural topologies with transmissible loads. Struct Multidiscip Optim 19(4):263– 273CrossRefGoogle Scholar
- Gallagher RH, Zienkiewicz OC (1973) Optimum structural design: theory and applications. Wiley, LondonzbMATHGoogle Scholar
- 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
- 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
- Hemp W (1973) Optimum structures, 1st edn. Oxford University Press, OxfordGoogle Scholar
- International Code Council (2015) 2015 International Building CodeGoogle Scholar
- Jiang Y (2015) Free form finding of grid shell structures. Master’s thesis, University of Illinois at Urbana-Champaign, USAGoogle Scholar
- Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4):373–395MathSciNetCrossRefzbMATHGoogle Scholar
- Kilian A, Ochsendorf J (2006) Particle spring systems for structural form finding. J Intern Assoc Shell Spatial Struct 46(2):77– 84Google Scholar
- 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
- 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
- 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
- Lewis WJ (2003) Tension structures: form and behaviour. Thomas Telford Publishing, LondonCrossRefGoogle Scholar
- Linkwitz K, Schek H (1971) Einige Bemerkungen zur Berechnung von vorgespannten Seilnetzkonstruktionen. Ingenieur-Archiv 40(3):145–158CrossRefGoogle Scholar
- Martinell C, Collins GR, Rohrer J (1975) Gaudí: his life, his theories, his work. MIT Press, CambridgeGoogle Scholar
- Maxwell JC (1890) In: Niven WD (ed) The scientific papers of James Clerk Maxwell. Library Collection, CambridgeGoogle Scholar
- Michell A (1904) The limits of economy of material in frame structures. Phil Mag 8(47):589–597CrossRefzbMATHGoogle Scholar
- 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
- Mitchell T (2013) A limit of eonomy of material in shell structures. PhD thesis, University of California Berkeley, USAGoogle Scholar
- Nouri-Baranger T (2004) Computational methods for tension-loaded structures. Arch Comput Meth Eng 11 (2003):143– 186CrossRefzbMATHGoogle Scholar
- 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
- 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
- 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
- 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
- Rozvany GIN (2001) On design-dependent constraints and singular topologies. Struct Multidiscip Optim 21 (2):164–172MathSciNetCrossRefGoogle Scholar
- 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
- Rozvany GIN, Prager W (1979) A new class of structural optimization problems: optimal archgrids. Comput Methods Appl Mech Eng 19(1):127–150MathSciNetCrossRefzbMATHGoogle Scholar
- 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
- Rozvany GIN, Wang CM (1983) On plane Prager-structures—I. Int J Mech Sci 25(7):519–527CrossRefzbMATHGoogle Scholar
- 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
- 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
- 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
- Siev A, Eidelman J (1964) Stress analysis of prestressed suspended roofs. J Struct Div 90(4):103–122Google Scholar
- Sokół T (2011) A 99 line code for discretized Michell truss optimization written in Mathematica. Struct Multidiscip Optim 43(2):181–190CrossRefzbMATHGoogle Scholar
- 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
- 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
- Tabarrok B, Qin Z (1992) Nonlinear analysis of tension structures. Comput Struct 45(5):973–984CrossRefzbMATHGoogle Scholar
- 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
- Thrall AP, Billington DP, Bréa KL (2012) The Maria Pia Bridge: A major work of structural art. Eng Struct 40:479–486CrossRefGoogle Scholar
- 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
- 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
- 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
- Wright M (2004) The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull Am Math Soc 42(01):39–56MathSciNetCrossRefzbMATHGoogle Scholar
- 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
- Zalewski W, Allen E (1997) Shaping structures: statics. Wiley, New YorkGoogle Scholar
- Zegard T (2014) Structural optimization: from continuum and ground structures to additive manufacturing. PhD thesis, University of Illinois at Urbana-Champaign, USAGoogle Scholar
- 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
- 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
- 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
- Zhang Y (1996b) Solving large-scale linear programs by interior-point methods under the Matlab Environment. Optim Methods Softw 10(1):1–31Google Scholar