Abstract
A new method of simultaneous optimization of geometry and topology is presented for plane and spatial trusses. Compliance under single loading condition is minimized for specified structural volume. The difficulties due to existence of melting nodes are successfully avoided by considering force density, which is the ratio of axial force to the member length, as design variable. By using the fact that the optimal truss is statically determinate with the same absolute value of stress in existing members, the compliance and structural volume are expressed as explicit functions of force density only. After obtaining optimal cross-sectional area, nodal locations, and topology, the cross-sectional areas and nodal coordinates are further optimized using a conventional method of nonlinear programming. Accuracy of the optimal solution is verified through examples of plane trusses and a spatial truss. It is shown that various nearly optimal solutions can be found using the proposed method.
Similar content being viewed by others
References
Achtziger W (2007) On simultaneous optimization of truss geometry and topology. Struct Multidisc Optim 33:285–304
Achtziger W, Bendsøe MP, Ben-Tal A, Zowe J (1992) Equivalent displacement based formulations for maximum strength truss topology design. Impact Comput Sci Eng 4(4):315–345
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer
Descamps B, Coelho RF (2014) The nominal force method for truss geometry optimization incorporating stability considerations. Int J Solids Struct 51:2390–2399
Dobbs W, Felton LP (1969) Optimization of truss geometry. J Struct Div ASCE 95(ST10):2105–2119
Dorn WR, Gomory R, Greenberg H (1964) Automatic design of optimal structures. J de Mecanique 3:25–52
Gill PE, Murray W, Saunders MA (2002) SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J Opt 12:979–1006
Guo X, Liu W, Li H (2003) Simultaneous shape and topology optimization of truss under local and global stability constraints. Acta Mechanica Solida Sinca 16(2):95–101
Hagishita T, Ohsaki M (2009) Topology optimization of trusses by growing ground structure approach. Struct Multidisc Optim 37(4):377–393
Hemp WS (1973) Optimum structures. Clarendon Press, Oxford
Kanno Y, Ohsaki M (2003) Minimum principle of complementary energy of cable networks by using second-order cone programming. Int J Solids Struct 40(17):4437–4460
Luh G-C, Lin C-Y (2008) Optimal design of truss structures using ant algorithm. Struct Multidisc Optim 36:365–379
McKeown JJ (1998) Growing optimal pin-jointed frames. Struct Opt 15:92–100
Ohsaki M (1998) Simultaneous optimization of topology and geometry of a regular plane truss. Comput Struct 66(1):69–77
Ohsaki M (2010) Optimization of finite dimensional structures. CRC Press
Ohsaki M, Arora JS (1993) A direct application of higher order parametric programming techniques to structural optimization. Int J Numer Meth Engng 36:2683–2702
Rojas-Labanda S, Stolpe M (2015) Benchmarking optimization solvers for structural topology optimization. Struct Multidisc Optim 52:527–547
Schek H-J (1974) The force density method for form finding and computation of general networks. Comput Methods Appl Mech Eng 3:115–134
Stolpe M (2007) On the reformulation of topology optimization problems as linear or convex quadratic mixed 0-1 programs. Opt Eng 8:163–192
Tibert AG, Pellegrino S (2011) Review of form-finding methods for tensegrity structures. Int J Space Structures 26(3):241–255
Yang Y, Soh CK (2002) Automated optimum design of structures using genetic programming. Comp Struct 80:1537–1546
Zhang JY, Ohsaki M (2015) Tensegrity structures: form, stability, and symmetry. Mathematics for Industry 6. Springer
Zhang JY, Ohsaki M (2006) Adaptive force density method for form-finding problem of tensegrity structures. Int J Solids Struct 43(18–19):5658–5673
Acknowledgments
This work is partially supported by JSPS KAKENHI No. 16H03014.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ohsaki, M., Hayashi, K. Force density method for simultaneous optimization of geometry and topology of trusses. Struct Multidisc Optim 56, 1157–1168 (2017). https://doi.org/10.1007/s00158-017-1710-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-017-1710-8