# Efficient size and shape optimization of truss structures subject to stress and local buckling constraints using sequential linear programming

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## Abstract

The advance in digital fabrication technologies and additive manufacturing allows for the fabrication of complex truss structure designs but at the same time posing challenging structural optimization problems to capitalize on this new design freedom. In response to this, an iterative approach in which Sequential Linear Programming (SLP) is used to simultaneously solve a size and shape optimization sub-problem subject to local stress and Euler buckling constraints is proposed in this work. To accomplish this, a first order Taylor expansion for the nodal movement and the buckling constraint is derived to conform to the SLP problem formulation. At each iteration a post-processing step is initiated to map a design vector to the exact buckling constraint boundary in order to facilitate the overall efficiency. The method is verified against an exact non-linear optimization problem formulation on a range of benchmark examples obtained from the literature. The results show that the proposed method produces optimized designs that are either close or identical to the solutions obtained by the non-linear problem formulation while significantly decreasing the computational time. This enables more efficient size and shape optimization of truss structures considering practical engineering constraints.

## Keywords

Truss structures Linear programming Sequential linear programming Shape optimization Local buckling## References

- Achtziger W (1997) Topology optimization of discrete structures. In: Topology optimization in structural mechanics. Springer, pp 57–100Google Scholar
- Achtziger W (1999a) Local stability of trusses in the context of topology optimization Part I: exact modelling. Structural Optimization 17(4):235–246Google Scholar
- Achtziger W (1999b) Local stability of trusses in the context of topology optimization Part II: a numerical approach. Structural Optimization 17(4):247–258Google Scholar
- Achtziger W (2007) On simultaneous optimization of truss geometry and topology. Struct Multidiscip Optim 33(4–5):285–304MathSciNetCrossRefzbMATHGoogle Scholar
- Achtziger W, Bendsøe M, Ben-Tal A, Zowe J (1992) Equivalent displacement based formulations for maximum strength truss topology design. IMPACT of Computing in Science and Engineering 4(4):315–345MathSciNetCrossRefzbMATHGoogle Scholar
- ApS M (2015) The MOSEK optimization toolbox for MATLAB manual. Version 7.1 (Revision 28). http://docs.mosek.com/7.1/toolbox/index.html. Accessed 1 Dec 2016
- Ben-Tal A, Bendsøe MP (1993) A new method for optimal truss topology design. SIAM J Optim 3 (2):322–358MathSciNetCrossRefzbMATHGoogle Scholar
- Ben-Tal A, Kočvara M, Zowe J (1993) Two nonsmooth approaches to simultaneous geometry and topology design of trusses. In: Topology design of structures. Springer, pp 31–42Google Scholar
- Bendsøe MP, Ben-Tal A, Zowe J (1994) Optimization methods for truss geometry and topology design. Structural Optimization 7(3): 141–159CrossRefGoogle Scholar
- Chang PS, Rosen DW (2013) The size matching and scaling method: a synthesis method for the design of mesoscale cellular structures. Int J Comput Integr Manuf 26(10):907–927CrossRefGoogle Scholar
- Dorn WS (1964) Automatic design of optimal structures. Journal de mecanique 3:25–52Google Scholar
- Fleron P (1964) Minimum weight of trusses. Bygningsstatiske Meddelelser 35(3):81Google Scholar
- Freund R (2004) Truss design and convex optimization. MIT Course notes, Massachusetts Institute of TechnologyGoogle Scholar
- Gilbert M, Tyas A (2003) Layout optimization of large–scale pin–jointed frames. Eng Comput 20(8):1044–1064CrossRefzbMATHGoogle Scholar
- Grant M, Boyd S (2008) Graph implementations for nonsmooth convex programs. In: Blondel V, Boyd S, Kimura H (eds) Recent advances in learning and control, lecture notes in control and information sciences. Springer, pp 95-110Google Scholar
- Grant M, Boyd S (2014) CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx. Accessed 1 May 2017
- Haftka RT, Gürdal Z (2012) Elements of structural optimization, vol 11. Springer Science & Business MediaGoogle Scholar
- Hansen S, Vanderplaats G (1990) Approximation method for configuration optimization of trusses. AIAA journal 28(1):161–168CrossRefGoogle Scholar
- He L, Gilbert M (2015) Rationalization of trusses generated via layout optimization. Struct Multidiscip Optim 52(4):677–694MathSciNetCrossRefGoogle Scholar
- Hemp W (1974) Michell framework for uniform load between fixed supports. Eng Optim 1(1):61–69CrossRefGoogle Scholar
- Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4:373–395MathSciNetCrossRefzbMATHGoogle Scholar
- Kočvara M (2002) On the modelling and solving of the truss design problem with global stability constraints. Struct Multidiscip Optim 23(3):189–203CrossRefGoogle Scholar
- Kocvara M, Zowe J (1995) How to optimize mechanical structures simultaneously with respect to topology and geometry. In: Olhoff N, Rozvany GIN (eds) Proceedings of the first world congress of structural and multidisciplinary optimization, vol 23, pp 135–140Google Scholar
- Kocvara M, Zowe J (1996) How mathematics can help in design of mechanical structures. Pitman Research Notes in Mathematics Series 11:76–93zbMATHGoogle Scholar
- Lamberti L, Pappalettere C (2000) Comparison of the numerical efficiency of different sequential linear programming based algorithms for structural optimisation problems. Comput Struct 76(6):713–728CrossRefGoogle Scholar
- Lamberti L, Pappalettere C (2003a) Move limits definition in structural optimization with sequential linear programming. part i: optimization algorithm. Comput Struct 81(4):197–213Google Scholar
- Lamberti L, Pappalettere C (2003b) Move limits definition in structural optimization with sequential linear programming. part ii: numerical examples. Comput Struct 81(4):215–238Google Scholar
- Lewiński T, Zhou M, Rozvany G (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
- Majid K, Tang X (1984) Optimum design of pin–jointed space structures with variable shape. Struct Eng 62:31–37Google Scholar
- Nesterov Y, Nemirovskii A (1994) Interior-point polynomial algorithms in convex programming. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
- Pedersen NL, Nielsen AK (2003) Optimization of practical trusses with constraints on eigenfrequencies, displacements, stresses, and buckling. Struct Multidiscip Optim 25(5):436–445CrossRefGoogle Scholar
- Rosen DW (2016) A review of synthesis methods for additive manufacturing. Virtual and Physical Prototyping 11(4):305–317CrossRefGoogle Scholar
- Schittkowski K, Zillober C, Zotemantel R (1994) Numerical comparison of nonlinear programming algorithms for structural optimization. Structural Optimization 7(1):1–19CrossRefGoogle Scholar
- Schmit L (1960) Structural design by systematic synthesis. In: Procee– dings of 2nd asce conference electronic computation. ASCE, New YorkGoogle Scholar
- Smith CJ, Gilbert M, Todd I, Derguti F (2016) Application of layout optimization to the design of additively manufactured metallic components. Struct Multidiscip Optim 54(5):1297–1313MathSciNetCrossRefGoogle Scholar
- Sokół T, Rozvany G (2013) On the adaptive ground structure approach for multi-load truss topology optimization. In: Tenth world congress on structural and multidisciplinary optimization, pp 19–24Google Scholar
- Topping B (1983) Shape optimization of skeletal structures: a review. J Struct Eng 109(8):1933–1951CrossRefGoogle Scholar
- Tyas A, Gilbert M, Pritchard T (2006) Practical plastic layout optimization of trusses incorporating stability considerations. Comput Struct 84(3–4):115–126CrossRefGoogle Scholar
- Vanderplaats G, Kodiyalam S, Long M (1990) A two-level approximation method for stress constraints in structural optimization. In: Thirtieth structures, structural dynamics and materials conference, p 1218Google Scholar
- Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57MathSciNetCrossRefzbMATHGoogle Scholar