Efficient size and shape optimization of truss structures subject to stress and local buckling constraints using sequential linear programming
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 wh...
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| Published in | Structural and multidisciplinary optimization Vol. 58; no. 1; pp. 171 - 184 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.07.2018
Springer Nature B.V |
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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. |
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| AbstractList | 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. |
| Author | Schwarz, Jonas Shea, Kristina Stanković, Tino Chen, Tian |
| Author_xml | – sequence: 1 givenname: Jonas surname: Schwarz fullname: Schwarz, Jonas email: schwarjo@ethz.ch organization: Engineering Design and Computing Laboratory, Department of Mechanical and Process Engineering, ETH Zurich – sequence: 2 givenname: Tian surname: Chen fullname: Chen, Tian organization: Engineering Design and Computing Laboratory, Department of Mechanical and Process Engineering, ETH Zurich – sequence: 3 givenname: Kristina surname: Shea fullname: Shea, Kristina organization: Engineering Design and Computing Laboratory, Department of Mechanical and Process Engineering, ETH Zurich – sequence: 4 givenname: Tino surname: Stanković fullname: Stanković, Tino organization: Engineering Design and Computing Laboratory, Department of Mechanical and Process Engineering, ETH Zurich |
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| Cites_doi | 10.1007/s10107-004-0559-y 10.1007/BF02579150 10.1016/S0045-7949(02)00443-1 10.1080/17452759.2016.1240208 10.1137/1.9781611970791 10.1007/978-3-7091-2566-3_2 10.1016/S0045-7949(02)00442-X 10.1016/0899-8248(92)90005-S 10.1061/(ASCE)0733-9445(1983)109:8(1933) 10.1007/s00158-003-0294-7 10.1137/0803015 10.1080/0951192X.2011.650880 10.1007/BF01206999 10.2514/3.10367 10.1016/S0045-7949(99)00185-6 10.2514/3.25147 10.1007/s00158-002-0177-3 10.1007/BF01742459 10.1007/s00158-006-0092-0 10.1007/978-94-011-1804-0_3 10.1007/s00158-015-1260-x 10.1007/978-1-84800-155-8_7 10.1016/0020-7403(94)90043-4 10.1007/s00158-016-1426-1 10.1016/j.compstruc.2005.09.032 10.1080/03052157408960577 10.1007/BF01207000 10.1108/02644400310503017 10.1007/BF01742498 |
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| Keywords | Local buckling Linear programming Sequential linear programming Truss structures Shape optimization |
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Accessed 1 Dec 2016 KocvaraMZoweJHow mathematics can help in design of mechanical structuresPitman Research Notes in Mathematics Series19961176930858.73059 Schmit L (1960) Structural design by systematic synthesis. In: Procee– dings of 2nd asce conference electronic computation. ASCE, New York KarmarkarNA new polynomial-time algorithm for linear programmingCombinatorica1984437339577990010.1007/BF025791500557.90065 Lamberti L, Pappalettere C (2003a) Move limits definition in structural optimization with sequential linear programming. part i: optimization algorithm. Comput Struct 81(4):197–213 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 31–42 FleronPMinimum weight of trussesBygningsstatiske Meddelelser196435381 WächterABieglerLTOn the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programmingMath Program200610612557219561610.1007/s10107-004-0559-y1134.90542 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 1218 Achtziger W (1999b) Local stability of trusses in the context of topology optimization Part II: a numerical approach. Structural Optimization 17(4):247–258 Achtziger W (1997) Topology optimization of discrete structures. In: Topology optimization in structural mechanics. 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| References_xml | – reference: MajidKTangXOptimum design of pin–jointed space structures with variable shapeStruct Eng1984623137 – reference: WächterABieglerLTOn the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programmingMath Program200610612557219561610.1007/s10107-004-0559-y1134.90542 – reference: Achtziger W (1999a) Local stability of trusses in the context of topology optimization Part I: exact modelling. Structural Optimization 17(4):235–246 – reference: RosenDWA review of synthesis methods for additive manufacturingVirtual and Physical Prototyping201611430531710.1080/17452759.2016.1240208 – reference: SmithCJGilbertMToddIDergutiFApplication of layout optimization to the design of additively manufactured metallic componentsStruct Multidiscip Optim201654512971313357117410.1007/s00158-016-1426-1 – reference: ChangPSRosenDWThe size matching and scaling method: a synthesis method for the design of mesoscale cellular structuresInt J Comput Integr Manuf2013261090792710.1080/0951192X.2011.650880 – reference: LewińskiTZhouMRozvanyGExtended exact solutions for least-weight truss layouts—part i: cantilever with a horizontal axis of symmetryInt J Mech Sci199436537539810.1016/0020-7403(94)90043-40805.73046 – reference: KarmarkarNA new polynomial-time algorithm for linear programmingCombinatorica1984437339577990010.1007/BF025791500557.90065 – reference: Lamberti L, Pappalettere C (2003b) Move limits definition in structural optimization with sequential linear programming. part ii: numerical examples. Comput Struct 81(4):215–238 – reference: AchtzigerWBendsøeMBen-TalAZoweJEquivalent displacement based formulations for maximum strength truss topology designIMPACT of Computing in Science and Engineering199244315345119635410.1016/0899-8248(92)90005-S0769.73054 – reference: Grant M, Boyd S (2014) CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx. Accessed 1 May 2017 – reference: 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 – reference: LambertiLPappalettereCComparison of the numerical efficiency of different sequential linear programming based algorithms for structural optimisation problemsComput Struct200076671372810.1016/S0045-7949(99)00185-6 – reference: Achtziger W (1997) Topology optimization of discrete structures. In: Topology optimization in structural mechanics. Springer, pp 57–100 – reference: 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 19–24 – reference: ToppingBShape optimization of skeletal structures: a reviewJ Struct Eng198310981933195110.1061/(ASCE)0733-9445(1983)109:8(1933) – reference: Schmit L (1960) Structural design by systematic synthesis. In: Procee– dings of 2nd asce conference electronic computation. ASCE, New York – reference: Freund R (2004) Truss design and convex optimization. MIT Course notes, Massachusetts Institute of Technology – reference: Haftka RT, Gürdal Z (2012) Elements of structural optimization, vol 11. Springer Science & Business Media – reference: NesterovYNemirovskiiAInterior-point polynomial algorithms in convex programming1994PhiladelphiaSIAM10.1137/1.97816119707910824.90112 – reference: FleronPMinimum weight of trussesBygningsstatiske Meddelelser196435381 – reference: KočvaraMOn the modelling and solving of the truss design problem with global stability constraintsStruct Multidiscip Optim200223318920310.1007/s00158-002-0177-3 – reference: 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 135–140 – reference: Lamberti L, Pappalettere C (2003a) Move limits definition in structural optimization with sequential linear programming. part i: optimization algorithm. Comput Struct 81(4):197–213 – reference: 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 95-110 – reference: HansenSVanderplaatsGApproximation method for configuration optimization of trussesAIAA journal199028116116810.2514/3.10367 – reference: PedersenNLNielsenAKOptimization of practical trusses with constraints on eigenfrequencies, displacements, stresses, and bucklingStruct Multidiscip Optim200325543644510.1007/s00158-003-0294-7 – reference: AchtzigerWOn simultaneous optimization of truss geometry and topologyStruct Multidiscip Optim2007334–5285304231058710.1007/s00158-006-0092-01245.90112 – reference: Ben-TalABendsøeMPA new method for optimal truss topology designSIAM J Optim199332322358121544710.1137/08030150780.90076 – reference: SchittkowskiKZilloberCZotemantelRNumerical comparison of nonlinear programming algorithms for structural optimizationStructural Optimization19947111910.1007/BF01742498 – reference: KocvaraMZoweJHow mathematics can help in design of mechanical structuresPitman Research Notes in Mathematics Series19961176930858.73059 – reference: TyasAGilbertMPritchardTPractical plastic layout optimization of trusses incorporating stability considerationsComput Struct2006843–411512610.1016/j.compstruc.2005.09.032 – reference: Ben-Tal A, Kočvara M, Zowe J (1993) Two nonsmooth approaches to simultaneous geometry and topology design of trusses. 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| SubjectTerms | Computational Mathematics and Numerical Analysis Computing time Engineering Engineering Design Euler buckling Iterative methods Linear programming Post-processing Research Paper Shape optimization Taylor series Theoretical and Applied Mechanics Trusses |
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| Title | Efficient size and shape optimization of truss structures subject to stress and local buckling constraints using sequential linear programming |
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