Abstract
Truss topology design for minimum external work (compliance) can be expressed in a number of equivalent potential or complementary energy problem formulations in terms of member forces, displacements and bar areas. Using duality principles and non-smooth analysis we show how displacements only as well as stresses only formulations can be obtained and discuss the implications these formulations have for the construction and implementation of efficient algorithms for large-scale truss topology design. The analysis covers min-max and weighted average multiple load designs with external as well as self-weight loads and extends to the topology design of reinforcement and the topology design of variable thickness sheets and sandwich plates. On the basis of topology design as an inner problem in a hierarchical procedure, the combined geometry and topology design of truss structures is also considered. Numerical results and illustrative examples are presented.
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References
Achtziger, W. 1992: Truss topology design under multiple loadings.DFG-Report (FSP Applied Optimization and Control), No. 367, Universität Bayreuth, FRG
Achtziger, W. 1993: Minimax compliance truss topology subject to multiple loadings. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 43–54. Dordrecht: Kluwer
Achtziger, W.; Bendsøe, M.P.; Ben-Tal, A.; Zowe, J. 1992: Equivalent displacement based formulations for maximum strength truss topology design.IMPACT of Computing in Science and Engineering. 4, 315–345
Allaire, G.; Kohn, R.V. 1993: Topology optimization and optimal shape design using homogenization. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 207–218. Dordrecht: Kluwer
Ben-Tal, A.; Bendsøe, M.P. 1993: A new method for optimal truss topology design.SIAM J. Optimization,3, 322–358
Ben-Tal, A.; Kocvara, M.; Zowe, J. 1993: Two non-smooth methods for simultaneous geometry and topology design of trusses. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 31–42. Dordrecht: Kluwer
Ben-Tal, A.; Nemirovskii, A. 1992: Interior point polynomial-time methods for truss topology design.Research Report 3/92, Optimization Lab., Technion (Israel Inst. of Technology)
Ben-Tal, A.; Nemirovskii, A. 1993: An interior point algorithm for truss topology design. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 55–70. Dordrecht: Kluwer
Ben-Tal, A.; Yuzefovich, I.; Zibulevsky, M. 1992b: Penalty/barrier multiplier methods for minmax and constrained smooth convex problems.Research Report 9/92, Optimization Lab., Technion (Israel Inst. of Technology)
Bendsøe, M.P.; Ben-Tal, A.; Haftka, R.T. 1991: New displacement-based methods for optimal truss topology design.Proc. AIAA/ASME/ASCE/AHS/ASC 32nd. Structures, Structural Dynamics and Materials Conference. (held in Baltimore, MD)
Bendsøe, M.P.; Guedes, J.M.; Haber, R.B.; Pedersen, P.; Taylor, J.E. 1992: An analytical model to predict optimal material properties in the context of optimal structural design.J. Appl. Mech. (to appear)
Bendsøe, M.P.; Kikuchi, N. 1988: Generating optimal topologies in structural design using a homogenization method.Comp. Meth. Appl. Mech. Engrg. 71, 197–224
Bendsøe, M.P.; Mota Soares, C.A. (eds.) 1993:Topology optimization of structures. Dordrecht: Kluwer
Cheng, G.; Jiang, Z. 1991: Study on topology optimization with stress constraints.Report No. 37, Inst. of Mech. Engrg., University of Aalborg, Denmark
Demyanov, V.F.; Malozemov, V.N. 1974:Introduction to minimax. New York: John Wiley and Sons
Diaz, A.; Belding, B. 1991: On optimum truss layout by a homogenization method.ASME Transactions of Mechanical Design. (to appear)
Diaz, A.; Bendsøe, M.P. 1992: Shape optimization of structures for multiple loading conditions using a homogenization method.Struct. Optim. 4, 17–22
Dorn, W.; Gomory, R.; Greenberg, M. 1964: Automatic design of optimal structures.J. de Mecanique. 3, 25–52
Fleron, P. 1964: The minimum weight of trusses.Bygnings statiske Meddelelser 35, 81–96
Fleury, C. 1993: Discrete valued optimal design problems. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 81–88. Dordrecht: Kluwer
Grierson, D.E.; Pak, W.H. 1993: Discrete optimal design using a genetic algorithm. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 89–102. Dordrecht: Kluwer
Hajela, P.; Lee, E.; Lin, C.Y. 1993: Genetic algorithms in structural topology optimization. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 117–134. Dordrecht: Kluwer
Hemp, W.S. 1973:Optimum structures. Oxford: Clarendon Press
Jog, C.; Haber, R.B. Bendsøe, M.P. 1994: Topology design with optimized, self-adaptive materials.Int. J. Num. Meth. Engng. (in press)
Kirsch, U. 1989: Optimal topologies of structures.Appl. Mech. Rev. 42, 223–239
Kirsch, U. 1993: Fundamental properties of optimal topologies. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 3–18. Dordrecht: Kluwer
Kocvara, M.; Zowe, J. 1991: Codes for truss topology design: a numerical comparison. Preprint, Universität Bayreuth
Lewinski, T.; Zhou, M.; Rozvany, G.I.N. 1993: Exact least-weight truss layouts for rectangular domains with various support conditions.Struct. Optim. 6, 65–67
Lewinski, T.; Zhou, M.; Rozvany, G.I.N. 1994: Extended exact solutions for least-weight truss layouts — Part I: cantilever with a horizontal axis of symmetry. Part II: unsymmetric cantilevers.Int. J. Mech. Sci. (proofs returned)
Michell, A.G.M. 1904: The limits of economy of material in frame structures.Phil. Mag. 8, 589–597
Nakamura, T.; Ohsaki, M. 1992: A natural generator of optimum topology of plane trusses for specified fundamental frequency.Comp. Meth. Appl. Mech. Engrg. 94, 113–129
Olhoff, N.; Taylor, J.E. 1983: On structural optimization.J. Appl. Mech. 50, 1134–1151
Pedersen, P. 1970: On the minimum mass layout of trusses.AGARD-CP-36-70
Pedersen, P. 1972: On the optimal layout of multi-purpose trusses.Comp. Struct. 2, 695–712
Pedersen, P. 1973: Optimal joint positions for space trusses.ASCE J. 99, 2459–2476
Pedersen, P. 1993: Topology optimization of three dimensional trusses. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 19–30. Dordrecht: Kluwer
Ringertz, U. 1985: On topology optimization of trusses.Engrg. Optim. 9, 21–36
Ringertz, U. 1986: A branch and bound algorithm for topology optimization of truss structures.Engrg. Optim. 10, 111–124
Ringertz, U. 1988: A mathematical programming approach to structural optimization.Research Report No. 88-24, Department of Leightweight Structures, The Royal Institute of Technology, Stockholm
Ringertz, U. 1992: On finding the optimal distribution of material properties.Struct. Optim. 5, 265–267
Rossow, M.P.; Taylor, J.E. 1973: A finite element method for the optimal design of variable thickness sheets.AIAA J. 11, 1566–1569
Rozvany, G.I.N. 1976:Optimal design of flexural systems. Oxford: Pergamon
Rozvany, G.I.N. 1989:Structural design via optimality criteria. Dordrecht: Kluwer
Rozvany, G.I.N. (ed.) 1992: Shape and lay-out optimization in structural design.CISM Lecture Notes No. 325. Vienna: Springer
Rozvany, G.I.N. 1993: Lay-out theory for grid-type structures. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 251–272. Dordrecht: Kluwer
Saka, M.P. 1980: Shape optimization of trusses.ACSE J. Struct. Engrg. 106, 1155–1174
Sankaranaryanan, S.; Haftka, R.; Kapania, R.K. 1993: Truss topology optimization with stress and displacement constraints. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 71–78. Dordrecht: Kluwer
Schramm, H.; Zowe, J. 1992: A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results.SIAM J. Optim. 1, 121–152
Suzuki, K.; Kikuchi, N. 1991: Shape and topology optimization for generalized layout problems using the homogenization method.Comp. Meth. Appl. Mech. Engrg. 93, 291–318
Svanberg, K. 1984: On local and global minima in structural optimization. In Atrek, A.; Gallagher, R.H.; Ragsdell, K.M.; Zienkiewicz, O.C. (eds.)New directions in optimum structural design, pp. 327–341. New York: Wiley
Svanberg, K. 1992a: On the global convergence of a modified stress ratio method for stress-constrained truss sizing and topology optimization. Preprint, Dept. of Optimization and Systems Theory, The Royal Institute of Technology, Stockholm
Svanberg, K. 1992b: On the global convergence of a modified optimality criteria method for compliance- constrained truss sizing and topology optimization. Preprint, Dept. of Optimization and Systems Theory, The Royal Institute of Technology, Stockholm
Taylor, J.E. 1969: Maximum strength elastic structural design.Proc. ASCE 95, 653–663
Taylor, J.E. 1993: Truss topology design for elastic/softening materials. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 451–468. Dordrecht: Kluwer
Taylor, J.E.; Rossow, M.P. 1977: Optimal truss design based on an algorithm using optimality criteria.Int. J. Solids Struct. 13, 913–923
Topping, B.M.V. 1992: Mathematical programming techniques for shape optimization of skeletal structures. In: Rozvany, G.I.N. (ed.)Shape and layout optimization in structural design, pp. 349–276. Vienna: Springer
Zhou, M.; Rozvany, G.I.N. 1991: The COC Algorithm, part II: topological, geometrical and generalized shape optimization.Comp. Meth. Appl. Mech. Engrg. 89, 309–336
Zhou, M.; Rozvany, G.I.N. 1992/1993: DCOC: an optimality criteria method for large systems. Part I: theory. Part II: algorithm.Struct. Optim. 5, 12–25:6, 250–262
Zibulevsky, M.; Ben-Tal, A. 1993: On a new class of augmented Lagrangian methods for large scale convex programming problems.Research Report 2/93, Optimization Lab., Technion (Israel Inst. of Technology)
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Bendsøe, M.P., Ben-Tal, A. & Zowe, J. Optimization methods for truss geometry and topology design. Structural Optimization 7, 141–159 (1994). https://doi.org/10.1007/BF01742459
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DOI: https://doi.org/10.1007/BF01742459