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Topology optimization in structural mechanics

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EN
Abstrakty
EN
Optimization of structural topology, called briefly: topology optimization, is a relatively new branch of structural optimization. Its aim is to create optimal structures, instead of correcting the dimensions or changing the shapes of initial designs. For being able to create the structure, one should have a possibility to handle the members of zero stiffness or admit the material of singular constitutive properties, i.e. void. In the present paper, four fundamental problems of topology optimization are discussed: Michell’s structures, two-material layout problem in light of the relaxation by homogenization theory, optimal shape design and the free material design. Their features are disclosed by presenting results for selected problems concerning the same feasible domain, boundary conditions and applied loading. This discussion provides a short introduction into current topics of topology optimization.
Rocznik
Strony
23--37
Opis fizyczny
Bibliogr. 109 poz., rys., tab.
Twórcy
autor
autor
autor
  • Department of Structural Mechanics and Computer Aided Engineering, Institute of Building Engineering, Faculty of Civil Engineering, Warsaw University of Technology, 16 Armii Ludowej St., 00-637 Warszawa, Poland
Bibliografia
  • [1] C. Maxwell, “On reciprocal figures, frames and diagrams of forces”, Scientific Papers II 26, 161-207 (1870).
  • [2] A.G.M. Michell, “The limits of economy of material in frame structures”, Phil. Mag. 8 (47), 589-597 (1904).
  • [3] L. Tartar, “An introduction to the homogenization method in optimal design”, in: Optimal Shape Design, ed. B. Kawohl, O. Pironneau, L. Tartar and J.P. Zolesio, pp. 47-156, Springer, Berlin, 2000.
  • [4] A.V. Cherkaev, Variational Methods for Structural Optimization, Springer, New York, 2000.
  • [5] G. Allaire, Shape Optimization by the HomogenizationMethod, Springer, New York, 2002.
  • [6] R. Lipton, “A saddle-point theorem with application to structural optimization”, J. Optim. Theory. Appl. 81 (3), 549-568 (1994).
  • [7] J. Sokołowski and J.P. Zolesio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer-Verlag, Berlin, 1992.
  • [8] Z.Wasiutyński, “On the congruency of the forming according to the minimum potential energy with that according to equal strength”, Bull. Pol. Ac.: Tech. VIII (6), 259-268 (1960).
  • [9] J. Sokołowski and A. Żochowski, “On topological derivative in shape optimization”, SIAM J. Control. Optim. 37 (4), 1251-1272 (1999).
  • [10] H.A. Eschenauer, V.V. Kobelev, and A. Schumacher, “Bubble method for topology and shape optimization of structures”, Struct. Optim. 8 (1), 42-51 (1994).
  • [11] M.P. Bendsoe, J.M. Guedes, R.B. Haber, P. Pedersen, and J.E. Taylor, “An analytical model to predict optimal material properties in the context of optimal structural design”, J. Appl. Mech. Trans. ASME 61 (4), 930-937 (1994).
  • [12] J. Haslinger, M. Koˇcvara, G. Leugering, and M. Stingl, “Multidisciplinary free material optimization”, SIAM J. Appl. Math. 70 (7), 2709-2728 (2010).
  • [13] M. Koˇcvara, M. Stingl, and J. Zowe, “Free material optimization: recent progress”, Optimization 57 (1), 79-100 (2008).
  • [14] S. Czarnecki and T. Lewiński, “The stiffest designs of elastic plates. Vector optimization for two loading conditions”, Comp. Meth. Appl. Mech. Eng. 200 (17-20), 1708-1728 (2011).
  • [15] S. Czarnecki, T. Lewiński, and T. Łukasiak, “Free material optimum design of plates of pre-defined Kelvin moduli”, 9th World Congress on Structural and Multidisciplinary Optimization, CD-ROM (2011).
  • [16] W. Zalewski and A. Allen, Shaping Structures. Statics, J. Wiley, New York, 1998.
  • [17] S. Kuś and W. Zalewski, “Shaping the structures”, Symp. Conceptual Designing-Structural Shaping. Corrugated IronStructures. Cable Structures 1, 11-26 (2000), (in Polish).
  • [18] W. Zalewski, “Strength and lightness - the muses of a structural designer”. Architecture 74 (11), 94-95 (2000), (in Polish).
  • [19] W. Zabłocki, “Optimization of structures and new forms of tall buildings”, Architecture 74 (11), 96-98 (2000), (in Polish).
  • [20] W. Zalewski and W. Zabłocki, “Engineering inspiration of forming tall buildings - Structural shapes of light tall buildings”, Symp. Conceptual Designing-Structural Shaping. CorrugatedIron Structures. Cable Structures 1, 91-106 (2000), (in Polish).
  • [21] H.L. Cox, The Design of Structures of Least Weight, Pergamon Press, Oxford, 1965.
  • [22] W. Hemp, Optimum Structures, Clarendon Press, Oxford, 1973.
  • [23] F. Demengel and P. Suquet, “On locking materials”, ActaAppl. Math. 6 (2), 185-211 (1986).
  • [24] T. Lewiński and J.J. Telega, “Michell-like grillages and structures with locking”, Arch. Mech. 53 (4-5), 457-485 (2001).
  • [25] A. Borkowski, “On dual approach to piecewise-linear elastoplasticity. Part I: Continuum models”, Bull. Pol. Ac.: Tech. 52 (4), 329-343 (2004).
  • [26] A. Borkowski, ”On dual approach to piecewise-linear elastoplasticity. Part II: Discrete models” Bull. Pol. Ac.: Tech. 52 (4), 345-352 (2004).
  • [27] G. Strang and R.V. Kohn, “Hencky-Prandtl nets and constrained Michell trusses”, Comp. Meth. Appl. Mech. Eng. 36 (2), 207-222 (1983).
  • [28] C. Graczykowski and T. Lewiński, “Michell cantilevers constructed within trapezoidal domains - Part I: Geometry of Hencky nets”, Struct. Multidisc. Optim. 32 (5), 347-368 (2006).
  • [29] C. Carathéodory and E. Schmidt, “˝Uber die Hencky- Prandtlschen Kurven”, Zeitschrift f ¨ur Angewandte Mathematikund Mechanik 3 (6), 468-475 (1923).
  • [30] C. Graczykowski and T. Lewiński, “The lightest plane structures of a bounded stress level transmitting a point load to a circular support”, Control and Cybernetics 34 (1), 227-253 (2005).
  • [31] T. Lewiński, M. Zhou, and G.I.N. Rozvany, “Extended exact solutions for least-weight truss layouts - Part I: Cantilever with a horizontal axis of symmetry”, Int. J. Mech. Sci. 36 (5), 375-398 (1994).
  • [32] T. Lewiński, M. Zhou, and G.I.N. Rozvany, “Extended exact solutions for least-weight truss layouts - Part II: Unsymmetric cantilevers”, Int. J. Mech. Sci. 36 (5), 399-419 (1994).
  • [33] C. Graczykowski and T. Lewiński, “Michell cantilevers constructed within trapezoidal domains - Part II: Virtual displacement fields”, Struct. Multidisc. Optim. 32 (6), 463-471 (2006).
  • [34] C. Graczykowski and T. Lewiński, “Michell cantilevers constructed within trapezoidal domains - Part III: Force fields”, Struct. Multidisc. Optim. 33 (1), 27-46 (2007).
  • [35] C. Graczykowski and T. Lewiński, “Michell cantilevers constructed within trapezoidal domains - Part IV: Complete exact solutions of selected optimal designs and their approximations by trusses of finite number of joints”, Struct. Multidisc. Optim. 33 (2), 113-129 (2007).
  • [36] C. Graczykowski and T. Lewiński, “Michell cantilevers constructed within a halfstrip. Tabulation of selected benchmark results”, Struct. Multidisc. Optim. 42 (6), 869-877 (2010).
  • [37] T. Lewiński and G.I.N. Rozvany, “Exact analytical solutions for some popular benchmark problems in topology optimization II: three - sided polygonal supports”, Struct. Multidisc. Optim. 33 (4-5), 337-349 (2007).
  • [38] T. Lewiński and G.I.N. Rozvany, “Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains”, Struct. Multidisc. Optim. 35 (2), 165-174 (2008).
  • [39] T. Lewiński and G.I.N. Rozvany, “Analytical benchmarks for topological optimization IV: square-shaped line support”, Struct. Multidisc. Optim. 36 (2), 143-158 (2008).
  • [40] T. Sokoł and T. Lewiński, “On the solution of the three forces problem and its application to optimal designing of a class of symmetric plane frameworks of least weight”, Struct. Multidisc. Optim. 42 (6), 835-853 (2010).
  • [41] G.I.N. Rozvany, “Some shortcomings in Michell’s truss theory”, Struct. Optim. 12 (4), 244-250 (1996); and Struct. Optim. 13 (2-3), 203-204 (1997).
  • [42] W. Dorn, R. Gomory, and M. Greenberg, “Automatic design of optimal structures”, J. de Mecanique 3, 25-52 (1964).
  • [43] W. Achtziger, “Topology optimization of discrete structures: an introduction in view of computational and nonsmooth aspects”, in: Topology Optimization in Structural Mechanics,CISM Courses and Lectures 374, ed. G.I.N. Rozvany, pp. 57-100, Springer, New York, 1997.
  • [44] T.J. Pritchard, M. Gilbert, and A. Tyas, “Plastic layout optimization of large-scale frameworks subject to multiple load cases, member self-weight and with joint length penalties”, 6th World Congress of Structural and Multidisciplinary Optimization, CD-ROM (2005).
  • [45] M. Gilbert and A. Tyas, “Layout optimization of largescale pin-jointed frames”, Eng. Comput. 20 (7-8), 1044-1064 (2003).
  • [46] A. Tyas, A.V. Pichugin, and M. Gilbert, “Optimum structure to carry a uniform load between pinned supports: exact analytical solution”, Proc. Roy. Soc. A 467 (2128), 1101-1120 (2011).
  • [47] T. Sokoł, “A 99 line code for discretized Michell truss optimization written in Mathematica”, Struct. Multidisc. Optim. 43 (2), 181-190 (2011).
  • [48] http://link.springer.com/content/esm/art:10.1007/s00158-010-0557-z/MediaObjects/ 158 2010 557 MOESM1 ESM.pdf. (Electronic Supplementary Material with Complete Code ofProgram by Sokoł [47]).
  • [49] R.E. McConnel, “Least-weight frameworks for loads across span”, J. Eng. Mech. Div. 100 (5), 885-901 (1974).
  • [50] T. Lewiński, G.I.N. Rozvany, T. Sokoł, and K. Bołbotowski, “Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains revisited”, Struct. Multidisc. Optimiz. (2013), (to be published).
  • [51] A.V. Pichugin, A. Tyas, and M. Gilbert, “Michell structure for a uniform load over multiple spans”, 9th World Congresson Structural and Multidisciplinary Optimization, CD-ROM (2011).
  • [52] H.S.Y. Chan, “Symmetric plane frameworks of least weight”, in: Optimization in Structural Design, ed. A. Sawczuk, Z. Mroz, pp. 313-326, Springer, Berlin, 1975.
  • [53] A.V. Pichugin, A. Tyas, and M. Gilbert, “On the optimality of Hemp’s arch with vertical hangers”, Struct. Multidisc. Optimiz. 46 (1), 17-25 (2012).
  • [54] T. Sokoł and G.I.N. Rozvany, “New analytical benchmarks for topology optimization and their implications. Part I: bisymmetric trusses with two point loads between supports”, Struct. Multidisc. Optim. 46 (4), 477-486 (2012).
  • [55] T. Sokoł and T. Lewiński, “Optimal design of a class of symmetric plane frameworks of least weight”, Struct. Multidisc. Optim. 44 (5), 729-734 (2011).
  • [56] T. Sokoł and T. Lewiński, “On the three forces problem in truss topology optimization. Analytical and numerical solutions”, 9th World Congress on Structural and MultidisciplinaryOptimization, Book of Abstracts 1, 76 (2011).
  • [57] G.I.N. Rozvany and T. Sokoł, “Exact truss topology optimization: allowance for support costs and different permissible stresses in tension and compression - Extensions of a classical solution by Michell”, Struct. Multidisc. Optimiz. 45 (3), 367-376 (2012).
  • [58] T. Lewiński, “Michell structures formed on surfaces of revolution”, Struct. Multidisc. Optim. 28 (1), 20-30 (2004).
  • [59] D. Bojczuk, “Sensitivity analysis and optimization of bar structures”, Monographs, Studies, Treatises, Kielce University of Technology, Kielce, 1999 (in Polish).
  • [60] W. Prager, “Optimal layout of trusses of finite number of joints”, J. Mech. Phys. Solids 26 (4), 241-250 (1978).
  • [61] W. Prager, “Nearly optimal design of trusses”, Comp. Struct. 8 (3-4), 451-454 (1978).
  • [62] A. Mazurek, W. Baker, and C. Tort, “Geometrical aspects of optimum truss like structures”, Struct. Multidisc. Optim. 43 (2), 231-242 (2011).
  • [63] Z. Rychter and A. Musiuk, “Topological sensitivity to diagonal member flips of two-layered statically determinate trusses under worst loading”, Int. J. Solids. Struct. 44 (14-15), 4942-4957 (2007).
  • [64] S. Czarnecki, “Compliance optimization of the truss structures”, Comp. Assist. Mech. Engrg. Sci. 10 (2), 117-137 (2003).
  • [65] S. Czarnecki, “Minimization of compliance of frame structures. Application of symbolic programme Maple to the analysis of sensitivity”, in: Theoretical Foundations of CivilEngineering, Polish-Ukrainian Transactions, ed. W. Szcześniak, vol. 12 (1), pp. 49-64, Warsaw University of Technology, Warsaw-Dnepropetrovsk, 2004, (in Polish).
  • [66] S. Czarnecki, “Application of the moving asymptotes method in the optimization of topology and geometry of spatial frames”, 50th Jubilee Scientific Conf. Commitee on Civil andHydroengineering and Scientific Commitee of Polish Associationof Civil Engineers and Technicians II, 45-52 (2004), (in Polish).
  • [67] S. Czarnecki, “Form finding of tulip-like space structures”, in: Int. IASS Symposium on Lightweight Structures in CivilEngineering, ed. J.B. Obrębski, pp. 119-128, J.B. Obrębski Publishing House, Warsaw, Poland, 2002.
  • [68] S. Czarnecki, “Application of the Strongin-Sergeyev global optimization method in the compliance minimization of latticed shells”, Comp. Assist. Mech. Eng. Sci. 16 (3/4), 291-307 (2009).
  • [69] G.I.N. Rozvany and C.M. Wang, “On plane Prager-structures- I”, Int. J. Mech. Sci. 25 (7), 519-527 (1983).
  • [70] C.M. Wang and G.I.N. Rozvany, “On plane Prager-structures- II. Non- parallel external loads and allowances for selfweight”, Int. J. Mech. Sci. 25 (7), 529-541 (1983).
  • [71] K. Hetmański and T. Lewiński, “Shaping the plane frames and arches for avoiding bending”, in: Theoretical Foundationsof Civil Engineering - XV, Proc. Polish-Ukrainian-LithuanianTransactions, ed. W. Szcześniak, pp. 231-246, Warsaw University of Technology Publishing House, Warszawa, 2007, (in Polish).
  • [72] G.I.N. Rozvany, Optimal Design of Flexural Systems, Pergamon Press, London, 1976.
  • [73] G.I.N. Rozvany, Structural Design via Optimality Criteria, Kluwer Academic Publishers, Dordrecht, 1989.
  • [74] O. Amir and O. Sigmund, “Reinforcement layout design for concrete structures based on continuum damage and truss topology optimization”, Struct. Multidisc. Optimiz. 47 (2), 157-174 (2013).
  • [75] http://www.som.com/project/transbay-transit-center-designcompetition.
  • [76] http://www.som.com/project/tianjin-high-speed-rail-station.
  • [77] T. Lewiński and J.J. Telega, Plates, Laminates and Shells. Asymptotic Analysis and Homogenization, World Scientific Publishing, London, 2000.
  • [78] T. Lewiński, “Homogenization and optimal design in structural mechanics”, in: Nonlinear homogenization and its appli- cation to composites, polycrystals and smart materials, eds. P.P. Casta˜neda, J.J. Telega and B. Gambin, pp. 139-168, NATO Science Series II, Mathematics, Physics and Chemistry 170, Dordrecht, 2004.
  • [79] G. Dzierżanowski, Optimum Layout of Materials Within ThinElastic Plates, Warsaw University of Technology Publishing House, Warszawa, 2010 (in Polish).
  • [80] G. Dzierżanowski, “Bounds on the effective isotropic moduli of thin elastic composite plates”, Arch. Mech. 62 (4), 253-281 (2010).
  • [81] M.P. Bendsoe and O. Sigmund, “Material interpolation schemes in topology optimization”, Arch. Appl. Mech. 69 (9-10), 635-654 (1999).
  • [82] M.P. Bendsoe, Optimization of Structural Topology, Shapeand Material, Springer, Berlin, 1995.
  • [83] M.P. Bendsoe and O. Sigmund, Topology Optimization, Theory,Methods and Applications, Springer, Berlin, 2003.
  • [84] M.P. Bendsoe, “Optimal shape design as a material distribution problem”, Struct. Optim. 1 (4), 193-202 (1989).
  • [85] M. Zhou and G.I.N. Rozvany, “The COC algorithm, Part II: Topological, geometrical and generalized shape optimization”, Comp. Meth. Appl. Mech. Eng. 89 (1-3), 309-336 (1991).
  • [86] G. Dzierżanowski, “On the comparison of material interpolation schemes and optimal composite properties in plane shape optimization”, Struct. Multidisc. Optim. 46 (5), 693-710 (2012).
  • [87] M. Stolpe and K. Svanberg, “An alternative interpolation scheme for minimum compliance topology optimization”, Struct. Multidisc. Optim. 22 (2), 116-124 (2001).
  • [88] O. Sigmund and J. Petersson, “Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima”, Struct. Optim. 16 (1), 68-75 (1998).
  • [89] B. Bourdin, “Filters in topology optimization”, Int. J. Numer. Meth. Eng. 50 (9), 2143-2158 (2001).
  • [90] R. Kutyłowski, “On an effective topology procedure”, Struct. Multidisc. Optim. 20 (1), 49-56 (2000).
  • [91] K. Kolanek and T. Lewiński, “Circular and annular two-phase plates of minimal compliance”, Comp. Assist. Mech. Eng. Sci. 10 (2), 177-199 (2003).
  • [92] S. Czarnecki, M. Kursa, and T. Lewiński, “Sandwich plates of minimal compliance”, Comp. Meth. Appl. Mech. Eng. 197 (51-52), 4866-4881 (2008).
  • [93] R. Studziński, Z. Pozorski, and A. Garstecki, “Optimal design of sandwich panels with a soft core”, J. Theor. Appl. Mech. 47 (3), 685-699 (2009).
  • [94] A.R. Díaz, R. Lipton and C.A. Soto, “A new formulation of the problem of optimum reinforcement of Reissner-Mindlin plates”, Comp. Meth. Appl. Mech. Eng. 123 (1-4), 121-139 (1995).
  • [95] G. Dzierżanowski, “Stress energy minimization as a tool in the material layout design of shallow shells”, Int. J. SolidsStruct. 49 (11-12), 1343-1354 (2012).
  • [96] L.A. Krog and N. Olhoff, “Topology and reinforcement layout optimization of disk, plate, and shell structures”, in: TopologyOptimization in Structural Mechanics, ed. G.I.N. Rozvany, pp. 237-322, Springer, Wien, 1997.
  • [97] G. Allaire, E. Bonnetier, G. Francfort, and F. Jouve, “Shape optimisation by the homogenisation method”, Numer. Math. 76 (1), 27-68 (1997).
  • [98] T. Borrvall and J. Petersson, “Large-scale topology optimization in 3D using parallel computing”, Comp. Meth. Appl. Mech. Eng. 190 (46-47), 6201-6229 (2001).
  • [99] S. Czarnecki and T. Lewiński, “Shaping the stiffest threedimensional structures from two given isotropic materials”, Comp. Assist. Mech. Eng. Sci. 13 (1), 53-83 (2006).
  • [100] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin 1984.
  • [101] M. Nowak, “Structural optimization system based on trabecular bone surface adaptation”, Struct. Multidisc. Optim. 32 (3), 241-249 (2006).
  • [102] T. Lewiński and J. Sokołowski, “Energy change due to the appearance of cavities in elastic solids”, Int. J. Solids Struct. 40 (7), 1765-1803 (2003).
  • [103] S. Czarnecki and T. Lewiński, “A stress-based formulation of the free material design problem with the trace constraint and one loading condition”, Bull. Pol. Ac.: Tech. 60 (2), 191-204 (2012).
  • [104] G. Dzierżanowski and T. Lewiński, “Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part I. Solving the local optimization problem”, Arch. Mech. 64 (1), 21-40 (2012); erratum: Arch. Mech. 64 (2), 121 (2012).
  • [105] G. Dzierżanowski and T. Lewiński, “Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part II. The effective boundary value problem and exemplary solutions”, Arch. Mech. 64 (2), 111-135 (2012).
  • [106] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C. The Art of Scientific Computing, Cambridge University Press, Cambridge, 1992.
  • [107] A. Długosz and T. Burczyński, “Multiobjective shape optimization of selected coupled problems by means of evolutionary algorithms”, Bull. Pol. Ac.: Tech. 60 (2), 215-222 (2012).
  • [108] M. Mrzygłod, “Multi-constrained topology optimization using constant criterion surface algorithm”, Bull. Pol. Ac.: Tech. 60 (2), 229-236 (2012).
  • [109] M. Szczepanik and T. Burczyński, “Swarm optimization of stiffeners locations in 2-D structures”, Bull. Pol. Ac.: Tech. 60 (2), 241-246 (2012).
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