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A critical review of established methods of structural topology optimization. (English) Zbl 1274.74004
Summary: The aim of this article is to evaluate and compare established numerical methods of structural topology optimization that have reached the stage of application in industrial software. It is hoped that our text will spark off a fruitful and constructive debate on this important topic.

MSC:
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74P10 Optimization of other properties in solid mechanics
74P15 Topological methods for optimization problems in solid mechanics
Software:
CONLIN; top.m; ESOFRAME
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[1] Allaire G (2002) Shape optimization by the homogenization method. Springer, New York · Zbl 0990.35001
[2] Allaire G, Kohn RV (1993a) Explicit bounds on elastic energy of a two-phase composite in two space dimensions. Q Appl Math 51:675–699 · Zbl 0805.73042
[3] Allaire G, Kohn RV (1993b) Topology optimization and optimal shape design using homogenization. Proceedings of NATO ARW Topology Design of Structures (Sesimbra 1992). Kluwer, Dordrecht, pp 207–218
[4] Allaire G, Jouve F, Toader AM (2002) A level set method for shape optimization. Comptes Rendus de l’Acad. des Sci 334:1125–1130 · Zbl 1115.49306
[5] Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comp Phys 104:363–393 · Zbl 1136.74368
[6] Bendsoe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202
[7] Bendsoe MP (1995) Optimization of structural topology, shape and material. Springer, Berlin · Zbl 0822.73001
[8] Bendsoe MP, Haber RB (1993) The Michell layout problem as a low volume fraction limit of the perforated plate topology optimization problem: an asymptotic study. Struct Optim 6:263–267
[9] Bendsoe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224 · Zbl 0671.73065
[10] Bendsoe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654 · Zbl 0957.74037
[11] Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin
[12] Bendsoe MP, Diaz A, Kikuchi N (1993) Topology and generalized layout optimization of elastic structures. Proceedings of NATO ARW Topology Design of Structures (Sesimbra 1992). Kluwer, Dordrecht, pp 207–218
[13] Bruns TE, Tortorelli DA (2003) An element removal and reintroduction strategy for topology optimization of structures and compliant mechanisms. Int J Numer Methods Eng 57:1413–1430 · Zbl 1062.74589
[14] Buhl T, Pedersen CBW, Sigmund O (2000) Stiffness design of geometrically nonlinear structures in topology optimization. Struct Multidisc Optim 19:93–104
[15] Chern JM, Prager W (1972) Optimal design of trusses for alternative loads. Ing Arch 41:225–231 · Zbl 0231.73036
[16] Diaz AR, Bendsoe MP (1992) Shape optimization of structures for multiple loading situations using a homogenization method. Struct Optim 4:17–22
[17] Diaz AR, Kikuchi N (1992) Solution to shape and topology eigenvalue problems using a homogenization method. Int J Numer Methods Eng 35:1487–1502 · Zbl 0767.73046
[18] Diaz AR, Lipton R (1997) Optimal material layout for 3D elastic structures. Struct Optim 13:60–64
[19] Diaz AR, Lipton R (2000) Optimal material layout in 3D elastic structures subjected to multiple loads. Mech Struct Mach 28:219–236
[20] Diaz AR, Sigmund O (1995) Checkerboard patterns in layout optimization. Struct Optim 10:40–45
[21] Du J, Olhoff N (2004a) Topological optimization of continuum structures with design-dependent surface loads–Part I: New computational approach for 2D problems. Struct Multidisc Optim 27:151–165 · Zbl 1243.74149
[22] Du J, Olhoff N (2004b) Topological optimization of continuum structures with design-dependent surface loads–Part II: Algorithm and examples for 3D problems. Struct Multidisc Optim 27:166–177 · Zbl 1243.74150
[23] Du J, Olhoff N (2007a) Minimization of sound radiation from vibrating bi-material structures using topology optimization. Struct Multidisc Optim 33:305–321
[24] Du J, Olhoff N (2007b) Topological design of freely vibrating continuum structures for maximum values of single and multiple eigenfrequencies and frequency gaps. Struct Multidisc Optim 34:91–110 · Zbl 1273.74398
[25] Edwards CS, Kim HA, Budd CJ (2006) Investigation on the validity of topology optimization methods. Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics Conference. Paper No. AIAA 2006-1984. Newport, Rhode Island pp 1–15, May 2006
[26] Edwards CS, Kim HA, Budd CJ (2007) An evaluative study on ESO and SIMP for optimising a cantilever tie-beam. Struct Multidisc Optim 34:403–414
[27] Fleury C (1982) Reconciliation of mathematical programming and optimality criteria approaches to structural optimization. In: Morris AJ (ed) Foundations of structural optimization: a unified approach. Wiley, Chichester, pp 363–404
[28] Fleury C (1989) CONLIN: an efficient dual optimizer based on convex approximation concepts. Struct Optim 1:81–89
[29] Fuchs MB, Jiny S, Peleg K (2005) The SRV constraint for 0/1 topological design. Stuct Multidisc Optim 30:320–326
[30] Gaspar Z, Logo J, Rozvany GIN (2002) Addenda and corrigenda to ”Aims, scope, methods, history and unified terminology of computer-aided optimization in structural mechanics”. Struct Multidisc Optim 24:338–342
[31] Haber RB, Bendsoe MP, Jog CS (1996) A new approach to variable topology shape design using constraint on the perimeter. Struct Optim 11:1–12
[32] Haftka RT, Gurdal Z (1992) Elements of structural optimization. Kluwer, Dordrecht
[33] Hajela P, Lee E (1995) Genetic algorithms in truss topology optimization. Int J Solids Struct 32:3341–3357 · Zbl 0919.73113
[34] Hammer VB, Olhoff N (1999) Topology optimization with design dependent loads. In: Bloebaum CL (ed) Proceedings of WCSMO-3 (CD-Rom)
[35] Hammer VB, Olhoff N (2000) Topology optimization of continuum structures subject to pressure loading. Struct Multidisc Optim 19:85–92
[36] Hegemier GA, Prager W (1969) On Michell trusses. Int J Mech Sci 11:209–215 · Zbl 0164.26302
[37] Hemp WS (1973) Optimum structures. Clarendon, Oxford · Zbl 0284.73061
[38] Huang X, Xie YM (2007) A new look at ESO and BESO optimization methods. Struct Multidisc Optim 35(1):89–92
[39] Jensen JS, Sigmund O (2004) Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends. Appl Phys Lett 84:2022–2024
[40] Jensen JS, Sigmund O (2005) Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide. J Opt Soc Am B Opt Phys 22:1191–1198
[41] Jog CS, Haber RB, Bendsoe MP (1994) Topology design with optimized self-adaptive materials. Int J Numer Methods Eng 37:1323–1350 · Zbl 0807.73044
[42] Kharmanda G, Olhoff N (2004) Reliability based topology optimization. Struct Multidisc Optim 26:295–307 · Zbl 1176.90142
[43] Kohn RV, Strang G (1986) Optimal design and relaxation of variational problems I, II, III. Commun Pure Appl Math 39:113–137, 139–182, 353–377 · Zbl 0609.49008
[44] Lewinski T, Rozvany GIN (2007) Exact analytical solutions for some popular benchmark problems in topology optimization II: three-sided polygonal supports. Struct Multidisc Optim 33:337–350 · Zbl 1245.74073
[45] Lewinski T, Rozvany GIN (2008a) Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains. Struct Multidisc Optim 35:165–174 · Zbl 1273.74400
[46] Lewinski T, Rozvany GIN (2008b) Exact analytical solutions for some popular benchmark problems in topology optimization IV: square-shaped line support. Struct Multidisc Optim DOI 10.1007/s00158-007-0205-4
[47] Lewinski T, Zhou M, Rozvany GIN (1994) Extended exact solutions for least-weight truss layouts. Part II: unsymmetric cantilevers. Int J Mech Sci 36:375–398 · Zbl 0805.73046
[48] Ling QQ, Steven GP (2002) A performance-based optimization method for topology design of continuum structures with mean compliance constraint. Comp Methods Appl Mech Eng 191:1471–1489 · Zbl 1098.74660
[49] Lurie K, Cherkaev AV (1986) Effective characteristics of composite materials and optimal design of structural elements. Uspekhi Mekhaniki 9:3–81 (in Russian)
[50] Martinez JM (2005) A note on the theoretical convergence properties of the SIMP method. Struct Multidisc Optim 29:319–323 · Zbl 1243.90210
[51] Mattheck C, Burkhardt S (1990) A new method of structural optimization based on biological growth. Int J Fatigue 12:185–190
[52] Maute K, Ramm E (1997) Adaptive topology optimization of shell structures. AIAA J 37:1767–1777 · Zbl 0899.73324
[53] Maute K, Ramm E (1998) Adaptive topology optimization. Struct Optim 10:100–112
[54] Maute K, Schwarz S, Ramm E (1998) Adaptive topology optimization of elastoplastic structures. Struct Optim 15:81–90
[55] McConnell RE (1974) Least weight frameworks for loads across a span. J Eng Mech ASCE, EM 5:885–891
[56] Michell AGM (1904) The limits of economy of material in frame structures. Phil Mag 8:589–597 · JFM 35.0828.01
[57] Mlejnek HP (1992) Some exploration of the genesis of structures. In: Bendsoe MP, Mota Soares CA (eds) Topology design of structures. Proceedings of NATO ARW (Sesimbra, Portugal). Kluwer, Dordrecht, pp 287–300
[58] Norato JA, Bendsoe MP, Haber RB, Tortorelli DA (2007) A topological derivative method for topology optimization. Struct Multidisc Optim 33:375–386 · Zbl 1245.74074
[59] Olhoff N, Du J (2006) Topology optimization of bi-material structures with respect to sound radiation. In: Bendsoe MP, Olhoff N, Sigmund O (eds) Proceedings of IUTAM Symposium on topological design optimization of structures, machines and materials, Copenhagen, 2005. Springer, Heidelberg, pp 43–52
[60] Olhoff N, Bendsoe MP, Rasmussen J (1991) On CAD-integrated structural topology and design optimization. Comp Methods Appl Mech Eng 89:259–279
[61] Pedersen P (2000) On optimal shapes in materials and structures. Struct Multidisc Optim 19:169–182
[62] Pedersen CBW, Allinger P (2006) Industrial implementation and applications of topology optimization and future needs. In: Bendsoe MP, Olhoff N, Sigmund O (eds) Topology design optimization of structures, machines and materials (Proceedings of IUTAM Symposium, Copenhagen, 2006). Springer, Dordrecht, pp 229–238
[63] Petersson J, Sigmund O (1998) Slope constrained topology optimization. Int J Numer Methods Eng 41:1417–1434 · Zbl 0907.73044
[64] Pomezanski V, Querin OM, Rozvany GIN (2005) CO-SIMP: extended SIMP algorithm with direct COrner COntact COntrol. Struct Multidisc Optim 30:164–168
[65] Poulsen TA (2002) A simple scheme to prevent checkerboard patterns and one node connected hinges in topology optimization. Struct Multidisc Optim 24:396–399
[66] Prager W, Rozvany GIN (1977) Optimization of the structural geometry. In: Bednarek AR, Cesari L (eds) Dynamical systems (Proceedings of the International Conference, Gainsville, Florida). Academic, New York, pp 265–293
[67] Rietz A (2001) Sufficiency of a finite exponent in SIMP (power law) methods. Struct Multidisc Optim 21:159–163
[68] Rossow MP, Taylor JE (1973) A finite element method for the optimal design of variable thickness sheets. AIAA J 11:1566–1569
[69] Rozvany GIN (1972a) Grillages of maximum strength and maximum stiffness. Int J Mech Sci 14:1217–1222
[70] Rozvany GIN (1972b) Optimal load transmission by flexure. Comput Methods Appl Mech Eng 1:253–263
[71] Rozvany GIN (1976) Optimal design of flexural systems. Pergamon, Oxford (Russian translation Stroiizdat, Moscow, 1980)
[72] Rozvany GIN (1989) Structural design via optimality criteria. Kluwer, Dordrecht · Zbl 0687.73079
[73] Rozvany GIN (1993) Layout theory for grid-like structures. In: Bendsoe MP, Mota Soares CA (eds) Topology design of structures (Proceedings of NATO ARW, Sesimbra, Portugal). Kluwer, Dordrecht, pp 251–272
[74] Rozvany GIN (1998) Exact analytical solution for some popular benchmark problems in topology optimization. Struct Optim 15:42–47 · Zbl 1245.74075
[75] Rozvany GIN (2001a) Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics. Struct Multidisc Optim 21:90–108
[76] Rozvany GIN (2001b) Stress ratio and compliance based methods in topology optimization–a critical review. Struct Multidisc Optim 21:109–119
[77] Rozvany GIN (2008) Traditional vs. extended optimality in topology optimization. DOI 10.1007/s00158-008-0231-x
[78] Rozvany GIN, Querin OM (2001) Present limitations and possible improvements of SERA (Sequential Element Rejections and Admissions) methods in topology optimization. Proceedings of WCSMO 4, Dalian, China
[79] Rozvany GIN, Querin OM (2002a) Combining ESO with rigorous optimality criteria. Int J Veh Des 28:294–299
[80] Rozvany GIN, Querin OM (2002b) Theoretical foundations of sequential element rejections and admissions (SERA) methods and their computational implementations in topology optimization. Proceedings of the 9th AIAA/ISSMO Symposium on Multidisc. Anal and Optim. (held in Atlanta, Georgia), AIAA, Reston, VA
[81] Rozvany GIN, Querin O (2004) Sequential element rejections and admissions (SERA) method: applications to multiconstraint problems. Proceedings of the 10th AIAA/ISSMO Multidisc. Anal. Optim. Conference, Albany, NY, Aug.–Sept. 2004
[82] Rozvany GIN, Zhou M (1991) Applications of the COC algorithm in layout optimization. In: Eschenauer H, Matteck C, Olhoff N (eds) Engineering optimization in design processes. Proceedings of the International Conference (Karlsruhe, Germany, Sept 1990). Springer, Berlin, pp 59–70
[83] Rozvany GIN, Zhou M (1993) Layout and generalized shape optimization by iterative COC methods. In: Rozvany GIN (ed) Optimization of large structural systems. Proceedings of NATO ASI (Berchtesgaden 1991). Kluwer, Dordrecht, pp 103–120
[84] Rozvany GIN, Zhou M (1994) Optimality criteria methods for large structural systems. In: Adeli H (ed) Advances in design optimization. Chapman & Hall, London, pp 41–108
[85] Rozvany GIN, Olhoff N, Bendsoe MP, Ong TG, Szeto WT (1985) Least-weight design of perforated elastic plates. DCAMM Report No. 306, Techn. Univ. Denmark, Lyngby
[86] Rozvany GIN, Olhoff N, Bendsoe MP, Ong TG, Szeto WT (1987) Least-weight design of perforated elastic plates I, II. Int J Solids Struct 23:521–536, 537–550 · Zbl 0606.73105
[87] Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Optim 4:250–254
[88] Rozvany GIN, Zhou M, Sigmund O (1994) Optimization of topology. In: Adeli H (ed) Advances in design optimization. Chapman & Hall, London, pp 340–399
[89] Rozvany GIN, Bendsoe MP, Kirsch U (1995) Layout optimization of structures. Appl Mech Rev ASME 48:41–119
[90] Rozvany GIN, Querin OM, Gaspar Z, Pomezanski V (2002a) Extended optimality in topology design. Struct Multidisc Optim 24:257–261
[91] Rozvany GIN, Querin OM, Gaspar Z, Pomezanski V (2002b) Checkerboards, element chains and hinges in topology optimization: causes and prevention. In: Gosling P (ed) Proceedings of the 4th ASMO-UK/ISSMO Conference. Univ. Newcastle-upon-Tyne, UK, pp 149–159
[92] Rozvany GIN, Querin OM, Gaspar Z, Pomezanski V (2003) Weight increasing effect of topology simplification. Struct Multidisc Optim 25:459–465
[93] Rozvany GIN, Pomezanski V, Querin OM, Gaspar Z, Logo J (2004) Corner contact suppression in topology optimization. In: Querin OM, Sienz J, Toropov VV, Goslong P (eds) Extended Abstracts 5th ASMO UK/ISSMO Conference, pp 33–40
[94] Rozvany GIN, Pomezanski V, Gaspar Z, Querin OM (2005a) Some pivotal issues in structural topology optimization. In: Herskovits J, Mazorche S, Canelas A (eds) Proceedings of WCSMO 6 (CD-ROM), Rio de Janeiro, Brasil, 2005, CD paper, ISBN 85-285-0070-5
[95] Rozvany GIN, Querin OM, Gaspar Z, Pomezanski V (2005b) Erratum for ”Extended optimality in topology design”. Struct Multidisc Optim 30:504
[96] Rozvany GIN, Lewinski T, Querin OM, Logo J (2006a) Quality control in topology optimization using analytically derived benchmarks. Proceedings of the 11th AIAA/ISSMO Multidisc. Anal. Opt. Conf., Portsouth, Virginia, USA, AIAA paper No.: AIAA 2006-6940, CD paper, 11 pp Collection of Technical Papers–11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 1, pp 484–494
[97] Rozvany GIN, Pomezanski V, Querin OM, Gaspar Z, Logo J (2006b) Some basic issues of topology optimization. In: Bendsoe MP, Olhoff N, Sigmund O. Proceedings of IUTAM Symposium on topology design optimization of structures, machines and materials. Springer, Heidelberg, pp 77–86
[98] Sankaranarayanan S, Haftka RT, Kapania RK (1992) Truss topology optimization with simultaneous analysis and design. Proceedings of 33rd AIAA/ASME/ASCE/AHS/ASC Struct Dyn Mat Conf (Dallas). AIAA, Washington, DC, pp 2576–2585
[99] Save M, Prager W (1985) Structural Optimization–vol 1: optimality criteria. Plenum, New York
[100] Schnack E, Spörl V, Iancu G (1988) Gradientless shape optimization with FEM. VDI Research Report 647/88. VDI-Verlag, Düsseldorf
[101] Sethian JA, Wiegman A (2000) Structural boundary design via level set and immersed interface methods. J Comp Phys 163:489–528 · Zbl 0994.74082
[102] Sigmund O (1994) Design of material structures using topology optimization. PhD thesis, Dept Solid Mech, Univ Denmark
[103] Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25:493–524
[104] Sigmund O (2001a) A 99 line topology optimization code written in Matlab. Struct Multidisc Optim 21:120–127
[105] Sigmund O (2001b) Design of multiphysics actuators using topology optimization. Part II: two-material structures. Comput Methods Appl Mech Eng 190:6605–6627 · Zbl 1116.74407
[106] Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33:401–424
[107] Sigmund O, Jensen JS (2003) Systematic design of phononic band gap materials and structures by topology optimization. Philos Trans R Soc A Math Phys Eng Sci 361:1001–1019 · Zbl 1067.74053
[108] Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16:68–75
[109] Sokolowski J, Zochowski A (1999) On the topological derivative in shape optimization. Siam J Control Optim 37:1251–1272 · Zbl 0940.49026
[110] Stolpe M, Bendsoe MP (2007) A non-linear branch and cut method for solving discrete minimum compliance problems to global optimality, WCSMO-7, Proceedings CD, pp 1–10
[111] Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidisc Optim 22:116–124
[112] Svanberg K (1987) The method of moving asymptotes–a new method for structural optimization. Int J Numer Methods Eng 24:359–373 · Zbl 0602.73091
[113] Svanberg K, Werme M (2005) Hierarchical neighbourhood search method for topology optimization. Struct Multidisc Optim 29:325–340 · Zbl 1243.74159
[114] Svanberg K, Werme M (2006a) Topology optimization by sequential integer linear programming. In: Bendsoe MP, Olhoff N, Sigmund O (eds) Proceedings of IUTAM Symposium on topological design optimization of structures, machines and materials. Springer, Dordrecht, pp 425–438
[115] Svanberg K, Werme M (2006b) Topology optimization by neighbourhood search method based on efficient sensitivity calculations. Int J Numer Methods Eng 67:1670–1699 · Zbl 1113.74056
[116] Svanberg K, Werme M (2007) Sequential integer programming method for stress constrained topology optimization. Struct Multidisc Optim 34:277–299 · Zbl 1273.74407
[117] Sved G (1954) The minimum weight of certain redundant structures. Aust J appl Sci 5:1–8 · Zbl 0058.39304
[118] Tanskanen P (2002) The evolutionary structural optimization method: theoretical aspects. Comput Methods Appl Mech Eng 191:5485–5498 · Zbl 1083.74572
[119] Tanskanen P (2006) A multiobjective and fixed element based modification of the evolutionary structural optimization method. Comput Methods Appl Mech Eng 196:76–90 · Zbl 1120.74708
[120] Tcherniak D, Sigmund O (2001) A web-based topology optimization program. Struct Multidisc Optim 22:179–187
[121] Wang MY, Wang XM, Guo DM (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246 · Zbl 1083.74573
[122] Wang X, Wang MY, Guo D (2004) Structural shape and topology optimization in level-set-based framework of region representation. Struct Multidisc Optim 27:1–19
[123] Werme M (2006) Globally optimal benchmark solutions to some small scale discretized continuum topology optimization problems. Struct Multidisc Optim 32:259–262
[124] Xie YM, Steven GP (1992) Shape and layout optimization via an evolutionary procedure. Proceedings of the International Conference Comput. Eng. (Hong Kong), Hong Kong University, p 421
[125] Xie YM, Steven GP (1997) Evolutionary structural optimization. Springer, London
[126] Yang XY, Xie YM, Steven GP, Querin OM (1998) Bi-directional evolutionary structural optimization. Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium Multidisc Anal. Optim (St. Louis), pp 1449–1457
[127] Zhou M, Rozvany GIN (1991) The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89:309–336
[128] Zhou M, Rozvany GIN (1992/1993) DCOC: an optimality criterion method for large systems. Part I: theory. Part II: algorithm. Struct Optim 5:12–25, 6:250–262
[129] Zhou M, Rozvany GIN (2001) On the validity of ESO type methods in topology optimization. Struct Multidisc Optim 21:80–83
[130] Zhou M, Shyy YK, Thomas HL (2001) Checkerboard and minimum member size control in topology optimization. Struct Multidisc Optim 21:152–158
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