Milestones in the 150-Year History of Topology Optimization: A Review

János Lógó; Hussein Ismail

doi:10.24423/cames.296

János Lógó Budapest University of Technology and Economics
Hussein Ismail Budapest University of Technology and Economics

DOI: http://dx.doi.org/10.24423/cames.296

Abstract

Structural optimization is one of the most intensively investigated research areas in engineering. Recently, topology optimization has become the most popular engineering subfield. The starting date of structural optimization cannot be precisely determined. Michell’s optimization paper, published in 1904, is considered as the first publication in this subfield. However, his paper starts with a statement that his work is a generalization
of Maxwell’s idea presented in the paper published in 1870.
The authors of this review paper consider that this date can be accepted as the starting date of topology optimization. This paper is an overview of subjectively selected state-of-art achievements in topology optimization during its history of 150 years. The selection of the achievements is a rather difficult task because, in the early period of the history of topology optimization, a lot of meetings were classified and the results were not available for the public. The optimization community has almost no knowledge about the publications in topology optimization in the 1950s. Around that time, one can find some information on workshops and meetings connected to the Cambridge University or Oxford University with researchers such as Foulkes, Cox, Hemp, and Shield, who published significant results and these communications are generally not known for the reason mentioned above. After the 1970s, this situation has changed and there were more possibilities to find publications due to the changes and thanks to digitalization. As indicated earlier here subjectively selected works are overviewed from the 150-year history focusing on the first hundred twenty years.

Keywords

topology optimization, optimal layout, optimality criteria method, level set method, heuristic optimal design,

References

1. J. Lógó, 150 Years History of topology Optimization, [in:] B. Błachowski, P. Tauzowski [Eds], Workshop on engineering optimization 2019, p. 19, Institute of Fundamental Technological Research, Polish Academy of Sciences, Warszawa, Poland, 2019, http://bluebox.ippt.pan.pl/~ptauzow/WEO/WEO2019_bookOfAbstracts.pdf.
2. J. Lógó, New type of optimal topologies by iterative method, Mechanics Based Design of Structures and Machines, 33(2): 149–172, 2005, doi: 10.1081/SME-200067035.
3. J. Lógó, E. Pintér, Numerical methods in probabilistic topology optimisation: a review, Computational Technology Reviews, 5: 79–108, 2012, doi: 10.4203/ctr.5.3.
4. J. Lógó, B. Balogh, E. Pintér, Topology optimization considering multiple loading, Computers & Structures, 207: 233–244, 2018, doi: 10.1016/j.compstruc.2017.03.018.
5. W.S. Hemp, Theory of Structural Design, NATO Studies, Report No. 214, Paris, 1958.
6. A.G.M. Michell, The limits of economy of material in frames structures, Philosophical Magazine, 8: 589–597, 1904, doi: 10.1080/14786440409463229.
7. J.C. Maxwell, On reciprocal figures, frames, and diagrams of forces, Transactions of the Royal Society of Edinburgh, 26(1): 1–40, 1870.
8. M.S. Bazaraa, C.M. Shetty, Nonlinear programming, theory and algorithms, John Wiley & Sons, New York, 1979.
9. G. Farkas, Theorie der einfachen Ungleichungen, Crelle Journal, 124: 1–27, 1902.
10. W. Karush, Minima of functions of several variables with inequalities as side constraints [M.Sc. thesis], Dept. of Mathematics, Univ. of Chicago, Chicago, Illinois, 1939.
11. H.W. Kuhn, A.W. Tucker, Nonlinear Programming Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, pp. 481–492, 1951.
12. V. Pareto, Cours d’Economie Politique, Droz, Geneva, 1896.
13. F.H. Cilley, The exact design of statically determinate frameworks and exposition of its possibility but futility, ASME Transactions, 43: 353–407, 1900.
14. J.D. Foulkes, The minimum design of structural frames, [in:] Proceedings of the Royal Society, Series A, 223: 482–494, 1954, https://www.jstor.org/stable/99453.
15. H.L. Cox, The theory of design, Aeronautical Research Council 19791, 1958.
16. H.L. Cox, Structures of minimum weight: the basic theory of design applied to the beam under pure bending, Aeronautical Research Council 19785, 1958.
17. H.L. Cox, The design of structures of least weight, Pergamon Press, 1965.
18. W.S. Hemp, Notes on the problem of the optimum design of structures, College of Aeronautics Note, 73, 1958, http://dspace.lib.cranfield.ac.uk/handle/1826/11525.
19. W.S. Hemp, Studies in the theory of Michell structures, [in:] Proceedings 11th International Congress of Applied Mechanics, Munich, pp. 621–628, 1964, doi: 10.1007/978-3-662-29364-5_83.
20. G. Sved, The minimum weight of certain redundant structures, Australian Journal of Applied Sciences, 5: 1–9, 1954.
21. J. Barta, On the minimum weight of certain redundant structures, Acta Technica Academiae Scientiarum Hungaricae, 18: 67–76, 1957.
22. D.C. Drucker, R.T. Shield, Bounds on minimum weight design, The Quarterly of Applied Mathematics, 12: 269–281, 1957, https://www.jstor.org/stable/43634462.
23. Z. Mroz, The load carrying capacity and minimum weight design of annular plates, Engineering Transactions, 114: 605–625, 1958, www.ippt.pan.pl/Repository/o60.pdf.
24. W. Prager, R.T. Shield, Optimal design of multi-purpose structures, International Journal of Solids and Structures, 4(4): 469–475, 1968, doi: 10.1016/0020-7683(68)90050-4.
25. R.T. Shield, Optimum design methods for multiple loading, Journal of Applied Mathematics and Physics (ZAMP), 14: 38–45, 1963, doi: 10.1007/BF01601144.
26. L.A. Schmit, Structural design by systematic synthesis, [in:] Proceeding 2nd ASCE Conference on Electronic Computation, pp. 105–132, 1960, http://www.vrand.com/wpcontent/uploads/2017/06/Struc_Design_Sys_Synthesis.pdf.
27. W. Lansing, W. Dwyer, R. Emerton, E. Ranalli, Application of fully stressed design procedure to wing and empennage structures, Journal of Aircraft, 8: 683–688, 1971.
28. R.A. Gellatly, R.H. Gallagher, A procedure for automated minimum weight structural design, Part I, Theoretical Basis, Aeronautical Quart, XVII: 216–230, 1966, doi: 10.1017/S000192590000384X.
29. R.H. Gallagher, Fully stressed design, [in:] R.H. Gallagher, O.C. Zienkiewicz [Eds], Optimum Structural Design: Theory and Applications, Wiley, London, pp. 19–32, 1973.
30. L. Berke, N.S. Khot, Use of optimality criteria methods for large scale system, AGARD Lec. 70, pp. 1–29, 1974, https://ci.nii.ac.jp/naid/20000182383/.
31. W. Prager, J.E. Taylor, Problems of optimal structural design, Journal of Applied Mechanics ASME, 35: 122–106, 1968, doi: 10.1115/1.3601120.
32. W. Prager, Optimization of structural design, Journal of Optimization Theory and Application, 6(1): 1–21, 1970, doi: 10.1007/BF00927037.
33. W. Prager, Foulkes mechanism in optimal plastic design for alternative loads, International Journal of Mechanical Sciences, 13: 971–973, 1971, doi: 10.1016/0020-7403(71)90083-X.
34. W. Prager, Condition for optimality, Computers & Structures, 2: 833–840, 1972.
35. W. Prager, Introduction to structural optimization, CISM Courses and Lectures Notes, 212, Springer Verlag, Wien, New York, 1974, doi: 10.1007/978-3-7091-2644-8.
36. W. Prager, G.I.N. Rozvany, Optimal layout of grillages, Journal of Structural Mechanics, 5: 1–18, 1977, doi: 10.1080/03601217708907301.
37. C. Nagtegaal, W. Prager, Optimal layout of a truss for alternative loads, International Journal of Mechanical Sciences, 15(7): 583–592, 1973, doi: 10.1016/0020-7403(73)90082-9.
38. A.S.L. Chan, On Foulkes mechanism in portal frame design for alternative loads, Journal of Applied Mechanics, 36(1): 73–75, 1969, doi: 10.1115/1.3564588.
39. G.I.N. Rozvany, N. Olhoff, M.P. Bendsøe, T.G. Ong, R. Sandler, W.T. Szeto, Leastweight design of perforated elastic plates [parts I and II], DCAMM Report No 306, International Journal of Solids and Structures, 23(4): 521–536, 537–550, 1987, doi: 10.1016/0020-7683(87)90015-1.
40. W. Achtziger, Truss topology optimization including bar properties different in tension and compression, Structural Optimization, 12: 63–74, 1996, doi: 10.1007/BF01270445.
41. W. Achtziger, Multiple load truss topology and sizing optimization: Some properties of minimax compliance, Journal of Optimization Theory and Applications, 98(2): 255–280, 1998, doi: 10.1023/A:1022637216104.
42. W. Achtziger, M. Stolpe, Global optimization of truss topology with discrete bar areas – Part I: theory of relaxed problems, Computational Optimization and Applications, 40(2): 247–280, 2008, doi: 10.1007/s10589-007-9138-5.
43. W. Achtziger, M. Stolpe, Global optimization of truss topology with discrete bar areas – Part II: Implementation and numerical results, Computational Optimization and Applications, 44(2): 315–341, 2009, doi: 10.1007/s10589-007-9152-7.
44. M.A. Save, Remarks on the minimum-volume design of three-bar truss, Journal of Structural Mechanics, 11(1): 101–110, 1982, doi: 10.1080/03601218308907434.
45. T. Sokół, T. Lewinski, On the solution of the three forces problem and its application in optimal designing of a class of symmetric plane frameworks of least weight, Structural and Multidisciplinary Optimization, 42: 835–853, 2010, doi: 10.1007/s00158-010-0556-0.
46. M.P. Rossow, J.E. Taylor, A finite element method for the optimal design of variable thickness sheets, AIAA Journal, 11: 1566–1569, 1973, doi: 10.2514/3.50631.
47. M.P. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering, 71: 197–224, 1988, doi: 10.1016/0045-7825(88)90086-2.
48. M.P. Bendsøe, Optimal shape design as a material distribution problem, Structural Optimization, 1: 193–202, 1989, doi: 10.1007/BF01650949.
49. G.I.N. Rozvany, Structural design via optimality criteria, Kluwer Academic Publisher, Dordrect, 1989, doi: 10.1007/978-94-009-1161-1.
50. G.I.N. Rozvany, Shape and layout optimization of structural systems and optimality criteria methods, CISM Courses and Lectures, No. 325, Springer Verlag, Wien, 1992, doi: 10.1007/978-3-7091-2788-9.
51. G. Kharmanda, N. Olhoff, A. Mohamed, M. Lemaire, Reliability-based topology optimization, Structural and Multidisciplinary Optimization, 26(5): 295–307, 2004, doi: 10.1007/s00158-003-0322-7.
52. J. Lógó, M. Ghaemi, A. Vásárhelyi, Stochastic compliance constrained topology optimization based on optimality criteria method, Periodica Polytechnica Civil Engineering, 51(2): 5–10, 2007, doi: 0.3311/pp.ci.2007-2.02.
53. J. Lógó, New type of optimality criteria method in case of probabilistic loading conditions, Mechanics Based Design of Structures and Machines, 35(2): 147–162, 2007, doi: 10.1080/15397730701243066.
54. J. Lógó, M. Ghaemi, M. Movahedi Rad, Optimal topologies in case of probabilistic loading: The influence of load correlation, Mechanics Based Design of Structures and Machines, 37(3): 327–348 , 2009, doi: 10.1080/15397730902936328.
55. J. Lógó, SIMP type topology optimization procedure considering uncertain load position, Periodica Polytechnica Civil Engineering, 56(2): 213–219, 2012, doi: 10.3311/pp.ci.2012-2.07.
56. P.D. Dunning, H.A. Kim, G. Mullineux, Introducing uncertainty in direction of loading for topology optimization, 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Orlando, USA, 2010, doi: 10.2514/6.2010-2843.
57. P.D. Dunning, H.A. Kim, G. Mullineux, Introducing loading uncertainty in topology optimization, AIAA Journal, 49(4): 760–768, 2012, doi: 10.2514/1.J050670.
58. J.K. Guest, T. Igusa, Structural optimization under uncertain loads and nodal locations, Computer Methods in Applied Mechanics and Engineering, 198: 116–124, 2008, doi: 10.1016/j.cma.2008.04.009.
59. A. Csébfalvi, A new theoretical approach for robust truss optimization with uncertain load directions, Mechanics Based Design of Structures and Machines, 42(4): 442–453, 2014, doi: 10.1080/15397734.2014.880064.
60. A. Csébfalvi, Structural optimization under uncertainty in loading directions: benchmark results, Advances in Engineering Software, 120: 68–78, 2018, doi: 10.1016/j.advengsoft.2016.02.006.
61. A. Csébfalvi, J. Lógó, A critical analysis of expected-compliance model in volumeconstrained robust topology optimization with normally distributed loading directions, using a minimax-compliance approach alternatively, Advances in Engineering Software, 120: 107–115, 2018, doi: 10.1016/j.advengsoft.2018.02.003.
62. G. Kazinczy, Experiments with clamped end beams [in Hungarian], Betonszemle [Concrete Review], 2(4, 5, 6): 68–71, 83–87, 101–104, 1914.
63. G. Kazinczy, Plastic behaviour of the materials regarding the load carrying capacity of structures [in Hungarian], Lecture Notes III, 13, Hungarian Royal Academic Press, Budapest, 1942.
64. E. Melan, Theorie statisch unbestimmter Systeme, [in:] Prelim. Publ. 2nd Congress Int. Ass. Bridge Struct. Eng. Berlin, pp. 43–64, 1936.
65. S.Y. Wang, K. Tai, Graph representation for structural topology optimization using genetic algorithms, Computers & Structures, 82(20): 1609–1622, 2004, doi: 10.1016/j.compstruc.2004.05.005.
66. T. Lewinski, T. Sokół, C. Graczykowski, Michell structures, Springer, Cham, 2019, doi: 10.1007/978-3-319-95180-5.
67. A.S.L. Chan, The design of Michell optimum structures, Aeronautical Research Council Reports and Memoranda, No. 3303, 1–40, 1960, http://resolver.tudelft.nl/uuid:b85292de-8943-4737-9f7b-bd907355eabe.
68. H.S.Y. Chan, Optimal Michell frameworks for three parallel forces, Report No. 167, College of Aeronautics, Cranfield, UK, 1963, http://hdl.handle.net/1826/2494.
69. G.I.N. Rozvany, Some shortcomings in Michell’s truss theory, Structural Optimization, 12: 244–250, 1996, doi: 10.1007/BF01197364.
70. A.M. Brandt [Ed.], Criteria and methods of structural optimization, PWN – Polish Scientific Publishers, Warsaw, 1984.
71. Z. Wasiutynski, Strength design. Part I: Methods of strength design, Part II: Design, for minimum potential, Part III: On the design of I-beams [in Polish: O kształtowaniu wytrzymałosciowym], Akademia Nauk Technicznych, Warsaw, 1939.
72. R.T. Shield, Optimum design of structures through variational principles, [in:] Proceedings of Optimization and Stability Problems in Continuum Mechanics, P.K.C. Wang [Ed.], Springer-Verlag, Berlin, Heidelberg, New York, pp. 13–37, 1973, doi: 10.1007/3-540-06214-9_2.
73. W. Kozlowski, Z. Mróz, Optimal design of disks subject to geometric constraints, International Journal of Mechanical Sciences, 12: 1007–1021, 1970, doi: 10.1016/0020-7403(70)90029-9.
74. L. Berke, An efficient approach to the minimum weight design of deflection limited design, AFFDL-TM-70-4-FDTR, Wright-Patterson AFB, 1970.
75. C. Fleury, M. Geradin, Optimality criteria and mathematical programming in structural weight optimization, Computers and Structures, 8: 7–17, 1978, doi: 10.1016/0045-7949(78)90155-4.
76. C. Fleury, G. Sander, Relations between optimality criteria and mathematical programming in structural optimization, USC Symposium on Applied Computer Methods in Engineering, LA, Vol. l, pp. 507–520, 1977.
77. N.S. Khot, L. Berke, V.B. Venkayya, Comparison of optimality criteria algorithms for minimum weight design of structures, AIAA Journal, 7: 182–190, 1979, doi: 10.2514/3.61093.
78. C. Fleury, Structural weight optimization by dual methods of convex programming, International Journal for Numerical Methods in Engineering, 14: 1761–1783, 1979, doi: 10.1002/nme.1620141203.
79. C. Fleury, L.A. Schmit, Primal and dual methods in structural optimization, Journal of the Structural Division (ASCE), 106(ST5): 1117–1133, 1980.
80. L.A. Schmit, C. Fleury, Structural synthesis by combining approximation concepts and dual method, AIAA Journal, 18: 1252–1254, 1980, doi: 10.2514/3.50877.
81. L.A. Schmit, C. Fleury, Discrete-continuous variable structural synthesis using dual method, AIAA Journal, 18: 1515–1524, 1980, doi: 10.2514/3.7739.
82. C. Fleury, V. Braibant, Structural optimization: a new dual method using mixed variables, International Journal for Numerical Methods in Engineering, 23: 409–428, 1986, doi: 10.1002/nme.1620230307.
83. G. Gerard, Minimum weight analysis of compression structures, University Press, New York, 1956.
84. W. Prager, An introduction to plasticity, Addison-Wesley, Reading, Massachusetts, 1959.
85. W. Johnson, Optimum design of mechanical elements, Wiley, New York, 1960.
86. M.Z. Cohn [Ed.], An introduction to structural optimization, Solid Mechanics Division Study No. 1, University of Waterloo, 1969.
87. R.H. Gallagher, O.C. Zienkiewicz [Eds], Optimum structural design: Theory and applications, Wiley, New York, 1973.
88. W.S. Hemp, Optimum structures, Clarendon, Oxford, 1973.
89. A.A. Cyras, A.E. Borkauskas, R.P. Karkauskas, Theory and methods of optimization of elastic-plastic systems [in Russian], Stroijzdat, Moscow, 1974.
90. A. Sawczuk, Z. Mróz [Eds], Optimization in structural design, Proceedings of IUTAM Symposium held in Warsaw, August 1973, Springer-Verlag, Berlin, 1975.
91. G.A. Hegemeir, W. Prager, On Michell trusses, International Journal of Mechanical Sciences, 11: 209–215, 1969, doi: 10.1016/0020-7403(69)90006-X.
92. M. Save, W. Prager, Structural optimization-optimality criteria, Plenum Press, New York, London, 1985, doi: 10.1007/978-1-4615-7921-2.
93. G.I.N. Rozvany, Optimal design of flexural systems: beams, grillages, slabs, plates and shells, Pergamon Press, Pergamon Press, Oxford, New York, Sydney, Toronto, 1976.
94. W. Prager, R.T. Shield, A general theory of optimal plastic design, Journal of Applied Mechanics, 34(1): 184–186, 1967, doi: 10.1115/1.3607621.
95. N. Olhoff, Optimization of vibrating beams with respect to higher order natural frequencies, Journal of Structural Mechanics, 4: 87–122, 1976, doi: 10.1080/03601217608907283.
96. N. Olhoff, Maximizing higher order eigenfrequencies of beams with constraints on the design geometry, Journal of Structural Mechanics, 5: 107–134, 1977, doi: 10.1080/03601217708907308.
97. K.T. Cheng, N. Olhoff, An investigation concerning optimal design of solid elastic plates, International Journal of Solids and Structures, 17: 305–323, 1981, doi: 10.1016/0020-7683(81)90065-2.
98. M.P. Bendsøe, N. Olhoff, J.E. Taylor, A variational formulation for multicriteria structural optimization, Journal of Structural Mechanics, 11: 523–544, 1983, doi: 10.1080/03601218308907456.
99. G. Cheng, N. Olhoff, Regularized formulation for optimal design of axisymmetric plates, International Journal of Solids and Structures, 18: 153–169, 1982.
100. G.I.N. Rozvany, M.P. Bendsøe, U. Kirsch, Layout optimization of structures, Applied Mechanics Reviews, 48(2): 41–119, 1995.
101. H.A. Eschenauer, N. Olhoff, Topology optimization of continuum structures: A review, Applied Mechanics Reviews, 54(4): 331–389, 2001.
102. O. Sigmund, K. Maute, Topology optimization approaches, Structural and Multidisciplinary Optimization, 48: 1031–1055, 2013.
103. S. Zargham, T.A. Ward, R. Ramli, I.A. Badruddin, Topology optimization: A review for structural designs under vibration problems, Structural and Multidisciplinary Optimization, 53: 1157–1177, 2016.
104. G.I.N. Rozvany, Optimization in structural mechanics, CISM Courses and Lectures Notes, 374, Springer-Verlag, Wien, New York, 1997.
105. G.I.N. Rozvany, T. Lewinski [Eds], Topology optimization in structural and continuum mechanics, CISM Courses and Lectures Notes 549, Springer-Verlag, Wien, New York, 2014.
106. G.I.N. Rozvany, T. Birker, Generalized Michell structures – exact least-weight truss layouts for combined stress and displacement constraints: Part I – General theory for plane trusses, Structural Optimization, 9: 178–188, 1995, doi: 10.1007/BF01743967.
107. G.I.N. Rozvany, Partial relaxation of the orthogonality requirement for classical Michell structures, Structural Optimization, 13: 271–274, 1997, doi: 10.1007/BF01197457.
108. G.I.N. Rozvany, W. Gollub, M. Zhou, Exact Michell trusses for various combinations of line supports, Part II, Structural Optimization, 14: 138–149, 1997.
109. G.I.N. Rozvany, Stress ratio and compliance based methods in topology optimization – a critical review, Structural and Multidisciplinary Optimization, 21: 109–119, 2001, doi: 10.1007/s001580050175.
110. M. Zhou, G.I.N. Rozvany, The COC algorithm, Part II: Topological, geometrical and generalized shape optimization, Computer Methods in Applied Mechanics and Engineering, 89: 309–336, 1991, doi: 10.1016/0045-7825(91)90046-9.
111. H.P. Mlejnek, Some aspects of the genesis of structures, Structural Optimization, 5: 64–69, 1992.
112. Y.M. Xie, G.P. Steven, A simple evolutionary procedure for structural optimization, Computers & Structures, 49: 885–896, 1993.
113. M. Zhou, G.I.N. Rozvany, DCOC: An optimality criterion method for large systems, Part I: Theory; Part II: Algorithm, Structural Optimization, 5: 12–25; 6: 250–262, 1992; 1993.
114. M.P. Bendsøe, O. Sigmund, Material interpolation schemes in topology optimization, Archive of Applied Mechanics, 69: 635–654, 1999.
115. J. Sokołowski, A. Zochowski, On the topological derivative in shape optimization, SIAM Journal on Control and Optimization, 37: 1251–1272, 1999.
116. G. Allaire, F. Jouve, A.M. Toader, A level-set method for shape optimization, Comptes Rendus Mathematique, 334(12): 1125–1130, 2002.
117. M.Y. Wang, X.M. Wang, D.M. Guo, A level set method for structural topology optimization, Computer Methods in Applied Mechanics and Engineering, 192(1–2): 227–246, 2003.
118. T. Yamada, K. Izui, S. Nishiwaki, A. Takezawa, A topology optimization method based on the level set method incorporating a fictitious interface energy, Computer Methods in Applied Mechanics and Engineering, 199(45–48): 2876–2891, 2010, doi: 10.1016/j.cma.2010.05.013.
119. B. Bourdin, A. Chambolle, Design-dependent loads in topology optimization, ESAIM: Control, Optimisation and Calculus of Variations, 9: 19–48, 2003, doi: 10.1051/cocv:2002070.
120. A.R. Díaz, O. Sigmund, Checkerboard patterns in layout optimization, Structural Optimization, 10: 40–45, 1995.
121. C.S. Jog, R.B. Haber, Stability of finite elements models for distributed-parameter optimization and topology esign, Computer Methods in Applied Mechanics and Engineering, 130: 203–226, 1996.
122. O. Sigmund, J. Peterson, Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Structural Optimization, 16: 68–75, 1998.
123. T.A. Poulsen, A simple scheme to prevent checkerboard patterns and one node connected hinges in topology optimization, Structural and Multidisciplinary Optimization, 24: 396–399, 2002, doi: 10.1007/s00158-002-0251-x.
124. M. Zhou, Y.K. Shyy, H.L. Thomas, Checkerboard and minimum member size control in topology optimization, Structural and Multidisciplinary Optimization, 21: 152–158, 2001, doi: 10.1007/s001580050179.
125. B. Balogh, J. Lógó, The application of drilling degree of freedom to checkerboards in structural topology optimization, Advances in Engineering Software, 107: 7–12, 2017, doi: 10.1016/j.advengsoft.2017.02.005.
126. A. Kaveh, Advances in metaheuristic algorithms for optimal design of structures, Springer International Publishing AG, Cham, Switzerland, 2017, doi: 10.1007/978-3-319-46173-1.
127. R.T. Haftka, Z. Gürdal, M.P. Kamat, Elements of structural optimization, Kluwer, Dordrecht, The Netherlands, 1990.
128. U. Kirsch, Structural optimization, Springer Verlag, Berlin, New York, 1992.
129. M.P. Bendsøe, Optimization of structural topology, shape and material, Springer-Verlag, Berlin, 1995, doi: 10.1007/978-3-662-03115-5.
130. M.P. Bendsøe, O. Sigmund, Topology optimization: Theory, methods and applications, Springer-Verlag, Berlin, 2003.
131. T. Lewinski, M. Zhou, G.I.N. Rozvany, Extended exact least-weight truss layouts – Part II: Cantilever with horizontal axis of symmetry, International Journal of Mechanical Sciences, 36: 399–419, 1994, doi: 10.1016/0020-7403(94)90043-4.
132. R.E. Melchers, On extending the range of Michell-like optimal topologies structures, Structural and Multidisciplinary Optimization, 29: 85–92, 2005.
133. T. Lewinski, G.I.N. Rozvany, Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains, Structural and Multidisciplinary Optimization, 35: 165–174, 2008, doi: 10.1007/s00158-007-0157-8.
134. N. Olhoff, J. Du, Structural topology optimization with respect to eigenfrequencies of vibration, [in:] G. Rozvany, T. Lewinski [Eds], Topology optimization in structural and continuum mechanics, International Centre for Mechanical Sciences, Udine, Italy, June 18–22, 2012, Springer-Verlag, Vienna, 2014, doi: 10.1007/978-3-7091-1643-2_11.
135. P.M. Clausen, O. Sigmund, The pressure load problem re-visited, [in:] M.P. Bendsøe, N. Olhoff, O. Sigmund [Eds], IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, Solid Mechanics and Its Applications, Vol. 137, Springer, Dordrecht, 2006, doi: 10.1007/1-4020-4752-5_26.
136. H.C. Rodrigues, J.M. Guedes, M.P. Bendsøe, Hierarchical optimization of material and structure, Structural and Multidisciplinary Optimization, 24(1): 1–10, 2002, doi: 10.1007/s00158-002-0209-z.
137. P.G. Coelho, P.R. Fernandes, H.C. Rodrigues, Multiscale modeling of bone tissue with surface and permeability control, Journal of Biomechanics, 44(2): 321–329, 2001.
138. K.P. Jayachandran, J.M. Guedes, H.C. Rodrigues, Optimal configuration of microstructure in ferroelectric materials by stochastic optimization, Journal of Applied Physics, 108(2), 024101, 2010, doi: 10.1063/1.3462450.
139. G. Pingen, A. Evgrafov, K. Maute, Topology optimization of flow domains using the lattice Boltzmann method, Structural and Multidisciplinary Optimization, 34(6): 507–524, 2007, doi: 10.1007/s00158-007-0105-7.
140. S. Kreissl, G. Pingen, K. Maute, Topology optimization for unsteady flow, International Journal for Numerical Methods in Engineering, 87(13): 1229–1253, 2011, doi: 10.1002/nme.3151.
141. T. Nomura, E.M. Dede, J. Lee, S. Yamasaki, T. Matsumori, A. Kawamoto, N. Kikuchi, General topology optimization method with continuous and discrete orientation design using isoparametric projection, International Journal for Numerical Methods in Engineering, 101(8): 571–605, 2015, doi: 10.1002/nme.4799.
142. K. Suzuki, N. Kikuchi, A homogenization method for shape and topology optimization, Computer Methods in Applied Mechanics and Engineering, 93: 291–318, 1991, doi: 10.1007/978-3-7091-2566-3_3.
143. J.M. Guedes, N. Kikuchi, Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods, Computer Methods in Applied Mechanics and Engineering, 83(2): 143–198, 1990, doi: 10.1016/0045-7825(90)90148-F.
144. S. Nishiwaki, M.I. Frecker, S. Min, N. Kikuchi, Topology optimization of compliant mechanisms using the homogenization method, International Journal for Numerical Methods in Engineering, 42(3): 535–559, 1998, doi: 10.1002/(SICI)1097-0207(19980615)42:3<535::AID-NME372>3.0.CO;2-J.
145. J. Lee, N. Kikuchi, Structural topology optimization of electrical machinery to maximize stiffness with body force distribution, IEEE Transactions on Magnetics, 46(10): 3790–3794, 2010, doi: 10.1109/TMAG.2010.2052365.
146. J. Lee, J.H. Seo, N. Kikuchi, Topology optimization of switched reluctance motors for the desired torque profile, Structural and Multidisciplinary Optimization, 42(5): 783–796, 2010, doi: 10.1007/s00158-010-0547-1.
147. K. Svanberg, The method of moving asymptotes – a new method for structural optimization, International Journal for Numerical Methods in Engineering, 24: 359–373, 1987, doi: 10.1002/nme.1620240207.
148. P. Duysinx, M.P. Bendsøe, Topology optimization of continuum structures with local stress constraints, International Journal for Numerical Methods in Engineering, 43(8): 1453–1478, 1998, doi: 10.1002/(SICI)1097-0207(19981230)43:8<1453::AIDNME480>3.0.CO;2-2.
149. M. Bruggi, P. Duysinx, Topology optimization for minimum weight with compliance and stress constraints, Structural and Multidisciplinary Optimization, 46(3): 369–384, 2012, doi: 10.1007/s00158-012-0759-7.
150. M. Bruggi, P. Duysinx, A stress-based approach to the optimal design of structures with unilateral behavior of material or supports, Structural and Multidisciplinary Optimization, 48(2): 311–326, 2013, doi: 10.1007/s00158-013-0896-7.
151. M. Bruggi, On an alternative approach to stress constraints relaxation in topology optimization, Structural and Multidisciplinary Optimization, 36(2): 125–141, 2008, doi: 10.1007/s00158-007-0203-6.
152. M. Bruggi Topology optimization with mixed finite elements on regular grids, Computer Methods in Applied Mechanics and Engineering, 305: 133–153, 2016, doi: 10.1016/j.cma.2016.03.010.
153. A. Cherkaev, L. Gibiansky, Coupled estimates for the bulk and shear moduli of a two dimensional isotropic elastic composite, Journal of the Mechanics and Physics of Solids, 41: 937–980, 1993, doi: 10.1016/0022-5096(93)90006-2.
154. M.P. Bendsøe, J.M. Guedes, R.B. Haber, P. Pedersen, J.E. Taylor, An analytical model to predict optimal material properties in the context of optimal structural design, Journal of Applied Mechanics, 61: 930–937, 1994, doi: 10.1115/1.2901581.
155. M.P. Bendsøe, J.M. Guedes, S. Plaxton, J.E. Taylor, Optimization of structure and material properties for solids composed of softening materials, International Journal of Solids and Structures, 33(12): 1799–1813, 1996, doi: 10.1016/0020-7683(95)00121-2.
156. J.M. Guedes, J.E. Taylor, On the prediction of material properties and topology for optimal continuum structures, Structural Optimization, 14: 193–199, 1997 doi: 10.1007/BF01812523.
157. J.E. Taylor, Addendum to: An energy model for the optimal design of linear continuum structures, Structural and Multidisciplinary Optimization, 19: 317–320, 2000.
158. G. Dzierzanowski, T. Lewinski, Compliance minimization of thin plates made of material with predefined Kelvin moduli, Part I. Solving the local optimization problem, Archives of Mechanics, 64: 21–40, 2012.
159. P. Hajela, E. Lee, C.Y. Lin, Genetic Algorithms in structural topology optimization, [in:] M.P. Bendsøe, C.A.M. Soares [Eds], Topology design of structures, NATO ASI Series (Series E: Applied Sciences), vol. 227, Springer, Dordrecht, 1993, doi: 10.1007/978-94-011-1804-0_10.
160. K. Svanberg, M. Werme, A hierarchical neighbourhood search method for topology optimization, Structural and Multidisciplinary Optimization, 29: 325–340, 2005, doi: 10.1007/s00158-004-0493-x.
161. K. Svanberg, M. Werme, Topology optimization by a neighbourhood search method based on efficient sensitivity calculations, International Journal for Numerical Methods in Engineering, 67: 1670–1699, 2006, doi: 10.1002/nme.1677.
162. M. Werme, Globally optimal benchmark solutions to some small-scale discretized continuum topology optimization problems, Structural and Multidisciplinary Optimization, 32(3): 259–262, 2006, doi: 10.1007/s00158-006-0015-0.
163. M. Stolpe, M.P. Bendsøe, A non-linear branch and cut method for solving discrete minimum compliance problems to global optimality, [in:] Proceedings of 7th World Congress on Structural and Multidisciplinary Optimization, pp. 2513–2522, 2007.
164. W. Gutkowski, J. Bauer, Discrete structural optimization, IUTAM Symposium, Zakopane, Poland, August 31 – September 3, 1993, Springer Verlag, Berlin, 249, 1994, doi: 10.1007/978-3-642-85095-0.
165. J. Bauer, A survey of methods for discrete optimum structural design, Computer Assisted Mechanics and Engineering Sciences, 1: 27–38, 1994.
166. W. Gutkowski, Z. Mróz [Eds], WCSMO-2 – Proceedings of the 2nd World Congress of Structural and Multidisciplinary Optimization, Institute of Fundamental Technological Research, Warsaw, 1997.
167. W. Gutkowski, T.A. Kowalewski [Eds], Mechanics of the 21st Century, Proceedings of the 21st International Congress of Theoretical and Applied Mechanics, Warsaw, Poland, 15–21 August 2004, Springer, Dordrecht, 2004, doi: 10.1007/1-4020-3559-4.
168. W. Gutkowski, J. Bauer, Discrete structural optimization, International Centre for Mechanical Sciences book series (CISM, vol. 373), CISM Udine, 1997, doi: 10.1007/978-3-7091-2754-4.
169. G.I.N Rozvany, O.M. Querin, Present limitations and possible improvements of SERA (sequential element rejections and admissions) methods in topology optimization, [in:] C. Gengdong [Ed.], 4th World Congress of Structural and Multidisciplinary Optimization, Dalian, China, Liaoning Electronic Press, 2001.
170. G.I.N. Rozvany, O.M. Querin, Combining ESO with rigorous optimality criteria, International Journal of Vehicle Design (IJVD), 28(4): 294–299, 2002, doi: 10.1504/IJVD.2002.001991.
171. G.I.N. Rozvany, O.M. Querin, J. Lógó, Sequential element rejections and admissions (SERA) method: application to multiconstraint problems, [in:] Proceedings of the 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Reston (VA), USA, American Institute of Aeronautics and Astronautics, pp. 2459–2464, 2004.
172. M. Zhou, G.I.N. Rozvany, On the validity of ESO type methods in topology optimization, Structural and Multidisciplinary Optimization, 21: 80–83, 2001, doi: 10.1007/s001580050170.
173. G.I.N. Rozvany, A critical review of established numerical methods of structural topology optimization, Structural and Multidisciplinary Optimization, 37: 217–237, 2009, doi: 10.1007/s00158-007-0217-0.
174. K. Marti, G. Stöckl, Optimal (topology) design under stochastic uncertainty, [in:] Safety and reliability, Vol. 2, G.I. Schüller, P. Kafka [Eds], Rotterdam-Brookfield, Balkema, pp. 1597–1602, 1999.
175. K. Marti, G. Stöckl, Topology and geometry optimization under stochastic uncertainty, [in:] Numerical methods of uncertainties, Proceedings EUROMECH 405 Colloquium, P. Level et al. [Eds], Presses Universitaires Valenciennes, Valenciennes, pp. 55–63, 2000.
176. K. Marti, Stochastic optimization methods, Springer-Verlag, Berlin-Heidelberg, 2005.
177. J. Lógó, M. Movahedi Rad, T. Tamássy, J. Knabel, P. Tauzowski, Reliability based optimal design of frames with limited residual strain energy capacity, [in:] Proceedings of the Twelfth International Conference on Civil, Structural and Environmental Engineering Computing, B.H.V. Topping, L.F. Costa Neves, R.C. Barros [Eds], Civil-Comp Press, Stirlingshire, United Kingdom, paper 52, 2009, doi: 10.4203/ccp.91.52.
178. B. Balogh, M. Bruggi, J. Lógó, Optimal design accounting for uncertainty in loading amplitudes: A numerical investigation, Mechanics Based Design of Structures and Machines, 46(5): 552–566, 2018, doi: 10.1080/15397734.2017.1362987.
179. D. Bojczuk, Z. Mróz, Topological sensitivity derivative and finite topology modifications, application to plates in bending, Structural and Multidisciplinary Optimization, 39: 1–15, 2009, doi: 10.1007/s00158-008-0333-5.
180. Z. Mróz, D. Bojczuk, Shape and topology sensitivity analysis and its application structural design, Archives of Applied Mechanics, 82: 1541–1555, 2012, doi: 10.1007/s00419-012-0672-y.
181. R. Stocki, K. Kolanek, S. Jendo, M. Kleiber, Continuous and discrete reliability-based optimization of truss structures, [in:] Safety and reliability of industrial products, systems and structures, C. Guedes Soares [Ed.], CRC Press/Balkema, pp. 215–227, 2010, doi: 10.1201/b10572-23.
182. R. Stocki, K. Kolanek, S. Jendo, M. Kleiber, Study on discrete optimization techniques in reliability-based optimization of truss structures, Computers & Structures, 79(22–25): 2235–2247, 2001, doi: 10.1016/S0045-7949(01)00080-3.
183. K. Dolinski, J. Knabel, Reliability-oriented shakedown formulation, [in:] ICOSSAR 2005, G. Augusti, G.I. Schueller, M. Ciampoli [Eds], Millpress, Rotterdam, pp. 2323–2330, 2005.
184. S. Jendo, K. Dolinski [Eds], Reliability-based design and optimization, Proceedings of AMAS Course – RBO’03, IPPT, Warsaw, 2003.
185. P. Tauzowski, B. Blachowski, J. Lógó, Functor-oriented topology optimization of elastoplastic structures, Advances in Engineering Software, 135, 102690, 11 pages, 2019, doi: 10.1016/j.advengsoft.2019.102690.
186. B. Blachowski, P. Tauzowski, J. Lógó, Yield limited optimal topology design of elastoplastic structures, Structural and Multidisciplinary Optimization, 61: 1953–1976, 2020, doi: 10.1007/s00158-019-02447-9.
187. B. Blachowski, P. Tauzowski, J. Lógó, Modal approximation based optimal design of dynamically loaded plastic structures, Periodica Polytechnica Civil Engineering, 61(4): 987–992, 2017, doi: 10.3311/PPci.11016.
188. A. Cherkaev, Variational methods for structural optimization, Springer, New York, 2000, doi: 10.1007/978-1-4612-1188-4.