Recent development in structural design and optimization. (English) Zbl 1270.74164
Summary: With the fast development of computational mechanics and the capacity as well as the speed of modern computers, simulation-based structural optimization has become an indispensable tool in the design process of competitive products. This paper presents a brief description of the current status of structural optimization by reviewing some significant progress made in the last decades. Potential research topics are also discussed. The entire literatures of the field are not covered due to the limitation of the length of paper. The scope of this review is limited and closely related to the authors’ own research interests.
MSC:
| 74P05 | Compliance or weight optimization in solid mechanics |
| 74P15 | Topological methods for optimization problems in solid mechanics |
| 74K99 | Thin bodies, structures |
| 74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |
References:
| [1] | Michell A.G.M.: The limits of economy of materials in frame structures. Philos. Mag. Ser. 6 8(47), 589–597 (1904) · JFM 35.0828.01 · doi:10.1080/14786440409463229 |
| [2] | Eshenauer H.A., Olhoff N.: Topology optimization of continuum structures: a review. Appl. Mech. Rev. 54(4), 332–390 (2001) |
| [3] | Bendsoe M.P., Sigmund O.: Topology Optimization-Theory, Methods and Applications. Springer, Berlin (2003) |
| [4] | Cheng K.T., Olhoff N.: An investigation concerning optimal design of solid elastic plates. Int. J. Solids Struct. 17, 305–323 (1981) · Zbl 0457.73079 · doi:10.1016/0020-7683(81)90065-2 |
| [5] | Bendsoe M.P., Kikuchi N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988) · Zbl 0671.73065 · doi:10.1016/0045-7825(88)90086-2 |
| [6] | Bendsore M.P.: Optimal shape design as a material distribution problem. Struct. Optim. 1, 193–202 (1989) · doi:10.1007/BF01650949 |
| [7] | Rozvany G.I.N., Zhou M., Birker T.: Generalized shape optimization without homogenization. Struct. Optim. 4, 250–252 (1992) · doi:10.1007/BF01742754 |
| [8] | Zhou M., Rozvany G.I.N.: The COC algorithm, Part II: Topological, geometrical and generalized shape optimization. Comput. Methods Appl. Mech. Eng. 89, 309–336 (1991) · doi:10.1016/0045-7825(91)90046-9 |
| [9] | Diaz A., Sigmund O.: Checkerboard patterns in layout optimization. Struct. Multidisc. Optim. 10, 40–45 (1995) · doi:10.1007/BF01743693 |
| [10] | Petersson J.: A finite element analysis of optimal variable thickness sheet. SIAM J. Numer. Anal. 36, 1759–1778 (1999) · Zbl 0938.74054 · doi:10.1137/S0036142996313968 |
| [11] | Sigmund O.: Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. Solids Struct. 31, 2313–2329 (1994) · Zbl 0946.74557 · doi:10.1016/0020-7683(94)90154-6 |
| [12] | Sigmund O.: On the design of compliant mechanisms using topology optimization. Mech. Struct. Mach. 25, 495–526 (1997) |
| [13] | Bourdin B.: Filters in topology optimization. Int. J. Numer. Methods Eng. 50, 2143–2158 (2001) · Zbl 0971.74062 · doi:10.1002/nme.116 |
| [14] | Guo X., Gu Y.X.: A new density-stiffness interpolation scheme for topology optimization of continuum structures. Eng. Comput. 21, 9–22 (2004) · Zbl 1063.74079 · doi:10.1108/02644400410511819 |
| [15] | Wang M.Y., Wang S.: Bilateral filtering for structural topology optimization. Int. J. Numer. Methods Eng. 63, 1911–1938 (2005) · Zbl 1138.74379 · doi:10.1002/nme.1347 |
| [16] | Borrvall T., Petersson J.: Topology optimization using regularized intermediate density control. Comput. Methods Appl. Mech. Eng. 190, 4911–4928 (2001) · Zbl 1022.74035 · doi:10.1016/S0045-7825(00)00356-X |
| [17] | Zhou J.X., Zou W.: Meshless approximation combined with implicit topology description for optimization of continua. Struct. Multidisc. Optim. 36, 347–353 (2008) · doi:10.1007/s00158-007-0168-5 |
| [18] | Zheng J., Long S.Y., Li G.Y.: The topology optimization design for continuum structures based on the element free Galerkin method. Eng. Anal. Bound. Elem. 34, 666–672 (2010) · Zbl 1267.74094 · doi:10.1016/j.enganabound.2010.03.001 |
| [19] | Guest J., Prevost J., Belytschko T.: Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int. J. Numer. Methods Eng. 61, 238–254 (2004) · Zbl 1079.74599 · doi:10.1002/nme.1064 |
| [20] | Sigmund O.: Morphology-based black and white filters scheme for topology optimization. Struct. Multidisc. Optim. 33, 401–424 (2007) · doi:10.1007/s00158-006-0087-x |
| [21] | Xu S.L., Cai Y.W., Cheng G.D.: Volume preserving nonlinear density filter based on Heaviside functions. Struct. Multidisc. Optim. 41, 495–505 (2010) · Zbl 1274.74419 · doi:10.1007/s00158-009-0452-7 |
| [22] | Qian Z.Y., Ananthasuresh G.K.: Optimal embedding of rigid objects in the topology design of structures. Mech. Des. Struct. Mach. 32, 165–193 (2004) · doi:10.1081/SME-120030555 |
| [23] | Zhu J.H., Zhang W.H., Beckers P. et al.: Simultaneous design of components layout and supporting structures using coupled shape and topology optimization technique. Struct. Multidisc. Optim. 36, 29–41 (2008) · Zbl 1273.74382 · doi:10.1007/s00158-007-0155-x |
| [24] | Zhu J.H., Zhang W.H., Beckers P.: Integrated layout design of multi-component system. Int. J. Numer. Methods Eng. 78, 631–651 (2009) · Zbl 1183.74215 · doi:10.1002/nme.2499 |
| [25] | Kumar A.V., Gossard D.C.: Synthesis of optimal shape and topology of structures. J. Mech. Des. 118, 68–74 (1996) · doi:10.1115/1.2826858 |
| [26] | Sethian J.A., Wiegmann A.: Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163, 489–528 (2000) · Zbl 0994.74082 · doi:10.1006/jcph.2000.6581 |
| [27] | Osher S.J., Santosa F.: Level set methods for optimization problems involving geometry and constraints I. Frequency of a two density homogeneous drum. J. Comput. Phys. 171, 272–298 (2001) · Zbl 1056.74061 · doi:10.1006/jcph.2001.6789 |
| [28] | Wang M.Y., Wang X.M., Guo D.M.: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192, 227–246 (2003) · Zbl 1083.74573 · doi:10.1016/S0045-7825(02)00559-5 |
| [29] | Allaire G., Jouve F., Toader A.M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194, 363–393 (2004) · Zbl 1136.74368 · doi:10.1016/j.jcp.2003.09.032 |
| [30] | Wang M.Y., Wang X.M.: ”Color” level sets: a multi-phase level set method for structural topology optimization with multiple materials. Comput. Methods Appl. Mech. Eng. 193, 469–496 (2004) · Zbl 1060.74585 · doi:10.1016/j.cma.2003.10.008 |
| [31] | Allaire G., Jouve F.: A level-set method for vibration and multiple loads structural optimization. Comput. Methods Appl. Mech. Eng. 194, 3269–3290 (2005) · Zbl 1091.74038 · doi:10.1016/j.cma.2004.12.018 |
| [32] | Amstutz S., Andrä H.: A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216, 573–588 (2006) · Zbl 1097.65070 · doi:10.1016/j.jcp.2005.12.015 |
| [33] | Cho S., Ha S.H., Park C.Y.: Topological shape optimization of power flow problems at high frequencies using level set approach. Int. J. Solids Struct. 43, 172–192 (2006) · Zbl 1119.74520 · doi:10.1016/j.ijsolstr.2005.04.033 |
| [34] | Wei P., Wang M.Y.: Piecewise constant level set method for structural topology optimization. Int. J. Numer. Methods Eng. 78, 379–402 (2009) · Zbl 1183.74222 · doi:10.1002/nme.2478 |
| [35] | Rong J.H., Liang Q.Q.: A level set method for topology optimization of continuum structures with bounded design domains. Comput. Methods Appl. Mech. Eng. 197, 1447–1465 (2008) · Zbl 1194.74274 · doi:10.1016/j.cma.2007.11.026 |
| [36] | Yamasaki S., Nishiwaki S., Yamda T. et al.: A structural optimization method based on the level set method using a new geometry-based re-initialization scheme. Int. J. Numer. Methods Eng. 83, 1580–1624 (2010) · Zbl 1202.74130 · doi:10.1002/nme.2874 |
| [37] | Eschenauer H.A., Kobelev V.V., Schumacher A.: Bubble method for topology and shape optimization of structures. Struct. Optim. 8, 42–51 (1994) · doi:10.1007/BF01742933 |
| [38] | Sokolowski J., Zochowski A.: On the topological derivative in shape optimization. SIAM J. Control Optim. 37, 1251–1272 (1999) · Zbl 0940.49026 · doi:10.1137/S0363012997323230 |
| [39] | Cea J., Garreau S., Guillaume P. et al.: The shape and topological optimizations connection. Comput. Methods Appl. Mech. Eng. 188, 713–726 (2000) · Zbl 0972.74057 · doi:10.1016/S0045-7825(99)00357-6 |
| [40] | Novotny A.A., Feijoo R.A., Taroco E. et al.: Topological-shape sensitivity analysis. Comput. Methods Appl. Mech. Eng. 192, 803–829 (2003) · Zbl 1025.74025 · doi:10.1016/S0045-7825(02)00599-6 |
| [41] | Lewinski T., Sokolowski J.: Energy change due to the appearance of cavities in elastic solids. Int. J. Solids Struct. 40, 1765–1803 (2003) · Zbl 1035.74009 · doi:10.1016/S0020-7683(02)00641-8 |
| [42] | Allaire G., Gournay F., Jouve F. et al.: Structural optimization using topological and shape sensitivity via a level set method. Control Cybern. 34, 59–80 (2005) · Zbl 1167.49324 |
| [43] | Berger M., Hackl B., Ring W.: Incorporating topological derivatives into level set methods. J. Comput. Phys. 194, 344–362 (2004) · Zbl 1044.65053 · doi:10.1016/j.jcp.2003.09.033 |
| [44] | Guo X., Zhao K., Wang M.Y.: A new approach for simultaneous shape and topology optimization based on dynamic implicit surface function. Control Cybern. 34, 255–282 (2005) · Zbl 1167.65433 |
| [45] | Gibson L.J., Ashby M.F.: Cellular Solids: Structure and Properties. Cambridge University Press, Cambridge (1997) · Zbl 0723.73004 |
| [46] | Rodrigues H., Guedes J.M., Bendsoe M.P.: Hierarchical optimization of material and structure. Struct. Multidisc. Optim. 24, 1–10 (2002) · doi:10.1007/s00158-002-0209-z |
| [47] | Coelho P.G., Fernandes P.R., Guedes J.M.: A hierarchical model for concurrent material and topology optimization of three -dimensional structures. Struct. Multidisc. Optim. 35, 107–115 (2008) · doi:10.1007/s00158-007-0141-3 |
| [48] | Liu L., Yan J., Cheng G.D.: Optimum structure with homogeneous optimum truss-like material. Comput. Struct. 86, 1417–1425 (2008) · doi:10.1016/j.compstruc.2007.04.030 |
| [49] | Yan J., Cheng G.D., Liu L.: A uniform optimum material based model for concurrent optimization of thermoelastic structures and materials. Int. J. Simul. Multidisc. Des. Optim. 2, 259–266 (2008) · doi:10.1051/ijsmdo/2008035 |
| [50] | Liu S.T., Su W.Z.: Topology optimization of couple-stress material structures. Struct. Multidisc. Optim. 40, 319–327 (2010) · Zbl 1274.74364 · doi:10.1007/s00158-009-0367-3 |
| [51] | Niu B., Yan J., Cheng G.D.: Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Struct. Multidisc. Optim. 39, 115–132 (2009) · doi:10.1007/s00158-008-0334-4 |
| [52] | Chellappa S., Diaz A.R., Bendsøe M.P.: Layout optimization of structures with finite-sized features using multiresolution analysis. Struct. Multidisc. Optim. 26, 77–91 (2004) · Zbl 1243.74136 · doi:10.1007/s00158-003-0306-7 |
| [53] | Zhou K.M., Li X.: Topology optimization for minimum compliance under multiple loads based on continuous distribution of members. Struct. Multidisc. Optim. 37, 49–56 (2008) · doi:10.1007/s00158-007-0214-3 |
| [54] | Royset J.O., Der Kiureghian A., Polak E.: Reliability-based optimal design of series structural systems. J. Eng. Mech. ASCE 127, 607–614 (2001) · doi:10.1061/(ASCE)0733-9399(2001)127:6(607) |
| [55] | Choi K.K., Tu J., Park Y.H.: Extensions of design potential concept for reliability-based design optimization to nonsmooth and extreme cases. Struct. Multidisc. Optim. 22, 335–350 (2001) · doi:10.1007/s00158-001-0154-2 |
| [56] | Jung D.H., Lee B.C.: Development of a simple and efficient method for robust optimization. Int. J. Numer. Methods Eng. 53, 2201–2215 (2002) · Zbl 1169.74522 · doi:10.1002/nme.383 |
| [57] | Papadrakakis M., Lagaros N.D.: Reliability-based structural optimization using neural networks and Monte-Carlo simulation. Comput. Methods Appl. Mech. Eng. 191, 3491–3507 (2002) · Zbl 1101.74377 · doi:10.1016/S0045-7825(02)00287-6 |
| [58] | Kharmanda G., Olhoff N., Mohamed A. et al.: Reliability-based topology optimization. Struct. Multidisc. Optim. 26, 295–307 (2004) · doi:10.1007/s00158-003-0322-7 |
| [59] | Lee K.H., Park G.J.: Robust optimization considering tolerances of design variables. Comput. Struct. 79, 77–86 (2001) · doi:10.1016/S0045-7949(00)00117-6 |
| [60] | Sandgren E., Cameron T.M.: Robust design optimization of structures through consideration of variation. Comput. Struct. 80, 1605–1613 (2002) · doi:10.1016/S0045-7949(02)00160-8 |
| [61] | Lee K.H., Park G.J.: Robust optimization in discrete design space for constrained problems. AIAA J. 40, 774–780 (2002) · doi:10.2514/2.1712 |
| [62] | Valdebenito M.A., Schuëller G.I.: A survey on approaches for reliability-based optimization. Struct. Multidisc. Optim. 42, 645–663 (2010) · Zbl 1274.90125 · doi:10.1007/s00158-010-0518-6 |
| [63] | Tu J., Choi K., Park Y.: Design potential method for robust system parameter design. AIAA J. 39, 667–677 (2001) · doi:10.2514/2.1360 |
| [64] | Lee I., Choi K., Du L. et al.: Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems. Comput. Methods Appl. Mech. Eng. 198, 14–27 (2008) · Zbl 1194.74250 · doi:10.1016/j.cma.2008.03.004 |
| [65] | Agarwal H., Mozumder C., Renaud J. et al.: An inverse-measure-based unilevel architecture for reliability-based design. Struct. Multidisc. Optim. 33, 217–227 (2007) · doi:10.1007/s00158-006-0057-3 |
| [66] | Kharmanda G., Mohamed A., Lemaire M.: Efficient reliabilitybased design optimization using a hybrid space with application to finite element analysis. Struct. Multidisc. Optim. 24, 233–245 (2002) · doi:10.1007/s00158-002-0233-z |
| [67] | Kaymaz I., Marti K.: Reliability-based design optimization for elastoplastic mechanical structures. Comput. Struct. 85, 615–625 (2007) · doi:10.1016/j.compstruc.2006.08.076 |
| [68] | Liang J., Mourelatos Z., Tu J.: A single-loop method for reliability-based design optimisation. Int. J. Prod. Dev. 5, 76–92 (2008) · doi:10.1504/IJPD.2008.016371 |
| [69] | Du X., Chen W.: Sequential optimization and reliability assessment method for efficient probabilistic design. J. Mech. Des. 126, 225–233 (2004) · doi:10.1115/1.1649968 |
| [70] | Cheng G.D., Xu L., Jiang L.: A sequential approximate programming strategy for reliability-based structural optimization. Comput. Struct. 84, 1353–1367 (2006) · doi:10.1016/j.compstruc.2006.03.006 |
| [71] | Yi P., Cheng G.D., Jiang L.: A sequential approximate programming strategy for performance-measure-based probabilistic structural design optimization. Struct. Saf. 30, 91–109 (2008) · doi:10.1016/j.strusafe.2006.08.003 |
| [72] | Chan K.Y., Skerlos S., Papalambros P.: An adaptive sequential linear programming algorithm for optimal design problems with probabilistic constraints. J. Mech. Des. 129, 140–149 (2007) · doi:10.1115/1.2337312 |
| [73] | Aoues Y., Chateauneuf A.: Benchmark study of numerical methods for reliability-based design optimization. Struct. Multidisc. Optim. 41, 277–294 (2010) · Zbl 1274.90115 · doi:10.1007/s00158-009-0412-2 |
| [74] | Ganzerli S., Pantelides C.P.: Optimum structural design via convex model superposition. Comput. Struct. 74, 639–647 (2000) · doi:10.1016/S0045-7949(99)00077-2 |
| [75] | Au F.T.K., Cheng Y.S., Tham L.G. et al.: Robust design of structures using convex models. Comput. Struct. 81, 2611–2619 (2003) · doi:10.1016/S0045-7949(03)00322-5 |
| [76] | Belegundu A.D., Chandrupatla T.R.: Optimization Concepts and Applications in Engineering. Prentice-Hall, New Jersey (1999) · Zbl 0941.90074 |
| [77] | Cao H.J., Duan B.Y.: A study on non-probabilistic reliability-based structural optimization (in Chinese). Chin. J. Appl. Mech. 22, 381–385 (2005) |
| [78] | Gurav S.P., Goosen J.F.L., VanKeulen F.: Bounded-but -unknown uncertainty optimization using design sensitivities and parallel computing: application to MEMS. Comput. Struct. 83, 1134–1149 (2005) · doi:10.1016/j.compstruc.2004.11.021 |
| [79] | Kang Z., Luo Y.J.: Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex models. Comput. Methods Appl. Mech. Eng. 198, 3228–3238 (2009) · Zbl 1230.74153 · doi:10.1016/j.cma.2009.06.001 |
| [80] | Kanno Y., Takewaki I.: Sequential semidefinite program for maximum robustness design of structures under load uncertainty. J. Optimiz. Theory App. 130, 265–287 (2006) · Zbl 1278.90300 · doi:10.1007/s10957-006-9102-z |
| [81] | Guo X., Bai W., Zhang W.S. et al.: Confidence structural robust design and optimization under stiffness and load uncertainties. Comput. Methods Appl. Mech. Eng. 198, 3378–3399 (2009) · Zbl 1230.74148 · doi:10.1016/j.cma.2009.06.018 |
| [82] | Guo, X., Du, J.M., Gao, X.X.: Confidence structural robust optimization by nonlinear semidefinite programming based single-level formulation. Int. J. Numer. Methods Eng. doi: 10.1002/nme.3083 (2010) |
| [83] | Chen S.H., Wu J., Yu Y.D. et al.: Interval optimization for uncertain structures. Finite Elem. Anal. Des. 40, 1379–1398 (2004) · doi:10.1016/j.finel.2003.09.006 |
| [84] | Jiang C., Han X., Liu G.R.: A sequential nonlinear interval number programming method for uncertain structures. Comput. Methods Appl. Mech. Eng. 197, 4250–4265 (2008) · Zbl 1194.74247 · doi:10.1016/j.cma.2008.04.027 |
| [85] | Jiang C., Han X., Liu G.R.: Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval. Comput. Methods Appl. Mech. Eng. 196, 4791–4800 (2007) · Zbl 1173.74364 · doi:10.1016/j.cma.2007.03.024 |
| [86] | Qiu Z.P.: Convex models and interval analysis method to predict the effect of uncertain-but-bounded parameters on the buckling of composite structures. Comput. Methods Appl. Mech. Eng. 194, 2175–2189 (2005) · Zbl 1091.74018 · doi:10.1016/j.cma.2004.07.018 |
| [87] | Qiu Z.P., Wang X.J., Chen J.Y.: Exact bounds for the static response set of structures with uncertain-but-bounded parameters. Int. J. Solids Struct. 43, 6574–6593 (2006) · Zbl 1120.74624 · doi:10.1016/j.ijsolstr.2006.01.012 |
| [88] | Kanno Y., Takewaki I.: Semidefinite programming for uncertain linear equations in static analysis of structures. Comput. Methods Appl. Mech. Eng. 198, 102–115 (2008) · Zbl 1194.74188 · doi:10.1016/j.cma.2008.04.003 |
| [89] | Kanno Y., Takewaki I.: Semidefinite programming for dynamic steady-state analysis of structures under uncertain harmonic loads. Comput. Methods Appl. Mech. Eng. 198, 3239–3261 (2009) · Zbl 1230.74090 · doi:10.1016/j.cma.2009.06.005 |
| [90] | Guo X., Bai W., Zhang W.S.: Extreme structural response analysis of truss structures under material uncertainty via linear mixed 0-1 programming. Int. J. Numer. Methods Eng. 76, 253–277 (2008) · Zbl 1195.74133 · doi:10.1002/nme.2298 |
| [91] | Du J., Olhoff N.: Minimization of sound radiation from vibrating bi-material structures using topology optimization. Struct. Multidisc. Optim. 33, 305–321 (2007) · doi:10.1007/s00158-006-0088-9 |
| [92] | Liu B.S., Zhao G.Z., Li A.: PEM based sensitivity analysis for acoustic radiation problems of random responses. J. Vib. Acoust. 132, 021012 (2010) · doi:10.1115/1.4000776 |
| [93] | Wang B., Cheng G.D., Jiang L.: Design of multi-tubular heat exchanger for optimum efficiency of heat dissipation. Eng. Optimiz. 40, 767–788 (2008) · doi:10.1080/03052150802054027 |
| [94] | Veselago V.G.: The electrodynamics of substances with simultaneously negative value of {\(\epsilon\)} and {\(\mu\)}. Sov. Phys. Usp. 10, 509–514 (1968) · doi:10.1070/PU1968v010n04ABEH003699 |
| [95] | Xu W.K., Liu S.T., Dong Y.Z.: Design of structural left-handed material based on topology optimization. J. Wuhan Univ. Tech. Mater. Sci. Ed. 25, 282–286 (2010) · doi:10.1007/s11595-010-2282-2 |
| [96] | Diaz A., Sigmund O.: A topology optimization method for design of negative permeability metamaterials. Struct. Multidisc. Optim. 43, 163–177 (2010) · Zbl 1274.74262 · doi:10.1007/s00158-009-0416-y |
| [97] | Sigmund O., Jensen J.S.: Systematic design of phononic band gap materials and structures by topology optimization. Philos. Trans. R. Soc. Lond. A 361, 1001–1019 (2003) · Zbl 1067.74053 · doi:10.1098/rsta.2003.1177 |
| [98] | Duhring M.B., Sigmund O., Feurer T.: Design of photonic band gap fibers by topology optimization. J. Opt. Soc. Am. B 27, 51–58 (2010) · doi:10.1364/JOSAB.27.000051 |
| [99] | Jensen J.S., Sigmund O.: Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide. J. Opt. Soc. Am. B 22, 1191–1198 (2005) · doi:10.1364/JOSAB.22.001191 |
| [100] | Stainko R., Sigmund O.: Tailoring dispersion properties of photonic crystal waveguides by topology optimization. Wave Random Complex 17, 477–489 (2007) · Zbl 1191.78049 · doi:10.1080/17455030701501851 |
| [101] | De Dood M.J.A., Snoeks E., Moroz A., Polman A.: Design and optimization of 2D photonic crystal waveguides based on silicon. Opt. Quant. Electron. 34, 145–159 (2002) · doi:10.1023/A:1013352814225 |
| [102] | Diaz A., Haddow A.G., Ma L.: Design of band-gap grid structures. Struct. Multidisc. Optim. 29, 418–431 (2005) · doi:10.1007/s00158-004-0497-6 |
| [103] | Guenneau S., Movchan A., Pétursson G., Ramakrishna S.A.: Acoustic metamaterials for sound focusing and confinement. New J. Phys. 9, 399–406 (2007) · doi:10.1088/1367-2630/9/11/399 |
| [104] | Laude V., Khelif A., Benchabane S., Wilm M.: Phononic band-gap guidance of acoustic modes in photonic crystal fibers. Phys. Rev. B 71, 045107 (2005) · doi:10.1103/PhysRevB.71.045107 |
| [105] | Luo Z., Yang J.Z., Chen L.P.: A new procedure for aerodynamic missile designs using topological optimization approach of continuum structures. Aerosp. Sci. Technol. 10, 364–373 (2006) · Zbl 1195.74136 · doi:10.1016/j.ast.2005.12.006 |
| [106] | Niu, F., Wang, B.: The topology optimization design of complex structure based on super-element. In: Conference on Structural and Multidisciplinary Optimization-Theory and Applications, 3–4 Sep 2009, Dalian, China (2009) |
| [107] | Maute K., Allen M.: Conceptual design of aeroelastic structures by topology optimization. Struct. Multidisc. Optim. 27, 27–42 (2004) · doi:10.1007/s00158-003-0362-z |
| [108] | Maute, K., Reich, G.W.: An aeroelastic topology optimization approach for adaptive wing design. In: 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 19–22 Apr 2004, Palm Springs, California (2004) |
| [109] | Schuhmacher, G.: Numerical optimization methods in the aerospace design process. In: 2nd European Hyperworks Technology Conference, Sep 30–Oct 1 2008, Strasbourg, France (2008) |
| [110] | Chang C.J., Zheng B., Gea H.C.: Automated design of thin-walled packaging structures. Struct. Multidisc. Optim. 35, 601–608 (2008) · doi:10.1007/s00158-007-0170-y |
| [111] | Chang C.H., Yang R.J., Li G. et al.: Multidisciplinary design optimization on vehicle tailor rolled blank design. Struct. Multidisc. Optim. 35, 551–560 (2008) · doi:10.1007/s00158-007-0152-0 |
| [112] | Kirsch U.: Reanalysis of Structures–a Unified Approach for Linear, Nonlinear, Static and Dynamic Systems. Springer, Berlin (2008) · Zbl 1141.74001 |
| [113] | Shin M.K., Park K.J., Park G.J.: Optimization of structures with nonlinear behavior using equivalent loads. Comput. Methods Appl. Mech. Eng. 196, 1154–1167 (2007) · Zbl 1120.74707 · doi:10.1016/j.cma.2006.09.001 |
| [114] | Kim Y., Park G.J.: Nonlinear dynamic response structural optimization using equivalent static loads. Comput. Methods Appl. Mech. Eng. 199, 660–676 (2010) · Zbl 1227.74045 · doi:10.1016/j.cma.2009.10.014 |
| [115] | Madsen J.I., Shyy W., Haftka R.T.: Response surface techniques for diffuser shape optimization. AIAA J. 38, 1512–1518 (2000) · doi:10.2514/2.1160 |
| [116] | Kirsch U.: On singular topologies in optimum structural design. Struct. Optim. 2, 133–142 (1990) · doi:10.1007/BF01836562 |
| [117] | Cheng G.D., Jiang Z.: Study on topology optimization with stress constraint. Eng. Optimiz. 20, 129–148 (1992) · doi:10.1080/03052159208941276 |
| [118] | Cheng G.D., Guo X.: {\(\epsilon\)}-Relaxed approach in structural topology optimization. Struct. Optim. 13, 258–266 (1997) · doi:10.1007/BF01197454 |
| [119] | Rozvany G.I.N.: On design dependent constraints and singular topologies. Struct. Multidisc. Optim. 21, 164–172 (2001) · doi:10.1007/s001580050181 |
| [120] | Sui Y.K., Peng X.R.: The ICM method with objective function transformed by variable discrete condition for continuum structure. Acta Mech. Sin. 22, 68–75 (2006) · Zbl 1200.74117 · doi:10.1007/s10409-005-0088-9 |
| [121] | Sui Y.K., Du J.Z., Guo Y.Q.: Independent continuous mapping for topological optimization of frame structures. Acta. Mech. Sin. 22, 611–619 (2006) · Zbl 1202.74134 · doi:10.1007/s10409-006-0031-8 |
| [122] | Gao T., Zhang W.H., Zhu J.H. et al.: Topology optimization of heat conduction problem involving design dependent heat load effect. Finite Elem. Anal. Des. 44, 805–813 (2008) · doi:10.1016/j.finel.2008.06.001 |
| [123] | Gao T., Zhang W.H.: Topology optimization involving thermo-elastic stress loads. Struct. Multidisc. Optim. 42, 725–738 (2010) · Zbl 1274.74333 · doi:10.1007/s00158-010-0527-5 |
| [124] | Stolpe M., Svanberg K.: Modelling topology optimization problems as linear mixed 0–1 programs. Int. J. Numer. Methods Eng. 57, 723–739 (2003) · Zbl 1062.74593 · doi:10.1002/nme.700 |
| [125] | Achtziger W., Stolpe M.: Truss topology optimization with discrete design variables-guaranteed global optimality and benchmark examples. Struct. Multidisc. Optim. 34, 1–20 (2007) · Zbl 1273.74396 · doi:10.1007/s00158-006-0074-2 |
| [126] | Rasmussen M.H., Stolpe M.: Global optimization of discrete truss topology design problems using a parallel cut-and-branch method. Comput. Struct. 86, 1527–1538 (2008) · doi:10.1016/j.compstruc.2007.05.019 |
| [127] | Kanno Y., Guo X.: A mixed integer programming for robust truss topology optimization with stress constraints. Int. J. Numer. Methods Eng. 83, 1675–1699 (2010) · Zbl 1202.74133 · doi:10.1002/nme.2871 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
