Topology optimization approaches

Aage N, Sigmund O (2013) Density interpolations for electromagnetic topology optimization schemes. in preparation

Aage N, Nobel-Jørgensen M, Andreasen C, Sigmund O (2013) Interactive topology optimization on hand-held devices. Struct Multidiscip Optim 47:1–6. doi: 10.1007/s00158-012-0827-z

Allaire G, Francfort GA (1993) A numerical algorithm for topology and shape optimization. In: Bendsøe MP, Soares Mota CA (eds) Topology optimization of structures. Kluwer, Boston, pp 239–248

Allaire G, Kohn RV (1993) Topology optimization and optimal shape design using homogenization. In: Bendsøe MP, Soares Mota CA (eds) Topology design of structures. Kluwer, Boston, pp 207–218

Allaire G, Jouve F (2008) Minimum stress optimal design with the level set method. Eng AnalBound Elemen 32(11):909–918. doi: 10.1016/j.enganabound.2007.05.007

Allaire G, Jouve F, Toader AM (2002) A level-set method for shape optimization. C R Math 334(12):1125–1130. doi: 10.1016/S1631-073X(02)02412-3

Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393

Allaire G, de Gournay F, Jouve F, Toader A (2005) Structural optimization using topological and shape sensitivity via a level set method. Control Cybern 34(1):59–80

Ambrosio L, Buttazzo G (1993) An optimal design problem with perimeter penalization. Calc Var 1:55–69

Amstutz S (2011) Connections between topological sensitivity analysis and material interpolation schemes in topology optimization. Struct Multidiscip Optim 43(6):755–765. doi: 10.1007/s00158-010-0607-6

Amstutz S, Novotny A (2010) Topological optimization of structures subject to von mises stress constraints. Struct Multidiscip Optim 41:407–420. doi: 10.1007/s00158-009-0425-x

Andreassen E, Clausen A, Schevenels M, Lazarov B, Sigmund O (2011) Efficient topology optimization in matlab using 88 lines of code. Struct Multidiscip Optim 43:1–16. doi: 10.1007/s00158-010-0594-7 . MATLAB code available online at: www.topopt.dtu.dk

Ansola R, Vegueria E, Canales J, Tarrago J (2007) A simple evolutionary topology optimization procedure for compliant mechanism design. Finite Elem Anal Des 44(1–2):53–62. doi: 10.1016/j.finel.2007.09.002

Ansola R, Vegueria E, Canales J (2010) 3d compliant mechanisms synthesis by a finite element addition procedure. Finite Elem Anal Des 46(9):760–769. doi: 10.1016/j.finel.2010.04.006

Arnout S, Firl M, Bletzinger KU (2012) Parameter free shape and thickness optimisation considering stress response. Struct Multidiscip Optim 45:801–814. doi: 10.1007/s00158-011-0742-8

Beckers M (1999) Topology optimization using a dual method with discrete variables. Struct Optim 17(1):14–24

Beckers M (2000) Dual methods for discrete structural optimization problems. Int J Numer Meth Eng 48:1761–1784

Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202

Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224

Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9–10):635–654

Bendsøe MP, Sigmund O (2004) Topology optimization-theory, methods and applications. Springer, Berlin

Bendsøe MP, Díaz AR, Kikuchi N (1993) Topology and generalized layout optimization of elastic structures. In: Bendsøe MP, Soares Mota CA (eds) Topology design of structures. Kluwer, Boston, pp 159–206

Bendsøe MP, Guedes JM, Haber RB, Pedersen P, Taylor JE (1994) An analytical model to predict optimal material properties in the context of optimal structural design. Trans ASME J Appl Mech 61(4):930–937

Bletzinger KU, Maute K (1997) Towards generalized shape and topology optimization. Eng Optim 29(1–4):201–216

Bonnet M, Guzina B (2004) Sounding of finite solid bodies by way of topological derivative. Int J Numer Methods Eng 61(13):2344–2373

Borrvall T (2001) Topology optimization of elastic continua using restriction. Arch Comput Methods Eng 8(4):351– 385

Borrvall T, Petersson J (2001) Topology optimization using regularized intermediate density control. Comput Methods Appl Mech Eng 190:4911–4928

Borrvall T, Petersson J (2003) Topology optimization of fluids in stokes flow. Int J Numer Methods Fluids 41:77–107. doi: 10.1002/fld.426

Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158

Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. ESAIM Control Optim Calc Var 9:19–48

Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459

Burger M, Osher S (2005) A survey in mathematics for industry—a survey on level set methods for inverse problems and optimal design. Eur J Appl Math 16:263–301. doi: 10.1017/S0956792505006182

Burger M, Stainko R (2006) Phase-field relaxation of topology optimization with local stress constraints. SIAM J Control Optim 45(4):1447–1466. doi: 10.1137/05062723X

Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194(1):344–362. doi: 10.1016/j.jcp.2003.09.033

Cea J, Garreau S, Guillaume P, Masmoudi M (2000) The shape and topological optimizations connection. Comput Methods Appl Mech Eng 188:713–726

Challis V, Guest J (2009) Level set topology optimization of fluids in Stokes flow. Int J Numer Methods Eng 79(10):1284–1308. doi: 10.1002/nme.2616

Challis V, Roberts A, Wilkins A (2008) Design of three dimensional isotropic microstructures for maximized stiffness and conductivity. Int J Solids Struct 45(14–15):4130–4146. doi: 10.1016/j.ijsolstr.2008.02.025

Challis VJ (2010) A discrete level-set topology optimization code written in matlab. Struct Multidiscip Optim 41(3):453–464. doi: 10.1007/s00158-009-0430-0

Chen B, Kikuchi N (2001) Topolgy optimization with design-dependent loads. Finite Elem Anal Des 37:57–70

Chen S, Chen W (2011) A new level-set based approach to shape and topology optimization under geometric uncertainty. Struct Multidiscip Optim 44(1):1–18. doi: 10.1007/s00158-011-0660-9

Cheng GD, Guo X (1997) 𝜖-relaxed approach in topology optimization. Struct Optim 13:258–266

Cheng GD, Jiang Z (1992) Study on topology optimization with stress constraints. Eng Optim 20:129–148

Cho S, Ha SH (2009) Isogeometric shape design optimization: exact geometry and enhanced sensitivity. Struct Multidiscip Optim 38:53–70. doi: 10.1007/s00158-008-0266-z

Christiansen A, Nobel-Jørgensen M, Aage N, Sigmund O, Bærentzen J (2013) Topology optimization using an explicit interface representation. in preparation

de Ruiter M, van Keulen F (2004) Topology optimization using a topology description function. Struct Multidiscip Optim 26(6):406–416. doi: 10.1007/s00158-003-0375-7

Deaton J, Grandhi R (2013) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 1–38. doi: 10.1007/s00158-013-0956-z

Díaz AR, Sigmund O (1995) Checkerboard patterns in layout optimization. Struct Optim 10(1):40–45

Du J, Olhoff N (2004a) Topological optimization of continuum structures with design-dependent surface loading-Part I: new computational approach for 2D problems. Struct Multidiscip Optim 27:151–165

Du J, Olhoff N (2004b) Topological optimization of continuum structures with design-dependent surface loading—Part II: algorithm and examples for 3D problems. Struct Multidiscip Optim 27:166–177

Dühring MB, Sigmund O, Feurer T (2010) Design of photonic-bandgap fibers by topology optimization. J Opt Soc Am B 27:51–58

Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress coinstraints. Int J Numer Methods Eng 43(8):1453–1478

Duysinx P, Sigmund O (1998) New developments in handling stress constraints in optimal material distributions. In: 7th symposium on multidiciplinary analysis and optimization, AIAA/USAF/NASA/ISSMO, AIAA-98-4906, pp 1501–1509

Erentok A, Sigmund O (2011) Topology optimization of sub-wavelength antennas. IEEE Trans Antennas Propag 59(1):58–69. doi: 10.1109/TAP.2010.2090451

Eringen A (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703. doi: 10.1063/1.332803

Eschenauer HA, Kobelev VV, Schumacher A (1994) Bubble method for topology and shape optimization of structures. Struct Optim 8:42–51

FE-Design (2011) Tosca structure manual. FE-Design GMBH, version 7.1 edn., www.fe-design.com

Fuchs MB, Shemesh NNY (2004) Density-based topological design of structures subjected to water pressure using a parametric loading surface. Struct Multidiscip Optim 28(1):11–19

Gersborg-Hansen A, Sigmund O, Haber RB (2005) Topology optimization of channel flow problems. Struct Multidiscip Optim 30(3):181–192. doi: 10.1007/s00158-004-0508-7

Gill P, Murray W, Saunders MA (2005) SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev 47(1):99–131

Gomes A, Suleman A (2006) Application of spectral level set methodology in topology optimization. Struct Multidiscip Optim 31(6):430–443. doi: 10.1007/s00158-006-0005-2

Guest J (2009a) Imposing maximum length scale in topology optimization. Struct Multidiscip Optim 37(5):463–473. doi: 10.1007/s00158-008-0250-7

Guest J (2009b) Topology optimization with multiple phase projection. Comput Method Appl Mech Eng 199(1–4):123–135. doi: 10.1016/j.cma.2009.09.023

Guest J, Prevost J, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254

Guest J, Asadpoure A, Ha SH (2011) Eliminating beta-continuation from heaviside projection and density filter algorithms. Struct Multidiscip Optim 44(4):443–453. doi: 10.1007/s00158-011-0676-1

Guo X, Zhang W, Zhang L (2013) Robust structural topology optimization considering boundary uncertainties. Comput Methods Appl Mech Eng 253:356–368. doi: 10.1016/j.cma.2012.09.005

Haber RB, Bendsøe MP, Jog CS (1996) Perimeter constrained topology optimization of continuum structures. In: IUTAM symposium on optimization of mechanical systems (Stuttgart, 1995). Kluwer, Dordrecht, pp 113–120

Hammer VB, Olhoff N (2000) Topology optimization of continuum structures subjected to pressure loading. Struct Multidiscip Optim 19:85–92

Huang X, Xie YM (2010a) A further review of ESO type methods for topology optimization. Struct Multidiscip Optim 41:671–683. doi: 10.1007/s00158-010-0487-9

Huang Y, Xie YM (2010b) Evolutionary topology optimization of geometrically and materially nonlinear structures under prescribed design load. Struct Eng Mech 34(5):581–595

Jansen M, Lazarov B, Schevenels M, Sigmund O (2013) On the similarities between micro/nano lithography and topology optimization projection methods. Struct Multidiscip Optim 1–14. doi: 10.1007/s00158-013-0941-6

Jensen JS, Sigmund O (2005) Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide. J Opt Soc Am B 22(6):1191–1198

Jensen JS, Sigmund O (2011) Topology optimization for nano-photonics. Laser Photonics Rev 5(2):308–321. doi: 10.1002/lpor.201000014

Jog CS, Haber RB (1996) Stability of finite element models for distributed-parameter optimization and topology design. Comput Methods Appl Mech Eng 130(3–4):203–226

Kawamoto A, Matsumori T, Yamasaki S, Nomura T, Kondoh T, Nishiwaki S (2011) Heaviside projection based topology optimization by a PDE-filtered scalar function. Struct Multidiscip Optim:1–6. doi: 10.1007/s00158-010-0562-2

Kawamoto A, Matsumori T, Yamasaki S, Nomura T, Kondoh T, Yamasaki S, Nishiwaki S (2013) Topology optimization by a time-dependent diffusion equation. Int J Numer Methods Eng 93(8):795–817. doi: 10.1002/nme.4407

Kim DH, Lee SB, Kwak BM, Kim HG, Lowther D (2008) Smooth boundary topology optimization for electrostatic problems through the combination of shape and topological design sensitivities. IEEE Trans Magn 44(6):1002–1005

Kita E, Toyoda T (2000) Structural design using cellular automata. Struct Multidiscip Optim 19(1):64–73

Klarbring A, Torstenfelt B (2010) Dynamical systems and topology optimization. Struct Multidiscip Optim 42(2):179–192. doi: 10.1007/s00158-010-0479-9

Kocvara M, Stingl M, Zowe J (2008) Free material optimization: recent progress. Optimization 57(1):79–100. doi: 10.1080/02331930701778908

Kreissl S, Maute K (2012) Levelset based fluid topology optimization using the extended finite element method. Struct Multidiscip Optim 3:311–326. doi: 10.1007/s00158-012-0782-8

Kreissl S, Pingen G, Evgrafov A, Maute K (2010) Topology optimization of flexible micro-fluidic devices. Struct Multidiscip Optim 42(4):495–516

Lazarov B, Sigmund O (2011) Filters in topology optimization as a solution to Helmholtz type differential equation. Int J Numer Methods Eng 86(6):765–781. doi: 10.1002/nme.3072

Lazarov B, Schevenels M, Sigmund O (2012a) Topology optimization considering material and geometric uncertainties using sparse grid stochastic collocation method. Struct Multidiscip Optim 46:597–612. doi: 10.1007/s00158-012-0791-7

Lazarov B, Schevenels M, Sigmund O (2012b) Topology optimization with geometric uncertainties by perturbation techniques. Int J Numer Methods Eng 90(11):1321–1336. doi: 10.1002/nme.3361

Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41:605–620. doi: 10.1007/s00158-009-0440-y

Le C, Bruns T, Tortorelli D (2011) A gradient-based, parameter-free approach to shape optimization. Comput Methods Appl Mech Eng 200(9–12):985–996. doi: 10.1016/j.cma.2010.10.004

Liu Z, Korvink JG, Huang R (2005) Structure topology optimization: fully coupled level set method via FEMLAB. Struct Multidiscip Optim 29(6):407–417

Luo Z, Tong L, Wang M, Wang S (2007) Shape and topology optimization of compliant mechanisms using a parameterization level set method. J Comput Phys 227(1):680–705. doi: 10.1016/j.jcp.2007.08.011

Luo J, Luo Z, Chen S, Tong L, Wang MY (2008a) A new level set method for systematic design of hinge-free compliant mechanisms. Comput Methods Appl Mech Eng 198(2):318–331

Luo Z, Wang M, Wang S, Wei P (2008b) A level set-based parameterization method for structural shape and topology optimization. Int J Numer Methods Eng 76(1):1–26. doi: 10.1002/nme.2092

Luo Y, Wang M, Kang Z (2013) An enhanced aggregation method for topology optimization with local stress constraints. Comput Methods Appl Mech Eng 254:31–41. doi: 10.1016/j.cma.2012.10.019

Mattheck C, Burkhardt S (1990) A new method of structural shape optimization based on biological growth. Int J Fatigue 12(3):185–190

Maute K, Ramm E (1995) General shape optimizationan integrated model for topology and shape optimization. In: Olhoff N, Rozvany G (eds) Proceedings of the first world congress of structural and multidisciplinary optimization. Pergamon, Oxford, pp 299–306

Maute K, Kreissl S, Makhija D, Yang R (2011) Topology optimization of heat conduction in nanocomposites. In: 9th world congress on structural and multidisciplinary optimization. Shizuoka, Japan

Mei Y (2004) A level set method for structural topology optimization and its applications. Adv Eng Softw 35(7):415–441. doi: 10.1016/j.advengsoft.2004.06.004

Michell AGM (1904) The limit of economy of material in frame structures. Philos Mag 8(6):589–597

Missoum S, Gurdal Z, Setoodeh S (2005) Study of a new local update scheme for cellular automata in structural design. Struct Multidiscip Optim 29(2):103–112. doi: 10.1007/s00158-004-0464-2

Misztal M, Bærentzen J (2012) Topology adaptive interface tracking using the deformable simplicial complex. ACM Trans Graph 31(3):24:1–24:12. doi: 10.1145/2167076.2167082

Mlejnek HP (1992) Some aspects of the genesis of structures. Struct Optim 5:64–69

Niordson FI (1983) Optimal design of plates with a constraint on the slope of the thickness function. Int J Solids Struct 19:141–151

Norato J, Haber R, Tortorelli D, Bendsoe M (2004) A geometry projection method for shape optimization. Int J Numer Methods Eng 60(14):2289–2312

Norato J, Bendsøe M, Haber R, Tortorelli D (2007) A topological derivative method for topology optimization. Struct Multidiscip Optim 33:375–386. doi: 10.1007/s00158-007-0094-6

Novotny A, Feijoo R, Taroco E, Padra C (2003) Topological sensitivity analysis. Comput Methods Appl Mech Eng 192(7–8):803–829. doi: 10.1016/S0045-7825(02)00599-6

Novotny A, Feijoo R, Taroco E, Padra C (2007) Topological sensitivity analysis for three-dimensional linear elasticity problem. Comput Methods Appl Mech Eng 196(41):4354–4364

Olhoff N, Bendsøe MP, Rasmussen J (1992) On CAD-integrated structural topology and design optimization. Comput Methods Appl Mech Eng 89:259–279

Olsson E, Kreiss G (2005) A conservative level set method for two phase flow. J Comput Phys 210(1):225–246. doi: 10.1016/j.jcp.2005.04.007

Olsson E, Kreiss G, Zahedi S (2007) A conservative level set method for two phase flow ii. J Comput Phys 225(1):785–807. doi: 10.1016/j.jcp.2006.12.027

Osher S, Fedkiw R (2003) Level set methods and dynamic implicit surfaces. Springer, New York

Osher S, Sethian J (1988) Fronts propagating with curvature-dependent speed-algorithms based on hamilton-jacobi formulations. J Comput Phys 79(1):12–49

Otomori M, Yamada T, Izui K, Nishiwaki S (2011) Level set-based topology optimisation of a compliant mechanism design using mathematical programming. Mech Sci 2(1):91–98

Papalambros PY, Chirehdast M (1993) Integrated structural optimization system. In: Bendsøe MP, Mota Soares CA (eds) Topology design of structures. Kluwer, Boston, pp 501–514

Pedersen CBW, Buhl T, Sigmund O (2001) Topology synthesis of large-displacement compliant mechanisms. Int J Numer Methods Eng 50(12):2683–2705

Pedersen NL (2000) Maximization of eigenvalues using topology optimization. Struct Multidiscip Optim 20(1):2–11

Petersson J (1999) A finite element analysis of optimal variable thickness sheets. SIAM J Num Anal 36:1759–1778

Petersson J, Sigmund O (1998) Slope constrained topology optimization. IInt J Numer Methods Eng 41(8):1417–1434

Pingen G, Waidmann M, Evgrafov A, Maute K (2010) A parametric level-set approach for topology optimization of flow domains. Struct Multidiscip Optim 41:117–131

Qian X, Sigmund O (2012) Topological design of electromechanical actuators with robustness toward over-and under-etching. Comput Method Appl Mech Eng 253:237–251

Raulli M, Maute K (2005) Topology optimization of electrostatically actuated microsystems. Struct Multidiscip Optim 30(5):342–359

Rozvany G (2009) A critical review of established methods of structural topology optimization. Struct Multidiscip Optim 37:217–237. doi: 10.1007/s00158-007-0217-0

Rozvany G, Sobieszczanski-Sobieski J (1992) New optimality criteria methods: forcing uniqueness of the adjoint strains by corner-rounding at constraint intersections. Struct Optim 4:244–246. doi: 10.1007/BF01742752

Ru C, Aifantis E (1993) A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech 101:59–68. doi: 10.1007/BF01175597

Schevenels M, Lazarov B, Sigmund O (2011) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Method Appl Mech Eng 200(49–52):3613–3627. doi: 10.1016/j.cma.2011.08.006

Schury F, Stingl M, Wein F (2012) Slope constrained material design. Struct Multidiscip Optim 46:813–827. doi: 10.1007/s00158-012-0795-3

Seo YD, Kim HJ, Youn SK (2010) Isogeometric topology optimization using trimmed spline surfaces. Comput Method Appl Mech Eng 199(49–52):3270–3296. doi: 10.1016/j.cma.2010.06.033

Sethian JA (1999) Level set methods and fast marching methods. Cambridge University Press, Cambridge

Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163(2):489–528

Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25(4):493–524

Sigmund O (2001a) A 99 line topology optimization code written in MATLAB. Struct Multidiscip Optim 21:120–127. doi: 10.1007/s001580050176 . MATLAB code available online at: www.topopt.dtu.dk

Sigmund O (2001b) Design of multiphysics actuators using topology optimization-part II: two-material structures. Comput Method Appl Mech Eng 190(49–50):6605–6627. doi: 10.1016/S0045-7825(01)00252-3

Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4–5):401–424

Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech Sinica 25(2):227–239. doi: 10.1007/s10409-009-0240-z

Sigmund O (2011) On the usefulness of non-gradient approaches in topology optimization. Struct Multidiscip Optim 43(5):589–596. doi: 10.1007/s00158-011-0638-7

Sigmund O, Clausen P (2007) Topology optimization using a mixed formulation: an alternative way to solve pressure load problems. Comput Method Appl Mech Eng 196(13–16):1874–1889

Sigmund O, Maute K (2012) Sensitivity filtering from a continuum mechanics perspective. Struct Multidiscip Optim 46(4):471–475. doi: 10.1007/s00158-012-0814-4

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(1):68–75

Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization method. J Mech Phys Solids 45(6):1037–1067

Sokołowski J, Zochowski A (1999) On the topological derivative in shape optimization. SIAM J Control Opt 37:1251–1272

Stolpe M, Bendsøe M (2010) Global optima for the Zhou-Rozvany problem. Struct Multidiscip Optim 43:151–164. doi: 10.1007/10.1007

Stolpe M, Svanberg K (2001a) An alternative interpolation scheme for minimum compliance optimization. Struct Multidiscip Optim 22(2):116–124. doi: 10.1007/s001580100129

Stolpe M, Svanberg K (2001b) On the trajectories of penalization methods for topology optimization. Struct Multidiscip Optim 21:128–139

Suresh K (2010) A 199-line matlab code for pareto-optimal tracing in topology optimization. Struct Multidiscip Optim 42:665–679. doi: 10.1007/s00158-010-0534-6

Svanberg K (1987) The method of moving asymptotes-a new method for structural optimization. Int J Numer Methods Eng 24:359–373

Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12(2):555–573. doi: 10.1137/S1052623499362822

Svanberg K, Werme M (2006) Topology optimization by a neighbourhood search method based on efficient sensitivity calculations. Int J Numer Methods Eng 67(12):1670–1699. doi: 10.1002/nme.1677

Talischi C, Paulino G, Pereira A, Menezes I (2012) Polytop: a matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Struct Multidiscip Optim 45:329–357. doi: 10.1007/s00158-011-0696-x

van Dijk N, Langelaar M, van Keulen F (2012) Explicit level-set-based topology optimization using an exact heaviside function and consistent sensitivity analysis. Int J Numer Methods Eng 91(1):67–97. doi: 10.1002/nme.4258

van Dijk N, Maute K, Langelaar M, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 1–36. doi: 10.1007/s00158-013-0912-y

Van Miegroet L, Duysinx P (2007) Stress concentration minimization of 2d filets using x-fem and level set description. Struct Multidiscip Optim 33(4–5):425–438. doi: 10.1007/s00158-006-0091-1

Vigdergauz SB (1999) Energy-minimizing inclusions in a planar elastic structure with macroisotropy. Struct Optim 17(2–3):104–112

Wächter A, Biegler L (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57

Wall W, Frenzel M, Cyron C (2008) Isogeometric structural shape optimization. Comput Methods Appl Mech Eng 197(33–40):2976–2988. doi: 10.1016/j.cma.2008.01.025

Wallin M, Ristinmaa M, Askfelt H (2012) Optimal topologies derived from a phase-field method. Struct Multidiscip Optim 45(2):171–183. doi: 10.1007/s00158-011-0688-x

Wang M, Wang X (2004a) Color level sets: a multi-phase method for structural topology optimization with multiple materials. Comput Methods Appl Mech Eng 193(6–8):469–496. doi: 10.1016/j.cma.2003.10.008

Wang M, Wang X (2004b) Pde-driven level sets, shape sensitivity and curvature flow for structural topology optimization. Comput Model Eng Sci 6(4):373–395

Wang M, Zhou S (2004) Phase field: a variational method for structural topology optimization. Comput Model Eng Sci 6(6):547–566

Wang S, Wang M (2006a) A moving superimposed finite element method for structural topology optimization. Int J Numer Methods Eng 65(11):1892–1922

Wang S, Wang M (2006b) Radial basis functions and level set method for structural topology optimization. Int J Numer Methods Eng 65(12). doi: 10.1002/nme.1536

Wang M, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246

Wang M, Zhou S, Ding H (2004a) Nonlinear diffusions in topology optimization. Struct Multidicip Optim 28(4):262–276. doi: 10.1007/s00158-004-0436-6

Wang X, Wang M, Guo D (2004b) Structural shape and topology optimization in a level-set-based framework of region representation. Struct Multidiscip Optim 27(1–2):1–19. doi: 10.1007/s00158-003-0363-y

Wang SY, Lim KM, Khoo BC, Wang MY (2007) An unconditionally time-stable level set method and its application to shape and topology optimization. Comput Model Eng Sci 21(1):1–40

Wang F, Lazarov B, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784. doi: 10.1007/s00158-010-0602-y

Wei P, Wang M (2009) Piecewise constant level set method for structural topology optimization. Int J Numer Methods Eng 78(4):379–402. doi: 10.1002/nme.2478

Wei P, Wang M, Xing X (2010) A study on x-fem in continuum structural optimization using a level set model. Comput Aided Des 42(8):708–719. doi: 10.1016/j.cad.2009.12.001

Wu CH, Tseng KY (2010) Topology optimization of structures using modified binary differential evolution. Struct Multidiscip Optim 42:939–953

Xia Q, Shi T, Liu S, Wang M (2012) A level set solution to the stress-based structural shape and topology optimization. Comput Struct 90–91:55–64. doi: 10.1016/j.compstruc.2011.10.009

Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896

Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on heaviside funtions. Struct Mulidiscip Optim 41:495–505

Yamada T, Izui K, Nishiwaki S, Takezawa A (2010) A topology optimization method based on the level set method incorporating a fictitious interface energy. Comput Methods Appl Mech Eng 199(45–48):2876–2891. doi: 10.1016/j.cma.2010.05.013

Yamada T, Izui K, Nishiwaki S (2011) A level set-based topology optimization method for maximizing thermal diffusivity in problems including design-dependent effects. J Mech Des 133(3):031011. doi: 10.1115/1.4003684

Yamasaki S, Nomura T, Kawamoto A, Sato K, Izui K, Nishiwaki S (2010) A level set based topology optimization method using the discretized signed distance function as the design variables. Struct Multidiscip Optim 41(5):685–698

Yamasaki S, Nomura T, Kawamoto A, Sato K, Nishiwaki S (2011) A level set-based topology optimization method targeting metallic waveguide design problems. Int J Numer Methods Eng 87(9):844–868. doi: 10.1002/nme.3135

Yoon GH, Sigmund O (2008) A monolithic approach for topology optimization of electrostatically actuated devices. Comput Methods Appl Mech Eng 197:4062–4075. doi: 10.1016/j.cma.2008.04.004

Yoon GH, Jensen JS, Sigmund O (2007) Topology optimization for acoustic structure interaction problems. Int J Numer Methods Eng 70(9):1049–1075

Young V, Querin OM, Steven GP, Xie YM (1999) 3D and multiple load case bi-directional evolutionary structural optimization (BESO). Struct Optim 18(2–3):183–192

Zhou M, Rozvany GIN (1991) The COC algorithm, part II: topological, geometry and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336

Zhou M, Rozvany GIN (2001) On the validity of ESO type methods in topology optimization. Struct Multidiscip Optim 21:80–83

Zuo Z, Xie Y, Huang X (2012) Evolutionary topology optimization of structures with multiple displacement and frequency constraints. Adv Struct Eng 15(2):359–372