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Structural and Multidisciplinary Optimization

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01.09.2013 | Review Article

Level-set methods for structural topology optimization: a review

verfasst von: N. P. van Dijk, K. Maute, M. Langelaar, F. van Keulen

Erschienen in: Structural and Multidisciplinary Optimization | Ausgabe 3/2013

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Abstract

This review paper provides an overview of different level-set methods for structural topology optimization. Level-set methods can be categorized with respect to the level-set-function parameterization, the geometry mapping, the physical/mechanical model, the information and the procedure to update the design and the applied regularization. Different approaches for each of these interlinked components are outlined and compared. Based on this categorization, the convergence behavior of the optimization process is discussed, as well as control over the slope and smoothness of the level-set function, hole nucleation and the relation of level-set methods to other topology optimization methods. The importance of numerical consistency for understanding and studying the behavior of proposed methods is highlighted. This review concludes with recommendations for future research.

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Some LSMs reverse the sign in (1) (e.g., Wang and Wang 2006b).

Some implementations can be found on the internet: The codes of G. Allaire, V.J. Challis, N.P. van Dijk and M.Y. Wang.

Abe K, Kazama S, Koro K (2007) A boundary element approach for topology optimization problem using the level set method. Commun Numer Methods Eng 23(5):405–416MathSciNetMATH

Allaire G, Jouve F (2005) A level-set method for vibration and multiple loads structural optimization. Comput Methods Appl Mech Eng 194(30–33):3269–3290MathSciNetMATH

Allaire G, Jouve F (2008) Minimum stress optimal design with the level set method. Eng Anal Bound Elem 32(11):909–918MATH

Allaire G, Bonnetier E, Francfort G, Jouve F (1997) Shape optimization by the homogenization method. Numer Math 76(1):27–68MathSciNetMATH

Allaire G, Jouve F, Toader AM (2002) A level-set method for shape optimization. CR Math 334(12):1125–1130MathSciNetMATH

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

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

Allaire G, Dapogny C, Frey P (2011) Topology and geometry optimization of elastic structures by exact deformation of simplicial mesh. CR Math 349(17–18):999–1003MathSciNetMATH

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

Amstutz S (2011) Connections between topological sensitivity analysis and material interpolation schemes in topology optimization. Struct Multidisc Optim 43(6):755–765MathSciNetMATH

Amstutz S, Andrä H (2006) A new algorithm for topology optimization using a level-set method. J Comput Phys 216(2):573–588MathSciNetMATH

Belytschko T, Xiao SP, Parimi C (2003) Topology optimization with implicit functions and regularization. Int J Numer Methods Eng 57(8):1177–1196MATH

Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):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 (2003) Topology optimization: theory, methods, and applications. Springer, Berlin

Bletzinger KU, Kimmich S, Ramm E (1991) Efficient modeling in shape optimal design. Comput Syst Eng 2(5–6):483–495

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

Brooks AN, Hughes TJR (1982) Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 32(1):199–259MathSciNetMATH

Bruyneel M, Duysinx P, Fleury C (2002) A family of MMA approximations for structural optimization. Struct Multidisc Optim 24(4):263–276

Burger M (2003) A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound 5(3):301–330MathSciNetMATH

Burger M, Osher S (2005) A survey on level set methods for inverse problems and optimal design. Eur J Appl Math 16(2):263–301MathSciNetMATH

Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194(1):344–362MathSciNetMATH

Céa J, Garreau S, Guillaume P, Masmoudi M (2000) The shape and topological optimizations connection. Comput Methods Appl Mech Eng 188(4):713–726MATH

Cecil T, Qian J, Osher S (2004) Numerical methods for high dimensional Hamilton–Jacobi equations using radial basis functions. J Comput Phys 196(1):327–347MathSciNetMATH

Challis VJ (2010) A discrete level-set topology optimization code written in Matlab. Struct Multidisc Optim 41(3):453–464MathSciNetMATH

Challis VJ, Guest JK (2009) Level set topology optimization of fluids in Stokes flow. Int J Numer Methods Eng 79(10):1284–1308MathSciNetMATH

Chen S, Chen W (2011) A new level-set based approach to shape and topology optimization under geometric uncertainty. Struct Multidisc Optim 44(1):1–18MATH

Chen S, Wang MY, Liu AQ (2008) Shape feature control in structural topology optimization. Comput-Aided Des 40(9):951–962MathSciNet

Chen S, Chen W, Lee S (2010) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidisc Optim 41(4):507–524MathSciNetMATH

Choi KK, Kim NH (2005) Structural sensitivity analysis and optimization: linear systems, vol 1. Springer, New York

Choi JS, Yamada T, Izui K, Nishiwaki S, Yoo J (2011) Topology optimization using a reaction-diffusion equation. Comput Methods Appl Mech Eng 200(29–32):2407–2420MathSciNetMATH

COMSOL (2011) COMSOL multiphysics user’s guide, version 4.2a

De Gournay F (2006) Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J Control Optim 45(1):343–367MathSciNetMATH

De Gournay F, Allaire G, Jouve F (2008) Shape and topology optimization of the robust compliance via the level set method. ESAIM Control Optim Calc Var 14(01):43–70MathSciNetMATH

De Ruiter MJ, Van Keulen F (2000) Topology optimization: approaching the material distribution problem using a topological function description. In: Topping BHV (ed) Computational techniques for materials, composites and composite structures, Edinburgh, United Kingdom, pp 111–119

De Ruiter MJ, Van Keulen F (2001) Topology optimization using the topology description function approach. In: Cheng G, Gu Y, Liu S, Wang Y (eds) 4th World congress on structural and multidisciplinary optimization, Dailan, China

De Ruiter MJ, Van Keulen F (2002) The topological derivative in the topology description function approach. In: Gosling P (ed) Engineering design optimization, product and process improvement, ASMO UK/ISSMO: University of Newcastle-upon-Tyne

De Ruiter MJ, Van Keulen F (2004) Topology optimization using a topology description function. Struct Multidisc Optim 26(6):406–416

Duan XB, Ma YC, Zhang R (2008) Shape-topology optimization for Navier–Stokes problem using variational level set method. J Comput Appl Math 222(2):487–499MathSciNetMATH

Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54:331

Eschenauer HA, Kobelev VV, Schumacher A (1994) Bubble method for topology and shape optimization of structures. Struct Multidisc Optim 8(1):42–51

Fleury C, Braibant V (1986) Structural optimization: a new dual method using mixed variables. Int J Numer Methods Eng 23(3):409–428MathSciNetMATH

Frei WR, Tortorelli DA, Johnson HT (2007) Geometry projection method for optimizing photonic nanostructures. Opt Lett 32(1):77–79

Frei WR, Johnson HT, Tortorelli DA (2008) Optimization of photonic nanostructures. Comput Methods Appl Mech Eng 197(41–42):3410–3416MATH

Fries TP, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3):253–304MathSciNetMATH

Fulmański P, Laurain A, Scheid JF, Sokołowski J (2007) A level set method in shape and topology optimization for variational inequalities. Int J Appl Math Comput Sci 17(3):413–430MathSciNetMATH

Fulmański P, Laurain A, Scheid JF, Sokołowski J (2008) Level set method with topological derivatives in shape optimization. Int J Comput Math 85(10):1491–1514MathSciNetMATH

Garreau S, Guillaume P, Masmoudi M (2001) The topological asymptotic for PDE systems: the elasticity case. SIAM J Control Optim 39:1756MathSciNetMATH

Gomes AA, Suleman A (2006) Application of spectral level set methodology in topology optimization. Struct Multidisc Optim 31(6):430–443MathSciNetMATH

Groenwold AA, Etman LFP (2008) On the equivalence of optimality criterion and sequential approximate optimization methods in the classical topology layout problem. Int J Numer Methods Eng 73(3):297–316MathSciNetMATH

Groenwold AA, Etman LFP (2010) A quadratic approximation for structural topology optimization. Int J Numer Methods Eng 82(4):505–524MathSciNetMATH

Guest JK (2009a) Imposing maximum length scale in topology optimization. Struct Multidisc Optim 37(5):463–473MathSciNetMATH

Guest JK (2009b) Topology optimization with multiple phase projection. Comput Methods Appl Mech Eng 199(1–4):123–135MathSciNetMATH

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

Guo X, Zhao K, Wang MY (2005) A new approach for simultaneous shape and topology optimization based on dynamic implicit surface function. Control Cybern 34(1):255–282MathSciNetMATH

Gurtin ME (1996) Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Phys D 92(3–4):178–192MathSciNetMATH

Ha SH, Cho S (2005) Topological shape optimization of heat conduction problems using level set approach. Numer Heat Transf, B Fundam 48(1):67–88

Ha SH, Cho S (2008a) Level set based topological shape optimization of geometrically nonlinear structures using unstructured mesh. Comput Struct 86(13–14):1447–1455

Ha SH, Cho S (2008b) Level set-based topological shape optimization of nonlinear heat conduction problems. Numer Heat Transf, B Fundam 54(6):454–475

Haber E (2004) A multilevel, level-set method for optimizing eigenvalues in shape design problems. J Comput Phys 198(2):518–534MathSciNetMATH

Haber RB, Bendsøe MP (1998) Problem formulation, solution procedures and geometric modeling–key issues in variable-topology optimization. In: 7th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and optimization, St. Louis, MO, pp 1864–1873

Haber RB, Jog CS, Bendsøe MP (1996) A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim 11(1):1–12

Hartmann D, Meinke M, Schröder W (2010) The constrained reinitialization equation for level set methods. J Comput Phys 229(5):1514–1535MathSciNetMATH

Hassani B, Hinton E (1998) A review of homogenization and topology optimization. I–Homogenization theory for media with periodic structure. Comput Struct 69(6):707–717MATH

He L, Kao CY, Osher S (2007) Incorporating topological derivatives into shape derivatives based level set methods. J Comput Phys 225(1):891–909MathSciNetMATH

Hirsch C (2007) Numerical computation of internal and external flows: fundamentals of computational fluid dynamics, vol 1. Butterworth-Heinemann, Oxford

Ho HS, Lui BFY, Wang MY (2011) Parametric structural optimization with radial basis functions and partition of unity method. Optim Methods Softw 26(4–5):533–553MathSciNetMATH

Ho HS, Wang MY, Zhou MD (2012) Parametric structural optimization with dynamic knot RBFs and partition of unity method. Struct Multidisc Optim 1–13. doi:10.1007/s00158-012-0848-7

Huang X, Xie M (2010) Evolutionary topology optimization of continuum structures: methods and applications. Wiley, New York

Iga A, Nishiwaki S, Izui K, Yoshimura M (2009) Topology optimization for thermal conductors considering design-dependent effects, including heat conduction and convection. Int J Heat Mass Transfer 52(11–12):2721–2732MATH

Kao CY, Osher S, Yablonovitch E (2005) Maximizing band gaps in two-dimensional photonic crystals by using level set methods. Appl Phys, B Lasers Opt 81(2):235–244

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 Multidisc Optim 44(1):19–24MATH

Khalil H, Bila S, Aubourg M, Baillargeat D, Verdeyme S, Jouve F, Delage C, Chartier T (2010) Shape optimized design of microwave dielectric resonators by level-set and topology gradient methods. Int J RF Microw Comput-Aided Eng 20(1):33–41

Kim MG, Ha SH, Cho S (2009) Level set-based topological shape optimization of nonlinear heat conduction problems using topological derivatives. Mech Des Struct Mach 37(4):550–582

Kreisselmeier G, Steinhauser R (1979) Systematic control design by optimizing a vector performance index. In: International federation of active controls symposium on computer-aided design of control systems, Zürich

Kreissl S, Maute K (2012) Level set based fluid topology optimization using the extended finite element method. Multidisc Optim 46(3):311–326MathSciNetMATH

Kreissl S, Pingen G, Maute K (2011) An explicit level set approach for generalized shape optimization of fluids with the lattice Boltzmann method. Int J Numer Methods Fluids 65(5):496–519MATH

Kwak J, Cho S (2005) Topological shape optimization of geometrically nonlinear structures using level set method. Comput Struct 83(27):2257–2268MathSciNet

Lazarov BS, Sigmund O (2011) Filters in topology optimization based on Helmholtz-type differential equations. Int J Numer Methods Eng 86(6):765–781MathSciNetMATH

Le C, Bruns T, Tortorelli DA (2011) A gradient-based, parameter-free approach to shape optimization. Comput Methods Appl Mech Eng 200(9–12):985–996MathSciNetMATH

Lim S, Yamada T, Min S, Nishiwaki S (2011) Topology optimization of a magnetic actuator based on a level set and phase-field approach. IEEE Trans Magn 47(5):1318–1321

Liu Z, Korvink JG (2008) Adaptive moving mesh level set method for structure topology optimization. Eng Optim 40(6):529–558MathSciNet

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

Luo Z, Tong L (2008) A level set method for shape and topology optimization of large-displacement compliant mechanisms. Int J Numer Methods Eng 76(6):862–892MathSciNetMATH

Luo Z, Tong L, Wang MY, Wang S (2007) Shape and topology optimization of compliant mechanisms using a parameterization level set method. J Comput Phys 227(1):680–705MathSciNetMATH

Luo J, Luo Z, Chen L, Tong L, Wang MY (2008a) A semi-implicit level set method for structural shape and topology optimization. J Comput Phys 227(11):5561–5581MathSciNetMATH

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

Luo Z, Wang MY, Wang S, Wei P (2008c) A level set-based parameterization method for structural shape and topology optimization. Int J Numer Methods Eng 76(1):1–26MathSciNetMATH

Luo Z, Tong L, Luo J, Wei P, Wang MY (2009a) Design of piezoelectric actuators using a multiphase level set method of piecewise constants. J Comput Phys 228(7):2643–2659MathSciNetMATH

Luo Z, Tong L, Ma H (2009b) Shape and topology optimization for electrothermomechanical microactuators using level set methods. J Comput Phys 228(9):3173–3181MathSciNet

Luo Z, Zhang N, Gao W, Ma H (2012) Structural shape and topology optimization using a meshless Galerkin level set method. Int J Numer Methods Eng 90(3):369–389MathSciNetMATH

Lyra PRM, Morgan K (2000a) A review and comparative study of upwind biased schemes for compressible flow computation. Part I: 1-D firstorder schemes. Arch Comput Methods Eng 7(1):19–55MathSciNetMATH

Lyra PRM, Morgan K (2000b) A review and comparative study of upwind biased schemes for compressible flow computation. Part II: 1-D higher-order schemes. Arch Comput Methods Eng 7(3):333–377MathSciNetMATH

Lyra PRM, Morgan K (2002) A review and comparative study of upwind biased schemes for compressible flow computation. Part III: Multidimensional extension on unstructured grids. Arch Comput Methods Eng 9(3):207–256MathSciNetMATH

Malladi R, Sethian JA, Vemuri BC (1995a) Shape modeling with front propagation: a level set approach. IEEE Trans Pattern Anal Mach Intell 17(2):158–175

Malladi R, Sethian JA, Vemuri BC (1995b) Shape modeling with front propagation: a level set approach. IEEE Trans Pattern Anal Mach Intell 17(2):158–175

Marchuk GI (1990) Splitting and alternating direction methods. Handb Numer Anal 1:197–462MathSciNet

Maute K, Schwarz S, Ramm E (1998) Adaptive topology optimization of elastoplastic structures. Struct Multidisc Optim 15(2):81–91

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

Mohamadian M, Shojaee S (2012) Binary level set method for structural topology optimization with MBO type of projection. Int J Numer Methods Eng 89(5):658–670MathSciNetMATH

Myśliński A (2008) Level set method for optimization of contact problems. Eng Anal Bound Elem 32(11):986–994MATH

Norato J, Haber R, Tortorelli DA, Bendsøe MP (2004) A geometry projection method for shape optimization. Int J Numer Methods Eng 60(14):2289–2312MATH

Norato JA, Bendsøe MP, Haber RB, Tortorelli DA (2007) A topological derivative method for topology optimization. Struct Multidisc Optim 33(4):375–386MATH

Novotny AA, Feijóo RA, Taroco E, Padra C (2003) Topological sensitivity analysis. Comput Methods Appl Mech Eng 192(7–8):803–829MATH

Olsson E, Kreiss G, Zahedi S (2007) A conservative level set method for two phase flow II. J Comput Phys 225(1):785–807MathSciNetMATH

Osher S, Fedkiw RP (2001) Level set methods: an overview and some recent results. J Comput Phys 169(2):463–502MathSciNetMATH

Osher S, Fedkiw RP (2003) Level set methods and dynamic implicit surfaces, vol 153. Springer, New YorkMATH

Osher S, Paragios N (2003) Geometric level set methods in imaging, vision, and graphics. Springer, New YorkMATH

Osher SJ, Santosa F (2001) Level set methods for optimization problems involving geometry and constraints: I. Frequencies of a two-density inhomogeneous drum. J Comput Phys 171(1):272–288MathSciNetMATH

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

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

Park KS, Youn SK (2008) Topology optimization of shell structures using adaptive inner-front (AIF) level set method. Struct Multidisc Optim 36(1):43–58MathSciNetMATH

Petersson J (1999) Some convergence results in perimeter-controlled topology optimization. Comput Methods Appl Mech Eng 171(1–2):123–140MathSciNetMATH

Pingen G, Waidmann M, Evgrafov A, Maute K (2010) A parametric level-set approach for topology optimization of flow domains. Struct Multidisc Optim 41(1):117–131MathSciNetMATH

Pironneau O (1989) Finite element method for fluids. Wiley, Chichester, England and New York/John Wiley and Sons, Paris

Rong JH, Liang QQ (2008) A level set method for topology optimization of continuum structures with bounded design domains. Comput Methods Appl Mech Eng 197(17–18):1447–1465MathSciNetMATH

Rozvany GIN (2001) Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics. Struct Multidisc Optim 21(2):90–108

Rozvany GIN (2009) A critical review of established methods of structural topology optimization. Struct Multidisc Optim 37(3):217–237MathSciNetMATH

Rozvany GIN, Zhou M (1991) The COC algorithm, part I: cross-section optimization or sizing. Comput Methods Appl Mech Eng 89(1–3):281–308

Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Multidisc Optim 4(3):250–252

Rozvany GIN, Bendsøe MP, Kirsch U (1995) Layout optimization of structures. Appl Mech Rev 48:41

Schleupen A, Maute K, Ramm E (2000) Adaptive FE-procedures in shape optimization. Struct Multidisc Optim 19(4):282–302

Schumacher A (1995) Topologieoptimierung von bauteilstrukturen unter verwendung von lopchpositionierungkrieterien. PhD thesis, Universität-Gesamthochschule Siegen, Siegen, Germany

Sethian JA (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Cambridge University Press, CambridgeMATH

Sethian JA (2001) Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J Comput Phys 169(2):503–555MathSciNetMATH

Sethian JA, Smereka P (2003) Level set methods for fluid interfaces. Annu Rev Fluid Mech 35(1):341–372MathSciNet

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

Shim H, Ho VTT, Wang S, Tortorelli DA (2008) Topological shape optimization of electromagnetic problems using level set method and radial basis function. Comput Model Eng Sci 37(2):175–202MathSciNet

Shojaee S, Mohammadian M (2011) A binary level set method for structural topology optimization. Int J Optim Civil Eng 1:73–90

Sigmund O (1994) Design of material structures using topology optimization. PhD thesis, Department of Solid Mechanics, Technical University of Denmark

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

Sigmund O (2011) On the usefulness of non-gradient approaches in topology optimization. Struct Multidisc Optim 43:589–596MathSciNetMATH

Sigmund O, Maute K (2012) Sensitivity filtering from a continuum mechanics perspective. Struct Multidisc Optim 46(4):471–475MathSciNetMATH

Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Multidisc Optim 16(1):68–75

Sokołowski J, Żochowski A (1999) On the topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272MathSciNetMATH

Sokołowski J, Żochowski A (2001) Topological derivatives of shape functionals for elasticity systems. Mech Struct Mach 29(3):331–349

Sokołowski J, Zolésio JP (1992) Introduction to shape optimization; shape sensitivity analysis. In: Springer series in computational mathematics, vol 16. Springer

Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidisc Optim 22(2):116–124

Strain J (1999) Semi-Lagrangian methods for level set equations. J Comput Phys 151(2):498–533MathSciNetMATH

Sussman M, Smereka P, Osher S (1994) A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 114(1):146–159MATH

Suzuki K, Kikuchi N (1991) A homogenization method for shape and topology optimization. Comput Methods Appl Mech Eng 93(3):291–318MATH

Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetMATH

Swan CC, Kosaka I (1997) Voigt–Reuss topology optimization for structures with nonlinear material behaviors. Int J Numer Methods Eng 40(20):3785–3814MathSciNetMATH

Takezawa A, Nishiwaki S, Kitamura M (2010) Shape and topology optimization based on the phase field method and sensitivity analysis. J Comput Phys 229(7):2697–2718MathSciNetMATH

Tikhonov AN, Goncharsky AV, Stepanov VV, Yagola AG (1995) Numerical methods for the solution of ill-posed problems. Springer, New YorkMATH

Van Dijk NP, Yoon GH, Van Keulen F, Langelaar M (2010) A level-set-based topology optimization using the element connectivity parameterization method. Struct Multidisc Optim 42(2):269–282MathSciNetMATH

Van Dijk NP, 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):67–97MathSciNetMATH

Van Keulen F, Haftka RT, Kim NH (2005) Review of options for structural design sensitivity analysis. Part 1: Linear systems. Comput Methods Appl Mech Eng 194(30–33):3213–3243MathSciNetMATH

Van Miegroet L, Duysinx P (2007) Stress concentration minimization of 2D filets using X-FEM and level set description. Struct Multidisc Optim 33(4):425–438

Van Miegroet L, Moës N, Fleury C, Duysinx P (2005) Generalized shape optimization based on the level set method. In: 6th World congress of structural and multidisciplinary optimization

Wang MY, 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):469–496MATH

Wang MY, Wang X (2004b) PDE-driven level sets, shape sensitivity and curvature flow for structural topology optimization. Comput Model Eng Sci 6:373–396MATH

Wang MY, Wang X (2005) A level-set based variational method for design and optimization of heterogeneous objects. Comput-Aided Des 37(3):321–337

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

Wang S, Wang MY (2006b) Radial basis functions and level set method for structural topology optimization. Int J Numer Methods Eng 65(12):2060–2090MATH

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

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

Wang X, Wang MY, Guo D (2004) Structural shape and topology optimization in a level-set-based framework of region representation. Struct Multidisc Optim 27(1):1–19

Wang MY, Chen S, Wang X, Mei Y (2005) Design of multimaterial compliant mechanisms using level-set methods. J Mech Des 127:941–956

Wang SY, Lim KM, Khoo BC, Wang MY (2007a) An extended level set method for shape and topology optimization. J Comput Phys 221(1):395–421MathSciNetMATH

Wang SY, Lim KM, Khoo BC, Wang MY (2007b) A geometric deformation constrained level set method for structural shape and topology optimization. Comput Model Eng Sci 18(3):155–181MathSciNet

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

Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidisc Optim 43:767–784MATH

Wei P, Wang MY (2006) Parametric structural shape and topology optimization method with radial basis functions and level-set method. In: Proceedings of international design engineering technical conferences & computers and information in engineering conference

Wei P, Wang MY (2009) Piecewise constant level set method for structural topology optimization. Int J Numer Methods Eng 78(4):379–402MATH

Wei P, Wang MY, Xing X (2010) A study on X-FEM in continuum structural optimization using a level set model. Comput-Aided Des 42(8):708–719

Xia Q, Wang MY (2008) Topology optimization of thermoelastic structures using level set method. Comput Mech 42(6):837–857MathSciNetMATH

Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49(5):885–896

Xing X, Wang MY, Lui BFY (2007) Parametric shape and topology optimization with moving knots radial basis functions and level set methods. In: 7th World congress on structural and multidisciplinary optimization, Seoul, Korea

Xing X, Wei P, Wang MY (2010) A finite element-based level set method for structural optimization. Int J Numer Methods Eng 82(7):805–842MathSciNetMATH

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):2876–2891MathSciNetMATH

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 133:1–9

Yamasaki S, Nishiwaki S, Yamada T, Izui K, Yoshimura M (2010a) A structural optimization method based on the level set method using a new geometry-based re-initialization scheme. Int J Numer Methods Eng 83(12):1580–1624MathSciNetMATH

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

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):844–868MathSciNetMATH

Yanenko NN (1971) The method of fractional steps. Springer, BerlinMATH

Yoon GH, Kim YY (2005) Element connectivity parameterization for topology optimization of geometrically nonlinear structures. Int J Solids Struct 42(7):1983–2009MathSciNetMATH

Yulin M, Xiaoming W (2004a) A level set method for structural topology optimization and its applications. Adv Eng Softw 35(7):415–441MATH

Yulin M, Xiaoming W (2004b) A level set method for structural topology optimization with multi-constraints and multi-materials. Acta Mech Sin 20(5):507–518MathSciNet

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

Zhou S, Li Q (2008) A variational level set method for the topology optimization of steady-state Navier–Stokes flow. J Comput Phys 227(24):10178–10195MathSciNetMATH

Zhou JX, Zou W (2008) Meshless approximation combined with implicit topology description for optimization of continua. Struct Multidisc Optim 36(4):347–353MATH

Zhou S, Wang MY (2007) Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition. Struct Multidisc Optim 33(2):89–111MATH

Zhou SW, Li W, Sun GY, Li Q (2010) A level-set procedure for the design of electromagnetic metamaterials. Opt Express 18(7):6693–6702

Zhu S, Liu C, Wu Q (2010) Binary level set methods for topology and shape optimization of a two-density inhomogeneous drum. Comput Methods Appl Mech Eng 199(45–48):2970–2986MathSciNetMATH

Zhuang C, Xiong Z, Ding H (2009) Structural shape and topology optimization based on level-set modelling and the element-propagating method. Eng Optim 41(6):537–555MathSciNet

Zhuang CG, Xiong ZH, Ding H (2007) A level set method for topology optimization of heat conduction problem under multiple load cases. Comput Methods Appl Mech Eng 196(4–6):1074–1084MathSciNetMATH

Zhuang C, Xiong Z, Ding H (2010) Topology optimization of multi-material for the heat conduction problem based on the level set method. Eng Optim 42(9):811–831MathSciNet

Titel: Level-set methods for structural topology optimization: a review
verfasst von: N. P. van Dijk
K. Maute
M. Langelaar
F. van Keulen
Publikationsdatum: 01.09.2013
Verlag: Springer Berlin Heidelberg
Erschienen in: Structural and Multidisciplinary Optimization / Ausgabe 3/2013
Print ISSN: 1615-147X
Elektronische ISSN: 1615-1488
DOI: https://doi.org/10.1007/s00158-013-0912-y

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