Skip to main content
Erschienen in: Structural and Multidisciplinary Optimization 2/2018

13.02.2018 | EDUCATIONAL ARTICLE

An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions

verfasst von: Peng Wei, Zuyu Li, Xueping Li, Michael Yu Wang

Erschienen in: Structural and Multidisciplinary Optimization | Ausgabe 2/2018

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

This paper presents a compact and efficient 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions (RBFs), which is applied to minimize the compliance of a two-dimensional linear elastic structure. This parameterized level set method using radial basis functions can maintain a relatively smooth level set function with an approximate re-initialization scheme during the optimization process. It also has less dependency on initial designs due to its capability in nucleation of new holes inside the material domain. The MATLAB code and simple modifications are explained in detail with numerical examples. The 88-line code included in the appendix is intended for educational purposes.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Anhänge
Nur mit Berechtigung zugänglich
Literatur
Zurück zum Zitat Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393MathSciNetCrossRefMATH Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393MathSciNetCrossRefMATH
Zurück zum Zitat 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, 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
Zurück zum Zitat Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim 43(1):1–16CrossRefMATH Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim 43(1):1–16CrossRefMATH
Zurück zum Zitat Bendsøe M, Sigmund O (2003) Topology optimization. Theory, methods and applications. Springer, BerlinMATH Bendsøe M, Sigmund O (2003) Topology optimization. Theory, methods and applications. Springer, BerlinMATH
Zurück zum Zitat Burger M, Osher SJ (2005) A survey on level set methods for inverse problems and optimal design. Eur J Appl Math 16(2):263–301MathSciNetCrossRefMATH Burger M, Osher SJ (2005) A survey on level set methods for inverse problems and optimal design. Eur J Appl Math 16(2):263–301MathSciNetCrossRefMATH
Zurück zum Zitat Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194(1):344–362MathSciNetCrossRefMATH Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194(1):344–362MathSciNetCrossRefMATH
Zurück zum Zitat Cecil T, Qian J, Osher S (2004) Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions. J Comput Phys 196(1):327–347MathSciNetCrossRefMATH Cecil T, Qian J, Osher S (2004) Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions. J Comput Phys 196(1):327–347MathSciNetCrossRefMATH
Zurück zum Zitat Challis VJ (2010) A discrete level-set topology optimization code written in MATLAB. Struct Multidiscip Optim 41(3):453–464MathSciNetCrossRefMATH Challis VJ (2010) A discrete level-set topology optimization code written in MATLAB. Struct Multidiscip Optim 41(3):453–464MathSciNetCrossRefMATH
Zurück zum Zitat Chan TF, Vese LA (2001) Active contours without edges. IEEE Trans Image Process 10(2):266–277CrossRefMATH Chan TF, Vese LA (2001) Active contours without edges. IEEE Trans Image Process 10(2):266–277CrossRefMATH
Zurück zum Zitat Choi KK, Kim NH (2005) Structural sensitivity analysis and optimization 1. Springer, Berlin Choi KK, Kim NH (2005) Structural sensitivity analysis and optimization 1. Springer, Berlin
Zurück zum Zitat Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38MathSciNetCrossRef Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38MathSciNetCrossRef
Zurück zum Zitat Dijk NPV, Maute K, Langelaar M, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48(3):437–472MathSciNetCrossRef Dijk NPV, Maute K, Langelaar M, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48(3):437–472MathSciNetCrossRef
Zurück zum Zitat Emre B, To AC (2015) Proportional topology optimization: a new non-sensitivity method for solving stress constrained and minimum compliance problems and its implementation in MATLAB. PLoS One 10(12):e0145041CrossRef Emre B, To AC (2015) Proportional topology optimization: a new non-sensitivity method for solving stress constrained and minimum compliance problems and its implementation in MATLAB. PLoS One 10(12):e0145041CrossRef
Zurück zum Zitat Gain AL, Paulino GH (2013) A critical comparative assessment of differential equation-driven methods for structural topology optimization. Struct Multidiscip Optim 48(4):685–710MathSciNetCrossRef Gain AL, Paulino GH (2013) A critical comparative assessment of differential equation-driven methods for structural topology optimization. Struct Multidiscip Optim 48(4):685–710MathSciNetCrossRef
Zurück zum Zitat Huang X, Xie YM (2010) Evolutionary Topology Optimization of Continuum Structures: Methods and Applications. Wiley, New YorkCrossRefMATH Huang X, Xie YM (2010) Evolutionary Topology Optimization of Continuum Structures: Methods and Applications. Wiley, New YorkCrossRefMATH
Zurück zum Zitat Kansa EJ, Power H, Fasshauer GE, Ling L (2004) A volumetric integral radial basis function method for time-dependent partial differential equations. I. Formulation. Eng Anal Bound Elem 28(10):1191–1206CrossRefMATH Kansa EJ, Power H, Fasshauer GE, Ling L (2004) A volumetric integral radial basis function method for time-dependent partial differential equations. I. Formulation. Eng Anal Bound Elem 28(10):1191–1206CrossRefMATH
Zurück zum Zitat Kharmanda G, Olhoff N, Mohamed A, Lemaire M (2004) Reliability-based topology optimization. Struct Multidiscip Optim 26(5):295–307CrossRef Kharmanda G, Olhoff N, Mohamed A, Lemaire M (2004) Reliability-based topology optimization. Struct Multidiscip Optim 26(5):295–307CrossRef
Zurück zum Zitat Liu K, Tovar A (2014) An efficient 3d topology optimization code written in MATLAB. Struct Multidiscip Optim 50(6):1175–1196MathSciNetCrossRef Liu K, Tovar A (2014) An efficient 3d topology optimization code written in MATLAB. Struct Multidiscip Optim 50(6):1175–1196MathSciNetCrossRef
Zurück zum Zitat Mei Y, Wang X (2004) A level set method for structural topology optimization and its applications. Adv Eng Softw 35(7):415–441CrossRefMATH Mei Y, Wang X (2004) A level set method for structural topology optimization and its applications. Adv Eng Softw 35(7):415–441CrossRefMATH
Zurück zum Zitat Morse BS, Yoo TS, Chen DT, Rheingans P, Subramanian KR (2001) Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions. Int Conf Shape Model Appl 15(2):89–98CrossRef Morse BS, Yoo TS, Chen DT, Rheingans P, Subramanian KR (2001) Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions. Int Conf Shape Model Appl 15(2):89–98CrossRef
Zurück zum Zitat Olesen LH, Okkels F, Bruus H (2006) A high-level programming-language implementation of topology optimization applied to steady-state navier-stokes flow. Int J Numer Methods Eng 65(7):975–1001MathSciNetCrossRefMATH Olesen LH, Okkels F, Bruus H (2006) A high-level programming-language implementation of topology optimization applied to steady-state navier-stokes flow. Int J Numer Methods Eng 65(7):975–1001MathSciNetCrossRefMATH
Zurück zum Zitat Osher S, Fedkiw R (2002) Level set methods and dynamic implicit surfaces. Springer, New YorkMATH Osher S, Fedkiw R (2002) Level set methods and dynamic implicit surfaces. Springer, New YorkMATH
Zurück zum Zitat Osher S, 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):272–288MathSciNetCrossRefMATH Osher S, 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):272–288MathSciNetCrossRefMATH
Zurück zum Zitat Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on HamiltonJacobi formulations. J Comput Phys 79(1):12–49MathSciNetCrossRefMATH Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on HamiltonJacobi formulations. J Comput Phys 79(1):12–49MathSciNetCrossRefMATH
Zurück zum Zitat Otomori M, Yamada T, Izui K, Nishiwaki S (2014) MATLAB code for a level set-based topology optimization method using a reaction diffusion equation. Struct Multidiscip Optim 51(5):1159–1172MathSciNetCrossRef Otomori M, Yamada T, Izui K, Nishiwaki S (2014) MATLAB code for a level set-based topology optimization method using a reaction diffusion equation. Struct Multidiscip Optim 51(5):1159–1172MathSciNetCrossRef
Zurück zum Zitat Rochafellar RT (1973) The multiplier method of Hestenes and Powell applied to convex programming. J Optim Theory Appl 12:555–562MathSciNetCrossRef Rochafellar RT (1973) The multiplier method of Hestenes and Powell applied to convex programming. J Optim Theory Appl 12:555–562MathSciNetCrossRef
Zurück zum Zitat Schmidt S, Schulz V (2011) A 2589 line topology optimization code written for the graphics card. Comput Vis Sci 14(6):249–256MathSciNetCrossRefMATH Schmidt S, Schulz V (2011) A 2589 line topology optimization code written for the graphics card. Comput Vis Sci 14(6):249–256MathSciNetCrossRefMATH
Zurück zum Zitat 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 (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Cambridge University Press, CambridgeMATH
Zurück zum Zitat Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163(2):489–528MathSciNetCrossRefMATH Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163(2):489–528MathSciNetCrossRefMATH
Zurück zum Zitat Sigmund O (2001) A 99 line topology optimization code written in MATLAB. Struct Multidiscip Optim 21(21):120–127CrossRef Sigmund O (2001) A 99 line topology optimization code written in MATLAB. Struct Multidiscip Optim 21(21):120–127CrossRef
Zurück zum Zitat Sokołowski J, Żochowski A (1999) On the topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272MathSciNetCrossRefMATH Sokołowski J, Żochowski A (1999) On the topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272MathSciNetCrossRefMATH
Zurück zum Zitat Sokołowski J, Zolésio JP (1992) Introduction to shape optimization: shape Sensitity analysis. Introduction to shape optimization : shape sensitivity analysis. Springer-Verlag, BerlinMATH Sokołowski J, Zolésio JP (1992) Introduction to shape optimization: shape Sensitity analysis. Introduction to shape optimization : shape sensitivity analysis. Springer-Verlag, BerlinMATH
Zurück zum Zitat Suresh K (2010) A 199-line MATLAB code for Pareto-optimal tracing in topology optimization. Struct Multidiscip Optim 42(5):665–679MathSciNetCrossRefMATH Suresh K (2010) A 199-line MATLAB code for Pareto-optimal tracing in topology optimization. Struct Multidiscip Optim 42(5):665–679MathSciNetCrossRefMATH
Zurück zum Zitat Talischi C, Paulino GH, Pereira A, Menezes IFM (2012) Polytop: a MATLAB implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Struct Multidiscip Optim 45(3):329–357MathSciNetCrossRefMATH Talischi C, Paulino GH, Pereira A, Menezes IFM (2012) Polytop: a MATLAB implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Struct Multidiscip Optim 45(3):329–357MathSciNetCrossRefMATH
Zurück zum Zitat Tavakoli R, Mohseni SM (2014) Alternating active-phase algorithm for multimaterial topology optimization problems: a 115-line MATLAB implementation. Struct Multidiscip Optim 49(4):621–642MathSciNetCrossRef Tavakoli R, Mohseni SM (2014) Alternating active-phase algorithm for multimaterial topology optimization problems: a 115-line MATLAB implementation. Struct Multidiscip Optim 49(4):621–642MathSciNetCrossRef
Zurück zum Zitat Van Dijk NP, Maute K, Langelaar M, Van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48(3):437–472MathSciNetCrossRef Van Dijk NP, Maute K, Langelaar M, Van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48(3):437–472MathSciNetCrossRef
Zurück zum Zitat Wang SY, Wang MY (2006a) Radial basis functions and level set method for structural topology optimization. Int J Numer Meth Engng 65:2060–2090MathSciNetCrossRefMATH Wang SY, Wang MY (2006a) Radial basis functions and level set method for structural topology optimization. Int J Numer Meth Engng 65:2060–2090MathSciNetCrossRefMATH
Zurück zum Zitat Wang SY, Wang MY (2006b) Structural shape and topology optimization using an implicit free boundary parameterization method. Comput Model Eng Sci 13(2):119–147MathSciNetMATH Wang SY, Wang MY (2006b) Structural shape and topology optimization using an implicit free boundary parameterization method. Comput Model Eng Sci 13(2):119–147MathSciNetMATH
Zurück zum Zitat Wang MY, Wei P (2005) Topology optimization with level set method incorporating topological derivative. 6th World Congress on Structural & Multidisciplinary Optimization, Rio de Janeiro, Brazil Wang MY, Wei P (2005) Topology optimization with level set method incorporating topological derivative. 6th World Congress on Structural & Multidisciplinary Optimization, Rio de Janeiro, Brazil
Zurück zum Zitat Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246MathSciNetCrossRefMATH Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246MathSciNetCrossRefMATH
Zurück zum Zitat Wang SY, Lim KM, Khoo BC, Wang MY (2007) An extended level set method for shape and topology optimization. J Comput Phys 221(1):395–421MathSciNetCrossRefMATH Wang SY, Lim KM, Khoo BC, Wang MY (2007) An extended level set method for shape and topology optimization. J Comput Phys 221(1):395–421MathSciNetCrossRefMATH
Zurück zum Zitat Wei P, Wang MY (2006) The augmented Lagrangian method in structural shape and topology optimization with RBF based level set method, The 4th China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems, Kunming, China Wei P, Wang MY (2006) The augmented Lagrangian method in structural shape and topology optimization with RBF based level set method, The 4th China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems, Kunming, China
Zurück zum Zitat Wei P, Wang MY (2009) Piecewise constant level set method for structural topology optimization. Int J Numer Methods Eng 78(4):379–402MathSciNetCrossRefMATH Wei P, Wang MY (2009) Piecewise constant level set method for structural topology optimization. Int J Numer Methods Eng 78(4):379–402MathSciNetCrossRefMATH
Zurück zum Zitat Wendland H (1995) Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math 4(1):389–396MathSciNetCrossRefMATH Wendland H (1995) Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math 4(1):389–396MathSciNetCrossRefMATH
Zurück zum Zitat Xia L, Breitkopf P (2015) Design of materials using topology optimization and energy-based homogenization approach in MATLAB. Struct Multidiscip Optim 52(6):1229–1241MathSciNetCrossRef Xia L, Breitkopf P (2015) Design of materials using topology optimization and energy-based homogenization approach in MATLAB. Struct Multidiscip Optim 52(6):1229–1241MathSciNetCrossRef
Zurück zum Zitat Xie X, Mirmehdi M (2011) Radial basis function based level set interpolation and evolution for deformable modelling. Image Vis Comput 29(2–3):167–177CrossRef Xie X, Mirmehdi M (2011) Radial basis function based level set interpolation and evolution for deformable modelling. Image Vis Comput 29(2–3):167–177CrossRef
Zurück zum Zitat 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):2876–2891MathSciNetCrossRefMATH 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):2876–2891MathSciNetCrossRefMATH
Zurück zum Zitat Zegard T, Paulino GH (2015) GRAND3 - ground structure based topology optimization for arbitrary 3D domains using MATLAB. Struct Multidiscip Optim 52(6):1161–1184CrossRef Zegard T, Paulino GH (2015) GRAND3 - ground structure based topology optimization for arbitrary 3D domains using MATLAB. Struct Multidiscip Optim 52(6):1161–1184CrossRef
Zurück zum Zitat Zhang W, Yuan J, Zhang J, Guo X (2016) A new topology optimization approach based on moving Morphable components (MMC) and the ersatz material model. Struct Multidiscip Optim 53(6):1243–1260MathSciNetCrossRef Zhang W, Yuan J, Zhang J, Guo X (2016) A new topology optimization approach based on moving Morphable components (MMC) and the ersatz material model. Struct Multidiscip Optim 53(6):1243–1260MathSciNetCrossRef
Zurück zum Zitat Zhao HK, Chan T, Merriman B, Osher S (1996) A variational level set approach to multiphase motion. J Comput Phys 127:179–195MathSciNetCrossRefMATH Zhao HK, Chan T, Merriman B, Osher S (1996) A variational level set approach to multiphase motion. J Comput Phys 127:179–195MathSciNetCrossRefMATH
Zurück zum Zitat Zhou S, Cadman J, Chen Y, Li W, Xie YM, Huang X et al (2012) Design and fabrication of biphasic cellular materials with transport properties – a modified bidirectional evolutionary structural optimization procedure and MATLAB program. Int J Heat Mass Transf 55(25–26):8149–8162CrossRef Zhou S, Cadman J, Chen Y, Li W, Xie YM, Huang X et al (2012) Design and fabrication of biphasic cellular materials with transport properties – a modified bidirectional evolutionary structural optimization procedure and MATLAB program. Int J Heat Mass Transf 55(25–26):8149–8162CrossRef
Zurück zum Zitat Zuo ZH, Xie YM (2015) A simple and compact python code for complex 3D topology optimization. Adv Eng Softw 85:1–11CrossRef Zuo ZH, Xie YM (2015) A simple and compact python code for complex 3D topology optimization. Adv Eng Softw 85:1–11CrossRef
Metadaten
Titel
An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions
verfasst von
Peng Wei
Zuyu Li
Xueping Li
Michael Yu Wang
Publikationsdatum
13.02.2018
Verlag
Springer Berlin Heidelberg
Erschienen in
Structural and Multidisciplinary Optimization / Ausgabe 2/2018
Print ISSN: 1615-147X
Elektronische ISSN: 1615-1488
DOI
https://doi.org/10.1007/s00158-018-1904-8

Weitere Artikel der Ausgabe 2/2018

Structural and Multidisciplinary Optimization 2/2018 Zur Ausgabe

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.