Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Structural and Multidisciplinary Optimization
Published in: Structural and Multidisciplinary Optimization 4/2011

01-10-2011 | Research Paper

Piecewise constant level set method for structural topology optimization with MBO type of projection

Authors: Saeed Shojaee, Mojtaba Mohammadian

Published in: Structural and Multidisciplinary Optimization | Issue 4/2011

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, we combine a Piecewise Constant Level Set (PCLS) method with a MBO scheme to solve a structural shape and topology optimization problem. The geometrical boundary of structure is represented implicitly by the discontinuities of PCLS functions. Compared with the classical level set method (LSM) for solving Hamilton–Jacobi partial differential equation (H-J PDE) we don’t need to solve H-J PDE, thus it is free of the CFL condition and the reinitialization scheme. For solving optimization problem under some constraints, Additive Operator Splitting (AOS) and Multiplicative Operator Splitting (MOS) schemes will be used. To increase the convergency speed and the efficiency of PCLS method we combine this approach with MBO scheme. Advantages and disadvantages are discussed by solving some examples of 2D structural topology optimization problems.
previous article Eliminating beta-continuation from Heaviside projection and density filter algorithms
next article Topology optimization using B-spline finite elements

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

inform now

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!

inform now
Literature
Allaire G, Kohn RV (1993) Optimal bounds on the effective behavior of a mixture of two well ordered elastic materials. Quat Appl Math 51:643–674MathSciNetMATH
Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level set method. J Comput Phys 194:363–393MathSciNetMATHCrossRef
Bendsoe MP, Kikuchi N (1988) Generating optimal topologies in structural design using homogenizeation method. Comput Methods Appl Mech Eng 71:197–224MathSciNetCrossRef
Bendsoe MP (1989) Optimal shape design as a material distribution problem. Struct Opt 1:193–202CrossRef
Bendsøe MP, Sigmund O (2003) Topology optimization. Theory, methods and applications. Springer, Berlin
Esedoglu S, Tsai YHR (2004) Threshold dynamics for the piecewise constant Mumford–Shah functional. Tech Rep CAM 04-63
Evans LC, Soner HM, Souganidis PE (1992) Phase transitions and generalized motion by mean curvature. Commun Pure Appl Math 45(9):1097–1123MathSciNetMATHCrossRef
Gilles A, Kornprobst P (2002) Mathematical problems in image processing. Applied Mathematical Sciences, vol 147. Springer, New York (Partial differential equations and the calculus of variations, With a foreword by Olivier Faugeras)
Lu T, Neittaanmaki P, Tai X-C (1991) A parallel splitting up method and its application to Navier–Stokes equations. Appl Math Lett 4:25–29MathSciNetMATHCrossRef
Luo JZ, Luo Z, Chen LP, Tong LY, Wang MY (2008a) A semi-implicit level set method for structural shape and topology optimization. J Comput Phys 227(11):61–81MathSciNetCrossRef
Luo Z, Wang MY, Wang SY, Wei P (2008b) A level set based parameterization method for structural shape and topology optimization. Int J Numer Methods Eng 76:1–26MathSciNetMATHCrossRef
Luo Z, Tong L, Luo J, Wei P, Wang MY (2009) Design of piezoelectric actuators using a multiphase level set method of piecewise constants. J Comput Phys 228:2643–2659MathSciNetMATHCrossRef
Marchuk GI (1990) Splitting and alternating direction methods. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, vol I. Elsevier Science Publishers B.V., North-Holland, pp 197–462
Merriman B, Bence JK, Osher SJ (1994) Motion of multiple functions: a level set approach. J Comput Phys 112(2):334–363MathSciNetCrossRef
Osher S, Fedkiw RP (2002) Level set methods and dynamic implicit surface. Springer, New York
Osher S, Sethian JA (1988) Front propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 78:12–49MathSciNetCrossRef
Qiang D, Chun L, Xiaoqiang W (2004) A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J Comput Phys 198(2):450–468MathSciNetMATHCrossRef
Rubinstein J, Sternberg P, Keller JB (1993) Front interaction and non-homogeneous equilibria for tristable reaction diffusion equations. SIAM J Appl Math 53(6):1669–1685MathSciNetMATHCrossRef
Samson C, Blanc-Feraud L, Aubert G, Zerubia J (2000a) A level set model for image classification. IJCV 40(3):187–198MATHCrossRef
Samson C, Blanc-Feraud L, Aubert G, Zerubia J (2000b) A variational model for image classification and restoration. TPAMI 22(5):460–472CrossRef
Sethian JA (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer version and material science. Cambridge Monograph on Applied and Computational Mathematics, Cambridge University Press, UK
Suzuki K, Kikuchi N (1991) A homogenization method for shape and topology optimization. Comput Methods Appl Mech Eng 93:291–318MATHCrossRef
Tai X-C, Christiansen O, Lin P, SkjÆlaaen I (2007) Image segmentation using some piecewise constant level set methods with MBO type of projection. Int J Comput Vis 73:61–76CrossRef
Wang MY, Wang XM, Guo DM (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:217–224CrossRef
Wang SY, Wang MY (2006a) Radial basis functions and level set method for structural topology optimization. Int J Numer Methods Eng 65:60–90
Wang SY, Wang MY (2006b) Structural shape and topology optimization using an implicit free boundary parameterization method. Comput Model Eng Sci 13:19–47
Wei P (2007) Level set methods for shape and topology optimization of structures. PhD thesis, The Chinese University of Hong Kong
Wei P, Wang MY (2009) Piecewise constant level set method for structural topology optimization. Int J Numer Methods Eng 78:379–402MathSciNetMATHCrossRef
Weickert J, Romeny BM, Viergever M (1998) Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans Image Process 7:398–410CrossRef
Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49(5):885–896CrossRef
Metadata
Title
Piecewise constant level set method for structural topology optimization with MBO type of projection
Authors
Saeed Shojaee
Mojtaba Mohammadian
Publication date
01-10-2011
Publisher
Springer-Verlag
Published in
Structural and Multidisciplinary Optimization / Issue 4/2011
Print ISSN: 1615-147X
Electronic ISSN: 1615-1488
DOI
https://doi.org/10.1007/s00158-011-0646-7

Other articles of this Issue 4/2011

Structural and Multidisciplinary Optimization 4/2011 Go to the issue

Industrial Application

Optimization of the fatigue life of threaded connections by the positioning method

Industrial Application

Improving machine dynamics via geometry optimization

Research Paper

Topology optimization using B-spline finite elements

Industrial Application

Die shape optimisation for net-shape accuracy in metal forming using direct search and localised response surface methods

Industrial Application

Optimum design of a controlled cable stayed bridge subject to earthquakes

Industrial Application

Optimization of laser butt welding parameters based on the orthogonal array with fuzzy logic and desirability approach

Premium Partners

    Image Credits