Efficient topology optimization in MATLAB using 88 lines of code

  • Erik Andreassen
  • Anders Clausen
  • Mattias Schevenels
  • Boyan S. Lazarov
  • Ole SigmundEmail author
Educational Article


The paper presents an efficient 88 line MATLAB code for topology optimization. It has been developed using the 99 line code presented by Sigmund (Struct Multidisc Optim 21(2):120–127, 2001) as a starting point. The original code has been extended by a density filter, and a considerable improvement in efficiency has been achieved, mainly by preallocating arrays and vectorizing loops. A speed improvement with a factor of 100 is obtained for a benchmark example with 7,500 elements. Moreover, the length of the code has been reduced to a mere 88 lines. These improvements have been accomplished without sacrificing the readability of the code. The 88 line code can therefore be considered as a valuable successor to the 99 line code, providing a practical instrument that may help to ease the learning curve for those entering the field of topology optimization. The paper also discusses simple extensions of the basic code to include recent PDE-based and black-and-white projection filtering methods. The complete 88 line code is included as an appendix and can be downloaded from the web site


Topology optimization MATLAB Education Computational efficiency 



This work was financially supported by the Eurohorcs/ESF European Young Investigator Award (EURYI), by a Center of Advanced User Support (CAUS) grant from the Danish Center of Scientific Computing (DCSC), and by an Elite Research Prize from the Danish Minister of Research. The third author is a postdoctoral fellow of the Research Foundation—Flanders and a member of K.U.Leuven-BOF PFV/10/002 OPTEC-Optimization in Engineering Center.


  1. Alberty J, Carstensen C, Funken S (1999) Remarks around 50 lines of Matlab: short finite element implementation. Numer Algorithms 20(2–3):117–137zbMATHCrossRefMathSciNetGoogle Scholar
  2. Alberty J, Carstensen C, Funken S, Klose R (2002) Matlab implementation of the finite element method in elasticity. Computing 69(3):239–263CrossRefMathSciNetGoogle Scholar
  3. Allaire G (2009) Shape and topology optimization by the level set method.
  4. Bendsøe M (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202CrossRefGoogle Scholar
  5. Bendsøe M, Sigmund O (2003) Topology optimization. Theory, methods and applications. Springer, BerlinGoogle Scholar
  6. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158zbMATHCrossRefMathSciNetGoogle Scholar
  7. Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459zbMATHCrossRefGoogle Scholar
  8. Challis VJ (2010) A discrete level-set topology optimization code written in Matlab. Struct Multidisc Optim 41(3):453–464CrossRefMathSciNetGoogle Scholar
  9. Dabrowski M, Krotkiewski M, Schmid D (2008) MILAMIN: MATLAB-based finite element method solver for large problems. Geochemistry Geophysics Geosystems 9(4). doi: 10.1029/2007GC001719
  10. Davis T (2007) Creating sparse finite-element matrices in MATLAB. Guest blog in Loren on the art of MATLAB.;
  11. Davis T (2008) User guide for CHOLMOD: a sparse Cholesky factorization and modification package. Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, USAGoogle Scholar
  12. Díaz A, Sigmund O (1995) Checkerboard patterns in layout optimization. Struct Optim 10(1):40–45CrossRefGoogle Scholar
  13. 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):238–254zbMATHCrossRefMathSciNetGoogle Scholar
  14. Jog C, Haber R (1996) Stability of finite element models for distributed-parameter optimization and topology design. Comput Methods Appl Mech Eng 130(3–4):203–226zbMATHCrossRefMathSciNetGoogle Scholar
  15. Lazarov B, Sigmund O (2009) Sensitivity filters in topology optimisation as a solution to helmholtz type differential equation. In: Proc. of the 8th world congress on structural and multidisciplinary optimizationGoogle Scholar
  16. Lazarov B, Sigmund O (2010) Filters in topology optimization based on Helmholtz type differential equations. Int J Numer Methods Eng. doi: 10.1002/nme.3072 Google Scholar
  17. Sigmund O (1994) Design of material structures using topology optimization. PhD thesis, DCAMM S-report S69, Department of Solid Mechanics, Technical University of DenmarkGoogle Scholar
  18. Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25(4):493–524CrossRefGoogle Scholar
  19. Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidisc Optim 21(2):120–127CrossRefGoogle Scholar
  20. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33(4–5):401–424CrossRefGoogle Scholar
  21. Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech Sin 25(2):227–239CrossRefGoogle Scholar
  22. 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):68–75CrossRefGoogle Scholar
  23. Suresh K (2010) A 199-line Matlab code for Pareto-optimal tracing in topology optimization. Struct Multidisc Optim 42(5):665–679. doi: 10.1007/s00158-010-0534-6 CrossRefGoogle Scholar
  24. Svanberg K (1987) Method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373zbMATHCrossRefMathSciNetGoogle Scholar
  25. The MathWorks (2010) MATLAB programming fundamentalsGoogle Scholar
  26. Wang F, Lazarov B, Sigmund O (2010) On projection methods, convergence and robust formulations in topology optimization (submitted)Google Scholar
  27. Zhou M, Rozvany G (1991) The COC algorithm. Part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Erik Andreassen
    • 1
  • Anders Clausen
    • 1
  • Mattias Schevenels
    • 2
  • Boyan S. Lazarov
    • 1
  • Ole Sigmund
    • 1
    Email author
  1. 1.Department of Mechanical Engineering, Solid MechanicsTechnical University of DenmarkLyngbyDenmark
  2. 2.Department of Civil EngineeringK.U. LeuvenLeuvenBelgium

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