Free material optimization for stress constraints

  • Michal KočvaraEmail author
  • Michael Stingl
Research Paper


Free material design deals with the question of finding the lightest structure subject to one or more given loads when both the distribution of material and the material itself can be freely varied. We additionally consider constraints on local stresses in the optimal structure. We discuss the choice of formulation of the problem and the stress constraints. The chosen formulation leads to a mathematical program with matrix inequality constraints, so-called nonlinear semidefinite program. We present an algorithm that can solve these problems. The algorithm is based on a generalized augmented Lagrangian method. A number of numerical examples demonstrate the effect of stress constraints in free material optimization.


Topology optimization Material optimization Stress based design Nonlinear semidefinite programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Achtziger W, Kanzow C (2005) Mathematical programs with vanishing constraints: Optimality conditions and constraint qualifications. Tech. rep. no. 263, Institute of Applied Mathematics and Statistics, University of Würzburg, GermanyGoogle Scholar
  2. Allaire G, Jouve F, Maillot H (2004) Topology optimization for stress design with the homogenization method. Struct Multidis Optim 28:87–98MathSciNetGoogle Scholar
  3. Ben-Tal A, Kočvara M, Nemirovski A, Zowe J (1997) Free material design via semidefinite programming. The multi-load case with contact conditions. SIAM J Optim 9:813–832CrossRefGoogle Scholar
  4. Bendsøe M, Sigmund O (2002) Topology optimization. Theory, methods and applications. Springer, Berlin Heidelberg New YorkGoogle Scholar
  5. Bendsøe MP, Guades JM, Haber R, Pedersen P, Taylor JE (1994) An analytical model to predict optimal material properties in the context of optimal structural design. J Appl Mech 61:930–937zbMATHGoogle Scholar
  6. Ciarlet PG (1978) The finite element method for elliptic problems. North-Holland, AmsterdamzbMATHGoogle Scholar
  7. Duff IS, Reid JK (1982) MA27—A set of Fortran subroutines for solving sparse symmetric sets of linear equations. Tech. Report R.10533, AERE, Harwell, Oxfordshire, UKGoogle Scholar
  8. Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43:1453–1478zbMATHCrossRefGoogle Scholar
  9. Fourer R, Gay DM, Kernighan BW (1993) AMPL: A modeling language for mathematical programming. Scientific, South San Francisco, CAGoogle Scholar
  10. Kirsch U (1990) On singular topologies in optimum structural design. Struct Optim 2:133–142CrossRefGoogle Scholar
  11. Kočvara M, Stingl M (2003) PENNON—a code for convex nonlinear and semidefinite programming. Optim Methods Softw 18(3):317–333CrossRefMathSciNetGoogle Scholar
  12. Kočvara M, Stingl M (2006) On the solution of large-scale SDP problems by the modified barrier method using iterative solvers. Math Program, Ser B (in press)Google Scholar
  13. Kočvara M, Zowe J (2002) Free material optimization: an overview. In: Siddiqi A, Kočvara M (eds) Trends in industrial and applied mathematics. Kluwer, Dordrecht, pp 181–215Google Scholar
  14. Lipton R (2002) Design of functionally graded composite structures in the presence of stress constraints. Int J Solids Struct 39:2575–2586zbMATHCrossRefMathSciNetGoogle Scholar
  15. Lipton R, Stuebner M (2006) Inverse homogenization and design of microstructure for pointwise stress control. Q J Mech Appl Math 59:139–161zbMATHCrossRefMathSciNetGoogle Scholar
  16. Pereira JT, Fancello EA, Barcellos CS (2004) Topology optimization of continuum structures with material failure constraints. Struct Multidisc Optim 26:50–66CrossRefMathSciNetGoogle Scholar
  17. Stingl M (2005) On the solution of nonlinear semidefinite programs by augmented lagrangian methods. PhD thesis, Institute of Applied Mathematics II, Friedrich-Alexander University of Erlangen-NurembergGoogle Scholar
  18. Vanderbei RJ, Shanno DF (1999) An interior point algorithm for nonconvex nonlinear programming. Comput Optim Appl 13:231–252zbMATHCrossRefMathSciNetGoogle Scholar
  19. Wächter A, Biegler LT (2006) On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math Program 106:25–57zbMATHCrossRefMathSciNetGoogle Scholar
  20. Werme M, Svanberg K (2005) A hierarchical neighbourhood search method for stress-constrained topology optimization. In: Herskovits J, Mazorche S, Canelas A (eds) 6th World Congress on Structural and Multidisciplinary Optimization, ISSMO–Internat. Society for Structural and Multidisciplinary Optimization, paper 3211.
  21. Zowe J, Kočvara M, Bendsøe M (1997) Free material optimization via mathematical programming. Math Program B 79:445–466CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  1. 1.Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicPraha 8Czech Republic
  2. 2.Faculty of Electrical EngineeringCzech Technical UniversityPrague 6Czech Republic
  3. 3.Institute of Applied MathematicsUniversity of Erlangen-NurembergErlangenGermany

Personalised recommendations