Structural and Multidisciplinary Optimization

, Volume 54, Issue 3, pp 619–639 | Cite as

A full-space barrier method for stress-constrained discrete material design optimization



In this paper, we present a full-space barrier method designed for stress-constrained mass minimization problems with discrete material options. The advantages of the full-space barrier method are twofold. First, in the full-space the stress constraints are provably concave, which facilitates the construction of convex subproblems within the optimization algorithm. Second, by using the full-space, it is no longer necessary to employ stress constraint aggregation techniques to reduce adjoint-gradient evaluation costs. The proposed optimization algorithm uses a Newton method where an approximate linearization of the KKT conditions is solved inexactly at each iteration using a preconditioned Krylov subspace method. Sparse constraints that arise in the discrete material parametrization are treated using a null-space method. Results of the proposed algorithm are demonstrated on a series of three topology and multimaterial optimization problems with selection between isotropic and orthotropic materials, as well as discrete ply-angle selection.


Stress constraints Discrete material optimization Full-space method Topology optimization 


  1. Akgun MA, Haftka RT, Wu KC, Walsh JL, Garcelon JH (2001) Efficient structural optimization for multiple load cases using adjoint sensitivities. AIAA J 39(3):511–516. doi:10.2514/2.1336 CrossRefGoogle Scholar
  2. Amir O, Stolpe M, Sigmund O (2009) Efficient use of iterative solvers in nested topology optimization. Struct Multidiscip Optim 42(1):55–72. doi:10.1007/s00158-009-0463-4. ISSN 1615– 1488CrossRefMATHGoogle Scholar
  3. Bathe K-J (1996) Finite element procedures, 2nd edn. Prentice HallGoogle Scholar
  4. Biros G, Ghattas O (2005a) Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part I: The Krylov–Schur solver. SIAM J Sci Comput 27(2):687–713. doi:10.1137/S106482750241565X MathSciNetCrossRefMATHGoogle Scholar
  5. Biros G, Ghattas O (2005b) Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part II: The Lagrange–Newton solver and its application to optimal control of steady viscous flows. SIAM J Sci Comput 27(2):714–739. doi:10.1137/S1064827502415661 MathSciNetCrossRefMATHGoogle Scholar
  6. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press. ISBN 9780521833783Google Scholar
  7. Bruns T E, Tortorelli D A (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459. doi:10.1016/S0045-7825(00)00278-4. ISSN 0045-7825CrossRefMATHGoogle Scholar
  8. Bruyneel M (2011) SFP - A new parameterization based on shape functions for optimal material selection: application to conventional composite plies. Struct Multidiscip Optim 43:17–27. doi:10.1007/s00158-010-0548-0. ISSN 1615-147XCrossRefGoogle Scholar
  9. Bruyneel M, Duysinx P (2006) Note on singular optima in laminate design problems. Struct Multidiscip Optim 31:156–159. doi:10.1007/s00158-005-0569-2. ISSN 1615-147XCrossRefGoogle Scholar
  10. Bruyneel M, Duysinx P, Fleury C, Gao T (2011) Extensions of the shape functions with penalization parametrization for composite-ply optimization. AIAA J 49(10):2325–2329. doi:10.2514/1.J051225 CrossRefGoogle Scholar
  11. Cheng G D, Guo X (1997) 𝜖-relaxed approach in structural topology optimization. Struct Multidiscip Optim 13:258–266. doi:10.1007/BF01197454. ISSN 1615-147XCrossRefGoogle Scholar
  12. Cramer E, Dennis J, Frank P, Lewis R, Shubin G (1994) Problem formulation for multidisciplinary optimization. SIAM J Optim 4(4):754–776MathSciNetCrossRefMATHGoogle Scholar
  13. Dembo RS, Eisenstat SC, Steihaug T (1982) Inexact Newton methods. SIAM J Numer Anal 19(2):400–408. doi:10.1137/0719025 MathSciNetCrossRefMATHGoogle Scholar
  14. Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43(8):1453–1478. ISSN 1097-0207MathSciNetCrossRefMATHGoogle Scholar
  15. Fiacco AV, McCormick GP (1990) Nonlinear Programming. Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611971316
  16. Fletcher R (2000) Practical methods of optimization, 2nd edn. Wiley. ISBN 9781118723180Google Scholar
  17. Forsgren A, Gill P, Wright M (2002) Interior methods for nonlinear optimization. SIAM Rev 44(4):525–597. doi:10.1137/S0036144502414942 MathSciNetCrossRefMATHGoogle Scholar
  18. Gibiansky LV, Sigmund O (2000) Multiphase composites with extremal bulk modulus. J Mech Phys Solids 48(3):461–498. doi:10.1016/S0022-5096(99)00043-5. ISSN 0022-5096MathSciNetCrossRefMATHGoogle Scholar
  19. Guest JK (2009) Topology optimization with multiple phase projection. Comput Methods Appl Mech Eng 199(1–4):123–135. doi:10.1016/j.cma.2009.09.023. ISSN 0045-7825MathSciNetCrossRefMATHGoogle Scholar
  20. Guo X, Zhang WS, Wang MY, Wei P (2011) Stress-related topology optimization via level set approach. Comput Methods Appl Mech Eng 200(47–48):3439–3452. doi:10.1016/j.cma.2011.08.016. ISSN 0045-7825MathSciNetCrossRefMATHGoogle Scholar
  21. Guo X, Zhang W, Zhong W (2014) Stress-related topology optimization of continuum structures involving multi-phase materials. Comput Methods Appl Mech Eng 268:632–655. doi:10.1016/j.cma.2013.10.003. ISSN 0045-7825MathSciNetCrossRefMATHGoogle Scholar
  22. Haftka R T (1985) Simultaneous analysis and design. AIAA J 23(7):1099–1103. doi:10.2514/3.9043 CrossRefMATHGoogle Scholar
  23. Hicken JE (2014) Inexact Hessian-vector products in reduced-space differential-equation constrained optimization. Optim Eng:1–34. doi:10.1007/s11081-014-9258-6
  24. Holmberg E, Torstenfelt B, Klarbring A (2013) Stress constrained topology optimization. Struct Multidiscip Optim 48(1):33–47. doi:10.1007/s00158-012-0880-7. ISSN 1615-147XMathSciNetCrossRefMATHGoogle Scholar
  25. Hvejsel C, Lund E (2011) Material interpolation schemes for unified topology and multi-material optimization. Struct Multidiscip Optim 43:811–825. doi:10.1007/s00158-011-0625-z. ISSN 1615–147XCrossRefMATHGoogle Scholar
  26. Hvejsel C, Lund E, Stolpe M (2011) Optimization strategies for discrete multi-material stiffness optimization. Structu Multidiscip Optim 44:149–163. doi:10.1007/s00158-011-0648-5. ISSN 1615-147XCrossRefGoogle Scholar
  27. Jones RM (1996) Mechanics of composite materials. Technomic Publishing Co.Google Scholar
  28. Kennedy GJ (2015a) Strategies for adaptive optimization with aggregation constraints using interior-point methods. Comput Struct 153:217–229. doi:10.1016/j.compstruc.2015.02.024. ISSN 0045-7949CrossRefGoogle Scholar
  29. Kennedy GJ (2015b) Discrete thickness optimization via piecewise constraint penalization. Struct Multidiscip Optim 51(6):1247–1265. doi:10.1007/s00158-014-1210-z. ISSN 1615-147XMathSciNetCrossRefGoogle Scholar
  30. Kennedy G J, Hicken J E (2015) Improved constraint-aggregation methods. Comput Methods Appl Mech Eng 289:332–354. doi:10.1016/j.cma.2015.02.017. ISSN 0045-7825MathSciNetCrossRefGoogle Scholar
  31. Kennedy GJ, Martins JRRA (2013) A laminate parametrization technique for discrete ply-angle problems with manufacturing constraints. Struct Multidiscip Optim:1–15. doi:10.1007/s00158-013-0906-9. ISSN 1615-147X
  32. Kennedy GJ, Martins JRRA (2014) A parallel aerostructural optimization framework for aircraft design studies. Struct Multidiscip Optim 50(6):1079–1101. doi:10.1007/s00158-014-1108-9. ISSN 1615-147XCrossRefGoogle Scholar
  33. 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. ZurichGoogle Scholar
  34. Lambe AB, Kennedy GJ, Martins JRRA (2014) Multidisciplinary design optimization of an aircraft wing via a matrix-free approach. In: 15th AIAA/ISSMO multidisciplinary analysis and optimization conference. AtlantaGoogle Scholar
  35. Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41:605–620. doi:10.1007/s00158-009-0440-y. ISSN 1615–147XCrossRefGoogle Scholar
  36. Lund E (2009) Buckling topology optimization of laminated multi-material composite shell structures. Compos Struct 91(2):158–167. doi:10.1016/j.compstruct.2009.04.046. ISSN 0263-8223CrossRefGoogle Scholar
  37. Martins JRRA, Lambe AB (2013) Multidisciplinary design optimization: a survey of architectures. AIAA J 51:2049–2075. doi:10.2514/1.J051895 CrossRefGoogle Scholar
  38. Martins JRRA, Alonso JJ, Reuther JJ (2004) High-fidelity aerostructural design optimization of a supersonic business jet. J Aircraft 41(3):523–530. doi:10.2514/1.11478 CrossRefGoogle Scholar
  39. Martins JRRA, Alonso JJ, Reuther JJ (2005) A coupled–adjoint sensitivity analysis method for high–fidelity aero–structural design. Optim Eng 6:33–62. doi:10.1023/B:OPTE.0000048536.47956.62 CrossRefMATHGoogle Scholar
  40. Nocedal J, Wright SJ (2006) Numerical optimization. Springer series in operations research and financial engineering, 2nd edn. SpringerGoogle Scholar
  41. Olhoff N, Bendsøe MP, Rasmussen J (1991) On CAD-integrated structural topology and design optimization. Comput Methods Appl Mech Eng 89(1):259–279. doi:10.1016/0045-7825(91)90044-7. ISSN 0045-7825CrossRefMATHGoogle Scholar
  42. Peterson P (2009) F2PY: a tool for connecting Fortran and Python programs. Int J Comput Sci Eng 4(4):296–305CrossRefGoogle Scholar
  43. Poon N, Martins JRRA (2007) An adaptive approach to constraint aggregation using adjoint sensitivity analysis. Struct Multidiscip Optim 34:61–73. doi:10.1007/s00158-006-0061-7. ISSN 1615-147XCrossRefGoogle Scholar
  44. Ramani A (2011) Multi-material topology optimization with strength constraints. Struct Multidiscip Optim 43(5):597–615. doi:10.1007/s00158-010-0581-z. ISSN 1615-147XMathSciNetCrossRefGoogle Scholar
  45. Saad Y (1993) A fexible inner-outer preconditioned GMRES algorithm. SIAM J Sci Comput 14(2):461–469. doi:10.1137/0914028 MathSciNetCrossRefMATHGoogle Scholar
  46. Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. PWS Pub. Co.Google Scholar
  47. Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7(3):856–869. doi:10.1137/0907058 MathSciNetCrossRefMATHGoogle Scholar
  48. Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization method. J Mech Phys Solids 45(6):1037–1067. doi:10.1016/S0022-5096(96)00114-7. ISSN 0022-5096MathSciNetCrossRefGoogle Scholar
  49. Sørensen SN, Lund E (2013) Topology and thickness optimization of laminated composites including manufacturing constraints. Struct Multidiscip Optim 48(2):249–265. doi:10.1007/s00158-013-0904-y. ISSN 1615-147XMathSciNetCrossRefGoogle Scholar
  50. Sørensen SN, Sørensen R, Lund E (2014) DMTO – a method for discrete material and thickness optimization of laminated composite structures. Struct Multidiscip Optim 50(1):25–47. doi:10.1007/s00158-014-1047-5. ISSN 1615- 147XCrossRefGoogle Scholar
  51. Stegmann J, Lund E (2005) Discrete material optimization of general composite shell structures. Int J Numer Methods Eng:2009–2027. doi:10.1002/nme.1259. ISSN 1097-0207
  52. Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22:116–124. doi:10.1007/s001580100129. ISSN 1615-147XCrossRefGoogle Scholar
  53. Wang S, de Sturler E, Paulino GH (2007) Large-scale topology optimization using preconditioned Krylov subspace methods with recycling. Int J Numer Methods Eng 69(12):2441–2468. doi:10.1002/nme.1798. ISSN 1097-0207MathSciNetCrossRefMATHGoogle Scholar
  54. Wesseling P (1992) An introduction to multigrid methods. John Wiley & SonsGoogle Scholar
  55. Xia Q, Shi T, Liu S, Wang MY (2012) A level set solution to the stress-based structural shape and topology optimization. Comput Struct 90–91:55–64. doi:10.1016/j.compstruc.2011.10.009. ISSN 0045-7949CrossRefGoogle Scholar
  56. Yang R, Chen C (1996) Stress-based topology optimization. Struct Optim 12(2–3):98–105. doi:10.1007/BF01196941. ISSN 0934-4373CrossRefGoogle Scholar
  57. Zhang WS, Guo X, Wang MY, Wei P (2013) Optimal topology design of continuum structures with stress concentration alleviation via level set method. Int J Numer Methods Eng 93(9):942–959. doi:10.1002/nme.4416. ISSN 1097-0207MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Aerospace EngineeringGeorgia Institute of TechnologyAtlantaUSA

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