A dual mesh method with adaptivity for stress-constrained topology optimization

  • Daniel A. WhiteEmail author
  • Youngsoo Choi
  • Jun Kudo
Research Paper


This paper is concerned with the topological optimization of elastic structures, with the goal of minimizing the compliance and/or mass of the structure, subject to a stress constraint. It is well known that depending upon the geometry and the loading conditions, the stress field can exhibit singularities, if these singularities are not adequately resolved, the topological optimization process will be ineffective. For computational efficiency, adaptive mesh refinement is required to adequately resolve the stress field. This poses a challenge for the traditional Solid Isotropic Material with Penalization (SIMP) method that employs a one-to-one correspondence between the finite element mesh and the optimization design variables because as the mesh is refined, the optimization process must somehow be re-started with a new set of design variables. The proposed solution is a dual mesh approach, one finite element mesh defines the material distribution, a second finite element mesh is used for the computation of the displacement and stress fields. This allows for stress-based adaptive mesh refinement without modifying the definition of the optimization design variables. A second benefit of this dual mesh approach is that there is no need to apply a filter to the design variables to enforce a length scale, instead the length scale is determined by the design mesh. This reduces the number of design variables and allows the designer to apply spatially varying length scale if desired. The efficacy of this dual mesh approach is established via several stress-constrained topology optimization problems.


Topopolgy optimization Stress Adaptive mesh refinement and Bernstein polynomials 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Amrosio L, Buttazaro G (1993) An optimal-design problem with perimeter penalization. Calc Var 1(1):55–69MathSciNetCrossRefGoogle Scholar
  2. Amstutz S, Novotny A (2010) Topological optimization of structures subject to Von Mises stress constraints. Struct Multidisc Optim 41(3):407–420MathSciNetzbMATHCrossRefGoogle Scholar
  3. Bendsoe M (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202CrossRefGoogle Scholar
  4. Bendsoe M, Sigmund O (2003) Topology optimization theory, methods, and applications. Springer, BerlinzbMATHGoogle Scholar
  5. Biswas A, Shapiro V, Tsukanov I (2004) Heterogeneous material modeling with distance fields. Comp Aided Geom Des 21:215–232MathSciNetzbMATHCrossRefGoogle Scholar
  6. Bourdin B (2001) Filters in topology optimization. Int J Num Meth Eng 50:2143–2158MathSciNetzbMATHCrossRefGoogle Scholar
  7. Bruggi M (2008) On an alternative approach to stress constraints relaxation in topology optimization. Struct Multidiscip Optim 36(2):125–141MathSciNetzbMATHCrossRefGoogle Scholar
  8. Bruggi M, Verani M (2011) A fully adaptive topology optimization algorithm with goal oriented error control. Comp Struct 89:1481–1493CrossRefGoogle Scholar
  9. Bruns T, Tortorelli D (2001) Topology optimization of nonlinear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190:26–27zbMATHCrossRefGoogle Scholar
  10. Bruns T, Tortorelli D (2003) An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms. Int J Numer Methods Eng 57(10):1413–1430zbMATHCrossRefGoogle Scholar
  11. Chen J, Shapiro V (2008) Optimization of continuous heterogenous models. Heterogen Object Model Appl Lect Notes Comput Sci 4889:193–213CrossRefGoogle Scholar
  12. Cheng G, Jiang Z (1992) Study on topology optimization with stress constraints. Eng Optim 20(2):129–148CrossRefGoogle Scholar
  13. Cheng G, Guo X (1997) ε-relaxed approach in structural topology optimization. Struct Optim 13(4):258–266CrossRefGoogle Scholar
  14. Duysinx P, Bendsoe M (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Meth Eng 43:1453–1478MathSciNetzbMATHCrossRefGoogle Scholar
  15. Frigo M, Johnson SG (2005) The design and implementation of fftw3. Proc IEEE 93(2):216–231CrossRefGoogle Scholar
  16. Gomes A, Suleman A (2006) Application of spectral level set methodology in topology optimization. Struct Multi Optim 31:430–443MathSciNetzbMATHCrossRefGoogle Scholar
  17. Guest TBJK, Prevost JH (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Meth Eng 61(2):238–254MathSciNetzbMATHCrossRefGoogle Scholar
  18. Guest JK, Genut LCS (2010) Reducing dimensionality in topology optimization using adaptive design variable fields. Int J Numer Meth Eng 81(8):1019–1045zbMATHGoogle Scholar
  19. Guo X, Zhang W, Wang MY, Wei P (2011) Stress-related topology optimization via level set approach. Comput Methods Appl Mech Eng 200(47-48):3439–3452MathSciNetzbMATHCrossRefGoogle Scholar
  20. Guo X, Zhang W, Zhong W (2014a) Doing topology optimization explicitly and geometrically-a new moving morphable components based framework. Journal Of Applied Mechanics-Transactions Of The ASME 81(8):081009Google Scholar
  21. Guo X, Zhang W, Zhong W (2014b) Stress-related topology optimization of continuum structures involving multi-phase materials. Comput Methods Appl Mech Eng 268:632–655MathSciNetzbMATHCrossRefGoogle Scholar
  22. Gupta DK, Langelaar M, van Keulen F (2018) Qr-patterns: artefacts in multiresolution topology optimization. Struct Multidiscip Optim 58(4):1335–1350CrossRefGoogle Scholar
  23. Haber R, Jog C, Bendsoe M (1996) A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim 11(1):1–12CrossRefGoogle Scholar
  24. Jensen KE (2016) Solving stress and compliance constrained volume minimization using anisotropic mesh adaptation, the method of moving asymptotes, and a global p-norm. Struct Mult Optim 54:831–841MathSciNetCrossRefGoogle Scholar
  25. Kang Z, Wang Y (2011) Structural topology optimization based on non-local Shepard interpolation of density field. Comp Meth Appl Mech Eng 200(49-52):3515–3525MathSciNetzbMATHCrossRefGoogle Scholar
  26. Kang Z, Wang YQ (2012) A nodal variable method of structural optimization based on shepard interpolant. Struct Mult Opt 90:329–342MathSciNetzbMATHGoogle Scholar
  27. Karp SN, Karal FC (1962) The elastic-field behavior in the neighborhood of a crack of arbitrary angle. Comm Pure Appl Math XV:413–421MathSciNetzbMATHGoogle Scholar
  28. Kim Y, Yoon G (2000) Multi-resolution multi-scale topology optimization - a new paradigm. Int J Solids Struct 37(39):5529–5559MathSciNetzbMATHCrossRefGoogle Scholar
  29. Körner TW (1988) Fourier analysis. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  30. Lazarov B, Sigmund O (2011) Filters in topology optimization based on helmholtz-type differential equations. Int J Numer Meth Eng 86:765–781MathSciNetzbMATHCrossRefGoogle Scholar
  31. Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidisc Optim 41:605–620CrossRefGoogle Scholar
  32. Lieu QX, Lee J (2017) Multiresolution topology optimization using isogeometric analysis. Int J Numer Methods Eng 112(13):2025–2047MathSciNetCrossRefGoogle Scholar
  33. Liu C, Zhu Y, Sun Z, Li D, Du Z, Zhang W, Guo X (2018) An efficient moving morphable component (mmc)-based approach for multi-resolution topology optimization. Struct Multidiscip Optim 58(6):2455–2479MathSciNetCrossRefGoogle Scholar
  34. Luo Z, Zhang N, Wang Y, Gao W (2013) Topology optimization of structures using meshless density variable approximants. Int J Numer Meth Eng 93(4):443–464MathSciNetzbMATHCrossRefGoogle Scholar
  35. Matsui K, Terada K (2004) Continuous approximation of material distribution for topology optimization. Int J Numer Meth Eng 59:1925–1944MathSciNetzbMATHCrossRefGoogle Scholar
  36. Maute K, Ramm E (1995) Adaptive topology optimization. Struct Opt 10:100–112zbMATHCrossRefGoogle Scholar
  37. MFEM (2019) Modular finite element methods, mfem.orgGoogle Scholar
  38. Nana A, Cuilliere JC, Francois V (2016) Towards adaptive topology optimization. Adv Eng Soft 100:290–307CrossRefGoogle Scholar
  39. Nguyen TH, Paulino GH, Song J, Le CH (2010) A computational paradigm for multiresolution topology optimization (mtop). Struct Multidiscip Optim 41(4):525–539MathSciNetzbMATHCrossRefGoogle Scholar
  40. Nguyen TH, Le CH, Hajjar JF (2017) Topology optimization using the p-version of the finite element method. Struct Multidiscip Optim 56(3):571–586MathSciNetCrossRefGoogle Scholar
  41. Niordson F (1983) Optimal-design Of elastic plates with a constraint on the slope of the thickness function. Int J Solids Struct 19(2):141–151zbMATHCrossRefGoogle Scholar
  42. Norato JA, Bell BK, Tortorelli DA (2015) A geometry projection method for continuum-based topology optimization with discrete elements. Comp Meth App Mech Eng 293:306–327MathSciNetzbMATHCrossRefGoogle Scholar
  43. Panesar A, Brackett D, Ashcroft I, Wildman R, Hague R (2017) Hierarchical remeshing strategies with mesh mapping for topology optimization. Int J Numer Meth Eng 111:676–700CrossRefGoogle Scholar
  44. Picelli R, Townsend S, Brampton C, Norato J, Kim H (2018) Stress-based shape and topology optimization with the level set method. Comput Methods Appl Mech Eng 329:1–23MathSciNetCrossRefGoogle Scholar
  45. Poulsen T (2002) Topology optimization in wavelet space. Int J Numer Meth Eng 53:567–582MathSciNetzbMATHCrossRefGoogle Scholar
  46. Qian X (2013) Topology optimization in b-spline space. Comp Meth Appl Mech Eng 265:15–35MathSciNetzbMATHCrossRefGoogle Scholar
  47. Rahmatalla S, Swan C (2004) A Q4/Q4 continuum structural topology implementation. Struct Mult Optim 27:130–135CrossRefGoogle Scholar
  48. Rozvany G (2001) Aims, scope, methods, history, and unified terminology of computer aided optimization in structural mechanics. Struct Multidiscip Opt 21(2):90–108MathSciNetCrossRefGoogle Scholar
  49. 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
  50. Salazar de Troya MA, Tortorelli D (2018) Adaptive mesh refinement in stress-constrained topology opotimization. Struct Mult Opt 58:2369–2386CrossRefGoogle Scholar
  51. Sayood K (2012) Introduction to data compression. Morgan KaufmannGoogle Scholar
  52. Sidmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16:68–75CrossRefGoogle Scholar
  53. Suresh K, Takalloozadeh M (2013) Stress-constrained topology optimizatioin: a topilogical level-set approach. Struct Mult Optim 48:295–309CrossRefGoogle Scholar
  54. 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(1):5–57zbMATHCrossRefGoogle Scholar
  55. Wang Y, Kang Z, He Q (2013) An adaptive refinement approach for topology optimization based on separated density field description. Comput Struct 117:10–22CrossRefGoogle Scholar
  56. Wang Y, Kang Z, He Q (2014) Adaptive topology optimization with independent error control for separated displacement and density fields. Comput Struct 135:50–61CrossRefGoogle Scholar
  57. Williams ML (1952) Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J App Mech ASME 19(4):526–528Google Scholar
  58. White DA, Stowell MS (2018) Topological optimization of structures using Fourier representations. Struct Multidisp Opt 58:1205–1220CrossRefGoogle Scholar
  59. Yang R, Chen C (1996) Stress-based topology optimization. Struct Optim 12:98–105CrossRefGoogle Scholar
  60. Yosibash Z (2012) Singularities in elliptic boundary value problems and elasticity and their connection with failure initiation. Springer, BerlinzbMATHCrossRefGoogle Scholar
  61. Zhang S, Norato JA, Gain AL, Lyu N (2016) A geometry projection method for the topology optimization of plate structures. Struct Multidiscip Optim 54(5, SI):1173–1190MathSciNetCrossRefGoogle Scholar
  62. Zhang W, Liu Y, Weng P, Zhu Y, Guo X (2017) Explicit control of structural complexity in topology optimization. Comp Meth Appl Mech Engrg 324:149–169MathSciNetCrossRefGoogle Scholar
  63. Zhang W, Li D, Zhou J, Du Z, Li B, Guo X (2018) A moving morphable void (Mmv)-based explicit approach for topology optimization considering stress constraints. Comput Methods Appl Mech Eng 334:381–413MathSciNetCrossRefGoogle Scholar

Copyright information

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Lawrence Livermore National LaboratoryLivermoreUSA

Personalised recommendations