Three-dimensional adaptive mesh refinement in stress-constrained topology optimization

Abstract

Structural optimization software that can produce high-resolution designs optimized for arbitrary cost and constraint functions is essential to solve real-world engineering problems. Such requirements are not easily met due to the large-scale simulations and software engineering they entail. In this paper, we present a large-scale topology optimization framework with adaptive mesh refinement (AMR) applied to stress-constrained problems. AMR allows us to save computational resources by refining regions of the domain to increase the design resolution and simulation accuracy, leaving void regions coarse. We discuss the challenges necessary to resolve such large-scale problems with AMR, namely, the need for a regularization method that works across different mesh resolutions in a parallel environment and efficient iterative solvers. Furthermore, the optimization algorithm needs to be implemented with the same discretization that is used to represent the design field. To show the efficacy and versatility of our framework, we minimize the mass of a three-dimensional L-bracket subject to a maximum stress constraint and maximize the efficiency of a three-dimensional compliant mechanism subject to a maximum stress constraint.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Notes

  1. 1.

    Such data structures are also required in parallel high-order finite difference simulation codes (Tegeler et al. 2017).

  2. 2.

    The normal gradient ∇n is defined such that ∇na = ∇an

References

  1. Aage N, Lazarov B S (2013) Parallel framework for topology optimization using the method of moving asymptotes. Struct Multidiscip Optim 47(4):493–505. https://doi.org/10.1007/s00158-012-0869-2

    MathSciNet  Article  Google Scholar 

  2. Aage N, Andreassen E, Lazarov B S (2015) Topology optimization using PETSc: an easy-to-use, fully parallel, open source topology optimization framework. Struct Multidiscip Optim 51(3):565–572. https://doi.org/10.1007/s00158-014-1157-0

    MathSciNet  Article  Google Scholar 

  3. Aage N, Andreassen E, Lazarov B S, Sigmund O (2017) Giga-voxel computational morphogenesis for structural design. Nature 550(7674):84

    Article  Google Scholar 

  4. Alexandersen J, Sigmund O, Aage N (2016) Large scale three-dimensional topology optimisation of heat sinks cooled by natural convection. Int J Heat Mass Transfer 100:876–891

    Article  Google Scholar 

  5. Amstutz S, Novotny A A (2010) Topological optimization of structures subject to von Mises stress constraints. Struct Multidiscip Optim 41(3):407–420. https://doi.org/10.1007/s00158-009-0425-x

    MathSciNet  Article  Google Scholar 

  6. Ayachit U (2015) The Paraview guide: a parallel visualization application

  7. Balay S, Abhyankar S, Adams MF, Brown J, Brune P, Buschelman K, Dalcin L, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Rupp K, Smith BF, Zampini S, Zhang H, Zhang H (2016) PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.7, Argonne National Laboratory

  8. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50 (9):2143–2158. https://doi.org/10.1002/nme.116

    MathSciNet  Article  Google Scholar 

  9. Brandts JH, Korotov S, Křížek M (2008) The discrete maximum principle for linear simplicial finite element approximations of a reaction–diffusion problem. Linear Algebra Appl 429(10):2344–2357

  10. Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Computer Methods Appl Mech Eng 190(26):3443–3459, https://doi.org/10.1016/S0045-7825(00)00278-4

  11. Burman E, Ern A (2004) Discrete maximum principle for Galerkin approximations of the laplace operator on arbitrary meshes. Comptes Rendus Math 338(8):641–646

    MathSciNet  Article  Google Scholar 

  12. Burman E, Claus S, Hansbo P, Larson M G, Massing A (2015) CutFEM: Discretizing geometry and partial differential equations. Int J Numer Methods Eng 104(7):472–501. https://doi.org/10.1002/nme.4823

    MathSciNet  Article  Google Scholar 

  13. De Leon D M, Alexandersen J, Fonseca J S, Sigmund O (2015) Stress-constrained topology optimization for compliant mechanism design. Struct Multidiscip Optim 52(5):929–943

    MathSciNet  Article  Google Scholar 

  14. Evgrafov A, Rupp C J, Maute K, Dunn M L (2008) Large-scale parallel topology optimization using a dual-primal substructuring solver. Struct Multidiscip Optim 36(4):329–345

    MathSciNet  Article  Google Scholar 

  15. Eymard R, Gallouët T, Herbin R (2000) Finite volume methods. Handb Numer Anal 7:713–1018

    MathSciNet  MATH  Google Scholar 

  16. Kirk BS, Peterson JW, Stogner RH, Carey GF (2006) libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng Comput 22(3–4):237–254. https://doi.org/10.1007/s00366-006-0049-3

  17. Lazarov B S, Sigmund O (2011) Filters in topology optimization based on Helmholtz-type differential equations. Int J Numer Methods Eng 86(6):765–781. https://doi.org/10.1002/nme.3072

    MathSciNet  Article  Google Scholar 

  18. Leader MK, Chin TW, Kennedy G (2018) High resolution topology optimization of aerospace structures with stress and frequency constraints. In: 2018 Multidisciplinary Analysis and Optimization Conference, pp 4056

  19. LLNL (2018) hypre: High Performance Preconditioners. Lawrence Livermore National Laboratory, http://www.llnl.gov/CASC/hypre/

  20. Meneghelli L R, Cardoso E L (2013) Design of compliant mechanisms with stress constraints using topology optimization. Springer International Publishing, Cham, pp 35–48. https://doi.org/10.1007/978-3-319-00717-5-3

  21. Munro D, Groenwold A (2017) Local stress-constrained and slope-constrained sand topology optimisation. Int J Numer Methods Eng 110(5):420–439. https://doi.org/10.1002/nme.5360

    MathSciNet  Article  Google Scholar 

  22. Najafi A R, Safdari M, Tortorelli D A, Geubelle P H (2017) Shape optimization using a NURBS-based interface-enriched generalized FEM. Int J Numer Methods Eng 111(10):927–954

    MathSciNet  Article  Google Scholar 

  23. Saxena A, Ananthasuresh G K (2001) Topology optimization of compliant mechanisms with strength considerations*. Mech Struct Mach 29(2):199–221. https://doi.org/10.1081/SME-100104480

    Article  Google Scholar 

  24. Schillinger D, Ruess M (2015) The finite cell method: a review in the context of higher-order structural analysis of CAD and image-based geometric models. Arch Comput Methods Eng 22(3):391–455. https://doi.org/10.1007/s11831-014-9115-y

    MathSciNet  Article  Google Scholar 

  25. Schwedes T, Ham D A, Funke S W, Piggott M D (2017) Mesh dependence in PDE-constrained optimisation: an application in tidal turbine array layouts. Springer Publishing Company, Incorporated

  26. Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22(2):116–124. https://doi.org/10.1007/s001580100129

    Article  Google Scholar 

  27. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373

    MathSciNet  Article  Google Scholar 

  28. Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12(2):555–573. https://doi.org/10.1137/S1052623499362822

    MathSciNet  Article  Google Scholar 

  29. Tegeler M, Shchyglo O, Kamachali R, Monas A, Steinbach I, Sutmann G (2017) Parallel multiphase field simulations with openphase. Comput Phys Commun 215:173–187

    MathSciNet  Article  Google Scholar 

  30. Salazar de Troya MA, Tortorelli DA (2018) Adaptive mesh refinement in stress-constrained topology optimization. Struct Multidiscip Optim 58:2369–2386. https://doi.org/10.1007/s00158-018-2084-2

    MathSciNet  Article  Google Scholar 

  31. Salazar de Troya MA, Oxberry GM, Petra CG, Tortorelli DA (Under review) Mesh independence in topology optimization. Structural and Multidisciplinary Optimization

  32. Verfürth R (1999) A review of a posteriori error estimation techniques for elasticity problems. Comput Methods Appl Mech Eng 176(1-4):419–440

    MathSciNet  Article  Google Scholar 

  33. Wang S, Ed S, Paulino G H (2010) Large-scale topology optimization using preconditioned Krylov subspace methods with recycling. Int J Numer Methods Eng 69(12):2441–2468. https://doi.org/10.1002/nme.1798

    MathSciNet  Article  Google Scholar 

  34. Wang F, Lazarov B S, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784. https://doi.org/10.1007/s00158-010-0602-y

    Article  Google Scholar 

Download references

Acknowledgments

The author thanks the Livermore Graduate Scholar Program for its support.

Funding

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Miguel A. Salazar de Troya.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Replication of results

The software used to generate the results shown in Section 5 is property of the US Department of Energy and has not yet been approved for public release, and therefore is not currently openly distributed. The computational meshes used to generate the results can be obtained by contacting the corresponding author. All the details necessary to reproduce the results in Section 5 (loads, boundary conditions, constraints, objectives, optimization parameters, etc.) have been defined in the paper. We summarize the most important parameters in Table 1.

Table 1 Relevant parameters

Responsible Editor: Fred van Keulen

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Salazar de Troya, M.A., Tortorelli, D.A. Three-dimensional adaptive mesh refinement in stress-constrained topology optimization. Struct Multidisc Optim (2020). https://doi.org/10.1007/s00158-020-02618-z

Download citation

Keywords

  • Topology optimization
  • Stress constrained
  • Adaptive mesh refinement
  • Large-scale design