Advertisement

On multigrid-CG for efficient topology optimization

  • Oded Amir
  • Niels Aage
  • Boyan S. Lazarov
RESEARCH PAPER

Abstract

This article presents a computational approach that facilitates the efficient solution of 3-D structural topology optimization problems on a standard PC. Computing time associated with solving the nested analysis problem is reduced significantly in comparison to other existing approaches. The cost reduction is obtained by exploiting specific characteristics of a multigrid preconditioned conjugate gradients (MGCG) solver. In particular, the number of MGCG iterations is reduced by relating it to the geometric parameters of the problem. At the same time, accurate outcome of the optimization process is ensured by linking the required accuracy of the design sensitivities to the progress of optimization. The applicability of the proposed procedure is demonstrated on several 2-D and 3-D examples involving up to hundreds of thousands of degrees of freedom. Implemented in MATLAB, the MGCG-based program solves 3-D topology optimization problems in a matter of minutes. This paves the way for efficient implementations in computational environments that do not enjoy the benefits of high performance computing, such as applications on mobile devices and plug-ins for modeling software.

Keywords

Topology optimization Preconditioned conjugate gradients Multigrid 

Notes

Acknowledgments

The authors wish to thank Ole Sigmund for fruitful discussions and helpful comments on the manuscript. The anonymous reviewers’ valuable remarks are gratefully acknowledged. The authors acknowledge the financial support received from the European Commission Research Executive Agency, grant agreement PCIG12-GA-2012-333647; and from the NextTop project, sponsored by the Villum foundation. The authors also thank Krister Svanberg for the MATLAB MMA code.

Supplementary material

158_2013_1015_MOESM1_ESM.m (7 kb)
(M 7.45 KB)

References

  1. Aage N, Lazarov B (2013) Parallel framework for topology optimization using the method of moving asymptotes. Struct Multidiscip Optim 47(4):493–505. doi: 10.1007/s00158-012-0869-2 CrossRefzbMATHMathSciNetGoogle Scholar
  2. Aage N, Nobel-Jørgensen M, Andreasen CS, Sigmund O (2013) Interactive topology optimization on hand-held devices. Struct Multidiscip Optim 47(1):1–6CrossRefGoogle Scholar
  3. Amir O, Bendsøe MP, Sigmund O (2009) Approximate reanalysis in topology optimization. Int J Numer Methods Eng 78:1474–1491CrossRefzbMATHGoogle Scholar
  4. Amir O, Stolpe M, Sigmund O (2010) Efficient use of iterative solvers in nested topology optimization. Struct Multidiscip Optim 42:55–72CrossRefzbMATHGoogle Scholar
  5. Amir O, Sigmund O (2011) On reducing computational effort in topology optimization: how far can we go? Struct Multidiscip Optim 44:25–29CrossRefzbMATHGoogle Scholar
  6. Amir O, Sigmund O, Schevenels M, Lazarov B (2012) Efficient reanalysis techniques for robust topology optimization. Comput Methods Appl Mech Eng 245–246:217–231CrossRefMathSciNetGoogle Scholar
  7. Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in matlab using 88 lines of code. Struct Multidiscip Optim 43:1–16CrossRefzbMATHGoogle Scholar
  8. Ashby SF, Falgout RD (1996) A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations. Nucl Sci Eng 124:145–159Google Scholar
  9. Baker A, Falgout R, Gamblin T, Kolev T, Schulz M, Yang U (2012a) Scaling algebraic multigrid solvers: on the road to exascale. In: Bischof C, Hegering H-G, Nagel WE, Wittum G (eds) Competence in high performance computing 2010. Springer, Berlin, pp 215–226Google Scholar
  10. Baker A, Falgout R, Kolev T, Yang U (2012b) Scaling hypre’s multigrid solvers to 100,000 cores. In: Berry MW, Gallivan KA, Gallopoulos E, Grama A, Philippe B, Saad Y, Saied F (eds) High-performance scientific computing. Springer, London, pp 261–279. doi: 10.1007/978-1-4471-2437-5_13 CrossRefGoogle Scholar
  11. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202CrossRefGoogle Scholar
  12. Bendsøe MP, Sigmund O (2003) Topology optimization—theory, methods and applications. Springer, BerlinGoogle Scholar
  13. Bogomolny M (2010) Topology optimization for free vibrations using combined approximations. Int J Numer Methods Eng 82(5):617–636. doi: 10.1002/nme.2778 zbMATHGoogle Scholar
  14. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50:2143–2158CrossRefzbMATHMathSciNetGoogle Scholar
  15. Bramble JH, Pasciak JE, Wang J, Xu J (1991) Convergence estimates for multigrid algorithms without regularity assumptions. Math Comput 57:23–45CrossRefzbMATHMathSciNetGoogle Scholar
  16. Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190:3443–3459CrossRefzbMATHGoogle Scholar
  17. Chow E, Falgout RD, Hu JJ, Tuminaro RS, Yang UM (2006) A survey of parallelization techniques for multigrid solvers. In: Heroux MA, Raghavan P, Simon HD (eds) Parallel processing for scientific computing, chapter 10. SIAM, Philadelphia, pp 179–201CrossRefGoogle Scholar
  18. Davis TA (2006) Direct methods for sparse linear system. SIAM, PhiladelphiaCrossRefGoogle Scholar
  19. Evgrafov A, Rupp CJ, Maute K, Dunn ML (2008) Large-scale parallel topology optimization using a dual-primal substructuring solver. Struct Multidiscip Optim 36:329–345CrossRefzbMATHMathSciNetGoogle Scholar
  20. Farhat C, Lesoinne M, LeTallec P, Pierson K, Rixen D (2001) FETI-DP: a dual–primal unified FETI method—part I: a faster alternative to the two-level FETI method. Int J Numer Methods Eng 50(7):1523–1544CrossRefzbMATHMathSciNetGoogle Scholar
  21. Guest JK, Smith Genut LC (2010) Reducing dimensionality in topology optimization using adaptive design variable fields. Int J Numer Methods Eng 81(8):1019–1045. doi: 10.1002/nme.2724 zbMATHGoogle Scholar
  22. Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Nat Bur Stan 49(6):409–436CrossRefzbMATHMathSciNetGoogle Scholar
  23. Kim JE, Jang G-W, Kim YY (2003) Adaptive multiscale wavelet-galerkin analysis for plane elasticity problems and its applications to multiscale topology design optimization. Int J Solids Struct 40(23):6473–6496CrossRefzbMATHGoogle Scholar
  24. Kim SY, Kim IY, Mechefske CK (2012) A new efficient convergence criterion for reducing computational expense in topology optimization: reducible design variable method. Int J Numer Methods Eng 90(6):752–783. doi: 10.1002/nme.3343 CrossRefzbMATHGoogle Scholar
  25. Kim YY, Yoon GH (2000) Multi-resolution multi-scale topology optimization—a new paradigm. Int J Solids Struct 37(39):5529–5559CrossRefzbMATHMathSciNetGoogle Scholar
  26. Lazarov BS (2013) Topology optimization using multiscale finite element method for high-contrast media. In reviewGoogle Scholar
  27. Maar B, Schulz V (2000) Interior point multigrid methods for topology optimization. Struct Multidiscip Optim 19:214–224CrossRefGoogle Scholar
  28. Nguyen TH, Paulino GH, Song J, Le CH (2010) A computational paradigm for multiresolution topology optimization (mtop). Struct Multidiscip Optim 41:525–539. doi: 10.1007/s00158-009-0443-8 CrossRefzbMATHMathSciNetGoogle Scholar
  29. Nguyen TH, Paulino GH, Song J, Le CH (2012) Improving multiresolution topology optimization via multiple discretizations. Int J Numer Methods Eng 92(6):507–530. doi: 10.1002/nme.4344 CrossRefMathSciNetGoogle Scholar
  30. Poulsen TA (2002) Topology optimization in wavelet space. Int J Numer Methods Eng 53(3):567–582CrossRefzbMATHMathSciNetGoogle Scholar
  31. Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  32. Sigmund O (1997a) On the design of compliant mechanisms using topology optimization. Mech Based Des Struct Mach 25:493–524CrossRefGoogle Scholar
  33. Sigmund O, Torquato S (1997b) Design of materials with extreme thermal expansion using a three-phase topology optimization method. J Mech Phys Solids 45(6):1037–1067CrossRefMathSciNetGoogle Scholar
  34. Sigmund O, Maute K (2012) Sensitivity filtering from a continuum mechanics perspective. Struct Multidiscip Optim 46:471–475. doi: 10.1007/s00158-012-0814-4 CrossRefzbMATHMathSciNetGoogle Scholar
  35. Stainko R (2006a) An adaptive multilevel approach to the minimal compliance problem in topology optimization. Commun Numer Methods Eng 22(2):109–118. doi: 10.1002/cnm.800 CrossRefzbMATHMathSciNetGoogle Scholar
  36. Stainko R (2006b) Advanced multilevel techniques to topology optimization. PhD thesis, Johannes Kepler Universitȧt LinzGoogle Scholar
  37. Suresh K (2013) Efficient generation of large-scale pareto-optimal topologies. Struct Multidiscip Optim 47:49–61. doi: 10.1007/s00158-012-0807-3 CrossRefzbMATHMathSciNetGoogle Scholar
  38. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373CrossRefzbMATHMathSciNetGoogle Scholar
  39. Tatebe O, Oyanagi Y (1994) Efficient implementation of the multigrid preconditioned conjugate gradient method on distributed memory machines. In: Proceedings of Supercomputing’94. IEEE, pp 194–203Google Scholar
  40. Trottenberg U, Oosterlee C, Schuller A (2001) Multigrid. Academic Press, LondonzbMATHGoogle Scholar
  41. Vassilevski PS (2008) Multilevel block factorization preconditioners: matrix-based analysis and algorithms for solving finite element equations. Springer, New YorkGoogle Scholar
  42. Wang S, de Sturler E, Paulino GH (2007) Large-scale topology optimization using preconditioned Krylov subspace methods with recycling. Int J Numer Methods Eng 69:2441–2468CrossRefzbMATHGoogle Scholar
  43. Zhou S, Wang MY (2007) Multimaterial structural topology optimization with a generalized cahn-hilliard model of multiphase transition. Struct Multidiscip Optim 33:89–111. doi: 10.1007/s00158-006-0035-9 CrossRefzbMATHMathSciNetGoogle Scholar
  44. Zuo W, Xu T, Zhang H, Xu T (2011) Fast structural optimization with frequency constraints by genetic algorithm using adaptive eigenvalue reanalysis methods. Struct Multidiscip Optim 43:799–810. doi: 10.1007/s00158-010-0610-y CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental EngineeringTechnion - Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of Mechanical EngineeringTechnical University of DenmarkCopenhagenDenmark

Personalised recommendations