Advertisement

Structural and Multidisciplinary Optimization

, Volume 51, Issue 1, pp 41–57 | Cite as

Revisiting approximate reanalysis in topology optimization: on the advantages of recycled preconditioning in a minimum weight procedure

  • Oded Amir
RESEARCH PAPER

Abstract

An efficient procedure for three-dimensional continuum structural topology optimization is proposed. The approach is based on recycled preconditioning, where multigrid preconditioners are generated only at selected design cycles and re-used in subsequent cycles. Building upon knowledge regarding approximate reanalysis, it is shown that integrating recycled preconditioning into a minimum weight problem formulation can lead to a more efficient procedure than the common minimum compliance approach. Implemented in MATLAB, the run time is roughly twice faster than that of standard minimum compliance procedures. Computational savings are achieved without any compromise on the quality of the results in terms of the compliance-to-weight trade-off achieved. This provides a step towards integrating interactive 3-D topology optimization procedures into CAD software and mobile applications. MATLAB codes complementing the article can be downloaded from the author’s personal webpage.

Keywords

Topology optimization Recycled preconditioning Reanalysis of structures Combined approximations 

Notes

Acknowledgments

The author is grateful to the anonymous reviewers for many insightful comments and for numerous suggestions that helped improved the article. The author wishes to thank Niels Aage and Boyan S. Lazarov for fruitful discussions on related topics. Financial support received from the European Commission Research Executive Agency, grant agreement PCIG12-GA-2012-333647, is gratefully acknowledged.

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):1–6. doi: 10.1007/s00158-012-0827-z CrossRefGoogle Scholar
  3. Amir O, Sigmund O (2011) On reducing computational effort in topology optimization: how far can we go? Struct Multidiscip Optim 44:25–29CrossRefzbMATHGoogle Scholar
  4. Amir O, Bendsøe MP, Sigmund O (2009) Approximate reanalysis in topology optimization. Int J Numer Methods Eng 78:1474–1491CrossRefzbMATHGoogle Scholar
  5. Amir O, Stolpe M, Sigmund O (2010) Efficient use of iterative solvers in nested topology optimization. Struct Multidiscip Optim 42:55–72CrossRefzbMATHGoogle 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. Amir O, Aage N, Lazarov BS (2013) On multigrid-cg for efficient topology optimization. Struct Multidiscip Optim 1–15. doi: 10.1007/s00158-013-1015-5
  8. Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in matlab using 88 lines of code. Struct Multidiscip Optim 43:1–16. http://link.springer.com/article/10.1007 CrossRefzbMATHGoogle Scholar
  9. Arioli M (2004) A stopping criterion for the conjugate gradient algorithm in a finite element method framework. Numer Math 97(1):1–24CrossRefzbMATHMathSciNetGoogle Scholar
  10. Ashby SF, Falgout RD (1996) A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations. Nucl Sci Eng 124:145–159Google 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. Borrvall T, Petersson J (2001) Large-scale topology optimization in 3d using parallel computing. Comput Methods Appl Mech Eng 190(46):6201–6229CrossRefzbMATHMathSciNetGoogle Scholar
  15. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50:2143–2158CrossRefzbMATHMathSciNetGoogle Scholar
  16. Braess D (1986) On the combination of the multigrid method and conjugate gradients. In: Hackbusch W, Trottenberg U (eds) Multigrid methods II. Springer, Berlin, pp 52–64CrossRefGoogle Scholar
  17. Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190:3443–3459CrossRefzbMATHGoogle Scholar
  18. Chen Y, Davis TA, Hager WW, Rajamanickam S (2008) Algorithm 887: cholmod, supernodal sparse cholesky factorization and update/downdate. ACM Trans Math Softw (TOMS) 35(3):22. http://dl.acm.org/citation.cfm?id=1391995 CrossRefMathSciNetGoogle Scholar
  19. Davis TA (2006) Direct methods for sparse linear system. SIAMGoogle Scholar
  20. 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
  21. 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
  22. Fleury C, Braibant V (1986) Structural optimization-a new dual method using mixed variables. Int J Numer Methods Eng 23(3):409–428CrossRefzbMATHMathSciNetGoogle Scholar
  23. Kettler R (1982) Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods. In: Multigrid methods. Springer, pp 502–534Google Scholar
  24. Kim YY, Yoon GH (2000) Multi-resolution multi-scale topology optimization-a new paradigm. Int J Solids Struct 37(39):5529–5559CrossRefzbMATHMathSciNetGoogle Scholar
  25. Kim JE, Jang GW, 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
  26. Kim TS, Kim JE, Kim YY (2004) Parallelized structural topology optimization for eigenvalue problems. Int J Solids Struct 41:2623–2641CrossRefzbMATHGoogle Scholar
  27. Kirsch U (1991) Reduced basis approximations of structural displacements for optimal design. AIAA J 29:1751–1758CrossRefzbMATHGoogle Scholar
  28. Kirsch U (2002) Design-oriented analysis of structures. Kluwer Academic Publishers, DordrechtzbMATHGoogle Scholar
  29. Kirsch U (2008) Reanalysis of structures. Springer, DordrechtzbMATHGoogle Scholar
  30. Kirsch U, Kočvara M, Zowe J (2002) Accurate reanalysis of structures by a preconditioned conjugate gradient method. Int J Numer Methods Eng 55:233–251CrossRefzbMATHGoogle Scholar
  31. Mahdavi A, Balaji R, Freckerand M, Mockensturm EM (2006) Topology optimization of 2D continua for minimum compliance using parallel computing. Struct Multidiscip Optim 32:121–132CrossRefGoogle Scholar
  32. MATLAB (2013) MATLAB version 8.1.0.604 (R2013a). Natick, MassachusettsGoogle Scholar
  33. 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
  34. 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
  35. Notay Y (2007) Convergence analysis of perturbed two-grid and multigrid methods. SIAM J Numer Anal 45(3):1035–1044. http://epubs.siam.org/doi/abs/10.1137/060652312 CrossRefzbMATHMathSciNetGoogle Scholar
  36. Rozvany GIN, Zhou M (1992) COC methods for a single global constraint. In: Rozvany GIN (ed) Shape and layout optimization of structural systems and optimality criteria methods. Springer, BerlinGoogle Scholar
  37. Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAMGoogle Scholar
  38. Schmidt S, Schulz V (2011) A 2589 line topology optimization code written for the graphics card. Comput Vis Sci 14(6):249–256CrossRefMathSciNetGoogle Scholar
  39. Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Based Des Struct Mach 25:493–524CrossRefGoogle Scholar
  40. Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidiscip Optim 21:120–127CrossRefGoogle Scholar
  41. 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. http://link.springer.com/article/10.1007/s00158-012-0814-4 CrossRefzbMATHMathSciNetGoogle Scholar
  42. 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):1037–1067CrossRefMathSciNetGoogle Scholar
  43. Stainko R (2006) 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
  44. Suresh K (2013) Efficient generation of large-scale pareto-optimal topologies. Struct Multidiscip Optim 47(1):49–61CrossRefzbMATHMathSciNetGoogle Scholar
  45. Svanberg K (1987) The method of moving asymptotes-a new method for structural optimization. Int J Numer Methods Eng 24:359–373CrossRefzbMATHMathSciNetGoogle Scholar
  46. Tatebe O (1993) The multigrid preconditioned conjugate gradient method. In: Nasa Conference PublicationGoogle Scholar
  47. Tatebe O, Oyanagi Y (1994) Efficient implementation of the multigrid preconditioned conjugate gradient method on distributed memory machines. In: Supercomputing’94. Proceedings, IEEE, pp 194–203Google Scholar
  48. Trottenberg U, Oosterlee C, Schuller A (2001) Multigrid. Academic Press, New YorkzbMATHGoogle Scholar
  49. Vemaganti K, Lawrence EW (2005) Parallel methods for optimality criteria-based topology optimization. Comput Methods Appl Mech Eng 194:3637–3667CrossRefzbMATHMathSciNetGoogle Scholar
  50. Wadbro E, Berggren M (2009) Megapixel topology optimization on a graphics processing unit. SIAM Rev 51(4):707–721CrossRefzbMATHMathSciNetGoogle Scholar
  51. 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
  52. Zegard T, Paulino GH (2013) Toward GPU accelerated topology optimization on unstructured meshes. Struct Multidiscip Optim 48(3):473–485. doi: 10.1007/s00158-013-0920-y CrossRefGoogle Scholar
  53. Zhou M, Rozvany G (1991) The coc algorithm, part ii: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1):309–336CrossRefGoogle Scholar
  54. 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 2014

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental EngineeringTechnion - Israel Institute of TechnologyHaifaIsrael

Personalised recommendations