Skip to main content
SpringerLink
Log in

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

  • RESEARCH PAPER
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  Google Scholar 

  • Amir O, Sigmund O (2011) On reducing computational effort in topology optimization: how far can we go? Struct Multidiscip Optim 44:25–29

    Article  MATH  Google Scholar 

  • Amir O, Bendsøe MP, Sigmund O (2009) Approximate reanalysis in topology optimization. Int J Numer Methods Eng 78:1474–1491

    Article  MATH  Google Scholar 

  • Amir O, Stolpe M, Sigmund O (2010) Efficient use of iterative solvers in nested topology optimization. Struct Multidiscip Optim 42:55–72

    Article  MATH  Google Scholar 

  • Amir O, Sigmund O, Schevenels M, Lazarov B (2012) Efficient reanalysis techniques for robust topology optimization. Comput Methods Appl Mech Eng 245–246:217–231

    Article  MathSciNet  Google Scholar 

  • 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

  • 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

    Article  MATH  Google Scholar 

  • Arioli M (2004) A stopping criterion for the conjugate gradient algorithm in a finite element method framework. Numer Math 97(1):1–24

    Article  MATH  MathSciNet  Google Scholar 

  • Ashby SF, Falgout RD (1996) A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations. Nucl Sci Eng 124:145–159

    Google Scholar 

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202

    Article  Google Scholar 

  • Bendsøe MP, Sigmund O (2003) Topology optimization-theory, methods and applications. Springer, Berlin

    Google Scholar 

  • 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

    MATH  Google Scholar 

  • Borrvall T, Petersson J (2001) Large-scale topology optimization in 3d using parallel computing. Comput Methods Appl Mech Eng 190(46):6201–6229

    Article  MATH  MathSciNet  Google Scholar 

  • Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50:2143–2158

    Article  MATH  MathSciNet  Google Scholar 

  • 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 52–64

    Chapter  Google Scholar 

  • Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190:3443–3459

    Article  MATH  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Davis TA (2006) Direct methods for sparse linear system. SIAM

  • Evgrafov A, Rupp CJ, Maute K, Dunn ML (2008) Large-scale parallel topology optimization using a dual-primal substructuring solver. Struct Multidiscip Optim 36:329–345

    Article  MATH  MathSciNet  Google Scholar 

  • 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):1523–1544

    Article  MATH  MathSciNet  Google Scholar 

  • Fleury C, Braibant V (1986) Structural optimization-a new dual method using mixed variables. Int J Numer Methods Eng 23(3):409–428

    Article  MATH  MathSciNet  Google Scholar 

  • Kettler R (1982) Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods. In: Multigrid methods. Springer, pp 502–534

  • Kim YY, Yoon GH (2000) Multi-resolution multi-scale topology optimization-a new paradigm. Int J Solids Struct 37(39):5529–5559

    Article  MATH  MathSciNet  Google Scholar 

  • 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):6473–6496

    Article  MATH  Google Scholar 

  • Kim TS, Kim JE, Kim YY (2004) Parallelized structural topology optimization for eigenvalue problems. Int J Solids Struct 41:2623–2641

    Article  MATH  Google Scholar 

  • Kirsch U (1991) Reduced basis approximations of structural displacements for optimal design. AIAA J 29:1751–1758

    Article  MATH  Google Scholar 

  • Kirsch U (2002) Design-oriented analysis of structures. Kluwer Academic Publishers, Dordrecht

    MATH  Google Scholar 

  • Kirsch U (2008) Reanalysis of structures. Springer, Dordrecht

    MATH  Google Scholar 

  • Kirsch U, Kočvara M, Zowe J (2002) Accurate reanalysis of structures by a preconditioned conjugate gradient method. Int J Numer Methods Eng 55:233–251

    Article  MATH  Google Scholar 

  • Liu K, Tovar A (2013) http://et.engr.iupui.edu/tovara/top3d/

  • Mahdavi A, Balaji R, Freckerand M, Mockensturm EM (2006) Topology optimization of 2D continua for minimum compliance using parallel computing. Struct Multidiscip Optim 32:121–132

    Article  Google Scholar 

  • MATLAB (2013) MATLAB version 8.1.0.604 (R2013a). Natick, Massachusetts

    Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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, Berlin

  • Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAM

  • Schmidt S, Schulz V (2011) A 2589 line topology optimization code written for the graphics card. Comput Vis Sci 14(6):249–256

    Article  MathSciNet  Google Scholar 

  • Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Based Des Struct Mach 25:493–524

    Article  Google Scholar 

  • Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidiscip Optim 21:120–127

    Article  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Suresh K (2013) Efficient generation of large-scale pareto-optimal topologies. Struct Multidiscip Optim 47(1):49–61

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Tatebe O (1993) The multigrid preconditioned conjugate gradient method. In: Nasa Conference Publication

  • Tatebe O, Oyanagi Y (1994) Efficient implementation of the multigrid preconditioned conjugate gradient method on distributed memory machines. In: Supercomputing’94. Proceedings, IEEE, pp 194–203

  • Trottenberg U, Oosterlee C, Schuller A (2001) Multigrid. Academic Press, New York

    MATH  Google Scholar 

  • Vemaganti K, Lawrence EW (2005) Parallel methods for optimality criteria-based topology optimization. Comput Methods Appl Mech Eng 194:3637–3667

    Article  MATH  MathSciNet  Google Scholar 

  • Wadbro E, Berggren M (2009) Megapixel topology optimization on a graphics processing unit. SIAM Rev 51(4):707–721

    Article  MATH  MathSciNet  Google Scholar 

  • Wang S, de Sturler E, Paulino GH (2007) Large-scale topology optimization using preconditioned Krylov subspace methods with recycling. Int J Numer Methods Eng 69:2441–2468

    Article  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Zhou M, Rozvany G (1991) The coc algorithm, part ii: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1):309–336

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Faculty of Civil and Environmental Engineering, Technion - Israel Institute of Technology, Haifa, Israel

    Oded Amir

Authors
  1. Oded Amir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oded Amir.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amir, O. Revisiting approximate reanalysis in topology optimization: on the advantages of recycled preconditioning in a minimum weight procedure. Struct Multidisc Optim 51, 41–57 (2015). https://doi.org/10.1007/s00158-014-1098-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-014-1098-7

Keywords

Search

Navigation