On multigrid-CG for efficient topology optimization
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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.
KeywordsTopology optimization Preconditioned conjugate gradients Multigrid
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.
- Ashby SF, Falgout RD (1996) A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations. Nucl Sci Eng 124:145–159Google Scholar
- 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
- 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
- Bendsøe MP, Sigmund O (2003) Topology optimization—theory, methods and applications. Springer, BerlinGoogle Scholar
- Lazarov BS (2013) Topology optimization using multiscale finite element method for high-contrast media. In reviewGoogle Scholar
- Stainko R (2006b) Advanced multilevel techniques to topology optimization. PhD thesis, Johannes Kepler Universitȧt LinzGoogle Scholar
- 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
- Vassilevski PS (2008) Multilevel block factorization preconditioners: matrix-based analysis and algorithms for solving finite element equations. Springer, New YorkGoogle Scholar