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On design-set restriction in SAND topology optimization

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Abstract

We study design-set restriction—i.e., the issue of nonexistence of solutions, mesh-dependence, and local minima—in the alternative SAND (‘simultaneous analysis and design’, or ‘direct’) setting of the classical SIMP minimum compliance topology optimization problem. Compared to some predominant filtering techniques, which are typically studied in the conventional NAND setting, so-called ‘slope constraints’—point-wise bounds on the gradient of the material distribution function—necessitate only a reasonable amount of computational resources in the alternative SAND setting. The numerical method is based on a standard finite element procedure and sequential approximate optimization (SAO) method. Numerical experiments with a random multistart strategy and a simple relaxation procedure suggest that ‘probably’ globally optimal 0-1 designs may be obtained in a reasonable amount of computation time.

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Notes

  1. The imposition of a minimum length scale.

  2. Nonconvexity; many local minima.

  3. Density filtering methods have also been based on image processing techniques (Sigmund 2007).

  4. To set up the matrix which describes the linear mapping may in fact be nontrivial, computationally speaking, if the FE mesh is not structured in some way.

  5. In 3D, with e z = 100, for example, this number is 3 090 903.

  6. The freely available Matlab codes (Sigmund 2001; Andreassen et al. 2011) may be used to generate these numbers.

  7. In 3D this number is 6.

  8. In 3D, this is \({e_{r}^{3}}-1\).

  9. In 3D, with e z = 100, this number is 2 299 631 832.

  10. In 3D, for an e x × e y × e z = 100 mesh, about 12 × e y × e x × e z additional constraints are required.

  11. In 3D, for e z = 100, about 24 000 000 additional nonzero terms are required—in the equivalent density-filtered case, this number is 2 299 631 832.

  12. In practice so-called ‘convergent’ algorithms are only guaranteed to terminate, perhaps at an infeasible point, particularly if the restoration procedure resorted to fails to produce a feasible iterate. Herein we employ a simple backtracking strategy as a rudimentary restoration procedure. Throughout the feasibility of some point is measured as the infinity norm of all the constraint violations, including the equilibrium constraints, as per the SAND problem (14).

  13. Results in Petersson and Sigmund (1998) suggest that the slope constraints have to be impractically restrictive in order to suppress checkerboards if Q4’s are used.

  14. Here we refer to ‘subproblems’ rather than ‘iterations’, because the convergent trust-region algorithm (Fletcher et al. 2002) involves ‘inner iterations’.

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Acknowledgements

The authors wish to thank Stellenbosch University’s Department of Mechanical and Mechatronic Engineering for financial support, and Charl Möller, administrator of the Rhasatsha HPC, for technical assistance and support. Gratitude is also extended to the reviewers of the manuscript, whose willingness to comment, criticize, and suggest improvement, contributed to a manuscript of increased quality.

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  1. Department of Mechanical and Mechatronic Engineering, University of Stellenbosch, Stellenbosch, South Africa

    Dirk Munro & Albert A. Groenwold

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  1. Dirk Munro
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  2. Albert A. Groenwold
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Correspondence to Dirk Munro.

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Munro, D., Groenwold, A.A. On design-set restriction in SAND topology optimization. Struct Multidisc Optim 57, 1579–1592 (2018). https://doi.org/10.1007/s00158-017-1827-9

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