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A combined parametric shape optimization and ersatz material approach

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Abstract

This article describes a parametric shape optimization approach using vertical or horizontal structures with a fine parametrization of their center lines and profiles. In this context horizontal means a lateral connection from left to right and vertical means a bottom-up connection. These structures are projected to a pseudo density field associated with a fixed mesh using a differentiable mapping. This enables the use of existing topology optimization tools with respect to the solution of the state problem based on the pseudo density field. The approach belongs to a class of geometry projection onto a fictitious domain methods. It therefore shares the property that sensitivity analysis is reduced to extend the well known gradient calculation from topology optimization by chain using the sensitivity of the mapping from shape variables to pseudo density. The contribution lies in the combination with our specific shape parametrization and the associated regularization. Optimization problems can be formulated concurrently in terms of shape variables and pseudo density. We discuss regularization, periodicity constraints, symmetry formulations and overhang constraints in terms of shape variables. Volume and perimeter constraints are easily formulated in terms of the pseudo density. We see our approach as being particularly beneficial for certain problem classes where it may be difficult to restrict the design space, e.g. restricting isolated structures or holes or where a strict control of solid to void transition is necessary. Consequently, we show examples for phononic band gap maximization, boundary driven heat optimization and perimeter maximization for a flow problem. We also present a formulation of overhang constraints for additive manufacturing in terms of shape variables.

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Notes

  1. Solid Isotropic Material with Penalization.

  2. Local mesh refinement, as in Maute and Ramm (1995), does not appear to be widely applied.

  3. We note that the periodic design constraint is no replacement for periodic boundary conditions on the state problem.

  4. See Gill et al. (2002).

  5. Slope constraints, curvature constraints and possibly overhang constraints.

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Acknowledgements

The authors gratefully acknowledge the support of the Cluster of Excellence ‘Engineering of Advanced Materials’ at the University of Erlangen-Nuremberg, which is funded by the German Research Foundation (DFG) within the framework of its ‘Excellence Initiative’. The implementation of the heat and flow problems was done by Bich Ngoc Vu.

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  1. Friedrich-Alexander-Universität Erlangen-Nürnberg, Zentralinstitut Scientific Computing, Martenstrasse 5a, 91080, Erlangen, Germany

    Fabian Wein & Michael Stingl

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  1. Fabian Wein
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  2. Michael Stingl
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Correspondence to Fabian Wein.

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Wein, F., Stingl, M. A combined parametric shape optimization and ersatz material approach. Struct Multidisc Optim 57, 1297–1315 (2018). https://doi.org/10.1007/s00158-017-1812-3

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  • DOI: https://doi.org/10.1007/s00158-017-1812-3

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