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Topology optimization guided by a geometrical pattern library

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Abstract

This work proposes an approach for structural Topology Optimization enforcing geometrical features on optimized designs using a predefined library of geometrical patterns. The approach applies a density-based Topology Optimization subject to a geometrical constraint guiding the design toward shapes matching the geometrical features found in the predefined pattern library. Multiple distance measures and suitable matching algorithms are studied to calculate local mappings between the design in each optimization iteration and the pattern library. An aggregated appearance constraint is evaluating the pattern matching. The gradient of the appearance constraint, objective function, and other constraints are applied in a gradient-based optimization scheme. A parameter for the appearance constraint dictates how closely the design should match the patterns defined in the library. The convergence behavior is studied on a variety of 2D and 3D optimization scenarios. The formulation is also applied to design variables controlling the material orientations alongside the material density as well as other optimization objectives such as stress minimization.

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Acknowledgements

Ole Sigmund would like to acknowledge the support of the Villum Foundation, Denmark through the Villum Investigator Project InnoTop. All authors acknowledge the reviewers for their comments helping to improve the manuscript.

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Author notes

  1. Tanguy Navez, Martin-Pierre Schmidt have contributed equally to this work.

Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark

    Tanguy Navez

  2. Laboratory of Mathematics, National Institute of Applied Sciences, Rouen, France

    Martin-Pierre Schmidt

  3. Department of Mechanical Engineering, Technical University of Denmark, Lyngby, Denmark

    Tanguy Navez & Ole Sigmund

  4. 22587, Hamburg, Germany

    Claus B. W. Pedersen

Authors
  1. Tanguy Navez
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  2. Martin-Pierre Schmidt
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  3. Ole Sigmund
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  4. Claus B. W. Pedersen
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Corresponding author

Correspondence to Tanguy Navez.

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On behalf of all authors, the corresponding author states that there is no conflict of interest.

Replication of results

Unless explicitly stated in the text, all results presented in this work apply the same default parameters: \(p= 3\), \(Y_{min}= 10^{-6}\), \(Y_0= 1\). We use a density filter of radius 1.5 relative to the element size in regular 2D or 3D grid meshes consisting of squares or cubes, respectively. The patch dimensions are \(l_{x}= 8\), \(l_{y}= 8,\) and \(l_{z}= 8\) in 3D cases and \(l_{x}= 0\), \(l_{y}= 8,\) and \(l_{z}= 8\) in 2D cases. All results are obtained in our C++ Topology Optimization prototyping framework with a dependency on the open-source Eigen library for solving linear systems of equations, although any other similar linear algebra library would be equally suitable.

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Navez, T., Schmidt, MP., Sigmund, O. et al. Topology optimization guided by a geometrical pattern library. Struct Multidisc Optim 65, 108 (2022). https://doi.org/10.1007/s00158-022-03197-x

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  • DOI: https://doi.org/10.1007/s00158-022-03197-x

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