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PolyMat: an efficient Matlab code for multi-material topology optimization

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Abstract

We present a Matlab implementation of topology optimization for compliance minimization on unstructured polygonal finite element meshes that efficiently accommodates many materials and many volume constraints. Leveraging the modular structure of the educational code, PolyTop, we extend it to the multi-material version, PolyMat, with only a few modifications. First, a design variable for each candidate material is defined in each finite element. Next, we couple a Discrete Material Optimization interpolation with the existing penalization and introduce a new parameter such that we can employ continuation and smoothly transition from a convex problem without any penalization to a non-convex problem in which material mixing and intermediate densities are penalized. Mixing that remains due to the density filter operation is eliminated via continuation on the filter radius. To accommodate flexibility in the volume constraint definition, the constraint function is modified to compute multiple volume constraints and the design variable update is modified in accordance with the Zhang-Paulino-Ramos Jr. (ZPR) update scheme, which updates the design variables associated with each constraint independently. The formulation allows for volume constraints controlling any subset of the design variables, i.e., they can be defined globally or locally for any subset of the candidate materials. Borrowing ideas for mesh generation on complex domains from PolyMesher, we determine which design variables are associated with each local constraint of arbitrary geometry. A number of examples are presented to demonstrate the many material capability, the flexibility of the volume constraint definition, the ease with which we can accommodate passive regions, and how we may use local constraints to break symmetries or achieve graded geometries.

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Acknowledgments

GHP and EDS acknowledge the financial support from the US National Science Foundation (NSF) under project #1663244, and the endowment provided by the Raymond Allen Jones Chair at the Georgia Institute of Technology. AP appreciates the financial support from the Carlos Chagas Filho Research Foundation of Rio de Janeiro State (FAPERJ) under grant E-26/203.189/2016, and from Tecgraf/PUC-Rio (Group of Technology in Computer Graphics), Rio de Janeiro, Brazil. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. The information provided in this paper is the sole opinion of the authors and does not necessarily reflect the views of the sponsoring agencies.

Author information

Authors and Affiliations

  1. School of Civil and Environmental Engineering, Georgia Institute of Technology, 790 Atlantic Drive NW, Atlanta, GA, 30332, USA

    Emily D. Sanders & Glaucio H. Paulino

  2. Department of Mechanical Engineering, Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rua Marquês de São Vicente, 225, Rio de Janeiro, RJ, 22451-900, Brazil

    Anderson Pereira

  3. Simulation and Modeling Sciences, Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM, 87185, USA

    Miguel A. Aguiló

Authors
  1. Emily D. Sanders
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  2. Anderson Pereira
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  3. Miguel A. Aguiló
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  4. Glaucio H. Paulino
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Corresponding author

Correspondence to Glaucio H. Paulino.

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Responsible Editor: Xu Guo

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Appendices

Appendix A: Convergence plots for selected problems

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Convergence plots: a 15-material Michell cantilever with continuation on the material interpolation parameters and with filter reduction in the final continuation step and b 1-material serpentine without continuation on the material interpolation parameters and with filter reduction after convergence

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Appendix B: Summary of filtering techniques

The three filtering techniques discussed in Section 5.1 are summarized in Table 8.

Table 8 Summary of filtering techniques
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Appendix C: Library of example problems

The Matlab files needed to run each of the problems discussed herein are provided with download of PolyMat. Table 9 summarizes the problems and indicates which files need to be called from PolyScript to run each.

Table 9 Examples provided with PolyMat
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Appendix D: PolyScript

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Appendix E: PolyMat

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Appendix F: InitialGuess

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Appendix G: MultiMatIntFnc

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Appendix H: ConvertColors

The ConvertColors function is used to enable a result visualization in which white represents void. To plot the multi-material density in a given element, we sum the RGB Colors associated with each material in the element, scaled by the corresponding material densities, to find a point on the unit RGB color cube representing the mixing in that element. In general, the unit RGB color cube is defined such that black represents void (0, 0, 0) and white represents full mixing (1, 1, 1). Thus, ConvertColors is called before the optimization loop in PolyMat to compute a rigid body motion matrix, RBM, that can be applied to the RGB color cube so that white is at the origin. Using RBM, the RGB Colors specified by the user are converted to the corresponding colors on the rotated unit RGB color cube (e.g., red (1, 0, 0) goes to cyan (0, 1, 1), green (0, 1, 0) goes to yellow (1, 1, 0), and blue (0, 0, 1) goes to magenta (1, 0, 1)). These converted colors are scaled by the density of each material and summed in line 30 of PolyMat to find a point on the rotated unit RGB color cube representing the mixing in a given element, where void elements have RGB value (0, 0, 0). When rotated back to the original unit RGB color cube, we re-gain the specified color scheme and the void elements are now at (1, 1, 1), which represents white. The code used to compute the RBM matrix and convert the user-specified colors is provided in this Appendix.

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Sanders, E.D., Pereira, A., Aguiló, M.A. et al. PolyMat: an efficient Matlab code for multi-material topology optimization. Struct Multidisc Optim 58, 2727–2759 (2018). https://doi.org/10.1007/s00158-018-2094-0

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