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A FreeFEM code for topological derivative-based structural optimization

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Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

This article presents an educational code written in FreeFEM, based on the concept of topological derivative together with a level-set domain representation method and adaptive mesh refinement processes, to perform compliance minimization in structural optimization. The code is implemented in the framework of linearized elasticity, for both plane strain and plane stress assumptions. As a first-order topology optimization algorithm, the topological derivative is in fact used within the numerical procedure as a steepest descent direction, similar to methods based on the gradient of cost functionals. In addition, adaptive mesh refinement processes are used as part of the optimization scheme for enhancing the resolution of the final topology. Since the paper is intended for educational purposes, we start by explaining how to compute topological derivatives, followed by a step-by-step description of the code, which makes the binding of the theoretical aspects of the algorithm to its implementation. Numerical results associated with three classic examples in topology optimization are presented and discussed, showing the effectiveness and robustness of the proposed approach.

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Acknowledgements

The authors would like to express deep gratitude to Professor Antonio André Novotny (LNCC) for his vision and technical guidance in this paper.

Funding

This research was partly supported by CNPq (Brazilian Research Council), CAPES (Brazilian Higher Education Staff Training Agency), and FAPESP (The São Paulo Research Foundation), grant 2021/049479-8. These financial supports are gratefully acknowledged.

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    Authors and Affiliations

    1. Coordenação de Matemática Aplicada e Computacional, Laboratório Nacional de Computação Científica LNCC/MCTI, Avenida Getúlio Vargas, 333, Petrópolis, Rio de Janeiro, 25651-075, Brazil

      Jorge Morvan Marotte Luz Filho & Raquel Mattoso

    2. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, São Paulo, 05508-090, Brazil

      Lucas Fernandez

    Authors
    1. Jorge Morvan Marotte Luz Filho
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    2. Raquel Mattoso
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    3. Lucas Fernandez
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    Contributions

    Jorge Morvan Marotte Luz Filho, Raquel Mattoso, and Lucas dos Santos Fernandez have contributed equally to this work.

    Corresponding author

    Correspondence to Lucas Fernandez.

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    Replication of results

    The source code is given in the Appendix. All the presented results are obtained by direct running of the code.

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    Responsible Editor: Xiaojia Shelly Zhang

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    Appendix: Source code

    Appendix: Source code

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    Luz Filho, J.M.M., Mattoso, R. & Fernandez, L. A FreeFEM code for topological derivative-based structural optimization. Struct Multidisc Optim 66, 74 (2023). https://doi.org/10.1007/s00158-023-03529-5

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    • DOI: https://doi.org/10.1007/s00158-023-03529-5

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