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PolyDyna: a Matlab implementation for topology optimization of structures subjected to dynamic loads

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Abstract

We present a Matlab implementation for topology optimization of structures subjected to dynamic loads. The code, which we name PolyDyna, is built on top of PolyTop—a Matlab code for static compliance minimization based on polygonal finite elements. To solve the structural dynamics problem, we use the HHT-α method, which is a generalization of the classical Newmark-β method. In order to handle multiple regional volume constraints efficiently, PolyDyna uses a variation of the ZPR design variable update scheme enhanced by a sensitivity separation technique, which enables it to solve non-self-adjoint topology optimization problems. We conduct the sensitivity analysis using the adjoint method with the “discretize-then-differentiate” approach, such that the sensitivity analysis is consistently evaluated on the discretized system (both in space and time). We present several numerical examples, which are explained in detail and summarized in a library of benchmark problems. PolyDyna is intended for educational purposes and the complete Matlab code is provided as electronic supplementary material.

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Notes

  1. This work uses a variation of this constraint, but imposed on sub-regions of the design domain. Details are provided in the next section.

  2. To be consistent with previous work, here we adopt the notation by Talischi et al. (2012b).

  3. The characteristic function, \(\chi _{{\varOmega }_{\ell }}(\mathbf {x})\), associated with element Ωℓ, is equal to 1 if x ∈Ωℓ and 0 otherwise (Talischi et al. 2012b).

  4. Details related to the filter operation when considering symmetry are given by Giraldo-Londoño and Paulino (2020b).

  5. The form of (18) is valid when \(\bar {\textbf {u}}(\textbf {x},t)=\textbf {0}\) (see Fig. 1). When ground accelerations are considered, fi should be replaced by − ag(ti)1, in which ag(ti) is the ground acceleration at t = ti.

  6. Although we use the threshold projection function to solve the examples in the present manuscript, PolyDyna has other volume interpolation functions built in, e.g., the Heaviside projection function by Guest (2009), as implemented in PolyTop (Talischi et al. 2012b).

  7. For dynamic topology optimization problems, we recommend the RAMP function because SIMP may lead to instabilities when the element densities approach zero.

  8. Without loss of generality, here we have dropped the explicit dependence of f on \(\mathbf {u}_{0},\ldots ,\mathbf {u}_{N_{t}}\).

  9. Although this example assumes the load duration to be equal to the time integration duration used to evaluate the objective function, they need not be equal. For example, the load duration could be set as t = ts and the load defined as \(f(t)=f_{0}\sin \limits (\pi t/t_{s})(1-H(t-t_{s}))\), in which H(⋅) is the unit step function, while the duration of the dynamic event, tf, could be set as tf = 2ts and use the displacement field up to t = tf to evaluate the objective function.

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Acknowledgements

We acknowledge Sandia National Laboratories, 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. We also thank the support provided by the National Science Foundation (NSF) under grant number 1663244. We are grateful to the insightful comments by Emily D. Sanders and Americo Cunha, which contributed to substantial improvements to the paper. The interpretation of the results of this work is solely that by the authors, and it does not necessarily reflect the views of the sponsors or sponsoring agencies.

Funding

The authors received support provided by the National Science Foundation (NSF) under grant number 1663244.

Author information

Authors and Affiliations

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

    Oliver Giraldo-Londoño & Glaucio H. Paulino

  2. Department of Civil and Environmental Engineering, University of Missouri, Columbia, MO, 65211, USA

    Oliver Giraldo-Londoño

Authors
  1. Oliver Giraldo-Londoño
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  2. Glaucio H. Paulino
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Correspondence to Glaucio H. Paulino.

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Responsible Editor: Gengdong Cheng

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All results presented in this manuscript can be replicated using the codes provided as electronic supplementary material.

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Appendices

Appendix A: Library of examples

Table 10 summarizes of all examples discussed in this manuscript. The first column depicts the domain geometries for all examples, while the second column shows the name of the domain files needed by PolyMesher to generate the finite element meshes. The last column provides a description for each of the problems, including dimensions of the domains, magnitude of the dynamic loads, material properties, and filter radius. These problems can be verified with the electronic supplementary material (ESM) provided with this paper. Moreover, the user can easily modify these examples or create new ones using PolyDyna.

Table 10 Examples provided with PolyDyna
Full size table

Appendix B: PolyScript

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Appendix C: PolyDyna

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figure m

Appendix D: FEM_Dyna

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Appendix E: AdjointProblem

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Giraldo-Londoño, O., Paulino, G.H. PolyDyna: a Matlab implementation for topology optimization of structures subjected to dynamic loads. Struct Multidisc Optim 64, 957–990 (2021). https://doi.org/10.1007/s00158-021-02859-6

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