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Virtual element method (VEM)-based topology optimization: an integrated framework

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Abstract

We present a virtual element method (VEM)-based topology optimization framework using polyhedral elements, which allows for convenient handling of non-Cartesian design domains in three dimensions. We take full advantage of the VEM properties by creating a unified approach in which the VEM is employed in both the structural and the optimization phases. In the structural problem, the VEM is adopted to solve the three-dimensional elasticity equation. Compared to the finite element method, the VEM does not require numerical integration (when linear elements are used) and is less sensitive to degenerated elements (e.g., ones with skinny faces or small edges). In the optimization problem, we introduce a continuous approximation of material densities using the VEM basis functions. When compared to the standard element-wise constant approximation, the continuous approximation enriches the geometrical representation of structural topologies. Through two numerical examples with exact solutions, we verify the convergence and accuracy of both the VEM approximations of the displacement and material density fields. We also present several design examples involving non-Cartesian domains, demonstrating the main features of the proposed VEM-based topology optimization framework. The source code for a MATLAB implementation of the proposed work, named PolyTop3D, is available in the (electronic) Supplementary Material accompanying this publication.

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Change history

  • 09 July 2020

    The original version of this article unfortunately contains several errors introduced by the typesetter during the publishing process and which have been corrected.

Notes

  1. Although the CVT meshes are generated from reflected seeds to ensure their symmetry, certain regions on the mesh boundaries, especially near the two circles, are not fully symmetric due to the limitation of the meshing software used by Thedin et al. (2014) in representing curved boundaries. Hence, results obtained from the proposed framework (i.e. Figs 18 and 19a) are slightly asymmetric in those regions. However, we note that this minor issue will not affect the overall quality of the designs (and can be resolved when an improved version of meshing software is used).

  2. Such mixed elements deserve further theoretical investigation to address their stability and the balance between the discrete spaces for the displacement field and the density field (see, for example, Jog and Haber 1996; Chi et al. 2016).

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Acknowledgement

This paper is dedicated to the memory of Prof. Luiz Eloy Vaz (January 24, 1947-April 15, 2014).

Funding

HC and GHP received financial support from the US National Science Foundation (NSF) under project no. 1663244. We also received an endowment provided by the Raymond Allen Jones Chair at the Georgia Institute of Technology. AP received financial support from the Carlos Chagas Filho Research Foundation of Rio de Janeiro State (FAPERJ) under grant E-26/203.189/2016. AP and IFMM received support from Tecgraf/PUC-Rio (Group of Technology in Computer Graphics), Rio de Janeiro, Brazil, and from National Council for Scientific and Technological Development (CNPq).

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Authors and Affiliations

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

    Heng Chi & Glaucio H. Paulino

  2. Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rua Marquês de São Vicente, 225, Rio de Janeiro, R.J., 22451-900, Brazil

    Anderson Pereira & Ivan F. M. Menezes

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  1. Heng Chi
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  2. Anderson Pereira
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  3. Ivan F. M. Menezes
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  4. Glaucio H. Paulino
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Correspondence to Glaucio H. Paulino.

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Appendix: PolyTop3D: an efficient MATLAB implementation of the proposed VEM-based topology optimization framework

Appendix: PolyTop3D: an efficient MATLAB implementation of the proposed VEM-based topology optimization framework

An implementation of the proposed VEM-based topology optimization framework into a modular MATLAB code named PolyTop3D, which can handle any non-Cartesian design domains (specified by the users) on general polyhedral discretizations (both structured and unstructured), is available in the Electronic Supplementary Material accompanying this publication. The PolyTop3D is modularized in a similar manner to the PolyTop code, presented in Talischi et al. (2012b), together with a similar naming convention for its variables. Thus, we refer the readers to Talischi et al. (2012b) for a thorough introduction of the structure of the code. We hope that the modularity and flexibility offered by PolyTop3D will motivate the community to explore the proposed VEM-based framework in other topology optimization problems.

In the sequel, we demonstrate the efficiency of the PolyTop3D code by benchmarking it with the Top3D code by Liu and Tovar (2014). For purpose of comparison, the cantilever example, presented in Table 4 of Liu and Tovar (2014), is solved on a set of three regular hexahedral meshes whose statistics are shown in Table 5. Each element in those meshes is a unit cube. Throughout this study, the filter radius is set as R = 1.5 and the volume constraint is taken to be \(\overline {V}=15\%\). For both computer codes, a constant penalty parameter of p = 3 is used and 200 optimization iterations are carried out on a desktop computer with an Intel(R) Xeon(R), 3.00 GHz processor and 256 GB of RAM running MATLAB R2016a. For all the meshes, the two codes produce almost identical final topologies and thus are not shown here for the sake of conciseness.

Table 5 Statistics of three meshes for the cantilever problem
Full size table

In Table 6, we present a comparison of the total runtimes of PolyTop3D and Top3D for the three meshes. In addition, Table 7 shows the breakdown of the total runtime of the PolyTop3D code into major steps. One immediate conclusion from Tables 6 and 7 is that the PolyTop3D code is able to achieve similar efficiency to the Top3D code using more than four times the number of DVs. The major runtime difference of the two codes comes from the steps of forming projection matrices, \(\mathbf {P}^{\mathcal {F}}\) and \(\mathbf {P}^{\mathcal {V}}\) (c.f. (56) and (60)), and VEM shape functions φi.

Table 6 Total runtime comparison of PolyTop3D with the Top3D code
Full size table
Table 7 Breakdown of the PolyTop3D run time from 200 optimization iterations
Full size table

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Chi, H., Pereira, A., Menezes, I.F.M. et al. Virtual element method (VEM)-based topology optimization: an integrated framework. Struct Multidisc Optim 62, 1089–1114 (2020). https://doi.org/10.1007/s00158-019-02268-w

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