Virtual element method (VEM)-based topology optimization: an integrated framework
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RESEARCH PAPER
Virtual element method (VEM)-based topology optimization: an integrated framework Heng Chi1 · Anderson Pereira2 · Ivan F. M. Menezes2 · Glaucio H. Paulino1 Received: 27 July 2018 / Revised: 29 January 2019 / Accepted: 21 March 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We present a virtual element method (VEM)-based topology optimization framework using polyhedral elements, which allows for easy 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 of the framework. In the structural problem, the VEM is adopted to solve the three-dimensional elasticity equation. Compared to the finite element method (FEM), the VEM does not require numerical integration 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 VEM basis functions. As compared to the standard element-wise constant one, the continuous approximation enriches geometrical representations 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, will be made available in the (electronic) Supplementary Material accompanying this publication. Keywords Topology optimization · Polyhedral meshes · Virtual element method · MATLAB software
Responsible Editor: Byeng D Youn Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00158-019-02268-w) contains supplementary material, which is available to authorized users. Glaucio H. Paulino
[email protected] Heng Chi [email protected] Anderson Pereira [email protected] Ivan F. M. Menezes [email protected] 1
School of Civil and Environmental Engineering, Georgia Institute of Technology, 790 Atlantic Drive, Atlanta, GA, 30332, USA
2
Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rua Marquˆes de S˜ao Vicente, 225, Rio de Janeiro, R.J. 22451-900, Brazil
1 Introduction Topology optimization is a powerful computational tool to design optimal structures under given loads and boundary conditions. Since the seminal work of Bendsøe and Kikuchi (1988), the field of topology optimization has experienced tremendous growth and had a major impact on several areas of engineering and technology. For an overview of this field, we refer the interested readers to textbooks (Christensen and Klarbring 2009; Haftka and G¨urdal 2012; Bendsoe and Sigmund 2013) and review paper (Rozvany 2009). Among various topology optimization approaches, the density-based approac
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