Efficient hybrid topology and shape optimization combining implicit and explicit design representations
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RESEARCH PAPER
Efficient hybrid topology and shape optimization combining implicit and explicit design representations Tuan T. Nguyen1
· J. Andreas Bærentzen1 · Ole Sigmund2 · Niels Aage2
Received: 9 March 2020 / Revised: 29 May 2020 / Accepted: 10 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper presents an interactive hybrid topology optimization method that (1) employs density for topology optimization and (2) in a seamless fashion uses a Deformable Simplicial Complex for shape optimization. Omitting hole insertions during the shape optimization allows us to utilize adaptive mesh coarsening, which reduces the mesh size with up to seven times. The result is a combined method which can reduce computation time up to ten times in comparison with pure Lagrangian methods, while still producing adaptive meshes of good quality for analysis and design. Given the robustness of the method, we are able to perform topology optimization by explicit meshing and shape optimization on a mobile device at frame rates that allow for real-time user interaction. The resulting “TopOpt Shape” app is available in the App Store for iOS devices. Keywords Deformable simplicial complex · Density · TopOpt shape
1 Introduction and related works An important aspect of structural optimization is how the material boundaries are represented. In general, two categories exist: implicit and explicit. Implicit representations make topology changes trivial, therefore they are largely adopted in topology optimization. Typical methods in this category are density methods, e.g., Bendsoe (1989) and Sigmund (2003), and certain fixed grid level set methods, e.g., Osher and Fedkiw (2006), Wang et al. (2003), and Allaire et al. (2004). On the other hand, explicit representations require complicated meshing schemes to deal with topology changes. This problem is closely related to the interface tracking topic in computer graphics, where deformable meshes that support split and merge operations have been studied intensively (Misztal and Bærentzen 2012; Wojtan et al. 2009; Da et al. 2014), since such methods Responsible Editor: Juli´an Andr´es Norato Tuan T. Nguyen
[email protected] 1
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark
2
Department of Mechanical Engineering, Technical University of Denmark, Lyngby, Denmark
allow the majority of the mesh to remain unchanged as the interface moves. Despite the difficulty in handling topology changes, explicit representations are still desired for their well defined interfaces and the ability to perform accurate finite element analysis. Early approaches (Ha and Cho 2008; Allaire et al. 2011; Yamasaki et al. 2011; Xia et al. 2012) utilize an underlying level set function to resolve topology change, and a new explicit interface is generated in each iteration. Other approaches try to combine shape and topology optimization by iteratively introducing holes and then optimizing the shape without mesh change (Eschenauer et al
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