Adaptive thermodynamic topology optimization
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RESEARCH PAPER
Adaptive thermodynamic topology optimization Andreas Vogel1
· Philipp Junker2
Received: 12 February 2020 / Revised: 6 May 2020 / Accepted: 24 June 2020 © The Author(s) 2020
Abstract The benefit of adaptive meshing strategies for a recently introduced thermodynamic topology optimization is presented. Employing an elementwise gradient penalization, stability is obtained and checkerboarding prevented while very fine structures can be resolved sharply using adaptive meshing at material-void interfaces. The usage of coarse elements and thereby smaller design space does not restrict the obtainable structures if a proper adaptive remeshing is considered during the optimization. Qualitatively equal structures and quantitatively the same stiffness as for uniform meshing are obtained with less degrees of freedom, memory requirement and overall optimization runtime. In addition, the adaptivity can be used to zoom into coarse global structures to better resolve details of interesting spots such as truss nodes. Keywords Thermodynamic topology optimization · Adaptivity · Geometric multigrid
1 Introduction Topology optimization has been introduced several decades ago and it has been established as a powerful tool during engineering design processes. Review papers are provided by Rozvany (2009), van Dijk et al. (2013), and Huang and Xie (2010). Different target functions can be defined, among which the optimization of the mechanical stiffness of a system is probably the most prominent one (Sigmund and Maute 2013). The goal of this optimization problem is to find the spatial description of the topology, i.e., a distinct void/full material distribution. This can be expressed in terms of the so-called material density χ = χ(x) = {χmin , 1}, where the spatial coordinate is given by x. Intermediate configurations, i.e., χ ∈]χmin , 1[, are difficult to be interpreted (foam) and even more challenging to be manifactured. Consequently, these “gray” solutions are to be avoided. A simple yet powerful strategy is the
Responsible Editor: Jianbin Du Andreas Vogel
[email protected] 1
High Performance Computing in the Engineering Sciences, Ruhr University Bochum, Bochum, Germany
2
Continuum Mechanics, Ruhr University Bochum, Bochum, Germany
multiplication of the compliance energy with a non-linear function in χ, e.g., = χ 3 0 is a very famous approach (Sigmund and Maute 2013). The effect of this non-linear interpolation between void and full material configurations turns the energy being non-convex, which, of course, renders the problem inherently ill-posed. A numerical artifact of the ill-posedness is the phenomenon of patterns of repeated black/white distributions that represent in average the gray solution. Due to its appearance, it is referred to as the checkerboard phenomenon (Diaz and Sigmund 1995). A prominent approach to prevent checkerboarding are filter schemes of which a huge variety can be found in literature (e.g., Sigmund and Petersson 1998; Bourdin 2001; Zhou et al. 2001; Lazarov and Sigmund 2011; Wadbro
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