Adaptive level set topology optimization using hierarchical B-splines
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RESEARCH PAPER
Adaptive level set topology optimization using hierarchical B-splines ¨ 1 · M. Schmidt1 · C. Messe1 · J.A. Evans1 · K. Maute1 L. Noel Received: 25 September 2019 / Revised: 24 March 2020 / Accepted: 26 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper presents an adaptive discretization strategy for level set topology optimization of structures based on hierarchical B-splines. This work focuses on the influence of the discretization approach and the adaptation strategy on the optimization results and computational cost. The geometry of the design is represented implicitly by the iso-contour of a level set function. The extended finite element method is used to predict the structural response. The level set function and the state variable fields are discretized by hierarchical B-splines. While first-order B-splines are used for the state variable fields, up to thirdorder B-splines are considered for discretizing the level set function. The discretizations of the design and the state variable fields are locally refined along the material interfaces and selectively coarsened within the bulk phases. For locally refined meshes, truncated B-splines are considered. The properties of the proposed mesh adaptation strategy are studied for level set topology optimization where either the initial design is comprised of a uniform array of inclusions or inclusions are generated during the optimization process. Numerical studies employing static linear elastic material/void problems in 2D and 3D demonstrate the ability of the proposed method to start from a coarse mesh and converge to designs with complex geometries, reducing the overall computational cost. Comparing optimization results for different B-spline orders suggests that higher interpolation order promote the development of smooth designs and suppress the emergence of small features, without providing an explicit feature size control. A distinct advantage of cubic over quadratic B-splines is not observed. Keywords Topology optimization · Level set · XFEM · Adaptive mesh refinement · Truncated hierarchical B-splines
1 Introduction Following the seminal work of Bendsøe and Kikuchi (1988), topology optimization has become, over the last decades, a reliable and efficient design tool in various application fields (see Sigmund and Maute (2013) and Deaton and Grandhi (2014)). In general, an optimization problem is formulated to find the optimal material distribution, within a design domain, that maximizes a given objective while satisfying some given constraints on the geometry and/or the physical response. In specific regions of the design domains, the state variable fields may Responsible Editor: Christian Gogu L. No¨el
[email protected] K. Maute
[email protected] 1
Ann and H. J. Smead, Department of Aerospace Engineering Sciences, University of Colorado at Boulder, 3775 Discovery Dr, Boulder, CO 80309-0429, USA
exhibit large gradients or discontinuities. Additionally, the design variable fields m
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