Consistent boundary conditions for PDE filter regularization in topology optimization
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RESEARCH PAPER
Consistent boundary conditions for PDE filter regularization in topology optimization Mathias Wallin1 · Niklas Ivarsson1 · Oded Amir2 · Daniel Tortorelli3,4 Received: 8 November 2019 / Revised: 8 February 2020 / Accepted: 21 February 2020 © The Author(s) 2020
Abstract Design variables in density-based topology optimization are typically regularized using filtering techniques. In many cases, such as stress optimization, where details at the boundaries are crucially important, the filtering in the vicinity of the design domain boundary needs special attention. One well-known technique, often referred to as “padding,” is to extend the design domain with extra layers of elements to mitigate artificial boundary effects. We discuss an alternative to the padding procedure in the context of PDE filtering. To motivate this augmented PDE filter, we make use of the potential form of the PDE filter which allows us to add penalty terms with a clear physical interpretation. The major advantages of the proposed augmentation compared with the conventional padding is the simplicity of the implementation and the possibility to tune the boundary properties using a scalar parameter. Analytical results in 1D and numerical results in 2D and 3D confirm the suitability of this approach for large-scale topology optimization. Keywords Topology optimization · PDE filter · Boundary effects
1 Introduction Topology optimization is a computational design methodology that is widely used in industry, in particular for aerospace and automotive applications. Common structural objectives are to find optimal trade-offs between weight, stiffness, strength, and natural frequency. One of the leading approaches to topology optimization, which is also the one followed in this article, is the density-based approach where the topology is described by a density, i.e. volume fraction Responsible Editor: Ole Sigmund Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00158-020-02556-w) contains supplementary material, which is available to authorized users. Mathias Wallin
[email protected] 1
Solid Mechanics, Lund University, Lund, Sweden
2
Faculty of Civil and Environmental Engineering, Technion-Israel Institute of Technology, Haifa, Israel
3
Center of Design and Optimization, Lawrence Livermore National Labolatory, Livermore, CA, USA
4
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA
of solid isotropic material variable at each discretization point in the design domain (Bendsøe 1989; Bendsøe and Sigmund 2003). As the underlying density variables are continuous between zero and one, penalizing intermediate density values using certain material interpolation schemes is necessary to obtain crisp 0/1 layouts. The most popular material interpolation functions are the SIMP (Solid Isotropic Material with Penalization, cf. Bendsøe (1989), Zhou and Rozvany (1991), and Mlejnek (1992)) and the RAMP stiffness interpolation (Rational
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