Topology Optimization of Flows: Stokes and Navier-Stokes Models
This lecture will discuss density-based approaches for solving flow topology optimization problems. The focus is on low-Reynolds number fluid models, namely Stokes and laminar Navier-Stokes models, at steady-state conditions.
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Department of Aerospace Engineering Sciences, University of Colorado, Boulder, USA
Abstract. This lecture will discuss density-based approaches for solving flow topology optimization problems. The focus is on lowReynolds number fluid models, namely Stokes and laminar NavierStokes models, at steady-state conditions.
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Introduction
Finding the geometry of systems to optimize the performance of internal and external flows is of great importance across a wide spectrum of applications. For example, fluids apply forces on bodies, such as lift and drag, and transport species and thermal energy. Manipulating these phenomena is a central issue for a large number of engineering systems, including aircraft aerodynamics, injection molding, and liquid cooling. Due to the complexity of modeling and predicting flows, historically the design of flow problems was driven by experimental studies. With the advent of computational fluid dynamics (CFD) in the late 1960s, the flow about more complex geometries could be simulated and analyzed numerically. Improved numerical schemes G. Rozvany, T. Lewiński (Eds.), Topology Optimization in Structural and Continuum Mechanics, CISM International Centre for Mechanical Sciences, DOI 10.1007/978-3-7091-1643-2_17, © CISM, Udine 2014
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K. Maute
and more powerful computing platforms enabled the systematic optimization of flow problems via mathematical optimization methods. Until recently, research on optimizing the geometry of flow problems solely focused on shape optimization, accounting for ever complex flow phenomena (see, for example, Gunzburger (2003) and Mohammadi and Pironneau (2001)). Fluid topology optimization was pioneered by Borrvall and Petersson (2003) adopting the concept of density methods, originally developed for problems in solid mechanics, to Stokes flows. For a general introduction to topology optimization, the reader is referred to the book by Bendsøe and Sigmund (2003). Guest and Pr´evost (2006) conducted fluid topology optimization using a Darcy-Stokes flow model. Duan et al. (2008b) and Challis and Guest (2009) used a level-set parametrization of the material distribution to solve Stokes flow problems. The work on Stokes models was extended to Navier-Stokes models, for example by Gersborg-Hansen et al. (2005); Okkels et al. (2005); Olesen et al. (2006); Evgrafov et al. (2007). Kreissl and Maute (2011) recently presented an approach to optimize the transient response of flows predicted by the incompressible Navier-Stokes equations. Othmer (2006); Othmer et al. (2007) optimized the layout of 3D air-duct manifolds for automotive applications, employing an incompressible Navier-Stokes model. As an alternative to the Navier-Stokes flow model, Pingen et al. (2007a, 2010) used the lattice Boltzmann method (LBM) for solving fluid topology optimization problems. Kreissl et al. (2011) employed the LBM in combination with a level-set based geometric interface representation for generalized shape optimization of fluids. In this lecture we will discuss the basic formulations and numerical tec
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