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A level-set method for steady-state and transient natural convection problems

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Abstract

This paper introduces a topology optimization method for 2D and 3D, steady-state and transient heat transfer problems that are dominated by natural convection in the fluid phase and diffusion in the solid phase. The geometry of the fluid-solid interface is described by an explicit level set method which allows for both shape and topological changes in the optimization process. The heat transfer in the fluid is modeled by an advection-diffusion equation. The fluid velocity is described by the incompressible Navier-Stokes equations augmented by a Boussinesq approximation of the buoyancy forces. The temperature field in the solid is predicted by a linear diffusion model. The governing equations in both the fluid and solid phases are discretized in space by a generalized formulation of the extended finite element method which preserves the crisp geometry definition of the level set method. The interface conditions at the fluid-solid boundary are enforced by Nitsche’s method. The proposed method is studied for problems optimizing the geometry of cooling devices. The numerical results demonstrate the applicability of the proposed method for a wide spectrum of problems. As the flow may exhibit dynamic instabilities, transient phenomena need to be considered when optimizing the geometry. However, the computational burden increases significantly when the time evolution of the flow fields needs to be resolved.

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Acknowledgments

The authors acknowledge the support of the National Science Foundation under grant EFRI-SEED 1038305 and CBET 1246854. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organization.

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  1. Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO, 80309, USA

    Peter Coffin & Kurt Maute

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  1. Peter Coffin
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Correspondence to Peter Coffin.

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Coffin, P., Maute, K. A level-set method for steady-state and transient natural convection problems. Struct Multidisc Optim 53, 1047–1067 (2016). https://doi.org/10.1007/s00158-015-1377-y

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