Topology Optimization of Fluid Problems by the Lattice Boltzmann Method

We consider the optimal design of flow domains for Navier-Stokes. The problem is solved by a topology optimization approach varying the effective porosity of a fictitious material. The boundaries of the flow domain are represented by potentially discontinuous material distributions. Navier-Stokes flows are traditionally approximated by finite element and finite volume methods. These schemes, however, are particularly sensitive to the discretization of the flow along the boundaries, leading to significant robustness issues in the case of non-smooth boundary representations. Therefore, we study the potential of the lattice Boltzmann method for approximating low Mach-number incompressible viscous flows for topology optimization. In the lattice Boltzmann method the geometry of flow domains is defined in a discontinuous manner, similar to the approach used in material based topology optimization. In addition, this nontraditional discretization method features parallel scalability and allows for high-resolution fluid meshes. In this paper, we show how the variation of the porosity can be used in conjunction with the lattice Boltzmann method for the optimal design of fluid domains. An adjoint formulation of the sensitivity equations will be presented and the potential of this topology optimization approach will be illustrated by a numerical examples.

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