A Level Set-Based Topology Optimization Using the Lattice-Boltzmann Method

Abstract

This paper presents a topology optimization method for fluid problems using the lattice Boltzmann method (LBM). In the field of computational fluid dynamics, the LBM is a new approach for calculating viscous fluid behavior that replaces the classical formulation employing the Navier-Stokes equation. Since the LBM is formulated with a linear equation rather than the nonlinear Navier-Stokes equation, the LBM algorithm can be easily constructed. Therefore, the explicit scheme of the LBM makes it especially suitable when implementing large-scale parallel computations. In conventional approaches, since the optimization formula is constructed using a discrete governing equation, the adjoint equation typically includes a large-scale asymmetric matrix in the sensitivity analysis, which severely inflates the computational cost of solving an optimization problem. To overcome this, we construct a structural optimization problem governed by the continuous Boltzmann equation so that an adjoint equation that uses the same framework as that of the Boltzmann equation can be derived. Due to the framework characteristics, the adjoint equation can be calculated explicitly, as with the lattice Boltzmann equation. In this paper, we construct an optimization formula based on the lattice Boltzmann equation and the adjoint lattice Boltzmann equation. Furthermore, in order to obtain clear boundaries in the design domain, we use level set boundary expressions.