Topology optimization using the lattice Boltzmann method incorporating level set boundary expressions

Abstract

This paper presents a topology optimization method for fluid dynamics problems, based on the level set method and using the lattice Boltzmann method (LBM). In this optimization method, the optimization problems are formulated based on the original Boltzmann equation, and the design sensitivities are precisely obtained without the time-consuming numerical operations encountered when dealing with a large-scale asymmetric matrix, in contrast to previous research in which the LBM uses the lattice Boltzmann equation (LBE) for the formulations of optimization problems and the derivation of their adjoint equations. That is, we newly derive sensitivity formulations from the original Boltzmann equation, not the LBE that can be said to be an approximated equation, and these formulations yield strictly correct sensitivities that are error free. Based on the above formulations, we construct a level set-based topology optimization method incorporating a fictitious interface energy for the design of a fluid channel that minimizes flow friction. Furthermore, two-and three-dimensional numerical examples are provided to confirm the validity and utility of the presented method.

Topology optimization of unsteady flow problems using the lattice Boltzmann method

“…The work is extended to multiphase flow problems by Makhija et al [8]. In addition, a number of studies have investigated a level-set based optimization approach using the LBM [17][18][19].…”

Local-in-time adjoint-based topology optimization of unsteady fluid flows using the lattice Boltzmann method

“…Due to the similarity of the locality properties, this approach can make use of highly efficient algorithms, while the design sensitivities can be obtained without using matrix operations. Yaji et al (2014) first proposed a topology optimization method based on the ALBM, and have recently proposed an improved method ) that deals with the discrete Boltzmann equation, whereas the original ALBM uses a continuous formulation. In this improved approach, various boundary conditions can be precisely introduced, since the discrete Boltzmann equation incorporates discrete particle velocities that respectively correspond to the boundary conditions in the LBM.…”

“…Pingen et al (2007) pioneered a topology optimization method using the LBM for steady-state viscous flows and clarified that optimized configurations can be obtained for the standard pressure drop minimization problems. Yaji et al (2014) proposed a topology optimization method applying the adjoint lattice Boltzmann method (Krause et al, 2013), based on a continuous adjoint sensitivity analysis that enables explicit calculation of the adjoint equation, to solve two-and three-dimensional pressure drop minimization problems. On the other hand, Liu et al (2014) applied a discrete sensitivity analysis to topology optimization problems dealing with steady-state viscous flows.…”

Recent progress on lattice Boltzmann simulation of nanofluids: A review

“…The equilibrium distribution functions f i eq and g i eq are chosen so that they satisfy the macroscopic equations via Chapman-Enskog expansion. They can be written as [30][31][32]:…”